Graded MINI mass initialization can destroy the smooth instantaneous pressure rate

We first establish the internal claims needed by the construction, and then make one simultaneous parameter choice and apply them. A conditional estimate in an internal claim is used only after its stated geometric or source premise is proved in a later phase. For bounded Hilbert sequences below, weak subsequences can be obtained without an unstated compactness supplier: work in the closure of the countable union of the finite spaces involved and the fixed data, choose a countable dense set, extract successively convergent scalar pairings by interval bisection, and take the diagonal subsequence. Its bounded limiting functional extends by density and is represented by Riesz; norm lower semicontinuity follows by testing against that representative. For an exterior exhaustion the countable nodal bases and bounded-radius cutoffs give the same separable subspace.

Two elementary inequalities used below have direct local proofs. For a rectangle or fixed convex container, $\mathrm{\parallel }v-\bar{v}{\mathrm{\parallel }}_2^2=(2\mathrm{\mid }C\mathrm{\mid }{)}^{-1}{\iint }_{C^2}\mathrm{\mid }v(x)-v(y){\mathrm{\mid }}^{2}dxdy\le C\mathrm{diam}(C{)}^{2s}[v{]}_{s,C}^2$. For zero-boundary smooth scalar functions on ${\mathbb{R}}^2$, integrate ${\mathrm{\partial }}_x\mathrm{\mid }v{\mathrm{\mid }}^2$ and ${\mathrm{\partial }}_y\mathrm{\mid }v{\mathrm{\mid }}^2$ along the coordinate lines, multiply the two bounds, integrate and apply Cauchy–Schwarz to obtain $\mathrm{\parallel }v{\mathrm{\parallel }}_4^2\le 2\mathrm{\parallel }v{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$; density and componentwise application give the same estimate for every $H_0^1$ vector, irrespective of its divergence. These prove the fractional Poincare and Ladyzhenskaya uses in the internal claims.

Fix a scale $r>0$, the rectangle $C=[-4r,4r]\times [0,4r]$, and the lower triangle with vertices $(-4r,0),(0,-4r),(4r,0)$. Their union is the fixed convex pentagon $D$. Keep its two lower triangles $K_{\pm }$ formed by the root $o=(0,0)$ unchanged. Inside $C$, choose a narrow positive-$x$ fan with consecutive outer vertices $v_j=R_j(\cos {\theta }_j,\sin {\theta }_j)$, $R_j\in \{r\mathrm{/}2,r\}$ alternating, ${\theta }_j=\delta +ja\mathrm{/}N$, and triangles $(o,v_j,v_{j+1})$, together with its reflection in the vertical axis. Use a fixed multiple of $N$ such sides and retain short end ramps in the same wedge. For $a\ll \delta \ll 1$, their interiors are disjoint, their total area is comparable to $a{r}^2$, and the distance from $o$ to each long straight side is comparable to $ar\mathrm{/}N$, by the exact determinant $R_j{R}_{j+1}\sin (a\mathrm{/}N)$. On each side's middle third, its longitudinal coordinate from $o$ is comparable to $r$ and its clearance from nonincident fan edges is at least a fixed multiple of $ar\mathrm{/}N$: scale a pair of neighboring alternating-radius triangles by that transverse distance; their fixed radial ratio and bounded middle-third parameter give this positive minimum directly. Place equally spaced attachment triples there at spacing $\mathrm{\ell }\asymp b_{0}ar\mathrm{/}N$, each child radius $\rho r\asymp \sigma \mathrm{\ell }$. Discard a fixed bounded number near ends and pair every choice by reflection. The total count is $B\asymp N^2\mathrm{/}(b_{0}a)$; the boxes and their comparison windows are disjoint when $b_0,\sigma$ and the fixed window multiplier are sufficiently small. This explicit fan supplies the side/count hypotheses of part 2. All subsequent insertions are the finite conforming collar, guard and macrointerface constructions in that phase.

The element matrices used throughout are exact. On a triangle $K$, barycentric integration gives ${\int }_K{\lambda }_i^2=\mathrm{\mid }K\mathrm{\mid }\mathrm{/}6$, ${\int }_K{\lambda }_i{\lambda }_j=\mathrm{\mid }K\mathrm{\mid }\mathrm{/}12$ for $i\mathrm{\ne }j$, ${\int }_{K}b=9\mathrm{\mid }K\mathrm{\mid }\mathrm{/}20$, ${\int }_{K}b{\lambda }_i=3\mathrm{\mid }K\mathrm{\mid }\mathrm{/}20$, and ${\int }_K{b}^2=81\mathrm{\mid }K\mathrm{\mid }\mathrm{/}280$, for $b=27{\lambda }_1{\lambda }_2{\lambda }_3$. These follow by mapping to $(x,y)\in \{x,y>0:x+y<1\}$ and integrating each monomial; equivalently ${\int }_K\prod {\lambda }_i^{m_i}=2\mathrm{\mid }K\mathrm{\mid }\prod m_i!\mathrm{/}(2+\sum m_i)!$, proved by successive elementary integrations. The mean-one bubble $\beta =20b\mathrm{/}9$ has mean square $\gamma =10\mathrm{/}7$. Eliminating the bubble in the actual mass/gradient mixed form yields nodal scalar blocks $C_{ii}=4\mathrm{\mid }K\mathrm{\mid }\mathrm{/}45$, $C_{ij}=\mathrm{\mid }K\mathrm{\mid }\mathrm{/}180$, $H_{ij}=\mathrm{\mid }K\mathrm{\mid }\mathrm{\nabla }{\lambda }_j\mathrm{/}10$, and pressure block $-7\mathrm{\mid }K\mathrm{\mid }\mathrm{\nabla }{\lambda }_i\cdot \mathrm{\nabla }{\lambda }_j\mathrm{/}10$. For example subtract $(3\mathrm{/}20{)}^2\mathrm{/}(81\mathrm{/}280)=7\mathrm{/}90$ from each affine mass entry, and subtract $(3\mathrm{/}20)(9\mathrm{/}20)\mathrm{/}(81\mathrm{/}280)=7\mathrm{/}30$ from $1\mathrm{/}3$ in its pressure coupling. The symmetric-gradient affine/bubble stiffness block is zero because ${\int }_K\mathrm{\nabla }b=0$. Thus none of these calculations replaces the actual strain form by a scalar Laplacian.

Part 2: Finite recursive geometry and all boundary-star budgets. 1. Scales and the true outer-radius restriction. Let the parent fan radius be H, aperture a_p, and tooth count N. A straight main tooth side has root distance D comparable to a_p H/N. Choose selected attachment centers on its middle thirds at spacing ell comparable to b0 D. The child half-container is, in local tangent/normal coordinates, C_r=[-4r,4r]x[0,4r], r comparable to sigma ell. Choose the already free dyadic strip height with 0<epsilon_strip<=c a_c. Its base contains exactly c_-=(-4r,0), o=(0,0), c_+=(4r,0); its other four boundary nodes are (±4r,epsilon_strip r),(±4r,4r). These seven nodes and all seven boundary edges remain unchanged under every descendant replacement. The clearance from an attachment to nonincident tooth sides is at least cD, but SAME-SIDE neighboring children are only distance comparable to ell apart. Thus a single-child ring must use outer scale R comparable to c0 ell, r/R comparable to sigma, (1) with c0 a sufficiently small fixed constant. It cannot use R comparable to D without additionally treating the neighboring children as part of the near field. Choose R as a dyadic multiple of r within a fixed factor of c0 ell. Make 8R smaller than both one quarter of the same-side spacing and the tooth-side clearance. Then the half-disc of radius 8R on the exterior side of this straight parent base lies in the available gap, misses every other child, lies inside the parent container, and lies in ONE affine macro-zone of every parent logarithmic pressure lift. The last condition follows from the protected near-base zone's angular margin, since R/H=O(b0 a_p/N)<<delta. All these are strict finite inequalities fixed before recursion depth.

2. The actual uniform ring and its transition. Use local square coordinates on the exterior half-plane. Let S_s=[-s,s]x[0,s]. Keep the actual child C_r inside S_(8r). Choose one uniform half-plane MINI triangulation, with square step d and its prescribed diagonal orientation. Take d=2^(-k)r, ; d<=c min{a_c r²/R, epsilon_strip r, ; epsilon_Q D, a_c r}, (2) where a_c is the CHILD fan aperture. Increase the integer k if needed; no lower bound on d is imposed. This in particular implies the required d<=Cr²/R. R/r is dyadic, so every boundary of S_(8r) and S_(4R) is a union of lattice edges. Make the actual mesh exactly this uniform mesh on S_(4R) minus S_(8r). The nonuniform child-to-uniform transition is confined to S_(8r)\C_r. In Euclidean radius it is contained in the half-disc of radius 12r. Thus, with the cutoff parameter r_tr=16r, the common annuli 2r_tr through 8r_tr and their one-cell enlargements are entirely uniform, provided R is chosen larger than a fixed multiple of r. This holds once sigma is sufficiently small. Distant cutoff drivers at distance comparable to R and their stencil enlargements are also on this exact common uniform mesh. There is no interpolation or projection from uniform cells through the coarse transition. The stronger first bound in (2) is useful for ideal layer energy: on a finite base extent O(R), either exact neutral layer has mass plus pressure-gradient energy squared O(dR|eta|²). Hence dR<=C a_c r² keeps that cost at the same scale as the child root fan star, rather than introducing an unexplained factor 1/a_c. A smaller d remains an allowed fixed finite template choice.

3. Keeping every long child edge unrefined. Construct a polygonal collar on the EXTERIOR side of the child's four side/top boundary chains; at the base corners it ends on the physical base outside [-4r,4r]. No node is inserted on a child boundary edge. For an edge of length L_e, a strip of positive height tau_e has area O(L_e tau_e). Choose all strip heights sufficiently small that the sum of collar triangle areas incident to each ORIGINAL child boundary vertex is at most zeta a_c r². At corners use smaller offset sectors and the same budget. These choices are possible simultaneously because the list is finite and each budget is positive. The heights may be dyadic and smaller than epsilon_strip r as needed. One explicit strip is the quadrilateral formed by the original edge and two nearby exterior offset endpoints, split into triangles. Its original-edge endpoints see total area O(L_e tau_e+tau_e²), tending to zero with tau_e. Near a base corner subdivide ONLY the outward physical base, not the child's vertical edge. Lattice nodes of spacing d on that outward base are permitted. Their first-edge area contribution is O(L_e d), which is within the budget after reducing d as in (2); for L_e<=8r the condition d<=ca_c r suffices. The remaining transition meets the offset polygon, not the original child vertices, so no later triangle enlarges their stars. Also put a thin physical-base collar along the transition base outside [-4r,4r]. Its edges may have lattice spacing d, but its height is at most c a_c r, tapering to height d where it meets the uniform ring. The corner cap adjacent to c_± has height at most epsilon_strip r<=c a_c r. Thus the TOTAL area of transition triangles incident to physical-base vertices is O(a_c r²), and each newly inserted parent guard has star at most C d a_c r (or C d² in the uniform part). This total budget is needed for the ideal neutral-driver lift; individual small-star bounds alone would not prove it. Triangulate the remaining finite polygon between this collar and S_(8r), retaining the exact outer lattice edges. All imposed macrointerfaces below are included in this finite planar straight-line subdivision. Holes/strips are disjoint after shrinking their widths. Triangulation by finite noncrossing diagonals and interior Steiner points then gives a finite conforming mesh. Every triangle is nondegenerate. Its smallest angle can be tiny and depends on the fixed data N,a_p,a_c,b0,sigma,epsilon_strip,d/r and collar widths, but is positive and independent of recursive depth. No uniform-in-these-parameters minimum angle is asserted.

4. The SHARP nonterminal root-star bound inside the child. Inside C_r retain the protected two mirrored radial fans. Their combined area is between fixed multiples of a_c r², and EVERY protected fan triangle contains the child root o. Refining a fan edge radially does not change this total root area. All additional triangles incident to o can be restricted to arbitrarily small polygonal sectors outside the protected fans. On a whole coarse base edge o-c use one arbitrarily thin adjacent triangle; its area is at most C r tau, with tau as small as desired. The remaining root sector can be closed by interior offset points near o and near the short initial/final fan edges, before filling the complementary polygon. Those offset distances are independently shrinkable. The only compulsory root edges outside the fans are the two fixed coarse base edges and finitely many short root-ramp edges. Their exterior adjacent triangles have area tending to zero with their chosen offset heights. The filler beyond the offset polygon has no root vertex. At another original fan vertex i the same separate collar construction makes all added nonfan triangles as small as any prescribed fraction of its positive incident fan area A_i. Connecting root-sector collar pieces may use new interior vertices; those pieces need not contain i. Therefore shrinking a root-sector cap does not force a large star at a short-ramp guard. This distinction is essential when its fan area is much smaller than r². For a fully explicit way to impose the last area budgets, apply the vertex surgery of Section 8 at o to EACH NONFAN triangle incident to o, after the finite pressure-interface overlay. Its root contribution becomes exactly 2e times its old area while all protected fan triangles and all old edges remain unchanged. Choose e so their total is at most zeta a_c r². Other original stars never increase. Applying the same surgery afterwards at the finitely many fan guards gives their required small exterior stars without increasing the already fixed root star. If a new long background-only edge crosses the mandatory inner buffer, subdivide it at O(ell_child) and refine the adjacent triangles; subdivision preserves their total original-vertex star bounds. Long original seven-node boundary edges are outside that buffer and are kept whole. This is an explicit finite implementation of the thin-collar area argument, rather than an assumed area floor. Consequently one can enforce simultaneously c a_c r² <= area(star_child(o)) <= C a_c r², ; area(star_child(c_±)) <= C a_c r², (3) and arbitrarily smaller bounds at the other inherited boundary vertices if desired. The lower bound is supplied by the protected fans; the upper bound is NOT the old whole-box O(r²) estimate. The external transition has no root triangle at all: a half-neighborhood of o is inside C_r. At c_± its extra star is controlled by Section 3. Every first-generation reserve and its full ring is a positive distance from o and from the outer boundary. Construct these root/boundary collars before any descendant replacement. They therefore stabilize once the finite current template is fixed. Terminal fills must obey the SAME sharp boundary-star budgets; do not fill a terminal box by a coarse root-to-all-corners fan of area O(r²). A finite terminal fill with the protected two fans retained and with the above thin root sectors, closing its own unexpanded reserves by fixed thin-collar fills, has (3). For an empty terminal reserve with no retained fan, start from any fixed pressure-macro-conforming fill and apply the same root surgery with a common e chosen so its TOTAL root-star area is exactly a_c r² (or a fixed comparable fraction); the original star area is O(r²), so such e lies in (0,1/2) for small a_c. This supplies both the lower and upper star bounds without an infinite recursion. All terminal boundary triangles are then fixed before descendants are considered; no depth-dependent cell is needed.

5. Parent star budgets, new uniform guards and exact old formulas. At a selected child root the only nonparent cells incident to it lie inside that child. Its two adjacent parent half-base edges have length 4r and are NEVER refined. Hence its parent star area is exactly 4rD. From (3), its nonparent-to-parent ratio is at most C a_c r/D <= C a_c sigma b0. (4) The base endpoints c_± also receive O(a_c r²) from the child and transition, while their parent star has area at least 2rD. They satisfy the same bound. This is the sharp endpoint/root mechanism; a whole-box upper bound would lose a_c. Uniform grid nodes on the physical base OUTSIDE the child's two base edges are new PARENT fan guards. Subdivide the corresponding parent long-side segments at those nodes and refine each affected fan triangle RADIALLY from the parent root. This preserves the radial fan form and its total area. A uniform base guard has exterior star O(d²), while its parent fan star is comparable to dD, so its ratio is O(d/D)<=C epsilon_Q by (2). Nonuniform transition guards have exterior/fan ratio at most C a_c r/D by the thin physical-base collar in Section 3; this is the same small quantity as (4). The uniform part's forced star is already within budget. All other old guards/ramps receive their prescribed thin collars. Thus allouter-star bounds hold simultaneously, not merely at selected centers. The radial guard refinement leaves all exact full-outer formulas valid: local condensed coefficients remain Dloc=4/45,Oloc=1/180, and each parent triangle still has root plus two outer vertices. The full outer normalized matrix still has diagonal 16 and off-diagonal row sum one, including every NEW guard; its weighted inverse bound is unchanged. The pressure-free hybrid nodal tests are also unchanged elementwise. The total outer star sum is exactly twice the fan area, independently of how many guards are added. Each selected root's two half-base edges remain intact, so its star weight and the selected fraction comparable to sigma are unchanged. Tooth-line distance D and the root--center longitudinal coordinate are unchanged; hence the exact half-slope continuity factor O(a_p/N) still applies. The full-fan weighted-Hardy slope estimate permits arbitrarily small ramp radii and all these radial refinements. Source triangles below the original coarse base and their actuator moments are untouched. All transition edges have length O(r)<=O(ell); all new ring edges have length O(d). They therefore introduce no long background-only shortcut through the parent's mandatory fine buffer. The child itself retains its own internal buffer, which must still be crossed before any exterior ring can be reached. Existing coarse-base-collar exceptions remain confined below their prescribed base-strip line.

6. Exact pressure lifts and no eighth boundary node. Use the established logarithmic corner lifts inside the child. Their interfaces are finite and miss the protected fan interiors and holes, where the functions are affine. Overlay those interfaces with the root/thin-star collars by CONFORMING REFINEMENT. Refinement subdivides already assigned star regions and never increases their total area; it preserves all pressure-lift energies. No child boundary edge is subdivided, because these interfaces meet it only at the prescribed seven vertices. No projection on skinny cells is used. For a base pressure kink use the single lift selected by the preceding kernel argument, F(t,n)=kappa |t-n|, or its reflected partner kappa|t+n|. See the internal proof in part 17. The line t=n goes from root to the EXISTING top-right corner. Its traces are affine on EACH of the seven boundary edges: base kappa|t|, right side kappa(4r-n), left side kappa(4r+n), top kappa(4r-t). Thus no eighth top-edge node is needed. Its gradient cost is C|kappa|r. Protected near-base fans avoid the line and see exact affine pieces. Constrain the transition to this same macroline. On the SWNE uniform stencil the line is already a union of diagonal edges; its reflected lift uses NWSE. Cutoffs vary ONLY on the common uniform annulus. All coarse-zone values remain exact. Every parent pressure lift is affine throughout the complete selected child's ring neighborhood by the margin in Section 1. Thus neither uniform diagonals nor radial parent-guard refinements require a nonaffine pressure interface in the uniform ring.

7. Reflection covariance and the proposed powers. Construct the two mirror zones as exact reflected finite subdivisions. Offset choices, dyadic spacings, terminal fills, gauges and macro lifts are reflected together. A SWNE diagonal in one ring is NWSE in its mirror. Do NOT impose a single unreflected global diagonal on both zones. Disjoint rings permit these independent orientations; their exterior fillers are then joined by a finite reflected subdivision. This preserves the exact covariance of the two-driver matrices and incoming extractors. For L large, the proposed powers delta=L^(-200), a_p=a_c=L^(-220), b0=L^(-20), ; sigma=L^(-150), epsilon_Q=L^(-380) give a_c sigma b0=L^(-390), leaving a factor L^(-10) to absorb fixed constants in (4). They also give r/R comparable to sigma. Choose the dyadic lattice so d/r<=c a_c(r/R), as well as the other conditions in (2). Then dR<=C a_c r², and d/D is smaller still. The tiny lattice/node count affects the finite shape constant, not its depth independence or the universal reference mass/bubble bounds. Collar widths can be selected AFTER all these parameters to meet every positive star budget; no area floor O(r²) is introduced. The ideal-driver neutral velocity lift has the exact weighted-star mass bound from the structured seven-node note; its coarse root/base-node and transition-base contribution is O(a_c r²|eta|²), and the finite uniform layers contribute O(dR|eta|²), which is of the same order by (2). One may implement the comparison by a cutoff of the exact uniform layer which is zero through the transition and changes only on the common uniform annulus, adding the neutral boundary lift for the missing near-base velocity. The normal layer pressure is zero on the base and O(d|eta|) elsewhere; its uniform cutoff adds gradient cost O(d|eta|), already bounded by O(sqrt(a_c)r|eta|). Thus no coarse interpolation or logarithmic boundary-pressure lift is required for the short-layer cutoff itself. Affine backgrounds and the stated kink lift are exact P1 on the transition. Uniform cutoff estimates use only fixed uniform reference cells outside it. Any strain constant on the finite transition is finite but is NOT invoked in this mass-only ring binding. This supplies the actual finite conforming geometry assumed by the passive-cutoff and two-driver embedding notes. It does not make the twelve free child ports small, identify the actual incoming macro normal component, or prove a recurrent channel from absolute error estimates. Those are separate analytic/recurrence obligations.

8. Arbitrarily smaller CHILD-SIDE base-corner stars. There is no fan-area floor at c_- or c_+. For the full vector unequal-base-velocity remainder, choose ANY epsilon_corner>0 and require S_corner(c_±)<=epsilon_corner V_parent(c_±), (5) where S_corner counts ONLY cells on the child/transition side of the straight parent fan boundary, and V_parent is the SINGLE adjacent protected parent triangle between the selected center and that endpoint. Its half-base length is 4r, so V_parent=2rD exactly. The parent fan cells are untouched; a whole endpoint star is not substituted for V_parent. This stronger bound follows by an explicit finite local surgery, AFTER every transition/macro/log-lift refinement. For any such triangle K=(c,A,B) choose 0<e<1/2 and the interior point P=c+e[(A-c)+(B-c)]. Replace K by (c,A,P),(c,P,B),(A,B,P). The first two triangles have area e|K| each, so their TOTAL c-star area is exactly 2e|K|. All three original edges remain whole. Every old affine pressure lift remains affine on the new pieces. All old vertex stars are nonincreasing under this subdivision; thus successive operations at both corners preserve every previously achieved original-vertex budget. Choosing e for the finite set of incident triangles yields (5) for arbitrarily small positive epsilon_corner. All these cells lie in the child boundary collar or the nonuniform transition, strictly inside the inner ring radius. No exact uniform stencil cell is changed. The protected child fans lie away from these base corners, and no parent fan triangle is refined. Possible new long background edges remain in the outer child collar/transition (outside its mandatory inner buffer); at the parent scale their length is O(r)<<ell_parent, so no parent buffer shortcut is created. The finite minimum angle may decrease with epsilon_corner but remains positive and fixed with depth. For the proposed powers one may take epsilon_corner=L^(-600), which is smaller than sigma² b0 a_c=L^(-540). The center root still retains its necessary fan floor c a_c r²; no claim suppresses that center star to the corner budget. This distinction permits choosing the incident base velocity to be the center/root value, so the residual has zero center value and only two endpoint errors. Their exact neutral MINI extension is then controlled by (5), including the NORMAL velocity component; no nonexistent normal pressure-slope bound is needed.

9. Binding the ENTIRE comparison base to one parent radial side. For the present the internal proof in part 16, use the entire finite-ring base interval, not merely the child base triple. Let t_i be the unit tangent of the chosen straight parent long side, pointing away from the parent's root, and n_i its normal pointing from that root toward the line. In coordinates relative to the PARENT root, the complete line is x=D_i>0. Write its selected center as (D_i,y_i). The two long-side endpoints have parent radii H/2 and H with angular separation comparable to a_p/N. Their tangential coordinates are between fixed positive multiples of H, and their difference is comparable to H. This follows directly from their cross product and their length: D_i is their determinant divided by side length, while the side direction differs by O(a_p/N) from a radial direction. Thus D_i comparable to a_p H/N, ; c H<=y_i<=C H at every middle-third selected center, and its distance along that straight side to BOTH endpoints is at least cH. These facts remain true for downward or reflected sides after orienting t_i as just specified. Take the WHOLE comparison base to be I_i=[y_i-4R_i,y_i+4R_i], L_i=8R_i, ; Y_i=y_i-4R_i, where R_i satisfies Section 1 and, additionally, 20R_i<=y_i and 4R_i is smaller than the center's distance to both side endpoints. These inequalities follow for sufficiently small fixed c0 in R_ic0 ell_i, since ell_ib0 D_i<<H. Therefore I_i lies on ONE straight parent long side, lies in [Y_i,2Y_i], and L_i<=Y_i/2. The entire finite common half-ring base, including its driver/cutoff endpoints, is included in I_i; no part crosses a tooth peak, valley, ramp, neighboring child, or change of supporting line. The below-base cells throughout I_i are exactly radial triangles from the SAME parent root. Uniform-base guard insertion subdivides only that straight side and radially refines these cells. Hence the present exact distributional identity applies to the actual pressure, with every added breakpoint retained. No affine actual-pressure assumption is made. The exact coefficient to carry into its trace-lift estimate is L_i D_i/Y_i². (6) Since L_ib0 D_i and Y_iH in this main-side placement, this is bounded above and below by fixed multiples of b0(a_p/N)². If radii differ between parent types or one changes the selected intervals, retain D_i,Y_i,L_i and (6) individually; do not replace Y_i by a global H without the displayed comparability. Normalized child averaging must likewise use the actual r_i² and the actual child fan area. Disjoint intervals have bounded overlap (indeed the chosen rings are disjoint), so their physical parent-patch energies can be summed. Finally choose ALL new physical-base guard abscissas within I_i on the same local d_i lattice as the uniform ring. The three child base nodes already lie on it because d_i divides r_i. There is no other old parent subdivision point inside I_i: the original center spacing is ell_i and 4R_i<<ell_i; only its protected three-node attachment triple is retained there. Transition collars may use arbitrarily small/nonuniform HEIGHTS while keeping their base endpoints on this lattice. The corner-star surgery adds interior points only and preserves all old edges. Conforming interior filler triangulation requires no new boundary point after the base chain is fixed. Thus the actual entire-base pressure trace is piecewise affine with breakpoints on that lattice, as required for the structured exact constant-normal extension on the uniform grid. The sole coarse-child kink is handled by the seven-node macro lift in Section 6, without an eighth boundary node or a skinny-cell projection.

10. The strengthened h² star budget for EVERY transition-base guard. Write h=d for the uniform lattice step. The preliminary thin-base collar in Section 3 bounds a transition guard star by O(h a_c r); the velocity guard estimate needs the STRONGER O(h²). Impose it explicitly after all pressure macrointerfaces and transition subdivisions are present. At every newly inserted physical-base guard z outside the child's two unrefined base edges, apply the finite vertex surgery of Section 8 to its NONUNIFORM EXTERIOR incident triangles ONLY. Pick the positive surgery factors so their total z-star area is at most h². Leave every uniform-ring triangle unchanged; its contribution is already at most C h² by the uniform stencil. Thus S_exterior(z)<=C h² (7) for every such transition or ring guard, including guards on the common boundary of uniform and nonuniform regions. The parent radial fan triangles below the base are never changed. Every original base edge, every grid abscissa and every pressure-lift value remains unchanged. Surgery inserts interior points only. Applying it at the finitely many guards in any order is legitimate because refining a triangle never increases another old vertex star. Corner surgeries can subsequently meet both their single-triangle budget (5) and, if wanted, an h² bound by choosing the smaller target. The new interior nodes remain in the nonuniform transition and its parent-scale O(r) region. Subdivide any new background-only edge that crosses a mandatory buffer at its prescribed fine spacing; this only refines triangles, so (7) and all old-vertex budgets remain valid. No uniform stencil is altered. The smallest resulting angle is positive for the fixed finite template, although it depends on h and the surgery targets. No strain estimate through these cells is used. For each such guard the parent fan area below is at least c hD (exactly hD for two equal neighboring base segments). At an outer patch endpoint the other neighboring segment may be longer; this only increases its parent area and improves the following ratio. Consequently its exterior-to-parent ratio is now O(h/D), rather than O(a_c r/D). This is the actual geometric input for the full-vector velocity-guard budget; the improved h² bound must be used when that analytic estimate is invoked. The selected center has its fan floor a_c r² and is NOT subject to (7); its velocity residual is zero when the incident base velocity is chosen as its actual root value.

11. Fixed interfaces for the ENTIRE guard-pressure lift. The the internal proof in part 19 uses a fixed child-scale macro function S with S=|t-n| inside C_r, S=|t| outside S_(8r), base trace |t| and uniformly bounded affine gradients. Include its fixed U-shaped transition macrotriangulation BEFORE the final star surgeries. It has no new node on a child edge: the inner seven-edge traces are affine, and its outer top-midpoint is on S_(8r), where lattice nodes are allowed. Its inner crease is the existing root-to-top-right line. Reflected zones use the reflected S and grid. Also include the vertical interfaces t=t_j for ALL possible new base guard abscissas in the transition, not just those with nonzero unknown coefficients. They lie on the fixed d lattice and have |t_j|>=4r. They therefore miss the child interior; the extreme interfaces t=±4r coincide with its sides and meet its top only at its existing corners. Conform the finite transition to these lines and the fixed S interfaces. On the common uniform ring the vertical lines are already lattice edges and require NO stencil change. For an actual base remainder g, its only child-base kink is at t=0, so g=a t+k|t| on [-4r,4r]. The full lifted expression g(t)+k[S-|t|] is exactly a t+kS in that strip. Therefore there is NO uncancelled vertical t=0 interface through the coarse child or its top edge. Do not add a vertical root-to-top-midpoint line through the child as an unnecessary refinement: it would create the forbidden eighth node. The fixed S macrostructure already carries the correct replacement crease. This finite interface set is chosen from the mesh and its trace space, BEFORE any pressure coefficients or driver values are known. Every resulting function is prescribed by exact affine pieces on its cells; there is no interpolation through skinny transition cells. The distant cutoff interpolation occurs only on the exact uniform ring as in the structured proof. Its added nodes/diagonals are those already present there. Now perform the root/nonfan, inherited-corner and all transition-base guard surgeries from Sections 4,8,10 LAST, and subdivide any remaining inadmissible background edge. Refinement preserves every affine piece of S, g and all logarithmic lifts and cannot enlarge an old vertex star. Thus the full-base pressure lift and the strengthened h²/epsilon_corner star budgets hold simultaneously. The protected fans and their reserves lie wholly in affine regions of S; their interiors are not crossed by its macrointerfaces. Descendant replacement therefore preserves this fixed lift exactly.

12. Binding the final analytic cutoff radii. The final structured entire-base lift uses an analytic radius R_an with bulk annulus R_an..2R_an, all bulk stencils inside S_(3R_an), local base cutoffs equal to one on S_(4R_an) and zero outside S_(8R_an), and tail support starting at |t|=4R_an. Its trace comparison interval has endpoints ±16R_an. Bind the geometry radius used above by R_geom=16R_an. The exact uniform region through S_(4R_geom) then reaches S_(64R_an), and the comparison interval lies on the ONE straight radial parent side guaranteed by Section 9. Choose R_an<=ell/1000 and the earlier fixed c0 small enough that the larger safety neighborhood of radius 128R_an lies inside ell/4. These are only fixed numerical reductions of the ring radius, not new scale exponents. The child ratio r/R_an remains comparable to sigma by fixed factors. All local cutoffs therefore avoid the nonuniform child transition; all bulk stencils avoid the distant-tail region. This binding applies to the final nested-radius version of the structured the internal proof in part 19.

Part 3: Logarithmic seven-node pressure extensions. Explicit right-corner construction. Scale first to r=1. Let the half-container be [-4,4]x[0,4], with root o=(0,0), right base corner c=(4,0), and side strip node s=(4,epsilon). Choose epsilon=2^(-J), J>=3. This is the previous epsilon=4*2^(-n) with J=n-2. Let p=(4-epsilon,epsilon), ; L(x,y)=(x-y)/4, ; t_j=epsilon*2^j, 0<=j<=J, ; A_j=(4-t_j, epsilon*(4-t_j)/(4-epsilon)), ; B_j=(4-t_j, epsilon+t_j). Thus A_0=p, A_J=(3,3epsilon/(4-epsilon)), B_J=(3,1+epsilon). All A_j lie on the INTERNAL segment o-p. All B_j lie on the INTERNAL segment s-B_J. No base or side boundary edge is subdivided. Triangulate the support polygon (o,c,s,B_J) by the following pieces: the base triangle (o,c,p), subdivided from c at the A_j on its edge o-p; the two small triangles (c,p,s) and (p,s,B_0); the quadrilaterals (A_j,A_(j+1),B_(j+1),B_j), 0<=j<J, each split into two triangles; the triangle (o,A_J,B_J). The open interiors are disjoint. This is directly seen by vertical sections: between x=3 and x=4-epsilon, the ladder occupies precisely the region between the line o-p and the line s-B_J; the two small triangles close the remaining corner; the base triangle is below the line o-p; and the last triangle closes the left end. The support polygon is below the line o-B_J and inside the right coarse macrotriangle y<=x. Set the function to zero outside this support polygon. Give the nodal values H(o)=0, H(c)=1, H(s)=0, ; H(A_j)=L(A_j), H(B_j)=0. Take H=L throughout the base triangle, and use the affine interpolant on each other displayed triangle. On the support boundary o-B_J and B_J-s all nodal values are zero. Thus extension by zero is continuous. Its trace on the actual container boundary is exactly the corner boundary hat: affine from zero at o to one at c along the WHOLE unrefined base edge; affine from one at c to zero at s along the short side edge; zero on all other container edges.

Energy bound. On the base triangle H=L, so its total energy is bounded by an absolute constant, regardless of its subdivisions. Each small corner triangle has area comparable to epsilon² and diameter comparable to epsilon, and nodal values bounded by one. Hence its energy is bounded by an absolute constant. For a ladder quadrilateral, its horizontal width is t_j. Its vertical side heights are comparable to t_j: indeed B_j,y-A_j,y =t_j + epsilon*(t_j-epsilon)/(4-epsilon), which lies between t_j and 2t_j for epsilon<=1/8. Its edge slopes are bounded absolutely. After scaling by t_j, either diagonal split therefore lies in a fixed compact family of nondegenerate triangles. The nodal values are in [0,1]; reference-element scaling in two dimensions bounds the energy of each of its two triangles by an absolute constant. There are J quadrilaterals. The last triangle (o,A_J,B_J) has a uniformly positive area and uniformly bounded affine coefficients, because its two nonzero vertices have x=3 and vertical separation bounded above and below by positive absolute constants. Its energy is bounded absolutely. Consequently ||grad H||_L2 <= C sqrt(1+J), ||H||_infinity<=1. (1) The maximum bound follows from affine interpolation of nodal values in [0,1].

Protected fan and recursion. The last triangle contains every point satisfying 0<x<2, epsilon*x/(4-epsilon) < y < ((1+epsilon)/3)*x. Choose the protected near-tangent fan and ALL of its reserve holes strictly inside this wedge. In particular a fixed positive tilt theta with epsilon/(4-epsilon)<theta and aperture lying below 1/3 is sufficient. This is exactly the admissible regime epsilon much smaller than the tilt. On this ENTIRE triangle H is one affine function, with gradient bounded absolutely. It need not equal L and need not vanish there. Every similarity-recursive refinement inside every protected hole therefore leaves H exactly P1 without any interpolation estimate. All ladder and small-corner cells lie in x>=3, outside the inner background buffer. The long edges of the base triangle have the active corner c or root o as endpoint. The two long edges of the protected triangle have active root o as endpoint, and its opposite edge is outside the buffer. Subdividing internal edges at the A_j does not add any boundary vertex; on the base-side pieces the pressure remains L. The construction is therefore compatible with the previous no-background-shortcut geometry after the same finite filler completion. Its finite minimum angle can depend on epsilon and the other fixed template parameters, but not on recursion depth. Mirror it for the other corner or for the reflected aligned template. Restore r by similarity. Formula (1) is unchanged in two-dimensional energy. Use the coarse macro corner hat L_c, whose actual side-strip value is 1-epsilon/4. Then H_strip=(L_c-H)/(1-epsilon/4) is the exact side-strip boundary nodal hat, with energy bounded by C sqrt(1+J). Both L_c and H are affine throughout the protected fan/holes, so their difference is too. The other five coarse boundary hats have fixed energy. Thus arbitrary seven-node pressure trace d has an actual P1 extension Phi satisfying ||grad Phi||2 <= C sqrt(1+log(1/epsilon)) osc(d), ; ||Phi-d_ref||2 <= C r osc(d). (2) The construction may be incorporated by a fixed conforming refinement respecting all its interfaces and all protected holes; pressure affinity makes refinements inside those pieces exact. The previous corner construction imposed H=0 on the protected fan, paying theta^(-1/2). That extra condition is unnecessary for (2) or recursive conformity.

Part 4: Actual mass spaces, strong recovery and autonomous Schur limit. 1. Explicit finite-template hypotheses and limit spaces. Let C be the root rectangle. At depth m its finitely many reserve rectangles are D_m; replacing each terminal rectangle by its next fixed template produces the depth m+1 mesh. Assume: The finite collection of oriented/scaled templates and terminal fills is fixed. Each reserve boundary has the same seven nodes/edges under replacement; no new boundary nodes are inserted. Each current annulus (container minus immediate reserve interiors) has a fixed finite triangulation. All triangles incident to prescribed container boundary nodes are fixed before replacing any strictly interior reserve. Terminal boundary-star weights and these stabilized boundary-cell weights have a finite ratio C_N, uniform under similarities and depth. The total child area is at most eta<1 times its container area. Thus |D_m|<=eta^m|C| and the fixed open annuli partition C up to a null set. These are finite-template premises, not quasiuniformity. C_N may depend on the fixed N and all fixed geometric parameters. No refinement/nesting of the old terminal triangles is assumed. Let V_m be the actual continuous MINI space on the finite mesh, viewed in H=L²(C)^2, and V_m^0 its zero-outer-trace subspace. Define V_infinity to consist of L² fields which restrict to actual MINI polynomials on every stabilized annulus, with the inherited polynomial traces matching on every fixed interface. Outer traces are read on the fixed outer boundary cells. These are closed linear constraints in L²: on each fixed cell polynomial coefficients are continuous linear functionals of its L² restriction. Hence V_infinity and V_infinity^0 are closed subspaces of H. Limit fields need not belong to global H1. Let Q_m be the actual continuous P1 pressure space, with constants included. Let Q_infinity be the closed subspace of H1(C) consisting of functions P1 on every stabilized annulus, with the same fixed seven-node P1 outer trace space. Use mean-zero representatives, or zero outer trace where specified. No claim Q_m is nested is needed.

2. Velocity Mosco convergence in L². Weak condition: if v_m in V_m is bounded and converges weakly in L², then its restriction to each fixed cell eventually lies in the same finite-dimensional polynomial space. Weak limits retain that polynomial form and every fixed trace match. Thus the limit belongs to V_infinity. The same holds for prescribed fixed outer trace, since outer boundary-cell coefficients converge weakly, hence strongly as finite-dimensional vectors. Strong condition: given v in V_infinity, keep it outside D_m. Inside each terminal rectangle H replace it by the actual mean-zero-cell boundary lift of its exact seven-node velocity trace g: E_g=lambda-boundary interpolant ; -(20/9)b_K mean_K(lambda-boundary interpolant). This fill is conforming, has the correct trace and zero cell means. Its L² norm squared is bounded by C sum_i w_i^terminal |g_i|². By the finite template ratio, and reference-polynomial extraction of nodal values on the STABILIZED boundary cells of H, ||E_g||_(L²(H))² <= C_N ||v||_(L²(boundary cells of H))². All such cells lie in D_m. Summing the disjoint holes gives an actual v_m in V_m with ||v_m-v||2²<=C_N ||v||_(L²(D_m))² ->0. (1) The last step follows from |D_m|->0 and absolute continuity of the L² integral. Outer traces are unchanged. This proves L² Mosco convergence, including homogeneous or fixed outer traces. For later use, Mosco convergence of closed subspaces implies strong convergence of their orthogonal projectors. A direct proof is short: projections of fixed f are bounded; every weak subsequential limit lies in the limit space; strong recovery of any limit competitor proves it is the unique nearest point; the projection norm identity then upgrades weak to strong convergence. In particular P_(V_m^0) f -> P_(V_infinity^0) f strongly for every f in H. (2)

3. Pressure H1 Mosco convergence WITHOUT nesting. Weak condition follows by H1 weak restriction to each fixed finite-dimensional P1 cell space, with fixed trace conditions preserved. For strong recovery keep q in Q_infinity outside D_m. On each hole H, fill its exact seven-node boundary trace by ANY fixed terminal P1 extension. This is a finite-dimensional boundary operator on a fixed terminal mesh. Its energy is controlled by the original pressure in H. Indeed every boundary node belongs to an unsubdivided long edge of length comparable to the diameter R_H. The exact edge P1 Gram estimate, rectangle trace and Poincare give sum_i |q(i)-mean_H q|² <= C ||grad q||_(L²(H))². (3) The fixed terminal extension of these seven values therefore has gradient norm squared at most C_N||grad q||H² and L² norm squared, after subtracting mean_H q, at most C_N R_H²||grad q||H². It matches q exactly on the boundary. Gluing it into all holes gives q_m in Q_m and ||q_m-q||_(H1(C))²<=C_N sum_H(1+R_H²)||grad q||H² ->0. (4) The original difference from its hole mean is handled by rectangle Poincare. As in (1), the sum is the gradient integral on D_m and tends to zero. Mean-zero representatives can be restored by one constant; zero outer trace remains exact. Thus Q_m/constant and Q_m^0 Mosco-converge in H1 to their corresponding limit spaces. By Poincare, gradient is an isometry when the quotient is equipped with its gradient norm. Consequently the closed gradient subspaces G_m=grad Q_m, G_m^0=grad Q_m^0 Mosco-converge in L²(C)^2. Their orthogonal projectors converge strongly. This observation is essential for nonnested pressure meshes.

4. Actual boundary moments and constrained velocity recovery. Let T be the fixed seven-node boundary P1 trace space. The Hilbert space of H1 functions with trace in T, modulo constants, has gradient norm. For any seven-vector ell of sum zero, the functional L_ell(q)=sum_i ell_i q(i) is continuous on this space by the edge-Gram/trace/Poincare estimate (3). Its Riesz representative has the form F_ell=grad chi with (F_ell,grad q)=L_ell(q). This is one FIXED ambient L² vector, independent of m; no point evaluation on arbitrary H1 traces is being claimed. The functional is defined only on the fixed finite-dimensional trace class. Every Q_m and Q_infinity lies in that class. The actual feasible sets for fixed boundary velocity g and ALL weak moments ell are K_m(g,ell)={v in V_m with trace g: ; P_(G_m)(v-F_ell)=0}, and analogously K_infinity. Weak limits pass these constraints: strong recovery of each gradient test, or strong projector convergence, gives the limit equations. For strong recovery take v in K_infinity and a velocity approximation w_m from (1), with its exact trace g. Then ||P_(G_m)(w_m-F_ell)||2 ->0, (5) because w_m->v strongly and P_(G_m)(v-F_ell)->P_(G_infinity)(v-F_ell)=0. This is the correct residual argument for NONNESTED Q_m; it does not assume v satisfies every finite-m pressure constraint. Solve the actual scalar Neumann problem (grad chi_m,grad q)=(w_m-F_ell,grad q), q in Q_m. Let e_m|K=beta_K grad chi_m|K, where beta_K=(20/9)b_K has mean one and mean square gamma=10/7. This is an actual zero-boundary MINI field and (e_m,grad q)=(w_m-F_ell,grad q), ; ||e_m||2=sqrt(10/7)||P_(G_m)(w_m-F_ell)||2 ->0. Thus v_m=w_m-e_m belongs to K_m(g,ell) and converges strongly to v. For interior moment constraints only, use Q_m^0 and its Dirichlet scalar problem. The same bubble correction works. Hence all these actual affine constrained sets Mosco-converge. They are nonempty: the established neutral-boundary lift plus full-moment Neumann bubble lift supplies bounded finite feasible vectors for every fixed compatible (g,ell); weak compactness supplies a limit feasible vector. The zero-interior-moment version is even simpler.

5. Strong mass-response convergence and pressure convergence. For fixed forcing f in L², the actual mass response with primal ports (g,ell) is the unique minimizer of J(v)=||v||2²/2-(f,v) over K_m(g,ell). Mosco convergence gives W_m->W_infinity strongly in L². Explicitly weak compactness and weak lower semicontinuity give the limit minimizer; strong recovery bounds the limsup energies; strict quadratic convexity then gives norm convergence. This is not a claim of nested Galerkin monotonicity. Its pressure phi_m is determined modulo constants by (W_m-f,v)+(v,grad phi_m)=0 for v in V_m^0. There is an especially direct actual proof of strong pressure convergence. Define the cellwise weighted-average operator T_m F|K=integral_K b_K F / integral_K b_K. The exact moments give ||T_m||_(L²->L²)<=sqrt(10/7). This operator stabilizes on every fixed annulus. Let T_infinity be its identical local definition on the union of those annuli; the remaining set has measure zero. For fixed F, ||(T_m-T_infinity)F||2 <=2sqrt(10/7)||F||_(L²(D_m)) ->0. (6) The actual bubble momentum equation, with its correct sign, is grad phi_m=T_m(f-W_m). (7) Uniform boundedness, (6), and strong W_m convergence therefore give strong convergence of the pressure gradients to T_infinity(f-W_infinity). After fixing the mean, Poincare gives strong H1 pressure convergence. The weak pressure Mosco condition identifies its limit in Q_infinity. Recovering fixed velocity tests proves the limit momentum equation, and the same exact bubble identity on stabilized cells gives uniqueness of its pressure gradient. No limit equation is applied to arbitrary varying finite-m tests. For fixed dual ports (boundary g, boundary pressure f_boundary), the established construction supplies a SIMULTANEOUS pressure lift p_b in EVERY Q_m: it is P1 on the root current annulus and affine inside every first-generation reserve box, hence is preserved by every terminal fill and descendant replacement. The neutral boundary velocity lift E_g likewise belongs to EVERY V_m, since its supporting outer boundary cells are fixed. It has zero moments against all pressure functions. Let Z_m^0={v in V_m^0:(v,grad q)=0 for all q in Q_m^0}. The constrained Mosco proof above gives Z_m^0 -> Z_infinity^0 and strong convergence of their L² orthogonal projectors. Thus the actual fixed-dual-port response has the explicit formula W_m=E_g+P_(Z_m^0)[f-E_g-grad p_b], (8) which converges strongly. Its normalized pressure gradient then converges by (7). The prescribed boundary pressure is retained because strong H1 convergence preserves its trace. Formula (8) uses actual fixed boundary lifts; it does not assume nesting of either velocity or pressure spaces. If one uses a finite template without a simultaneous lift, strongly converging fixed-trace pressure/velocity lifts from Sections 2--3 give the same conclusion with a strongly varying right side.

6. The actual finite mass DtN matrices converge. For each fixed boundary velocity test trace h choose strong velocity recovery h_m with that trace, and for each boundary pressure test trace k choose strong H1 pressure recovery k_m. Boundary reactions are independent of the chosen extensions by the interior equations. Their formulas involve only (W_m-f,h_m)+(h_m,grad phi_m), ; (W_m,grad k_m) (and any specified source moment subtractions). The strong convergences above therefore give convergence of every reaction entry for every fixed boundary port. There are only finitely many ports (twenty after pressure gauge), so the actual mass DtN matrices converge in matrix norm. Fixed L² source responses converge as well. The conclusion holds for the actual nonnested MINI recursive family satisfying Section 1; no assertion of finite-depth monotonicity was used. Section 7 proves that this limit satisfies the actual autonomous finite Schur fixed-point equations, by checking invertibility of the limiting retained interface blocks. Neither convergence nor that fixed point alone implies spectral amplification, source coupling to a growing eigenchannel, or a positive pressure residue. No real-parameter strain Mosco assertion is made. The neutral terminal fill has controlled L² cost but not a depth-uniform strain cost; replacing the mass argument by H1 energy would be unjustified.

7. The actual limiting finite interface block is invertible. The fixed-point passage uses the actual recursive mixed assembly. Keep the current annulus and all its interface nodal velocity/pressure states. Replace each child finite-depth block by its mass DtN limit just proved, with its correct similarity pullback, pressure gauge and gradient-convention reactions. There are finitely many interfaces and retained coefficients, so this gives a finite matrix H_infinity. Its exterior-to-interior decomposition is exactly the limit of the corresponding finite-depth H_m matrices. Constants are fixed by one pressure gauge; for the zero-exterior-pressure problem the gauge is already fixed. Claim: its interior-interface block is invertible. Suppose an interior-interface state with zero exterior boundary velocity AND zero exterior boundary pressure is in its kernel. Solve every limit child problem with these interface traces, and use the retained current-annulus coefficients. Equal traces glue their velocities into W in V_infinity and their pressures into p in Q_infinity, with zero outer traces. The fixed-cell definition of these spaces makes this gluing legitimate; no global H1 velocity is required. Local child equations and current-annulus interior equations give the global mixed equations against tests with zero interface trace. The zero interface reactions add the equations against every retained interface nodal test. Any limit test decomposes into these finitely many interface lifts plus zero-interface child tests, so the FULL equations are (W,v)+(v,grad p)=0, v in V_infinity^0, ; (W,grad q)=0, q in Q_infinity^0. Take v=W and q=p. They are admissible limit tests and yield ||W||2²=0. Then every stabilized-cell bubble test gives grad p=0 on that cell. The annuli cover C up to a null set, so grad p=0 globally, and zero outer pressure gives p=0. Thus every retained interface coefficient is zero. The finite square interface block has trivial kernel and is invertible. It follows by ordinary finite-dimensional matrix continuity that H_m,II^(-1) converges to H_infinity,II^(-1). Therefore its Schur complement converges to the actual Schur complement of H_infinity. Since the exact finite-depth recursion is D_(m+1)=R(D_m), and both consecutive sequences have the DtN limit D_infinity, one obtains D_infinity=R(D_infinity). (9) For finitely many oriented template types use the vector of their DtN matrices; the same argument proves the simultaneous fixed-point equations. All pressure scaling/gauge maps and current-annulus crossblocks are retained inside R. This is an actual mass fixed point, not an assumed autonomous limit or a consequence of boundedness alone. Equation (9) supplies no eigenvalue or source-visible residue by itself. Differentiation, strain perturbation, renewal/front estimates, and evolutionary pressure observation require their own proofs. In particular mass-space Mosco convergence does not justify differentiating a strain-energy limit or replacing finite-depth normalized strain response by its mass derivative.

Part 5: Source-free affine-Darcy fractional detection. 1. Actual source-free bubble identity and the correct background. Let C be one actual child container of diameter comparable to R, containing no support of the smooth source u. It may contain arbitrarily many recursive descendants. Let z=P_Zh u and let phi_h be its actual mass-Schur potential, with (z,v_h)+(grad phi_h,v_h)=(u,v_h). Write, using the mean-one bubble beta_K, z|K=l_h|K+beta_K c_K, ; gamma=m_K(beta_K²)=10/7, ; p_K=grad phi_h|K. Testing with beta_K e gives, on every cell of C, p_K=-m_K(l_h)-gamma c_K. (1) Define exactly the background selected by the mixed Schur quotient, b_C=-|C|^-1 integral_C grad phi_h. This is NOT the mean of z. Since m_Kz=m_Kl+c, mean_C(z)-b_C=-(gamma-1)mean_C(c). (2) Thus the discrepancy between the pressure-selected constant background and the velocity mean is itself controlled by the bubble coefficient. This identity retains the actual exterior and all its affine potential components. The affine Darcy state z=b, phi=-b dot x+constant has c=0, b_C=b, and zero quotient. Conversely, if the physical fractional seminorm of z on connected C is zero, z is a constant b, c=0, and (1) gives grad phi=-b. Thus the kernel of that seminorm on actual source-free mass solutions is precisely the affine Darcy background.

2. Uniform-in-depth quotient coercivity. Fix 0<s<1. There is a constant depending on s, the fixed container shape and the prescribed mesh shape bound, but NOT on its recursion depth or smallest cell, such that ||z-b_C||L²(C) + ||grad phi_h+b_C||L²(C) <= C_s R^s |z|H^s(C). (3) If the seminorm is infinite this is automatic; for each finite MINI mesh it is finite for s<1 because z is conforming H1. Here |.|H^s is the physical Slobodeckij seminorm, not an intrinsic discrete operator norm. First, finite-dimensional polynomial independence on the reference simplex, followed by affine scaling, gives |l+beta_K c_K|H^s(K)² >= c_s h_K^(-2s)|K||c_K|². (4) The coefficient functional c vanishes on constants, while the fractional seminorm is a norm modulo constants on P1 plus the bubble; hence the reference bound is positive. Shape regularity makes its constant uniform. Since every h_K<=diam C<=CR, ||c||L²(C) <= C_s R^s |z|H^s(C). (5) Local cell seminorms sum to at most the complete C seminorm, so no minimum cell scale occurs. The scaled fractional Poincare inequality on the fixed connected Lipschitz container gives ||z-mean_C z||L²(C)<=C_s R^s |z|H^s(C). Together with (2) and (5) this proves the first term in (3). Finally grad phi_h+b_C=-(m_Kl_h-b_C)-gamma c_K, ; ||l_h-b_C||2<=||z-b_C||2+sqrt(gamma)||c||2, and cell averaging is an L² contraction. These facts prove the second term of (3). The same argument works on a fixed connected annulus template, with its own shape-dependent Poincare constant. For the actual Schur quotient one should use the whole child container and its b_C, as above. In particular a large pressure-selected quotient velocity or quotient pressure gradient cannot be hidden by adding the exact affine Darcy exterior state. At any one depth, child containers are disjoint, so their squared physical seminorm lower bounds from (3) may be summed without counting a nested region repeatedly.

Part 6: Actual uniform mass symbol and its two neutral layers. 1. Actual uniform bulk stencil. Take the square lattice of spacing d>0, splitting every square along its southwest-to-northeast diagonal. Use the actual bubble-eliminated local matrices C_ii=(4/45)|K| I, C_ij=(1/180)|K| I, ; H_ij=(|K|/10) grad lambda_j, ; G=(7/10)K, K_ij=|K|grad lambda_i dot grad lambda_j. At a vertex at least one complete element star from an interface, the assembled equations are C v+H p=0, H^T v-(7/10)K p=0. (1) C is a scalar stencil times the vector identity. Its diagonal entry is 4d^2/15, and its six off-diagonal entries, at offsets +/-e_1, +/-e_2, +/-(e_1+e_2), are d^2/180. At Fourier frequency theta=(x,y), put c(theta)=4/15+[cos x+cos y+cos(x+y)]/90, ; k(theta)=4-2cos x-2cos y, ; A(theta)=2sin x-sin y+sin(x+y), ; B(theta)=-sin x+2sin y+sin(x+y). Then exactly C_hat=d^2 c(theta), ; H_hat=(i d/10)(A(theta),B(theta)), ; K_hat=k(theta). (2) For example the H coefficients at +e_1,+e_2,+(e_1+e_2) are respectively d(1/10,-1/20), d(-1/20,1/10), d(1/20,1/20), with opposite coefficients at the negative offsets. This independently fixes the signs and normalization in (2). All bulk stencils commute. Multiplying the pressure equation by C and using the velocity equation gives the actual compact pressure equation L p=0, L=(7/10)C K+H^T H. (3) The factor 7/10 must be retained with the stated definition of K. Its exact symbol is L_hat=d^2[(7/10)c(theta)k(theta) +(A(theta)^2+B(theta)^2)/100]. (4) Since -3/2<=cos x+cos y+cos(x+y)<=3, 1/4<=c(theta)<=3/10. Also A^2+B^2<=C k, by elementary sine inequalities. Hence (7/40)d^2 k(theta)<=L_hat<=C d^2 k(theta). (5) The only zero on the Fourier torus is theta=0. Near zero, L_hat=(3/10)d^2(x^2+y^2)+O(d^2|theta|^4). Thus this is a genuinely elliptic compact scalar stencil derived from the actual MINI saddle system. It is not an identification with the P1 Poisson Ritz matrix. For the full infinite lattice, (5) gives a rigorous global difference estimate. If L_d=d^(-4)L and D_i^+ is the forward difference divided by d, Plancherel gives, for every finitely supported pressure sequence, sum_ij ||D_i^+ D_j^+ p||_ell2(d^2)^2 <=C ||L_d p||_ell2(d^2)^2. (6) Indeed the square-root multiplier on the left is k(theta)/d^2, whereas L_d has multiplier bounded below by (7/40)k(theta)/d^2. All lattice differences commute with L_d, so the corresponding higher global difference estimates follow by applying (6) to difference sequences. These are full-lattice estimates; applying them across a flat boundary requires the actual boundary rows or a justified extension, not merely the bulk formula.

2. Exact tangential constant-base layer. Take the flat base y=0, and first impose an x-independent tangential nodal velocity v_j=U_j e_1 and pressure p=p_t x+constant. The complete bulk equations reduce exactly to U_(j+1)+25 U_j+U_(j-1)+27 p_t=0. (7) Every scalar pressure row vanishes: H^T v is zero and K p=0. Equivalently, the P1 vector is U(y)e_1, so its gradient pressure pairing with every compactly supported P1 test vanishes. No scalar pressure constraint is omitted. The bounded solution tending to -p_t at infinity is U_j=-p_t+A_t r_t^j, ; r_t=(-25+sqrt(621))/2, -1<r_t<0. (8) Its pressure remains exactly affine. Thus the tangential mismatch between a constant imposed base velocity and the Darcy far field produces a mesh-width velocity layer, not an additional long-range pressure mode. The actual bubbles are recovered from their eliminated equations; the velocity in (8) is the nodal component, not the full MINI field.

3. Exact normal constant-base layer, including its pressure. Now take x-independent normal nodal velocity U_j e_2 and pressure values P_j. For a geometric mode (U_j,P_j)=(U,P)r^j, put s=r+r^(-1). The complete bulk equations are [d^2(25+s)/90] U+[3d(r-r^(-1))/20]P=0, ; -[3d(r-r^(-1))/20]U+[(7/10)(s-2)]P=0. (9) The determinant is exactly d^2(s-2)(109s+862)/3600. (10) Besides the affine/constant branch s=2, there is one decaying branch s_n=-862/109, ; r_n=(-862+sqrt(862^2-4*109^2))/218, ; -1<r_n<0, P=d k_n U, ; k_n=(3/14)[r_n-r_n^(-1)]/[s_n-2]. (11) In particular its pressure amplitude is O(d|U|), and its pressure differences divided by a child scale rho are O((d/rho)|U|). Unlike the tangential layer, this actual weakly constrained normal layer has a nonzero pressure; deleting it would lose a genuine MINI mode. For arbitrary affine far-field pressure p_0+g dot x, the constant nodal velocity -g, together with its actual zero bubble correction, solves the bulk equations. Add (8) and (11) to match either component of a constant base nodal velocity. A gauge adjustment fixes the desired constant pressure value on the base. The base pressure remains affine in the tangential coordinate. Thus the affine/constant incoming background is an exact finite-dimensional family of the actual uniform equations, accompanied by two explicit exponentially decaying mesh layers. At positive heights y>=eta rho, the layer factors are bounded by exp(-c eta rho/d). The normal pressure layer has the additional prefactor d/rho in the scaled child pressure units. These assertions describe the bulk half-plane recurrence with its j=0 nodal values prescribed. They do not assert that an arbitrarily coarsened boundary transition has the same first-row stencil. Coarse boundary edges and the seven-node transition have their own exact local mixed matrices and must be included in the ideal near-field self-response.

3a. Actual uniform-base reaction moments. The first boundary row can also be assembled exactly, rather than discarded. There are three incident half-plane triangles. For x-independent normal data it gives the pressure reaction at one base vertex r_p=-(3d/20)(U_0+U_1)-(7/10)(P_0-P_1). (12) The normal velocity reaction is r_v,n=d^2(25U_0+2U_1)/180+(3d/20)(P_1-P_0). (13) For the decaying normal mode (11), r_p=0 exactly. Indeed (r-r^(-1))/(s-2)=(r+1)/(r-1), so the two terms of (12) cancel. Its velocity reaction is the positive scalar r_v,n=d^2 U sqrt(862^2-4*109^2)/5040. (14) For the tangential mode (8), the pressure reaction is likewise zero and its tangential velocity reaction is r_v,t=d^2 A_t sqrt(621)/180. (15) For the affine background p=p_0+g dot x, v=-g, the boundary velocity reaction is zero and the pressure reaction is d g_n, with g_n the upward pressure derivative. It is the retained normal-flux channel in the elementwise gradient convention. For a finite base segment, the end reactions and tangential nonconstant data must also be retained; (12)--(15) are per-vertex translation-invariant base moments, not a declaration that a finite segment has no endpoint residual. Thus both mesh-width mismatch layers are exactly neutral in the actual base pressure rows. Their boundary response is an explicit positive velocity impedance. The affine normal-flux reaction, constant pressure gauge, endpoint moments and prescribed kink remain separate, as required for an eventual source-free monopole check.

Part 7: Exact characteristic, stable boundary amplitudes and high arcs. 1. Full normalized saddle symbol and exact characteristic polynomial. Take the square lattice of spacing d, every square split southwest-to-northeast, with base y=0 and complete upper half-plane stars. Write a geometric mode at vertex (id,jd) as v_(i,j)=v z^i r^j, p_(i,j)=d P z^i r^j, ; z=exp(i theta). Thus P=p/d is the normalized pressure, while v retains its physical velocity units. Put c=[48+z+z^(-1)+r+r^(-1)+zr+z^(-1)r^(-1)]/180, ; k=4-z-z^(-1)-r-r^(-1), ; h_x=[2(z-z^(-1))-(r-r^(-1))+(zr-z^(-1)r^(-1))]/20, ; h_y=[-(z-z^(-1))+2(r-r^(-1))+(zr-z^(-1)r^(-1))]/20. After dividing the velocity rows by d² and pressure row by d, the FULL actual symbol is F(z,r)=[ cI_2 h ; -h^T -(7/10)k ]. (1) The lower-left sign is the formal lattice adjoint: h(z^(-1),r^(-1))=-h(z,r). It is not a complex conjugation of a complex radial root. Its determinant is exactly det F=-c L, L=(7/10)c k-h_x²-h_y². (2) On the Fourier torus h is purely imaginary, so (2) is the positive pressure factor in the present bulk-symbol note. The factors d disappear ONLY because pressure has been normalized by d. The full unnormalized mixed determinant has its corresponding powers of d. The tangential factor has characteristic quadratic (1+z)r²+(48+z+z^(-1))r+(1+z^(-1))=0. (3) For the pressure factor, write Q=3600 r² L. Its exact quartic is Q(z,r)=q_2(z)r^4+q_1(z)r^3+q_0(z)r² +q_1(z^(-1))r+q_2(z^(-1)), (4) where q_2(z)=-18z²-32z-59, ; q_1(z)=-32z²+100z-612-100z^(-1), ; q_0(z)=2848-612(z+z^(-1))-59(z²+z^(-2)). One direct exact expansion is 3600L=14 Cbar k-9(Abar²+Bbar²), where Cbar=180c and (Abar,Bbar)=20h. This fixes all coefficients of (4) without a numerical root calculation. At z=1, (3)=2(r²+25r+1), ; Q(1,r)=-(r-1)²(109r²+862r+109). (5) Thus the two strictly stable mesh roots have exact limits r_t=(-25+sqrt(621))/2, ; r_n=(-862+sqrt(862²-4*109²))/218, (6) both between -1 and 0. Their reciprocal roots are unstable. The remaining double root at 1 is the scalar physical/gauge branch.

2. The physical branch and the two mesh branches. Set r=exp(i eta) temporarily. The exact bulk symbol gives, for complex frequencies near zero, L(exp(i theta),exp(i eta)) =(3/10)(theta²+eta²)+O((|theta|+|eta|)^4). (7) There is no cubic term: c and k are even and h odd under simultaneous frequency reversal. For theta=t>0, put r=exp(-t omega). Divide (7) by t². The result is analytic at t=0, equals (3/10)(1-omega²) there, and has nonzero derivative -3/5 at omega=1. The analytic implicit-function theorem yields omega(t)=1+O(t²), ; r_ph(t)=exp(-t+O(t³)) =1-t+t²/2+O(t³). (8) For negative theta use conjugation of the actual real stencil. Therefore each side has a one-sided analytic stable branch with the common limit 1 and r_ph(theta)=1-|theta|+O(theta²), ; |r_ph(theta)|<=exp(-c|theta|) (9) for sufficiently small nonzero real theta. There is exactly one stable physical root locally; the other physical root is unstable. Equations (6) continue analytically in theta by simplicity, with absolute value bounded strictly below one in the same neighborhood. There is a useful exact expression for the tangential branch. Put r=exp(-i theta/2)R. Equation (3) becomes 2cos(theta/2)R²+(48+2cos theta)R+2cos(theta/2)=0. Choose the real negative stable root R near theta=0. In particular r_t'(0)=-(i/2)r_t. For the normal mesh branch, differentiating L at (z,r)=(1,r_n) gives partial_theta L=-(17i/900)(r-r^(-1))(r+r^(-1)-2), ; partial_r L=-(109/3600)(r+r^(-1)-2)(1-r^(-2)), and hence r_n'(0)=-(68i/109)r_n. (10) These formulas are optional for the leading boundary amplitude, but verify its ordinary analytic O(theta) corrections.

3. Exact stable mode vectors and invertible boundary matrix. Normalize the tangential mesh vector by v_t=1, the normal mesh vector by v_n=1, and the physical vector by P=1. They can be chosen as e_t(theta)=(1,-h_x/h_y,0) at r=r_t(theta), ; e_n(theta)=(h_x/h_y,1,-c/h_y) at r=r_n(theta), ; e_ph(theta)=(-h_x/c,-h_y/c,1) at r=r_ph(theta). (11) All denominators in (11) are nonzero near theta=0 on the stated branches. Crucially the tangential mesh branch has pressure exactly ZERO at every sufficiently small theta, not just at zero. This follows directly from c=0 and h^T v=0 in (1). Put s_n=r_n+r_n^(-1)=-862/109 and k_n=(3/14)[r_n-r_n^(-1)]/[s_n-2]. (12) The normal vector's pressure is -c/h_y=k_n at theta=0, using the exact characteristic equation. Thus the common zero-frequency limits are e_t(0)=(1,0,0), ; e_n(0)=(0,1,k_n), ; e_ph(0)=(0,0,1). (13) For the physical vector (7)--(8) also give e_ph(theta)=(-i theta,|theta|,1)+O(theta²) in its velocity entries. With columns ordered (t,n,ph), the stable boundary matrix therefore has limit B_0=[[1,0,0],[0,1,0],[0,k_n,1]], det B_0=1. (14) It has a uniformly bounded inverse for sufficiently small theta on BOTH sides of zero. Its entries and inverse are one-sided analytic with the same limit (14). This verifies the actual low-frequency complementing condition with all three base data retained. If the prescribed base Fourier data are (v_base,t,v_base,n,p_base), expand their normalized vector (v_base,t,v_base,n,p_base/d) in these columns. The normalized physical pressure coefficient is A_ph=p_base/d-k_n v_base,n +O(|theta|[|p_base|/d+|v_base|]). (15) Multiplying by d gives the PHYSICAL pressure coefficient a_ph=p_base-d k_n v_base,n +O(|theta|[|p_base|+d|v_base|]). (16) In particular a_ph=p_base+O(d|v_base|)+O(|theta||p_base|), as requested. Both velocity components enter the harmless d-scaled error; the exact leading normal contribution is retained in (16). This is a boundary-mode identity, not a choice of a freely prescribed inner-child pressure field.

4. Actual boundary reactions and the zero-frequency flux distinction. The three triangles incident to one base vertex give exact mode reactions. Let c_b=2/15+(z+z^(-1))/360+(1+z)r/180, ; h_b,x=[z-z^(-1)+r(z-1)]/20, ; h_b,y=(z+2)(r-1)/20, ; eta_b,x=-h_b,x, ; eta_b,y=-[2+z^(-1)+(z+2)r]/20, ; k_b=2-(z+z^(-1))/2-r. Then the actual two velocity reactions and pressure reaction at that base vertex are R_v=d²[c_b v+h_b P], ; R_p=d[eta_b dot v-(7/10)k_b P]. (17) These are uncondensed-in-ports reactions of the actual condensed element assembly; pressure reaction has NOT been set to zero as an extra boundary condition. At theta=0 the two mesh modes (13) satisfy R_p=0 exactly. The tangential identity is immediate. For the normal mode it is exactly -(3/20)(1+r_n)-(7/10)(1-r_n)k_n=0, using (12). The limiting physical/gauge mode has constant pressure and likewise R_p=0. Thus a constant normal boundary velocity mismatch produces the genuine mesh pressure layer of amplitude d k_n v_base,n, but no hidden zeroth-mode base-flux reaction. For the small-frequency physical vector, (17) gives R_p=-d |theta| A_ph+O(d theta²|A_ph|) =-|theta| a_ph+O(theta²|a_ph|). (18) The mesh-mode pressure reactions vanish at theta=0 and are analytic O(d|theta|) times their velocity-normalized amplitudes near zero. They are not declared zero at nonzero theta. At exact theta=0 the bounded physical mode is a pressure gauge, not a decaying pressure mode. A separate affine normal background p=p_0+g_n y, v=-g_n e_n has the actual reaction R_p=d g_n, as in the present exact flat-layer note. This retained normal-flux channel cannot be removed by a pressure gauge. A bounded decaying half-plane calculation with zero net flux excludes that affine growing-pressure branch only when the actual source/interface flux compatibility supplies that exclusion. Finite-segment endpoint reactions likewise remain separate.

5. Away from zero and the descriptor point. On the full Fourier torus c>=1/4 and L is positive except at the simultaneous zero frequency. Hence for real theta different from zero, no root of the full symbol lies on |r|=1. This rules out a unit-circle collision of stable and unstable modes away from zero. Stable roots can collide WITHIN the disk; individual global analytic labels are not needed for the preceding kernel argument's stable-subspace argument. There is one explicit descriptor feature: at theta=pi, 1+z=0, so the tangential characteristic loses its vertical leading coefficient. The stable tangential root tends to zero. In the limiting half-plane recurrence this is a velocity mode supported in the boundary row, with h proportional to (-2,1) and velocity in the orthogonal direction (1,2). It has no pressure. It should be retained as a boundary-supported stable mode, rather than inferred absent from a reduced-degree bulk characteristic. The low-frequency analysis (8)--(16) does not approach that point. For homogeneous zero base velocity AND pressure, a decaying half-plane solution is zero by the actual mixed mass energy identity: testing with its velocity and pressure gives its mass norm zero, then actual bubble tests give zero pressure gradient and decay removes the constant. Consequently the stable boundary map has no kernel wherever the full stable bundle is used, including its boundary-supported mode at the descriptor point. Establishing global stable-bundle smoothness/bounds on compact nonzero-frequency sets may use this energy formulation without choosing individually analytic colliding roots. All recovered bubbles are still the actual ones, obtained from their local mass equations. No zero-cell-mean assumption was introduced in the characteristic or boundary calculation.

6. Uniform high-arc analyticity despite the descriptor point. The stable boundary pressure map can also be justified without a companion matrix at theta=pi. Work in the FULL nodal-plus-bubble half-line spaces at fixed Fourier frequency, after extending the three prescribed boundary values into the first strip. On the homogeneous boundary spaces, the actual mass matrix is uniformly positive and equivalent to the sequence norm. Its scalar pressure Schur operator is S_theta=D_theta M_theta^(-1)D_theta^*, ; S_theta >= (7/10)K_theta. The inequality follows from the actual element bubble capture of pressure gradients; it is the same exact 7/10 used in the condensed symbol. The Dirichlet half-line scalar stiffness satisfies (K_theta q,q)>=(2-2cos theta)||q||_ell2². Thus on every closed real arc separated from theta=0 modulo 2pi, the full homogeneous mixed operator is boundedly invertible, with a uniform bound. Its bounded sequence-space coefficients depend analytically on exp(i theta), as do the finitely supported boundary extensions. Ordinary inverse-operator analyticity supplies an analytic pressure solution map in a complex neighborhood of each point of that arc. Finite covering gives uniform derivative bounds and includes theta=pi. This argument retains all actual bubble and boundary rows; it does not invert the disappearing leading coefficient in (3). For the pressure component, the leading coefficient q_2(z) of (4) never vanishes on |z|=1: the two roots of 18z²+32z+59 have modulus sqrt(59/18)>1. The reciprocal coefficient is likewise nonzero. Therefore its fourth-order scalar recurrence has an ordinary analytic companion matrix on the whole real-frequency circle. Its stable/unstable separation on compact nonzero-frequency arcs follows from the absence of unit-circle roots. Riesz projections onto the stable pressure bundle and its powers then give exponential vertical decay uniform on each such arc, including all fixed-order theta derivatives. Collisions inside that bundle do not affect this conclusion. The missing boundary-supported tangential mode has identically zero pressure and was already included in the full boundary solvability argument. These facts justify the high-arc pressure multiplier regularity used by the preceding kernel argument's Fourier integration, without assuming globally analytic individual root labels.

For this phase $c,h,k,L,r_t,r_n,k_n$ and the stable boundary matrix are those of part 7. The source is at layer $m\ge 1$, $P=p\mathrm{/}d$, and $F=(F_t,F_n)$ and $Q$ are normalized row impulses: $cv+hP=F{\delta }_{j,m}$ and $-h^{T}v-(7\mathrm{/}10)kP=Q{\delta }_{j,m}$. For actual row moments $f_v,f_p$, $F=f_v\mathrm{/}{d}^2$ and $Q=f_p\mathrm{/}d$. Prescribe $v_0=P_0=0$; the base reactions are outputs. Part 8: Actual interior residues, reflection and velocity contact orders. 1. Exact full-line physical residues. Elimination on the full vertical line gives the pressure multiplier P(z,r)=[-c(z,r)Q-h(z,r) dot F]/L(z,r). (2) Let r_+(theta) be the stable physical root and r_-(theta) the unstable physical root. For real theta near zero, r_-=1/conjugate(r_+). Put D_+=r_+ partial_r L(z,r_+). The exact pressure residues above and below the source are A_+=[-c(z,r_+)Q-h(z,r_+) dot F]/D_+, A_-=-[-c(z,r_-)Q-h(z,r_-) dot F] /[r_- partial_r L(z,r_-)]. (3) The corresponding contributions are A_+ r_+^(j-m) for j>=m and A_- r_-^(j-m) for j<=m. The signs follow directly from the Laurent coefficient contour; for the positive scalar model 2-r-r^(-1), its stable coefficient is +1/[r partial_r L]. Write t=|theta| and sigma=sign(theta). The already proved physical root and symbol expansions give c(z,r_+)=3/10+O(t²), ; h(z,r_+)=(3/10)(i theta,-t)+O(t³), ; D_+=(3/5)t+O(t³). Consequently A_+=-Q/(2t)-(i sigma/2)F_t+(1/2)F_n +O(t|Q|+t²|F|), ; A_-=-Q/(2t)-(i sigma/2)F_t-(1/2)F_n +O(t|Q|+t²|F|). (4) All constants in these expansions are independent of m, because m enters only the propagating powers. Thus a scalar pressure-row impulse has pressure order t^(-1); a velocity-row impulse has pressure order zero. For the PHYSICAL pressure residues, multiply (4) by d: a_+ =-f_p/(2t)-(i sigma/2d)f_v,t+(1/2d)f_v,n +O(t|f_p|+t²|f_v|/d), (5) and similarly with the negative normal term below. These are per-node row moments, not density-normalized functions.

2. Exact reflected physical coefficient. Let b(theta) be the last row of B(theta)^(-1): it extracts the stable physical pressure amplitude from a base vector (v_t,v_n,P). Let e_-(theta)=(-h(z,r_-)/c(z,r_-),1) denote the downward physical mode, with pressure normalized to one. Define the exact reflection coefficient R(theta)=b(theta)e_-(theta). (6) Since b e_+=1 for the upward physical column and e_--e_+=(0,-2t,0)+O(t²), ; b(0)=(0,-k_n,1), one obtains R(theta)=1+2k_n t+O(t²). (7) Thus imposing both zero velocity and pressure base values is NOT identical to scalar pressure-only odd reflection. Its leading Dirichlet reflection is one, but its first correction is fixed by the normal MINI mesh layer. Let X_m(theta) be the COMPLETE full-line base vector generated by the source at layer m. It is the sum of the downward physical vector A_- r_-^(-m)e_- and all downward mesh contributions. Write J_m(theta)=b(theta)[X_m-A_- r_-^(-m)e_-]. (8) No mesh component is dropped here. For j>=m the exact outgoing PHYSICAL pressure of the zero-base half-plane solution is P_ph,j=A_+ r_+^(j-m) ; -[R A_- r_-^(-m)+J_m]r_+^j. (9) This is full-line Green minus the actual homogeneous half-plane response matching its complete base vector. Mesh pressure branches in the solution are additional to (9), as required to recover both base conditions. The source's full-line velocity also has a pressure-free scalar-mass branch. At a c-root its residue vector is obtained from v=hQ/L + [F+h(h dot F)/L]/c. Thus its c-root numerator is the exact projection F-h(h dot F)/(h dot h), divided by the appropriate radial derivative. A scalar pressure source never excites this pressure-free branch. A velocity source can excite it, and its velocity enters the matching functional (8). Omitting it would give the wrong tangential first-order coefficient below.

3. Dirichlet cancellation and exact fixed-layer source moments. For the scalar source Q, the leading part of (9), after factoring r_+^(j-m), is -Q/(2t) [1-R(theta)(r_+/r_-)^m]-J_m r_+^m. (10) Since log(r_+/r_-)=-2t+O(t³), (10) has the claimed Dirichlet cancellation [1-exp(-2mt)]/t, with its actual reflection and exponentially localized mesh corrections retained. In particular the bare full-line t^(-1) singularity cancels for every FIXED source layer m as t tends to zero. One must not replace the m-dependent exponential by its Taylor series if m t is not small. Here are the exact first moments for fixed m. They give a particularly sharp check of signs and matching. Put q_t=r_t-r_t^(-1), q_n=r_n-r_n^(-1), ; s_t=-25, s_n=-862/109. Then q_t²=621, ; q_n²=1120*621/109², ; k_n=-109 q_n/5040. (11) At theta=0 the downward full-line NORMAL mesh coefficients are: scalar Q: v_n=-(Q/2)r_n^m, ; P=(k_n Q/2)r_n^m; ; normal force: v_n=-[F_n/(2k_n)]r_n^m, ; P=(F_n/2)r_n^m. (12) Applying b(0)=(0,-k_n,1) therefore gives J_m^Q(0)=k_n Q r_n^m, ; J_m^Fn(0)=F_n r_n^m. (13) The tangential source needs first order. The downward tangential mass mode has velocity amplitude (90/q_t)F_t r_t^m at theta=0. Its contribution to b e_t^- has derivative 14i k_n/q_t. The downward normal pressure residue generated by F_t has derivative -i(s_n+4)F_t/(6q_n). Consequently J_m^Ft(theta) =i theta F_t [1260 k_n r_t^m/q_t² ; -(s_n+4)r_n^m/(3q_n)] +O(theta²|F_t|). (14) For clarity, these derivative identities follow from h_x(theta=0,r)=0 for every r, so its total branch derivative is simply i(s+4)/20; the radial-root derivative term vanishes. The stable tangential mode's normal-velocity derivative is +7i/q_t, and its downward partner's is -7i/q_t. Their difference under b supplies the factor 14i k_n/q_t. The normal pressure coefficient in (14) follows from r_n partial_r L=3q_n/10. Define the outgoing physical coefficient after factoring r_+^(j-m) in (9) by A_eff,m. Equations (4),(7),(13)--(14) give, for each FIXED m, scalar source: A_eff,m -> -Q H_Q(m), ; normal force: A_eff,m -> F_n H_n(m), ; tangent force: A_eff,m =-i theta F_t H_t(m) +O(theta²|F_t|), (15) where H_Q(m)=m-k_n(1-r_n^m), ; H_n(m)=1-r_n^m, ; H_t(m)=m-k_n[1-(140/69)r_t^m+(71/69)r_n^m]. (16) The scalar and normal limits in (15) have O_m(t) errors. The tangential coefficient uses the exact identities in (11), which convert 1260/q_t² to 140/69 and the normal contribution to -71k_n/69. All three H-functions vanish at m=0, as a consistency check of exact boundary matching; m=0 is not an imposed interior source in this problem. Thus the scalar effective height is NOT exactly m: the normal MINI mismatch layer shifts it by -k_n(1-r_n^m). The normal force has a nonzero order-zero pressure moment; the tangent force has a tangential derivative moment with coefficient H_t. These statements concern Fourier pressure amplitudes, not a scalar replacement of the coupled operator.

4. Base-neutral mesh corrections and the orders needed by the proof. The half-plane correction of a downward physical input e_- with unit pressure differs from the stable physical column only by a velocity mismatch O(t). Its stable mesh amplitudes are therefore O(t). Multiplying by the full-line physical source amplitudes (4) gives the reflected NORMAL mesh-pressure amplitude orders scalar row source: O(|Q|) exp(-c m t), ; velocity row source: O(t|F|) exp(-c m t). (17) The stable tangential mesh mode itself has pressure identically zero. Direct full-line source-mesh contributions are exponentially small with the source-to-base layer distance and are included in (8); for low frequencies their physical-row orders are bounded by O(|Q|+|F|)gamma^m with gamma<1. They must not be confused with the reflected physical contribution in (17). Restoring physical pressure and actual row units, (17) is scalar: O(|f_p|) exp(-c m|theta|), ; velocity: O(|theta||f_v|/d) exp(-c m|theta|). (18) This is precisely the scalar order-zero and velocity order-one boundary-layer input needed for the preceding kernel argument's all-port pressure VALUE estimate. A thin pressure mesh layer need not have a small Hessian; its value order (18), not a false uniform second-derivative estimate, is the relevant input.

5. Pressure reaction, compatibility and source scope. The base is zero in v and P, but its pressure reaction is not generally zero. At zero tangential frequency, summing every pressure row (equivalently testing by the pressure partition of unity and taking a far-layer limit) gives R_p,base(0)=-Q (normalized), ; actual R_p,base(0)=-f_p. (19) For a pure velocity row source Q=0 this net reaction is zero. For a scalar pressure impulse it exactly balances the imposed interior pressure moment. The theta=0 solutions in (15) have a constant limiting pressure at large height, with decaying velocity/pressure differences; a pressure gauge does not remove the reaction (19). An assembled source with zero net weak pressure load may cancel this channel after summing its actual pressure impulses and boundary/interface reactions. It may NOT be removed from each individual impulse calculation by assumption. For an actual compact bubble velocity source, static elimination supplies BOTH a velocity load and induced pressure-row loads with their exact coefficients; (1)--(19) apply by linear superposition to that coupled source. They do not authorize independent choice of its pressure load or omission of its mass-potential subtraction in an eventual evolutionary pressure observation. The full-plane and halfplane kernel bounds are established in parts 9–11; the present residue calculation retains their actual source orders.

6. Uniform source-layer versions of the leading amplitudes. The reflected formulas can also be stated uniformly for ALL integers m>=1, which avoids using a fixed-m Taylor expansion when the physical source height m d is large. With t=|theta|, sigma=sign(theta), and the zero-frequency constants r_t,r_n,k_n, equations (3),(6),(8)--(14) give A_eff,m^Q =-Q[1-exp(-2mt)]/(2t) +k_n Q[exp(-2mt)-r_n^m exp(-mt)] +O(t|Q|), A_eff,m^Fn =(F_n/2)[1+exp(-2mt)] ; -F_n r_n^m exp(-mt) +O(t|F_n|), A_eff,m^Ft =-(i sigma F_t/2)[1-exp(-2mt)] +i theta k_n F_t[exp(-2mt) ; -((140/69)r_t^m-(71/69)r_n^m)exp(-mt)] +O(t²|F_t|). (20) The error constants are independent of m. To see this directly, replacing (r_+/r_-)^m by exp(-2mt) costs at most C m t³ exp(-cmt)<=C t². The scalar coefficient has one factor 1/t, leaving O(t|Q|). Replacing a mesh root by its zero-frequency value costs at most C t m gamma^(m-1), uniformly bounded after increasing gamma<1. Its amplitude is O(Q), O(F_n), or O(tF_t), respectively. Likewise r_+^m differs from exp(-mt) by O(t²) uniformly. The expansions of the source residues and reflection coefficient have the orders stated earlier. These observations prove each remainder in (20), with no source-depth restriction. In particular |A_eff,m^Q|<=C|Q|[min(m,t^(-1))+1]. Its singular full-line factor is reduced by the actual base reflection, while the two explicit bounded corrections retain the base-neutral mesh contribution. Equations (15)--(16) are recovered by fixing m in (20).

7. Velocity residues, source-height contacts and shallow reflected mesh inputs. This section independently checks the preceding kernel argument's velocity/neutral-layer use of the pressure residues. It corrects a possible overly strong reading of Section 4: a DIRECT mesh input reflected from a shallow source does not have a purely order-one velocity-force coefficient. There is also an analytic order-zero coefficient. It is harmless at far horizontal separation, but must be separated explicitly. On the physical branches, the leading velocity vectors are (-i theta,+t) above and (-i theta,-t) below. Multiplying (4) gives the leading DIRECT full-line velocity residues: scalar Q: ; v_+=(i sigma Q/2,-Q/2), ; v_-=(i sigma Q/2,+Q/2); velocity force: ; v_+=(-t F_t/2-i theta F_n/2, ; -i theta F_t/2+t F_n/2), ; v_-=(-t F_t/2+i theta F_n/2, +i theta F_t/2+t F_n/2). (21) Thus the scalar jump is the smooth constant vector (0,-Q). The velocity-force jumps are the smooth polynomials (-i theta F_n,-i theta F_t). A first vertical difference across j=m produces these CONTACT terms plus scalar order-one or velocity order-two nonsmooth remainders. A smooth frequency cutoff times a polynomial has arbitrarily rapidly decaying horizontal Fourier coefficients, so the contacts do not change the distant-source difference orders. Away from the source-height crossing, a first target difference directly adds one power of t. The exact higher symbol corrections have the same one-sided analytic structure and the displayed higher orders. In actual row units, for a target separated from the source support by R with d<=R, the physical velocity therefore obeys the precise first-difference scale |D v_ph| <= C [A/R²+B/R³] = C K_*, (22) where A is the sum of absolute actual pressure-row moments and B the corresponding actual velocity-row moments. Here D is any spatial first lattice difference divided by d. For example the scalar coefficient is Q times order one; division by d and inverse Fourier decay (d/R)² give (f_p/d²)(d/R)²=f_p/R². The velocity coefficient is F times order two; its scaling is (f_v/d³)(d/R)³=f_v/R³. Smooth contacts satisfy the same estimates by taking sufficiently many integrations by parts. At large vertical separation exponential factors supply the same bounds; at a source-height crossing the required separation is horizontal. The discrete gradient relation likewise yields |v_ph+grad_d p_ph|<=C d K_*. (23) Indeed h/c equals the central discrete gradient symbol through first order, with order-three error; a forward gradient incurs an additional order-two term. After multiplication by the source pressure orders t^(-1) and order zero, these give the required orders one and two. Contact discrepancies are smooth Fourier polynomials and are rapidly localized horizontally. Thus (23) holds also across a source-height crossing in the distant-source region, without imposing a false continuous global C2 extension. Now let X_m,mesh(theta) be the downward DIRECT source-mesh base vector in (8). Every mesh root and mode is analytic THROUGH theta=0, whereas the physical branch is only one-sided analytic there. For each fixed derivative order k, the direct mesh vector and its derivatives satisfy ||partial_theta^k X_m,mesh|| <=C_k gamma_1^m (|Q|+|F|), (24) with gamma_1<1; factors polynomial in m are absorbed into gamma_1. The pure velocity C-root contribution is included. Multiplying (24) by the stable boundary inverse gives reflected source-mesh coefficient = an analytic O((|Q|+|F|)gamma_1^m) part + a piecewise analytic O(t(|Q|+|F|)gamma_1^m) part. (25) To interpret "analytic part" for Fourier estimates, subtract a common finite Taylor jet and multiply it by a smooth cutoff. Its Fourier coefficients decay faster than any prescribed finite power; the remaining one-sided term vanishes to order one at zero. This is sufficient and avoids asserting a globally analytic cutoff. For a shallow VELOCITY source the analytic O(F gamma_1^m) term in (25) can be nonzero. It must not be claimed absent. Its nonanalytic remainder is, however, exactly order one, as required. Taking a tangential first difference of that remainder produces order two. Replacing the actual analytic mesh roots/vectors by their theta=0 values multiplies by an analytic O(theta) factor: the common analytic contribution remains rapidly localized, while the nonanalytic remainder becomes order two. Consequently the velocity root/vector replacement error is bounded by C d K_* (26) at distant ports. In detail, the shallow velocity term gives F(d/R)³=d B/R³; the scalar term gives Q(d/R)²=d A/R² (or a better power for the secondary term in (25)). Factors j gamma^j from root differentiation are uniformly bounded. The physical-input reflected mesh terms in (17) have scalar order zero and velocity order one and obey the same bounds. Their first tangential amplitude variation is at most K_. DIRECT source-mesh fields themselves use r_beta^(|j-m|), with the proper opposite-side eigenvectors, rather than one common r_beta^j profile across the source height. At far horizontal separation they are smooth Fourier multipliers with the bounds (24), and hence are an O(dK_) velocity remainder after sufficiently many integrations by parts. At large vertical separation their exponential decay gives the same conclusion. Their physical pressure and its first differences are respectively O(d²K_) and O(dK_). They may therefore be put in this explicit remainder instead of being incorrectly factored into a single target-base layer amplitude.

8. Consequences for the actual two-neutral-layer extraction. For the decomposition used in the preceding kernel argument's incoming-velocity note, choose c=v_ph at the base origin. Equation (22) gives |v_ph-c|<=C rho K_* in a neighborhood of radius rho, while (23) and a lattice path give |grad_d p_ph+c|<=C rho K_* for rho>=d. The target mesh amplitudes have tangential variation at most C rho K_* and their constant-root/vector replacement error is (26). The exact zero base velocity then gives b_t(0)=-c_t+O(dK_*), ; b_n(0)=-c_n+O(dK_*). (27) The exact zero base pressure, normal mesh pressure d k_n b_n, and its O(d²K_) root/vector/source-mesh remainder give p_ph(0)=d k_n c_n+O(d²K_*). (28) Taking a TANGENTIAL lattice difference of this zero-base pressure identity shows |D_t p_ph(0)|<=C dK_: the mesh-amplitude variation contributes d K_, and the O(d²K_) pressure remainder contributes the same bound by its neighboring values. Combining with (23) gives |c_t|<=C dK_*. (29) Hence the normal component remains the independent incoming regular component; the tangent component is small for this zero-base distant-interior-source problem. This is not an assumption of zero normal forcing or zero pressure reaction. Equations (27)--(29) give exactly the constant affine field plus the two neutral incident layers in the preceding kernel argument's note, with velocity error C rho K_* and pressure error C rho²K_. A normal pressure-layer remainder of amplitude O(d rho K_) has gradient O(rho K_), not O(rho K_/d). The O(d²K_) source/root corrections have gradient O(dK_). Thus the collar L² pressure-gradient error has scale rho²K_. Finally, for the RAW bubble b_K=27 lambda_0 lambda_1 lambda_2, the exact coefficient in v=l+b_K beta_K is beta_K=-(14/27)sum_(i in K) v_i-(14/9)grad p|K. For the mean-one bubble bbar_K=(20/9)b_K=60 lambda_0 lambda_1 lambda_2, the coefficient is instead c_K=(9/20)beta_K=-(7/10)(mean_K l+grad p|K), since mean_K l=(sum_i v_i)/3. Thus this is exactly the preceding kernel argument's c_K=-gamma^(-1)(mean_K l+grad p), gamma=10/7; the two formulas differ only by the stated bubble normalization. The nodal and pressure-gradient error estimates therefore transfer to the full MINI velocity with collar L² error rho²K_. Nonzero cell means are retained throughout. The only required repair to the informal velocity-force order assertion is the explicit analytic-plus-order-one decomposition (25) and the direct source-height remainder treatment. They introduce no missing leading incoming channel and no violation of K_* or dK_*. Actual finite/coarse transitions and arbitrary near-hole self-response remain outside this uniform distant-source calculation.

Take the uniform upper-halfplane lattice of parts 6–7, prescribe both base nodal velocity $g$ and pressure $f$, enforce all mixed rows at $j\ge 1$, and retain the base rows as reactions. The bounded Fourier solution uses the stable branch at every nonzero tangential frequency; summable data fix its pressure gauge by decay at infinity. Part 9: Distant base kernels. 1. Statement with actual source norms. Suppose f_n and g_n are supported at |nd|>=R. Put F=d sum_n |f_n|+d^2 sum_n |g_n|. Assume 0<d<=rho<=R/8. There is a real affine polynomial a(x,y) such that at all uniform-grid pressure nodes with |x|+y<=rho, |p_h(x,y)-a(x,y)| <=C [rho^2/R^3+d/R^2] F. (1) The constant is independent of d,rho,R and the data. Thus d<=rho^2/R gives max_ports |p_h-a| <=C (rho/R)^2 (F/R). (2) For pressure differences divided by rho, the gain relative to the natural larger-scale gradient F/R^2 is O(rho/R). Any fixed finite set of uniform-grid pressure ports is allowed, including base ports. The theorem concerns distant base drivers in an otherwise uniform half-plane. A curved outer interface, a finite base endpoint, a pressure kink at the child, or a coarse seven-node transition must be separately represented in those drivers or retained in the near-field operator; they are not covered by declaring them absent.

2. Stable-mode facts for the actual coupled equations. Here are the precise mode facts used, with their derivation. Normalize pressure by P=p/d and Fourier transform in n. Write z=exp(i theta) and use a vertical mode r^j. The full mixed determinant factors into the scalar mass factor c(theta,r) and the compact pressure factor. At theta=0 the stable limiting modes are: tangential mesh: (v,P)=(e_1,0), r=r_t, ; normal mesh: (v,P)=(e_2,k_n), r=r_n, ; physical: (v,P)=(0,1), r=1, with r_t,r_n,k_n exactly as in the preceding note. Their boundary matrix has determinant one. The double physical pressure factor at (theta,r)=(0,1) has leading part proportional to theta^2-(log r)^2, by the exact isotropic low-frequency expansion of (4) in that note. Analytic root perturbation on each side of theta=0 therefore gives one physical decaying root satisfying log r_ph(theta)=-|theta|+O(theta^2), ; |r_ph(theta)|<=exp(-c|theta|). (3) Its normalized velocity is O(|theta|), and its pressure is normalized to one. The two other stable modes have modulus uniformly less than one near zero. Their boundary matrix remains invertible and all mode amplitudes are analytic separately for theta>0 and theta<0, extending continuously to theta=0. In particular the physical pressure amplitude in physical units is f_hat-d k_n g_hat,n +O(|theta|[|f_hat|+d|g_hat|]). (4) This retains the effective normal-velocity monopole; zero flux has not been assumed. For completeness, the uniform high-frequency solution bound has a direct operator proof that avoids individually labeled roots. On a closed tangential-frequency arc away from zero, the zero-base half-line matrix has block form [C_theta,H_theta;H_theta^,-G_theta]. Here C_theta is uniformly positive, and G_theta=(7/10)K_theta is bounded below by a positive multiple of the identity, since K_theta includes the tangential contribution 4sin^2(theta/2). The Schur complement G_theta+H_theta^ C_theta^(-1)H_theta is therefore uniformly positive. This proves a uniform inverse bound on the half-line l2 space. Every stencil is finite-band, so conjugating by the weight exp(epsilon j) changes its operator norm by O(epsilon), uniformly on that arc. A Neumann inverse for sufficiently small fixed epsilon proves exponential vertical decay for a boundary-supported forcing. Differentiating the inverse identity in theta gives uniform bounds for all theta derivatives in the same weighted space. Prescribed base data are extended by zero to j>=1, producing just such a forcing in the first few rows. Thus the complete pressure solution multiplier is smooth and exponentially decaying on each closed nonzero-frequency arc, without any hypothesis on globally separate analytic roots. Equivalently, the actual full-torus positivity forbids a unit-circle root away from zero. The stable boundary map has no kernel by the actual mixed mass energy identity. The velocity-only root reaches r=0 at theta=pi, becoming a one-row boundary-supported mode, but the operator proof above includes it automatically. This is why the apparent descriptor singularity does not affect the pressure bounds. The low-frequency facts follow from the exact algebraic proof in the internal proof in part 7, local subsections 1–5. At theta=0 its pressure factor is proportional to -(r-1)^2(109r^2+862r+109), which agrees with the independent normal recurrence and the simple stable normal root in the preceding note. No numerical root calculation or mesh experiment is used.

3. Pressure-kernel bounds. Choose a smooth frequency cutoff equal to one near zero. Split the pressure solution into its cutoff physical branch p_ph and the remaining part p_m. The latter contains both mesh branches near zero and every branch away from zero. All its vertical factors are bounded by exp(-cj). Its multiplier from the input vector (f,dg) is continuous and piecewise smooth at theta=0, with finite one-sided derivatives of every order. Two elementary Fourier coefficient bounds suffice. For the mesh part, integration by parts twice on (-pi,0) and (0,pi) gives |K_m,j(n)|<=C exp(-cj)/(1+|n|)^2. (5) There is no jump in the multiplier itself at zero; the possible derivative jump contributes the displayed n^(-2) term. Derivatives of vertical factors produce polynomial factors in j, absorbed by reducing c. Contributions away from zero are smooth periodic multipliers with the same exponential vertical bound. Extend the cutoff physical branch to real (x,y) with Fourier factors exp(i theta x/d) exp((y/d)log r_ph). Every second spatial derivative introduces a multiplier O(d^(-2)|theta|^2). Its value and first theta derivative vanish at zero. Integration by parts three times, separately on the two half-arcs, therefore gives |D^2 K_ph,y(n)|<=C d^(-2)/(1+|n|)^3. (6) The bound is uniform for y>=0. Indeed the third derivative of theta^2 exp(-c(y/d)|theta|) has uniformly bounded L¹ norm, after rescaling theta by 1+y/d. Analytic amplitudes and the logarithm in (3) have bounded derivatives on each half-arc; the same rescaling bounds every product-rule term. The possible jump of the second derivative at zero gives a bounded n^(-3) boundary term. This proof does not require a maximum principle for the compact stencil. For |x|+y<=rho and sources at |nd|>=R, tangential separation is at least R/2. Convolving (5) with the actual input gives |p_m(x,y)|<=C d F/R^2. (7) Convolving (6) gives sup_(|x|+y<=rho) |D^2 p_ph(x,y)|<=C F/R^3. (8) The powers of d cancel explicitly: in (8), d^(-2)(d/R)^3 times the unweighted input sum F/d equals F/R^3. In (7), (d/R)^2 times F/d equals d F/R^2. Vector components of g are included through their individual pressure multipliers; their normalization is exactly the d in (4). Take a to be the affine Taylor polynomial of p_ph at (0,0). Taylor's theorem, (7) and (8) prove (1). In particular the mesh contribution is controlled even at base ports, where it cancels the small nonlocal physical-branch base value so that the actual prescribed base pressure is recovered.

4. Scope of the completed estimate. This is an actual separated-source half-plane pressure estimate with a concrete power gain, not a universal projector or Rellich hypothesis. It keeps the effective monopole (4) and every actual base reaction. It supplies one component of an incoming/background reduction: far base drivers yield affine pressure near a small child, with the explicit error (1). Still needed for the proposed recursive application are an actual decomposition of the parent's outer/coarse-boundary forcing into such incoming drivers with the stated F bound; treatment of a small kink at the child base; the matching curved/finite-outer-interface estimate; and passage through the coarse seven-node transition while retaining the full ideal child/environment self-response. In particular (1) does not declare the total twenty-dimensional child state close to affine or suppress arbitrary inner-boundary self-excitation.

This phase uses the same uniform upper-halfplane lattice, zero prescribed base velocity and pressure, and actual interior row moments. A nonzero affine or constant base background is removed with the exact two-layer family before this estimate is applied. Part 10: Interior mixed-load kernels and exact cutoff moments. 1. Physical-load statement. Let the actual assembled interior equations be C_phys v+H_phys p=L_v, ; H_phys^*v-(7/10)Kp=L_p, (1) where C_phys=d^2 C_0 and H_phys=d H_0. The arrays L_v,L_p are the actual row moments, not densities. They are supported at distance at least R from the origin in the upper half-plane. Define A=sum |L_p|, B=sum |L_v|. For d<=rho<=R/8 there is one affine polynomial a such that at all uniform pressure nodes |x|+y<=rho, |p-a|<=C[(rho^2/R^2+d/R)A +(rho^2/R^3+d/R^2)B]. (2) In particular d<=rho^2/R gives max_ports |p-a|<=C(rho/R)^2(A+B/R). (3) The scalar-load sum need not vanish. Any additive pressure gauge is absorbed by a. A logarithmic fundamental term from a nonzero scalar moment is smooth near the child because its source is distant, and its second derivative has the R^(-2) bound used in (2). No outgoing inner monopole has been set to zero.

2. Actual low-frequency source orders. Divide the velocity rows by d^2 and pressure rows by d, so the normalized loads are f_v=L_v/d^2, f_p=L_p/d. On the full plane the exact block inverse gives S P=H_0^* C_0^(-1)f_v-f_p, ; S=(7/10)K+H_0^* C_0^(-1)H_0, ; v=C_0^(-1)(f_v-H_0P). (4) After tangential Fourier transform, integrate the vertical Fourier variable by its stable/unstable roots. Near zero the physical pressure factor is a nonzero analytic multiple of theta^2+eta^2. Its simple vertical residue therefore has order |theta|^(-1). The exact H_0 factor vanishes to first order, while C_0 tends to 3/10. Consequently the full-plane physical residue orders are source f_p: pressure order -1, velocity order 0; ; source f_v: pressure order 0, velocity order 1. (5) Here order q means a one-sided analytic symbol bounded by C|theta|^q, with its corresponding differentiated bounds. The leading scalar-pressure residue is -exp(-|theta||j-m|)/(2|theta|), (6) for a unit scalar row impulse at layer m. The leading velocity-impulse pressure residues are the appropriate first derivatives of this physical Green factor. The exact discrete roots replace the displayed exponent by their analytic root factors and change the amplitudes by lower orders; (5) is unchanged. All nonphysical source residues are analytic and bounded near zero, with exponential decay in |j-m|. To enforce zero base data, add the actual half-plane Poisson response to minus the full-plane boundary trace. The physical response amplitude from base data is P_base-k_n v_base,n plus analytic order-one corrections, as independently derived from the three-column boundary matrix. Thus a scalar source retains physical order -1, and a velocity source physical order 0, also for the reflected term. The scalar leading physical factor is the Dirichlet image combination -[exp(-|theta||j-m|) ; -exp(-|theta|(j+m))]/(2|theta|). (7) The difference in (7) makes the zero-frequency half-plane value finite for each fixed j,m. Neither its two separate logarithmic terms nor an arbitrary inner flux is individually discarded. One order distinction is essential for mesh pressure layers. The mesh pressure amplitude from prescribed pressure alone is order one: at theta=0 the pure pressure datum is exactly the physical/gauge column, with no mesh pressure. Combining this with (5) gives the target-side mesh pressure amplitudes from f_p: order 0 times exp(-c m|theta|), ; from f_v: order 1 times exp(-c m|theta|), (8) plus source-mesh terms exponentially small in m. Each target mesh branch has the additional exp(-cj). Pressure derivatives across this mesh-width layer need not obey a macroscopic Hessian bound. Its VALUE is estimated below, rather than differentiating it twice and losing powers of d. The high-frequency part is controlled directly by the uniformly invertible half-line saddle operator and exponential conjugation, exactly as in the preceding half-plane note. On the full line the same argument applies in either vertical direction. This avoids selecting individual roots through collisions or through the velocity-only descriptor root at theta=pi. These source orders and the actual nontrivial reflection coefficient were checked against the exact proof in the internal proof in part 8, local subsections 1–5. In particular the exact reflection is 1+2k_n|theta|+O(theta^2), not identically one; its bounded lower-order contribution is retained in the estimates here.

3. Kernel bounds and scaling. Let n be tangential separation in lattice units. Split the pressure into a cutoff physical part and the remaining mesh part. For sources outside radius R and targets inside radius rho, all direct and reflected physical separations are at least a fixed multiple of R/d. For the physical part, two target spatial derivatives multiply its symbols by order two. The orders -1 and 0 in (5) therefore become orders one and two. Fourier integration by parts, separately on either side of zero, gives |D_lattice^2 G_ph,p|<=C distance^(-2), ; |D_lattice^2 G_ph,v|<=C distance^(-3). (9) For horizontal separation dominating the vertical separation, integrate twice or three times respectively; possible jumps of the first or second derivative at zero give exactly those powers. If vertical separation dominates, rescale theta by that separation in the exponentially decaying physical factor. This is the same elementary kernel argument as for the distant-base note, with one additional inverse power of |theta| in the scalar-source case. There is an additional detail when a target vertical difference crosses the SOURCE HEIGHT j=m. The switched full-line factor can jump there. Insert the two residues on either side before taking that difference. For a scalar source, the resulting leading contact term is a constant times Q, followed by an order-one remainder. For a normal velocity source it is a constant times F_n, followed by an order-two remainder; for a tangential velocity source it is a constant times i theta F_t, followed by an order-two remainder. These facts follow immediately from the residue expansions in the supplier's (4): the apparent order-one normal-force terms cancel between A_+ and A_-, while the tangential sign(theta)|theta| term is the smooth theta term. Smooth cutoff polynomials decay faster than any prescribed inverse power, and the remaining orders give exactly (9). Such a source is far horizontally. This permits discrete second differences; it does not assume a continuously C2 physical part through every source-height line. From (8), the mesh VALUE kernels are bounded by |G_m,p|<=C exp(-cj)/(1+m+|n|), ; |G_m,v|<=C exp(-cj)/(1+m+|n|)^2, (10) plus C_M exp(-c|j-m|)(1+|n|)^(-M) direct source-mesh terms and C_M exp(-c(j+m))(1+|n|)^(-2) source-mesh/base-correction terms. For the first terms in (10), use one or two integrations by parts when |n| dominates m, and direct rescaling when m dominates. One-sided jumps are retained; an order-zero symbol need only have a 1/|n| tail. This is sufficient. Choose M>=3 for the remaining terms. Return now to physical loads and physical pressure. The prefactor is d f_p=L_p for scalar loads and d f_v=L_v/d for velocity loads. Two physical spatial derivatives add d^(-2). Hence (9), convolved with the sources, gives |D^2 p_ph|<=C(A/R^2+B/R^3). (11) Equation (10) gives, without differentiating the target layer, |p_m|<=C(d A/R+d B/R^2). (12) The exponentially localized remaining terms satisfy the same bounds, whether the source is far vertically or horizontally. Use the affine polynomial determined by p_ph at the origin and its two coordinate forward differences there. The discrete Taylor estimate along lattice paths bounds its error by C rho^2 times the supremum of physical second differences in (11). Together with (12), this proves (2). This discrete Taylor argument also covers source-height crossings and makes no continuously C2 extension claim for that part of the Green decomposition.

4. Exact cutoff moments and ring-energy bound. Suppose (v,p) solves the homogeneous actual uniform equations on a half-annular ring of radii comparable to R, with zero residual base velocity and pressure. Choose a smooth nodal cutoff eta equal to one inside the ring and zero outside, with |D eta|<=C/R, |D^2 eta|<=C/R^2. Its exact commutator loads on the ring are L_v=[C_phys,eta]v+[H_phys,eta]p, ; L_p=[H_phys^*,eta]v-(7/10)[K,eta]p. (13) Let V=||v||_L2(ring), E=||grad p||_L2(ring), using the P1 nodal reconstruction for v and the actual P1 pressure. The nodal reconstruction norm is bounded by the full MINI velocity mass norm on a one-cell enlargement, by the uniformly positive local reduced mass form; thus V may be replaced by that actual full-velocity norm. Shape equivalence on this fixed uniform lattice gives sum |L_p|<=C(V+E), ; sum |L_v|<=C(d V+R E). (14) Here is the direct scaling argument, including the gauge issue. The zero residual pressure trace on the flat portions of the half-ring gives ||p||_L2<=C R E by anchored Poincare. C_phys has off-diagonal size O(d^2), so its commutator contributes at most C(d/R)||v||_L1<=C d V. H_phys has coefficients O(d), so its commutator contributes C R^(-1)||p||_L1<=C R E to the velocity load, and C R^(-1)||v||_L1<=C V to the scalar load. Finally, the five-point K commutator is the sum of p K eta and products of first differences. Its scalar-load l1 norm is at most C[R^(-2)||p||_L1+R^(-1)||grad p||_L1]<=C E. Every coefficient is from the actual coupled operator; no pressure row or bubble-induced term is dropped. Combining (3) and (14), for d<=rho^2/R, gives the concrete incoming estimate max_ports |p_incoming-a|<=C(rho/R)^2(V+E). (15) The scalar commutator moment is not presumed zero in deriving (14)--(15).

5. Near-field self-response and remaining application boundary. For a child plus coarse transition inside a radius comparable to rho, extend its trace arbitrarily onto a uniform fictitious interior. Applying the full uniform operator records a finite near-field mixed source array. Keep that source, equivalently its complete actual child/transition DtN, as the ideal self-response. A distant cutoff produces the distinct commutator sources (13). Formula (15) applies only to their incoming pressure response. It does not replace the coarse patch by its fictitious uniform extension or discard its reaction rows. An actual source-free coupled child has zero net inner weak pressure moment only after its own interior rows and all interface-equilibrium rows are summed correctly. Base and outer pressure reactions must be included in that identity. That identity can remove an outgoing monopole when supplied, but is not needed to manufacture (15), which also treats nonzero distant scalar moments. The pressure gauge and any remaining logarithmic channel stay explicit in (6)--(7).

Use the distant-load notation of part 10, put $K_{\ast }=A\mathrm{/}{R}^2+B\mathrm{/}{R}^3$, and assume $d\le \rho \le R\mathrm{/}8$. The base data here are zero. All constants are independent of these scales and the row arrays. Part 11: Full incoming field and both neutral columns. 1. Incoming extraction and the required estimates. There is a linear vector extractor c from the distant loads, namely the physical-velocity value at the base origin, with the following property. Let U_t,U_n be the exact two unit-base-velocity neutral layers, including their bubbles and pressure, as in the flat-layer note. Their nodal velocities are r_t^j e_t and r_n^j e_n. Their pressures, gauged to vanish on the base, are respectively zero and d k_n(r_n^j-1). The same order-zero scalar-source/order-one velocity-source kernels give |c|<=C(A/R+B/R^2). Thus this extraction has the natural incoming gradient scale, independently of the child self-response. The stronger tangential estimate |c_t|<=CdK_* below is a consequence of the exact zero base pressure trace, not a general normal-component estimate. The incident comparison field is v_id=c-c_t r_t^j e_t-c_n r_n^j e_n, ; p_id=-c dot(x,y)-d k_n c_n(r_n^j-1). (1) It is one affine mass solution plus the two neutral incident layers with eta=-c. In particular c_n is the actual incoming regular normal component, not a parent-fan slope substituted for it. At uniform nodal ports with |x|+y<=rho, |v-v_id|<=C rho K_*, ; |p-p_id|<=C rho^2 K_*. (2) For the actual P1 pressure reconstruction in a child-scale collar, also ||grad(p-p_id)||_L2(collar)<=C rho^2 K_*. (3) The collar has diameter O(rho), avoids the distant load support, and uses the uniform mesh on the incident side. The continuous expression in (1) is evaluated at nodes before P1 reconstruction; the exact layers use their actual recovered bubbles for velocity. The nodal estimate in (2) gives ||v-v_id||_L2(collar)<=C rho^2 K_* (4) for its full MINI reconstruction, by the same local bubble formulas and the pressure first-difference bounds described below. The velocity residues, contact cancellations, shallow-source correction and extraction identities below follow from local subsections 7–8 of the internal proof in the internal proof in part 8. Those sections retain the analytic shallow-source term and the pressure-free source-mass branch explicitly.

2. Physical-velocity kernel order. The exact homogeneous mode relation is v=-hP/c. On the physical branches, h/c is the discrete gradient symbol plus an order-three correction. In physical units this gives v_ph=-grad p_ph up to an O(d K_) local error. Source-height contacts must again be treated by finite differences. The scalar pressure-row source has physical velocity residue of order zero; its first target difference has order one, apart from a smooth contact polynomial. A velocity-row source has physical velocity residue of order one; its first target difference has order two, apart from smooth contact polynomials. To check the contact assertion explicitly, the leading velocity residues follow by multiplying the supplier's A_+,A_- by -h/c. For a normal force the normal residues agree to leading |theta| order, while the tangential jump is a smooth multiple of i theta. For a tangential force the tangential residues agree to leading |theta| order, while the normal jump is again a smooth multiple of i theta. For a scalar source the normal jump is constant and the tangential leading residue is the same on both sides. Thus the non-smooth remainders after one target difference have orders one and two respectively. The same Fourier integrations used for the pressure second-difference estimate give sup |D v_ph|<=C K_*, ; |v_ph-c|<=C rho K_* (5) throughout the child neighborhood. Local C_0^(-1) forcing terms are exponentially or rapidly small at this distance and satisfy the same estimates. The pressure physical first differences satisfy |D p_ph+c|<=C rho K_* (6) after allowing the O(dK_) symbol error. Statements (5)--(6) use discrete differences, including source-height crossings, rather than a spurious global continuous C2 extension.

3. Mesh-layer amplitudes and zero-base matching. Write the low-frequency target-side mesh velocity as b_t(theta)e_t(theta)r_t(theta)^j +b_n(theta)e_n(theta)r_n(theta)^j. Its coefficient kernels have the scalar-source order-zero and velocity-source order-one structure established in the residue supplier, together with exponentially localized source-mesh terms. Taking one tangential difference therefore gives the same K_* bound as in (5). Replacing e_beta(theta),r_beta(theta) by their theta=0 values makes an additional O(dK_) nodal error: their differences are O(|theta|), and j r_^j is bounded for a fixed r_<1. The exact mesh roots and vectors are ordinary analytic functions of theta near zero, including their O(theta) phase corrections. High-frequency pieces are handled by the weighted inverse argument. The shallow-source terms need a more precise argument than simply calling every velocity-source amplitude order one. A downward source-mesh base vector has an ORDINARY analytic symbol X_m(theta), with all derivatives bounded by C_k gamma_1^m for some gamma_1<1; derivatives of its root powers introduce only m^k gamma^m, absorbed by gamma_1^m. The boundary inverse is piecewise analytic, with common value at zero. Thus its product with X_m is an ordinary analytic leading part, plus a first nonsmooth term O(|theta||F|gamma_1^m) for a velocity source. The ordinary analytic part, multiplied by the smooth cutoff, has arbitrarily fast tangential coefficient decay. The first nonsmooth term has n^(-2) decay; its tangential derivative has n^(-3) decay. In actual velocity units these are respectively B/R^2 and B/R^3, precisely the required macro amplitude and K_ variation. Multiplication by a target eigenvector/root correction supplies an additional ordinary theta: its product with the analytic part still has rapid decay, while theta|theta| has n^(-3) decay. This gives d B/R^3, the required dK_* profile error. Scalar-source analytic mesh vectors obey the same argument and are no worse than their required bounds. If the source is far vertically, gamma_1^m instead supplies the bound directly. This explicitly includes shallow m and the pressure-free source-mass branch in the residue supplier; neither is omitted. The DIRECT source-mesh field is not factored as a boundary amplitude times r_beta^j across j=m. Its cutoff symbol is ordinary analytic on either side, with exponential source-height separation. At a source-height crossing the source is far horizontally; every target value or first difference then has arbitrarily rapid tangential decay by the analytic-symbol derivative bounds. Choosing that inverse power at least four, in the actual row units, bounds this entire direct source-mesh contribution by O(dK_) in velocity and O(d^2K_) in pressure, with pressure gradient O(dK_). It is included as this remainder, not in the target-side factored layers. Consequently the layer amplitudes vary by at most C rho K_ over the child-scale interval. At the base origin, the exact condition v=0 implies b_t(0 in physical space)=-c_t+O(dK_*), ; b_n(0 in physical space)=-c_n+O(dK_*). (7) Here “0 in physical space” means the inverse-Fourier coefficient evaluated at x=0, not a claim that arbitrary source data are constant along the base. The O(dK_) errors arise from the nonzero-frequency eigenvector corrections just estimated. Direct far-source mesh fields satisfy the same error bounds. Thus the mesh velocity is within C rho K_ of the two constant-amplitude layers in (1). Together with (5), this proves the velocity part of (2). The normal layer pressure is d k_n times its velocity amplitude; the tangential layer has pressure identically zero. Amplitude variation contributes at most C d rho K_* to pressure. Root/vector corrections contribute C d^2 K_. The exact base condition p=0, and (7), give p_ph(0,0)=d k_n c_n+O(d^2 K_*). (8) Moreover the zero base trace forces |c_t|<=C dK_: the tangential derivative of the physical base pressure is canceled by a mesh pressure derivative of this size. Thus replacing the physical affine pressure by the gauged expression (1) gives at most C rho^2 K_* error. Equations (6)--(8), with discrete Taylor, prove the pressure part of (2). One pressure derivative across a mesh layer costs d^(-1), but its remainder amplitude is O(d rho K_), so the derivative remainder is O(rho K_), not O(rho K_/d). Pressure root/profile corrections have the same bound. The shallow-source analytic/nonsmooth split just given also applies after this single normal derivative: the normal layer pressure carries the prefactor d, canceling the difference quotient's d^(-1), and the remaining source orders are the velocity orders already bounded. The pressure physical gradient remainder is bounded by (6). Integrating over collar area O(rho^2) proves (3). For clarity the final full-bubble step is an exact local equation, not a velocity-only reconstruction assumption. Normalize beta_K to have mean one and mean square gamma_2=10/7. In every source-free cell of the target collar, the full actual bubble coefficient satisfies c_K=-gamma_2^(-1)[m_K(l_h)+grad p_h|K]. (10) The ideal affine-plus-layer field obeys the SAME equation. Subtracting it gives |delta c_K|<=C(|m_K(delta l_h)|+|grad delta p_h|K|). Reference scaling yields ||delta v_h||_L2(K) <=C[||delta l_h||_L2(K)+||grad delta p_h||_L2(K)]. The nodal velocity bound in (2) controls the first term over the collar by C rho^2K_, and (3) controls the second. Summing proves (4) for the full MINI velocity, including every cell bubble. No second derivative of the thin pressure remainder was assumed.

4. Ring-energy form and child-scale lift. For the exact distant cutoff loads, A<=C(V+E), B<=C(dV+RE), hence K_<=C(V+E)/R^2. Therefore the complete incoming trace error has velocity amplitude C rho(V+E)/R^2 and pressure amplitude C rho^2(V+E)/R^2. Its child-scale extension mass and pressure-gradient norm are both <=C rho^2(V+E)/R^2. (9) Construct the velocity extension by the actual cell-mean-neutral nodal/bubble lift on the child-scale collar; its L² norm is at most C rho times the velocity amplitude. For pressure, use the incident remainder on that collar and a child-scale cutoff; the additional cutoff derivative costs rho^(-1) times the pressure amplitude and obeys the same bound. This is the scale required by the present passive-child mixed-energy estimate. Extending the velocity error over radius R instead would lose one rho/R factor and is not used. The base trace requires an explicit check. The comparison velocity in (1) is exactly zero at j=0, matching the actual zero base velocity. Its pressure there is -c_t x, while the actual base pressure is zero. Since |c_t|<=CdK_, lift the residual c_t x by that exact affine function throughout the child/coarse transition, cutting it off only on the common uniform annulus. Its pressure-gradient cost is at most C rho|c_t|<=C rho^2K_, including the cutoff term. No interpolation estimate on a skinny coarse cell is involved. Alternatively reset the extracted macro tangential component to zero, its exact supplied base pressure slope in this zero-base problem. This changes (1) by O(dK_) in velocity and O(rho dK_*) in pressure, already covered by (2), and gives exactly matching zero base pressure and velocity. The normal component remains the independently extracted c_n. For an affine nonzero base pressure trace use its exact negative tangential slope in this same reset. The proof the internal proof in part 13, now supplies the correct gluing mechanism. Use its cutoff equal to one near the actual child/transition and varying only on the COMMON uniform annulus. The exterior remainder is extended by zero near every coarse element. Its exact compact mixed commutator has the norms in (9); the passive mass estimate controls the complete scattered correction without any depth-dependent child inverse. Thus no separate coarse-transition interpolation constant is needed for this localization, provided the required common uniform annulus has actually been constructed. The geometric construction and its simultaneous compatibility with the surrounding fixed template are still separate obligations.

5. Binding to the retained mixed map without dropping a channel. Use exactly the radiation reaction J_infty and compact incident driver F_infty of part 14, defined by the finite-energy scattered correction on the auxiliary halfplane. These are not finite far-Dirichlet Schur data. The incident layers solve the uniform local rows and their compact patched residual defines F_infty. The cutoff is an identity on the actual near-field response. Its far commutator load enters the same ring solve as an additional linear right-hand side; its near-field load is retained in the child/transition self-response. The extraction above supplies c, with its independent normal component. After adding a constant base velocity g and affine base pressure background, by exact superposition the layer extractor is eta=g-c. The comparison A(c)+E_infty(D_child)eta retains the complete ideal child/environment feedback. To bound the actual difference, lift the incoming error on the CHILD scale using (9) and then use the present passive-child mixed-energy estimate, including any retained compact pressure moments. No global child-D inverse or false universal fractional projector is invoked. The affine normal reaction is carried by A(c), and both neutral incident layers carry their actual pressure reactions. If a net inner weak pressure moment survives the actual interface sum, its macroport remains; neither (7) nor the pressure gauge deletes it. Still required for a fully assembled application are the precise chosen transition's lift constants, local pressure-kink and unequal-base-velocity terms, and the final recurrent/nonextinction estimates. This note does not claim those or the original smooth pressure conclusion.

6. Distant pressure-only base data: the full-field corollary. For use with the endpoint cutoff, the same proof has a simple base-source version. Let the prescribed base velocity be zero, let the prescribed base pressure h be supported at |x|>=R, and put F=d sum|h|. Then replace K_* everywhere by K_base=F/R^3. The extracted c has |c|<=C F/R^2. All estimates (1)--(9), including the two neutral layers, the O(rho K_base) velocity error, O(rho^2K_base) pressure error and child-scale norm C rho^2K_base, hold. Here are the source-order details, rather than an appeal to a pressure-only estimate. The stable physical pressure multiplier from base pressure is order zero with limit one; its normalized velocity is order one, so one velocity derivative and two pressure derivatives give order two and the R^(-3) separated kernel. Both target mesh VELOCITY amplitudes from a pressure datum are order one: the constant pressure column at theta=0 is the physical/gauge mode, with zero mesh amplitudes. Tangential amplitude variation therefore has the same R^(-3) bound. Root/vector replacement adds another theta, giving the dK_base error after actual pressure normalization. There is no interior source-height contact or direct source-mesh term in this case. High frequencies again use the weighted inverse. Exact zero base data near the child fix the layer amplitudes and pressure gauge exactly as in (7)--(8); full bubbles follow from (10). This corollary retains all far-base pressure reactions and its extracted affine normal component. It does not replace a far pressure datum by zero, nor infer a zero weak flux from its gauge. It also extends to a distant prescribed BASE VELOCITY h_v, with pressure zero, by replacing F with d F_v, where F_v=d sum|h_v|. The physical pressure amplitude is -d k_n h_v,n+O(d|theta||h_v|), and its velocity is order |theta||h_v|. The normal mesh pressure amplitude is O(d|h_v|). Mesh VELOCITY amplitudes have a nonzero common analytic order-zero boundary value, unlike a pressure-only datum. At far tangential separation that analytic part is rapidly localized; their first nonsmooth term is order |theta|, yielding velocity value d F_v/R^2 and first variation d F_v/R^3. Multiplication by the analytic root/vector theta correction gives a smooth linear term plus nonsmooth order two and hence error d^2F_v/R^3. One normal pressure derivative cancels its prefactor d and obeys the same velocity order. Thus the full-field and neutral-layer comparison has K_base=d F_v/R^3 and |c|<=C d F_v/R^2. Combined distant pressure and velocity data use F_total=F_pressure+dF_v. Their actual local base traces are zero before the affine/two-layer comparison; the tiny extracted tangent component may be reset to the exact zero local base pressure slope. Normal macro, all base reactions and endpoint inputs remain retained.

Part 12: Passive mixed mass stability with all weak moments retained. 1. Exact finite glued problem. Let a finite conforming MINI mesh on a connected polygonal region Omega include an arbitrary recursively refined child. Let V_h have prescribed velocity trace g on the entire geometric boundary and let Q_h have prescribed scalar pressure values p_B on a specified nonempty boundary set B. Write V_h^0 for the zero-velocity-trace space and Q_h^0 for the scalar P1 functions vanishing on B. The scalar pressure gauge is fixed by B. Suppose the actual source-free mass equations are (W,z)+(grad p,z)=0 (z in V_h^0), ; (W,grad q)=0 (q in Q_h^0). (1) In particular no pressure reaction at B is set to zero. Those reactions are retained outputs. All child-interface variables are free and satisfy the actual glued equilibrium; no abstract arbitrary saddle matrix is substituted for the child. Suppose E_g is an actual MINI extension of g with ZERO vector mean on EACH cell, and P_B is an actual continuous P1 extension of the prescribed pressure values. Such E_g is obtained from any conforming nodal trace extension by the cell-mean bubble correction. The point here is the exact bound conditional on the actual extension norms, not an assertion that every boundary geometry has a uniform extension constant. Take z=W-E_g and q=p-P_B in (1). Since grad p is cellconstant, (grad p,E_g)=0, ; (W,grad p)=(W,grad P_B), ; ||W||_2^2=(W,E_g)-(W,grad P_B). Therefore ||W||_2 <= ||E_g||_2+||grad P_B||_2. (2) The sign of the pressure term in this identity is negative. The estimate is independent of the number of child refinements and contains no norm of the child's DtN inverse. Local bubble tests z=beta_K c in (1), with mean-one beta_K and gamma_d=mean(beta_K^2), also give |K| grad p|K = -integral_K beta_K W, ; ||grad p||_2 <= sqrt(gamma_d)||W||_2. (3) Every such bubble test lies in V_h^0 even on boundary cells. Thus (2) controls the velocity AND the actual pressure gradient, without a scalar Poisson replacement. The pressure reactions at B remain unconstrained and generally nonzero. The same result holds for the proved passive infinite child limit: use finite glued responses and the already established strong mass/pressure-gradient convergence. No strain estimate or pressure-front limit is used.

2. Scaling for a genuine incoming remainder. Suppose the ring/Green decomposition produces a remainder with velocity boundary values of size e and pressure potential boundary values of size R e on a region of diameter R. If its actual mesh extensions satisfy ||E_g||_2+||grad P_B||_2 <= K R e (d=2), (4) then the complete glued remainder has ||W_rem||_2+||grad p_rem||_2 <= C K R e. (5) Here K must be proved for the actual ring/transition/boundary set. The checked logarithmic pressure lift is one candidate for the pressure part; neither a continuous extension nor an interpolant may be substituted silently. If the decomposition leaves a weak-moment source, the second equation of (1) changes and (2) must be modified. In particular sum of all weak moments zero is not enough to discard a nonzero moment vector. For an incoming affine Taylor remainder e <= C(r/R)^2 A at a child of radius r, (5) gives total response norm at most C K (r^2/R) A. This uses the child's whole passive response, rather than attempting to suppress its inner trace directly.

3. Root velocity extraction, with the sharp star hypothesis stated. On each reference simplex, the MINI local mass form is positive definite in its P1 nodal and bubble coefficients. Consequently, after minimizing over the bubble and all other nodal coefficients, integral_K |v_h|^2 >= c_d |K| |v_h(o)|^2 for any vertex o of K. Summing the cells at a child root o yields |v_h(o)| <= C_d |star(o)|^(-1/2)||v_h||_L2(star(o)). (6) If the actual nonterminal template has |star(o)| >= c a r^2 (the root fan alone gives this lower bound), then (5) implies |W_rem(o)| <= C K e R/(r sqrt(a)). (7) With the affine remainder above, |W_rem(o)| <= C K (r/R)/sqrt(a) A. (8) This is a root-VELOCITY estimate only. It does not control arbitrary pressure coefficients on very thin boundary stars and is not a twenty-state norm equivalence. The background used to define the two layer amplitudes must still be extracted from the incoming regular field; (3) cannot replace that extraction with a raw normal pressure trace.

4. Pressure Taylor error versus velocity error. An incoming pressure affine remainder of size O(r^2/R^2) alone does NOT imply a velocity remainder of that same size. On the physical Darcy branch, W=-grad p, so the corresponding velocity variation is generally first order O(r/R). Equations (4) and (8) are conditional bounds for the stated two boundary amplitudes; they must not be applied to that physical branch with a falsely squared velocity amplitude. A second, potentially applicable scaling uses an actual CHILD-SCALE transition/collar. If its incoming velocity remainder is O((r/R) A), its pressure remainder is O(r(r/R) A), and actual extensions supported on that child-scale region have norm at most C K r(r/R) A, then (2) and (6) give the SAME useful root bound C K(r/R) A/sqrt(a). This requires those actual localized extensions and a valid split from the selfresponse; extending the velocity error over radius R instead loses this factor. Neither the localization nor the physical-branch derivative estimate is asserted here. For branching attachment distances R comparable to the parent tooth separation scale, insert the actual r/R and driver normalization before summing. No branching count or parameter power is inferred in this note. The root-star UPPER bound needed to make the exterior neutral reaction small is a different assertion from the LOWER bound used in (6), and remains a separate geometric obligation.

5. Uniform mixed stability for a retained compact moment residual. The omission of a nonzero moment vector in section 2 can be avoided by the following exact mass estimate. On the same actual finite mesh, with homogeneous velocity and pressure boundary values, suppose (v,z)+(grad p,z)=F(z) (z in V_h^0), ; (v,grad q)=ell(q) (q in Q_h^0), where |F(z)|<=f||z||_2 and |ell(q)|<=l||grad q||_2. There is no assumption ell=0 or a replacement by its total sum. Let chi in Q_h^0 solve (grad chi,grad q)=ell(q). Define the actual cellbubble field b_chi|K=beta_K grad chi|K. It has zero velocity trace, satisfies (b_chi,grad q)=ell(q), and ||b_chi||_2<=sqrt(gamma_d) l. Then v-b_chi lies in the kernel of all the pressure constraints. Testing momentum with v-b_chi gives ||v-b_chi||_2 <= f+||b_chi||_2, ; ||v||_2 <= f+2sqrt(gamma_d) l. (9) Testing momentum with beta_K grad p|K and summing, or using the same bubble right inverse of the gradient pairing, gives ||grad p||_2 <= sqrt(gamma_d)(f+||v||_2). (10) The scalar system for chi is positive definite after the actual prescribed pressure values/gauge, and all steps are finite dimensional. The constants are independent of grading, depth, the child DtN and the location of the residual. In particular a compact mixed-load representation F(z)=(f_loc,z), ell(q)=(v_loc,grad q) costs only ||f_loc||_2+||v_loc||_2. No physical boundary flux is substituted for ell. This permits a scattering/cutoff proof to retain its exact compact moment source rather than force it to zero. To use it for the ring complement, one must STILL construct an actual compact residual representation on the child-scale transition, with both norms O(r(r/R)A), after subtracting the ideal incoming field and retaining selfresponse. A pointwise pressure Taylor estimate by itself does not establish that representation.

Part 13: Localized incoming cutoff and restored scattering. 1. Precise geometric and boundary interface. Assume the actual coarse transition/child lies inside a half-disc of radius r. Outside it there is a common uniform MINI mesh through radii 2r to 8r and on to the outer drivers. Let the geometric base be flat there. Prescribed pressure data on that base are treated as Dirichlet data; their pressure reactions are outputs. This is the local embedding problem, not a claim that the original global pressure has Dirichlet boundary conditions. Compare two incoming fields, one actual and one affine-background-plus-exact-short-layers, both satisfying the SAME uniform mixed equations away from their distant drivers. Their difference d=(v,p) is homogeneous on the common annulus. Its actual pressure and velocity traces on the base need not vanish. After subtracting an additive pressure gauge, assume that its reconstructed fields satisfy on the common annulus and a one-element enlargement ||v||_2+||grad p||_2+r^(-1)||p||_2 <= e r. (1) In addition, the actual boundary traces of d near the child must have actual MINI/pressure lifts of cost at most C e r. This last condition is automatic for zero residual base data, but is a separate interface obligation for an incoming kink or unequal base values. No trace hypothesis is inferred merely from an L² volume norm. The desired application of the independently derived incoming field estimate has e=O((r/R)A), where R is the distant-driver scale. The pressure potential error is O(r e), whereas velocity and pressure-gradient errors are O(e). This distinction avoids assigning a squared Taylor gain to velocity.

2. Exact compact residual on the uniform annulus. First suppose d has ZERO residual base traces, so all comparison unknowns have homogeneous prescribed boundary values after subtracting the incident data. Choose a nodal cutoff zeta equal to one through radius 2r and zero beyond radius 4r, with local first differences at most C h/r and second differences at most C h^2/r^2. Use the same diagonal cutoff on every velocity nodal coefficient and pressure nodal coefficient; on a cell bubble coefficient use the cutoff value at the cell barycentre. Denote the resulting diagonal coefficient operator by J. The field (I-J)d is an actual comparison field outside the coarse transition and is identically zero on a neighbourhood of that transition. Thus it can be extended by zero into the actual child without defining a uniform interpolant on any skinny coarse element. All its nonzero cutoff stencils lie in the common uniform annulus. Let A be the full symmetric mixed mass matrix, with bilinear form a((v,p),(z,q))=(v,z)+(grad p,z)+(v,grad q). Let F be the difference of the true distant-driver loads, including prescribed far trace elimination. Outside the child, d solves the actual uniform equations with load F. Subtracting (I-J)d from the actual full glued difference leaves an unknown with homogeneous far data and residual R = F-A(I-J)d = [A,J]d (2) on the cutoff stencils, and zero elsewhere. Equation (2) is a LOCAL identity: no equation for d inside the coarse transition is required. On stencils wholly inside J=1 the subtracted field is zero; on those outside J=0 the incident equation cancels F. The only remaining stencils lie where the actual and ambient uniform meshes coincide. The exact residual tested against an actual pair (z,q) is R(z,q)=a(Jd,(z,q))-a(d,J(z,q)). (3) Consequently both the weak-moment residual and the velocity residual are retained. There is no inference that their scalar sum is zero or that their reactions vanish.

3. Dual norm of the actual commutators. On the fixed uniform reference cell, coefficient multiplication by the smooth cutoff satisfies ||J_v w-zeta w||_K <= C(h/r)||w||_K, ; ||grad J_p q-zeta grad q||_K <= C[r^(-1)||q||_K+(h/r)||grad q||_K], with the same bounds on a one-cell patch when the barycentre bubble cutoff and the nodal cutoff differ. These follow directly by subtracting the constant cutoff value on the cell: every remaining coefficient difference is O(h/r), and the finite-dimensional reference-cell mass and gradient norms have fixed equivalence constants. The analogous estimate for p has the same form. Summing the bounded-overlap patches gives the corresponding global estimates. Separate (3) into velocity tests z and pressure tests q. For z, insert and subtract pointwise multiplication by zeta. The mass commutator is bounded by C||v||_2||z||_2. The gradient-pressure commutator is bounded by C[||grad p||_2+r^(-1)||p||_2]||z||_2. For q, the velocity-pressure commutator is bounded by C||v||_2[||grad q||_annulus+r^(-1)||q||_annulus]. The annulus is a fixed-shape half-annulus touching a base interval on which q=0. Its anchored Poincare inequality gives ||q||_annulus<=C r||grad q||_annulus. This boundary condition is essential: without it one must retain the constant/monopole moment port, rather than assume the same estimate. Thus R=(F_loc,ell_loc), with exact dual bounds |F_loc(z)| <= C e r ||z||_2, ; |ell_loc(q)| <= C e r ||grad q||_2. (4) No derivative of the velocity is used, and no inverse estimate across the coarse transition is used.

4. Stable scattered response, including nonzero weak moments. Apply the actual mixed mass stability proved in the internal proof in part 12, section 5, to (4). It yields for the whole glued correction, regardless of the child's depth, ||W_corr||_2+||grad p_corr||_2 <= C e r. (5) Near the child the subtracted field (I-J)d is zero, so this correction IS the actual difference of the two full self-consistent child responses. In particular the child's outgoing selfresponse has not been omitted or replaced by a small trace assumption. For residual base traces with the actual lifts required in section 1, subtract those lifts first. Their exact velocity and weak-moment loads are bounded by their L² and pressure-gradient norms, so they add at most C e r to (4)-(5). The prescribed-base pressure reactions remain free outputs. A mere pointwise pressure estimate does not establish the required velocity trace lift; the full incoming-field estimate must supply it. If the root star has area at least c a r^2, the reference-cell mass lower bound gives |W_corr(o)| <= C e/sqrt(a). With e<=C(r/R)A this is the desired root-velocity complement bound C(r/R)A/sqrt(a). This is not an arbitrary twenty-port coefficient estimate. The incoming normal macro background must still be identified from the regular incident field, as specified in the radiation Schur construction in part 14. What remains for application is the actual uniform annulus/transition geometry, the incoming velocity-plus-short-layer estimate with its trace lift, and the quantitative recursive two-driver/nonextinction argument. No original instantaneous pressure claim is certified by this note alone.

Part 14: Radiation-consistent two-driver embedding. 1. Exterior mesh and scattered class. Use an actual child C with seven boundary nodes, its actual finite coarse transition, and a common uniform half-annulus outside that transition. Extend the uniform mesh beyond that annulus to the whole upper halfplane. This extension is an AUXILIARY exterior mesh; the actual parent field is localized into its distant commutator/base drivers. The uniform mesh and its reflected copy use the preceding actual MINI symbols. Let I_eta=(v_eta,p_eta) be the exact two neutral plane layers with base velocity eta and base pressure zero. They solve all uniform interior equations and retain their actual base pressure reactions. They have infinite mass because their constant tangential extent is infinite. We NEVER put I_eta itself in the global L² energy space. Seek an actual composite child/exterior solution whose difference from I_eta outside the coarse transition has velocity in L² and pressure gradient in L², with base velocity eta and base pressure zero. All child exterior-interface velocity and pressure coefficients are equilibrated. Its trace on the child's seven-node boundary, gauged at the root, will define E_infty(D)eta.

2. Compact actual construction. Choose a smooth nodal cutoff zeta equal to one on the entire child/coarse transition through radius 2r and zero beyond 4r, varying ONLY in the common uniform annulus. Define outside the coarse transition v_ext=(1-J_v)v_eta, p_ext=(1-J_p)p_eta, and extend them by zero inside the actual child. This is an actual field: no uniform interpolant is defined on a skinny coarse cell. On the base, v_ext equals (1-zeta)eta and p_ext is zero. Take the actual cell-mean-neutral velocity lift E_g with base trace zeta eta and zero other prescribed trace. It has exactly zero mean on every cell. Under the sharp coarse base-star geometry, ||E_g||_2 <= C(sqrt(a)r+sqrt(h r))|eta|. (1) Only base length O(r) is involved here because the infinite plane layer is already carried by v_ext outside 4r. This is not the earlier finite-ring far-boundary extension calculation. The pair (v_ext+E_g,p_ext) has the desired full base traces. Its residual on free test rows is compactly supported in the common annulus and in the finite support of E_g. Since I_eta solves the uniform equations away from the cutoff, it is an exact mixed commutator residual plus the mass load of E_g. For the neutral layers on a region of tangential extent O(r), their velocity norm and pressure-gradient norm are at most C sqrt(h r)|eta|: the layer coefficient decays geometrically in the mesh height j. Their pressure VALUE is at most C h|eta_n|; the cutoff derivative therefore costs at most C h|eta|. The full-MINI reference-cell cutoff estimates already proved in the internal proof in part 13, with pressure test functions vanishing on the geometric base, imply that the compact residual (F,ell) satisfies |F(z)|<=f||z||_2, |ell(q)|<=l||grad q||_2, ; f+l<=C[r sqrt(a)+sqrt(h r)+h]|eta|. (2) The lift contributes no weak moment because its cell means are zero. The cutoff weak moment ell is retained, not assumed zero.

3. Finite-energy mixed correction on the infinite mesh. For a finite-depth child, define V_0 as the L² closure of actual MINI velocities with BOUNDED SPATIAL SUPPORT and zero geometric-base trace. Define G_0 as the gradient-norm completion of continuous P1 pressures with bounded spatial support and zero geometric-base trace. For the limiting child, use the already proved V_infty/Q_infty INSIDE the child and the actual uniform exterior spaces, glued along their fixed finite interface traces, before taking this exterior spatial completion. Bounded spatial support allows all stabilized child annuli; it does NOT mean finitely many child cells or degrees of freedom. No cutoff is made at the interior refinement accumulation set. All radial far cutoffs below are identically one on the whole child and change only in the uniform exterior. Finite-depth recovery inside the child comes from the prior actual Mosco proof. Every cell bubble lies in V_0. The actual map q -> beta_K grad q|K extends to the pressure completion, has norm sqrt(gamma_2), and satisfies (Bq,grad q)=||grad q||_2^2. Thus the gradient pairing has the same uniform mass inf-sup as on every finite mesh. Solve the residual moment by gradient Riesz and its cellbubble lift, and solve the remaining velocity equation on the closed kernel of the pairing by L² Riesz. Recover the pressure-gradient component by the same inf-sup. This is exactly the finite mixed-stability proof in Hilbert completions; it uses no continuous Leray regularity or projector theorem. It yields a unique correction (delta v,grad delta p) with homogeneous base traces and ||delta v||_2+||grad delta p||_2<=C(f+l). (3) For this pressure completion the halfplane Hardy inequality is explicit: for a smooth finite-support function vanishing on y=0, integration by parts in y and Cauchy-Schwarz give integral |p|^2/y^2 <=4 integral |partial_y p|^2. Density gives it for continuous P1 pressures and their gradient completion. It supplies local L² control on every bounded strip (y bounded above), hence local H1 potentials and the unique zero-base trace/gauge. For a radial far cutoff varying on radius R, its error term integral |p|^2/R^2 on the annulus is bounded by C times the Hardy-weighted tail integral there, because y<=C R. That tail tends to zero. The ordinary uniform-mesh P1 cutoff-product estimate therefore proves density of bounded-SPATIAL-radius pressures in gradient norm, rather than merely postulating it. Velocity coefficient cutoffs are dense in L² by their bounded local mass norms. One may consequently exhaust the auxiliary exterior by nested uniform halfrectangles, impose zero CORRECTION values on their far and base boundaries, and extend each correction by zero. The uniform bubble inf-sup and Hilbert mixed Galerkin stability give convergence of velocity and pressure gradient to the same infinite solution. The child trace and reaction converge because they are finitely many bounded local coefficient/functionals. The interior limiting child is glued through its already proved finite boundary response and strong mass/gradient convergence; this does not require a nonexistent locally finite triangulation at an accumulation set. No L² norm of the infinite plane layer is taken in either construction. Choose the correction with the negative residual. Then W=v_ext+E_g+delta v, p=p_ext+delta p is sourcefree in the actual child and exterior ON BOUNDED-SPATIAL-SUPPORT TESTS (equivalently every local assembled free row) and has the prescribed base traces. Neither total W nor I_eta is asserted globally L², so their full homogeneous equations are not tested against arbitrary global L² fields. Only the compact-residual correction problem is the global Hilbert variational problem. Child/interface reactions use compact local trace lifts. Outside the coarse perturbation, W-v_eta=-(J_v v_eta)+E_g+delta v, ; grad(p-p_eta)=-grad(J_p p_eta)+grad delta p, so the scattered differences have finite energy. Uniqueness follows by applying the homogeneous mixed equations to the difference of two finite-energy scattered solutions, where global Hilbert testing IS legitimate. This retains all interface and base pressure reactions; no extra monopole compatibility equation is imposed or discarded. The same construction works with the proved passive limiting child D: use its actual finite-dimensional boundary response in the composite variational form, or pass from the finite child approximants by the established strong mass/gradient response convergence. Its interior carries no source; all new residuals remain outside or are exactly restored lifts.

4. Child energy, trace and exact operator choice. Inside the child v_ext and p_ext are zero, hence its physical response is (E_g+delta v,delta p). By (1)-(3), ||W||_L2(C)<=C[r sqrt(a)+sqrt(h r)+h]|eta|. (4) Choose h<=c a r. Then (4) is at most C r sqrt(a)|eta|. Its root velocity is exactly eta, because E_g has that trace and the correction has zero base trace. The actual root fan gives the matching lower bound. Thus the sharp ideal-driver energy equivalence holds for this radiation-consistent embedding. Its seven-node trace is a linear 20x2 map E_infty(D), and L_t(E_infty(D)e_t)=1. Reflection covariance holds when the entire auxiliary exterior grids are mirrored. The exterior response defines the corresponding RADIATION Schur J_infty and incident driver F_infty in the scattered class. All earlier exact finite Schur algebra, quotient projection, pairing, and restoration arguments apply with this consistently chosen operator; no equality J_infty=J_finite is claimed. The infinite-halfplane Green/localized-cutoff decomposition must therefore bind to E_infty(D), J_infty and F_infty throughout the actual per-child proof. The finite E_R(D) remains valid for its separate finite Dirichlet problem but cannot be substituted for E_infty without an additional comparison theorem.

Part 15: Compatible exterior Darcy limit and parity repair. 1. Domain, compatible ports and the h-family. Let C_r=[-4r,4r]x[0,4r] and Omega={n>0}\C_r. Its inner boundary Gamma consists of the child's two sides and top. On the flat physical base outside C_r prescribe pressure zero. Only inner pressure traces which vanish at BOTH base corners are considered. The four free side-strip/top pressure coefficients provide exactly this compatible trace space. Their fixed piecewise-affine traces have finite H1/2 extensions. Nonzero independent pressure values at a base corner against zero exterior base pressure would create a continuum corner jump and logarithmically divergent energy; no full twenty-port exterior convergence is claimed. The outer child mesh and its limiting DtN D_h may vary with h, but the exterior is constructed as follows. Keep the same seven inner boundary vertices and all their edges WHOLE. Choose positive collar widths tau_h->0 along Gamma; all exterior triangles incident to those seven vertices are contained in a collar of area tending to zero. Refine EVERY remaining bounded part of the exterior transition away from Gamma, not just the old uniform ring, so its mesh diameter tends to zero locally. On compact sets a positive distance from Gamma choose shape-regular refinements with constants depending only on the fixed finite macrogeometry, not h. Outside the fixed transition use the exact uniform half-plane SWNE mesh of step h. The reflected family uses NWSE. The pressure macrointerfaces below are retained by conforming refinement. Physical-base guards remain on the h lattice, and their exterior stars are O(h²). Such meshes follow from the finite collar construction: place the unrefined Gamma edges against arbitrarily thin strip triangles, then conformingly mesh the remaining fixed macro-polygons at scale h. No additional Gamma node is needed. Preserve a fixed finite macro-P1 extension p_f of each compatible inner pressure trace, compactly supported in a bounded exterior neighborhood and zero on the flat base. This is possible because its endpoint values match zero there. Overlay its finite interfaces before refining. The unrefined-edge anisotropy is confined to the shrinking collar. Shape constants may deteriorate as h->0; for the original recursive construction one eventually chooses ONE positive h and then keeps its finite template fixed with depth. For homogeneous prescribed velocity data, all boundary nodal values are zero. For a fixed inner velocity trace g, its actual cell-mean-neutral nodal/bubble extension E_h g has ||E_h g||2²<=C sum_(inner boundary i) w_i^exterior(h)|g_i|² ->0, (1) since every original exterior boundary star tends to zero. Its moments against ALL actual pressures are exactly zero. This is a mass assertion, not an H1 bound.

2. Continuous energy space and far density. Let Q_0 be the gradient-norm completion of C_c^infinity(Omega), with zero trace on BOTH Gamma and the flat base. Extension by zero into C_r reduces its half-plane Hardy inequality to integral |q|²/n²<=4 integral |partial_n q|². (2) This follows by one-dimensional integration by parts and Cauchy--Schwarz, then density. It supplies local H1 potentials and removes constants. A radial far cutoff equal to one through radius R and zero beyond 2R has gradient error bounded by the Hardy-weighted tail, because n<=2R on that annulus. Thus bounded-spatial-support zero-trace pressures are dense in Q_0. Ordinary bounded-domain H1_0 density then reduces to smooth compact support away from the boundary. No pressure cutoff is taken at an interior child accumulation set: Omega is the exterior domain only. For a compatible inner trace f, define p_0(f) as the unique finite-gradient harmonic function in p_f+Q_0: (grad p_0,grad q)=0, q in Q_0. Its velocity is W_0(f)=-grad p_0(f). This is the continuous mass/Darcy response; prescribed velocity traces have no continuous L² meaning and disappear in this limit.

3. Exterior velocity and gradient-space Mosco convergence. Let V_h^0 be the L² closure of actual bounded-spatial-support MINI velocities with zero prescribed traces on Gamma/base. Then V_h^0 Mosco-converges to ALL L²(Omega)^2. The weak condition is automatic. For strong recovery first approximate any L² field by a smooth compactly supported field away from the boundary. For h small its support misses the shrinking unrefined-edge collar and its affine nodal interpolant converges in L². In fact convexity of barycentric coordinates gives the local L-infinity interpolation error from the smooth field's Lipschitz norm and cell diameter, without a gradient inverse estimate. Far cutoffs give the same conclusion for the completed spaces. Let Q_h^0 be the gradient completion of actual continuous P1 pressures of bounded spatial support, with zero trace on Gamma/base. Their gradients form closed subspaces G_h^0 of L². They Mosco-converge to G_0=grad Q_0. Strong recovery follows from the smooth compact-support density just proved and locally shape-regular refinement away from Gamma. For the weak condition, bounded gradients give local H1 bounds by (2). Any weak limit has zero boundary trace, finite gradient and Hardy control; the far cutoff argument puts it in Q_0. Consequently P_(G_h^0) -> P_(G_0) strongly on L². (3) Let Z_h={v in V_h^0:(v,grad q)=0 for all q in Q_h^0}. Then Z_h Mosco-converges to Z_0=G_0^perpendicular. Weak passage uses strong gradient-test recovery. For strong recovery of v in Z_0 choose w_h in V_h^0 strongly approaching it. By (3), P_(G_h^0)w_h->0. Solve the scalar Dirichlet Riesz equation for this residual and set e_h=beta_K grad chi_h cellwise, with beta_K mean one. It is an actual zero-trace MINI correction and (e_h,grad q)=(w_h,grad q), ; ||e_h||2=sqrt(10/7)||P_(G_h^0)w_h||2 ->0. Thus w_h-e_h is a strong Z_h recovery. In particular the kernel projectors converge strongly.

4. Compatible homogeneous exterior DtNs converge. For inner velocity trace g and compatible pressure trace f, the exact exterior mass response is W_h=E_hg+P_(Z_h)[-E_hg-grad p_f]. (4) The fixed actual macro lift p_f belongs to every exterior pressure space. Equations (1),(3),(4) give W_h -> -P_(Z_0)grad p_f=-grad p_0(f) strongly in L². (5) The pressure gradients converge strongly too. Define T_h F on each cell as (integral b_K F)/(integral b_K). Its norm is at most sqrt(10/7). On every smooth compact support away from the boundary, its mesh diameters tend to zero, so T_h F->F strongly; uniform boundedness and L² density give strong operator convergence T_h->I on L². The exact bubble momentum equation is grad p_h=-T_h W_h. Therefore grad p_h->grad p_0(f). The zero-base trace/Hardy normalization fixes the potential. For an inner velocity test trace k use its actual neutral lift E_hk. Its pressure pairing is exactly zero and its mass norm tends to zero, so its reaction tends to zero. For a compatible pressure test trace j use the fixed macro lift p_j; its reaction tends to (W_h,grad p_j)->-(grad p_0(f),grad p_j). Thus the finite compatible exterior port matrix converges in matrix norm to J_0=[0 0; 0 -E_0], ; E_0(f,j)=(grad p_0(f),grad p_0(j)). (6) This includes the eight free inner velocities and four compatible free pressures used in the present embedding. Velocity crossloads generated by fixed base-corner velocities also vanish, by the same shrinking-star lift; any pressure-corner variation against the grounded exterior base is excluded. The domain and continuous problem are reflection invariant. Hence SWNE and NWSE exterior matrices converge to the SAME reflected-invariant J_0. They are not asserted equal at any positive h.

5. The radiation incident-layer forcing tends to zero. Use the present the internal proof in part 14, not the old finite grounded far-ring problem. Its infinite constant-along-base neutral layers themselves never belong to L². Cut them off only on a fixed common uniform annulus, subtracting that cutoff from the infinite incident field, and use an actual neutral boundary lift for the missing compact base trace and the specified inner-interface velocity values. For the EXTERIOR-only construction all its original boundary-star areas tend to zero. The compact physical-base portion has length O(r), its uniform and transition guard stars are O(h²), and its count is O(r/h). Hence the boundary lift has norm tending to zero. The exact uniform layer and pressure-gradient norms on the cutoff region are O(sqrt(h r)|eta|), and the pressure value/cutoff cost is O(h|eta|). The exact mixed commutator estimate consequently has velocity and weak-pressure dual norms tending to zero. The actual exterior bubble inf-sup is uniformly sqrt(7/10), independent of transition angles and h. Riesz lifting of the retained weak moment and kernel solution therefore gives a finite-energy scattered correction tending to zero. The total incident-plus-scattered solution tends to zero locally near Gamma after its infinite incident tail is excluded; the incident layer itself tends to zero on every fixed bounded neighborhood as well. Testing its reactions with the shrinking neutral velocity lifts and the fixed compatible pressure lifts proves that the restricted radiation driver vector F_h tends to zero. All weak moments are retained. No infinite-layer L² norm is taken. This argument also handles the finite-h corner convention in which a retained corner value and the first exterior-base nodal value differ: its continuous P1 velocity taper lies in the shrinking exterior corner/base stars and has vanishing mass. There is no analogous permission for incompatible pressure-corner data.

6. Uniform child coercivity and glued inverses. After fixing N and every required small-star budget, retain positive INTERNAL child boundary-star areas for all original seven velocity nodes, independent of h. Their stars may be extremely small but fixed. The protected root fan's TOTAL area, even when radially subdivided, is at least c a r²; no single fixed root triangle is necessary. The other boundary nodes can retain fixed positive collar cells inside the child, away from all subsequent exterior ring refinements. Do not impose an h² bound on these ORIGINAL internal corner cells as h->0: the endpoint budget is fixed first, then their positive internal cells are kept. The h² condition applies to NEW exterior-base guards. Exterior corner stars may still tend to zero. Write the actual child DtN, pressure gauge fixed at its root, as D_h=[A_h H_h; H_h^T -E_h]. For velocity- and pressure-generated child responses W_g,W_p, one has W_p in the zero-boundary weak kernel and W_g orthogonal to that kernel. Thus W_g perpendicular W_p, and for S=diag(I_velocity,-I_pressure), s^T S D_h s=g^T A_h g+p^T E_h p=||W_g+W_p||2². (7) Reference-polynomial nodal extraction on the preserved positive boundary stars controls every velocity coefficient by C_N||W||2. The exact bubble equation controls grad p by sqrt(10/7)||W||2; the seven long-edge Gram estimate and rectangle Poincare control the gauged pressure coefficients by C_N||grad p||2. Therefore s^T S D_h s>=c_N |s|², (8) with c_N>0 independent of h and child depth. All child port matrices remain bounded by the fixed internal neutral velocity and pressure macro lifts. The mirror construction makes D_h exactly reflection invariant. Restrict now to the twelve FREE embedding coefficients; the three base pressures are zero, so all its exterior pressure ports are compatible. Finite-h J_h and continuous J_0 have nonnegative signed energy. Combining with (8) gives ||[(D_h+J_h)_II]^-1||<=1/c_N, ; ||[(D_h+J_0)_II]^-1||<=1/c_N. (9) Indeed x^T S_I H_II x>=c_N|x|² and Cauchy--Schwarz yields ||H_II x||>=c_N||x||. This also checks the limiting homogeneous gluing kernel explicitly; it cannot acquire an unpenalized child velocity or pressure port.

7. Uniform comparison of the actual embeddings and symmetry. Let B_0 prescribe eta at all three base velocity nodes, zero base pressure, and let I_0 inject the twelve free variables. With the exact restricted radiation Schur data, the present embedding is E_h(D_h)=B_0+I_0[(D_h+J_h)_II]^-1 ; [F_h,I-I_0^T(D_h+J_h)B_0]. Equations (6),(9), bounded D_h, vanishing velocity crossblocks and F_h->0 imply ||E_h(D_h)-E_0(D_h)|| ->0, ; E_0(D)=B_0-I_0[(D+J_0)_II]^-1 I_0^T D B_0. (10) This comparison is UNIFORM along the actual varying child matrices D_h; it does not assume D_h itself converges. One may equivalently extract a bounded subsequence, but the signed coercivity already gives the required common inverse bound. More explicitly, with K_N bounding ||D_h|| and c_N from (8), the inverse resolvent identity gives ||E_h(D_h)-E_0(D_h)|| <=C(c_N,K_N,||B_0||,||I_0||) ; [||J_h-J_0||+||F_h,I||]. (11) Here the J_h B_0 term is controlled by the shrinking velocity-column lift, even though B_0 contains the prescribed base-corner velocities. All corner PRESSURES are zero. Thus no compactness of a family of singular glued inverses, and no convergence assumption on D_h, is hidden in (10). Since D_h and J_0 are reflection invariant, E_0(D_h) has the exact reflection parity of the two unit drivers. Consequently the undesired odd normal-column projection of E_h(D_h) tends to zero. This is a qualitative after-fixed-N statement. With no lower bound on h, choose one sufficiently small positive h AFTER all N-dependent finite driver/complement constants have been fixed, so its parity defect is below the required tolerance. Then fix that h and its finite actual transition for the subsequent depth family. No uniform rate or original pressure conclusion is asserted. The actual parent only needs the common local uniform ring to match this auxiliary exterior; the latter may extend past neighboring children after localization. All distant fields and outgoing self-response remain in the present exact radiation/cutoff decomposition. Convergence here repairs the exterior parity operator; it does not itself prove the recurrent two-driver estimate or mass fractional divergence.

Part 16: Entire straight guard-base affine remainder. 1. Exact distributional identity. Put the common fan root at (0,0) and the straight base at x=d>0. Let I be an interval in the y coordinate of length L, with Y<=y<=2Y on I and L<=Y/2. Every fan triangle meeting that interval has the root as its third vertex. The base may have arbitrarily many nonuniform subdivisions; introduce the interval endpoints as subdivision points if necessary. Let the actual continuous P1 pressure have constant normal/tangential gradients (N_j,T_j) on these triangles. Across the radial edge joining the root to (d,y_j), pressure continuity gives exactly d [N]_j+y_j [T]_j=0. (1) There is no additive intercept jump because all triangles have the same pressure value at the common root. Regard N,T as piecewise constant functions of y. In the sense of distributions on I, D T=-(d/y) D N, ; D(T+d N/y)=-d N/y^2. (2) The right side is an ordinary L² function. Thus G=T+dN/y has an absolutely continuous representative even when N and T have arbitrarily large individual jumps. This cancellation is the key distinction from summing the absolute values of the jumps. Let C_G be the mean of G over I. The one-dimensional Poincare inequality and y comparable to Y give ||G-C_G||_L2(I) <= C_0 d L Y^(-2)||N||_L2(I), ; ||T-C_G||_L2(I) <= C_1(d/Y)||N||_L2(I). (3) Now choose the actual affine extractor C=mean_I T, rather than C_G. Since the mean minimizes the L² distance to constants, ||T-C||<=||T-C_G||, so the same bound holds. This choice is exactly equivariant under affine pressure additions. Indeed for ANY constant N_0, apply the same argument to G_0=T+d(N-N_0)/y. Its derivative is -d(N-N_0)/y^2; hence ||T-mean_I T||_L2(I) <= C(d/Y)||N-N_0||_L2(I). (3a) The constants are independent of the base subdivision. C is the chord slope [f(right)-f(left)]/|I|, a LINEAR functional of the whole pressure trace. Adding an affine pressure adds exactly its tangential gradient to C and leaves the anchored trace remainder below unchanged. This exact affine covariance is required before working in the Darcy quotient. The intermediate C_G is used only in the proof and is not the final extractor.

2. Pressure trace and its actual energy weight. The base trace f(y)=p(d,y) is continuous and piecewise affine, with derivative T. Fix any y_* in I and define a_I(y)=f(y_*)+C(y-y_*), g=f-a_I. Then g(y_*)=0, and by (3) ||g'||_L2(I) <= C(d/Y)||N||_L2(I), ; ||g||_L2(I) <= C L(d/Y)||N||_L2(I). (4) Point evaluation here is only for a one-dimensional continuous piecewise affine trace; it is not point evaluation on arbitrary H1 functions in two dimensions. The physical parent fan triangles have area d Delta y/2. Consequently ||N||_L2(I)^2=(2/d) sum_K |K| |N_K|^2, (5) with fractional triangles at the two interval endpoints interpreted by the same radial-area formula. Combining (3)-(5) yields L ||g'||_L2(I)^2 + L^(-1)||g||_L2(I)^2 <= C (L d/Y^2) ||grad p||_L2(parent fan patch)^2. (6) For a pressure with an affine background removed, replace grad p on the right by that residual gradient. No bound on the absolute affine base slope C is claimed or needed: that slope belongs to the incoming macro background. The left side is the standard scale-correct bound for the squared H1 energy of a pressure extension from this base interval into a fixed-shape collar of width L, after an endpoint treatment/cutoff. For the actual uniform comparison collar, the corresponding discrete lift still needs to be chosen and its boundary endpoints accounted for. Equation (6) itself is the proved trace estimate; it does not silently interpolate g through the coarse child.

Part 17: Seven-node kink and neutral velocity lifts. 1. Exact decomposition of the three base pressures. In child coordinates let the container be [-4r,4r]x[0,4r], with base vertices c_-=(-4r,0), o=(0,0), c_+=(4r,0). The other four boundary vertices are the two side-strip nodes and the two top corners. Let p_-,p_0,p_+ be the actual continuous P1 base values. Define s=(p_+-p_-)/(8r), ; k=(p_++p_--2p_0)/(8r). On the two entire unrefined base edges the exact trace is p_base(x)=p_0+s x+k|x|. (1) The constant p_0 is the gauge and s is the mean tangential slope. Neither determines the incoming normal macro component. The remaining pressure kink has endpoint amplitude 4r|k|. Its size is not inferred from the center velocity or from normal continuity alone. For the common-annulus base trace, (1) continues to the required O(r) interval if the two parent affine pieces have no other breakpoint there. This is the separated-attachment geometry requirement. It must be checked; if additional breakpoints occur, their actual traces are additional data rather than being suppressed by the three-node formula.

2. A cheap kink lift preserving the seven-node boundary. The literal function |x| would put a new node at the middle of the top boundary. Instead use the tilted crease F(x,y)=|x-y|. (2) Its base trace is |x|. Its sole crease inside the box is the segment from the EXISTING root to the EXISTING top-right corner. On the other boundary edges it is affine: left side: 4r+y, ; right side: 4r-y, ; top: 4r-x. Thus (2) requires no eighth boundary node and no subdivision of an original seven-node boundary edge. Side-strip nodes simply carry the values of these affine traces. The other four pressure ports need not be zero: they are free interface variables in the glued comparison, and all induced internal loads are retained. Requiring them to be zero would be an unnecessary extra condition leading to the logarithmic corner-hat cost. The actual protected cones must lie on fixed sides of this crease. In the current convention, the right protected fan and its reserve holes lie in 0<y<x/3, while the mirrored left ones lie at x<0,y>0 with their corresponding near-horizontal aperture. Hence x-y is positive throughout the former and negative throughout the latter. The crease misses all protected fan interiors and all recursive holes. More invariantly, choose one of the existing top-corner rays from the root that avoids both protected cones; use the corresponding affine linear form that equals x on the base. This is the only macrozone condition used. Make that ray conforming by a finite intersection refinement outside the protected holes, if it is not already a macro edge. Intersections are interior to original boundary edges except at its two existing endpoints. Old P1 pressure lifts remain P1 under this refinement. The protected fans are untouched, so their actual sharp root-star lower bounds persist; refinement elsewhere cannot increase a vertex's original star area. Because F is affine on each whole recursive hole, every deeper copy is exact P1 there without further crease refinements. Once the template parameters are fixed there are finitely many new triangle shapes and the recursion-depth shape bound remains fixed. Extend kF through the child/coarse transition and turn it off only on the COMMON uniform annulus of radii comparable to r. Its crease is aligned with the southwest-to-northeast uniform diagonal in these local coordinates. Nodal interpolation of the cutoff product on that annulus has gradient bounded by C|k|: F=O(r), its gradient is bounded, and the cutoff derivative is O(1/r). Inside the child/transition no cutoff or interpolation is needed. Consequently the actual pressure lift Phi satisfies Phi|base=k|x|, ; ||grad Phi||_2<=C r|k|, ; ||Phi||_2<=C r^2|k|. (3) The constants are independent of recursion depth, aperture, side-strip height, and coarse-element angles. The estimate is obtained from the explicit affine gradients, not an inverse inequality. A transition not yet conformed to the crease must be refined as above; that is an explicit geometric requirement, not an assumed estimate on its old mesh.

3. Actual neutral lift for unequal base velocities. Let g_i at i in {-,0,+} be the actual base nodal velocities and let g_bar be the chosen driver velocity. Write delta g_i=g_i-g_bar. On the actual LOCAL child-plus-ring mesh, define the continuous P1 vector l by these three base nodal values and zero at every other vertex. No ancestor-side elements below the local base are part of this construction. Let S_i=sum_(K incident to i in this local domain)|K|. Reference mass integration, which is independent of triangle angles, gives ||l||_2^2<=C sum_i S_i |delta g_i|^2. (4) Normalize each actual MINI bubble beta_K to have cell mean one. Define Eg=l-sum_K beta_K m_K(l). (5) Its trace is exactly the required P1 boundary velocity trace, since bubbles vanish on every facet. Every cell mean is zero. Thus b(Eg,q_h)=0 for EVERY actual continuous P1 pressure, including pressure extensions of the constrained base hats. Moreover, with gamma_2=m_K(beta_K^2)=10/7, ||Eg||_2^2=||l||_2^2 +(gamma_2-2)sum_K |K||m_K(l)|^2 <=||l||_2^2. (6) This is an exact L² neutral lift; it claims no H1 stability. In particular ||Eg||_2<=C [sum_i S_i|delta g_i|^2]^(1/2) <=C r max_i|delta g_i| (7) when the local stars lie in a child-scale neighborhood. Retain the individual S_i when sharper star sums are available. A small center error does not imply small errors at both endpoints, and no weighted parent estimate is converted to (7) without checking all three actual indices and their weights. If the comparison needs unequal prescribed velocities on additional uniform BASE nodes of the common annulus, use the SAME construction with those actual values included in the nodal sum. Since sum_i S_i over a child-scale supported nodal set is at most C r^2, its mass cost is at most C r times the maximum actual boundary error. In the intended two-affine-piece parent trace, with no further breakpoint on the collar, linear continuation from the three base values bounds these additional errors by a fixed multiple of their three-node maximum. Otherwise keep their individual values and weights. Thus the lift does not silently replace the whole common-annulus base trace by only its center value.

4. Passive correction and exact scale. Subtract Eg and Phi to homogenize the actual residual base data. Their exact mixed loads are retained. The velocity lift is neutral even for base pressure extensions; the pressure lift contributes the velocity load -(grad Phi,z). Therefore their total mass forcing norm is bounded by L_base=C{[sum_i S_i|delta g_i|^2]^(1/2)+r|k|}. (8) The passive-child mixed mass estimate bounds the AUXILIARY homogeneous-base solution (w_aux,p_aux) with forcing -(Eg,z)-(grad Phi,z). The PHYSICAL error pair is (w_aux+Eg, p_aux+Phi). (8a) Its momentum equation is sourcefree by exact cancellation of those two lifted forces, and its weak equation is sourcefree because b(Eg,q)=0. Its base traces are the actual unequal velocity and pressure kink. The restored pair, not the forced auxiliary pair, is the actual child DtN response used in aggregation. Its velocity mass plus pressure-gradient norm is at most C L_base by the triangle inequality and the same passive bound, independently of child depth. Its constrained base reactions, including the restored lift contributions, remain outputs to be added in the parent assembly. If the actual child's root star has the verified lower area c a r^2, the standard local mass lower bound gives |delta v(o)|<=C L_base/(r sqrt(a)). (9) Thus a crude amplitude bound e_g for all three velocity errors yields root error C(e_g+|k|)/sqrt(a). The weighted version (8)--(9) is stronger and should be used if the three local star areas are O(a r^2). No such star estimate is inferred here from a center-only calculation. Combining the uniform distant-driver field remainder with this base lift adds its already-proved child-scale cost to (8). This supplies the local perturbation inputs required by the exact two-driver embedding, while retaining the independently extracted incoming normal macro field and every weak pressure moment.

5. Charging the actual tangential pressure kink. The internal proof in the radial continuity calculation in part 16, Sections 2--3, applies directly to (1). If p_-,p_+ now denote the two parent CELL pressure gradients and t the straight child-base tangent, then k=(p_+-p_-) dot t /2. With comparable attachment halves and parent two-triangle area T_i, the supplied exact continuity identity gives sum_i T_i |k_i|^2 <=C(a_parent/N)^2 ||W-c||_L2(parent fan)^2. (10) The fixed constant background c is arbitrary. Every inherited scalar kink is tangential in this formula; no normal pressure-gradient jump is declared small. The squared pressure-lift costs are r_i^2|k_i|^2. Therefore the exact geometric conversion is sum_i r_i^2|k_i|^2 <=C max_i(r_i^2/T_i)(a_parent/N)^2 ; ||W-c||_L2^2. (11) For a straight parent side at root distance d_i, a base of length 8r_i has T_i=4d_i r_i. Thus r_i^2/T_i=r_i/(4d_i), not one. In the separated design r_i comparable to sigma b_0 d_i, (11) has the additional factor C sigma b_0. This is the actual quantitative input for the kink part of (8); it does not need a logarithmic pressure-extension loss. For velocity errors the fully read the internal proof in part 20 retains all parent outer rows and gives its stated weighted component error D. To insert it into (8), retain the actual ratio S_i/A_i for EVERY selected child center AND endpoint and the actual multiplicity with which an endpoint belongs to selected bases. A center-only estimate cannot supply that triple bound. Nor can a bound on the tangent component be silently applied to the entire vector error; any other component required by the two-driver state must be included with its own actual row/energy control. This note establishes the lift interface, not that missing inheritance estimate.

Remaining use obligation. The parent-row estimates must still bound the actual k and ALL three delta g_i with the correct selected-root/endpoint counts and parent-to-child star ratios. This note proves the lifts and their propagation cost, not those inherited-data bounds or the final recurrent/nonextinction estimate. It does not certify the original instantaneous pressure claim.

Part 18: Full normal and tangential corner/guard velocity budget. 1. Exact shape-independent parent nodal and trace inequalities. On ANY triangle K of area V let the actual MINI vector be W=sum_(i=0)^2 lambda_i a_i+(27lambda_0lambda_1lambda_2)beta. The raw local Gram matrix, divided by V, has P1 diagonal 1/6, off-diagonal 1/12, P1-bubble entries 3/20 and bubble diagonal 81/280. Eliminating its bubble gives the nodal Schur matrix C=(1/12)I+(1/180)11^T, ; C^(-1)=12I-(2/3)11^T. (1) Therefore for ANY constant vector c, ANY pair of vertices i,j, V |a_i-a_j|² <=24 ||W-c||_(L²(K))², (2) ; V |a_i-c|² <=(34/3)||W-c||_(L²(K))². (3) These constants are exact dual Gram norms: the difference vector has squared norm24 and a coordinate vector has squared norm34/3 in C^(-1). Bubble coefficients and nonzero cell means remain arbitrary. Since affine pullback multiplies the entire mass Gram by V, neither estimate depends on the triangle's angle or aspect ratio. Equation (2) holds for the whole vector, including its normal component. There is also an exact edge trace inequality. Suppose edge e has length L and its opposite root has perpendicular distance d from its line; thus V=Ld/2. Eliminate first the bubble and then the root nodal coefficient from (1). The remaining two-node edge mass matrix is [[17,1],[1,17]]/192. The edge-trace matrix divided by L is [[1/3,1/6],[1/6,1/3]]. Its largest generalized eigenvalue against the preceding matrix is16/3 (the constant edge mode; the difference ratio is2). Hence, with g=W|e, integral_e |g-c|² <=(32/(3d))||W-c||_(L²(K))². (4) This is an actual P1-plus-bubble mass/trace statement, not a gradient trace theorem on skinny elements.

2. Child-center choice and arbitrarily cheap endpoint errors. For child i let its base root be o_i and its two corners e_i^-,e_i^+. Its unrefined base edges are actual straight parent fan edges after the child corner insertion. Let K_i^-,K_i^+ be the parent triangles on those edges, of areas V_i^-,V_i^+. Choose gbar_i=W(o_i), ; delta g_i(o_i)=0, ; delta g_i(e_i^h)=W(e_i^h)-W(o_i). (5) The definition uses the entire actual vector; no tangential projection or pressure-gradient replacement occurs. Let S_i^h be the total CHILD-SIDE local star area at that corner, including its ring/transition cells but excluding every ancestor/parent cell. Impose the explicit independent corner budget S_i^h <=epsilon_corner V_i^h. (6) The child root has its unavoidable fan star; it does not matter because its error in (5) is exactly ZERO. Child base corners are not fan roots and have no such fan-volume floor. The geometric construction must keep both entire base edges and all seven boundary nodes while enforcing (6); this proof does not derive (6) from a selected-center star condition. Let Eg be the actual cell-mean-neutral nodal/bubble lift of these endpoint errors, as in the structured seven-node lift note. Reference P1 mass positivity and the mean-neutral correction give ||Eg||² <=(1/3)sum_(i,h) S_i^h |delta g_i(e_i^h)|². Using (2) on the SINGLE small parent triangle K_i^h yields ||Eg||² <=8 epsilon_corner ; sum_(i,h)||W-c||_(L²(K_i^h))². (7) In particular one must compare S_i^h with the small base-triangle V_i^h, rather than with the larger FULL parent endpoint star. The latter comparison can insert an avoidable factor1/sigma. Equation (7) has no such loss. The chosen attachment windows have bounded overlap, so summing their parent patches gives sum_i ||Eg_i||² <=C epsilon_corner ; ||W-c||_(L²(parent fan))². (8) All normal antisymmetric endpoint patterns are included. They may have order-one amplitude. Equation (8) charges their lift energy correctly without claiming that pressure continuity makes them small.

3. Extra uniform-base guards: an explicit h/sigma loss. Let the common comparison window on a straight tooth side have length L_i<=C R_i. Its parent-root height is d_i; let r_i be the child radius, so the root-to-corner base length is4r_i and V_i^h=2r_i d_i. Added uniform base guards outside the unrefined child base lie at spacing h, with CHILD-SIDE stars at most C_star h². Their exact actual inherited trace is g(s), piecewise affine on the matching parent boundary subdivision. Define every guard error by delta g(s)=g(s)-g(o_i). Do not interpolate this data from only the three coarse base nodes if there are additional breakpoints. The exact one-dimensional P1 mass matrix implies h sum_guards |delta g(s_n)|² <=6 integral_window |delta g(s)|². Here each guard is incident to a matching uniform base edge of length h; endpoint multiplicities can be included in a fixed constant. Combining this with the star bound gives sum_guards S_n |delta g(s_n)|² <=C h integral_window |g-g(o_i)|². Choose any constant c. Equation (4) on all parent triangles in the window, and (3) on either small center base triangle K_i^0, give sum_guards S_n |delta g(s_n)|² <=C[(h/d_i)||W-c||_(L²(parent window patch))² +(h L_i/V_i^0)||W-c||_(L²(K_i^0))²]. (9) Thus if L_i comparable to R_i and V_i^0 comparable to r_i d_i, guard budget <=C[h/d_i+h R_i/(r_i d_i)] ; times parent energy. The geometry R_i comparable to b0 d_i and r_i comparable to sigma R_i gives h/d_i+h R_i/(r_i d_i) <=C h/(sigma d_i). (10) The extra1/sigma comes from subtracting the fixed center value over the larger comparison window. It is explicit, and can be paid by the independently freely chosen uniform width h. It must not be omitted merely because the ordinary uniform guard stars are O(h²). Window and center patches have bounded overlap. The complete full-vector neutral-lift budget is therefore sum_i ||Eg_i,allbase||² <=C eta_base ||W-c||_(L²(parent fan))², eta_base=epsilon_corner+max_i h/(sigma d_i). (11) A nonuniform uniform-ring width may instead use the exact ratios in (9). Source-free parent equations or pressure estimates were not needed for this purely kinematic mass bound.

4. Actual averaging scale and independent geometric choices. Let the parent radius be R. There are B comparable children of radius r, with B comparable to N²/(b0 a), ; r/R comparable to sigma b0 a/N, ; S_0=B(r/R)² comparable to sigma² b0 a. (12) The actual passive child correction estimate converts each lift cost in (11) to its full mixed response energy, retaining its induced reactions. After isometrically identifying the actual child template copies (or averaging separately within each matching type), Jensen and scaling in the actual unit-child energy norm give the averaged-error operator bound ||mean child correction||_(unit energy) <=C sqrt(eta_base/S_0) ; ||parent state||_(unit energy). (13) Indeed physical child energies are r² times unit-child forms; summing (11), dividing by B r², and multiplying the parent unit-energy normalization R² yields exactly eta_base/S_0. The aperture in S_0 is present. No pressure-invisible velocity channel is inferred from this estimate; it is a local perturbation input for the present actual state recurrence. Thus the sufficient geometric requirements are explicitly epsilon_corner <<sigma² b0 a, ; h/(sigma d_i) <<sigma² b0 a. (14) The latter is h<<sigma³ b0 a d_i. Both choices concern corner collars and uniform ring width, which are independently shrinkable once the finite template is fixed, provided the geometric construction ensures conformity and preserved interface nodes. Their finite minimum angles may deteriorate with that one template choice, but remain fixed as recursion depth grows. The previously verified sharp center star S_o<=C a r² is compatible with this argument and is not falsely made arbitrarily small: its multiplier is zero by (5). All endpoint and guard normal differences are paid by (11), regardless of their amplitude.

Part 19: Whole refined-base pressure, nested tails and physical restoration. 1. Inputs and the exact local trace remainder. Use one straight parent tooth side with root distance D, longitudinal coordinate s comparable to Y (the parent radius), and the comparison window I=[-16R,16R] in child-centered tangential coordinates. Here R is a sufficiently small FIXED multiple of b0 D, r is comparable to sigma R, and r/R is small. Choose R<=ell/1000, with ell the same-side attachment spacing. Every parent triangle over I still has the common parent root, including all new radial guard subdivisions. The child occupies [-4r,4r]x[0,4r]. The cutoff hierarchy is fixed throughout this note: the BULK commutator is supported between scales R and 2R, with its one-cell enlargement inside S_(3R); the PRESSURE lift cutoff equals one on S_(4R) and zero outside S_(8R); the retained BASE TAIL therefore starts at |x|>=4R. To bind the existing geometry notation, use R_geom=16R in the uniform-ring construction. It supplies exact uniform cells through S_(4R_geom)=S_(64R), while its clearance condition 8R_geom=128R<ell/4 is enforced by the chosen small fixed factor. The enlarged interval I and all cutoff stencils stay on the same straight side, avoid neighboring children, and remain away from side endpoints. None of these fixed factors changes the powers of b0,sigma,a,N. Use the AFFINE-EQUIVARIANT extractor C_I=mean_I T=[f(right)-f(left)]/|I| and a_I(x)=f(0)+C_I x. Thus g=f-a_I is unchanged under addition of ANY affine pressure, including a nonzero normal affine gradient. The internal proof in the internal proof in part 16 gives g(0)=0 and R ||g'||_L2(I)^2+R^(-1)||g||_L2(I)^2 <=C (R D/Y^2) E_I^2, (1) where E_I is the actual parent-patch pressure-gradient norm after an arbitrary selected affine background is removed. To verify the mean-T choice explicitly, for ANY constant N_0 use G=T+D(N-N_0)/s, whose derivative is -D(N-N_0)/s^2. The supplied bound for T-meanG and the contraction of mean subtraction give ||T-meanT||<=2||T-meanG||<=C(D/Y)||N-N_0||. The exact physical area D ds/2 then gives (1). The numerical value meanG is NOT used as the macro extractor. Constants do not depend on the number or sizes of base guard intervals. The actual child base [-4r,4r] retains just its three nodes, so on that subinterval alone g(x)=a x+k|x|. (2) Outside it, g has ALL its actual extra guard breakpoints. None is assigned the value of the continuation of (2). Use the allowed geometry choice confirmed by part 2: every new physical-base guard in this comparison window has tangential coordinate in the common d-lattice, with r/d dyadic. The child three nodes already have this property. Exterior collar heights can still be arbitrarily small and nonuniform. Corner-star surgery inserts only interior points and preserves these base edges. This choice does not bound pressure values or remove any pressure degree of freedom.

2. A fixed macro correction around the seven-node child. Construct a continuous piecewise-affine S, on a fixed child-scale macro subdivision, with S=|x-y| throughout the child box, ; S=|x| outside S_(8r), ; S|base=|x|, ; |S|<=C r, |grad S|<=C on its correction support. Here the last size bound is only required where S differs from |x|. The construction is elementary: triangulate the fixed U-shaped polygon between the child rectangle and S_(8r), assign the inner trace |x-y| and outer trace |x|, and interpolate on a fixed nondegenerate macro triangulation. The top midpoint of S_(8r) is allowed as a uniform-grid node; no midpoint is inserted on the CHILD top edge. The child side-strip nodes lie on affine macro edges; inserting them does not change S on those macro triangles. Its gradients therefore remain bounded independently of the side-strip height. Inside the child use only the root-to-existing-top-right crease. This S is affine on each protected fan and whole reserve hole, as in the cheap kink-lift proof. Overlay its finite interfaces with the transition, never subdividing a child boundary edge. Refinement preserves old affine pressure lifts and does not enlarge any old vertex star. Outside S_(8r), the exact uniform grid is unchanged and S-|x| vanishes. Define before the distant cutoff Psi(x,y)=g(x)+k[S(x,y)-|x|]. (3) Inside the child, (2) makes this exactly a x+k|x-y|. Thus the spurious vertical kink at x=0 cancels there, all seven boundary traces are affine, and no eighth node appears. Outside the child, the extra vertical g-breakpoint lines have |x|>=4r and never enter the child interior. On the uniform ring they are actual vertical lattice edges. The transition is conformed to those finitely many vertical interfaces and to the fixed S interfaces; it can be chosen before the pressure coefficients, since this is a finite linear trace space. No interpolation through a skinny triangle is invoked.

3. Distant endpoint cutoff and energy. Choose a smooth cutoff eta equal to one on S_(4R) and zero outside S_(8R), with first derivative C/R and second derivative C/R^2. In particular eta=1 on the coarse transition AND on the ENTIRE bulk commutator annulus R..2R and its stencil enlargement. On the common uniform annulus use the nodal P1 interpolant of eta Psi; inside it keep Psi exactly. Call the actual pressure function Phi. Its base trace equals the actual g throughout |x|<=4R, including EVERY guard breakpoint there. There is no base-tail source on a bulk cutoff stencil. Direct integration on the uniform grid, with its fixed reference shape, gives ||grad Phi||_2^2+R^(-2)||Phi||_2^2 <=C[R||g'||_L2(I)^2+R^(-1)||g||_L2(I)^2+r^2 k^2]. (4) The first two terms are the ordinary vertical extrusion and cutoff costs. For the correction, S-|x| has support area O(r^2), size O(r), and gradient O(1). There is no cutoff derivative there because eta=1. Its cost is C r^2k^2. On the two child base halves, r^2 k^2<=C r ||g'||_L2(-4r,4r)^2 <=C R||g'||_L2(I)^2. (5) Thus (1),(4),(5) prove the actual full-breakpoint lift bound ||grad Phi||_2^2+R^(-2)||Phi||_2^2 <=C(R D/Y^2) E_I^2. (6) Every constant here is independent of the guard count, smallest base spacing, side-strip height, fan aperture and recursion depth. The finite conforming macro refinement may change the fixed template's minimum angle, but (6) follows from the explicitly prescribed affine gradients and the uniform-grid interpolation only. It is not an assertion of H1 stability of interpolation on arbitrary skinny elements. In the intended scales this is ||grad Phi||_2^2 <=C b0(a_parent/N)^2 E_I^2. (7) This replaces the unsuccessful b0 delta^2 estimate by the actual small root-distance factor, with no sigma loss from the guard breakpoint count.

4. Endpoint values and the retained far-base driver. The cutoff is not declared equal to one at arbitrary outer endpoints. The actual unlifted tail (1-eta)g starts at |x|>=4R, outside the whole bulk commutator annulus and separated from its stencil enlargement inside S_(3R) by at least a fixed multiple of R. It remains a prescribed far-base driver, not zero or affine. For the part on I its nodal L¹ size satisfies F_tail=d sum |(1-eta)g| <=C R^(3/2)||g'||_L2(I). (8) Endpoint values obey the same estimate by the one-dimensional anchored inequality |g(x)|<=C sqrt(R)||g'||. If a fictitious half-plane representation is used, taper the last endpoint value to zero on one further base cell. Its L¹ cost is already covered by (8); no H1 norm of that distant taper is needed. Actual data beyond I and actual outer-interface reactions remain in the independently retained distant cutoff/interface driver, rather than being bounded by an unproved continuation of g outside I. The actual distant-base Poisson estimate gives affine pressure near the child plus error C(r^2/R^3+d/R^2)F_tail. Its FULL field version is proved explicitly in Section 6 of the internal proof in part 11: from a prescribed base pressure, the physical pressure multiplier has order zero and its velocity multiplier order one; both target mesh VELOCITY amplitudes have order one because the constant base pressure column is purely the physical gauge column. Thus velocity value has separated scale F_tail/R^2 and its first variation F_tail/R^3. Root/vector replacement adds theta and gives error d F_tail/R^3. The actual base velocity and pressure of the tail are zero near the child; exact matching fixes the two constant target-layer amplitudes, p_ph(0)=d k_n c_tail,n+O(d^2 F_tail/R^3), and |c_tail,t|<=C d F_tail/R^3. Reset that tiny tangential component to its exact zero local base slope if exact base matching is desired. The normal component is independently extracted and retained, not set to zero. The resulting child-scale residual cost is at most C r^2 F_tail/R^3 after choosing d<=r^2/R. By (8), this is at most C(r/R)^2 times the right-hand square root in (6). The extracted affine field and both neutral layers are retained as macro/incident data; only this residual is charged as error. For the BULK commutator estimate one also needs the whole tail field on its larger annulus, not merely its child-scale remainder. Every bulk stencil lies inside S_(3R), whereas the tail base starts outside |x|=4R. The same exact pressure/velocity multipliers give the pointwise bounds |v_tail|+|grad p_tail|<=C F_tail/R^2 there, including the neutral mesh layers: one derivative of their O(d) pressure amplitude cancels the d, leaving the same separated kernel. The actual pressure base is zero throughout that annulus, so anchored Poincare, or its explicit gauge oscillation bound, yields ||v_tail||_L2(bulk annulus) +||grad p_tail||_L2(bulk annulus) +R^(-1)||p_tail||_L2(bulk annulus) <=C F_tail/R<=C sqrt(R)||g'||<=C Q. (8a) Here Q denotes the trace-lift bound in (6), and one-cell enlargements satisfy the same estimate. Therefore subtracting the actual tail before forming the bulk commutator adds at most C Q to its ring norm. There is no pressure tail source, distant endpoint taper, or logarithmic near-source layer on those stencils. This is precisely why the pressure cutoff radii 4R..8R are OUTSIDE the bulk cutoff radii R..2R. Consequently endpoint cutoff does not hide an uncontrolled endpoint hat or a logarithmic energy factor. It divides the comparison into an actual local pressure lift and an actual distant-base input with its own reactions and incoming macro extraction.

4a. The unequal-velocity lift uses the SAME nesting. Choose the base velocity driver to be the actual center value g_center. The neutral boundary lift Eg includes eta(x,0)[g_base(x)-g_center] at EVERY local base node, not merely the child's three nodes. The same eta is one through 4R and zero beyond 8R. Its weighted-star norm is the one in the independent full-vector corner/guard budget; enlarging a fixed-size comparison window changes only its fixed constant. Thus the residual velocity trace is zero on every bulk cutoff stencil. The unlifted velocity tail h_v=(1-eta)[g_base-g_center] starts at |x|>=4R and is retained as a distant BASE VELOCITY input. Put F_v=d sum|h_v|. In the actual stable boundary map the normal velocity input gives physical pressure amplitude O(d h_v), while the tangential velocity input gives O(d|theta|h_v). The target mesh velocity has an ordinary analytic constant part, rapidly localized tangentially (including its local boundary value), plus a nonsmooth order-one part. Thus away from its support its full incoming-field bounds are those above with F_tail replaced by d F_v. In particular its whole-field bulk-annulus norm is at most C d F_v/R and its child-scale residual is at most C r^2 d F_v/R^3. Both its affine macro and its two neutral layers are retained. The L¹ norm need not be assumed small. The actual reference-cell parent trace inequality and center-star area comparable to rD give, after subtraction of any constant velocity c, ||g_base-g_center||_L2(I)^2 <=C[1/D+R/(rD)]||W-c||_L2(parent patch)^2 <=C/(sigma D)||W-c||_L2(parent patch)^2. Consequently F_v<=C sqrt(R/(sigma D))||W-c||, and the squared whole-tail ring cost is at most C d^2/(sigma D R)||W-c||^2 =C(d/R)[d/(sigma D)]||W-c||^2. (8b) This is no larger than the already retained guard-velocity lift budget C d/(sigma D)||W-c||^2, since d<=R. The center and corner contributions keep their separate actual star weights; (8b) does not substitute a normal pressure-slope estimate. Endpoint extension of the distant velocity tail is treated by the same L¹ construction as for pressure, and its reactions remain outputs. Thus neither pressure nor velocity is silently set to zero on an overlapping or unspecified bulk-base interval.

5. Coupling and aggregate normalization. Subtract Phi from the actual glued comparison. Its exact load is the velocity functional -(grad Phi,z); its norm is bounded by (6). The passive-child mixed mass estimate applies with the complete child/transition self-response and every constrained-base reaction retained. No inner pressure moment is deleted. The affine a_I and its independently extracted normal macro component remain in the retained state; a_I's slope is not forced back to a global parent slope without the separate reset estimate. For disjoint safe windows with bounded-overlap parent patches, squared costs sum using (6). With Rb0 D, D/Ya/N, child radius rsigma b0 a/N in unit parent scale, and BN^2/(b0 a) children, the normalized factor is exactly sqrt[b0(a/N)^2]/[sigma sqrt(b0 a)] =sqrt(a)/(N sigma). (9) This is the admissible polynomial-N gain. It treats the entire refined base trace and does not rely on only two affine pieces outside the child base. The full vector unequal-velocity comparison is handled separately by center matching and child-side corner/guard star budgets; it is not inferred from this scalar lift.

6. Exact decomposition into the retained state and errors. Use the exact radiation operators J_infty,F_infty,E_infty(D) of part 14 in the present decomposition. The ideal child DtN D and its radiation exterior self-response J_infty are unchanged; no finite grounded E_R is substituted. The full base trace is split linearly as actual base pressure = a_I + eta g + (1-eta)g. (10) First, a_I contributes its actual affine macro trace, with tangential macro velocity -a_I'. Its incoming NORMAL macro component is still extracted from the regular exterior field. Second, eta g is lifted by the actual Phi just constructed. Solve the AUXILIARY homogeneous-base problem (w_corr,z)+(grad p_corr,z)=-(grad Phi,z), ; b(w_corr,q)=0. The PHYSICAL pressure-error pair is (w_corr,p_corr+Phi), and R_Phi means the trace of THIS RESTORED PAIR. Its momentum equation is sourcefree because the two Phi-gradient terms cancel exactly; its weak constraint is unchanged; its prescribed base pressure is eta g. This distinction matters on child cells where Phi is nonzero: the un-restored auxiliary pair is not a sourcefree child response and must not be used in aggregation. Passive stability gives ||w_corr||_2+||grad p_corr||_2<=C Q, where Q is the square root in (6), and hence the restored pair obeys ||w_corr||_2+||grad(p_corr+Phi)||_2<=C Q as well. All its actual interface and constrained-base reactions, including the restored pressure contribution, remain in the parent equations. Thus R_Phi is an actual sourcefree D-child response with quantified norm, not a third ideal driver or a forced child field passed off as homogeneous. Third, solve the actual half-plane tail problem, including its prescribed distant base pressure and every base reaction. Decompose its incoming field as the macro c_tail plus the two neutral layers of amplitudes -c_tail, with its small remainder from Section 4. After the actual passive child is glued, this tail contributes A(c_tail)+E_infty(D)(-c_tail)+R_tail, (11) with the exact base-slope reset included in R_tail and mass norm at most C(r/R)^2 Q for that remainder. The additive pressure constant is a gauge; the normal affine weak-flux reaction in A(c_tail) is retained. If the previous retained macro and layer amplitudes were c and eta_driver, update them to c+c_tail and eta_driver-c_tail. Their sum, the prescribed local base velocity, is unchanged. Therefore the tail stays within the same affine-macro-plus-two-layer state, with its remainder charged explicitly. Together with the independently bounded bulk-driver and unequal-velocity errors, the actual trace has the retained form s_actual=A(c_total)+E_infty(D)eta_total +R_Phi+R_tail+R_bulk+R_velocity. (12) This is linear superposition of actual mixed solutions and lifts, keeping the complete D+J_infty solve. Equation (12) does not set any free child pressure coefficient to an affine value and does not replace the outgoing self-response by an incoming estimate. If an additional genuine inner monopole survives the actual interface-equilibrium sum, its existing macroport must still be retained. It is not manufactured or deleted by the guard trace split (10). The full refined comparison-base pressure introduces no uncontrolled additional port once its error is charged by (6)--(9). The same restoration convention applies to the separately constructed neutral velocity lift Eg. The auxiliary pair has velocity forcing -(Eg,z) and zero base velocity; the PHYSICAL error pair is (w_aux+Eg,p_aux). Since b(Eg,q)=0, its weak equation is sourcefree, and its momentum source cancels exactly after restoration. With simultaneous pressure and velocity lifts, restore (w_aux+Eg,p_aux+Phi). Its velocity mass and pressure-gradient norms remain bounded by C(||Eg||_2+||grad Phi||_2). For the bulk/tail cutoff errors, the compact residual is confined to the common annulus and the subtracted exterior field vanishes near the child, so the error pair is already sourcefree inside the actual child. In all cases the e_c used in parent/child aggregation is the RESTORED physical pair and is a legitimate actual child DtN response. Finally, the entire split is exactly compatible with the affine Darcy quotient. Adding a constant velocity c_0 and pressure -c_0 dot X plus a gauge changes the chord slope C_I by -c_0,t and the anchored affine trace by precisely that affine pressure; g, k, Phi and the pressure tail are unchanged. The velocity difference g_base-g_center and hence Eg and its tail are unchanged. The regular incoming macro adds c_0, while eta_driver=g_center-c_macro is unchanged. Thus all restored error responses are invariant under the background addition; only the retained A(c_total) and its actual normal weak-flux reactions change. No estimate depends on a non-equivariant mean-G extractor. The remaining work is the actual recurrent/nonextinction argument and its source/observation realization. This lift alone does not certify the original pressure claim.

Part 20: Actual full outer-fan weighted inverse. Exact actual mixed test and full outer-node equations. Let F be the entire radial parent fan with root 0, prescribed actual root velocity a, and outer nodes i, including ramps, guards and reserve-base endpoints. Every parent triangle contains the root and two outer vertices. Let W=P_Zh u with mass multiplier phi be an actual mass projection, and assume the supports of the tests below are disjoint from the fixed smooth source u. Fix one constant unit direction t and one scalar incoming pressure component p_0. Neither is selected separately at every node. For outer node i use its actual nodal hat, corrected by -(14/27)b_K on parent-fan cells, but by -(20/27)b_K on its EXTERIOR cells. This is a continuous actual MINI test: the bubble corrections do not change its trace. Its parent bubble moment is zero, so parent mass elimination is exactly the condensed C block. Its exterior unweighted cell means vanish, so the exterior pressure pairing is exactly zero. The equation is therefore sum_(K parent incident i)|K| ; [16 b_i+b_j+a_t+18 p_(K,t)]/180 +R_i=0, (1) where b_i=W(i) dot t, j is the other outer vertex of K, a_t=a dot t, p_(K,t)=grad phi|K dot t, and R_i=(W,E_i^ext), ; E_i^ext|K=(lambda_i-(20/27)b_K)t on exterior cells incident to i, zero elsewhere. Thus pressure impedance is not estimated or omitted in R_i: its pairing with this specific exterior test vanishes exactly. For ANY constant vector c, R_i=(W-c,E_i^ext), by its cell means. This subtraction does not discard a boundary velocity or pressure reaction: it is a literal vanishing integral against this actual test. Define A_i=sum_(K parent incident i)|K|, ; W_ij=sum_(K parent containing i,j)|K|, ; P_i=A_i^(-1)sum_(K parent incident i)|K|p_(K,t), ; w_i=b_i+p_0, A=a_t+p_0. Equation (1) is exactly 16w_i+sum_j (W_ij/A_i)w_j+A=e_i, ; e_i=-18(P_i-p_0)-180R_i/A_i. (2) All outer rows remain in this system. In particular guard values are solved with the block rather than inserted as an arbitrary uncontrolled constant d_b.

Geometry-independent full-block inverse. Put B_ij=W_ij/A_i. Each parent triangle gives exactly one other outer neighbor, so B has nonnegative entries and row sum one, including end/ramp vertices. Moreover A_iB_ij=W_ij=W_ji=A_jB_ji. Hence B is self-adjoint in the weighted space ||v||_A^2=sum_i A_i|v_i|^2. Jensen and the row/column balance give ||B||_(A->A)<=1, so every eigenvalue lies in [-1,1]. Thus 16I+B is positive definite, has inverse norm at most 1/15, and (16I+B)^(-1)1=1/17. Consequently the exact actual solution of (2) satisfies ||w+(A/17)1||_A <=(1/15)||e||_A. (3) No minimum angle, graph distance, minimum cell size, or N-dependent inverse constant enters (3). These enter separately in geometry/reset estimates, not in this finite scalar block. Let E_p=||grad phi dot t-p_0||L²(F) and E_R^2=sum_i R_i^2/A_i. Jensen, counting the two outer vertices of each parent triangle, yields sum_i A_i|P_i-p_0|^2<=2 E_p^2. Therefore the error in (3) obeys the concrete bound D:=||w+(A/17)1||_A <=[18sqrt(2)E_p+180E_R]/15. (4) Pressure-gradient deviations on guard/ramp cells are included in E_p; they have not been assigned zero or inferred small from their area alone.

Exact exterior star-budget estimate. Let E_i be the total exterior triangle area incident to outer node i, and suppose the actual geometry gives E_i<=epsilon_Q A_i. The scalar exterior functions zeta_i=lambda_i-(20/27)b have local Gram diagonal 13/126 and off-diagonal 5/252. Its largest eigenvalue is exactly 1/7. Thus for arbitrary real nodal coefficients xi_i, ||sum_i xi_i E_i^ext||2^2 <=(1/7)sum_i E_i xi_i^2 <=(epsilon_Q/7)||xi||_A^2. Duality now proves, for the actual velocity and any constant background c, E_R <=sqrt(epsilon_Q/7)||W-c||L²(exterior stars). (5) This is an actual mass reaction estimate. It needs no bound on the exterior pressure impedance, and does not assume the exterior velocity equals its boundary trace extension. Every exterior cell in these node stars is included; source overlap, if present at an initial coarse level, would add its actual load rather than disappear.

Designated child roots, guards and inherited pressure resets. Let J be the designated child-root subset, with total star weight A_J=sum_(i in J)A_i. Restricting (3) to J gives ||w||_(A,J)>=|A|sqrt(A_J)/17-D. (6) If the actual child-root contrast of interest is d_i=w_i+h_i, where h_i contains inherited half-trace pressure/reset errors and the difference between the fixed t and each true base tangent, then ||d||_(A,J)>=|A|sqrt(A_J)/17-D-||h||_(A,J). (7) For example D+||h||(A,J)<=|A|sqrt(A_J)/34 gives the quantitative designated-root lower bound ||d||(A,J)>=|A|sqrt(A_J)/34. The negative transfer is thus established on the selected roots, not merely in an aggregate that could live on guards. In the actual toothed geometry A_J is a fraction comparable to the reserve-width factor sigma of the full parent outer star weight. Thus squared relative error estimates cost 1/sigma when (7) is normalized by the full fan area. This factor is explicit; it is not claimed universal if sigma changes. Converting A_i to each child's physical area adds the actual fixed ratio r_i/d_i, which must also remain in the energy recurrence. The normal component of a pressure gradient is not multiplied by s/d here. Rotation from the fixed t to a true edge tangent produces the ordinary angular error O(delta)|grad phi|; that term is part of h or E_p and must be charged to actual local energy by the present reset estimates. No pressure reset or affine-background quotient is assumed without that estimate. Formula (7) is ready for the full inherited-state recurrence, but does not prove its pressure/reset hypotheses itself.

Exact finite full-side-chain diagnostic. For N=4,8,16, the optimized child quotient energy divided by total child area is approximately 0.000503012, 0.000501922, 0.000503487. These are finite exact-arithmetic diagnostic values, not a limiting theorem. They differ from the shrinking two-child Dirichlet-window diagnostic, whose normalized quotient energy decreased with N. Hence the latter must not be used as evidence that a complete fixed-length chain loses its transfer. Neither diagnostic is the recursive toothed template or an original pressure counterexample.

Guard-safe localization when the small star budget is not global. The inverse proof does not require the small exterior-star hypothesis. In particular G1 proved only for selected attachment centers is NOT enough by itself to use (5) for the entire outer block. Retain every exceptional guard/ramp/base row as follows. Partition the outer vertices into G and H, with errors e_G and e_H extended by zero. Let L be the graph distance from the designated roots J to H in the actual outer adjacency graph (an absent path has infinite distance). For every k<L, the matrix block from H to J of B^k vanishes. The convergent exact Neumann series and ||B||_A<=1 therefore give ||1_J(16I+B)^(-1)e_H||_A <= (16^(-L)/15)||e_H||_A. (8) Hence the actual designated-root error obeys the stronger localized bound ||w+(A/17)1||_(A,J) <= [||e_G||_A+16^(-L)||e_H||_A]/15. (9) All bad-row pressure and exterior reactions remain in e_H with their exact values from (2). No sign, cancellation, or guessed guard velocity is needed. For example if E_i/A_i<=epsilon_Q on G and <=Q_H on H, the same Gram duality, applied to coefficients supported in the respective sets, bounds the reaction terms by sqrt(epsilon_Q/7)||W-c|| and sqrt(Q_H/7)||W-c||. The bad contribution is then multiplied by 16^(-L). Similarly a bad-row pressure error is bounded by its actual local pressure norm, followed by the same attenuation. Thus either a global small star budget or an actual separation/budget proof suffices. Neither is assumed automatically from selected-center G1. For a given finite template L and Q_H are explicitly geometric quantities; the existence of a large L and controlled Q_H must be verified on that template before claiming a small error coefficient. The estimate applies to all graph components and nonuniform areas without modification. For the currently proposed template, the exterior-star geometry of part 2 and the neutral tests of part 21, the section "A simultaneous thin-collar refinement", supplies an explicit OPTIONAL template choice giving E_i<=epsilon_Q A_i on ALL fan vertices, including guards and ramps. It preserves the assigned three-node child bases and keeps additional long background edges below the mandatory fine-buffer line. This is the appropriate stated construction to invoke in (5); it is stronger than the original selected-center G1. The small pressure-reset estimate must likewise cover all fan triangles entering E_p, including short ramps. A mainfan-only reset estimate cannot be inserted into (4) without either its ramp extension or the explicit localization (9).

Part 21: Bounded odd selector from the two affine base traces. Reflection-odd rank-one selector: the tangential pressure term. There is a concrete bound for the tangential base-slope selector that avoids a skinny-cell inverse estimate. Let the actual child be the fixed rectangle of diameter O(r), with its three unrefined base vertices c_-,o,c_+. For any actual continuous P1 pressure p, define its average base slope p_t=[p(c_+)-p(c_-)]/(8r). The base trace is continuous and affine on each of its two halves. On this fixed two-element one-dimensional trace space, its H^(1/2) seminorm is at least c r |p_t|. The ordinary rectangle trace theorem, after subtracting the volume mean and using scaled Poincare, bounds that seminorm by C||grad p||_L2(C). These constants are properties of the fixed rectangle and the two base edges, independent of the interior triangulation and of the thin strip. Therefore |p_t| <= C r^(-1)||grad p||_2 <= C r^(-1)||W||_2 (6) for an actual source-free mass response, by its local bubble equations. Point evaluation in H1 is NOT being used: the pressure difference is controlled through the whole affine base trace. Set L_t(s)=g(o)_t+p_t. The root-star lower bound and (6) give r sqrt(a)|L_t(s)| <= C||W||_2. (7) For the ideal tangent column E_t=E_infty(D)e_t, the prescribed base pressure is zero and root velocity is e_t, so L_t(E_t)=1 exactly. Hence the rank-one operator P_t s=E_t L_t(s) is an exact projection and is uniformly bounded in the actual mass response energy whenever the sharp ideal energy estimate (3)-(4) holds. On a reflection-odd state its affine tangential background is removed by this root-plus-base-slope contrast. This does not identify the full incoming normal macro component, and it is not a two-driver projection on arbitrary states. The remaining task is still the actual averaged complementary transfer (I-P_t) T (I-P_t) on the reflection-odd state space. The estimate above supplies a bounded selector/projection, not a small transfer block. For the actual single-diagonal radiation exterior, its tangent column need not itself be reflection odd. Use E_t^-=(E_t-U E_t)/2 in the odd projection instead. The actual child template and its response energy are reflection invariant, so this is an actual source-free odd response, its energy is no greater than the original sharp bound, and L_t(E_t^-)=1 exactly. The same trace/root lower bound applies. Thus P_t=E_t^- L_t preserves the odd sector. The normal column's odd part is a separate parity error requiring the compatible-port exterior Darcy comparison; no exact cancellation is claimed for it.

Part 22: Actual all-state odd tangent top row. 1. State, quotient norm and the exact inherited base slope. Let s be ANY reflection-odd state of the actual limiting mass template, with recovered source-free response (W,p) on parent C of radius R. Its base has exactly the three retained vertices c_-,o,c_+. Gauging p(o)=0, oddness makes the continuous P1 base trace exactly p0 x, where p0=[p(c_+)-p(c_-)]/(8R). Let a=W(o). Put q=a_t+p0=L_t(s). Let E(s)=inf_b ||W-b||_L2(C), with affine-Darcy states quotiented. Oddness allows a tangential minimizing constant b; the normal mean is zero. The actual bubble relation gives ||grad p+b||<=sqrt(gamma_2)||W-b||. The fixed-rectangle H1 trace estimate on the two AFFINE base edges gives R|p0+b_t|<=C||grad p+b||. Therefore ||W+p0 t||+||grad(p-p0 x)|| <= C E(s). (1) This avoids replacing the quotient norm by an uncontrolled chosen background. No point evaluation in H1 is used. For completeness the all-guard reset used in this phase has the following direct proof. In coordinates normal/tangent to the incoming base, put $\zeta =\varphi -\varphi (o)-p_{0}y$; every point of the entire fan, including ramps and radial guards, lies in $0\le x\le \delta y$, $y>0$. On a triangle set $R_K$ equal to the larger outer $y$ coordinate. Its exact moment is ${\int }_K{y}^2\ge \mathrm{\mid }K\mathrm{\mid }{R}_K^2\mathrm{/}6$, with no comparability of its two radii required. Since $\zeta =N_{K}x+(T_K-p_0)y$, dividing the triangle $L^2$ estimate by $R_K$ and square-summing bounds the reset by $C\mathrm{\parallel }\zeta \mathrm{/}y{\mathrm{\parallel }}_{L^2(F)}+C\delta \mathrm{\parallel }{\mathrm{\partial }}_x\varphi {\mathrm{\parallel }}_{L^2(F)}$. The zero base trace and the one-dimensional fundamental theorem give ${\int }_0^{\delta y}\mathrm{\mid }\zeta (x,y){\mathrm{\mid }}^{2}dx\le (\delta y{)}^2{\int }_0^{\delta y}\mathrm{\mid }{\mathrm{\partial }}_x\varphi {\mathrm{\mid }}^{2}dx\mathrm{/}2$. Divide by $y^2$ and integrate: the reset is at most $C\delta \mathrm{\parallel }\mathrm{\nabla }\varphi {\mathrm{\parallel }}_{L^2(C)}$. The bubble equation gives $\mathrm{\parallel }\mathrm{\nabla }\varphi {\mathrm{\parallel }}_2\le \sqrt{\gamma }\mathrm{\parallel }W{\mathrm{\parallel }}_2$; subtract any affine Darcy background first to obtain exactly the quotient-energy version. This proves the all-triangle estimate, without an inverse bound or a minimum ramp radius.

2. Full outer-fan inverse, with ALL guards. Use the actual full outer-fan scalar weighted inverse proved in the internal proof in part 20. It applies componentwise to the actual vector velocity and retains every guard/ramp/endpoint. Let A_i be the total parent fan area at outer vertex i. Write b_i=W(i), and let pbar_i be its adjacent fan-area-weighted pressure gradient. After eliminating the bubble mass block, the actual outer equations have weighted matrix 16I+B, where B is reversible Markov in the A_i inner product and ||(16I+B)^(-1)||<=1/15. For the tangential component use the constant reference b_ref,t=-(a_t+18p0)/17. The exact weighted inverse implies ||b_t-b_ref,t||_A <=(18sqrt(2)/15) E_p+(180/15)E_R, (2) with E_p the actual weighted tangential pressure-gradient deviation from p0, and E_R the full exterior reaction norm. The exact constants/conventions are those in the internal calculation; changing harmless fixed constants below has no mathematical effect. The actual all-triangle Hardy reset, including radially added guards and arbitrary base subdivisions, gives E_p<=C delta E(s). (3) It uses the entire fan in its single narrow wedge, the base trace p0 x, and (1). It does not assert that additional guard pressure nodes are affine. For every outer vertex use the hybrid test: parent fan coefficient lambda_i-14 b/27, EXTERIOR coefficient lambda_i-20 b/27. The exterior part has exact zero cell mean and is continuous across interfaces. Thus ALL exterior pressure terms cancel. The checked joint star budget, including current root/child-side stars and all fine uniform guards, gives E_R<=C sqrt(epsilon_Q) E(s). (4) It is a mass test. No strain term, ancestor-star reassignment, pressure-only DtN smallness, or numerical root assumption enters (4). Combining (2)-(4) and adding the actual pressure deviation itself yields sum_i A_i |b_i,t+pbar_i,t+q/17|^2 <=C(delta^2+epsilon_Q) E(s)^2. (5)

3. Exact selected child contrast and orientation. Use the following explicit type transport. On each unoriented long tooth-side line, choose its physical child tangent t_i to have positive dot product with the PARENT tangent t; hence |t_i-t|<=C delta. Label its base endpoints c_±^i by this increasing tangent coordinate, irrespective of the direction in which the fan boundary was drawn. Choose its physical normal n_i to point into its child container. The frame O_i=[t_i,n_i] may have determinant -1; reflection embeddings are allowed, since the actual MINI mass equations and the later strain operator are orthogonally invariant. Its intrinsic child coordinates are O_i^T(X-X_i)/r. The actual template type is transported with this same frame; a down-facing container is not silently assigned the opposite tangent. For a reflected partner under the parent S=diag(-1,1), choose t_i'=-S t_i and n_i'=S n_i, so O_i'=S O_i S. The additional intrinsic reflection is an actual symmetry of the mirrored child template, including its paired uniform grids. Thus paired recoveries satisfy T_i' U_parent=U_child T_i in the precise reflected coordinate convention. For an ODD parent state, T_i' s=-U_child T_i s. Since L_t U_child=-L_t, their transported tangent contrasts are EQUAL, not opposite; the normal driver coefficient changes sign. Its actual normal column need not be reflection even: the odd column defect is retained and charged by part 15, not cancelled by this frame identity. This is the sign convention underlying the exact S-conjugated two-driver covariance. It applies to left/down-facing fan types as well as right/up-facing ones. All comparisons below are AFTER this exact intrinsic transport. Raw physical tangent vectors with an arbitrary drawing orientation need not be close to t and must not be inserted in the mean map. A selected child base has endpoints at equal distances 4r from its center i. Its two base edges are UNREFINED; all new uniform guards are outside those two edges. The two adjacent parent fan triangles therefore have equal area. Consequently its actual average base pressure slope is exactly [p(c_+^i)-p(c_-^i)]/(8r) =pbar_i dot t_i, (6) where t_i is its local tangent. This uses the AVERAGE of both halves, not a selected half; no pressure-kink bound or unsupported normal estimate is needed. Its root velocity is b_i. Thus its actual child contrast is q_i=(b_i+pbar_i) dot t_i. For a global constant background b, actual local MINI mass positivity and the bubble pressure relation give sqrt(A_i)|b_i+pbar_i|<=C||W-b||_parent_patch(i). This is invariant under the affine Darcy shift. Since |t_i-t|<=C delta, bounded patch overlap charges the orientation change to C delta^2 E(s)^2. Hence (5) implies on selected centers J sum_J A_i |q_i+q/17|^2 <=C(delta^2+epsilon_Q) E(s)^2. (7) This is the full vector orientation bound only where needed; it does not promote tangential slope inheritance to a full normal variation theorem. The fixed template chooses selected centers with comparable A_i~d_tooth r and sum_J A_i~sigma a R^2, ; |J|=B~N^2/(b0 a). The guarded middle-third geometry supplies these comparisons. Fixed-factor discrepancies can instead be retained as explicit reflection-invariant weights, provided the same weights define the actual mean map. For equally weighted, comparable-area centers, Cauchy-Schwarz in (7) gives |B^(-1)sum_J q_i+q/17| <=C (delta+sqrt(epsilon_Q))/(R sqrt(sigma a)) E(s). (8)

4. Normalized top row of the actual mean map. Let U be the actual mean child recovery on the reflection-odd affine-Darcy quotient, using child pressure/r scaling and the stated local rotations. The average in (8) is precisely L_t(U s). It is not the root velocity alone. For the actual single-diagonal radiation exterior, use the ODD PART E_t^- of its tangent column, rather than assuming that column itself is odd. Reflection is an exact symmetry of the child D and its response energy, so E_t^- is an actual odd child response, has L_t(E_t^-)=1, and inherits the sharp upper/lower energy bounds. Put g=||W_Et^-||_quotient on the unit parent template. The sharp ideal-energy proof gives c sqrt(a)<=g<=C sqrt(a). Normalize the tangent ideal vector by e=E_t^-/g and the selector by ell=g L_t, so ell(e)=1. Equations (8) give |ell(U s)+(1/17)ell(s)| <= e1 E(s), ; e1<=C (delta+sqrt(epsilon_Q))/sqrt(sigma). (9) Thus, with P=e ell and Q=I-P, the actual block entries satisfy |alpha+1/17|<=C e1, ||P U Q||<=C e1, (10) where the fixed constants include the already bounded nonorthogonal projection norm. This holds for ALL odd states, not merely the ideal two-driver family. The complementary row Q U is supplied separately by the actual incoming-field/remainder decomposition; (9) does not assert it. For delta=L^-200, sigma=L^-150, epsilon_Q=L^-380, a=L^-220, the two left-row contributions are L^-125 and L^-115. After dividing by the initial normalized source scale sqrt(a)=L^-110 they are L^-15 and L^-5. With the quantified inverse-logarithmic four-source lower bound, these are compatible with the asymmetric source-visibility condition for sufficiently large fixed N. This arithmetic does not replace the separate all-remainder Q U proof.

Part 23: Odd quotient aggregation and asymmetric block argument. 1. Actual quotient space and bounded projection. Use the actual limiting mass DtN and recovered responses on the fixed child template, already obtained by nonnested mass Mosco convergence. Quotient its twenty-state space by the exact affine Darcy family W=c,p=gamma-c dot X. The response energy norm is inf_c ||W-c||_2 on the fixed reference container. Its kernel is exactly the quotiented family, by the established mass affine-Darcy detection result. Pressure gauges are removed before forming the state space; their weak reactions remain in the actual assembly. Restrict to the reflection-odd sector U_ref s=-s. Let L_t be root tangential velocity plus average base pressure slope. It vanishes on affine Darcy states, is reflection odd, and satisfies r sqrt(a)|L_t|<=C inf_c||W-c||. The ODD PART E_t^- of the radiation tangent column has L_t(E_t^-)=1 and quotient energy at most C r sqrt(a). Thus P=E_t^- L_t is an exact bounded rank-one projection on this quotient; Q=I-P is bounded. These are the actual projection bounds established in the internal proof in part 21, not generic coefficient-norm assertions. All child states below use their own reference scaling: physical velocities are unchanged, pressure differences are divided by child radius, and local vector frames are transported by the actual child embedding. In reference coordinates their response L² norm equals physical norm divided by the child radius. Reflection covariance requires paired grids and all local frames/gauges to be chosen as in the orthogonal frame transport in part 22.

2. Exact averaging implication. Let R be the parent radius, r=rho R the common child radius, B the number of equally weighted children, and T_c their actual full state recoveries. Let U=B^(-1)sum_c T_c, after the exact reference coordinate transports. Suppose the actual child fields have been decomposed as T_c s = A(c_c)+E eta_c+e_c, (1) where A(c_c) is their incoming affine background and e_c is a source-free actual child response, including all retained base/guard/interface effects. The ideal E is the radiation embedding. A single-diagonal exterior is NOT itself reflection invariant. Pair covariance gives an averaged tangent ideal term in range E_t^- and also a possible odd normal term. For one stencil/type write that term E_n^- N(s), where N(s) is the linear normal-driver functional on one representative of each reflection pair. For finitely many oriented types sum these terms; all bounds below use their combined functional norm. Therefore the actual quotient identity is Q U s=Q [B^(-1)sum_c e_c]+Q E_n^- N(s). (2) There is no exact normal cancellation claim. The compatible-free12 exterior Darcy comparison supplies ||E_n^-|| ->0 after fixing N, while the ring/root driver estimates give ||N||<=C_N independent of h. Choose h small enough that the second term's operator norm is at most epsilon_sym. This after-fixed-N choice does not require a rate or an unrestricted full twenty-port exterior limit. Every e_c in (1) is the RESTORED source-free pair on the child, not an auxiliary loaded solve. In particular, subtracting a pressure lift Phi gives an auxiliary momentum load -grad Phi; its contribution must be recorded as (w_corr,p_corr+Phi). The velocity and its energy are unchanged by that restoration, and the restored pair is source free inside the child. For a neutral velocity lift E_g, record (w_corr+E_g,p_corr), with its full restored velocity norm bounded by the actual trace-response estimate. Compact scattering loads supported on the uniform annulus lie outside the child, so their restricted pairs are already source free there. These restorations are required before applying the child response map, its bubble pressure bound, or physical fractional detection. The averaged remainder is an actual response because the same limiting template response map is linear. Consequently ||Q U s||_reference <= C/(B r) sum_c ||W_e,c||_physical+epsilon_sym||s|| <= C/(r sqrt(B)) [sum_c ||W_e,c||_physical^2]^(1/2) +epsilon_sym||s||_reference. (3) This controls Q U on ALL reflection-odd states, not only on range P, PROVIDED the complete remainder budget in (1) holds for all those states. It then controls both Q U P and Q U Q without bounding the raw whole transfer matrix. For the distant incoming ring remainder, localized passive scattering gives ||W_e,c|| <= C sigma^2 V_c, ; sum_c V_c^2 <= C ||W_parent-background||^2, (4) using actual disjoint rings and the bubble gradient estimate. Thus its operator contribution is C sigma^2 R/(r sqrt(B)) = C sigma R/(ell sqrt(B)) ; ~ C sigma/sqrt(b0 a), (5) where r=sigma ell, ellb0 a R/N, BN^2/(b0 a). The polynomial N cancels because the local rings are averaged before taking the parent norm. Applying a single-ring bound without this summation does not prove (5). Additional proved/targeted squared budgets must be added before using (2): Full vector unequal base velocities, with the ideal constant matched to the actual child-center value, have center residual exactly zero. Independent corner stars and uniform guards give sum lift-energy^2 <= C eta_base parent-energy^2, eta_base=epsilon_corner+C h/(sigma d_tooth), per the established full-normal budget. Their contribution to (3) is C sqrt(eta_base/S0), S0=B rho^2~sigma^2 b0 a. The entire refined straight-base pressure remainder modulo its OWN affine slope has trace/lift target sum-energy^2 <= C b0(a/N)^2 parent-energy^2. If its actual collar lift is established, its contribution is C sqrt(a)/(N sigma). The affine slope belongs to A(c_c) and the root/tangent inheritance calculation; it is not discarded. Any remaining pressure kink, orientation or inherited prescribed data must be included in e_c with its actual energy budget. A three-node kink estimate does not cover arbitrary additional guard breakpoints without the entire-base trace lemma. Together with the explicitly retained epsilon_sym parity term, these items specify the error e2 in ||Q U||<=e2. They do not assume the still-required actual trace/decomposition steps.

3. Asymmetric block lemma. Normalize the odd ideal tangent vector E_t^- to unit quotient energy. In the direct sum range P plus range Q, suppose the actual reflection-odd U has block form U = [ alpha b ; c D ], ; |alpha+theta|<=e1, ||b||<=e1, ; ||c||<=e2, ||D||<=e2, theta=1/17. (6) The top row estimate must be the actual all-state tangent-inheritance bound. It is not inferred from the small complementary row. Choose e1,e2 sufficiently small compared with theta, with fixed constants accounting for the bounded nonorthogonal projection norm. For real lambda near -theta, lambda-D is invertible, and the scalar equation lambda=alpha+b(lambda-D)^(-1)c (7) is a contraction on a small interval around alpha: its displacement is at most C e1e2/theta and derivative at most C e1e2/theta^2. Thus it has a unique real solution with |lambda|>=theta/2. The exact left eigenfunctional, with first coordinate one, is ell_lambda=(1, b(lambda-D)^(-1)), ; ||ell_lambda-(1,0)||<=C e1/theta. (8) This is the ASYMMETRIC fact needed for source visibility: e2 does not replace e1 in (8). For any initial state s, ||U^m s|| >= |lambda|^m |ell_lambda(s)|/||ell_lambda||. (9) The AVG-functional four-source normalization in the internal proof in part 25 (not the historical right-half lemma) gives an initial tangent signal at least c sqrt(a)/(1+K_log)^2 in this unit-energy selector convention. Their initial state energies are bounded by fixed coarse-source mass bounds. Therefore the sufficient visibility condition is e1=o(sqrt(a)/(1+K_log)^2), rather than e2 satisfying that stronger requirement. All constants must be bound to the actual source norm and projection normalization before use.

Part 24: Fixed smooth coarse curl actuators. 1. Fixed admissible domain and coarse cells. Fix r>0 and let C=[-4r,4r]x[0,4r] be the recursive half-container, with o=(0,0), c_-=(-4r,0), c_+=(4r,0). Attach the lower triangle with vertices c_-, d=(0,-4r), c_+. Its union with C is a fixed convex pentagon D. Split that lower triangle into the TWO fixed cells K_-=(d,c_-,o), K_+=(d,o,c_+). The base of C has exactly the three nodes c_-,o,c_+, so these cells conform to every recursive C mesh. The nodes d,c_-,c_+ lie on the physical boundary and their no-slip velocities are zero. Thus on each fixed K the P1 part of ANY global no-slip MINI velocity is exactly lambda_o a, with the SAME root vector a at o. The two vectors grad lambda_o on K_- and K_+ are nonparallel. The complete C triangulation may vary arbitrarily with depth under the previously specified shape bound; nothing below assumes its terminal DtN converges. Write b_K=27 lambda_1 lambda_2 lambda_3. Let V_h,Q_h be the actual global MINI pair on D, pressure modulo constants. For any smooth solenoidal source u define its mass mixed solution (v,phi) by (v,z)+(z,grad phi)=(u,z), z in V_h, ; (v,grad q)=0, q in Q_h. (1) Thus v=P_Zh u. The potential phi is the actual mass-Schur potential, with the same positive-gradient convention as the branch calculations. Set t=(1,0), and define the actual incoming root contrast L(v,phi)=v(o) dot t + [phi(c_+)-phi(o)]/(4r). (2) The second term is the exact pressure slope on the right half of the coarse C base. Formula (2) is gauge invariant. The aligned positive-x fan uses this half-base.

2. Six explicit fixed smooth actuators. On each K choose a nonnegative smooth bump omega_K supported in its interior and positive on some open ball. The three scalar polynomials f_0=1, f_1=partial_x b_K, f_2=partial_y b_K are linearly independent. On the reference triangle b=27xy(1-x-y), comparison of the quadratic coefficients proves that a linear combination of its two derivatives cannot be a nonzero constant; affine transport preserves independence. Consequently the 3x3 Gram matrix G_(ij)=integral_K omega_K f_i f_j is positive definite: a nonzero polynomial cannot vanish on the open positivity ball. Define psi_(K,j)=omega_K sum_i (G^(-1))_(ji) f_i, ; u_(K,j)=(-partial_y psi_(K,j), partial_x psi_(K,j)), ; j=0,1,2, and extend each by zero to D. These SIX sources are fixed, smooth, compactly supported in the unchanged coarse cells, exactly divergence free, and no-slip. They have fixed finite H² norms independent of the recursive mesh. Their defining moments are integral_K psi_(K,j) f_k=delta_jk. (3) No zero-mean condition on psi is imposed here: the constant-curl actuator is essential. For a MINI velocity w|K=lambda_o a+b_K c_K, curl w = gamma_K + (c_K)_2 f_1-(c_K)_1 f_2, ; gamma_K=(partial_x lambda_o)a_2-(partial_y lambda_o)a_1. Integration by parts, with the displayed rotation convention, gives (u_(K,0),w)=-gamma_K, ; (u_(K,1),w)=-(c_K)_2, ; (u_(K,2),w)= (c_K)_1. (4) These are exact identities, not density or qualitative nonannihilation statements.

Part 25: Actual average-contrast four-source moment mismatch. 1. Exact functional and adjoint. Keep the unchanged bottom coarse triangles K_±, each of area 8r^2, and the four coarse bubbles w_j=b_K e_i with their fixed smooth solenoidal mass realizations u_j, R_hu_j=w_j. The desired contrast is L_avg(v,phi)=v(o)_t+[phi(c_+)-phi(c_-)]/(8r). Let its actual mixed mass adjoint be (w,z)+(grad p,z)=z(o)_t, ; (w,grad q)=[q(c_+)-q(c_-)]/(8r). (1) The finite problem exists by the actual MINI mass mixed inf-sup, and the limiting response follows from the proved DtN/Mosco convergence. Set e=max_j |(w_j,w)|. Reciprocity gives (u_j,w)=L_avg(P_hu_j,phi_j). (2) No assertion about L_right is used. The scalar functional in (1) is odd under reflection; the full contrast is odd as required by the recurrence.

2. Consequences of small four coarse bubble moments. On K_± write w=lambda_o a+b_K c_K, with b_K=27 lambda_1 lambda_2 lambda_3. The actual bubble momentum equations have zero right-hand side and imply grad p|K=O(e), ; c_K=-(14/27)a+O(e), ; mean_K w=a/10+O(e). (3) The implied constants depend only on the fixed coarse triangles and r. The bottom pressure hat q_d has zero right-hand side in (1); its gradient is (0,-1/(4r)) on both coarse triangles. Thus a_n=O(e). Gauge p(o)=0. Equation (3) makes all three prescribed base pressure values O(e). On the upper rectangle C, let w_ref be the source-free local mass response with root velocity a_t t, zero velocity at all other boundary nodes, and zero pressure trace on its entire base. Its pressure tests vanish on the base; all other pressure boundary coefficients are free. The actual sharp root-star geometry and cell-mean-neutral root lift give ||w_ref||_L2(C)<=C r sqrt(a_ap)|a_t|, (4) where a_ap is the fan aperture (distinguished from the vector a). The actual adjoint restriction has root normal trace O(e) and base pressure trace O(e). A neutral normal root lift and the established seven-node pressure extension therefore give ||w-w_ref||_L2(C)<=C K_log e, ; ||w||_L2(C)<=C[r sqrt(a_ap)|a_t|+K_log e]. (5) Here K_log>=1 is the depth-uniform pressure lift bound. Physical side/top pressure coefficients are free in this local comparison. No raw mesh pressure hat gradient, point pressure value estimate on an arbitrary H1 function, or right-half parity argument is used.

3. First exact linear equation: cell-mean-neutral root velocity test. Use the GLOBAL actual MINI field z_0 with nodal part lambda_o t and, on every cell incident to o, bubble correction -(20/27)b_K t. It is continuous, vanishes on the physical domain boundary, has root value t, and has exact zero vector mean on every cell. Hence its pressure pairing in the first equation of (1) is zero and (w,z_0)=1. (6) The sharp upper root star gives ||z_0||_L2(C)<=C r sqrt(a_ap). On a bottom coarse cell, (w,b_K)=O(e); therefore (w,z_0)_K=(w,lambda_o t)_K+O(e) =(4/45)|K| a_t+O(e). (7) The coefficient follows explicitly from integral lambda_o^2=|K|/6, integral b_K lambda_o=3|K|/20, and c_K=-(14/27)a+O(e). Summing the two coarse cells and using (5) above C yields 1=C_0 a_t+err_1, ; C_0=64r^2/45, ; |err_1|<=C[a_ap r^2|a_t| +(1+r sqrt(a_ap)K_log)e]. (8) For sufficiently small fixed a_ap this implies |a_t|<=C(1+K_log e), ; |a_t-C_0^(-1)|<=C[a_ap+K_log e]. (9) All constants here are independent of recursion depth and the fine boundary grids.

4. Second exact linear equation: global affine pressure test. Take q(X)=x, with a mean correction if the pressure space is mean-zero. It is an actual continuous P1 pressure function on every affine mesh. Its right-hand side in (1) is exactly one. Thus 1=integral_D w_t. (10) The two bottom coarse means in (3) give integral_(K_- union K_+) w_t=M_0 a_t+O(e), ; M_0=8r^2/5. The upper rectangle contribution is bounded by (5) and its fixed area. Hence 1=M_0 a_t+err_2, ; |err_2|<=C[r^2 sqrt(a_ap)|a_t|+K_log e]. (11) But M_0/C_0=9/8 exactly. Substituting (9) into (11) gives 1/8<=C[sqrt(a_ap)+K_log e]. (12) Choose the aperture sufficiently small that the first term is at most 1/16. Then max_j |L_avg(P_hu_j,phi_j)|=e>=c/K_log. (13) In particular the weaker convenient bound c/(1+K_log)^2 also holds. This is the genuine AVG/ODD four-source normalization. It relies on the actual sharp root-star geometry, which was not needed in the earlier right-half proof.

5. Reflection-odd source states and the pressure bridge. The contrast L_avg is odd. Project each source/state onto the reflection-odd sector. Its L_avg value is unchanged, its norm stays bounded, and the full coarse-bubble span is invariant under reflection. At least one of these odd projected source states therefore has the lower bound (13). Roughness of a fixed odd combination of the four coarse sources forces roughness of at least one individual source by the triangle inequality. Their full mass projections remain fixed coarse H1 bubbles/combinations, so the established pressure-observable coarse-projection bridge applies. The constants in the six-actuator mass realizations and their H2 bounds involve only the unchanged coarse cells. K_log grows at most like sqrt(1+log(1/epsilon_strip)), and the chosen aperture a_ap=L^-220 tends to zero. Thus (13) is compatible with the asymmetric left-eigenfunctional error bounds; no source residue is inferred merely from a nonzero finite-dimensional coupling.

Part 26: Pressure-observable coarse-projection bridge. Hypotheses and conclusion. On one fixed two-dimensional polygon D and the actual MINI mesh family, let R_h be the L² projector onto the full velocity space V_h, P_h the L² projector onto Z_h, and A_h the positive kernel Stokes operator for the true symmetric-gradient form a_nu. Let u_h^* be smooth exactly solenoidal no-slip sources with uniformly bounded H² norms. Assume R_h u_h^*=w_h, sup_h ||w_h||H1<infinity. (1) Suppose, for SOME 0<s<1/4, ||P_h u_h^*||H^s(D) is unbounded. (2) Then there are actual norm-one q_h, lambda_h -> infinity, and any chosen s' with s<s'<1/4, along a subsequence, such that lambda_h^(s') |(u_h^*,W_(lambda_h)q_h-W_hq_h)| -> infinity. (3) The lifts are the actual strain Brinkman and mass divergence lifts. Therefore (3) meets the complete finite-amplitude nonlinear/BE realization criterion for sufficiently large FIXED extraction order m. It would refute the ORIGINAL smooth pressure clause with its original data scope. This note proves the implication, not hypothesis (2).

1. An actual bounded divergence load and decomposition. Define q_h as the pressure-space Riesz representative of the functional eta -> (w_h,grad eta)=(-div w_h,eta). Thus ||q_h||2<=||div w_h||2 is uniformly bounded. Let L_hq_h be the energy-minimum divergence lift and W_hq_h the mass-minimum divergence lift, with this SAME actual moment convention. Uniform MINI inf-sup on the prescribed fixed shape bound gives ||L_hq_h||H1<=C||q_h||2. Set k_h=L_hq_h-W_hq_h=P_h L_hq_h, ; s_h=w_h-L_hq_h in Z_h. Since P_hu_h^*=P_hw_h=w_h-W_hq_h, P_hu_h^*=s_h+k_h, sup_h||s_h||H1<infinity. (4) The energy-minimum property gives a_nu(L_hq_h,z)=0 for every z in Z_h. The uniform L² bound on k_h follows from L² contraction of P_h and the H1 bound on L_hq_h.

2. Exact positive pressure observation for k_h. Let z(t)=exp(-t A_h)k_h, and let pi_k(t) be its actual homogeneous mixed Stokes pressure: (z_t,v)+a_nu(z,v)+(v,grad pi_k)=0, v in V_h. Test with L_hq_h. Because z,z_t belong to Z_h and the energy lift is energy-orthogonal to Z_h, (pi_k(t),q_h)=-(z_t,L_hq_h) =(A_h exp(-t A_h)k_h,k_h)>=0. (5) Consequently its infinite-time Laplace observation is exactly G_h(lambda):= integral_0^infinity exp(-lambda t)(pi_k(t),q_h) dt =(A_h(lambda+A_h)^(-1)k_h,k_h) =sum_i [a_i/(lambda+a_i)] |k_(h,i)|², (6) where a_i are the actual positive kernel eigenvalues. There is no modal cancellation in (6). The divergence load is bounded in the REQUIRED spatial L² norm. For the H1-bounded part s_h in (4), the homogeneous kernel solution has integral_0^infinity ||z_t||2² dt =a_nu(s_h,s_h)/2, ; integral_0^infinity ||z||H1² dt <=C||s_h||2². Full mixed inf-sup then bounds its actual pressure pi_s uniformly in L²(0,infinity;L²). This uses the actual full derivative z_t in L², not an extension from the solenoidal dual. Therefore |integral_0^infinity exp(-lambda t)(pi_s(t),q_h)dt| <=C lambda^(-1/2). (7) Equations (5)--(7) show that the actual pressure of initial P_hu_h^* has Laplace observation at least G_h(lambda)-C lambda^(-1/2).

3. Physical roughness forces slower-than-quarter decay. By (2) and (4), k_h is unbounded in physical H^s. The embedding ||k_h||H^s(D)<=C[||k_h||2+||A_h^(s/2)k_h||2] (8) has a constant independent of the mesh. Only the inclusion direction is needed; no uniform stability of an orthogonal projector or converse fractional norm theorem is used. One direct proof is the zero-extension translation/K-functional argument: for any decomposition k=x+y in Z_h, ||T_a k-k||2<=2||x||2+C|a|nu^(-1/2)||A_h^(1/2)y||2. Minimize the squared bound and integrate against |a|^(-2-2s). The finite spectral quadratic K-functional gives a constant times sum a_i^s|k_i|² for 0<s<1. Fractional Poincare and the already bounded L² term give (8). This is the same direct argument used for the intrinsic quarter-power Sobolev estimate in the nonlinear realization note. Hence sum_i a_i^s|k_(h,i)|² is unbounded. But the exact moment identity is integral_0^infinity lambda^(s-1) G_h(lambda) d lambda =c_s sum_i a_i^s |k_(h,i)|², ; c_s=integral_0^infinity t^(s-1)/(1+t)dt in (0,infinity). (9) If lambda^(s')G_h(lambda) were uniformly bounded for lambda>=1 and all h, (9) would be uniformly bounded: the part below one uses G_h<=||k_h||2², and the part above one integrates lambda^(s-1-s'). This contradicts (8)--(9). Thus one can select lambda_h -> infinity with lambda_h^(s') G_h(lambda_h) -> infinity. (10) Bounded lambda cannot produce this divergence since ||k_h||2 is uniformly bounded.

4. Actual source crossentry and original-scope realization. By (7), since s'<1/4<1/2, the positive term (10) dominates the s_h pressure. Normalize q_h to norm one whenever needed; its norms are uniformly bounded above, and the selected G_h is nonzero. The actual mixed Laplace identity identifies this pressure observation, up to the fixed sign convention, with (u_h^*,W_lambda q_h-W_hq_h). This identity uses the smooth source u_h^* and its actual projected initial value; no Kq is substituted as admissible initial data. Equivalently the pressure is phi_lambda-phi_0, where both normalized resolvent and mass problems have full source u_h^*. This proves (3). Choose one fixed integer m with s'<m/[4(m+1)]. Then (3) implies the strict-power hypothesis of the completed nonlinear realization theorem. Uniform H² source bounds allow its Section 6 varying-source form. It supplies exact stationary smooth NS data, actual finite-T pressures, and an actual BE root with arbitrarily fine permitted time steps. Thus the only additional hypothesis in this bridge is the MASS roughness (2) for sources satisfying (1).

Realizing (1) with fixed smooth coarse sources. Use the two unchanged coarse triangles K_-,K_+ of part 24. The six compact curl actuators there give an invertible pairing matrix on the six-dimensional local velocity coefficient space (a,c_-,c_+): if all pairings vanish, both bubble coefficients vanish and the two nonparallel coarse curl equations force a=0. Thus they realize ANY full velocity mass load supported on this space. In particular prescribe w_h=w_j to be one of the FOUR fixed coarse cell bubbles, zero on every other cell and with zero nodal values. Choose a fixed linear combination u_j of the six smooth actuators satisfying (u_j,v_h)=(w_j,v_h) for EVERY global v_h. The equality reduces to those six coarse coefficients, because both sides vanish outside the two unchanged triangles. Its fixed 6x6 matrix is invertible as just proved. Therefore R_hu_j=w_j exactly for every refinement depth. The four u_j are fixed smooth solenoidal sources with uniformly bounded H² norms; the four w_j have uniformly bounded H1 norms. Their divergence loads are fixed coarse L² functions before pressure projection. This is an actual admissible source family with property (1).

Part 27: Nonlinear finite-amplitude realization and genuine backward Euler roots. Final criterion and retained first proof. The final sufficient hypothesis is (13): for ANY fixed integer m>=1, lambda_j^[m/(4(m+1))] a_j→infinity implies the stated original-scope nonlinear/BE refutation. Consequently ANY actual strict power lower bound a_j>=c lambda_j^(−gamma), gamma<1/4, suffices. Section5 proves this by finite real-amplitude cancellation. The first m=1 proof, with exponent 1/8, is retained in Sections1–4 as a corollary and as the complete underlying small-data/Laplace/BE construction. No merely logarithmic critical-endpoint claim is made. Fix dimension two, one original admissible polygon D, viscosity nu>0 and T>0, one fixed smooth compactly supported exactly divergence-free u_* and a shape-regular sequence of actual MINI meshes. Let P_j be their actual weak-kernel mass projections and let W_j,W_(lambda,j) be their actual mass/Brinkman divergence lifts. Suppose there are lambda_j→infinity and q_j in Q_j with ||q_j||2=1 such that a_j=|(u_*,W_(lambda_j,j)q_j−W_jq_j)|, ; lambda_j^(1/8) a_j → infinity. (1) For this retained m=1 sufficient hypothesis there are amplitudes epsilon_j bounded by one fixed arbitrarily small number, stationary ORIGINAL smooth solutions u_j=epsilon_j u_*, p_j=0 with forces f_j=−nu epsilon_j Delta u_*+epsilon_j² N_s(u_*), (2) and genuine backward-Euler MINI solution branches initialized by P_j u_j whose instantaneous pressure ell^(4/3)L² norms diverge. All exact-solution data norms, M, M_infinity and M_E in the current smooth theorem stay uniformly bounded; the original explicit timestep condition can be met. The final Section5 relaxes this first sufficient hypothesis to every fixed m. Its premise is supplied in the final assembly.

2. Uniform small-data semidiscrete velocity bound. The full skew nonlinear vector is N_s(z)=z dot grad z+(div z)z/2. Holder and Ladyzhenskaya give ||N_s(z)||_(4/3)<=C||z||2^(1/2)||z||V^(3/2). (4) Combining (3) with (4) gives the kernel-dual bound ||A^(−1/4)P N_s(z)||2 <=C_nu||z||2^(1/2)||z||V^(3/2). (5) Here P N means only the finite Riesz representative of the nonlinear functional restricted to Z_h; P is not applied to a purported full L² vector when the latter is unavailable. The difference estimate follows from the actual bilinear nonlinear form and has constant C times the sum of the corresponding norms. For the mild semidiscrete equation use ||z||X_T=sup_(0<=t<=T)||z(t)||2 +sup_(0<t<=T) t^(1/2)||z(t)||V. Spectral calculus gives ||A^beta exp(−tA)||<=C_beta t^(−beta). The initial state epsilon P u_* and the constant force (2) have linear mild X_T norm at most C_(D,nu,T,u*) epsilon for epsilon<=1. For example the force contributes at most T||f||2 to the L² bound and C sqrt(t)||f||2 to the V bound before multiplication by sqrt(t). If z is in an X_T ball of radius R, (5) is at most C R² t^(−3/4). The nonlinear mild convolution has L² bound C R² integral_0^t (t−s)^(−1/4)s^(−3/4) ds=C R², and V bound C R² integral_0^t (t−s)^(−3/4)s^(−3/4) ds =C R² t^(−1/2). Both beta integrals are finite. The same calculation for differences makes the mild map a contraction on a ball of radius C epsilon when epsilon<=epsilon_0, with epsilon_0 independent of the mesh. This proves the UNIFORM actual velocity estimates ||z(t)||2<=C epsilon, ; ||z(t)||V<=C epsilon t^(−1/2), 0<t<=T. (6) For each fixed mesh this is the ordinary smooth finite-dimensional Navier–Stokes ODE solution; global existence on [0,T] also follows directly from its skew energy identity, and ordinary finite-dimensional local uniqueness identifies the mild solution. No uniqueness of arbitrary large-step roots is asserted or needed for a counterexample to an any-root conclusion.

4. Choosing amplitude and passing to actual BE roots. The actual uniform L² lift bound implies 0<=a_j<=C||u_||2. Choose epsilon_j=c a_j, with c>0 so small that epsilon_j<=epsilon_0 and C c<=1/4 in (10). Assumption (1) implies lambda_j a_j→infinity and exp(−lambda_jT)/a_j→0. Equations (8)–(11) therefore imply, for all sufficiently large j, |(p_hat_j,q_j)| >=(c/2) a_j². Holder on [0,T] gives ||p_j||L^(4/3)(0,T;Q) >= [4lambda_j/(1−exp(−4lambda_jT))]^(1/4) ; |(p_hat_j,q_j)| >=c' lambda_j^(1/4)a_j² →infinity. (12) The last divergence is precisely the square of (1). This is why the criterion deliberately demands the stronger exponent 1/8, rather than converting a merely linear 1/4 counterexample by unsupported linearization. For completeness the passage to backward Euler is only fixed-mesh finite-dimensional analysis. For each j the ODE vector field is polynomial and Lipschitz on a ball around its compact trajectory. For all sufficiently small steps the implicit map is a contraction on the neighboring ball. Taylor consistency and the recurrence e_n<=(1−L tau_n)^−1(e_(n−1)+C tau_n²) give uniform convergence of one genuine BE root branch to the ODE as tau_max→0. Its forcing is constant and hence equals the required step average exactly. Its initialization is the required epsilon_j P_j u_. The mixed pressure is a continuous finite-dimensional function of the current velocity: the kernel equation replaces the discrete time derivative by P f−A z−P N_s(z), and inf-sup recovers the unique meanzero pressure from the full residual. The same formula holds for the ODE. Hence the constructed BE pressures converge to p_j uniformly in time along the nodes, and their ell^(4/3)L² norms converge to the integral norm in (12). Choose the mesh-dependent steps sufficiently small to keep that norm within one of (12), meet tau_max Lambda<=1/2, and send tau_max to zero. Finally all exact solutions epsilon_j u_* have uniformly bounded approved data quantities; (2) is smooth and uniformly bounded as epsilon_j stays bounded. The mesh maxima are bounded by diam(D), so the original first-order RHS with its explicit data-dependent constant is uniformly bounded, whereas (12) diverges. This proves the internal order-one criterion under (1); its order-four extension below is the one applied in part 28.

5. Finite real-amplitude cancellation approaches the linear threshold. There is a stronger version of the criterion. Fix ANY integer m>=1. Replace (1) by lambda_j^[m/(4(m+1))] a_j →infinity. (13) Then the same original-scope nonlinear/BE refutation follows. In particular any actual fixed-source lower bound a_j>=c lambda_j^(−gamma) with gamma<1/4 suffices, by choosing a fixed finite m with gamma<m/[4(m+1)]. This does not settle a merely critical logarithmic violation of the linear threshold. Here is a complete uniform power-series justification, without deriving a linearized bound by dividing an estimate whose M has a constant floor. Let B_X be the bounded bilinear mild map established in Section2, with norm C_B. The linear coefficient z_1 is the Stokes response to initial P u_* and force −nu Delta u_, and the quadratic forcing response g_2 is the Stokes response to N_s(u_), both uniformly bounded in X_T. The velocity coefficients satisfy z_1=linear response, ; z_2=g_2−B_X(z_1,z_1), ; z_k=−sum_(r+s=k)B_X(z_r,z_s), k>=3. (14) Choose constants L_1,L_2 bounding the two linear responses. The scalar majorant w=L_1 x+L_2 x²+C_B w² has a power series with nonnegative coefficients, obtained by the convergent binomial series of w(x)=[1−sqrt(1−4C_B(L_1x+L_2x²))]/(2C_B) (with the evident linear formula if C_B=0). It converges for some fixed positive rho, independent of the mesh. Induction in (14) shows ||z_k||X is dominated by the coefficient of w. At any strictly smaller positive rho, positivity gives ||z_k||X<=C rho^(−k). Thus sum epsilon^k z_k converges absolutely in X_T for |epsilon|<rho, solves the actual mild equation by continuity of B_X, and equals its unique small-data solution. This proves the required uniform real power series; a complex analytic continuation is not assumed from an inequality for real data. For the full bilinear skew vector, Holder gives ||B_s(v,w)(t)||_(4/3) <=C[||v(t)||4||w(t)||V+||v(t)||V||w(t)||4] <=C||v||X||w||X t^(−3/4). (15) Equations (7),(15) consequently give a bilinear Laplace observation norm at most C||v||X||w||X for every lambda>=1 and every unit q. The terminal coefficient has bound C exp(−lambda T)||z_k||X; the only external forcing coefficients are orders one and two, bounded by C/lambda. The finite-time pressure observation therefore has the convergent scalar expansion P_(j,lambda)(epsilon)=(p_hat(epsilon),q_j) =epsilon d_(j,lambda)+sum_(k>=2)epsilon^k d_(k,j,lambda), ; |d_(k,j,lambda)|<=C R^(−k), k>=2, (16) for fixed R>0, uniformly in mesh and lambda. To see the geometric bound explicitly, (14) bounds each product in the kth nonlinear coefficient by C rho^(−k); there are k−1 terms, and k rho^(−k)<=C(rho/2)^(−k). The forcing and terminal terms obey the same enlarged bound. The linear coefficient obeys |d_(j,lambda)−(u_*,W_lambda q_j−Wq_j)| <=C/lambda+C exp(−lambda T). (17) All estimates follow along the actual velocity solution; no assertion about the full nonlinear instantaneous pressure norm is used in (16). Choose distinct fixed numbers t_1,...,t_m in (0,1], for example t_l=l/m. The finite matrix (t_l^k)(1<=k,l<=m) is invertible (a Vandermonde matrix times a nonzero diagonal), so there are fixed real weights w_l satisfying sum_l w_l t_l=1, ; sum_l w_l t_l^k=0, 2<=k<=m. (18) The zero-order condition is unnecessary because pressure at epsilon=0 is zero. Let W_m=sum|w_l|. For 0<epsilon<=R/2, (16),(18) show |sum_l w_l P_(j,lambda)(t_l epsilon) ; −epsilon d_(j,lambda)| <=C_m epsilon^(m+1). (19) Take epsilon_j=c_m a_j^(1/m), choosing the fixed c_m sufficiently small that epsilon_j<=min(epsilon_0,R/2) and C_m c_m^m<=1/4 for every j; the uniform upper bound for a_j makes this possible. Assumption (13) implies lambda_j a_j→infinity and exp(−lambda_jT)/a_j→0. Hence (17),(19), with the observation q_j reversed if necessary, give |sum_l w_l P_(j,lambda_j)(t_l epsilon_j)| >=(1/2)epsilon_j a_j. (20) At least one of the m actual smooth stationary-data amplitudes t_l epsilon_j therefore has pressure Laplace observation of absolute value at least epsilon_j a_j/(2W_m). Holder as in (12) yields, for that data choice, ||p_j||L^(4/3)(0,T;Q) >=c_m' lambda_j^(1/4)a_j^(1+1/m) →infinity. (21) The divergence is exactly (13) raised to power 1+1/m. The amplitudes remain uniformly small, all prescribed smooth data quantities remain uniformly bounded, and Section4 passes the selected actual trajectory to a genuine backward-Euler branch with arbitrarily small original-admissible steps. This proves the extended criterion. The construction uses a finite linear COMBINATION of observations only to identify one actual data amplitude with a large pressure. It does not combine different solutions into a purported Navier–Stokes solution. Each candidate amplitude individually has exact stationary solution t_l epsilon_j u* and its own force (2). The final assembly supplies an actual coupled crossentry satisfying (13) on one fixed admissible shape family. Neither the ideal normal transfer factor nor an Hs-only projection counterexample supplies it.

6. Uniformly bounded varying smooth sources are also admissible. The fixed source in Sections 1–5 is convenient, not required by the original pressure claim. Let u_j be any family of smooth exactly solenoidal no-slip velocities on the SAME fixed two-dimensional domain, with uniformly bounded H² norms. They may, for example, belong to a fixed finite-dimensional space of compact smooth coarse-patch sources with uniformly bounded coefficients. Suppose the actual entries a_j=|(u_j,W_(lambda_j) q_j-W_h q_j)|, ; ||q_j||2=1, satisfy the Section 5 strict-power criterion for one fixed integer m. Then the same original-scope pressure refutation follows. Indeed replace u_* everywhere by u_j. Their L² norms, the stationary linear forces -nu Delta u_j, and their quadratic forces N_s(u_j) are uniformly bounded by the assumed H² bound and the fixed-domain two-dimensional Sobolev inequalities. Thus the Kato analytic radius, the velocity coefficient majorant, the pressure coefficient bounds, the finite real extraction weights, the finite-T forcing error, and the terminal error all have the SAME mesh- and j-independent constants. The finitely many amplitudes chosen for each j therefore give exact stationary smooth data with uniformly bounded original M and all approved constants. Fixed-mesh BE approximation is still selected separately for each j, exactly as before. No convergence of u_j or fixed-source subsequence is needed. This observation does not establish a source-control lower bound. D_j-dependent coefficients are useful only after proving they remain uniformly bounded and produce the actual pressure-observed entry. A source family annihilated by the relevant adjoint channel cannot be used merely because its dimension is large.

For the actual strain Brinkman lift, let $Lq$ be the energy-minimum discrete divergence lift and $W_{\lambda }q$ the minimum of $\lambda \mathrm{\parallel }v{\mathrm{\parallel }}_2^2+a_{\nu }(v,v)$ under the same moments $(v,\mathrm{\nabla }r)=(q,r)$. Fixed-shape inf-sup gives $\mathrm{\parallel }Lq{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }q{\mathrm{\parallel }}_2$. Comparing the Brinkman minimum with $Lq$, for $\lambda \ge 1$, gives $\mathrm{\parallel }{W}_{\lambda }q{\mathrm{\parallel }}_2\le C\mathrm{\parallel }q{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }{W}_{\lambda }q{\mathrm{\parallel }}_{H^1}\le C\sqrt{\lambda }\mathrm{\parallel }q{\mathrm{\parallel }}_2$, with constants depending on the fixed viscosity and shape bound. Step 1.2 then gives $\mathrm{\parallel }{W}_{\lambda }q{\mathrm{\parallel }}_4\le C{\lambda }^{1\mathrm{/}4}\mathrm{\parallel }q{\mathrm{\parallel }}_2$. The mass lift $Wq$ obeys the first bound by comparison with $Lq$, and $W_{\lambda }q-Wq\in Z_h$. These are actual full mixed lifts, not extensions of a restricted time derivative.

1. Uniform positive fractional embedding without a projector theorem. For each mesh let A be the positive mass Stokes operator on Z_h. Its form obeys a(z,z)>=nu||grad z||2². The following endpoint estimate is uniform: ||z||4 <= C nu^(−1/4)||A^(1/4)z||2, z in Z_h. (3) Here is a direct proof that does not assume physical fractional stability of any orthogonal projector. Extend all zero-boundary functions by zero to R2. For z=x+y with x,y in Z_h, translation by a vector of length r obeys ||z(.+a)−z||2 <=2||x||2+r||grad y||2. Therefore its square is bounded by a fixed constant times the quadratic K-functional K2(r/sqrt(nu),z)^2 for the finite-dimensional pair (L²(Z_h), form-norm(Z_h)). In a mass-orthonormal eigenbasis of A, minimizing separately gives K2(t,z)^2=sum_k [t² alpha_k/(1+t² alpha_k)]|z_k|². Integrate the translation estimate over a with kernel |a|^(−3). Polar coordinates and the nonnegative finite spectral sum give [z_zero]_(1/2,2)^2 <= C nu^(−1/2) integral_0^infinity t^(−2)K2(t,z)^2 dt = C' nu^(−1/2) sum_k alpha_k^(1/2)|z_k|². The last integral uses integral_0^infinity (1+t² alpha)^−1 dt=pi/(2 sqrt(alpha)). Apply the actual published thm-fractional-sobolev-inequality-on-euclidean-space, with d=2,p=2,theta=1/2, to obtain (3). Countable Choice is inherited from that supplier. No Fourier/Slobodeckij identification or interpolation-space equality is needed. The ordinary two-dimensional Ladyzhenskaya estimate ||z||4<=C||z||2^(1/2)||z||V^(1/2) is the scalar/componentwise inequality proved in step 1.2; it does not require exact divergence freedom.

Part 28: One fixed template, source-visible growth and the original pressure claim. 2. One parameter receipt and actual mesh family. Put L=log N and take sufficiently large integer N (dyadic rounding of the following small parameters is harmless): delta=L^-200, a=L^-220, b0=L^-20, ; sigma=L^-150, epsilon_Q=L^-380, ; epsilon_corner=L^-600. Fix epsilon_strip DYADIC in [a/8,a/4] at this stage, independently of every later h refinement. The physical side-strip vertices and their positive internal port cells are therefore fixed before h. The logarithmic lift has K_log^2<=C(1+log(1/epsilon_strip))=O(1+log L), in particular O(L). Here a is fan aperture, not a velocity coefficient or the final nonlinear amplitude. Use the mirrored two-fan template. The actual straight long tooth sides have root distance D_i comparable to a/N and longitudinal coordinate Y_i comparable to one. Reserve middle-third attachment windows with spacing ell comparable to b0 a/N, common child radius rho comparable to sigma ell, and reflection-paired counts B comparable to N^2/(b0 a), S0=B rho^2 comparable to sigma^2 b0 a. Use the exact seven-node unrefined child boundary. Every child is a rotated or reflected homothetic copy of the same template. Frame tangents are labelled in the direction nearest the parent tangent; normals point into the child. The reflected-frame convention is O_i'=S O_i S, as proved in the internal proof in part 22. This preserves the sign of paired tangent contrasts. The actual geometry, matching uniform rings and radial refined parent guards are those in the internal proof in part 2. In particular: Every local comparison window lies wholly on ONE straight tooth side, with all below-base triangles radial from the same parent root and Y_i comparable to the parent radius. Safe radii are fixed small multiples of ell (one may use ell/1000). Nested cutoffs fit inside these windows: bulk commutator annulus R..2R, stencil enlargement below3R, base pressure and velocity lifts equal the actual remainder through4R and vanish beyond8R, trace window through16R. Tail inputs therefore start beyond the bulk annulus. The common actual uniform region includes every cutoff stencil. Physical child base edges and all seven child boundary edges remain whole. Extra uniform guard nodes occur outside the two child-base edges. Their exterior stars are O(h^2). The root has fan area comparable to a times its container area. All nonfan exterior stars in the outer-row tests satisfy epsilon_Q. The unavoidable selected center ratio is O(a sigma b0)=O(L^-390), strictly below epsilon_Q. Original child boundary-port stars remain positive independently of the later h choice. Child-side CORNER stars satisfy the independent single-parent-triangle ratio epsilon_corner. This is possible by finite interior star surgery, leaving both original boundary edges whole. The center mismatch is made exactly zero by matching the constant driver to its actual velocity, so its fan-area floor is not used for this smaller error. Mirror grids have opposite diagonal orientations. Include BOTH child kink-macro crease families t=n and t=-n, and both U-transition overlays, not only the right-top crease. These rays end at existing root/top-corner nodes, avoid the protected near-base cones, and add no eighth child boundary node. All collar/pressure-macro refinements and h-dependent changes are reflected together. Pressure lift affine macrozones contain the entire protected fans and all recursive holes. Choose one uniform width h sufficiently small AFTER N. Besides all finite geometric grid conditions, require h/(sigma D_i) <= c L^-600, ; h <= c a rho, ; h <= c rho^2/R for every finitely many first-generation side type. Make the exterior transition vary as h decreases in the explicit compatible exterior Mosco construction, rather than keep a coarse transition fixed. This shrinking EXTERIOR transition does not remove the positive INTERNAL child port triangles. Further reduce h below the symmetry tolerance in section 4. Once selected, fix this ONE h and its finite actual template forever. Repeating the finite pattern to depth m gives conforming meshes with a finite shape-regularity constant depending on this fixed N,h, but independent of m. No condition of the original theorem bounds that constant by a preselected numerical value.

3. Actual mass limit, quotient and correct source functional. For each fixed chosen template, the actual nonnested mass Mosco proof gives the limiting child DtN D_infinity, actual response maps and finite-depth convergence, including pressure gradients and all interface coefficients. This is the internal proof in part 4. Its exact Schur fixed point gives actual child recoveries T_i; no terminal map is iterated in place of the recursive one. Quotient the finite retained boundary-state space by the exact affine Darcy family W=c,p=gamma-c dot X. The quotient norm is inf_c ||W-c||_L2(C). Its kernel is exactly that family by the actual source-free mass bubble/affine detection proof. Affine pressure gauges are removed; affine weak reactions are retained. Let U=B^-1 sum_i T_i be the actual mean recovery after the exact child scaling and type transport. Reflection preserves its odd sector. The functional used throughout is the AVERAGE contrast L_t(s)=W(o)_t+[p(c_+)-p(c_-)]/8. It is reflection odd. The earlier RIGHT-HALF functional is not substituted for it. Odd source states have exactly affine pressure on their three-node parent base.

4. Radiation ideal, bounded odd projection and parity repair. Use ONLY the radiation-consistent embedding from the internal proof in part 14. It is NOT the historical finite-ring far-Dirichlet embedding. Its auxiliary uniform halfplane is used after actual localization. The exact constant-base tangential/normal layers need not be globally L². Subtract them outside a compact cutoff, restore the actual neutral base lift, and solve the compact mixed residual in velocity L² and zero-flat-base pressure-gradient Hilbert spaces. Halfplane Hardy proves exterior density; the actual mean-one bubble gives mass inf-sup sqrt(7/10). The total plane-plus-scattered equations are local; the finite-energy correction is the global variational object. The Hilbert construction in part 14 proves these statements, with the mass limit and interface gluing of part 4. Its two ideal child responses have mass at most C sqrt(a), and root trace gives the matching lower bound. Let E_t^- be the actual ODD PART of its tangent column. Then L_t(E_t^-)=1 and its quotient norm g satisfies c sqrt(a)<=g<=C sqrt(a). The root mass lower bound and fixed-rectangle H1/2 trace estimate on the two affine base edges give g |L_t(s)| <= C ||s||_quotient. Thus e=E_t^-/g, ell=g L_t define a uniformly bounded exact rank-one projection P=e ell on the odd quotient; Q=I-P is bounded. A single SWNE exterior is not reflection invariant. Its normal column's odd part is retained. The internal proof in part 15 proves that this defect tends to zero as h tends to zero AFTER N, with the actual transition changed. Only the compatible free twelve interface ports are involved: three base pressures zero, four free outer pressure traces matching zero at base corners, and eight free velocities. No arbitrary incompatible corner-pressure convergence is asserted. Uniform actual child signed-energy coercivity bounds the coupled inverse despite varying child D_h; original child port triangles stay positive. The SW/NW exterior matrices converge to the SAME continuous Darcy map, and the radiation driver tends to zero. Therefore E_h(D_h)-E_0(D_h) tends to zero, uniformly along the actual family D_h, and E_0 is reflection equivariant. The normal-driver functional of an arbitrary parent state has a finite C_N bound INDEPENDENT of h: selected center values are bounded by their unrefined parent fan triangles, and incoming regular velocity by the actual uniform kernel value and ring energy. Choose h so small that the combined odd normal-column contribution is at most L^-30 times parent quotient norm. This is noncircular: all N-dependent constants and the positive child port floors are fixed first; no convergence rate is required. The finite h chosen afterward is then fixed with recursion depth.

5. Complete actual per-child decomposition and complementary row. All remainders below are RESTORED source-free child responses. For a pressure lift Phi the physical pair is (w_corr,p_corr+Phi); for a neutral velocity lift Eg it is (w_corr+Eg,p_corr). Auxiliary loaded pairs are not passed to the child response map or fractional detection. Start with an arbitrary odd parent response and subtract an arbitrary affine Darcy shift. On each safe side window extract the affine pressure CHORD slope C_I=mean(T), where T is the tangential gradient. The exact radial identity D_y T=-(D_i/y)D_y N, ; D_y[T+D_i(N-N0)/y]=-D_i(N-N0)/y^2 for every constant N0 gives ||T-mean T||_L2(I)<=C(D_i/Y_i)||N-N0||_L2(I). The base area measure is D_i dy/2. Thus all guard-breakpoint pressure remainders, not only the child three nodes, have an actual collar pressure lift of squared energy at most C R D_i/Y_i^2 times parent-patch quotient energy, namely C b0(a/N)^2. The explicit lift is g(x)+k(S-|x|), equal inside the child to an affine function plus k|x-y|; it preserves all seven edges, whole protected affine zones and all actual extra guard values. All gradient estimates are from prescribed affine pieces or uniform-grid products, not skinny-cell interpolation. Its nested far tail is retained as a distant base input, with actual full-field bound and all base reactions. For velocity choose the constant incident value equal to the actual child-center velocity, making its center residual zero. Actual coarse P1+b mass positivity gives |K||W(e)-W(o)|^2<=24||W-c||_K^2 on each parent base triangle. Independent corner stars and all uniform guard stars therefore give squared neutral-lift budget eta_base <=C[epsilon_corner+h/(sigma D_i)]<=C L^-600. This is a FULL VECTOR estimate, including normal endpoint patterns; no normal pressure-gradient regularity is assumed. The nested distant base-velocity tail is also retained. Its effective physical pressure input has an h gain and its ring-energy cost is no larger than the guard budget. After removing the local lifts and tails, the uniform bulk cutoff sees zero residual base data. Subtract the local affine field and the exact constant-base short layers before using its driver norm. The chord-slope cost on a ring is at most C(R/D_i) times parent patch energy; the constant shortlayer cost is C h/(sigma D_i) times that energy. The pressure/velocity tails have the previously bounded local ring norms, since they start beyond4R while bulk cutoff stencils lie below3R. Thus the sum of squared actual bulk-driver energies over disjoint rings is at most C times parent quotient energy. The actual incoming Green proof decomposes this far driver into its regular affine field, TWO exact neutral layers, and residual with velocity/pressure-gradient collar norm at most C(rho/R)^2 times ring energy. Its independently extracted normal affine component is not inferred from a parent normal slope. Child-scale compact cutoff, actual mixed bubble stability and restoration bound the scattered residual with the complete radiation child selfresponse. The exact same radiation operator is used here as in section 4. All outgoing child moments/selfresponse remain inside that solve; prescribing the actual base velocity/pressure traces leaves their weak reactions as OUTPUTS, not independent incoming parameters. No additional monopole is assumed zero. Consequently the actual child trace is affine macro plus the radiation two-driver response plus restored errors, with total squared physical error budgets as above. Equal averaging, child-reference rescaling and Cauchy-Schwarz give ||Q U|| <= e2, ; e2 <= C[sigma/sqrt(b0 a) +sqrt(eta_base/(sigma^2 b0 a)) +sqrt(a)/(N sigma)] + L^-30. (A) The final term is the explicitly retained normal parity defect. The cancellation of polynomial N is exact: ell sqrt(B) is comparable to sqrt(b0 a) in unit parent scale. The induced mean norm is the actual source-free child response norm; its linearity justifies averaging states. All affine macro states vanish in the quotient. The normal ideal parity defect is charged, not declared cancelled. For the fixed tuple, both first terms in (A) are O(L^-30); the third is L^40/N. Thus e2 tends to zero with N.

6. Actual all-state tangent row and asymmetric spectral channel. The complete proof is the internal proof in part 22. Use parent global tangent in the full reversible outer-fan inverse, with exact constant reference b_ref,t=-(a_root,t+18p0)/17. The all-guard tangential reset is C delta times parent quotient energy; hybrid parent/exterior tests have ZERO exterior cell means, so all exterior pressure terms cancel and the remaining reaction norm is C sqrt(epsilon_Q) times that energy. No pressure-only exterior block is assumed small. Selected child half-base triangles have equal areas and unrefined edges, so their average pressure slope is exactly the row's pbar dotted with the transported child tangent. Intrinsic nodal/bubble mass controls the O(delta) frame error. The selected parent star-area sum is comparable to sigma a. Therefore the normalized selector satisfies for ALL odd states |ell(U s)+(1/17)ell(s)|<=e1||s||, ; e1<=C(delta+sqrt(epsilon_Q))/sqrt(sigma) =O(L^-125+L^-115). (B) In the splitting range P plus range Q, (A)-(B) give the actual block U=[alpha b;c D], ; |alpha+1/17|,||b||<=C e1, ||c||,||D||<=C e2. For sufficiently large N the scalar Schur contraction has a real eigenvalue lambda with |lambda|>=1/34. Its left eigenfunctional is (1,b(lambda-D)^-1), differing from ell by O(e1), NOT O(e2). This is the asymmetric block lemma already proved in the internal proof in part 23. No generic high-dimensional determinant or raw whole-state norm estimate is used.

7. Genuine average-source visibility. Use the internal proof in part 25. For the adjoint to the AVERAGE contrast, let e_source be the maximum of the four unchanged coarse bubble moments. Small moments give coarse grad p=O(e_source), bubble coefficient -14a_root/27+O(e_source), and coarse mean a_root/10+O(e_source); the bottom pressure hat makes a_root,n=O(e_source). The GLOBAL cell-mean-neutral root velocity test gives the exact linear coefficient C0=64/45. The GLOBAL affine pressure test q=x gives M0=8/5. The upper rectangle's root-only reference has mass O(sqrt(a)|a_root,t|) by the sharp star, and the perturbation is O(K_log e_source). Thus the two equations have the incompatible ratio M0/C0=9/8 unless e_source>=c/K_log. In particular the convenient lower bound c/(1+K_log)^2 holds. This proof does not use the invalid even-root-pressure argument for the historical right-half functional. The fixed six coarse curl actuators realize the four coarse bubbles as FULL mass projections, with H2 bounds independent of mesh depth. Odd projection preserves this average functional; the coarse-bubble span is reflection invariant. Thus one odd source state has normalized selector at least c sqrt(a)/(1+K_log)^2 and bounded quotient energy. K_log^2=O(L) for the fixed dyadic strip choice. Dividing the two left-row terms in (B) by the source scale sqrt(a)=L^-110 gives L^-15 and L^-5, which beat the inverse-logarithmic source bound. Hence the actual left eigenfunctional is NONZERO on one odd coarse-source combination for sufficiently large fixed N. A rough odd combination of the four source responses implies roughness of at least one individual source. No unproved source-cyclicity assumption is made.

8. Actual mass fractional divergence and transfer back to finite depth. Fix s=1/8. At depth m, actual mean recovery is U^m, because D_infinity is the actual autonomous fixed point, not a terminal approximation. The nonzero eigenfunctional gives mean quotient energy bounded below by c|lambda|^m. Jensen in the actual response quotient energy makes the sum of squared depth-m energies at least c B^m|lambda|^(2m). Write r_m=rho^m. On each disjoint depth-m child, the established mass affine-Darcy fractional detection is explicitly E_phys(C_(r_m))<=C_N r_m^s |W|_Hs(C_(r_m)), ; E_ref=E_phys/r_m <=C_N r_m^(s-1)|W|_Hs(C_(r_m)). Therefore sum_child |W|_Hs^2 >=c_N r_m^(2-2s) sum_child E_ref^2. Using the Jensen lower bound above gives a lower growth factor [B rho^(2-2s)|lambda|^2]^m. For s=1/8, B rho^(7/4) is comparable to N^(1/4) times a fixed negative power of log N, hence tends to infinity. Choose N once so that the bracket exceeds one. The infinite-depth restricted source response therefore has infinite physical H^(1/8) norm. The actual finite-depth responses converge strongly in L² to that response. Fractional seminorm lower semicontinuity then forces unbounded H^(1/8) norms along finite depths. This does not require sharp terminal-leaf stars: omit the terminal generation or use the actual infinite limit and lower semicontinuity. Every finite mesh is a genuine conforming shape-regular MINI mesh of the SAME fixed template.

9. Actual pressure observation and original nonlinear/time-discrete realization. Apply the internal proof in part 26 to the resulting individual source u_j. Its FULL unconstrained mass projection is the fixed coarse H1 bubble w_j. With q_h=P_Q(-div w_j), uniformly L² bounded, the kernel initial state splits as uniformly H1 s_h plus Kq_h. The actual homogeneous Kq pressure observation is the positive spectral function G_h(lambda); its Mellin moment turns the physical H^(1/8) divergence into a sequence with lambda^(3/16) times the actual pressure crossentry unbounded. The H1 remainder contributes only O(lambda^-1/2) and cannot cancel it. Normalize q_h where nonzero. This is real pressure output, not an inference from an arbitrary pressure-free rough channel. Finally use the internal proof in part 27 with finite real-amplitude Vandermonde order m=4: m/[4(m+1)]=1/5>3/16. It supplies uniformly smooth exact stationary solenoidal velocities supported on the unchanged coarse cells, exact pressure zero, actual Navier–Stokes forcing, and a sequence of actual semidiscrete pressure norms growing without bound. The real amplitudes and all data obey the ORIGINAL hypotheses with uniform bounds. For each finite mesh, select an actual backward-Euler root arbitrarily close to the semidiscrete solution by choosing time steps sufficiently small; impose also the original tau_max Lambda<=1/2. The theorem's any-root quantifier is refuted by this legitimate branch. No additional mesh-time coupling is an assumption of the counterexample; the original universal statement allows these choices. Global maximum diameter and all approved data/viscosity constants stay bounded while the actual instantaneous ell^(4/3)L² pressure error is unbounded. Thus no constant of the originally claimed form can give its first-order bound on this fixed shape-regular mesh family.

3. Full mixed Laplace estimate along that actual trajectory. The actual lift bounds proved in step 2.1 give for lambda>=1 ||W_lambda q||2<=C||q||2, ; ||W_lambda q||V<=C sqrt(lambda)||q||2, ; ||W_lambda q||4<=C lambda^(1/4)||q||2. (7) The last is the two-dimensional Ladyzhenskaya inequality, not a fractional projector statement. Let p_h(t) be the actual semidiscrete mixed pressure. Integrate its full mixed equation against exp(−lambda t) on [0,T], and test by W_lambda q. The integrated velocity remains in Z_h, so its lambda-mass-plus-viscosity pairing against this lift vanishes. With hats denoting finite-time Laplace integrals, one obtains exactly (p_hat,q)=(epsilon P u_*+f_hat−N_hat ; −exp(−lambda T)z(T),W_lambda q). (8) Mass minimality gives (P u_,Wq)=0 and W_lambda−Wq in Z_h. Thus the initial contribution in (8) is exactly epsilon (u_*,W_lambda q−Wq). (9) By (4),(6),(7), the nonlinear contribution satisfies |(N_hat,W_lambda q)| <=C epsilon² lambda^(1/4)||q||2 ; integral_0^T exp(−lambda t)t^(−3/4)dt <=C epsilon² ||q||2. (10) This controls only the actual finite-time Laplace observation; it does not assert a uniform nonlinear instantaneous pressure bound. The smooth forcing and terminal contributions satisfy, respectively, |(f_hat,W_lambda q)|<=C epsilon lambda^(−1)||q||2, ; exp(−lambda T)|(z(T),W_lambda q)| <=C epsilon exp(−lambda T)||q||2. (11) All constants are independent of the mesh and lambda; they may depend on the fixed nu,D,T,u_. The divergence part of the skew nonlinearity is included in (4),(10).

Every local implication used in this last phase has now been discharged on the same fixed domain, finite template and parameter order. The mesh shape bound is fixed before recursive depth varies. The exact solutions and forces satisfy the unchanged original smooth assumptions with uniform bounds, and the selected pressures belong to genuine backward Euler roots with the required mass initialization. Their pressure norm diverges while $H$, the approved data quantities and the original right-hand side remain bounded. This refutes the stated universal instantaneous rate.