Same-trace unit-window normalization of finite marked Floer families

Normalize three flags of a smooth ordered disk to $\mathrm{\infty },0,1$ on its real boundary. Its remaining flag positions have the fixed cyclic inequalities. For a degenerating sequence, take their limits in ${\mathbb{R}\mathbb{P}}^1$; each coincident consecutive cluster is resolved by subtracting its first coordinate and dividing by its positive extremal separation. Its attachment is the child's infinity flag. Repeat inside each remaining cluster. Each resolution splits a finite cluster into smaller nonempty clusters, so it terminates. The identity ${\sum }_v(\mathrm{val}(v)-2)=N-2$ bounds a stable tree's vertices and edges. Quadruple cross-ratios determine its components and clusters recursively, so the closure in the finite cross-ratio cube is exactly this compact stable-tree space. Transition marks are included in $N$ and in every cluster, even on a transition-only child. Thus the construction does not replace the actual marked family by its ordinary unmarked stabilization.

Around a fixed stable tree choose three distinct flags on each component, and restrict its free positions to a relatively compact neighborhood with all its component flags still distinct. For a parent attachment $a$ and child coordinate $x$, use $z=a+\rho x$ and $w=-1\mathrm{/}x$; then $(z-a)w=-\rho$. At $\rho =0$ keep both normalized components. Finite products of these fixtures, with the complements of fixed disjoint flag disks, are actual holomorphic local families. Different anchor choices are rational Möbius maps with denominators bounded away from zero on these fixed cores and transition annuli. Their coordinate derivatives and inverse derivatives are bounded there. Compactness in step 1.1 gives finitely many such tree neighborhoods for each finite context. This is a finite cover of normalized component parameters, not a positive lower bound on distances of colliding marks in a single unnormalized root coordinate.

On a real fixture put $z=e^{-\pi s+i\pi t}$ and $w=e^{-\pi (L-s)+i\pi (1-t)}$, $0\le t\le 1$, so $zw=-e^{-\pi L}$. The change between its two logarithmic coordinates is $(s,t)\mapsto (L-s,1-t)$, an isometry. Choose fixed attachment annuli, independent of $L$. Their overlap with component core charts lies at fixed nonzero radii, so the logarithm and its inverse have bounded derivatives there. Long middles use translated width-one strips. Short fixtures have $\rho$ bounded away from zero and belong to an ordinary compact nonsingular family, covered by finitely many charts. The parent core omits its child cluster disk: the small affine derivative $\rho$ is therefore never used as a unit-window transition between two retained compact cores. They are joined through the just-constructed logarithmic fixture. A partition over fixed attachment annuli gives a complete conformal metric with uniformly bounded unit charts. Unit strips have a bounded-overlap rectangular cover; the finite core cover does too. Choosing $r_0<1\mathrm{/}8$ and shrinking the finitely many core charts if needed, an interior disk or a boundary-centered half-disk of a fixed comparable radius covers every point. A half-disk in a strip meets at most one of $t=0,1$. At a zero node work on its two punctured normalized branches; it is not a finite boundary corner of one window.

The protected Moore models are closed restrictions of this stable-tree family by $q_i\le e^{-\pi T_i}$. On a spine component, logarithmic adjacent mark coordinates give ${\tau }_{i+1}={\tau }_i+T_i+{\mathrm{\ell }}_i$. At a finite merge ${\mathrm{\ell }}_i=0$, the internal marks remain at their summed atomic offsets; the concatenated block has the sum of its durations. At a node $q_i$ is a positive smooth unit times the product of the actual edge parameters along its spine path. Consequently the long geometry is covered edge by edge by step 3.1, even when a total mark gap is not an independent coordinate. Active finite faces have $q_i\in (0,1)$ on the same normalized component, so their logarithmic differences and inverses have bounded derivatives in its compact flag neighborhood. At intersections of finite faces, summing the same ordered offsets is literally associative. No factor $1\mathrm{/}{\mathrm{\ell }}_i$ is introduced at ${\mathrm{\ell }}_i=0$. These restrictions inherit all the normalized windows.

We specify the relative problem on the actual finite face diagram. Include every stable tree of the given labelled word, every finite consecutive-block substitution, and every external parameter face; this is a finite diagram for the fixed context, by step 1.1 and the finite number of atoms and external cells. Near a corner its base coordinates are $(x_1,\dots ,x_c,y)$, with $x_j\ge 0$: ordinary node coordinates are the independent ${\rho }_e$, active protected finite coordinates can be the ${\mathrm{\ell }}_i$, and external coordinates are the distances to the selected cube or cell faces. An active finite constraint cannot cross a zero node, since its ratio is positive; it concerns marks on a single normalized component. Thus its independent logarithmic difference is a valid component coordinate as in step 4.1. Setting a subset of these coordinates to zero defines commuting local retractions ${\pi }_I$, with ${\pi }_I{\pi }_J={\pi }_{I\cup J}$. Disjoint node fixtures glue in either order. Nested nodes are different fixtures on the retained normalized components, so their quotient identifications also commute. Overlapping finite substitutions reconstruct the same original atomic offsets in either order. Finite substitutions and node restrictions commute because their atoms are either on the same surviving component or are partitioned among the node components, never copied. If $T_i$ varies with an external parameter, use the coordinates ${\mathrm{\ell }}_i={\tau }_{i+1}-{\tau }_i-T_i(p)$; retraction holds these ${\mathrm{\ell }}_i$ fixed, so it still commutes with setting an external face coordinate to zero. A genuinely shared parameter remains one coordinate throughout. Away from removed ordinary punctures and nodes, the actual universal family is smooth in these coordinates; all transition-mark neighborhoods are retained. Its normalized cores are product charts over the corner; on ends use the logarithmic strip products of step 3.1. No transition mark is converted into a removed strip puncture.

Here is the needed elementary relative extension calculation, rather than an appeal to contractibility. In one such product chart let $f_i$ be a smooth section on $x_i=0$, with values in a fixed linear coordinate space, and assume its restrictions agree on every face intersection under the actual identifications of step 5.1. Write $f_I$ for that common restriction to $x_i=0$ for $i\in I$. Define $\mathcal{E}(f)(x,y)={\sum }_{\mathrm{\varnothing }\mathrm{\ne }I\subseteq \{1,\dots ,c\}}(-1{)}^{\mathrm{\mid }I\mathrm{\mid }+1}{f}_I({\pi }_I(x,y))\mathrm{.}$ On $x_j=0$, the terms for $I$ not containing $j$ cancel with those for $I\cup \{j\}$; the one remaining term is $f_j$. This proves every face restriction simultaneously, including all mixed tangential derivatives. The formula is a finite sum of smooth functions and has no quotient by $x_i$, ${\rho }_e$, a gap, or their sum. For sections use a local frame on the normalized source product before making the same calculation. If the required values form an open convex subset of that coordinate space, the expression has the required value at every point of the corner, and continuity keeps it in that subset after shrinking about that point. Over a compact source/parameter subchart the shrink and every fixed derivative bound are uniform. This proves a local extension into an open convex set; it makes no false extension assertion for an arbitrary closed positivity cone.

Local extensions into an affine or open convex fiber can be joined without losing the restrictions. Choose finitely many smaller base charts covering the compact base, smooth nonnegative Euclidean bump functions ${\eta }_{\alpha }$ supported in the larger charts and positive on the smaller ones, and set ${\chi }_{\alpha }={\eta }_{\alpha }\mathrm{/}{\sum }_{\beta }{\eta }_{\beta }$. The denominator has a positive minimum on the compact set. These functions depend on the base, not on a clock's source coordinate. The weighted sum of the local sections has the prescribed restriction on every face because all summands have that restriction; convexity preserves the allowed fiber. Smooth coordinate changes are used to regard the summands as sections of the same actual source bundle before adding them. A prescribed genuine collar germ is preserved more directly: represent it on neighborhoods $U_i$ of the relevant faces, shrink them so the representatives agree on all actual overlaps, and hence obtain one section $f_U$ on $U={\bigcup }_i{U}_i$. For clocks first remove the source-end tails where both prescriptions are already the same constant. The remaining normalized cores over a compact face neighborhood are compact. Consequently $U$ contains every such core over some smaller base neighborhood: otherwise points in the closed complement with bases tending to the prescribed closed face set would have a convergent core subsequence ending in $U$. A base-only cutoff is therefore available. Choose it equal to one on a smaller base neighborhood and supported in this one; replace any extension by $\chi f_U+(1-\chi )f_0$. The term $\chi f_U$ extends by zero outside its support, so the formula is smooth and equals the whole prescribed germ on a smaller collar. The same construction is local in the source/target variables when necessary; on any preselected compact target window finitely many patches suffice. Thus values on lower faces are extended by step 6.1, whereas an already supplied compatible germ is preserved exactly rather than replaced by an independently chosen constant-normal germ.

We first make coherent choices in an open cone, before constructing monotone clocks. On the real boundary, oriented in the logical transition direction, choose a positive vertical vector field $V_i$ near each original transition mark. This is opposite the positive source-boundary traversal on its transition arc, as fixed by the actual word convention; thus the original monotonicity sign is retained. Such fields exist locally by the product charts in step 5.1, and positive multiples of the oriented coordinate vector form an open convex half-line. Their prescribed restrictions from lower faces are extended by steps 6.1–7.1 in that open half-line; genuine prescribed field germs are kept by the last formula of step 7.1. Choose these fields inductively over the finite face diagram, starting at its zero-dimensional strata and then increasing the dimension of the parameter cell. The face restrictions of a new cell are already chosen, and their multiple intersections agree by the preceding stages and the commuting actual identifications of step 5.1. The local formula extends all of them at once; it is not a succession of corrections that destroys a previous face. In a finite normalized cover, positivity and compactness give a common positive lower bound and upper derivative bounds for $V_i$ on a smaller neighborhood of its mark. Shrink these neighborhoods to be disjoint for distinct marks on each retained component and away from node and ordinary-puncture fixtures. This is possible in the normalized family: marked sections never hit a node on their own normalized component, even if their parent-coordinate images collide. For a fixed finite diagram take the minimum of the finitely many resulting positive widths. This constructs the coherent marking neighborhoods and vector fields, rather than assuming them. The construction also applies to an off-spine transition-only component, which retains its marks and its node ends.

Fix a smooth nonnegative bump $\beta$ supported in $(1\mathrm{/}4,3\mathrm{/}4)$ with integral one, and put $B(t)={\int }_{-\mathrm{\infty }}^t\beta (u)du$. The positive boundary field in step 8.1 gives an explicit local time coordinate: if $V_i=v_i(x){\mathrm{\partial }}_x$ and its mark has coordinate $a_i$, set $u_i(x)={\int }_{a_i}^x{v}_i(\xi {)}^{-1}d\xi$. Then $d{u}_i(V_i)=1$; the inverse function theorem [F3] gives its inverse near $u_i=0$. Choose one fixed $\delta >0$ small enough that $0\le u_i\le \delta$ remains in the chosen mark neighborhood for every atom and parameter in the finite diagram. In the protected model also choose it small enough that this neighborhood lies inside the assigned $\tau$-interval of duration $T_i$; the given positive lower duration bound and the finite upper bound for $d\tau (V_i)$ permit that choice. Define the actual atomic clock on this oriented source segment by $c_i=T_{i}B(u_i\mathrm{/}\delta )$, extending it by zero before and by $T_i$ after the segment. Its derivative in the logical boundary orientation is nonnegative, and it is constant on neighborhoods of both segment endpoints. Apply the given path ${\gamma }_i(p,c_i)$, with its actual transported primitive, to obtain the moving embedding. No target embeddings or primitives are averaged. The derivatives of $v_i^{-1}$ are bounded polynomials in the derivatives of $v_i$ and its positive lower bound; derivatives of the inverse coordinate follow by differentiating its identity. Hence all fixed source derivatives of $c_i$ are bounded by finite constants involving ${\delta }^{-r}$, bounded $T_i$ and the already obtained coordinate bounds. No duration tends to zero here; formal identities and fixed Morse/PSS fixtures are treated separately as in the Statement.

This choice has the required literal restrictions. On a finite merged block, use the same original atomic marks and on its $i$th segment use ${\sum }_{h<i}{T}_h+c_i$ as its block clock. Between the atomic segments it has the corresponding constant partial sum. These formulas are smooth at every seam since each clock, and each actual path, is sitting there. They describe the original Moore-concatenated path, with its atomwise clock, rather than replacing it by a new single bump over the total duration. Associativity is literal addition of the same offsets. On a node face, the normalized components inherit the restrictions of the very same marking vector fields, so the time integral in step 9.1 restricts to the lower component's time coordinate; uniqueness of that integral with value zero at its mark gives exact equality, not merely homotopy. Each component owns precisely its original atoms, and the clocks are constant at its before/after node ends. Zero-node branches are separate source windows. Smoothing identifies only those stationary end germs and does not delete a marked component. Thus unrestricted transition-only vertices, chains of simultaneous finite merges, intersections of several node faces, and mixed node/finite/external faces all obey the same formula. No duplicated external or diagonal variable is introduced.

The relative assertion also preserves every genuine already prescribed coherent clock germ. Here a germ is an actual smooth clock on a neighborhood of its face in the normalized source family, not only endpoint values or a boundary restriction. Compatibility means equality on the actual collar overlaps after the iterated identifications of step 5.1. Finitely many such germs have representatives on a common smaller neighborhood $U$ of their prescribed face set: shrink pairwise overlaps where their representatives agree and use the finite chart cover. Join that clock $c_U$ to the freely constructed clock $c_0$ of steps 9.1–10.1 by $c=\chi (p)c_U+(1-\chi (p))c_0$, with the base-only cutoff of step 7.1. The endpoint and atomic offset constraints are linear, and the oriented source derivative is $\chi {\mathrm{\partial }}_s{c}_U+(1-\chi ){\mathrm{\partial }}_s{c}_0\ge 0$ because ${\mathrm{\partial }}_s\chi =0$. Hence the result is a valid monotone clock, equals the prescribed whole germ on a smaller collar, and is sitting on the common smaller sitting neighborhoods. Apply the same given atomic paths to it. For clocks not prescribed as such germs the positive-field induction of steps 8.1–10.1 supplies their coherent choice; arbitrary facewise nonnegative densities are not extended by the open-cone formula. This distinction is necessary: equality of endpoint values alone does not make arbitrary transverse collar prescriptions compatible. It adds no uniform estimate as an assumption and excludes none of the genuine compatible germs in the relative problem.

This resolves the simultaneous induction on actual faces. At a new parameter cell use the already fixed lower component/word data on its ordinary node, finite substitution and external faces. When only lower face values have been fixed, steps 6.1–8.1 construct the new marking fields and therefore the coherent clocks of step 9.1. When an actual coherent collar germ has been fixed, step 11.1 preserves it. At a corner a shared atom or parameter appears on its unique owned component and is the same coordinate in every restriction; the common restriction is therefore independent of the order in which the faces are taken. Charts with different anchor choices need not have commuting global retractions: the calculation uses commuting retractions within each actual product chart, and step 7.1 joins their resulting sections intrinsically. There is no assumption of a global product-cube gap map. Nor is a product of arbitrarily chosen independent constant-normal face extensions used as if it already agreed. This constructs data on every retained face and every interior cell in the finite context while preserving its prescribed coherent subdiagram.

Extend the actual Floer coefficients by this same explicit relative construction. Hamiltonian one-forms on each normalized core form an affine space with the already prescribed linear trace, support and end conditions. In the retained history these include vanishing on ordinary boundary tangents and beyond the fixed target radius, and the stationary strip trace $H_{t}dt$. A moving boundary remains the actual ${\gamma }_i(c_i)$ condition; its Hamiltonian lift is not added as a new forcing on the target end. Take a reference one-form with these fixed traces using fixed source cutoffs, and extend its zero-trace difference from the lower-face data by step 6.1; step 7.1 preserves genuine already prescribed collar germs. All these operations preserve the linear vanishing/support constraints, and do not alter any primitive normalization. At a node the two end germs have the same recorded stationary pair $J_t,H_{t}dt$ for that node/chord; keep this pair unchanged past one fixed end depth, chosen as the maximum of the finitely many prescribed depths. For the tame almost-complex structures, use the compatible reference $J_0$ from [F1] and $g_0(v,w)=\omega (v,J_{0}w)$; write $A^{\ast }$ for the $g_0$-adjoint and $\mathrm{\parallel }A{\mathrm{\parallel }}_{g_0}$ for its induced operator norm. Write $J=J_{0}Z$, so $Z=J_0^{-1}J$; the equation $J^2=-1$ is equivalent to $Z^{-1}=J_0^{-1}Z{J}_0$, and tameness is $g_0(v,Zv)>0$ for every nonzero $v$. Set $W=(I-Z)(I+Z{)}^{-1}$. The positive symmetric part of $Z$ makes $I+Z$ invertible. Since $W(I+Z)v=(I-Z)v$, we have $\mathrm{\parallel }(I+Z)v{\mathrm{\parallel }}_{g_0}^2-\mathrm{\parallel }W(I+Z)v{\mathrm{\parallel }}_{g_0}^2=4g_0(v,Zv)>0$; compactness of the unit sphere gives $\mathrm{\parallel }W{\mathrm{\parallel }}_{g_0}<1$. The complex-structure relation gives $J_0^{-1}W{J}_0=-W$. Conversely, if $J_0^{-1}W{J}_0=-W$ and $\mathrm{\parallel }W{\mathrm{\parallel }}_{g_0}<1$, then $Z=(I-W)(I+W{)}^{-1}$ satisfies $Z^{-1}=J_0^{-1}Z{J}_0$ and has positive symmetric part, since $Z+Z^{\ast }=2(I+W^{\ast }{)}^{-1}(I-W^{\ast }W)(I+W{)}^{-1}>0$, so $J=J_{0}Z$ is $\omega$-tame. Thus this is an open unit ball in the real vector space of $J_0$-antilinear endomorphisms, and supplies the convex coordinate for the tame class. Extend the prescribed face germs in this coordinate using steps 6.1--7.1. Near each compact face/source/target chart, continuity of the local extension preserves a fixed strict margin from one; convex patching preserves that margin, and the prescribed germ is unchanged on its smaller collar. Outside the relative neighborhoods use the reference data. Choose the source cutoffs equal to the stationary end pair before a fixed depth and extend only the remaining compact cores. The constructions are local in target charts; a locally finite target partition in the common tame $W$ coordinate preserves convexity and prescribed data, and every fixed compact target window uses only finitely many of these terms. Source/target end data already prescribed are held fixed. This gives actual simultaneous relative coefficients on the finite face diagram, not an invocation of a nonexistent nonlinear corner theorem. Every fixed spatial derivative is bounded on the finite normalized core/compact-target cover; derivatives of $(I+W{)}^{-1}$ are bounded by its strict margin and repeated differentiation. The clocks and moving embeddings have the bounds of steps 9.1–12.1. On a growing neck nothing is interpolated across its length: both ends are already exactly stationary and only that stationary pair is sewn. Hence all the claimed pulled-back coefficient bounds are independent of neck length and translated-window position.

The absence of a hidden singular scale can be read in the concrete charts. On each node put $z=e^{-\pi s+i\pi t}$ as in step 3.1; then $\mathrm{\mid }z\mathrm{\mid }=e^{-\pi s}$ and $\mathrm{\mid }{\mathrm{\partial }}_s^a{\mathrm{\partial }}_t^{b}z\mathrm{\mid }\le {\pi }^{a+b}{e}^{-\pi s}$. A compact-core one-form $a(z)dz+\bar{a}(z)d\bar{z}$ therefore pulls back with coefficients and fixed source derivatives bounded by constants times $e^{-\pi s}$ wherever that expression is used. The stationary part is already $H_{t}dt$ and is not obtained by dividing such a one-form by $z$. Transitions between the two strip coordinates remain $(s,t)\mapsto (L-s,1-t)$, so their spatial derivatives have no $L$ factor. Cutoffs occur at fixed end depths; their derivatives have fixed bounds. On a finite merge, differentiation uses the actual additive atomic offsets and the fixed positive durations, not $1\mathrm{/}{\mathrm{\ell }}_i$. At simultaneous nodes use one such calculation per independent fixture, rather than differentiating a total gap monomial as if it were one normal coordinate. Thus the estimates just obtained concern precisely the normalized spatial and compact nonsingular coefficient parameters claimed in the Statement; no derivative bound in a singular raw plumbing coordinate is asserted. These facts complete the relative marked-face bridge needed by the later same-trace chart and blow-up steps.

This extension is compatible with the actual permitted relative perturbations. Take a countable smooth bump list on the finite normalized compact cores, with the prescribed zero support near ends and fixed faces. Rescale its $j$th member so all coefficient derivatives through order $j$ have norm at most $2^{-j}$ in the finite source/target/compact-parameter charts. For coefficients $a$ in a bounded ${\mathrm{\ell }}^2$ ball and each fixed order $r$, the tail ${\sum }_{j\ge r}{a}_j{e}_j$ has $C^r$ norm at most $\mathrm{\parallel }a{\mathrm{\parallel }}_2({\sum }_{j\ge r}{4}^{-j}{)}^{1\mathrm{/}2}$, plus the finitely many preceding terms. This proves the uniform bounds, and tails tend to zero uniformly while each finite head ranges in a compact finite-dimensional set. Diagonal extraction gives precompactness in every fixed $C^r$ norm. Small balls preserve tameness. Ordinary translation-invariant strip perturbations depend only on $t$ and the target and obey the same calculation. They do not acquire an $s$ derivative under translation. This construction precedes the regular-value choice, so the estimates do not assume regularity or compactness of the current top-dimensional moduli.

Uniform target boundary charts follow from the actual geometric data, not from individual embeddedness alone. For an actual compact family of Hamiltonian lifts ${\mathrm{\Phi }}_p$, its inverse is a smooth family of diffeomorphisms. Thus ${\bigcup }_p{\mathrm{\Phi }}_p^{-1}(C)$ is the compact image of the compact parameter set times $C$. Properness of each fixed initial brane makes its inverse image of this compact set compact. All moving-brane points that map into $C$ therefore come from one compact subset of its fixed source brane. The homogeneous end formulas give the same conclusion for the actual fixed end germs: on a compact parameter set the radial shift and all contact derivatives are bounded. Finite concatenation, inverse and refinement letters use finitely many compositions/inverses of these same smooth maps, hence still have compact inverse-image windows. Their actual transported primitives are smooth on this compact set; no end constant or seam term is reset.

In finitely many coordinate patches of that compact brane source, write the actual moving embedding as $i_p(x)$. With a fixed smooth target metric, define ${\mathrm{\Psi }}_p(x,y)={\exp }_{i_p(x)}(J_{p}d{i}_p(y))$. At $y=0$ its derivative sends $(v,w)$ to $d{i}_p(v)+J_{p}d{i}_p(w)$. This map is invertible: a vector in both the Lagrangian tangent and its $J_p$ image would contradict $\omega$-tameness. Its least singular value is a positive continuous function on the compact unit-frame parameter set, so it has a positive minimum. All derivatives of $i_p,J_p$ and the exponential on the chosen compact target neighborhoods are bounded by steps 13.1–14.3. The inverse theorem gives a uniform smaller radius. To check injectivity is not lost from distant brane points, a contrary sequence would have two preimages separated from the local diagonal in the same compact source, converging to distinct points of an embedded limiting brane with equal image. This is impossible. Therefore a finite cover has a fixed positive chart radius and inverse differential bound. Repeatedly differentiating ${\mathrm{\Psi }}^{-1}\circ \mathrm{\Psi }=\mathrm{id}$ expresses every higher inverse derivative as a finite polynomial in derivatives of $\mathrm{\Psi }$ and its bounded inverse first derivative. This proves all inverse-chart bounds.

Compose these charts with the actual clocks of steps 9.1–11.1. All source derivatives of the pulled-back embeddings and inverse charts are bounded by the chain rule and the finite clock bounds, including through marked seams and finite merges. At a smooth boundary point the pulled-back $J$ is standard on its real slice by the formula for $D\mathrm{\Psi }$ in step 15.1. Differentiation of this source-dependent chart adds the genuine term $-D{\mathrm{\Psi }}^{-1}{\mathrm{\partial }}_s\mathrm{\Psi }$ to the equation forcing; it is bounded with all required derivatives, rather than discarded. The transformed Hamiltonian forcing has the same bounds by step 13.1. On a long strip each finite-radius half-window sees one of its two edge embeddings; it never has to flatten both intersecting labels in one boundary chart. At a retained transition mark the trace is the smooth moving embedding constructed in step 9.1, and at a zero node one first selects a normalized branch as in step 3.1. These are the required complete same-trace windows. No corner Fredholm or nonlinear gluing theorem has entered this construction.

The remaining retained fixtures do not force a division by a vanishing positive duration. The unstable two-punctured ordinary strip has its translation gauge and the same translated unit windows. A formal identity has its separate constant-profile/forgetful fixture, so its clock derivatives are zero. A fixed-scale Morse/PSS cap has a smooth bordered core and one of the same ordinary strip ends; its attached gradient line is a separate one-dimensional equation and an evaluation constraint, not a jump in the two-dimensional boundary trace. Finite products of these fixtures retain their own parameter blocks once. Fixed transverse-copy sizes are held fixed in each finite context, as in the retained construction; no uniformity as an unretained transverse gap or copy scale tends to zero is inferred. The bounds in steps 3.1 and 13.1–16.1 thus apply to all these actual ordinary, merged, marked, node and fixed-unit/spike windows.

Here is the resulting exact applicability to local elliptic estimates. In a same-trace chart a linearized operator is $v_s+A(s,t)v_t+B(s,t)v$, with the quantitative total-reality and chart/inverse bounds of steps 15.1–16.1. A uniformly bounded complex frame puts it in the form $w_s+i{w}_t+Cw$ with real flat-edge trace. Integration by parts for compactly supported real-trace $w$ gives $\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_2^2=\mathrm{\parallel }{w}_s+i{w}_t{\mathrm{\parallel }}_2^2$; the flat-edge cross term vanishes because the tangential trace is real. Cutoffs give the local $H^1$ estimate. Tangential difference quotients preserve the trace, and the equation recovers normal derivatives, proving the higher estimate with constants determined by the just-derived coefficient bounds. There is no high-order reflection condition on the forcing. The two-dimensional product/composition estimates in these fixed charts give the $H^k$ nonlinear mapping chart and its derivative for $k\ge 4$. Finite evaluation and external blocks then remain their actual finite-dimensional blocks. This proves the applicability needed for the order-lifting at29.1 without assuming the nonlinear compactification.

For gradient blow-up select points in this uniform atlas, not in a collapsing root coordinate. Choose a fixed larger disk or half-disk window around each proposed unbounded-gradient point, and regard only its curved or lateral cutoff edges as artificial boundary. The original source boundary is retained, not counted as an artificial edge. On the closed window maximize the continuous function $d(z)\mathrm{\mid }d{u}_j(z)\mathrm{\mid }$, where $d$ is distance to that artificial boundary. Its value tends to infinity because the initial point has a fixed positive distance from it. At its maximizing point $z_j$ put $M_j=\mathrm{\mid }d{u}_j(z_j)\mathrm{\mid }$ and $d_j=d(z_j)$. On the radius-$d_j\mathrm{/}2$ disk or half-disk, $d\ge d_j\mathrm{/}2$ and hence $\mathrm{\mid }d{u}_j\mathrm{\mid }\le 2M_j$; meanwhile $d_j{M}_j\to \mathrm{\infty }$. This gives expanding rescaled windows and keeps the selected derivative equal to $M_j$. In normalized local coordinates put $v_j(\zeta )=u_j(z_j+\zeta \mathrm{/}{M}_j)$. An explicit source derivative of $J$ of order $a$ has factor $M_j^{-a}$, and a derivative of the transformed forcing has factor $M_j^{-1-a}$. The same factors apply to the source-dependent embedding and inverse-chart terms in step 16.1. Target derivatives of $J$ do not disappear. By steps 13.1–17.1, all these unscaled norms were uniformly bounded on the actual retained family. At bounded rescaled distance from the edge one obtains a half-plane with one frozen exact brane; otherwise one obtains a plane. The opposite strip edge is at distance comparable to $M_j$ and cannot produce a second edge in the limit. A transition mark remains a smooth point of its moving trace, while a domain cluster/node was already assigned its normalized marked component before this selection.

For completeness the normalized gradient bound is sufficient for the nonlinear bootstrap, without presupposing derivative compactness. Flatten the one moving boundary by $\mathrm{\Psi }$ and write $w_s+A(z,w)w_t=h(z,w)$. Its tangential derivative $v=w_s$ satisfies $v_s+A{v}_t+(D_{w}A[v])w_t={\mathrm{\partial }}_{s}h+D_{w}h[v]-({\mathrm{\partial }}_{s}A)w_t$. The bounded gradient and the coefficient bounds above make the coefficients and right side uniformly controlled. The local $H^1$ estimate of step 18.1, first in bounded Lipschitz frames and then by smooth approximation, gives $H^2$ for $w$; the original equation recovers normal derivatives. At each higher tangential order isolate the top term $(D_{w}A[{\mathrm{\partial }}_s^{j}w])w_t$ as zeroth order. Every other term contains derivatives of $w$ of order at most $j$; Hölder and the two-dimensional differentiated product estimate bound it by the previous Sobolev norm and the gradient bound. Induction on smaller nested patches gives all fixed derivative bounds, with constants from the actual normalized data. The same argument holds on interior windows. Consequently a diagonal subsequence converges smoothly on each finite rescaled window, and the selected derivative remains nonzero.

If the original solutions have a common image in $C$ and a Floer energy bound $E_0$, then on each fixed rescaled compact window the normalized forcing tends to zero. Equivalence of the tame metric energy and $\mathrm{\mid }du-X_K{\mathrm{\mid }}^2$ on $C$ bounds the limiting holomorphic energy there by a fixed multiple of $E_0$; the extra forcing-square term is $O(M_j^{-2})$ on a fixed rescaled window. Exhausting the limiting plane/half-plane proves finite total energy. Compactness of the inverse brane windows in step 14.3 also preserves the exact limiting brane and its primitive, with uniform continuity on that window. These are precisely the hypotheses of the exhaustion-Stokes contradiction at79.1. This lemma provides the normalization and quantitative applicability, including78.1–80.1; it neither assumes nor proves a separate corner/gluing/virtual theory, and it changes no history labels, full Morse complex or relative homotopy sector.