Actual weighted nodal CR gluing and continuous linear kernels

Let $C_0$ be a stable marked nodal curve with the disk-component and boundary-label conventions of def-gts-stable-j-holomorphic-disks-with-boundary-markings. Use either its smooth-boundary branch with a continuous smooth trace bundle, or the explicit weighted corner extension specified below. The latter is a corrected moduli problem, not an assertion that the original everywhere-smooth ordinary-Sobolev corner interface already defined it. It is supplied with an actual smooth local universal plumbing family, one complex coordinate $q_e$ at each interior node ($zw=q_e$), one real nonnegative smoothing parameter at each boundary node, smooth remaining parameters, and fixed cylindrical ends. Write $\mathcal{B}$ for this finite-dimensional local domain-parameter space and $d_{\mathcal{B}}={\dim }_{\mathbb{R}}\mathcal{B}$. Let $X$ be a compact smooth tame almost-complex target of real dimension $2m$, and let $u_0:C_0\to X$ be the nodal solution. On each disk boundary arc with label $L_j$, sections have boundary trace in $u_0^{\ast }T{L}_j$.

For some fixed $k\ge 6$, the smooth-boundary branch uses finite-chart $H^k$ spaces on each compact normalization component. In the corner branch supply the fixed-asymptote or prescribed clean moving-asymptote data, positive admissible spectral cuts, capping matrices and weighted spaces of def-gts-capped-weighted-label-changing-corner-data at every genuine label-changing mark. Arbitrary embedded label pairs are allowed at a fixed asymptote; clean intersection is required only for the claimed unrestricted smooth moving-asymptote chart. An explicitly prescribed smooth submanifold of the intersection is allowed as a declared restriction. All corner marks remain separate from nodes, persist without collision or relabeling, and have uniform spectral/coefficient bounds on the local family. Matching at nodes still means equality of limiting values. The corner topology is the weighted topology on persistent corner ends, together with the original graph topology away from those ends and the given finite domain/asymptote/forcing coordinates. It does not assert automatic positive exponential decay of every finite-energy or graph-near solution. Let $E$ be a finite-dimensional real obstruction space and $\lambda (e)$ a smooth linear equivariant forcing supported away from nodes. Impose $\mathrm{\ell }$ fixed core incidence constraints $u(p_i)\in D_i$ at distinct smooth-core marked points, where each $D_i\subset X$ is a smooth real codimension-two submanifold met transversely by $u_0$. Temporary transverse intersections used only to stabilize components are separate data and are smoothly recovered. Dependence on these recovered intersections is allowed. At $(u_0,e_0)$ assume its actual augmented vertical linearization, on matched sections satisfying the boundary and fixed incidence conditions together with $E^{\mathrm{\prime }}\subset E$, is onto. The fixed incidence conditions have codimension $2\mathrm{\ell }$.

Let $P_v$ denote the real Cauchy--Riemann operator on each normalization component $C_v$ with its stated boundary condition, and let $D_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$ be their direct sum restricted to sections whose values match at all nodes. If $N_{\mathrm{i}\mathrm{n}\mathrm{t}}$ and $N_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}$ are the numbers of interior and boundary nodes, respectively, its index is ${\mathrm{ind}}_{\mathbb{R}}{D}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}={\sum }_v{\mathrm{ind}}_{\mathbb{R}}{P}_v-2m{N}_{\mathrm{i}\mathrm{n}\mathrm{t}}-m{N}_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}\mathrm{.}$ For a closed component, ${\mathrm{ind}}_{\mathbb{R}}{P}_v=2m(1-g_v)+2⟨c_1(u_0^{\ast }TX),[C_v]⟩$; for a smooth-boundary disk component it is $m\chi (C_v)+\mu (u_0^{\ast }TX,u_0^{\ast }TL)$, with the smooth-pair conjugate-double degree. For a component with corners it is $m\chi (C_v)+{\mu }_{v,\delta }+c_v$, where ${\mu }_{v,\delta }$ is the degree of the explicitly capped smooth pair and $c_v$ is the real dimension of its prescribed moving-asymptote parameters. Fixed asymptotes contribute zero to $c_v$. Here $P_v$ includes these asymptote tangent variables. No conjugate double of discontinuous old trace planes is used.

Then the thickened regular solution locus is open in the stated branch topology and is locally a product of the given domain base with a finite-dimensional kernel ball, with local coordinates consisting of the domain parameters and the kernel of that actual augmented derivative. Its dimension is $d_{\mathcal{B}}+{\mathrm{ind}}_{\mathbb{R}}{D}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}+\dim E-2\mathrm{\ell }\mathrm{.}$ Projection to the domain base and $E\mathrm{/}{E}^{\mathrm{\prime }}$ is a topological submersion. The actual weighted residual, uniformly bounded right inverses, quadratic estimate and inverse-gluing statements are proved below; none is an additional hypothesis. The construction supplies determinant-line exact-sequence isomorphisms. It does not orient bordered determinant lines; coherent orientations require additional boundary orientation data and are treated separately. No differentiability in $q_e$ at zero is asserted.

The linear version holds for a complex bundle on the given plumbing family with smooth core transition functions and fixed holomorphic matching frames on the node fixtures: if its central matched Cauchy--Riemann operator is onto, the construction gives uniformly bounded right inverses and a continuous basis of holomorphic kernels through all partial nodal strata. In the smooth branch these are ordinary smooth sections; in the corner branch they are smooth away from the marks and belong to the declared weighted spaces, with the capping comparison and asymptote parameters just specified. This branch does not require a compact total-space target. The theorem assumes the specified family, not a compactification or atlas existence theorem.

Historical source evidence records reading the saved Pardon Appendix B; this continuation independently checked the local arguments and exact item suppliers, and does not claim a fresh reading of that Appendix. The explicit Fourier end proof replaces its external Lockhart–McOwen input. The corrected result supplies actual local analysis over a given family in its explicitly stated branch topology and does not by itself supply determinant orientations, global domain compactness or a virtual class. The determinant exact-sequence isomorphisms constructed here do not choose orientations of bordered determinant lines; a coherent orientation requires additional boundary orientation data and its compatibility under the node maps.

The following retained authoring components supply the detailed smooth-boundary subarguments used above. Their statements about not covering label jumps refer to these subarguments alone; the spectral-cut lemma supplies the separate declared corner branch. Native scientific review is still required for the whole proposed replacement.

Retained smooth-boundary nodal gluing components

Authoring only. These arguments have not been independently accepted. They complete proposed analytic components for a smooth totally real boundary bundle, including the actual single-L Reader13 consumers. They do not prove the label-changing-marking branch of the current supplier. corner-framework-and-exact-obstruction.md explains that separate public-interface issue. Neither this document nor another author's proposal is an installed logical supplier.

Use the original supplier's local universal family, supported forcing, regular augmentation, incidence slices and finite tuple data. On every normalization component assume the boundary condition is a smooth totally real subbundle. Same-label real nodes satisfy this hypothesis. Write n for complex bundle rank and $k\ge 6$. Normalize real ends to [0,infinity)×[0,pi], $z=\exp (-s-it)$; closed ends have angular period2pi. Fixed finite coordinate contexts have bounded smooth coefficient and inverse-chart norms. All constants below depend on that context; no uniformity over unrelated targets is claimed.

A. Actual strip operator; no high-order reflection restriction on forcing

For $A=i$ d/dt on the REAL Hilbert space $L2([0,pi],C^n)$, with real trace at both edges, the eigenvectors are exp(-ijt)v, j integer, v real, eigenvalue j. Real orthogonality follows by integrating cos((j-l)t). Completeness follows by conjugate-reflecting an L2 function to a period2pi function; its Fourier coefficients are real. The operator domain is H1 with these real traces. Thus for $0<\delta <1$, Fourier transform in s gives the inverse multiplier $(itau+j-\delta {)}^{-1}$ for d/ds+A-delta. Both its denominator and its first-derivative quotient are bounded away from zero/infinity by constants depending only on delta. It gives H1 solvability for every L2 forcing. Reality of the coefficient functions is preserved by the conjugation symmetry in tau.

For a target f in ordinary $H^{k-1}$, apply s difference quotients to this H1 estimate, through order k-1. They commute with the operator and preserve the two real traces. The equation $w_t=-i(f-w_s+\delta w)$ recovers all remaining mixed/normal derivatives by induction on their number of t factors. Consequently the inverse is bounded $H^{k-1}\to Hk$ for the unrestricted ordinary forcing space. No false assertion that conjugate reflection preserves arbitrary high forcing jets is used. A finite-end extension uses the ordinary real-coefficient Vandermonde extension in s, where there is no boundary condition. It is bounded on the required finite Sobolev order. Cutoff errors have fixed compact support.

Choose a smooth real boundary frame, complexify it by J, and extend over each branch half-disk. Its value at the node is fixed on both branches. Cylindrical normalization makes the principal operator exactly d/ds+i d/dt; the remaining coefficient and all fixed cylindrical derivatives are O(exp(-s)). Real scalar cutoffs preserve both edge traces. A target chart flattening the one Lagrangian L makes the map cutoffs L-valued; arbitrary exponential interpolation need not do so. These are one-sided smooth constructions; no smoothness of a reflected nonlinear J is asserted.

B. Global realization and index

Apply the actual smooth-F bordered Fredholm supplier on boundary components, and the closed supplier on closed ones. In the direct sum of ordinary Hk component spaces, node difference evaluation Δ is onto $V=direct$ sum of $C^n$ at interior nodes and $R^n$ at real nodes. Disjoint cutoff sections give a bounded right lift R of Δ. In the splitting kerΔ+imR, P-PRΔ is P restricted to kerΔ plus a zero finite-dimensional block. Thus

ind $D_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$ $=$ sum ind $P_v$ - dim V $=$ sum ind $P_v$ -2n $N_{\mathrm{i}\mathrm{n}\mathrm{t}}$ -n $N_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}$.

A boundary smoothing joins two boundary half-disks by a strip and reduces total Euler characteristic by one; an interior smoothing reduces it by two. The boundary double clutching is unchanged away from these trivial matching fixtures. Its Chern degree is additive; equivalently glue the fixed complex frames on the two trivial fixtures, whose transition is identity. The disk formula nχ+μ therefore changes by exactly -n at a boundary node, and the closed/relative formula by -2n at an interior node. This agrees with the constrained central index. The same E augmentation and 2ell real incidence losses occur on every stratum. This calculation does not apply a closed-node 2n loss at a real node.

The weighted spaces retain a real constant at each real node and complex constant at each interior node, with exponentially weighted decaying remainder. The end inverse in A and the usual closed Fourier inverse give parametrices. Core boundary patches use the real-trace H1 identity followed by tangential difference quotients and recovery of normal derivatives, exactly as the read bordered supplier. Fixed-support errors gain one derivative, and the end O(exp(-s)) coefficients tend to zero in operator norm after truncation; Fourier truncation on finitely many charts proves their compactness. Weighted homogeneous solutions have continuous common limits, extend weakly over branch punctures without a boundary atom, and acquire one-sided smoothness by the bordered estimates. Conversely a smooth matched section has O(exp(-s)) remainder. Smooth ordinary target coefficients have the same cylindrical factor. Cutting off weighted targets far down the ends gives dense images in the finite-dimensional quotient, so the compact and weighted kernel/cokernel agree, with the stated finite constraints. The assumed augmented regularity therefore gives a genuine bounded central weighted right inverse Q0.

C. Complement and separated-cutoff inverse

Let K be the kernel of the full E-augmented central operator. Its E-image B satisfies $B+E^{\mathrm{\prime }}=E$ by assumed surjectivity with E'. Choose $E=B+E^{\mathrm{\prime }\mathrm{\prime }}$ as a direct sum with E'' contained in E'. Split K into the zero-E kernel and a lift of B. Project onto B, subtract its kernel lift, then choose finitely many core evaluations dual to the remaining kernel. Unique continuation makes those evaluations detecting; finite-dimensionality supplies a finite list. This gives $L0\mathrm{\mid }K:K\to W$ an isomorphism and a complement kerL0 whose E component lies in E''. Its inverse Q0 has E component in E'. This specifies the complement rather than assuming it from a dimension count.

For each node, write its neck length6S. Preglue smooth kernel elements by cutting their O(exp(-s)) remainders to the common constant near S and5S. Their errors are O(exp(-(1-delta)S)). Keep Lq equal to the same fixed core evaluations and E projection, so Lq $kapp{a}_q=L0$ kappa exactly.

Break a target forcing with complementary cutoffs near3S, send each half to its normalized branch, apply Q0, and transport back. Retain the left decaying section remainder until4S, the right from2S; in their overlap add both and count their common constant once. The extension beyond the target cutoff is a bounded ordinary collar extension, not an asserted smooth zero extension of unmatched jets. The targets are first multiplied by cutoffs flat at their support boundary. On the two places where a section remainder is cut, the ratio of the glued weight to the old end weight is O(exp(-2delta S)). There are no differentiated target-breaking terms in DQ0: the already-solved target equations add to the original forcing in the overlap. The O(exp(-s)) coefficient multiplying a retained constant costs O(exp(-(1-delta)S)), not O(exp(-S)) in the weighted target. The decaying remainder coefficient error is O(exp(-S)). Therefore the approximate inverse T satisfies

||Dq T-I|| $\le$ $C(\mathrm{\mid }y\mathrm{\mid }+su{m}_e[\exp (-2\delta S_e)+\exp (-(1-\delta )S_e)])$, $\mathrm{\mid }\mathrm{\mid }T\mathrm{\mid }\mathrm{\mid }\le C$.

It has image in kerLq and E component in E'. Neumann inversion gives $Qq=T(DqT{)}^{-1}$, with the same complement and uniform bound. Equality of the Fredholm indices in B makes $dimkerDq=dimK$. Since Lq is onto on the entire domain and kills imQq, the decomposition $x=QqDqx+(x-QqDqx)$ makes Lq|kerDq onto W, hence an isomorphism. Thus $imQq=kerLq$ exactly. No claimed periodic-cylinder boundary condition is hidden here.

D. Nonlinear pregluing and contraction

Cut map remainders to the node in an L-flattening chart on each real neck; use the usual complex chart on closed necks. The node-constant region solves the unforced equation. The forcing and incidence data remain on fixed cores. Product differentiation on the fixed cutoff regions gives residual C(|y|+sum exp(-(1-delta)S)). Incidence slices at different source marks are straightened in their disjoint source neighborhoods, even if their target images coincide. Temporary transverse intersections are recovered by the finite-dimensional inverse theorem with the actual transverse derivative; these are uniformly smooth core functions.

Cover strips by unit rectangles/half-rectangles of bounded overlap, independent of S. Ordinary finite reflection extension for estimating Sobolev norms, followed by Fourier Cauchy-Schwarz, gives uniform $H^2\to L\mathrm{\infty }$ and products on these fixed patches. This extension is not a reflected equation. The inverse square weight has uniformly bounded longitudinal integral. Separate the real zero mode $a=p{i}^{-1}integral$ Re xi at the middle section; the imaginary mean is not a free constant mode. The weighted derivatives plus a control the global unweighted supremum. On boundary patches use smooth boundary-preserving bundle charts. Second variations of the CR map have terms h·Dg, g·Dh, $hg\cdot D{u}_q$, and supported forcing terms. There is no pure constant-only unforced neck term. Thus

||F(xi)-F(eta)-Dq(xi-eta)|| $\le$ C(||xi||+||eta||)||xi-eta||

with C independent of all neck lengths. For sufficiently small fixed radius, xi $\to$ xi-Qq F(xi) is a contraction on the affine fiber $kapp{a}_q+imQq$. It produces a unique solution with Lq $xi=L0kappa$. Its E/E' coordinate is the prescribed kernel coordinate because Qq's E image is in E'. The $B+E^{\mathrm{\prime }}=E$ splitting makes projection to E/E' a submersion. All statements are uniform when any subset of nodes remains nodal.

E. Boundary energy and differentiated decay

On a small L-chart there is a primitive theta of the symplectic form with $\theta \mathrm{\mid }TL=0$: subtract the exact pullback of any local primitive to the contractible L-chart. For each endpoint a in L, subtract the differential of a function vanishing on L to make ${\theta }_a(a)=0$. Its norm is bounded by C distance to a uniformly. For a vertical arc gamma of length pi with both endpoints on L, this alteration has zero endpoint contribution, and

|integral gamma*theta| $\le$ C pi $\mathrm{\mid }\mathrm{\mid }{\gamma }_t\mathrm{\mid }{\mathrm{\mid }}_L{2}^2$.

Stokes on [-r,r]×[0,pi] has no horizontal boundary term. Tameness gives $A(r)=integral$ u*omega comparable to $integral\mathrm{\mid }{u}_t{\mathrm{\mid }}^2$, and $A(r)\le C$ A'(r). Integration, without dividing by A where it is zero, gives $A(r)\le C$ exp(-c(R-r))(b(-R)+b(R)). This is a strip action argument, not a loop Parseval assertion for an open arc. Let $\mu =c\mathrm{/}2$ and take $\delta <\min (1,\mu )$. The local boundary small-energy estimate actually read in lem-boundary-energy-regularity-for-totally-real-strips now gives |du| $\le$ C times the square root of the local energy on a fixed smaller half-patch. Away from the boundary use the interior estimate.

For higher derivatives use the separately read proposed stopped_floer/uniform-boundary-bootstrap.md: its quantitative chart/inverse norms and tangential Hk argument apply to the fixed compact smooth family here. On a neck J is source-independent and forcing is zero. Differentiating in s gives $v=u_s$ with equation $v_s+J(u)v_t+(dJ(u)[v])u_t=0$ and real tangent boundary trace, straightened in a uniformly bounded frame. This is homogeneous in first derivatives. After the gradient estimate all coefficient norms needed at a fixed induction order are bounded by the previous bootstrap. The tangential estimate, equation recovery of normal derivatives and tame products consequently give $\mathrm{\mid }\mathrm{\mid }{D}^j$ u|| on a smaller unit patch $\le C_j\mathrm{\mid }\mathrm{\mid }du\mathrm{\mid }{\mathrm{\mid }}_{L}2$ on the original patch; there is no additive constant forcing. Induction retains the SAME exponential energy square-root factor. Translated half-patch constants do not depend on neck length. This supplies the differentiated decay actually needed by inverse gluing. The shared bootstrap remains an authoring proposal pending review, not an accepted input.

F. Graph-near solutions; no-neck and inverse gluing

On a fixed core regard the inhomogeneous solution as its graph in a product with source. The graph almost-complex structure has block form $(a,b)\to (ja,Jb+Aa)$, $Aj+JA=0$, with A chosen from the forcing convention. For sufficiently large fixed N, N sigma+omega tames it. Its graph boundary lies on R×L, a Lagrangian for that product form. Choose the local primitive vanishing on this Lagrangian and centered at a boundary point, as in E. For a half-disk cutoff chi the boundary term in Stokes is zero. The estimate $\mathrm{\mid }\theta \mathrm{\mid }\le C$ osc(graph) and Young's inequality give

integral chi² |d graph|² $\le$ C osc(graph)² integral |dchi|².

No reflection of the nonlinear structure is invoked. The interior version uses an ordinary local primitive. Graph convergence and shrinking the fixed patch make these energies small. Apply the cited boundary small-energy supplier without changing its fixed-boundary-structure hypothesis. On a smaller coordinate rectangle embed the source coordinates into a flat complex torus $T^2$, with its real source edge on a coordinate circle Gamma. Extend the smooth anti-complex-linear block A by a cutoff from that rectangle to $T^2$ times X, and set it to zero elsewhere. The block formula defines an autonomous smooth almost-complex structure on this compact target, because $Aj+JA=0$. The fixed submanifold Gamma times L is totally real: if a vector and its block-J image are tangent to it, the source component lies in both T Gamma and j T Gamma, hence is zero, and the target component lies in TL intersect JTL, hence is zero. Thus the supplier applies to the graph as an autonomous map into $T^2$ times X, with $J_b$ equal to this one fixed target structure along Gamma times L; it has no explicit source dependence. A sufficiently large product form N sigma+omega tames this structure uniformly on the finite coefficient neighborhood. The Caccioppoli bound makes its graph energy small after shrinking the patch, so the supplier gives the actual boundary gradient bound. Apply the proved tangential bootstrap on the smaller patch to obtain core C-infinity convergence. Finite source/forcing parameters give uniform compact-target coefficient bounds. This uses neither nonlinear reflection nor an unproved extension of the boundary energy theorem.

Fix a large normalized end depth R0 and then let parameters tend to zero. Core convergence makes endpoint slice derivatives O(exp(-R0)) plus a quantity tending to zero. Apply E on every remaining small-image neck. Summing the differentiated decay against weights $\delta <\mu$ yields a tail bounded by C exp(-(1-delta)R0) plus a quantity tending to zero for this fixed R0. Integrating the longitudinal derivative, rather than asserting a zero mean, controls the section remainder; the real middle mean converges to the node value by graph convergence. On the fixed core the correction converges in every required norm. First take the parameter limit, then $R0\to infinity$. The full weighted correction norm tends to zero, with its finite E/mean coordinates. This proves no-neck energy and the inverse-gluing norm assertion for the smooth-boundary case. Let $kappa=(L0\mathrm{\mid }K{)}^{-1}Lq$ xi. Then $xi-kapp{a}_q$ lies in imQq, so contraction uniqueness recovers the original actual solution. The graph topology on the given proper finite local family is metrizable; the sequential neighborhood conclusion is valid. Comparing the contraction equations proves continuity across all partial strata. This proves the local product chart and openness of regularity in this branch, not a new domain-family existence theorem.

G. Determinant exact sequences and smoothing compatibility

Write P for the direct sum of normalization operators with smooth boundary conditions and Δ for its onto node-difference map to V. The actual exact sequence is

0 $\to$ ker(P|kerΔ) $\to$ kerP $\to$ V $\to$ coker(P|kerΔ) $\to$ cokerP $\to$ 0.

The map $kerP\to V$ is Δ. The connecting map sends v to [PRv], where $\mathrm{\Delta }R=I$; changing R changes that representative by the matched range. Exactness follows directly: $\mathrm{\Delta }x=v$ and $Px=0$ is precisely the vanishing of [PRv]; every kernel of coker $matched\to cokerP$ is represented by Px and then by PRΔx. This construction gives, with the fixed component/node order,

det $D_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$ $=$ $(tenso{r}_v\det P_v)$ tensor $\det (V{)}^{\ast }$.

Use det $D=top$ exterior kerD tensor $(topexteriorcokerD{)}^{\ast }$. The finite-dimensional determinant isomorphism for an exact sequence is defined by lifting ordered quotient bases and cancelling identical lifted wedges. Independence of a lift is the triangular determinant-one identity. Here V has real n-dimensional factors for boundary nodes and complex n-dimensional $(=real2n)$ factors for interior nodes. Incidence conditions and E augmentations use the same exact-sequence construction; recovered intersection derivatives are finite-rank core terms.

For smoothing compatibility choose one finite-dimensional smooth core forcing space W making the central matched augmented operator onto. The cokernel spanning statement of the read bordered supplier permits this with support away from all node fixtures. The separated-cutoff construction in C gives onto augmentations for all nearby smoothings and continuous kernel identification

Uq:kappa $\to$ $(I-QqDq)kapp{a}_q$.

Its fixed core coordinate is L0kappa, hence it is injective and, by the index calculation, onto the augmented kernel. Determinant of the unaugmented operator is detker(Dq+W) tensor detW*, via the exact sequence $0\to kerDq\to \ker (Dq+W)\to W\to cokerDq\to 0$. Transport an ordered augmented-kernel basis by Uq and cancel the SAME ordered W basis. This constructs a continuous determinant isomorphism to the central determinant through every partial nodal stratum; not just an orientation chosen on each fiber. If W is enlarged, the extra kernel/forcing factor cancels by its identical order. Compare arbitrary W by W+W'; compare complements through their contractible affine space. Therefore the isomorphisms agree up to the canonical positive continuous homotopy and induce the same orientation transport.

At a real node the finite matching block is $(v1,v2)\to v2-v1$ with lift $w\to (0,w)$. The concatenated diagonal/lift basis has matrix [[I,0],[I,I]], determinant+1. At interior nodes it is the complex version, preserving complex orientation. Eliminating two matching coordinates in either order gives the same determinant cancellation with precisely the graded interchange of their factors. Thus the nodal exact-sequence identification agrees with smoothing and iteration under the stated ordering convention. It does not supply a spin structure or canonically orient a real factor without one. This is the missing linear determinant input in Reader13 relative-spin Step3.1.

H. Differentiated lengths and actual scope

For any fixed parameter derivative order j, do the preceding construction on k+j+3 source derivatives. In translated unit charts, differentiating a cutoff or weight with respect to S produces bounded coefficients; its tail has the same exp(-cS) factor. A parameter derivative of pullback loses at most one spatial derivative. Tangential boundary estimates give those extra derivatives on one-sided patches. Repeated differentiation of $(DqT{)}^{-1}$ by $-A^{-1}{A}^{\mathrm{\prime }}{A}^{-1}$, and of the fixed-point equation, gives exponential correction bounds on fixed cores. This justifies applying the already written common-profile calculation of the disk-atlas consumer to smooth-F boundaries. It does not assert norm differentiability of translation on bare Hk, smoothness in naive q, or estimates across a label jump. Corner spectral/capping theory, ordinary-versus-weighted map objects, and arbitrary nonclean intersections are not resolved by H or by an energy estimate.

Retained quantitative boundary bootstrap

Local quantitative estimate; authoring status retained

This note supplies the SAME smooth totally real boundary estimate needed by history steps 1.1, 29.1 and 78.1–80.1. It does not supply estimates across a jump of boundary label, change an index formula, or prove nonlinear nodal gluing/determinant compatibility.

Uniform hypotheses

Use nested half-disks $D_r^{+}=\{s^2+t^2<r^2,t\ge 0\}$, r in [1/2,1], or interior disks. A compact target set C has finitely many coordinate balls with metric and coordinate norms uniformly equivalent. For every integer l needed, $J_z(x)$, the forcing H(z,x), and moving embeddings $i_s:L\to M$ have bounded $C^{l+2}$ norms, uniformly in the actual compact parameter family; $J^2=-1$. Boundary coordinate charts and their inverses have uniformly bounded $C^{l+2}$ norms on a fixed radius, and the least singular value of the real-linear map $(v,w)\to d{i}_s(v)+J$ $d{i}_s(w)$ is at least $c>0$. This quantitative total reality, a fixed chart radius and inverse bounds are essential; smoothness of each member alone is insufficient. For a fixed bounded geometry family they follow from compactness and total reality by taking a finite subcover. No uniformity over infinitely many unrelated finite contexts is claimed.

A boundary chart can be made J-complex along its real slice: choose local real coordinates x on $L_s$ and use the map ${\mathrm{\Psi }}_s(x,y)={\exp }_{i_s(x)}(Jd{i}_s(x)y)$, after shrinking its radius uniformly. At $y=0$ its differential sends (v,w) to $d{i}_s(v)+Jd{i}_s(w)$; hence the pulled-back J is standard there. The inverse function theorem, uniform inverse derivative bound and compactness give the stated uniform chart/inverse bounds. Differentiation in s uses the actual moving embedding; this is not a simultaneous geodesic metric for intersecting branes.

Linear estimate with its constants

For L $v=v_s+A(s,t)v_t+B(s,t)v$ on $D_1^{+}$, trace v(s,0) in $F_s$, assume uniformly bounded $C^l$ coefficients and smooth complex frames P(s,t) of uniformly bounded $C^{l+1}$ norm/inverse with A $P=P$ i and $P(s,0)R^n=F_s$. Such frames exist locally by choosing a real frame of $F_s$ on the boundary, extending it, and using (I-Ai)/2 in a normalized neighborhood. The inverse bound is controlled by quantitative total reality and the coefficient bounds. Write $v=Pw$. Then $P^{-1}$ $L(Pw)=w_s+i{w}_t+Cw$, with real trace w. For compactly supported w in a half-disk,

$\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }w\mathrm{\mid }{\mathrm{\mid }}_2^2=\mathrm{\mid }\mathrm{\mid }(\mathrm{\partial }s+i\mathrm{\partial }t)w\mathrm{\mid }{\mathrm{\mid }}_2^2$.

Indeed the mixed term is a boundary divergence; the flat-boundary contribution is zero because w and its tangential derivative are real, and the curved-boundary term vanishes by support. This identity, a cutoff and absorption of bounded C give

$\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }{\mathrm{\mid }}_{H^1(D_r^{+})}$ $\le$ $K[(\mathrm{\mid }\mathrm{\mid }(\mathrm{\partial }s+i\mathrm{\partial }t+C)w\mathrm{\mid }{\mathrm{\mid }}_{L^2(D_R^{+})})+(R-r{)}^{-1}\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }{\mathrm{\mid }}_{L^2(D_R^{+})}]$.

For higher orders apply this $H^1$ estimate to $\mathrm{\partial }{s}^j$ w, which still has real trace. The commutator is exactly $su{m}_{a=1}^j$ $\left(\genfrac{}{}{0px}{}{(}{j}\right),a)(\mathrm{\partial }{s}^{a}C)\mathrm{\partial }{s}^{j-a}w$. Every term has lower derivative order of w. The equation $w_t=i$ $w_s-i(L^{\mathrm{\prime }}w-Cw)$ recovers derivatives containing t, inductively differentiating this equality. Cutoffs on nested radii therefore give

$\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }{\mathrm{\mid }}_{H^{l+1}(D_1\mathrm{/}{2}^{+})}$ $\le$ $K_l[\mathrm{\mid }\mathrm{\mid }Lv\mathrm{\mid }{\mathrm{\mid }}_{H^l(D_1^{+})}+\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }{\mathrm{\mid }}_{L^2(D_1^{+})}]$. (L)

$K_l$ depends only on the listed frame/coefficient bounds, c, l and the nested radii. Difference quotients in s preserve the real trace and satisfy the same estimates; weak compactness then proves (L) for weak solutions, not just already smooth ones. Normal derivatives follow distributionally from the equation. Interior estimates use ordinary cutoffs. Fixed-rank kernel/evaluation blocks and end weights are separate from this local estimate.

Nonlinear bootstrap after a gradient bound

Let u solve $u_s+J(z,u)u_t=H(z,u)$, u(s,0) in $L_s$, image in C, and $\mathrm{\mid }\mathrm{\mid }du\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}\le K$. In ${\mathrm{\Psi }}_s$ coordinates w has real trace and obeys $w_s+A(z,w)w_t=h(z,w)$; $A^2=-1$ and $A(s,0,x,0)=i$. Here h includes $-D{\mathrm{\Psi }}_s^{-1}$ $\mathrm{\partial }s{\mathrm{\Psi }}_s$ and the actual transformed forcing. Its derivatives are controlled by the hypotheses; they do not vanish merely because one flattens the boundary.

Tangential differentiation gives $v=w_s$, with real trace, and the EXACT equation

$v_s+A(z,w)v_t+[D_{w}A(z,w)v]w_t$ $=$ ∂s $h(z,w)+D_w$ $h(z,w)v-(\mathrm{\partial }sA)(z,w)w_t$. (T)

Here ∂s means explicit source derivative, not total derivative. With $\mathrm{\mid }\mathrm{\mid }dw\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}$ controlled, all coefficients and right side in (T) are bounded. A frame for A along w is $W^{1,infty}$, with bounds depending on K. The $H^1$ estimate above remains valid for such a frame by approximation or directly by the product rule. Hence $w_s$ is $H^1$ on a smaller half-disk. The original equation yields $w_t$ in $H^1$, so w is H² uniformly.

For higher orders differentiate tangentially j times. The two highest terms are $\mathrm{\partial }s(\mathrm{\partial }{s}^{j}w)+A$ $\mathrm{\partial }t(\mathrm{\partial }{s}^{j}w)$; isolate $(D_{w}A\mathrm{\partial }{s}^{j}w)w_t$ as a bounded zeroth-order coefficient. All remaining terms are products of derivatives of A,h and derivatives of w of order at most j; no uncontrolled derivative of order j+1 occurs on the right. The tame product/composition inequality on a two-dimensional disk with $\mathrm{\mid }\mathrm{\mid }dw\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}\le K$ bounds their L² norm by $C_j(1+\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }{\mathrm{\mid }}_{H^j})$; the same holds for mixed derivatives recovered from $w_t=A$ $w_s-Ah$. One proof of the tame inequality is to extend to a fixed disk, apply the Leibniz and chain rules, use Hölder with exponents 2(j-1)/(a-1) for a-th derivatives of w, and the interpolation inequality $\mathrm{\mid }\mathrm{\mid }{D}^a$ $w\mathrm{\mid }{\mathrm{\mid }}_{L^{2(j-1)\mathrm{/}(a-1)}}$ $\le$ C $\mathrm{\mid }\mathrm{\mid }dw\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}^{(j-a)\mathrm{/}(j-1)}$ $\mathrm{\mid }\mathrm{\mid }{D}^j$ $w\mathrm{\mid }{\mathrm{\mid }}_2^{(a-1)\mathrm{/}(j-1)}$ plus lower norms. The interpolation inequality follows by integration by parts and Hölder, iterating from adjacent integer derivative orders; cutoffs contribute lower terms. These exponents make the sum of reciprocal Hölder exponents at most 1/2 for every differentiated product. The $H^1$ tangential estimate and the normal equation now give a uniform $H^{j+1}$ bound by induction on j on nested radii. Thus every $C^r$ norm is uniformly bounded on a smaller disk by Sobolev embedding, with constants depending on K and the actual uniform data norms. This bootstrap is after a gradient bound and does not assume gradient compactness.

Actual blow-up derivatives and nonconstant limit

At a maximizing-radius selected point $z_j$ choose $M_j=\mathrm{\mid }d{u}_j(z_j)\mathrm{\mid }\to infty$ and rescale $v_j(\zeta )=u_j(z_j+\zeta \mathrm{/}{M}_j)$, with $\mathrm{\mid }d{v}_j\mathrm{\mid }\le 2$ on disks of radius $R_j\to infty$ and $\mathrm{\mid }d{v}_j(0)\mathrm{\mid }=1$. For a boundary limit use boundary-centered coordinates; the selected point has bounded rescaled distance from the real edge and tends to a finite interior or boundary point there. Its derivative remains one in the limit. If the distance tends to infinity the limit domain is the plane.

The rescaled equation is

$(v_j{)}_{\sigma }+J_j(z_j+\zeta \mathrm{/}{M}_j,v_j)(v_j{)}_{\tau }=M_j^{-1}$ $H_j(z_j+\zeta \mathrm{/}{M}_j,v_j)$.

For explicit source derivatives and target multi-index β,

$\mathrm{\partial }{\zeta }^{\alpha }$ $\mathrm{\partial }{x}^{\beta }$ $J_j(z_j+\zeta \mathrm{/}{M}_j,x)=M_j^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ $(\mathrm{\partial }{z}^{\alpha }\mathrm{\partial }{x}^{\beta }{J}_j)(z_j+\zeta \mathrm{/}{M}_j,x)$,

$\mathrm{\partial }{\zeta }^{\alpha }$ $\mathrm{\partial }{x}^{\beta }$ $[M_j^{-1}{H}_j(z_j+\zeta \mathrm{/}{M}_j,x)]=M_j^{-1-\mathrm{\mid }\alpha \mathrm{\mid }}(\mathrm{\partial }{z}^{\alpha }\mathrm{\partial }{x}^{\beta }{H}_j)(z_j+\zeta \mathrm{/}{M}_j,x)$.

The moving boundary is $i_{s_j+\sigma \mathrm{/}{M}_j}$, with $\mathrm{\partial }{\sigma }^a$ $\mathrm{\partial }{x}^{\beta }$ $i=M_j^{-a}$ $\mathrm{\partial }{s}^a$ $\mathrm{\partial }{x}^{\beta }$ i, and the same scaling for Ψ and its inverse. Target derivatives of J DO NOT tend to zero; the limit is generally $J_{\mathrm{\infty }}(x)$, not a constant complex structure. The transformed boundary-motion term is $O(M_j^{-1})$, including every fixed explicit source derivative. Quantitative total reality and chart radii persist. The bounded normalized gradient and finite target cover let the preceding bootstrap apply on fixed small rescaled disks, with constants independent of j. Diagonal extraction consequently gives $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$ convergence to a nonconstant $J_{\mathrm{\infty }}-holomorphic$ plane or half-plane with boundary on the limiting ONE exact brane. The data themselves have a smoothly convergent subsequence by the compact parameter family (or the explicitly summable Hilbert perturbation tails at history step 88.1). Uniform energy and target confinement pass to the limit on every compact domain, so the limit has finite energy. The existing exhaustion Stokes argument at 79.1 then excludes it. No derivative bound was used before the maximizing-radius normalization provided $\mathrm{\mid }d{v}_j\mathrm{\mid }\le 2$.

The same proof on translated unit half-strips with the same-label trace has constants independent of strip length; summing a bounded-overlap cover gives higher weighted estimates. Derivatives of a fixed exponential weight are bounded multiples of that weight, so the weight exponent is retained. This does not remove the real mean mode or manufacture an inverse on a long neck. Label-changing corners and nodes must first be represented by the appropriate strip/domain normalization; they are not half-disk points with one smooth trace.

Exact consumer fit and remaining boundary

History 29.1 concerns a FIXED smooth solution and a fixed finite domain context: its coefficients, tubular charts, finite cover and stationary strip ends supply the uniform hypotheses locally in parameter; (L) lifts the linear Fredholm realization, while $H^k$ composition and its remainder follow from the same product estimates. A point evaluation is used only after $H^k$ regularity, $k\ge 4$. History 78.1 uses finite compact target confinement from 77.1, normalized gradients, compact smooth boundary-family bounds, and the actual rescaling formulas above. History 80.1 follows after the Stokes contradiction. At a source degeneration use the existing normalized component/strip charts; this note supplies no estimate across a multi-label corner and does not certify that a claimed domain normalization has already been constructed. Any application to such a corner remains conditional on that separate construction.

This is a local authoring proposal for an established analytic prerequisite. It is not an open-problem theorem, an external import, or certification of the other 364 proof steps. No external full text was fetched/read during this repair; the argument was reconstructed from the actual local carriers and its stated estimates must receive native scientific review.