Finite exact stopped Floer operations, coherent wrapping histories and localization

Chapter 1: Local estimates and unique continuation. For the actual applicability, first use [F8] to normalize components and construct the smooth one-trace core/strip windows, including marked seams, finite concatenations and every retained nodal component. In a smooth complex frame the principal operator is $D={\mathrm{\partial }}_s+i{\mathrm{\partial }}_t$. For compactly supported smooth functions with real flat-edge trace, integration by parts gives $\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2=\mathrm{\parallel }Dv{\mathrm{\parallel }}_2^2$; its boundary cross term is zero because the tangential trace is real. Cutoffs give the local $H^1$ estimate with a lower-order $L^2$ term. Differentiate tangentially by difference quotients, which preserve the trace, and recover normal derivatives from the equation. This proves higher estimates for arbitrary ordinary forcing jets; no high-order reflection condition on the target is imposed. The actual uniform frame/coefficient/inverse bounds and nonlinear gradient-normalized application are the ones proved in [F8], with the quantitative local argument retained below. Fixed transverse ends still use exactly [F2], including their real-mean constraint and actual finite kernel/cokernel. Fourier truncation on fixed core charts proves compact inclusion; the same Fourier Cauchy–Schwarz and convolution estimates give the two-dimensional Sobolev products and $H^k\subset C^1$ for $k\ge 4$. This applies after actual source normalization, not in a collapsing root coordinate.

For completeness weak unique continuation needed for smooth solution/cokernel sections has a local direct proof. If ${\mathrm{\partial }}_{\bar{z}}v=Cv$ with a bounded real-linear zero-order coefficient, replace that coefficient by the bounded complex-linear matrix $A(z)=(C(z)v(z))v(z{)}^{\ast }\mathrm{/}\mathrm{\mid }v(z){\mathrm{\mid }}^2$ where $v\mathrm{\ne }0$, and zero where $v=0$. Then $Av=Cv$. The Cauchy transform $Tf(z)={\pi }^{-1}\int f(w)\mathrm{/}(z-w)dA(w)$ satisfies ${\mathrm{\partial }}_{\bar{z}}Tf=f$ distributionally and $\mathrm{\parallel }Tf{\mathrm{\parallel }}_{\mathrm{\infty }}\le 4r\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ on a radius-$r$ disk. The identity follows by polar integration against a test function; continuity follows by deleting a small kernel disk and uniformly translating the remaining integrable kernel. Thus $P=I+T(AP)$ has a continuous solution with $\mathrm{\parallel }P-I\mathrm{\parallel }<1\mathrm{/}2$ on a sufficiently small disk, by its geometric iteration, and $P$ is invertible. Its weak derivatives are locally $L^2$: for smooth compact approximants $f_j$ the identity $\mathrm{\parallel }\mathrm{\partial }T{f}_j{\mathrm{\parallel }}_2=\mathrm{\parallel }{\mathrm{\partial }}_{\bar{z}}T{f}_j{\mathrm{\parallel }}_2=\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_2$ follows from the preceding integration by parts after a large-radius cutoff (the transform is $O(1\mathrm{/}\mathrm{\mid }z\mathrm{\mid })$ and its derivative $O(1\mathrm{/}\mathrm{\mid }z{\mathrm{\mid }}^2)$); approximate bounded compact $f$ in both $L^2$ and $L^q$, $q>2$. The kernel is locally in $L^{q^{\mathrm{\prime }}}$, so the transforms converge uniformly on compact sets, and their derivatives converge in $L^2$. Consequently $P,P^{-1}$ are continuous $W^{1,2}$ matrices and $P^{-1}v$ is weakly holomorphic, by the product rule, first for smooth approximants and then in $W^{1,2}$. Mollifying this equation on smaller disks and using the mean-value identity shows its continuous representative is holomorphic. It is either zero or has a first nonzero Taylor coefficient and isolated zeros. Vanishing on an open set therefore propagates across each overlapping disk. A smooth variable principal complex structure is reduced to this one by $\mathrm{\Phi }=(I-Ji)\mathrm{/}2$, since $J\mathrm{\Phi }=\mathrm{\Phi }i$ and $\mathrm{\Phi }$ is invertible near a point with $J=i$. Subtracting two smooth nonlinear solutions gives such a linear equation with bounded zeroth-order coefficient. This proves every unique-continuation application below, including adjoint sections and equality of parametrized maps.

The countable generic choices below use the Baire property of complete Hilbert balls and the given Sard--Smale theorem [F3], not an uncountable residual intersection. The Baire property follows by choosing recursively closed balls of radii tending to zero, each inside the preceding ball and the next dense open set; completeness gives a point of all of them. Countable choices and arbitrary vector-space splittings are allowed by AC [F1]. A finite-dimensional restriction of [F3] supplies the ordinary Sard assertion about critical values used for regular clock times. All other geometric data and coefficient conventions are those of [F1,F5--F7]; no virtual theory, 6D attachment, generation, or strict geometric unit is an additional premise.

Chapter 2: Ordered real domains and face orientations. Fix $N\ge 3$ cyclically ordered leaf labels. A stable disk tree means a finite planar tree of disks, each with at least three boundary flags, whose external flags are exactly those labels. Write $K_N$ for the compactification constructed here, and distinguish the output flag. Double a smooth marked disk to a marked projective line with its real circle, or identify it directly with the upper half-plane by its three chosen boundary anchors. Positive real Möbius transformations normalize three anchors to∞,0,1. For a sequence, compactness of RP1 extracts limiting positions of its finitely many labels. There are at least three distinct limiting groups because of the anchors. They define the first stable component.

If a limiting position carries more than one label, use a finite real coordinate there (at∞ use−1/x). For that group choose its two extremal points in the induced small interval, subtract the first and divide by their positive separation. All points of that group now lie in[0,1], and at least two limiting groups lie at0 and1; the parent attachment is∞. This defines its stable child component. Recurse on every group still containing more than one coincident label. At every step a group splits into at least two strictly smaller nonempty groups; the recursion is therefore finite. It produces a planar labeled tree, component point configurations, and positive relative scales tending to zero. Counting flags gives ${\sum }_v(\mathrm{val}(v)-2)=N-2$; val(v)≥3 bounds its vertices by N−2 and its internal edges by N−3. Thus no infinite ghost tree is admitted by this construction.

Uniqueness/topology can be checked by finitely many real cross-ratios. For every ordered quadruple of leaves take its normalized cyclic cross-ratio in[0,1]. On a stable tree define it by contracting to its four-leaf stabilization, where it is0,1 or a component cross-ratio. These determine the tree and its coordinates. To verify this rather than assume it, choose any stable vertex and three leaves in three of its branches; normalize those leaves to∞,0,1. The cross-ratios with a fourth leaf determine its limiting position, hence the partition into the remaining branches. Recurse inside every nontrivial branch by choosing three of its flags. This recovers exactly the finite cluster construction 5.1; if no branching remains it recovers the smooth marked line itself. In particular two limits with identical quadruple data are isomorphic. The configuration closure in the finite cross-ratio cube is therefore precisely this stable-tree set, compact Hausdorff and metrizable. The original cyclic order determines its planar order and the choice of disk half on every real component.

Fix such a stable tree. Root it at one component. Choose three flags there and normalize to∞,0,1. On every child normalize its parent flag to∞ and two child flags to0,1. Remaining flag positions are the val(v)−3 component parameters; their cyclic inequalities persist in a small open neighborhood. Select representative leaves recursively for these flag anchors. For an edge attached at finite parent coordinate a, the generic-coordinate relation is $x_{p}arent=a+{\rho }_e{x}_{c}hild\mathrm{.}$ In the child coordinate $w=-1\mathrm{/}{x}_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}}$ this is $(x_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}-a)w=-{\rho }_e$. Attachments at parent∞ use the real coordinate $-1\mathrm{/}{x}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}$. These formulas compose recursively, expressing every generic root-coordinate marked point as a real rational function of the component parameters and the positive scales. Conversely the same normalized leaf coordinates recover a andρ and then all child coordinates. Thus for positive small scales they are actual coordinates, not merely formal deformation parameters.

At $\rho =0$ retain the two component charts and the fixture $zw=-\rho$. Charts away from $z=w=0$ glue by the displayed identities; they are actual holomorphic coordinate changes after complexifying the finite base parameters, and preserve the chosen real halves on the nonnegative real chamber. The fixtures and the complements of fixed smaller coordinate disks form a finite cover of each family fiber. Over a compact base subset their closed core pieces and their closed bidisk fixtures are compact; the glued inverse image is their continuous image, hence compact. Disjoint smaller fixtures and the product collars separate distinct points, so the total family is Hausdorff. This proves the required proper universal local family and smooth thick-part trivialization directly.

The coordinate inversions of 7.1, and the cross-ratio recovery of 6.1, show that after including zero scales this gives a homeomorphism from the orthant chart to an open stable-tree neighborhood. Its inverse on smooth configurations is the explicit normalized cluster calculation; as scales vanish it has exactly the limits from 5.1, and there are no other cross-ratio limits. Different anchor choices are rational Möbius coordinate changes on the components, nondegenerate at the fixed distinct flags; the fixture formulas then give their smooth family comparisons. Ordered labeled stable genus-zero domains have no automorphism: the leaf-labeled tree is fixed, and each component automorphism fixes at least three flags. Thus there is no hidden orbifold stabilizer or extra finite quotient.

For z,w in the upper half-plane the relation $zw=-\rho$ is exactly compatible with the disk halves. Put $z=e^{-\pi s+i\pi t}$ and $w=e^{-\pi (L-s)+i\pi (1-t)}$, $0\le t\le 1$. Their product is $-e^{-\pi L}$, so $L=-{\pi }^{-1}\log \rho$. Away from the fixture the component coordinates are fixed; near every remaining flag use the same product disk chart. This supplies the uniform cores/strip ends used by the polygon estimates. The unstable two-punctured strip is treated separately with its translation gauge, as required in the actual polygon problem.

Write $N=k+1$, with output∞ and inputs $0=x_1<x_2<\cdots <x_k=1$. Orient the open domain chart by ${\mathrm{\Omega }}_k=d{x}_2\wedge \cdots \wedge d{x}_{k-1}$. A one-edge split separates one consecutive block of s inputs after r and before t, $r+s+t=k$, with both component arities at least2. An interior block has $x_{r+j}=a+\rho b_j,b_1=0,b_s=1$. Its Jacobian wedge is ${\mathrm{\Omega }}_k=(-1{)}^{r+st}{\rho }^{s-2}d\rho \wedge {\mathrm{\Omega }}_{o}uter\wedge {\mathrm{\Omega }}_{i}nner\mathrm{.}$ To compute the sign, the block wedge contributes $(-1{)}^{s-2}da\wedge d\rho \wedge d{b}_2\wedge \cdots \wedge d{b}_{s-1}$. Move dρ past the r preceding coordinates, then move the t−1 outer after-coordinates past the s−2 inner coordinates. The total exponent is $s-2+r+(t-1)(s-2)=r+t(s-2)\equiv r+st(\mathrm{m}\mathrm{o}\mathrm{d}2)$.

If $r=0$, put $\rho =x_s$ and $x_j=\rho b_j$ inside the first block; its wedge contributes exponent $(s-2)+(s-2)(t-1)=st(\mathrm{m}\mathrm{o}\mathrm{d}2)$. If $t=0$, put $\rho =1-x_{r+1}$; the first differential−dρ and its movement past r−1 before-coordinates give exponent r. Thus the same formula covers every stable facet. The outward normal is−∂ρ, adding its one common minus sign to the oriented boundary product. Products of faces have the induced ordered normal signs; swapping two normal directions changes sign, so codimension-two corner contributions cancel in the boundary-of-boundary formula. This supplies the actual associahedral orientation convention needed for the Stasheff identity.

Chapter 3: Protected multimark/Moore domains and actual atom labels. Let $d\ge 0$ and $k\ge 1$. There are $d$ ordinary test punctures, distinguished module punctures $I,O$, and $k$ individually labelled transition marks $m_1,\dots ,m_k$ on the module edge. The positive cyclic order is $O$, the test punctures in their order, $I,m_k,\dots ,m_1,O$. The logical transition word is $1,\dots ,k$, opposite the positive boundary traversal on this edge. Treat every transition mark as an actual boundary special point. Put ${\mathcal{X}}_{d,k}=K_{d+2+k}$ in this labelled chamber of the ordered real-disk compactification. It is compact of dimension $d+k-1$, with the actual universal real-plumbed family and stable planar tree faces. Every original mark survives as a labelled flag, and a vertex can carry several marks. In particular a transition-only bubble with at least two marks and its attaching node is a genuine marked vertex even when it carries no original module or test puncture. It must not be removed as an ordinary exact disk bubble.

There is also a protected compact Moore model. On a smooth disk put $I=0,O=\mathrm{\infty }$ in an upper-half-plane coordinate $z$. Its module edge is the positive real axis, with $z_1>z_2>\cdots >z_k>0,\tau =-{\pi }^{-1}\log z\mathrm{.}$ The coordinate $\tau$ increases in logical transition order and is opposite the positive source-boundary orientation. The ratios $q_i=z_{i+1}\mathrm{/}{z}_i$ are intrinsic four-flag cross-ratios with $I,O$, independent of the residual dilation. They extend smoothly as real cross-ratios to ${\mathcal{X}}_{d,k}$. Choose positive atom durations $T_1,\dots ,T_k$ and set $a_i=e^{-\pi T_i}$ for $i<k$. Then ${\mathcal{T}}_{d,k}(\mathbf{T})=\{x\in {\mathcal{X}}_{d,k}:q_i(x)\le a_i\text{ for }1\le i<k\}$ is a compact manifold with corners of the same dimension, the closure of the corresponding smooth configurations. Besides its ordinary real-node faces, its new finite faces are $q_i=a_i$. No transition mark lies off the tree path from $I$ to $O$ in this protected model: a marked off-spine subtree would contain adjacent marks with $q_i=1$, which the inequalities exclude. Vertices on that spine can still carry multiple marks, and unmarked ordinary vertices between marked ones are retained.

On smooth configurations the actual Moore gaps are ${\mathrm{\ell }}_i={\pi }^{-1}\log (z_i\mathrm{/}{z}_{i+1})-T_i\ge 0,{\tau }_{i+1}={\tau }_i+T_i+{\mathrm{\ell }}_i\mathrm{.}$ At ${\mathrm{\ell }}_i=0$ replace adjacent atoms by the ordered concatenated word block. Its duration is the sum of its atom durations, and every internal atomic mark is recovered from its first mark by the fixed offsets of those durations. This gives an actual finite-face identification with the corresponding word-decorated lower-mark family. No atomic history or source point is silently discarded. Multiple such identifications are literally associative on their intersections because the same offsets are added and the same atomic profiles are Moore-concatenated.

At infinity use actual real-node fixtures. For a direct separation between adjacent marked components one may take $u=z\mathrm{/}{z}_i$ near the parent's zero flag and $v=-z_{i+1}\mathrm{/}z$ near the child's infinity flag. Then $uv=-q_i,q_i=e^{-\pi (T_i+{\mathrm{\ell }}_i)},$ so the actual strip length is $T_i+{\mathrm{\ell }}_i$. In a general tree with unmarked intermediate vertices, a total mark-gap ratio instead has the local form $q_i=u_i{\prod }_{e\in {\mathcal{P}}_i}{\rho }_e,u_i>0\text{ smooth},$ where ${\mathcal{P}}_i$ is the relevant ordered spine-edge path and ${\rho }_e$ are its independent real plumbing coordinates. Thus its infinite-gap locus can be a union of nodal faces. The global gap map is not asserted to be an independent product-cube coordinate at such corners.

All faces retain their actual labelled component configurations and the native domain orientation. An explicit smooth-chart orientation conversion is available: for $d\ge 1$ fix the first negative test coordinate $y_1=-1$, write $-1<y_2<\cdots <y_d<0$, and put ${\tau }_1=A$. In the native domain orientation, ${\mathrm{\Omega }}_{d,k}=(-1{)}^{k(k-1)\mathrm{/}2}Fd{y}_2\wedge \cdots \wedge d{y}_d\wedge dA\wedge d{\mathrm{\ell }}_1\wedge \cdots \wedge d{\mathrm{\ell }}_{k-1},F>0.$ For $d=0$ use the dilation to fix $z_1=1$, omit the test and $A$ factors, and obtain the same sign times a positive function times $d{\mathrm{\ell }}_1\wedge \cdots \wedge d{\mathrm{\ell }}_{k-1}$. Finite-gap faces use outward $-{\mathrm{\partial }}_{{\mathrm{\ell }}_i}$, infinite real-node faces outward $-{\mathrm{\partial }}_{{\rho }_e}$, equivalently increasing neck length. The ordinary domain-node Jacobians are those of the ordered real-domain construction in Chapter 2; they are not counted again as an independent sign. If gaps are placed before the dimension-$d$ ordinary single-mark domain factor, the additional reorder is $(-1{)}^{d(k-1)}$. Thus the native-to-word-cell-first orientation conversion is $(-1{)}^{k(k-1)\mathrm{/}2+d(k-1)}$, independent of external parameter factors already placed first.

Finally label atom $i$ by an actual fixed-end Moore path family ${\gamma }_i:P_i\to \mathrm{Path}(L_{i-1},L_i)$ with sitting duration $T_i$ and an independent external cube block $P_i$. On a finite face its substituted path is the literal Moore concatenation over the ordered product of those blocks. At a broken marked spine the atom blocks are partitioned among vertices, each atom owned once; fixed ordinary vertices have no external block. This supplies literal parameter/word face data without duplicated shared parameters. If one chooses a truly shared parameter family instead, its equality constraint and diagonal/prism comparison remain additional data. Endpoint modules and all operations must be transported to actual fixed bar data before applying a count identity.

This is a domain/word/profile construction only. It proves no regularity, analytic count compactification, gluing orientation, strict-unit count or higher count identity. Positive durations do not by themselves identify a stationary profile with the zero-length strict Moore unit; the actual forgetful/stabilization unit fixture is separate.

Apply the ordered-domain construction to the $d+2+k$ individually labelled points in the specified cyclic order. The total is at least three. Its tree construction retains all those flags and makes each vertex stable by its full flag count, including transition marks. A vertex with one attaching node and two or more transition marks is therefore stable. It is a transition-only vertex, not an ordinary unmarked disk component. The number of vertices and nodes is finite by the source flag bound. The case $d=0,k=1$ is the three-flag disk, equivalently the anchored two-punctured strip with its transition mark fixing translation. This proves the unrestricted actual compactification and its complete typed tree inventory.

On a smooth disk, fixing $I=0,O=\mathrm{\infty }$ leaves exactly positive real dilations, so each $q_i$ is intrinsic. It is the four-flag cross-ratio of $I,m_{i+1},m_i,O$. In a real plumbing chart, substitute the rational flag formulas of the ordered-domain construction in Chapter 2 and cancel common scale factors. If its four-flag stabilization is smooth, the denominator remains nonzero in the component flag chart. If it splits, the ratio or its complement is a product of the plumbing parameters along the intervening path, times a positive smooth unit. Thus $q_i$ extends smoothly in $[0,1]$. In particular a subtree off the $I$--$O$ spine containing transition marks lies entirely on the transition arc, by planarity. It has at least two consecutive marks because one mark plus its sole attaching node would be unstable. Their four-flag stabilization has coincident mark cluster away from $I,O$, giving $q_i=1$. Hence none of these off-spine marked subtrees belongs to the protected subset.

Compactness follows because the protected inequalities define a closed subset of ${\mathcal{X}}_{d,k}$. At any point of it, marks on different spine vertices have separating ratio zero. Therefore an active finite constraint $q_i=a_i\in (0,1)$ concerns two consecutive marks on the same spine vertex. Give that component its spine flags $0,\mathrm{\infty }$. Its logarithmic adjacent mark differences are independent coordinates, modulo the one common dilation; any ordinary test flags supply the remaining independent component coordinates. For every active constraint its derivative in the corresponding difference is nonzero. Distinct active differences are independent, including in a chain of marks. Keep the native node parameters fixed and apply the inverse theorem [F4] to these component variables. The resulting corner coordinates consist of the independent ${\rho }_e\ge 0$ and $a_i-q_i\ge 0$, plus unconstrained variables. They show both the manifold-with-corners claim and that every point is approachable by strictly positive nodes and strict finite inequalities. Thus the closed subset is exactly the claimed closure, rather than an extra unproved compactification.

In smooth logarithmic coordinates put ${\tau }_1=A$ and ${\tau }_{i+1}={\tau }_i+T_i+{\mathrm{\ell }}_i$. A chain of finite faces sets the internal ${\mathrm{\ell }}_i$ of its consecutive blocks to zero. Its atomic points have the uniquely reconstructed offsets ${\tau }_j={\tau }_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}+{\sum }_{h<j,h\text{ in block}}{T}_h$. The next block starts at the preceding block anchor plus the sum of its durations and its surviving gap. Forgetting the redundant atomic flags is only a coordinate description: reconstruct them by this formula and retain their labels. No component destabilizes, since a marked spine vertex has two spine flags and at least one word mark after substitution. The maps extend across the native node charts with those same formulas on each component. Merging overlapping blocks in either order gives exactly the same anchor offsets and ordered atomic history. This proves literal finite-face substitution and associativity on every intersection.

The stated direct node coordinates give $uv=-z_{i+1}\mathrm{/}{z}_i$ identically, so the ordered-domain construction in Chapter 2's strip formula gives length $-{\pi }^{-1}\log q_i=T_i+{\mathrm{\ell }}_i$. Along a longer spine, apply the same normalized coordinate relation at each edge. Ratios telescope through the intermediate component scalings, leaving the product of the edge ${\rho }_e$ times nonvanishing positive anchor factors. This proves the monomial formula. Each actual edge has its own independent plumbing parameter. Near its finite cutoff ${\rho }_e={\epsilon }_e$ use Moore collar length $-{\pi }^{-1}\log ({\rho }_e\mathrm{/}{\epsilon }_e)$; the zero collar face is that actual finite smoothing, retaining all marks. Gluing disjoint node fixtures in either order gives the same underlying family quotient. A total mark gap crossing several edges is not a substitute for those independent parameters.

For $d\ge 1$, the native positive real input order is $y_1,\dots ,y_d,0,z_k,\dots ,z_1$, output $\mathrm{\infty }$. To use its normalized first/last inputs, send $z$ to $(z+1)\mathrm{/}(z_1+1)$. Wedge its free input differentials; the fixed module point contributes $d(1\mathrm{/}(z_1+1))=-d{z}_1\mathrm{/}(z_1+1{)}^2$. All other terms containing $d{z}_1$ then vanish in the wedge. The result is $(-1{)}^k$ times a positive factor times $d{y}_2\wedge \cdots \wedge d{y}_d\wedge d{z}_k\wedge \cdots \wedge d{z}_1$. Since $z_j=e^{-\pi (A+{\sum }_{h<j}(T_h+{\mathrm{\ell }}_h))}$, its ordered logarithmic Jacobian gives another sign $(-1{)}^{k(k+1)\mathrm{/}2}$ and a positive factor. Their product is $(-1{)}^{k(k-1)\mathrm{/}2}$. For $d=0$, set $z_1=1$; the free native inputs are $z_k,\dots ,z_2$, and the remaining logarithmic Jacobian has sign $(-1{)}^{k(k-1)\mathrm{/}2}$ directly. Finite-gap outward orientation is decreasing ${\mathrm{\ell }}_i$ at zero; node outward orientation is decreasing ${\rho }_e$ at zero. Moving the $k-1$ gaps before the dimension-$d$ single-mark domain factor gives the further sign $(-1{)}^{d(k-1)}$. These calculations and the ordered-domain construction in Chapter 2's ordinary node Jacobian fix the full face orientation, including all intersections by ordered normals.

Assign actual transition path ${\gamma }_i$ to its atom. Increasing $\tau$ follows the positive logical word, while positive boundary orientation decreases $\tau$; hence a positive path profile has the nonincreasing boundary convention used by the moving no-escape construction. Sitting ends make the Moore concatenation smooth. On a finite face the word substitution of step 23.1 concatenates exactly those same profiles with their summed durations; a second substitution changes nothing pointwise because Moore concatenation and addition of its lengths are associative. Ordered external blocks are carried along as $P_1\times \cdots \times P_k$. Node vertices own disjoint consecutive atom subsets; the resulting parameter product is an actual partition, not copies of one diagonal variable. If parameters really are shared, retain their equality exact sequence with difference map $a_2-a_1$; its diagonal/normal block matrix has determinant $+1$ but does not imply matched regularity. Nor does it identify geometric diagonal counts with a cellular diagonal expression.

The universal family and every shared face are the same actual labelled real-domain family, restricted by the proved corner equations. Compatible word/profile labels and independent external block products restrict literally there. For unrestricted multi-mark families keep the transition-only vertices from step 20.1, with their before/after brane labels and actual node chord. The protected model avoids that particular collision through its explicit inequalities, not through an exactness argument. None of these finite-dimensional domain constructions supplies a Floer perturbation, count transversality or compactification of maps. Strict units require a separate actual identity-profile/forgetful fixture; a positive-duration constant sitting atom is not literally the zero-length Moore unit. This proves the complete stated domain supplier with its exact scope.

Chapter 4: Universal regularity and actual nonlinear matched gluing. 1. The finite actual problem and the order of construction Fix a finite collection of the exact conical surfaces, primitives, gradings, relative Spin structures and finite-rank flat systems in the finite exact stopped Lefschetz datum of [F1]. Enlarge by actual transverse Hamiltonian copies, including copies used for self-Homs. Ordinary stationary end generators are then discrete nondegenerate chords. Keep every generator of the proper self-Morse model, not only its minimum. Fix the end Hamiltonians and their transverse planes. External history parameters do not change these end generators. Let $k\ge 4$. For a fixed word use exponentially weighted $H^k$ maps, $H^{k-1}$ equation targets and its actual finite domain/external parameters. The parameter vector occurs ONCE in this Banach chart.

Apply [F8] to the finite actual disk/marked context of28.1: its atom durations, actual Hamiltonian path families and compact external blocks are precisely those constructed in Chapters2–3, and each unbounded neck has the prescribed stationary product data. The construction retains unrestricted transition-only vertices and all marked node components. Its component cores and translated width-one strip windows have a common radius, and each boundary half-window has one smooth trace. In those actual source coordinates, the pulled-back $J$, forcing, moving embeddings, total-reality inverses and boundary-adapted tubular charts have the derived uniform finite-order bounds. At an atomic seam the trace follows its smooth sitting path; at a finite merge it is the literal smooth concatenation; at a node one selects the normalized strip edge. Source-dependent inverse charts contribute their bounded explicit source-derivative forcing terms and are not discarded. Consequently the local tangential estimates and two-dimensional products of1.1 apply to these actual charts: the equation is smooth on weighted $H^k\to H^{k-1}$, $H^k\subset C^1$ for $k\ge 4$, and the end/core estimates lift the [F2] weak realization to $H^k$. Compact transition errors remain compact. Finite external/evaluation blocks remain their actual finite-rank blocks, with each shared parameter once. This derives the application instead of assuming smooth-F estimates across an unnormalized multi-label seam. No uniform derivative in a raw plumbing parameter at zero is asserted or needed.

This proves the Banach/Fredholm hypotheses for the ACTUAL equation; its cokernel remains until the following perturbation argument. The base ordinary strips are regularized first. Ordinary polygon data are then chosen by domain complexity, and marked cells by arity, block complexity and face dimension. Lower product collars are fixed before each interior choice. At a broken face each atom block belongs to its own marked vertex; a shared source parameter is not copied to make a direct sum appear surjective.

2. Ordinary differentials, including self-pairs Use source-translation-invariant $J_t$ and fixed transverse copy boundaries. A stationary Hamiltonian end can first be removed by its actual symplectic flow, transporting J, the branes, their primitive action bases and orientations. The resulting strip equation is $u_s+J_t(u)u_t=0$.

For a nonconstant strip, $u_s$ solves the homogeneous differentiated real Cauchy--Riemann equation. Local similarity/unique continuation implies its nonzero set is dense. If a local time-slice overlap is $u(s,t)=u(\sigma (s,t),t)$, differentiating the equality and comparing the two strip equations gives

$u_s(\sigma )({\sigma }_s-1)+J_t{u}_s(\sigma ){\sigma }_t=0$.

Where $u_s$ is nonzero its two displayed real directions are independent. Hence ${\sigma }_s=1$ and ${\sigma }_t=0$: the overlap is a translation. Unique continuation propagates that equality. A nonzero translation would make the strip periodic and force positive energy on infinitely many translated periods, contrary to finite energy. Therefore no distinct regular overlap branch contains an open set.

Here are the needed density quantifiers. Exhaust the other longitudinal preimages by compact intervals and separate them from s by 1/j. At a regular other preimage a nonzero target coordinate derivative gives its local inverse branch sigma; countably many rational rectangles cover those branches. Each equality set has empty interior by the preceding argument. Other critical preimages have zero-length image on each time slice: cover their parameter set by intervals with speed below epsilon, estimate their image length by epsilon times total interval length, and let epsilon decrease to zero. Such an image cannot contain a regular image arc. Exclude the finitely many asymptotic-generator images as well. Baire then gives dense time-slice injective points. Near any such point other preimages cannot escape to longitudinal infinity, because those generator values were excluded.

A nonzero cokernel section is smooth and cannot vanish on an open set by adjoint elliptic regularity and unique continuation. Choose an injective point where it and $u_s$ are nonzero. A compatible $J_t$ variation there can prescribe its action on the nonzero $u_t$ arbitrarily: in a unitary frame the anti-complex-linear symmetric variations are complex symmetric matrices, whose evaluation on one nonzero vector is onto. Extend in a small (t,target) patch with only that one preimage, and cut off. Its cokernel pairing is nonzero. Thus the universal SECTION derivative is onto while the strip data remain translation invariant. Constant transverse strips have the invertible reference operator; they have no nonzero translation vector and receive no artificial translation gauge.

A nonconstant strip must leave the disjoint fixed chord neighborhoods, since a strip remaining in one has the same generator at both ends and zero exact energy. Variations can consequently avoid all the frozen end neighborhoods. This is also the differential argument for self-complexes: the two branes are their actual transverse copies, not identical boundary planes declared invertible.

3. Marked cells: actual compact source-dependent perturbations Choose $R$ above the fixed generators and compact perturbation supports, with $K=0$ on $r\ge R$ and contact-type $J$ there. The elementary no-escape estimate is available before regularity: $a=\log r$ is subharmonic, and on a moving edge its outward derivative is $\alpha (V){\rho }^{\mathrm{\prime }}\le 0$, since the positive contact path is traversed nonincreasingly in boundary orientation. The interior/boundary maximum principle and convergence to the lower asymptotic chords give $r\le R$. This argument needs no compactification or gluing theorem. A new marked source core has an accessible interior point with image $r<R$. Otherwise its open core lies at $r=R$; the contact equation and the radial equality imply $du=0$ there. The constant at that level solves the same equation since K vanishes there. Subtracting gives a homogeneous linear real Cauchy--Riemann equation, so unique continuation makes the component constant, contradicting its end/node below R. No uniform margin below R is claimed.

At an accessible disk vary $K_t$ by beta(z)h(x), with compact source support away from edges, ends and already fixed collars. Its equation forcing is -J $X_{\beta h}(u)$. The value dh at a target point is arbitrary and the symplectic musical map is invertible, so these forcings prescribe arbitrary vectors. If a nonzero cokernel section annihilated every such variation, unique continuation would give a nonzero value on some accessible disk and a bump could pair nontrivially there, a contradiction. This works at constant maps and at repeatedly covered target images, because the perturbation depends on the SOURCE point. With several marked vertices the bumps have independent source supports; actual external parameters and equality equations still occur once.

For finite point/node evaluation constraints, apply this argument to the JOINT operator, not only to its components. An adjoint annihilator is smooth off the constrained marked points. Interior forcing kills that smooth equation part on each connected component. The remaining multiplier then annihilates all permitted independent local evaluation values; for the actual equality of two freely variable evaluation values this forces it to vanish. For asymptotic clean parameters retain their finite-dimensional $a_{+},a_{-}$ variables before imposing $a_{+}-a_{-}=0$; the same last argument uses those actual variables. Any algebraically redundant evaluation rows must first be expressed in their true independent target. An arbitrary finite block is Fredholm, but an evaluation block with a genuine structural cokernel does NOT become onto merely by naming a generic perturbation. The ordinary root-operation construction uses discrete stationary chords, so these additional clean rows are unnecessary there.

4. Smooth separable parameters and the correct Sard--Smale projection Choose countably many smooth allowed bumps $e_j$ dense on rational source/target charts. Rescale so $\mathrm{\mid }\mathrm{\mid }{e}_j\mathrm{\mid }{\mathrm{\mid }}_{C^m}\le 2^{-j}$ for $m\le j$. For a in ell² the series sum $a_j{e}_j$ converges in every fixed $C^m$ norm by Cauchy--Schwarz. Its finite span is unchanged; its dense image in each finite cokernel is the whole cokernel. For almost-complex-structure parameters, use the tame Cayley $W$-chart proved in [F8, proof13.1]; a sufficiently small Hilbert ball remains in the open unit ball $\mathrm{\parallel }W{\mathrm{\parallel }}_{g_0}<1$ and preserves all fixed support/end constraints. Mapping charts are second countable: finitely many compact target charts embed the common compact image neighborhood into Euclidean space, and the weighted Sobolev charts and finite source charts are separable. Relative homotopy classes of the finite handlebody and brane cores are countable.

Let $D:Z\to Y$ be the actual family derivative and $V:H\to Y$ the bump derivative. D is Fredholm and D+V onto. Finitely many bump columns span coker D; combine them with the inverse on a closed complement of ker D to obtain a bounded right inverse of D+V. The Banach implicit theorem gives the universal zero manifold. On its tangent kernel the projection to H has kernel ker D and image

ker[H $\to$ coker D],

where the quotient map is induced by V and is onto. Thus the projection is Fredholm of index ind D, and its derivative is onto exactly when D is onto. Sard--Smale and Baire give regular family data on the countably many required index and relative-homotopy charts. No universally onto projection is assumed. Regularity is for the family including its external parameters, not every fixed-parameter fibre.

For the tame almost-complex structures in this construction, fix a compatible reference $J_0$ and $g_0(v,w)=\omega (v,J_{0}w)$. Write $J=J_{0}Z$, so $Z=J_0^{-1}J$; $J^2=-1$ is equivalent to $Z^{-1}=J_0^{-1}Z{J}_0$, and tameness is $g_0(v,Zv)>0$ for every nonzero $v$. Set $W=(I-Z)(I+Z{)}^{-1}$. Positivity makes $I+Z$ invertible. Since $W(I+Z)v=(I-Z)v$, $\mathrm{\parallel }(I+Z)v{\mathrm{\parallel }}_{g_0}^2-\mathrm{\parallel }W(I+Z)v{\mathrm{\parallel }}_{g_0}^2=4g_0(v,Zv)>0,$ so 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, for any $J_0$-antilinear $W$ with $\mathrm{\parallel }W{\mathrm{\parallel }}_{g_0}<1$, $Z=(I-W)(I+W{)}^{-1}$ satisfies that relation and $Z+Z^{\ast }=2(I+W^{\ast }{)}^{-1}(I-W^{\ast }W)(I+W{)}^{-1}>0,$ hence $J=J_{0}Z$ is $\omega$-tame. Thus the tame class is the open unit ball in the real vector space of $J_0$-antilinear endomorphisms. The strict norm estimate above is the convention-specific consequence of positive tameness needed for this extension. Extend the prescribed tame face germs in this convex coordinate, and $K$ in its affine space. Keep stationary necks and differential strips unchanged. On uncountably many labelled cells use AC after fixing lower faces; do not intersect uncountably many residual sets in one space.

Degenerate external cubes are exact pullbacks from their uniquely ordered reduced cells, never independently perturbed. Their moduli are the reduced moduli times the unused coordinates. A putative rigid coefficient would require a regular negative-index reduced solution and is therefore empty. This is only the normalized EXTERNAL cubical quotient; it does not delete bar unit words, Moore gaps or positive-duration stationary atoms.

5. Actual matched pregluing and nonlinear contraction Fix a compact family of regular matched broken configurations, with its actual component compactness and all broken levels already extracted. Each ordinary matching chord is nondegenerate, giving a fixed transverse spectral gap $\nu >0$. Keep the shared finite parameter once, all true node evaluation rows, and any actual component translation slices in the matched core operator $D_{\mathrm{b}\mathrm{r}}$. Its onto derivative has finite kernel E and an inverse on a fixed complement, uniformly after shrinking this compact component family. This is the operator made regular above; no inverse is inferred from Fredholmness alone.

Use exactly stationary product necks and boundary-adapted nonlinear charts. Component tails and their derivatives decay as exp(-nu s). Cut and preglue at length 2T. The actual equation residual in the small positive weighted $H^{k-1}$ norm is O(exp(-cT)); so are its parameter derivatives on compact core families. A parameter-dependent chord is compared in its actual finite asymptotic-base variable, not represented as a decaying vertical section. In the ordinary root construction the end generators are fixed.

Patch the component inverse and the stationary neck inverse with cutoffs varying over width W. The commutator is bounded by C/W, the coefficient/tail error by C exp(-cT). Choose W large, the weight delta small enough that its fixed cutoff factor exp(2delta W) preserves this bound, then T large. The right-inverse error has norm below 1/2; its geometric-series correction gives the ACTUAL glued right inverse. No source parameter is multiplied into a fictitious cutoff-dependent parameter: the one common variable remains, with the actual equality rows on degluing. Its tail discrepancy is exponentially small.

For a nonconstant translation-invariant strip component use a core evaluation nonzero on its translation vector. Preglue that vector ${\tau }_T$, correct by ${\tau }_T-Q_T{D}_T{\tau }_T$, and invert its finite gauge matrix I+O(exp(-cT)). Augment only the true relative translation directions. Domain-node lengths are already fixed by their source smoothing coordinates. Correct preglued E sections by the same inverse and gauge correction; their core coordinate matrix is also I+O(exp(-cT)). Consequently the fixed-length operator has exactly E as kernel and is invertible on its core-coordinate-zero complement.

In the actual nonlinear chart write $F_T(\xi )=F_T(0)+D_T\xi +N_T(\xi )$. The $H^k$ algebra estimate and bounded local geometry give

$\mathrm{\parallel }{N}_T(\xi )-N_T(\zeta )\mathrm{\parallel }\le C(\mathrm{\parallel }\xi \mathrm{\parallel }+\mathrm{\parallel }\zeta \mathrm{\parallel })\mathrm{\parallel }\xi -\zeta \mathrm{\parallel }$.

The constant is uniform in T because unit-strip Sobolev estimates have bounded overlap; all source/target coefficients are stationary on the growing neck. For each small E-coordinate solve the complement equation by

$\xi =-Q_T[F_T(0)+N_T(\xi )]$.

On a ball of radius 2C exp(-cT) this is a contraction for large T. Thus it gives an actual smooth family for finite lengths with every prescribed evaluation and parameter constraint retained. It is not an application of the inverse theorem before an inverse has been built.

If a real zero-mode neck is deliberately introduced, retain its equation $a_0^{\mathrm{\prime }}=f_0$ and endpoint drift. Fixing both endpoints requires $\int f_0=0$, a real codimension-m row. The ready neck inverse has norm O(1+T), not a false uniform bound. After the actual matched evaluation matrix is onto, the finite Schur complement and that inverse give at worst a fixed polynomial $T^N$ for a fixed finite number of necks. Multiplication by the exponential pregluing residual still gives a contraction on a radius exp(-cT/2) ball. This proves forward gluing near that supplied regular clean match, with its finite obstruction equation solved. It does not exclude additional Morse--Bott cascade limits by itself. The ordinary transverse-copy construction above does not require this extra assertion.

6. Local converse and its exact compactness input For the nondegenerate ordinary match, suppose ordinary core convergence is known and every translated stationary unit-window subsequence in the intervening neck converges to a constant matching chord after all nonconstant strip levels have been extracted. If a central window stayed away from chord neighborhoods it would have a nonconstant complete translated limit, contradicting that hypothesis. Finitely many disjoint chord neighborhoods and connectedness fix the common chord label.

In that chart choose a local Liouville gauge ${\lambda }_{\ast }$ vanishing on both transverse branes, with $\mathrm{\mid }{\lambda }_{\ast }\mathrm{\mid }\le C\mathrm{\mid }u\mathrm{\mid }$. Transversality gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L_t^2}\le C\mathrm{\parallel }{u}_t{\mathrm{\parallel }}_{L_t^2}$. The local neck action $a(s)=-\int {\lambda }_{\ast }(u_t)$ has $a^{\mathrm{\prime }}=-e$ and $\mathrm{\mid }a\mathrm{\mid }\le Ce$. For the energy on the neck with depth R removed this gives $E(R)\le -C{E}^{\mathrm{\prime }}(R)$, hence $E(R)\le E(0)\exp (-R\mathrm{/}C)$. Local small-energy regularity and elliptic bootstrapping give the corresponding derivative decay. The weighted tails therefore tend to zero; ordinary core convergence upgrades to the small weighted gluing chart.

Recover component translations with the actual nonzero core gauges and recover the relative lengths from those shifts (or from stable-domain smoothing). The complement estimate and contraction uniqueness identify every sufficiently close solution with its glued member having those lengths and E-coordinate. This proves a local inverse collar; no extra zero-translation gauge is attached to a constant strip. Corners restrict to the same fixtures. The resulting collar is continuous at $\exp (-T)=0$ and smooth at positive length; smoothness in that exponential coordinate at zero is not needed for the oriented one-dimensional boundary count.

The hypothesis that all nonconstant levels and all bubbles/escapes have already been extracted is the global compactification input. This local proof does not supply it. For clean matching, its analogue must also exclude or explicitly retain cascades and tangential levels. Calling those missing components zero modes would not remove them.

Chapter 5: Primitive-sensitive compactification with stationary growing gaps. Precise finite-stage claim Take finitely many exact conical finite-type branes and their actual transverse Hamiltonian copies, with the original end primitives and finite-rank flat systems. Fix one finite normalized ordinary/marked word and a compact external parameter cell. Source fixtures are actual stable punctured disks, anchored marked strips and their finite products/Moore gaps. They have bounded-geometry unit charts on component cores and stationary long-neck charts, and their fixed face germs are the actual merged words/products. There are finitely many marked atom blocks in this word. Stationary end generators are fixed, finite and nondegenerate; their Hamiltonians and end bases do not vary with the external parameters. Include the zero-, one- and negative-index component problems in the regular-data construction.

Assume the data are constructed as follows. On the target end $\lambda =r\alpha$, $J(r{\mathrm{\partial }}_r)=R_{\alpha }$ and J is compatible on the contact distribution. The domain Hamiltonian one-form K vanishes on $r\ge R$ and on ordinary boundary tangents. All specified generators lie below that R. A moving boundary is the actual homogeneous lift of a positive contact path, traversed nonincreasingly along its ORIENTED source boundary. Compactly supported boundary isotopies with stationary ends are also allowed. Its primitive $f_t$ is the given actual transported primitive, not renormalized separately on different ends. On every finite word the profiles have bounded total variation, all new source perturbations lie on bounded component cores, and EVERY part whose length can grow without bound is exactly stationary with $K=H_{t}dt$. Interpolations are allowed only on fixed-width anchored cores. Atomic positive durations in this finite stage are fixed, or range over a compact interval bounded away from zero; formal identities are separate source data. The fixed finite parameter/domain families and all coefficient derivatives on these charts are uniformly smooth.

Then the union of solutions of this word over every relative homotopy sector has a common compact target image and energy bound. Every sequence has, after its actual domain normalizations and extraction of finitely many ordinary strip levels, a broken convergent subsequence. All components are the recorded ordinary/marked component problems, with each external block owned once. Nonconstant sphere or single-label disk bubbles do not occur. Constant unstable strips are deleted as reparametrization artifacts. With the actual joint regular data and nonlinear inverse gluing of the analytic construction in this Proof, zero-dimensional counts are finite and the one-dimensional compactification has precisely its ordinary/domain/parameter broken boundary. No weighted convergence or no-missing-component hypothesis is being inserted into this conclusion.

Fixed-scale spiked Morse/Floer unit domains are included: their proper Morse functions and matching evaluations are fixed, Morse--Smale, and their spikes begin in the compact target window. This claim does not include an unproved limit in which the stationary transverse gap tends to zero; such a Morse--Bott/adiabatic family would need its own extraction theorem.

1. Actual compact image, including moving boundary On $r\ge R$ the equation is unforced and $a=\log r$ satisfies $\mathrm{\Delta }a=d\alpha ({\pi }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}{u}_s,J{\pi }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}{u}_s)\ge 0$. The one-form identities $d\log r\circ J=-\alpha$ and $\alpha \circ J=d\log r$ make this calculation valid for the chosen domain-dependent contact-type J as well. For an oriented boundary tangent tau and outward normal n, $jn=\tau$, so

${\mathrm{\partial }}_{n}a=\alpha (du(\tau ))={\rho }^{\mathrm{\prime }}\alpha (V)\le 0$.

The conical Lagrangian tangent contributes zero; V is the actual positive contact velocity and ${\rho }^{\mathrm{\prime }}\le 0$. Stationary boundaries give zero. Interior and boundary Hopf maximum principles therefore exclude a maximum above the end-generator radius. End convergence places any such maximum on a compact source subset, and a constant maximum component cannot connect to its lower-level interface. Hence $r\le R$ for every curve, uniformly over this actual finite cell. The complement of this end is a compact core, so this is a common compact target set C.

Compactly supported boundary isotopies are stationary on this end and obey the same argument. A transformed/gauged problem may instead be bounded by its actual proper finite Hamiltonian gauge image of C; it is not legitimate to assert the same original contact form remains adapted after an arbitrary gauge. No stop crossing is inverted in this calculation.

2. Full energy identity and uniformity in every neck length Use ${\iota }_{X_H}\omega =-dH$ and $v_s=u_s-X_{K_s},v_t=u_t-X_{K_t}$. Direct expansion gives

$\omega (v_s,v_t)=u^{\ast }\omega -d(u^{\ast }K)+R_K,R_K={\mathrm{\partial }}_s{K}_t-{\mathrm{\partial }}_t{K}_s+\omega (X_{K_s},X_{K_t})$.

Along a moving edge $u(\sigma )=i_{\rho (\sigma )}(\mathrm{\ell }(\sigma ))$,

$\lambda (du\mathrm{/}d\sigma )=d[f_{\rho }(\mathrm{\ell })]\mathrm{/}d\sigma +B_{\rho }{\rho }^{\mathrm{\prime }},B_t=\lambda ({\mathrm{\partial }}_t{i}_t)-{\mathrm{\partial }}_t{f}_t$.

Differentiating $i_t\ast \lambda =d{f}_t$ by Cartan shows $B_t=G_t$ composed with $i_t-\kappa (t)$, with one actual scalar kappa per connected brane component. Keep it. Stokes on the truncated domain yields

$E={\sum }_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}}A(x_i)-A(x_{\mathrm{o}\mathrm{u}\mathrm{t}})+\int R_K+{\sum }_{\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{s}}\int B_{\rho }{\rho }^{\mathrm{\prime }}+J_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{m}}$,

where $A(x)=\int (\lambda (\dot{x})-H_t(x))dt+f_{\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}}(x(0))-f_{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}(x(1))$, and $J_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{m}}$ is each actual before-minus-after primitive jump. There is no duplicate puncture seam contribution. For K nonzero on an ordinary boundary tangent its additional negative boundary integral must be included; the construction here takes it zero.

All functions B are bounded on the actual finite family restricted to C, as are the end actions and seam terms. Thus each edge's work is bounded by ${\sup }_C\mathrm{\mid }B\mathrm{\mid }\times$ its total variation. On bounded component cores curvature has a common integral bound. On $K=H_{t}dt$ stationary necks curvature is zero. On an anchored fixed-width interpolation $H_{q(s),t}dt$ it is sum $q_j^{\mathrm{\prime }}{\mathrm{\partial }}_{qj}H$, bounded in $L^1$ by its total variation. The same energy estimate alone would allow a profile stretched over an unbounded neck, but such a family is NOT part of this compactification theorem: its adiabatic/chord-drift limits would require a separate analysis. Consequently $0\le E\le E_0$ independently of neck lengths and relative homotopy class. A constant curvature/work error integrated over an arbitrary neck would invalidate this conclusion; the constructed stationary/product data have no such term.

No sign of an interior Hamiltonian is needed for the absolute bound. Under the actual normalized primitive and nonnegative Hamiltonian its oriented moving work is nonpositive, but this sign cannot be claimed after an independent primitive reset. All original compact/core/tail constants survive the displayed identity.

3. Gradient blow-up cannot hide a bubble in the compact window. Use the common image $C$ from67.1 and energy bound from76.1. Before choosing a rescaling, apply [F8] to the actual retained finite/merged/node family: normalize every stable marked component, retain transition-only and other multi-label polygon vertices, and cover its core and all translated stationary strip windows by the uniform disks/one-edge half-disks. A smooth transition mark is a point of the actual sitting-path trace, not an instantaneous label corner. On such a larger window maximize the derivative times the distance to its artificial cutoff edge. The original flat boundary is not an artificial edge. If gradients are unbounded this gives points of size $M_j\to \mathrm{\infty }$ with gradient at most $2M_j$ on rescaled windows of radius tending to infinity, and selected normalized derivative one. For $v_j(\zeta )=u_j(z_j+\zeta \mathrm{/}{M}_j)$, every explicit source derivative of $J$ or the moving boundary has factor $M_j^{-\mathrm{\mid }\alpha \mathrm{\mid }}$; the forcing and the actual source-dependent chart correction have factor $M_j^{-1-\mathrm{\mid }\alpha \mathrm{\mid }}$. Their unscaled norms and inverse-chart norms were derived uniformly in [F8]. Target derivatives of $J$ survive. The quantitative one-trace tangential bootstrap now gives all fixed derivative bounds after the normalized gradient bound, and diagonal extraction yields a nonconstant limiting $J_{\mathrm{\infty }}(x)$-holomorphic plane or half-plane. A boundary limit has one frozen exact brane: the other strip edge is at rescaled distance tending to infinity, and every actual marked seam has the proved smooth trace. Domain clusters were assigned their normalized recorded components before this selection; they are not discarded as one-label bubbles. The fixed-window energy comparison in [F8] makes this limit finite energy in the compact target window, with its actual transported primitive and nonzero selected derivative.

Such a plane or one-brane half-plane is impossible by direct exhaustion Stokes; no removable-singularity theorem is inserted. For a finite-energy plane choose radii $R_j\to \mathrm{\infty }$ with $R_j$ times the circle integral of $\mathrm{\mid }du{\mathrm{\mid }}^2$ tending to zero, since otherwise the radial energy integral diverges logarithmically. Cauchy–Schwarz makes those target loops short. The primitive $\lambda$ is bounded on the compact window, so its boundary integral tends to zero and Stokes forces the entire nonnegative area to vanish. For a half-plane use half-circle radii. Its brane boundary integral is the actual primitive difference at the two endpoints. The normalization in [F8] places these in the compact inverse-image window of the one limiting proper embedded exact brane and retains its primitive. Uniform continuity there makes that difference tend to zero with the short half-circle. Again area is zero, contradicting the normalized nonzero derivative. No primitive or seam constant is reset.

A sphere or an additional one-label compact disk is excluded directly by exactness in the same manner. Thus after the complete same-trace component/strip normalization of [F8], no lost derivative/energy concentration remains on the actual unit windows. The resulting gradient bounds allow the quantitative nonlinear bootstrap and smooth convergence there. This does not discard any recorded multi-label marked component, finite merged-word face, or nonconstant translated stationary Floer level; the latter remain to be extracted as stationary Floer levels. All constants concern the fixed finite bounded-geometry context and its actual compact target window, not an uncontrolled end or uniformity over every unrelated word.

4. Normalize domains first, then extract every strip level The actual finite marked domain cell has its stable compactification. After a subsequence its finite external parameters and its normalized component cores converge; a gap either has a finite limit with the prescribed merged-word fixture or tends to infinity with exactly stationary component ends. Derivative bounds from the locality construction give smooth convergence on every fixed core and every translated stationary unit window. A diagonal subsequence supplies the limiting component solutions. Independent blocks and fixed end labels restrict to their actual faces; none is duplicated.

A complete nonconstant stationary strip level has positive exact energy equal to its end-action difference. Because the word uses a fixed finite end-generator set, its positive differences have a positive minimum $e_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}}$. Zero-action strips are stationary, by the energy identity. More generally compact uniformly nondegenerate chord branches can use a uniform geometric gap: an arbitrarily small-energy complete strip would on every translated unit window converge to a stationary chord, and connectedness could not change its label between separated chord neighborhoods. The fixed-end version suffices for this construction.

Extract a nonconstant strip whenever one translated intervening window has a nonstationary complete limit. Each extraction consumes at least $e_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}}$ of the total E0. There are therefore at most $\mathrm{floor}(E_0\mathrm{/}{e}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}})$ such levels. Recenter them by actual nonzero core translation gauges and repeat on the remaining intervals. After finitely many extractions EVERY translated limit on every remaining central neck is constant. This proves the no-missing-stationary-component input of local inverse gluing, rather than assuming it. A zero-length merger is a prescribed finite-domain face; it does not create an unrecorded extra strip equation.

The number of nonstationary marked components is bounded by the fixed atom blocks; a block is anchored and cannot be moved to infinitely many stationary levels. Stable ordinary disk vertices have their finite boundary punctures; after deleting unstable constant components the stable tree has finitely many such vertices bounded by its leaf/marked-vertex count. Trivial constant two-ended strips are deleted, not assigned translation gauges. At a source degeneration a multi-label disk is a retained ordinary vertex. These observations cover every possible component after the no-bubble argument; no marked, stationary, or zero-length sector is discarded by its name.

For a fixed proper Morse spike, downward trajectories stay in the compact sublevel through their bounded starting evaluation. A Morse--Smale gradient trajectory can break only at the finite critical list; the strictly decreasing function bounds its number of nonconstant levels. Constant levels are the prescribed matching/identity configurations. Thus fixed-scale Morse spikes add their true finite Morse faces, without requiring a Floer transverse spectral gap at an artificial zero-Hamiltonian end.

5. Actual neck containment and exhaustion of the local gluing charts After Section4 every sufficiently central neck window lies close to a stationary matching chord: otherwise a translated limit outside the disjoint chord neighborhoods would be nonconstant or nonstationary. Connectedness fixes its chord label. Transverse chord charts then give a local Liouville form ${\lambda }_{\ast }$ vanishing on both edge branes. With e(s) the slice energy and a(s) its signed neck action, one has $a^{\mathrm{\prime }}=-e$ and $\mathrm{\mid }a\mathrm{\mid }\le Ce$ by the transverse path Poincare estimate. Therefore the middle-tail energy satisfies $E(R)\le -C{E}^{\mathrm{\prime }}(R)$, yielding exponential decay. Local elliptic estimates lift it to derivative and small positive weighted norms.

Ordinary convergence thus lies in the actual matched weighted charts of the nonlinear-gluing construction in Chapter 4. Its core inverse, parameter/equality blocks and translation gauges supply the unique glued solution for the recovered lengths and component coordinates. Every sequence near a regular matched limit is in one of those collars. No claim of a uniform inverse for an unresolved real mean mode is needed: the ordinary end model is nondegenerate. If a clean/Morse--Bott end or an extra tangential cascade is inserted, its extraction and equality constraint must be added; the present proof cannot delete it by calling it a parameter.

For varying supported Hilbert perturbations in a closed bounded ball, the rapidly rescaled smooth basis has uniformly summable $C^m$ tails. The resulting data are precompact in every $C^m$ on the finite core charts. A diagonal subsequence converges smoothly, with the unchanged product ends. Thus the same extraction applies in relative-collar induction before choosing a generic current interior value. There is no circular assumption that the current top moduli are already regular in order to prove their compactness.

6. Finiteness, all sectors and the actual boundary Energy, image and extraction bounds did not depend on a relative homotopy label. The union over all sectors is consequently sequentially compact in the broken topology. Near an unbroken limit the class is locally constant in its mapping chart; near a broken regular limit the actual inverse-gluing collars have the recorded finite gluing branches. An infinite rigid sequence cannot evade compactness by changing its relative class indefinitely. For joint regular zero-dimensional data, each point is isolated; a putative broken/parameter-face limit would have negative expected dimension after the actual matching/face codimension, and is absent by the required component regularity. Hence its rigid set is finite.

For a one-dimensional problem the same argument gives a compact one-manifold with boundary after the prescribed corners/unstable components have been accounted for. Its boundary is precisely the ordinary strip/polygon breaks, actual parameter faces, finite merged-word faces, and the recorded Morse faces. Higher-codimension strata have negative expected dimension in this range. The actual local inverse collar gives multiplicity one for each regular boundary branch. With the coherent relative-Spin determinant system, parameter-first/outward-normal-first convention and parallel coefficient transport, its signed boundary count is the desired ordinary or history identity. External degeneracy cells retain their unused free coordinates and contribute no rigid counts, as in the regular-data supplier.

This is a full finite-word compactification result for the declared ordinary/positive marked data with exactly stationary growing necks. The history model may and here does choose all unbounded Moore gaps in this manner; a bound on total profile variation alone is not a substitute. It is not a derived-localization proof, a uniform compactness claim across all unbounded word lengths at one target radius, or a 6D critical trace/cap comparison. Coherent localization can consume these finite compatible counts and their histories; the independent attachment and generation holds are unchanged.

Chapter 6: Complete self complexes, proper Morse/PSS units and relative Spin determinants. 1. The full self model and its proper Morse data For each connected finite-type brane $L$, choose a finite Morse handle function on its compact core and extend it properly, bounded below, and strictly increasingly along each cylindrical end, retaining the prescribed end germ and all critical points. On a compact component keep the top handles. The end metric may first be chosen so the gradient is complete, and is then fixed, as are the function and metric near every critical point. For example start with a complete cylindrical metric $g_0$ and on the ends multiply it by a smooth factor at least $1+\mathrm{\mid }df{\mathrm{\mid }}_{g_0}^2$; the resulting metric is complete and its gradient speed is bounded, without changing the function or its end constants. The Morse--Smale choice required here has the following compact-support construction. Choose a regular value $a$ above all critical values. Every negative-gradient trajectory connecting two critical points lies in the compact sublevel $K=\{f\le a\}$, because $f$ decreases along it and is proper and bounded below. Fix disjoint small critical neighborhoods. Allow metric perturbations supported inside $K$ and outside those neighborhoods; all end germs stay fixed. Choose a countable dense list $s_j$ of smooth symmetric tensors with such supports, and positive weights $w_j$ with $w_j\mathrm{\parallel }{s}_j{\mathrm{\parallel }}_{C^k}\le 2^{-j}$ for $k\le j$. The map $(c_j)\in {\mathrm{\ell }}^1\mapsto {\sum }_j{c}_j{w}_j{s}_j$ takes values continuously in every $C^k$ space, hence in smooth tensors. The open subset on which $g+{\sum }_j{c}_j{w}_j{s}_j$ is positive definite is a separable Banach parameter space of smooth metrics, and its tangent directions contain every finite weighted linear combination of the chosen tensors. For each ordered pair of critical points use trajectories with those fixed limits in the $W^{1,2}$ path chart and the equation $\dot{\gamma }+{\mathrm{\nabla }}_{g}f=0$ with $L^2$ target. Its linearization is a first-order operator ${\mathrm{\partial }}_s+A(s)$ with hyperbolic limiting Hessians. On the two half-lines the positive/negative eigenprojections and variation-of-constants integrals give bounded inverses after their finite-dimensional initial-value spaces are retained; matching across a finite interval is a finite-dimensional linear map. Thus the trajectory derivative is Fredholm, and its cokernel consists of solutions of the adjoint ordinary differential equation. If such a nonzero adjoint solution annihilated all metric variations, choose a regular point of the trajectory away from the fixed critical neighborhoods. The trajectory crosses a small neighborhood of that point only once, since $f$ is strictly decreasing there. At a noncritical point $\mathrm{\nabla }f\mathrm{\ne }0$, a symmetric metric variation can change ${\mathrm{\nabla }}_{g}f$ in any tangent direction: the linear map $b\mapsto b(\mathrm{\nabla }f,\cdot )$ from symmetric tensors to covectors is onto. A smooth tensor supported in that neighborhood can therefore pair nontrivially with the adjoint solution. The dense tensor list has a finite linear combination with the same nonzero pairing, a contradiction. The universal trajectory derivative is consequently onto. The universal solution space and its projection to the Banach metric parameters satisfy [F3]; regular parameters make the fixed-metric trajectory derivative onto, equivalently the stable and unstable manifolds transverse. There are finitely many ordered critical pairs; separable path charts require at most countably many regular-value conditions. Their intersection is residual in the Banach parameter neighborhood by [F3] and Baire, and is nonempty. It supplies smooth Morse--Smale data with exactly the fixed end and critical germs, rather than applying a closed-manifold metric theorem to $L$. Constant trajectories have the usual complementary Hessian stable/unstable spaces. All relevant downward spikes from a fixed compact evaluation set likewise remain in one compact sublevel. One may arrange one minimum when compatible with the prescribed germs, but this is unnecessary: keep all minima and all other handles. Disconnected branes use this construction componentwise. These choices do not assert $H^{\ast }(L)=\mathbb{C}$.

In an exact conical Weinstein neighborhood the graph of a sufficiently small differential gives the actual Hamiltonian copy. The neighborhood is built from the symplectic normal identification with T*L and relative Moser correction; on an end use the homogeneous extension of the compact Legendrian chart. No global framing of TL is required. The original cotangent/symplectic sign is transported explicitly so the copy is the chosen positive Morse convention. Its end slope can be less than the positive short-self-Reeb-chord bound: compactness and the contact flow-box construction give that bound for an embedded Legendrian. Thus there are no accidental end intersections, and the self generators are the full finite critical list with their orientation and End(E) lines.

The small-copy strip problem splits into its constant tangential Fourier mode and its nonzero transverse modes. The nonzero modes have their fixed strip Fourier gap; the tangential equation after rescaling is the actual Morse gradient equation. Retain that gradient equation and its matching evaluations, rather than calling its mode invertible. The Morse--Smale derivative and the nonzero-mode inverse form the matched operator. Pregluing its trajectories/constant pieces gives a small residual, contraction on the complement gives the actual nearby strips, and the rescaled gradient compactness gives the converse with all Morse breakings. Equivalently use the actual Morse-to-Floer half-strip construction below; it avoids asserting that a selected finite-copy differential literally equals the Morse differential without its comparison.

2. The closed identity with arbitrary flat coefficients On the full Morse complex with coefficient End(E), let $e_M$ be the sum of all minimum generators with coefficient Id on a connected component. It is closed. Every one-dimensional Morse cell has two downward endpoints among those minima with opposite boundary orientations. Transport acts on End(E) by conjugation, so both endpoint coefficients are Id in their respective endpoint lines even when E has nontrivial holonomy. The two contributions cancel. This uses neither rank one nor triviality of the flat system. Proper downward flow cannot escape to an end since it stays in a compact sublevel.

Define the Morse-to-Floer map by spiked half-strips: one gradient edge from the selected Morse input meets the actual boundary evaluation of a Floer half-strip; on its positive strip end use the fixed self-copy equation. Choose exactly the same unit copy and primitive at its seam, so the action identity includes their actual Hamiltonian and Morse terms. The nonzero Fourier modes and the Morse evaluation matching are the operators in Section1. For a fixed finite problem their zero- and one-dimensional moduli are regularized by the joint source-dependent construction, with the end equation and Morse faces fixed. Properness and the ordinary compact-image/energy bounds keep the spikes in compact sublevels. The compactification has exactly Morse breaks, Floer breaks and its fixed end/collar faces. Counting its one-dimensional boundary gives

$d_F{\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}={\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}{d}_M$.

Set $e_F={\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}(e_M)$. Then $d_F{e}_F=0$. This is an ACTUAL closed cochain, not a declaration that an arbitrary isolated minimum of an unknown differential is closed. Any small-copy corrections in ${\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}$ are retained. Applying the reversed half-strip and the same marked-length interpolation gives the usual two comparison homotopies by the same broken-face count; no non-unit Morse generators are contracted away in this process.

3. Two genuine unit homotopies For a finite pair X,Y fit the chosen Morse data so all its finitely many transverse evaluation generators lie in the union of the minimum basins; the higher-codimension Morse stable manifolds can be avoided by generic choices without changing the fixed end germs; the higher stable manifolds can be moved off this finite set. Use the whole self complex and the same ${\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}$, not a new complex for each generator.

To compare ${\mu }_2(e_Y,x)$ with x, use the ordinary binary disk with its Y-side self-copy input, replace that input by the spiked half-strip, and vary the length of the resulting unit flag. Give every broken face its already fixed data. At one end of this ONE-parameter domain cell it is the actual product ${\mu }_2({\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}(e_M),x)$. At the other it is the strip with the negative-gradient line from x to its unique basin minimum and constant holomorphic pieces. That line is unique because x is in exactly one minimum basin. Its normal Fourier problem is invertible, its Morse evaluation is transverse and its matched gluing gives one thin cap with coefficient +1 and identity coefficient transport. It is the actual identity reference for the input/output orientation line. Thus the second end is the identity chain map.

The other faces are precisely the input/output differential breaks and Morse breaks. The latter are the already included chain-map term ${\mathrm{\Phi }}_{\mathrm{M}\mathrm{F}}(d_M{e}_M)$, which is zero. All nonconstant ordinary polygon and strip breaks remain in these operations and homotopies; they are not discarded by saying their area is positive. The compact one-dimensional boundary equation is

${\mu }_2(e_Y,-)-id=d{K}_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}+K_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}d$,

in the fixed unsuspended unit convention. The analogous cell with the flag on the other side gives

${\mu }_2(-,e_X)-id=d{K}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}+K_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}d$.

Its local determinant reference is transported from the SAME unit strip; the convention fixes its sign, rather than choosing an independent unit sign on the two sides. This proves the two cohomological unit equations on every morphism complex.

For disconnected branes take the sum of component identities. Inputs between different components have the usual matching block zero, so the sum acts as the identity on the full complex. For any finite collection the data and the marked faces can be selected coherently; additional unit flags use the same product domain collars and joint regularization. No arbitrary individual fixed external parameter value is asserted regular.

These constructions require the actual global compactification already specified at the start. Local gluing alone cannot prove that the displayed list of faces is exhaustive. In particular the argument does not transfer a 4D ordinary unit to the unresolved 6D critical attachment by notation.

4. Real relative-Spin determinant orientations and labelled caps. Work on the actual smooth realizations of order $k\ge 3$, so the node and Morse evaluations used below are bounded; their smooth kernels and cokernels identify with the weak realizations by [F2,F10]. For a real Fredholm operator put $\lambda (D)={\mathrm{\Lambda }}^{\max }\ker D\otimes ({\mathrm{\Lambda }}^{\max }\mathrm{coker}D{)}^{\ast }$, of parity ${\mathrm{ind}}_{\mathbb{R}}D$ modulo two, and let $\mathfrak{o}(D)$ be the free rank-one orientation module with opposite orientations identified by a minus sign. Tensor transpositions use these real parities and the already declared grading shifts. Determinant transports constructed with auxiliary choices are compared up to positive homotopy; the resulting maps on orientation modules are literal, and no choice of a norm or positive scalar is asserted canonical. At every actual fixed transverse chord choose once a one-punctured disk cap: its strip model is the actual fixed end model of [F2], and its remaining boundary arc is a smooth path ${\mathrm{\Gamma }}_x$ of totally real planes connecting the two end planes, with sitting ends and a specified extension of the endpoint Spin lifts. Such a cap is a choice, not an orientation inferred from complex linearity: in the product-angle frame connect the planes by the coordinate rotations, choose their integer turns and the given grading shifts as part of this reference, and interpolate the operator on its compact core. The compact bordered parametrix of [F10] and the transverse strip inverse of [F2], patched by cutoffs, make this actual cap operator real Fredholm. Its orientation line, together with the declared shift, is the reference $o_x$; an arbitrary generator is not made positive by notation. Glue the chosen complementary caps to the strip ends in the fixed order: original component first, then caps in the actual boundary cyclic order starting at the output. The needed TRANSVERSE STRIP-END sewing needs no zero-mode matching factor. On its actual fixed-angle end each Fourier mode is $a^{\mathrm{\prime }}+(\pi j-\theta )a$, with spectral gap $\gamma >0$ from [F2]. Its full-strip inverse has uniformly bounded norm. For this strip comparison first use the weak $H^1\to L^2$ realization of [F2]. Smooth compactly supported targets are dense; their images span the finite cokernel, so finitely many give a common core $W$ making the disjoint operators onto. Restriction of an augmented kernel vector to its fixed core, together with its $W$ coordinate, is injective: zero core data make every homogeneous end Fourier coefficient zero. Choose finitely many core dual functionals as coordinates $L_0$ on that finite-dimensional kernel $K$. On a neck of length $T$ use nested core/neck cutoffs whose derivatives are $O(T^{-1})$, the disjoint core inverses and that full-strip inverse. Extend all cutoff sections in their actual real trace spaces. The patched operator for $(D_T+W,L_T)$ has two-sided inverse errors bounded by $C\mathrm{/}T+{\sup }_{s\ge T\mathrm{/}8}\mathrm{\mid }B(s,t)\mathrm{\mid }$; multiplication by a cutoff commutes with the principal part up to its first derivative, and all core cutoffs are one on the support of $W,L_0$. This estimate follows on each piece from the already proved inverse estimates, then from the finite partition whose squared or nested cutoff products sum to one. The same localization applied to a putative kernel with $L_T=0$ gives the left estimate, so it does not assume the desired kernel dimension. The finite augmentation identity is $\lambda (A)=\det \ker (A+W)\otimes \det W^{\ast }$, obtained by cancelling ordered quotient lifts in $0\to \ker A\to \ker (A+W)\to W\to \mathrm{coker}A\to 0$. Here $A_W(x,w)=Ax+w$ and the map from its kernel to $W$ is $-w=Ax$, with the usual class map from $W$ to the cokernel. For sufficiently large $T$ the Neumann series makes $(D_T+W,L_T)$ an isomorphism onto $Y_T\oplus K$. Equivalently $U_{T}k=(D_T+W,L_T{)}^{-1}(0,k)$ is the actual augmented-kernel identification. Its core coordinate is exactly $k$; errors on preglued kernel sections also decay by the mode gap. Dividing out the same ordered $W$ constructs the strip sewing determinant map. Weak kernel/cokernel regularity and the common smooth forcing identify these determinant lines with the actual higher realizations; no point evaluation on $H^1$ was used in this strip argument. This applies to the chosen caps and to actual broken transverse Floer strips, with no clean moving-asymptote or zero-mode inverse silently assumed. This produces a compact bordered bundle pair $(\widehat{E},\widehat{F})$; its boundary is still REAL. Pull the actual common background bundle back to the disk, stabilize by its complexification with real boundary values, and use the actual relative-Spin lifts on the brane arcs and the chosen cap lifts to give the stabilized boundary loop its Spin structure. Add an oriented trivial summand if needed to reach rank at least three. The relevant construction in [F10, relative-Spin determinant proof2.1–6.1] is a construction on this bundle pair: it does not require all the arcs to be images in one globally framed Lagrangian. Here are its actual factors. A Spin section along the boundary projects to an oriented frame of the stabilized real boundary bundle. Extend that frame across a collar and pinch an INTERIOR circle of that collar. The normalization is a standard real-boundary disk, with boundary plane ${\mathbb{R}}^N$, and a closed complex piece carrying the remaining clutching; their matching fiber is ${\mathbb{C}}^N$. The real disk operator is surjective and has kernel the REAL constants ${\mathbb{R}}^N$, oriented by evaluation in that boundary frame. The closed-piece determinant is oriented by deformation to a complex-linear operator, and the matching fiber has its complex orientation. Thus the node exact sequence orients the compact stabilized determinant from the ordered factors $\lambda (D_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}})\otimes \det ({\mathbb{R}}^N)\otimes \det ({\mathbb{C}}^N{)}^{\ast }$. It is not a complex orientation of the original real boundary problem. The auxiliary real operator has real constant kernel equal to the oriented background fiber and zero cokernel; divide its determinant out in that fixed order. The end-cap determinant identity then defines the original operator's orientation with its external cap lines left visible. This reuses only the real disk/frame/pinching and determinant construction of [F10]; it does not import that supplier's separate Morse degree $q=n-r$ or its insertion-sign convention. The retained proper Morse cochains, minimum degree zero, suspensions remain unchanged. Choose the positive output reference in which the output-strip face contracts to $+D_{N}c$ and the source-module face to $-c{D}_M$.

Choice independence and actual cap comparison. The preceding construction is independent of its auxiliary analytic choices in the following precise sense. For a real Fredholm family $A_b$, choose finitely many smooth core forcings $W$ making $A_b+W$ onto near one base point. The exact sequence $0\to \ker A_b\to \ker (A_b+W)\to W\to \mathrm{coker}{A}_b\to 0$ identifies $\lambda (A_b)$ with $\det \ker (A_b+W)\otimes \det W^{\ast }$. The map $\ker (A_b+W)\to W$ is minus the forcing coordinate, equal to $A_{b}x$; this fixes the plus-augmentation convention. Explicitly put $I=W\cap \mathrm{im}{A}_b$, choose a kernel basis $a$, lifts $k_i$ of a basis $i$ of $I$, and vectors $c$ in $W$ representing a cokernel basis, with $W$ basis $(i,c)$. The map sends $(a\wedge k_i)\otimes (i\wedge c{)}^{\ast }$ to $(-1{)}^{\dim I\dim \mathrm{coker}{A}_b}a\otimes c^{\ast }$: the dual order is $(c^{\ast },i^{\ast })$, and moving $i^{\ast }$ past $c^{\ast }$ gives that sign. Changing a lift is triangular determinant one; changes of $i$ occur once and dually and cancel. The quotient in the enlargement sequence below is likewise minus the added forcing coordinate. If $W\subset W^{\mathrm{\prime }}$ use $0\to \ker (A_b+W)\to \ker (A_b+W^{\mathrm{\prime }})\to W^{\mathrm{\prime }}\mathrm{/}W\to 0$ and cancel the added ordered factor with its dual. Arbitrary $W,W^{\mathrm{\prime }}$ compare through $W+W^{\mathrm{\prime }}$. These cancellation identities are associative, so charts with kernel/cokernel jumps still have the same determinant-line transport. The smooth forcings and joint evaluation augmentations exist at this scope by [F10, bordered lemma8.1–12.1]. Changing collars, compact coefficients or pinching radii is handled by the same finite augmentation over the parameter interval and preserves the chosen real disk, complex closed-piece and matching orientations. Two Spin-compatible stabilized boundary frames differ by a closed loop in $\mathrm{Spin}(N)$, not by an arbitrary loop of oriented frames. This loop is nullhomotopic for $N\ge 3$: the quaternion description gives $\mathrm{Spin}(3)=S^3$, and the elementary rotation charts give $\mathrm{Spin}(N-1)\to \mathrm{Spin}(N)\to S^{N-1}$; induction contracts loops. Thus the projected frames are homotopic and induce the same real disk orientation. A real background trivialization extends over the disk, so its boundary change is nullhomotopic; a further oriented stabilization contributes the same real reference factor and its dual, which cancel. For families one needs no background bundle over the whole target or a global tangent framing: disks and one-parameter loops use precisely the given three-skeleton background and boundary two-skeleton Spin lift, as in [F10, relative-Spin proof1.1,4.1]. Every loop of the determinant local system is such a one-parameter test; the pulled-back Spin bundle over its boundary torus has a section because its structure group is connected and simply connected. The orientation therefore patches over finite cubical families and their face charts. Changing the actual transported Spin datum is not one of these auxiliary changes.

Actual Fredholm sewing and ordered finite matches. For normalization operators $P={⨁}_v{P}_v$ and the onto difference evaluation $\mathrm{\Delta }$ to the ordered finite real matching space $V$, choose disjoint smooth core lifts $R$ with $\mathrm{\Delta }R=1$. Such lifts exist in the actual higher realization by [F10]; at a real boundary point use a real trace frame, and at an interior point use a complex frame. Then the REAL exact sequence is $0\to \ker (P{\mathrm{\mid }}_{\ker \mathrm{\Delta }})\to \ker P\to V\to \mathrm{coker}(P{\mathrm{\mid }}_{\ker \mathrm{\Delta }})\to \mathrm{coker}P\to 0$, with connecting map $v\mapsto [PRv]$. A different $R$ changes that vector by the matched range. Lifting ordered quotient bases gives $\lambda (P{\mathrm{\mid }}_{\ker \mathrm{\Delta }})\otimes \det V\cong {⨂}_v\lambda (P_v)$, with all factors in their recorded order. In a common-$W$ reduction the ambient expression is $(k,n)\otimes w^{\ast }$; contracting a dual normal moved past $w^{\ast }$ contributes $(-1{)}^{\dim V\dim W}$. An enlargement by $U$ contributes the compensating $(-1{)}^{\dim V\dim U}$ when its new kernel lifts move past $V$, so the matching map is independent of the chosen common reduction; obstruction, incidence and shared-parameter blocks are included by the same sequence, each once. Retain the history's $\mathrm{\Delta }(v_1,v_2)=v_2-v_1$. For the real constant reference, the diagonal basis $(e_i,e_i)$ followed by the lift $(0,e_i)$ has block matrix $\left(\begin{array}{cc}I& 0\\ I& I\end{array}\right)$ and determinant $+1$. Using $v_1-v_2$ instead changes the matching normal by $-I_V$, and costs $(-1{)}^{\dim V}$; it is not the same convention for free. At an interior node $V$ is complex and has even real dimension; at a boundary/Morse match it is genuinely real. The smooth-node determinant transport, including all partial nodal strata, is the actual augmented-kernel map of [F10, weighted nodal lemma23.1 and supplementary determinant construction]: add a common core $W$, preglue a kernel vector and correct it by $I-Q_q(D_q+W)$; its fixed core coordinate gives the kernel identification and the SAME ordered $W$ is divided out. No orientation is inferred merely from equality of index or from the words complex operator. For transverse strip ends use the actual fixed-gap construction in108.1, with no finite zero-mode matching factor. Complements, cutoffs, sufficiently long lengths and smooth fixed-end coefficient deformations compare by the common-augmentation homotopy. For several nodes/ends add all $W$ and matching spaces at once, in their fixed order. Joint elimination uses the ordered quotient list $(N_1,N_2)$; sequential elimination in the opposite order gives $(N_2,N_1)$ and the explicit wedge factor $(-1{)}^{\dim N_1\dim N_2}$. Braiding the two dual matching factors contributes the same factor and restores the fixed joint order. In that fixed order block Gaussian elimination is associative. The corresponding analytic kernel maps have the same limiting core-coordinate map and remain in its positive component for sufficiently long necks. This proves associativity and naturality of the orientation-module sewing, also for mixed real/complex nodes and strip caps. Reordering independent odd factors gives exactly their Koszul sign; it is not declared to be orientation-preserving.For two independently chosen references $C_x^a,C_x^b$ for the SAME labelled, graded, fixed end datum, define the comparison ${\kappa }_x^{b\leftarrow a}$ by gluing $C_x^a$ to the complementary cap of $C_x^b$ and applying the compact REAL pinching construction of108.1, with the same ordered background and evaluation factors. This defines an isomorphism of orientation modules, not an unexplained positive scalar. A cap followed by its complementary cap has the doubled boundary path $\mathrm{\Gamma }\mathrm{\#}{\mathrm{\Gamma }}^{-1}$; contract that path with its actual Spin lift to the constant boundary path. The resulting compact disk is the real reference disk, whose determinant is oriented by real evaluation. For this nondegenerate strip-end sewing there is no finite matching normal: $\lambda (C_x^{-})\otimes \lambda (C_x^{+})\cong \lambda (R_x)$, where $R_x$ is that compact real reference disk. Its evaluation identifies $\lambda (R_x)$ with $\det {\mathbb{R}}^n$, so $\mathrm{ind}{C}_x^{-}+\mathrm{ind}{C}_x^{+}=n$ and $\lambda (C_x^{+})\cong \lambda (C_x^{-}{)}^{\ast }\otimes \det {\mathbb{R}}^n$, with the recorded graded order. Thus the normalized opposite cap is $\lambda (C_x^{+})\otimes (\det {\mathbb{R}}^n{)}^{\ast }$, not the raw opposite cap declared dual without its real disk factor. Only after this factor and its degree have been retained and cancelled is the prescribed dual evaluation available; reversing its order incurs the actual graded braid. If one instead represents a cap sewing by a marked evaluation normal $N$, its exact sequence contains the additional $\det N$ factor, which must also be retained, not counted as zero. In the retained four-dimensional target the physical rank is two; auxiliary ranks cancel before this grading statement, and the given cohomological shifts are not changed. The ordered associativity just proved cancels the middle cap against its complementary cap and proves ${\kappa }^{c\leftarrow b}{\kappa }^{b\leftarrow a}={\kappa }^{c\leftarrow a}$, with the actual declared grading permutations. If $b_x^a,b_x^b$ are separately selected generators, write ${\kappa }^{b\leftarrow a}(b_x^a)={ϵ}_{ba}{b}_x^b$, ${ϵ}_{ba}\in \{\pm 1\}$. This sign is retained; neither an interval nor an endpoint name proves it is $+1$.