Toric line-bundle subcategory mirror theorem

Let N be the fan lattice, M its dual, and Sigma a smooth complete projective fan. Let Q in $M_R$ be a simple moment polytope with normal fan Sigma. For definiteness its facet $F_{\rho }$ has outward primitive normal $v_{\rho }$ and inequality $<x,v_{\rho }>\le {\lambda }_{\rho }$; changing all normal conventions changes the displayed pole function accordingly, but not the construction. Put $k=C$ (the construction also works over any field). For a torus-invariant Cartier divisor $D=\sum$ $a_{\rho }$ $D_{\rho }$ write $L_D=O(D)$. Its weight-m rational section is ${\chi }^m$, with order In formulas: ord_(D_rho)(chi^m as a section of L_D)=<m,v_rho>+a_rho..

Thus the positive/pole function is $h_{D,m}(F_{\rho })=-a_{\rho }-<m,v_{\rho }>$. Set In formulas: B(D,m)=union of facets F_rho on which h_(D,m)>0..

It is a closed subcomplex of the face decomposition, and B(D+E,m+n) is contained in B(D,m) union B(E,n). The strict inequality is essential: order zero is permitted, so it is not part of the pole locus.

Weight sections on every intersection. For a cone sigma the invariant affine open is $U_{\sigma }=Spec$ $k[{\sigma }^{\vee }capM]$. Since Sigma is smooth, choose a basis extending its primitive ray generators. Then its ring is a polynomial ring in dim sigma coordinates times Laurent polynomials in the remaining coordinates. On $U_{\sigma }$ the divisor D has a Cartier character $m_{\sigma }$ with $<m_{\sigma },v_{\rho }\ge -a_{\rho }$ for $\rho$ in sigma(1). Consequently In formulas: Gamma(U_sigma,L_D)=chi^(m_sigma) k[sigma^vee cap M],.

and its weight-m subspace is k ${\chi }^m$ if $<m,v_{\rho }>+a_{\rho }\ge 0$ for EVERY $\rho$ in sigma(1), and zero otherwise. This formula follows by writing a monomial in those polynomial and invertible coordinates; linear independence of Laurent monomials shows that no cancellation can make a prohibited weight regular. A face inclusion of cones localizes a subset of those coordinates. The restriction on an allowed weight is the identity $k\to k$, and a zero weight module has its unique zero restriction. Products of sections send ${\chi }^m$ tensor ${\chi }^n$ to ${\chi }^{m+n}$, with coefficient 1.

Intersections are $U_{\sigma }$ intersection $U_{\tau }=U_{\sigma cap\tau }$, also for finitely many cones. In particular the weight condition on an INTERSECTION is the condition for its COMMON CONE. It is not the condition that the weight extend to each original affine open separately. This follows immediately from localization, and fixes every restriction map needed by the ordered Cech differential.

Every algebraic line bundle is represented this way, up to a character. To verify this without assuming an equivariant linearization theorem, restrict it to the dense torus. The Laurent polynomial ring is a factorial domain (Gauss's primitive polynomial argument followed by localization); on a regular factorial affine scheme every Cartier divisor is principal, so this restriction is trivial. Choose that trivialization as a rational section. Its divisor on X is supported on the complement of the torus, hence is an integer combination of the finitely many $D_{\rho }$. Conversely the displayed local characters make each such divisor Cartier because the rays in every smooth cone extend to a basis. Two boundary divisors differing by a character have isomorphic line bundles; conversely a rational function with divisor supported only on the boundary is an invertible function on the torus, hence c ${\chi }^m$ by comparing the units of the Laurent polynomial ring. This proves precisely the required object classification.

The actual Cech dg category. Index the invariant affine cover by all nonempty faces tau of Q, including Q itself: $U_{\tau }$ means the affine open associated to the normal cone of tau. All these opens are permitted; the vertex opens already cover X. An intersection $U_{\tau 0}$ intersection ... intersection $U_{taup}$ is $U_{\tau 0join\mathrm{.}\mathrm{.}\mathrm{.}jointaup}$, where join is the smallest face containing the listed faces. It is connected, affine, and its weight-m sections are either the one copy of k above or zero. Define In formulas: C^p_X(D,E)=direct sum_(tau0<...<taup); Gamma(U_tau0 intersection ... intersection U_taup,L_(E-D))..

Use alternating deletion followed by the actual localization maps for the differential. Composition is the ordered Alexander--Whitney cup product followed by monomial multiplication; an input of degree p and one of degree q evaluates on the front p+1 and back q+1 indices, sharing the middle index. Restrictions commute with multiplication. Expanding alternating deletion gives $d(ab)=(da)b+(-1{)}^p$ a(db); splitting an ordered index string at two successive positions gives associativity. Thus this is an actual dg category, not only a cohomology multiplication. It splits as a direct sum over M in each morphism complex, and products add weights.

This cover gives the usual line-bundle dg enhancement: finite intersections are affine and the line bundles are quasi-coherent there. The elementary affine acyclicity check used here is the localization Cech contraction. For a finite principal cover $D(f_i)$ of an affine, clear denominators in a given cocycle and choose N with $1=\sum$ $b_i$ $f_i^N$; insertion of sum $b_i$ $f_i^N$ in the first index contracts the alternating localization complex. Such principal covers form a basis and refine finite affine intersections. The augmented local Cech resolution is stalkwise split by inserting any index containing the stalk. Its double complex therefore identifies the displayed Cech complex with derived global sections of $\mathrm{Hom}(L_D,L_E)$; the preceding contraction gives acyclicity on its intersections. The same restriction double complex proves refinement invariance. These are finite-diagonal cochain eliminations, so no assertion that a bare cohomology isomorphism preserves dg products is being used. The products are already the same restriction-and-multiplication products in the double complex.

Replace algebraic weights by a sheaf on Q. For a union B of facets let $S_B$ be the extension by zero of the constant sheaf k from Q minus B, equivalently the relative locally constant sheaf. On a connected open star $O_{\tau }$ of a face tau, it has sections k if tau is not contained in B and zero if tau is contained in B. Indeed $O_{\tau }$ consists of relative interiors of faces containing tau. It and its intersection with B are contractible when the latter is nonempty: choose a point in the relative interior of tau and contract each incident face linearly to it. Only bad facets containing tau occur; these are preserved by the contraction. In the empty case only the first contraction is needed. A locally constant function on the connected complement which must vanish near a nonempty incident boundary facet is zero. The same description works for intersections of stars, since their intersection is the star of the join face.

The relative singular cochains of $(O_{\tau },O_{\tau }\bigcap B)$ therefore have cohomology k in degree zero if the second space is empty and zero otherwise. One can prove this directly with the integral prism homotopy of the stated contractions and then tensor with k. Their restriction diagrams identify this degree-zero sheaf with $S_B$. Consequently the open-star cover is acyclic for $S_B$. Its ordered Cech complex is EXACTLY the weight-m Cech complex of $L_D$: the common join face is prohibited precisely when some bad facet contains that face, which says precisely that the relevant common cone contains a ray with negative order. The nonzero modules, restrictions and products all agree.

The canonical sheaf multiplication is In formulas: S_(B(D,m)) tensor S_(B(E,n)) -> S_(B(D+E,m+n))..

It sends 1 tensor 1 to 1 wherever both factors are present, and is zero where either is absent. It is well defined because B(D+E,m+n) is contained in the union of the other two pole loci. These maps are associative on stalks and compatible with restriction. This also proves the multiplication comparison, not just the weightwise additive one.

A relative barycentric cochain model with the correct indexing. Let P be the barycentric subdivision of Q. A simplex is a chain tau0 properly contained in ... properly contained in taup. Let $B^b$ be the barycentric subcomplex of B. Choose a sufficiently small thickened dual-cell cover $W_{\tau }$ refining $O_{\tau }$. Its nonempty intersections are indexed by chains of faces; each intersection is contractible. It meets B precisely when the entire chain is in B, equivalently when its largest face is contained in B. This follows either from the dual-cell local coordinate model or by successively making the thickening widths smaller than the distance from a dual cell's essential part to every nonincident facet. There are finitely many faces, so all these distances can be enforced simultaneously. A nested-chain intersection meeting B contracts to the corresponding flag inside B, so it too is an acyclic relative intersection.

Its $S_B$ Cech complex is therefore In formulas: C^*(P,B^b;k)={cochains vanishing on simplices entirely in B^b}..

The differential is the relative simplicial differential and the product is precisely the front/back Alexander--Whitney product. If a simplex is contained in $B_1$ union $B_2$, it is contained in one of these subcomplexes, so at least one of the relevant front/back factors vanishes; thus the relative product maps to the union-relative complex. Restriction to the smaller bad locus B(D+E,m+n) gives the desired composition. On flag generators the Cech and simplicial formulas are literally identical, including their alternating signs.

The SAME face index and ordering give a strict dg functor, identity on divisors, In formulas: Cech_X = Cech_(O_tau)(S_B) -> Cech_(W_tau)(S_B) = C_rel..

It is literal componentwise restriction, using $W_{\tau }$ contained in $O_{\tau }$. Its index map is the identity and therefore order preserving; no multiplicativity assertion for a sorted, non-order-preserving refinement map is used.

Every morphism map is a quasi-isomorphism. To verify this, form the two-cover Cech double complex; in each fixed intersection the augmentation is a quasi-isomorphism by the explicitly contractible relative pairs just described. Eliminate columns, or rows, in each finite total degree. The comparison respects the sheaf products and their ordered cup products, so it is a dg comparison. Thus the dg category of toric line bundles is quasi-equivalent to the category on invariant divisor data with morphisms In formulas: direct sum_(m in M) C^*(P,B(E-D,m)^b;k),. and with the relative cup product and additive lattice weight. Character shifts identify the corresponding redundant divisor representatives. This is the exact algebraic interface needed by the Morse/shrub construction.

Diagnostic for the printed formula. Actual Abouzaid v4 PDF printed page 46, Eq. (6.4), says that every listed face must separately avoid the pole locus. For an intersection this is too strong. On $X=P1$ take the trivial line bundle and a character with a pole at one toric endpoint. It is not a section over that endpoint's affine chart; it IS a section over the intersection of that chart with the other chart, namely the torus. The overlap weight module is therefore k, not zero. On a nested face chain the correct criterion is that the LARGEST face avoid the bad facets, or equivalently that NOT ALL faces lie in the bad subcomplex. This is the relative-subcomplex rule already described in the source's Section 6.2. The present proof derives the criterion from the actual rings and localizations and does not use Eq. (6.4)'s printed individual-face test as a premise.

The polynomial and selected tropical region here are the source's adapted representative, followed by its relative symplectic transport when the original datum is expressed in a different member of that source family. We do not assert that every smooth Laurent zero fibre has a bounded component of its literal logarithmic amoeba: that assertion is false, already for certain quadratic Laurent polynomials in one variable. The actual datum names the source tropical region and section conventions.

Fix the lattice N, its dual M, and a Euclidean metric g from a basis of M. The logarithmic base and angle space are $M_R$, with angles modulo $2\pi$ M. Cotangent coordinates use the metric lowering $g:M_R\to N_R$; this is a chosen isomorphism and not an intrinsic identification of N with M. Take Q with OUTWARD fan normals $v_{\rho }$ and inequalities $<v_{\rho },x>\le {\lambda }_{\rho }$. The choice of moment-map sign is fixed by this convention. The coefficient of the constant monomial is -1. The adapted source polynomial is In formulas: W_t=-1+sum_rho exp(<v_rho,x+i theta>-lambda_rho log(t))..

Its constant-dominance region is log(t) Q. Projectivity supplies the strictly convex support data and hence this simple polytope. For a divisor $D=\sum$ $a_{\rho }$ $D_{\rho }$, let $m_{\sigma }$ be the Cartier character with $<m_{\sigma },v_{\rho }\ge -a_{\rho }$ on the facets incident to a vertex sigma. The tropical section will have angle $\mathrm{grad}(f_D)=2\pi$ $m_{\sigma }$ near that vertex. We suppress the common $2\pi$ in intermediate gradient formulas, retaining it in the deck period. An algebraic weight m corresponds to the geometric lift $f_E-f_D-g(m,-)$; this fixes the sign of the weight-to-deck correspondence.

Localize the polynomial by actual monomial bounds. Choose successive small face neighborhoods $O_{\tau }$, with product form along the tangent directions to each face. The active exponents near tau are 0 and the rays of the facets containing tau. They are linearly independent apart from 0, because the polytope is simple and the fan is smooth. In a neighborhood reduced slightly from $O_{\tau }$, every inactive term and each of its fixed derivatives is bounded by $C_k$ $t^{-\eta }$ relative to the active ones, for some $\eta >0$. Indeed its logarithmic exponent is smaller by eta log(t); differentiation in x or $\theta$ merely multiplies by the fixed vector $v_{\rho }$. Choose smooth face-compatible cutoffs with transition width a fixed multiple of log(t), and delete the inactive terms. The resulting $W_{t,1}$ has EXACT product form along tau, and $W_{t,s}$ interpolates to $W_t$ with $C^k$ error $C_k$ $t^{-\eta }$ in the region of interest. The cutoffs preserve the positive real locus.

The zero hypersurfaces are regular and symplectic in that region. To check this quantitatively, change to the active basis of monomials; their equation is $-1+z_1+\mathrm{.}\mathrm{.}\mathrm{.}+z_c$ plus the stated small error. On its zero set at least one active monomial has modulus at least 1/c. The inverse of the fixed exponent matrix then bounds the norm of the complex differential below by a fixed positive constant. Its real kernel is a complex hyperplane. A sufficiently small $C^1$ error retains its surjectivity and the nondegeneracy of the restricted symplectic form. The finitely many active matrices give uniform constants. The ordinary implicit function theorem gives the same $C^k$-small normal graph comparison on overlapping charts.

A relative symplectic transport can be built directly: view the moving hypersurface as that small section in its symplectic normal bundle, correct its pulled-back two-form by integrating the exact derivative along the normal graph, and solve the resulting linear contraction equation for its tangential vector field. Its sum with the graph velocity has contraction with omega equal to an exact one-form along the hypersurface. Extend its primitive constantly in symplectic-normal coordinates and cut it off at normal radius delta. Hamiltonian flow then carries the hypersurfaces to each other and has $C^k$ displacement at most $C_k$ $t^{-\eta }{\delta }^{-k}$. All the constructions can be chosen invariant under real conjugation, so the positive real locus is preserved. The flow is supported in the chosen compact tube of the relevant hypersurface region. These statements follow by differentiating the graph, contraction equation and ordinary flow equation; the lower differential bound above controls every inverse. There is no imported tropical isotopy theorem in this step.

We make the convexity of the localized central region explicit, since arbitrary cutoff patching does not preserve it. Put $l_{\rho }=<v_{\rho },x>-{\lambda }_{\rho }$ log(t), choose $R=\eta$ log(t) smaller than the gaps on the face charts, and let kappa(s) be smooth, nonnegative, zero for $s\le 0$ and one for $s\ge 1$. Define In formulas: beta_R(s)=integral_(-infinity)^s (s-r) exp(r) kappa(r+R) dr..

Then ${\beta }_R$ is nonnegative, increasing and convex, vanishes for $s\le -R$, and ${\beta }_R^{\mathrm{\prime }\mathrm{\prime }}(s)=\exp (s)\kappa (s+R)$. For $s\ge -R+1$ it differs from $\exp (s)$ by an affine function with coefficients $O((R+1)\exp (-R))$; on the bounded-above central region all its fixed differentiated errors are bounded by $C_k(R+1{)}^2$ $\exp (-R)$. Use In formulas: W_(t,1)=-1+sum beta_R(l_rho) exp(i<v_rho,theta>)..

This is an allowed smooth semi-tropical localization, not claimed to remain a Laurent polynomial. Its $C^k$-smallness is exactly the preceding derivative estimate. Near tau all inactive beta terms are zero and each remaining $l_{\rho }$ is constant in tangent directions to tau, so the asserted product is EXACT. On the positive real locus its central boundary is the level sum ${\beta }_R(l_{\rho })=1$ of a convex function. The central core has value0, so this is a regular level; its normal Hessian is positive in the genuinely curved directions. The triangle inequality shows this is also the boundary of the central logarithmic-amoeba complement: equality requires all nonzero active phases to agree with the constant term, and hence their phases are zero in the active basis. The normal basis is integral and unimodular near a vertex. Thus the positive real boundary lies in the symplectic zero hypersurface and supplies the required smooth convex $Q_t$, with its exact face products. Dividing logarithmic coordinates by log(t), $Q_t$ is as close to Q as desired, by the monomial dominance gaps. A finite cover can be used to make an orthogonal product compatible with the lattice; no integral orthogonal splitting is falsely assumed.

Retain this modification only in a small potential tube: multiply $W_{t,1}-W_t$ by a cutoff of $\mathrm{\mid }{W}_t\mathrm{\mid }$ which is1 for $\mathrm{\mid }{W}_t\mathrm{\mid }\le ϵ\mathrm{/}8$ and0 for $\mathrm{\mid }{W}_t\mathrm{\mid }\ge ϵ\mathrm{/}4$. First choose $ϵ>0$ and then t so large that every fixed error ${\delta }_t$ is much smaller than ${ϵ}^2$. No new zero is created in the transition annulus, since its error is less than $ϵ\mathrm{/}16$. The central zero boundary and its face products are unchanged, but the localized potential now agrees EXACTLY with the holomorphic $W_t$ near the outer annulus $\mathrm{\mid }W\mathrm{\mid }=ϵ$. The Hamiltonian transport of its zero hypersurface can be supported in $\mathrm{\mid }{W}_t\mathrm{\mid }<ϵ\mathrm{/}8$ by the normal graph displacement bound. Thus its transported almost complex structures are the flat one near the outer annulus. This precise cutoff is needed for the W-plane argument; closeness alone would not make a nonholomorphic localized potential holomorphic there.

For the source generic Laurent representative, choose the small positive coefficients generically within these strict estimates; every relevant face polynomial can simultaneously be made nondegenerate. Indeed the failure is the elimination locus of its value and logarithmic derivatives on that face torus. For each face it is a proper algebraic condition (a generic linear combination of its distinct characters is a submersion on its zero set, by the elementary universal evaluation derivative), and there are finitely many faces. A compact path in this generic coefficient locus can be obtained by successive complex line segments avoiding their finitely many discriminant intersections. This connects the generic source members under consideration to an adapted member; it is not an assertion about every smooth but nongeneric-at-infinity zero fibre.

The relative symplectic transport along such a path is complete in logarithmic coordinates. To verify the needed bound, divide its finitely many monomials by their maximum modulus. On each asymptotic face chamber the limiting normalized active coefficients and angles belong to a compact set; a vanishing limiting normalized logarithmic gradient on a zero would give a singular zero of the corresponding face polynomial. Generic face nondegeneracy excludes it, uniformly on the compact coefficient path. Thus $\mathrm{\mid }dW\mathrm{\mid }\ge c$ $max\mathrm{\mid }{c}_a$ $z^a\mathrm{\mid }$ on its zero set, while every fixed logarithmic derivative and its parameter derivative are bounded by $C_k$ times that maximum. The symplectic-normal velocity solving $dW(V)=-{\mathrm{\partial }}_s$ W is therefore bounded, with its fixed logarithmic derivatives. Its contraction with omega restricts to zero on the zero hypersurface. Extend its zero value and prescribed normal derivative to a Hamiltonian in the uniform normal tubes, and cut off normally. The same normalized derivative bounds give uniformly bounded second fundamental forms and a uniform graph radius by the quantitative inverse theorem. The resulting Hamiltonian vector field has bounded logarithmic speed, so its finite-time flow is complete. It carries the zero hypersurfaces and admissible collars symplectically. Transported selected objects are precisely the source's transported tropical sections; no literal bounded amoeba component for a nongeneric choice is asserted. This discharges the relative transport used in this construction at the actual source-data scope.

Construct every divisor section, rather than assume it exists. For vertices sigma and sigma' of a common face tau, the vectors $m_{\sigma }-m_{\sigma }^{\mathrm{\prime }}$ are tangent to tau: their pairing with every normal of a facet containing tau is zero. Choose a partition of unity ${\chi }_{\sigma }$ on the boundary collar, indexed by vertices, such that near tau only its vertices occur and each ${\chi }_{\sigma }$ depends only on the tangential coordinate of tau. This is a finite inductive construction: at a vertex use its constant weight1; on each edge interpolate its endpoint weights, constant in the normal coordinates; on a higher-dimensional face extend the already constructed boundary partition by a nonnegative partition with sum1 in its interior. Extend each step constantly along its product normal collar before continuing. Convex combinations preserve all the previously fixed restrictions.

Put $f_D=\sum$ ${\chi }_{\sigma }(g(m_{\sigma },x)+c_{\sigma })$, for any fixed constants $c_{\sigma }$, and extend smoothly into the interior. On the collar near tau, the derivative terms involving $dch{i}_{\sigma }$ are tangential to tau and all $m_{\sigma }$ have the same normal projection. Thus $\mathrm{grad}(f_D)$ belongs to $m_{\sigma }+T$ tau there. Near vertices it is exactly the fixed integral vector. This proves the required first-jet compatibility on EVERY face, rather than trying to glue arbitrary Neumann data without corner compatibility. After exponentiation the integral translate disappears, and the normal phase of the section is the positive-real normal phase. The product description in the Localize the polynomial by actual monomial bounds construction therefore places its boundary in the zero fibre. Its graph is exact and Lagrangian.

Make it admissible as follows. The boundary graph is Lagrangian inside the symplectic zero hypersurface. Its symplectic parallel transport along a small left-pointing segment of the W-plane is Lagrangian: omega is zero on its boundary tangent spaces and on the horizontal transport direction paired with them. At the real model the logarithmic projection in the new inward direction is nonzero, so the inverse function theorem makes this a graph over a collar; the same holds with the tangential factor and the small deformations above. Its primitive $g_D$ has the same value and full first derivative as $f_D$ on the boundary. Hence $g_D-f_D$ vanishes to second order in collar distance. Interpolate the two primitives with a collar cutoff, using $g_D$ on a smaller collar. The resulting graph has the old boundary and the required parallel-transport collar. The interpolation remains exact and Lagrangian throughout.

Equip the divisor graphs with the canonical spin structure and canonical graph grading: $Q_t$ is a disk, its tangent graph trivializes by projection, and the graph path from Hessian0 to $\mathrm{Hessian}(f_D)$ lifts its squared phase continuously starting at zero. In the fixed coordinates this lift is ${\pi }^{-1}\mathrm{tr}(\arctan Dp)$, since $\det (I+itDp)$ never vanishes for real symmetric $Dp$ and its continuous argument is the sum of the principal eigenvalue arctangents. Any other continuous grading differs by one constant integer $k$, because $Q_t$ is connected. Retain that integer as part of the object, rather than resetting it by continuation. For integral angular lifts with the canonical gradings, these choices give the coindex degree n-$\mathrm{index}(f_E-f_D-m)$. This follows also by diagonalizing a Hessian difference: each negative eigenvalue changes the positive angle count by one, with the zero graph as reference. Orientation lines may be chosen as their Morse positive-Hessian determinant lines. In the subsequent adiabatic comparison the nonzero transverse modes have their canonical complex orientations, so this fixes coherent polygon orientations for this graph class without importing a general relative-spin orientation theorem.

Every admissible section has Cartier data. It remains to prove the converse for an arbitrary object of the full section sub-pre-category. Work first in the adapted localized model of steps 28.1–30.1, before the stipulated transport. Its logarithmic base $Q_t$ is a convex disk. Any smooth section has a lift of its angular map to $M_{\mathbb{R}}$, since $Q_t$ is contractible; call the lift $p(q)$. Its Lagrangian condition is $d({\sum }_j{p}_{j}d{q}_j)=0$, so $p=df$ for a smooth real function $f$ on $Q_t$. At a boundary point the numbers $b_{\rho }={\beta }_R(l_{\rho })\ge 0$ sum to $1$, and the zero-fibre equation is ${\sum }_{\rho }{b}_{\rho }{e}^{i⟨v_{\rho },p⟩}=1$. Taking real parts gives ${\sum }_{\rho }{b}_{\rho }(1-\cos ⟨v_{\rho },p⟩)=0$. Every term is nonnegative, so every active phase is $0$ modulo $2\pi$. On a connected vertex patch the active $v_{\rho }$ form an integral basis, hence $p=2\pi m_{\sigma }$ for one constant $m_{\sigma }\in M$. On the connected patch of a face $\tau$, each active normal pairing $⟨v_{\rho },p⟩\mathrm{/}(2\pi )$ is an integer-valued continuous function and is therefore constant. Its values at the incident vertex patches agree. Adjacent $m_{\sigma }$ thus differ by a vector tangent to their common face. In particular, for each facet $F_{\rho }$ the integer $a_{\rho }=-⟨m_{\sigma },v_{\rho }⟩$ is independent of the incident vertex. These are precisely the integral Cartier data of steps 4.1–7.1. Changing the lift of $p$ adds one common lattice vector, as required. Compare $f$ with the constructed divisor primitive $f_D$ for these data. Their boundary normal pairings agree on each active face patch, so every linear interpolation of their differentials still satisfies the displayed zero-fibre equation on the boundary. It is a smooth exact Lagrangian graph throughout. A graph need not already satisfy the full product condition on an open collar: apply the parallel-transport collar replacement of step 36.1 to this compact interpolation family, keeping the original admissible collars at the two endpoints. The required logarithmic normal derivative of the horizontal lift is nonzero on the positive boundary model, since the real gradient of ${\sum }_{\rho }{\beta }_R(l_{\rho })$ is an outward normal to its regular convex level. The graph and its transported collar have the same boundary first jet; their primitives differ to second order in collar distance, permitting the same cutoff replacement uniformly in the interpolation parameter. The resulting family consists of admissible sections. Its Hamiltonian extension preserving the zero hypersurface is the one described next. Thus every original object is isomorphic by that admissible Hamiltonian continuation to one of the divisor sections; integrality has been deduced from its boundary condition, not imposed as a new object restriction. The stipulated Hamiltonian transports in steps 27.1 and 33.1 carry this classification to the transported source pre-category of the Statement. If two primitives have the same vertex integral vectors, their linear interpolation still satisfies the face condition and has boundary in the zero hypersurface. Perform the collar replacement in families to retain admissibility. The resulting exact Lagrangian isotopy is Hamiltonian: on its graph the Hamiltonian is minus the derivative of the primitive; extend it in the normal cotangent coordinates. At the hypersurface boundary the variation is tangent to the hypersurface, so extend its prescribed first jet constantly in symplectic-normal directions. The resulting Hamiltonian vector field preserves the hypersurface there. Cut off away from the compact family. This proves isotopy classification by the Cartier vectors modulo a common integral vector. The divisor classification therefore gives precisely all unshifted line-bundle objects for the canonical graph gradings. An arbitrary grading carries its constant integer offset throughout this graph interpolation and its admissible continuation. Write $L_D^{(a)}$ for the divisor section with phase lift equal to its canonical lift plus $a\in \mathbb{Z}$. For complementary tangent planes with phase lifts ${\theta }_0,{\theta }_1$ and positive rotation angles ${\varphi }_j\in (0,\pi )$, the absolute intersection degree in the coindex convention of step 37.1 is $i=n+{\theta }_1-{\theta }_0-{\pi }^{-1}{\sum }_j{\varphi }_j$. The squared-phase relation makes this integer; diagonalizing a graph Hessian gives exactly the coindex in step 37.1. Adding $a$ at the source and $b$ at the target therefore changes each generator degree by $-a+b$. Consequently $C{F}^r(L_D^{(a)},L_E^{(b)})=C{F}^{r+a-b}(L_D^{(0)},L_E^{(0)})$ as graded spaces with the usual shifted differential and orientation-line signs. With the cochain convention $C[k{]}^r=C^{r+k}$, the corresponding objects are thus ${\mathcal{L}}_D[-a]$ and ${\mathcal{L}}_E[-b]$, whose degree-$r$ Hom cohomology is ${\mathrm{Ext}}^{r+a-b}({\mathcal{L}}_D,{\mathcal{L}}_E)$. Equivalently a categorical shift $[k]$ subtracts $k$ from the phase lift; it does not add $k$. On every composable word these Hom shifts telescope, while suspension transports the same Koszul signs to the shifted cochain operations, so this extends the canonical comparison in every arity. Thus extending the already constructed canonical-graph comparison by shifts gives the claimed comparison for every graded section, with shifted line bundles as target; no grading shift is silently identified with degree zero.

Positive copies and the Morse boundary condition. For each finite ordered collection choose distinct collar slopes $K_i$ in counterclockwise order in the left W-half-plane. On a facet in a coordinate $x\le 0$ normal to its boundary, a primitive $K_i$ $x^2\mathrm{/}2$ gives angle $K_i$ x, so W is asymptotic to $x(1+i{K}_i)$. Increasing $K_i$ is exactly counterclockwise order. Choose the parallel-transport collars with these slopes, and arrange in a somewhat larger face neighborhood that the primitive has form In formulas: f_i,tangent+g(psi_i,x)+rho_i,.

where ${\psi }_i$ is an integral vector with its fixed face-normal projection, ${\rho }_i$ has small derivative, and ${\rho }_j-{\rho }_i$ has strictly inward gradient on a normal cone whenever its integral normal projection is zero. The allowed inward inequalities form an open convex cone; the product-face partition construction extends them inductively, with constants decreased at each finite step if needed. All modifications can be confined to the collar and all critical points made transverse by interior perturbations.

Use supporting hyperplanes to choose nested face strips $P_{\tau }$ with $O_{\tau }$ inside $P_{\tau }$ inside its corresponding dual-cell neighborhood, so that the complement of the strips up to each dimension is convex. A negative-gradient edge with both ends in the interior core cannot leave it and return. In a strip normal to tau its leading vector is the constant integral projection. If that projection vanishes, the inward $\rho$ ordering fixes its direction and prevents a return. Otherwise a $C^1$-small $\rho$ perturbation cannot change the sign relative to the appropriate supporting face; it either exits at the boundary or enters a lower-dimensional face strip, where induction applies. At a tree vertex the sum of the incoming edge gradients equals the outgoing one in the collar. If such a vertex first reached the boundary, reverse the outgoing edge: the inward normal pairings of all incident edges are nonpositive and sum to zero, so all vanish. The product boundary condition then keeps all these edges on the boundary, contradicting an interior external critical limit. Hence EVERY relevant gradient tree is uniformly separated from the boundary. The same holds for perturbations supported on the interior edge pieces used in the shrub construction.

The positive exit set for $f_E-f_D-g(m,-)$ has outward integer facet value In formulas: -a_E,rho+a_D,rho-<m,v_rho>,.

exactly the pole function in the algebraic component. A zero integer value has no positive exit, by the chosen inward correction. At a rounded face tau its normal part is a fixed vector ${\psi }_{\tau }$; the Gauss normals range over the convex spherical normal cone. The positive exit is the portion where $<{\psi }_{\tau },normal>>0$, with the zero boundary shifted inward. This is a convex spherical cap, hence empty or contractible. Normal projections for a higher face containing tau are its orthogonal projections, so the same linear inequality holds on all the adjacent product pieces of a sufficiently small dual-cell neighborhood. The contraction along normal slices therefore agrees on their intersections. The union of positive facet pieces gives its deformation retract: in a simplicial normal cone the positive rays form a nonempty face subset exactly when this cap is nonempty, and the straight radial contraction to their convex hull stays in the positive half-space. Product with the tangential face coordinates gives the required local deformation retraction on each dual cell. Thus the precise boundary hypothesis of the Morse--cellular component is discharged, including integer equality cases.

Extra positive copies before and after any finite list are obtained by collar slopes before and after its finitely many slopes. They have the same divisor labels. Their relative bad set for the appropriate identity lift is empty; their degree-zero Morse class maps to1 in $C^0(Q)$, and hence is a cohomological unit by the shrub comparison. Small interior perturbations make every required finite sequence regular. This is the unital extension property of the pre-category; no self-intersection transversality is assumed.

First holomorphic barrier: the W-plane. For a finite positive sequence adjust the collars on an annulus $\mathrm{\mid }W\mathrm{\mid }=ϵ$ so that their images there are distinct fixed left-pointing radial segments in their positive order. Interior generators map outside this circle. The $C^k$-small symplectic transports from the Localize the polynomial by actual monomial bounds construction can be supported inside it, by first choosing t large and then their tube radius. On each face this is the active model -1+sum $z_i$; all $\mathrm{\mid }{z}_i\mathrm{\mid }$ in its central logarithmic component are bounded by their positive boundary values, so W sends the selected collar graphs into the left half-plane. The potential-tube cutoff above retains this property. Quantitatively ${\beta }_R\le exp$, so on $Q_t$ the original positive amplitude sum is at most $1+{\delta }_t$, whereas the beta sum is at most1. If $\mathrm{\mid }{W}_t\mathrm{\mid }\ge ϵ\mathrm{/}8$ and $\mathrm{Re}(W_t)\ge 0$, then its imaginary part has size at least c $ϵ$; Cauchy--Schwarz implies sum $\exp (l_{\rho })(1-\cos ({\theta }_{\rho }))\ge c_{prime}$ ${ϵ}^2$. This contradicts $\mathrm{Re}(W_t)\ge 0$ when ${\delta }_t$ is chosen smaller than $c_{prime}$ ${ϵ}^2$. Therefore $W_t$ already has negative real part in the transition region, while the beta potential has nonpositive real part by the triangle inequality. Their cutoff convex combination also does. The segment collars retain the same left-half-plane condition.

No holomorphic polygon with those interior corners can meet $\mathrm{\mid }W\mathrm{\mid }<ϵ$, for any member of the relative almost-complex homotopy that agrees with a W-holomorphic structure near this annulus. If it did, take a regular value z on the annulus attained in its interior. Its local degree is positive by holomorphicity there. Move z along a path through the annulus gap and then the right half-plane to a point outside the compact polygon image. The change of degree is the intersection of that path with the boundary image. It is zero: along each Lagrangian arc both endpoints lie outside the circle, and its excursions along its fixed annular segment cross the path in opposite pairs. Outside the annulus all boundary images lie in the left half-plane, so the path in the right half-plane encounters none. Degree is therefore zero at z, a contradiction. This argument works for each of the finitely many boundary arcs; it is not restricted to a two-brane bigon. The constant-corner exception is excluded by the interior corners.

The symplectic-transport support was inside that circle, so these polygons lie where its transported almost complex structure is the original flat one. Their boundary logarithms lie inside a smaller convex region $Q_t(\delta )$. Each real logarithmic coordinate is harmonic for the flat torus complex structure, so every supporting affine function satisfies the maximum principle. Their interiors lie in the same convex region. This proves no escape for the original relative counts and the first complex-structure homotopy, not merely a boundary condition on the Lagrangians.

Second barrier: rescaling the cotangent complex structure. The W-map is not presumed holomorphic after cotangent metric rescaling. Instead choose a convex inner rounding $V=\{h=0\}$, lying inside the preceding excluded collar and outside all the gradient trees and generators. Near a face tau its normal slice is convex and its tangential factor is constant. Choose a common inward vector by projecting $q_0-x$ to the normal space of tau, with $q_0$ a fixed interior point. Its projections agree on incident faces. In a neighborhood of V replace each ${\rho }_i$ by a function with this CONSTANT normal gradient times its chosen small slope; extend it along the tangential directions and patch by the same face partition. The space of functions with prescribed constant normal gradient is affine, so the patching preserves it and the inward inequalities. Shrinking these corrections preserves the two previous separations.

On every resulting section $d^c$ h restricts to zero near V. In fact h depends only on the curved normal variables, while the derivative of the angle along the graph varies only in the flat tangential directions; their pairing is zero. The statement remains true after the cotangent metric is rescaled, since this merely multiplies the normal $d^c$ by a positive scalar. For a rescaled holomorphic polygon, h of its logarithm is subharmonic because h is convex and the logarithmic coordinates are harmonic. At a boundary tangency to V its normal derivative is zero by $d^c$ $h\mathrm{\mid }{L}_i=0$. The strong maximum principle and the boundary Hopf argument exclude a first tangency from the inside, unless the whole map lies in V; the interior corners exclude that case. The Hopf argument follows directly by subtracting a small exponential radial barrier on an interior tangent ball and applying the scalar maximum principle.

It follows that the component of polygon moduli confined to $\{h<0\}$ is closed as well as open under all the rescalings and continuation parameters under consideration. Its curves cannot acquire an end at V. At the initial complex structure all curves with these corners are in this component by the First holomorphic barrier: the W-plane construction. At sufficiently small rescaling the adiabatic compactness construction identifies every relevant graph-tree polygon in it; additional unrelated components outside the barrier are neither counted nor transported into this sub-pre-category. Boundary breaking preserves the same barrier, so this restriction is compatible with ALL operations and their functor equations. Compactly supported perturbations for transversality are chosen off a neighborhood of V and therefore preserve the argument.

To use the actual closed-exact-polygon supplier without extending its scope to boundary branes, embed the compact core of $Q_t$ in a large flat torus and extend the finitely many primitives smoothly to it. Their full cotangent graphs are now compact closed exact Lagrangians. The ambient cotangent bundle is Liouville, and outside a fixed fiber ball the standard complex structure and compactly supported data give no escape by the maximum principle. Thus the actual theorem thm-gts-r4-10c-exact-regular-polygon-count-a-infinity-branch applies to their regular moduli. Its local compactness and collars restricted to $\{h<0\}$ are exactly those of our relative polygons; no artificial exterior generator can occur on a limiting edge in that region. This is the legitimate local use of its closed-brane interface. The graph adiabatic component must still supply regularity for its rigid solutions, and coherent supported forcing supplies the other regular parameterized moduli. This note does not assume those assertions without that component.

Let Q be an oriented smooth compact n-disk with a finite triangulation P, including its boundary, and let P-dual be a dual cell decomposition. Consider functions $f_i$ with fixed admissible collar jets. For every allowed ordered pair put $f_{ij}=f_j-f_i$, and allow an additional integral affine function m. In the toric application the base is $M_R$ and the deck lattice is M. Fix a Euclidean metric g from a lattice basis, identify the angle vector m with the covector $m-flat=g(m,-)$, and by the affine function m mean q $\to$ g(m,q). This is an explicit metric lowering, not an intrinsic identification of the fan lattice N with M. The hypotheses to be discharged by the tropical construction are: their interior critical points are nondegenerate and separated from the boundary; all trees connecting those critical points stay in a fixed interior compact set; their positive-gradient exit set retracts locally on each dual cell to the specified bad boundary subcomplex $B_{ij},m$; and all perturbations below may be chosen supported off the fixed collar. There are only finitely many relevant lattice labels for each finite collection of functions, because a critical point requires $\mathrm{grad}(f_{ij})=-m$ in the bounded gradient image. Morphisms are direct sums over these lattice labels, with composition adding the labels.

We use cohomological degree $\mathrm{\mid }p\mathrm{\mid }=n-{\mathrm{index}}_p(f_{ij}+m)$, the dimension of the positive-Hessian subspace. Its orientation line is the determinant of that subspace. Geometrically an edge is directed from inputs toward the output and follows the NEGATIVE gradient. This convention gives a differential raising |p| by one. Tree operations have degree 2-k and shrub functor components degree 1-k. All sign statements below use the degree-minus-one suspension s, with $b_k=s$ ${\mu }_k(s^{-1}{)}^{\otimes }$ k.

Actual finite-dimensional flow spaces. For each planar tree, take one copy of Q for each vertex, one nonnegative length for each finite edge, and stable/unstable manifolds at its infinite ends. The matching equations identify an edge endpoint with the time-length negative-gradient flow of its other endpoint. Local existence of the flow is the integral equation and Picard contraction; the collar hypothesis ensures that every solution relevant here stays on a common compact interior set. Differentiating the same integral equation gives smooth dependence on initial point, length and function. At an infinite critical end, diagonalize the real symmetric Hessian. On its positive and negative eigenspaces the ODE inverses are the integrals from respectively the positive or negative infinite endpoint, with exponentially decaying kernels. The nonlinear remainder is quadratic, so contraction on a sufficiently small weighted ball constructs the stable and unstable manifolds and their exponential estimates. This also provides their tangent spaces, without assuming an infinite-dimensional transversality result.

Transversality can be arranged coherently. On the interior of an edge choose a tiny interval disjoint from vertices and a time-dependent compactly supported function. Its gradient at any specified point can be any vector (use a linear coordinate function times a bump). Variation of the endpoint is its integral transported by the invertible flow derivative. Thus these variations span the normal to every matching diagonal. Use finitely many such variations on a compact part of a flow space to obtain a finite-dimensional submersion. Ordinary Sard then gives transversality there. Exhaust by compact edge-length and end-cutoff ranges and choose successively smaller $C^r$ perturbations, preserving the already achieved open transverse conditions. The convergent smooth sum is transverse on the exhaustion. At boundary faces the edge data are already fixed to the product data of the smaller trees. Exponential ODE gluing makes transversality automatic near those regular broken faces, so the new variations can be confined away from them. Do this by arity and then on the countable exhaustion. The same argument applies to cell/shrub evaluation conditions. It is a construction of coherent perturbations with the prescribed collar, not a claim that independently chosen pairwise Morse--Smale data automatically suffice for every tree. Unperturbed regular sequences may instead be retained as a full transverse sub-pre-category; interpolating the compactly supported edge data gives the continuation comparison by the same construction.

The edge length cubes $[0,\mathrm{\infty }{]}^E$, with infinity compactified by $\exp (-length)$, describe all flow-tree limits. Zero lengths identify the two vertices and glue adjacent planar tree cells; infinity breaks at a critical point. The bound on the number of breaks follows from the finite critical set and the positive drop in f along a nonconstant trajectory of any fixed edge label. Finite-edge limits follow from ordinary ODE compactness. On a long edge entering a critical neighborhood, the explicit stable/unstable integral kernels above give exponential approach; outside those neighborhoods a uniform positive gradient norm bounds its time by the available variation of f. This proves compactness and accounts for every end. To glue a broken edge, retain the two trajectories up to length R, cut their exponentially small deviations from the critical point, and solve the ODE by the same stable/unstable inverse; its cutoff error is C $\exp (-cR)$. The finite-dimensional diagonal matching derivative is onto by construction. Neumann inversion and the quadratic remainder therefore give an actual collar, unique after the chosen end translations. Several breaks use disjoint cutoffs. No additional boundary is present by the given collar barrier.

Orient these spaces by their actual matching exact sequence: parameter and endpoint tangent spaces, followed by the dual of the matched diagonal tangent spaces. At an interior critical point contract its incoming determinant line with its dual on the outgoing edge. Choose the ordered real disk-domain orientation on each planar length cell; across a four-valent zero edge the two signed length coordinates are negatives, so the orientations agree. The outward normal at an infinite-length boundary is the positive length direction. In suspended determinant lines the inner operation has degree one; moving it past the first r inputs gives exactly $(-1{)}^{{\sum }_{j\le r}(\mathrm{\mid }{x}_j\mathrm{\mid }-1)}$. All other matched lines occur adjacent to their duals and cancel. The domain face orientation is the consecutive-block orientation obtained by expanding the ordered real marked-point coordinates. Thus the oriented boundary of a one-dimensional flow-tree space is precisely the shifted bar expression b squared. Its signed boundary count is zero. Rigid tree counts therefore define ALL the $b_k$, with b squared zero; no finite truncation of this conclusion is being substituted for the all-arity category.

The relative cellular dg model as an intersection model. Translate the dual subdivisions slightly by times $t_0<\mathrm{.}\mathrm{.}\mathrm{.}<t_k$ along a field respecting the total vertex order of P. Locally at a codimension-d dual cell use the standard fan in $R^d$ whose chambers record which of $0,x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_d$ is greatest. Translate by (-t/d,...,-t/2,-t). Direct comparison of the inequalities shows that the duals of $[v_0,\mathrm{.}\mathrm{.}\mathrm{.},v_p]$ and $[w_0,\mathrm{.}\mathrm{.}\mathrm{.},w_q]$ meet only when $v_p\le w_0$. If equality holds their transverse intersection represents the dual of $[v_0,\mathrm{.}\mathrm{.}\mathrm{.},v_p=w_0,\mathrm{.}\mathrm{.}\mathrm{.},w_q]$; if the inequality is strict it is collapsed to a cell one dimension lower. The comparison inequalities also show that these translations and collapse maps agree on faces. A partition of unity in the successive essential parts of the dual cells produces the global field with this local form. It can be chosen tangent to the boundary and to its prescribed subcomplexes.

Consequently intersection followed by this collapse is EXACTLY Alexander--Whitney cup product under cellular duality $C_{n-\ast }(P-\text{dual}minusB-\text{dual})=C^{\ast }(P,B)$. Its signs are fixed by orienting the dual cell so that simplex orientation followed by dual-cell orientation is the orientation of Q. This is checked on the ordered coordinate simplex, where the transverse diagonal equations remove the common middle vertex and leave the ordered front/back vertex strings. The differential is the cellular boundary in this dual grading; moving its outward normal past a codimension-p factor gives the usual $(-1{)}^p$ Leibniz sign. Thus the intersection/collapse product is strictly associative, because its dual product is literally the front/back simplex product. The relation $B_{02}$ contained in $B_{01}$ union $B_{12}$ makes this a dg pre-category for the varying relative pairs.

The translated dual decompositions are redundant copies of the same dg category. Forgetting their time is an isomorphism on every corresponding cellular/cochain module and product. Given a finite ordered list, extra translations can always be inserted before and after it, so these copies provide the extension property needed for a pre-category.

Shrubs and the necessary intersection homotopies. A k-input shrub is a planar metric tree with finite incoming edges, one infinite outgoing edge, and equal total distance from its root vertex to every input. Its finite parameters have dimension k-1. A shrub with inputs on dual cells $C_1,\mathrm{.}\mathrm{.}\mathrm{.},C_k$ and outgoing limit p is the transverse inverse image of their product under the flow-evaluation map from the outgoing stable manifold times the shrub parameters. Its dimension is In formulas: |p|-sum_i codim(C_i)+k-1..

It has a compactification by zero and infinite edge lengths as above. The equal-height condition means that infinite-height faces are a root tree with a collection of shrubs on its inputs, including the one-shrub case with an outgoing break. Zero-height faces collide adjacent incoming cells. Moving a cell input to its own boundary gives the remaining cellular-differential faces.

Counting the uncorrected shrubs alone would compare raw intersections, not the actual collapsed cellular cup product. Here are the required correction chains. For times $t_1<\mathrm{.}\mathrm{.}\mathrm{.}<t_a$ take the intersection of the translated cells and sweep their first a-1 times through In formulas: K(t_1,...,t_a)={t_i<=s_i<=t_a, s_1<=...<=s_(a-1)}..

Use the explicit local fan coordinates above to identify each intermediate intersection with the initial one. This gives an actual chain $K(C_1,\mathrm{.}\mathrm{.}\mathrm{.},C_a;t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_a)$ in Q. Its boundaries are: a boundary of an input cell; $s_i=s_{i+1}$, which is the collapsed adjacent intersection/cup product; and $s_i=t_i$, which is the raw intersection interface. Their orientations are the outward-normal orientations of the displayed convex parameter polytope, times the ordered intersection orientation. This describes all faces, including their corners; faces of codimension at least two do not enter a one-dimensional count.

For an ordered partition $E=(E_1,\mathrm{.}\mathrm{.}\mathrm{.},E_e)$ of the k inputs into consecutive nonempty blocks, put the chain $K(C_i:i\in E_j)$ on the j-th input of an e-input shrub. Count its rigid solutions with outgoing p, and sum over ALL ordered partitions. Denote the resulting map $F^k$. The total dimension is still the displayed dimension, since each block contraction loses one shrub input and adds one sweep parameter. Thus $F^k$ has degree 1-k. For $k=1$ there is no sweep and $F^1$ is the signed intersection of the input cell with the appropriate flow manifold.

The compactification is an actual fiber product of the already constructed flow/shrub compactification with these finite convex sweep chains. The preceding transversality construction includes every such product and every face. Its one-dimensional boundaries fall into precisely four groups: (i) an infinite-length tree at the root with shrubs on its branches, giving $b_M$ F; (ii) a cell boundary, giving F $b_1$; (iii) $s_i=s_{i+1}$, giving F $b_2$ for the strict cellular cup product; (iv) zero-height raw-intersection faces and the sweep faces $s_i=t_i$. The two faces in (iv) are the SAME evaluation/matching problem from adjacent ordered partitions, with opposite outward normals, and cancel. For a split into multiple branches every branch is itself one of the same partition-corrected shrubs; hence the sum in (i) uses precisely every $F^j$ already defined, with no omitted higher correction.

Orient the fiber products using the same matching determinant rule as the Actual finite-dimensional flow spaces construction and put the sweep coordinates in increasing input order. For (ii), moving a degree-one cellular differential past the preceding suspended input lines gives $(-1{)}^{\sum \text{preceding}(\mathrm{\mid }{C}_i\mathrm{\mid }-1)}$. For (iii) the ordered sweep face replaces two successive input lines by their product with its inherited intersection orientation. In (i) contract the intermediate p orientation line with its dual and move each root degree-one map past its preceding inputs, giving exactly the bar signs. Group (iv) cancels before any reordering since the same ordered matching determinant and evaluation chains occur on its two sides. These observations compute all the signs by one ordered determinant rule, rather than assuming an omitted shrub orientation theorem. The signed boundary identity is therefore In formulas: b_M composed with F = F composed with b_cell.

as maps of the full tensor bar modules. This is the all-A-infinity functor equation, with every component $F^k$ included. At corners the same determinant cancellation is associative, so the orientations used here are coherent with simultaneous breaks.

Quasi-isomorphism of the linear component. There is a finite relative Morse handle filtration. Remove small disjoint Morse balls and use the flow wherever its gradient is nonzero. Its time-to-level parametrization identifies the bands without critical points with products. Near an index-j critical point the coordinate function is $f(p)-\mathrm{\mid }{x}_{-}{\mathrm{\mid }}^2+\mathrm{\mid }{x}_{+}{\mathrm{\mid }}^2$; crossing its level attaches exactly one j-handle. The local flow and boundary-convex hypothesis make this a relative handle description, with the relevant exit boundary included. Reversing to the coindex convention gives one relative cohomology generator of degree |p| and its actual orientation line. The attaching differential is the signed intersection with the next flow handle, because following the product flow identifies its boundary evaluation with precisely the connecting negative-gradient trajectories. This follows directly in the coordinates of the two handles and the transverse flow boxes between them.

Use the corresponding handle CHAIN complex for positive-gradient flow, with one degree-|p| unstable cell (the stable cell for negative gradient). Its boundary counts the positive-gradient trajectories and its dual cochain differential is precisely the chosen negative-gradient Morse differential. The compactified handle cells define a chain map to the relative chains of (Q,exit); the preceding handle filtration shows that this map induces an isomorphism, since each attached handle quotient has one generator and the map on it is its oriented fundamental cell. This assertion is about the handle complex, not a false claim that arbitrary original dual cells are filtered by critical values.

A cellular cochain represented by its oriented dual cell evaluates on that handle chain by transverse intersection. This evaluation is exactly $F^1$. The cellular-to-singular cochain comparison and subdivision to make intersections transverse are obtained by the simplex prism contraction; their boundaries are the cellular boundary and the intersection boundary, so the evaluation commutes with the two differentials. Dualizing the handle chain comparison over the chosen field is exact and yields a cohomology isomorphism. Thus $F^1$ is a quasi-isomorphism. The local deformation-retract hypothesis identifies the cellular boundary locus with the flow exit locus and is used here. This proof is for the target's field coefficients; no perfect integral pairing in the presence of torsion is assumed.

Relation to the toric dg model and scope. Apply this construction separately to each character m and retain the object divisor labels. Every edge label is the sum of its consecutive input differences; thus the incoming character weights add to the outgoing weight. The compactness, sweep chains, orientations, and all functor equations respect this grading. Only finitely many character labels occur in any fixed generator word. For outward fan normals, choose $\mathrm{grad}(f_D)$ at a vertex equal to the Cartier character $m_{\sigma }$ with $<m_{\sigma },v_{\rho }\ge -a_{\rho }$. The geometric lift for an algebraic weight m is $f_j-f_i-M_m$, where $M_m(q)=g(m,q)$; its outward slope is therefore $-a_j+a_i-<m,v_{\rho }>$. This fixes the minus sign relating the deck lift to the algebraic weight. The toric boundary construction supplies $B_{ij},m=B(D_j-D_i,m)$, so the cellular dg category is exactly the one constructed in the algebraic construction. The partition-corrected shrub functor is a quasi-isomorphism on ALL these morphism complexes. It is the identity on the selected function/divisor object labels; every divisor object has a corresponding tropical function once that geometric construction is supplied. It is therefore a quasi-equivalence at precisely this object scope.

The linear component takes the local degree-zero relative unit to its Morse class. The higher functor equation for $k=2$ then shows that it acts as an identity on cohomology, since the cellular identity does so. Positive collar copies of each function before and after any finite sequence provide the required pre-category quasi-isomorphisms; their boundary divisor labels are unchanged. This proves the unital extension assertion once those collar copies have been constructed geometrically, without pretending every repeated-object sequence is transverse.

Fix a finite positive sequence of exact sections. Choose lifts to T*Q of the form $\mathrm{graph}(d{f}_i)$, where affine deck shifts are INCLUDED in $f_i$. The logarithmic cotangent model uses $\omega =\sum$ $d{q}_a$ wedge $d{p}_a$, $\lambda =-\sum$ $p_a$ $d{q}_a$, and standard complex coordinate q+i p. The corresponding metric-rescaled complex structure is equivalent, by fiber multiplication by $ϵ$, to the standard complex structure with boundary $\mathrm{graph}(ϵd{f}_i)$. Let $v=q+i$ p be the latter map. Its equations are In formulas: q_s=p_t, p_s=-q_t..

Orient edges from inputs toward the output, opposite the increasing strip s coordinate. Their limit equation is therefore minus the gradient of $f_j-f_i$. Coherent compactly supported Morse-edge perturbations can be represented by Hamiltonian one-forms of size $ϵ$; on long strips their coefficients depend on $\tau =ϵ$ s. The averaged equation below then has exactly the corresponding perturbed gradient vector field. These perturbations vanish near the tropical barrier and are product data on every broken end.

The following proves a signed bijection of RIGID moduli for every fixed finite positive collection and every arity allowed within it, and absence of negative-dimensional solutions, when $ϵ$ is sufficiently small. It does not assert one universal $ϵ$ for infinitely many objects or arities. the Build the genuine all-A-infinity pre-functor construction constructs the actual all-A-infinity functor without making that assertion.

The two explicit local inverses. On a strip subtract the linearly interpolated imaginary boundary values and decompose into the real t-average and its complementary modes. At epsilon0 the complementary modes have the elementary sine/cosine expansion, with transverse frequencies $\pi ,2\pi$,... . The Green factors for a frequency a are $\exp (-a\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid })$ on its allowed half-direction. Thus its inverse and every fixed derivative estimate are bounded independently of the strip length, also with exponential weight $\exp (\delta \mathrm{\mid }s\mathrm{\mid })$, $0<\delta <\pi$. Fourier orthogonality proves the L2 bounds; integrating the exponential kernels and using their first derivatives gives the corresponding fixed Sobolev/Lp bounds. Perturbing the graph boundary by its $ϵ-Hessian$ changes these complementary coefficients by $O(ϵ)$; Neumann inversion retains the bounds. The finite-dimensional real zero mode is NOT inverted with this gap.

At an unbroken vertex use a disk with boundary punctures. The zero-boundary leading operator is ordinary dbar with real boundary values. Reflect across its boundary. A decaying forcing is solved by the Cauchy transform on the doubled sphere after removing its constant mode; the reflected solution has real boundary values. The missing real constant is its n-dimensional kernel. Near each puncture the same Fourier expansion gives a real end constant and exponentially decaying complementary modes. These end constants are bounded linear functionals of the forcing and the one selected kernel constant. One may equivalently obtain the inverse by the compact bordered Fredholm supplier: the reflected trivial double has index n, its kernel consists exactly of real constants (the imaginary part is harmonic with zero boundary), and the same count gives zero cokernel. Adding one real evaluation makes it invertible. This explains precisely why the real vertex constants must be retained rather than claiming invertibility for coincident real planes.

The inhomogeneous leading vertex model is also explicit. Let $g_i=\mathrm{grad}(f_i)(q_v)$. On the upper half-plane put piecewise constant imaginary boundary value $g_i$ on the i-th interval. The holomorphic function with this imaginary part is a sum of logarithms of the puncture coordinates, with coefficient $(g_i-g_{i-1})\mathrm{/}\pi$ and the appropriate common sign. The sum of the jumps is zero. Its real part has the linear strip asymptotes given by the differences $g_j-g_i$. Add the real constant $q_v$. Consequently In formulas: v_app=q_v+epsilon w_v.

has the correct leading boundary values and matches the first-order Taylor expansion of each adjacent gradient edge. The source strip coordinate fixes the sign by $q_s=p_t$. Changing logarithm branches adds only a common imaginary constant already fixed by the boundary values. The derivatives of the decaying remainder at depth R are bounded by $C_k$ $\exp (-\pi R)$; the growing zero part has size C(1+R).

The actual slow graph inverse. On the zero modes put $\tau =ϵ$ s. For an edge gamma of a limiting gradient tree, linearization is In formulas: a_tau+Hess(f_j-f_i)(gamma(tau)) a = forcing,.

with its input/output sign changed if the edge coordinate is reversed. Include its length variation and its endpoint matching at each vertex. At critical infinite ends diagonalize the nondegenerate real Hessian; the inverse on a positive or negative eigen-direction is integration from the end at which its exponential decays. The kernels are bounded by C $\exp (-c\mathrm{\mid }\tau -{\tau }^{\mathrm{\prime }}\mathrm{\mid })$. A variation-of-constants formula on each finite edge expresses all remaining variables in terms of endpoint constants and its forcing. The vertex conditions are then a FINITE matrix, exactly the derivative of the gradient-tree flow-evaluation map to its diagonals. Regularity means that this matrix is onto. Choose an inverse on a complement of its kernel by the ordinary finite-dimensional inverse theorem. For a rigid regular tree the kernel is zero after domain/translation gauges. Over the finite compact rigid moduli this gives one common bound. This is the slow inverse, including the vertex constants left out of the fast inverse.

Use an adapted norm: for the real strip average use $W^{k,p}$ in tau, for its complement use $W^{k,p}$ in physical strip coordinates, and on the vertex cores use the physical norm plus their real constant evaluation. Use the corresponding target norm, dividing the averaged physical forcing by $ϵ$ before measuring it in tau. Matching end constants use their ordinary finite-dimensional norm. Here $p=4$ and $k\ge 3$ are fixed. The Sobolev constants are uniform: slow pieces are measured in tau, fast pieces in unit strip charts, and the fixed vertex cores have fixed charts. A zero-mode interval collapsing to a vertex is handled as its matching constant, not by a claimed uniform Sobolev estimate on a vanishing interval.

The fast/slow decomposition of the full linearized operator has blocks In formulas: [ epsilon D_graph epsilon B ]; [ epsilon C D_fast+O(epsilon) ].

before the displayed target normalization. In these formulas B,C have bounded coefficients on the prescribed finite graph pieces; the core end-constant maps are included in $D_{\text{graph}}$. The CHANGE OF MEASURE in the adapted norms must also be retained: on a long strip the normalized fast-to-slow coefficient is ${ϵ}^{1\mathrm{/}4}B$ and the slow-to-fast coefficient is ${ϵ}^{3\mathrm{/}4}C$. Their Schur product is $O(ϵ)$. Core end-constant forcings have an O(1) map into the matching equation, while their return to a fast equation has an $O(ϵ)$ coefficient. Including the possible cross terms gives a conservative $O({ϵ}^{1\mathrm{/}4})$ correction to the normalized slow graph operator. Eliminate the fast block using its actual inverse from the The two explicit local inverses construction and then the finite graph inverse; Neumann inversion applies to that small correction. This proves the actual combined inverse; it does not incorrectly treat all modes as having a gap independent of $ϵ$.

Pregluing estimates and the nonlinear solution. Take $R={ϵ}^{-1\mathrm{/}4}$. Cut the explicit logarithmic vertex solutions and the gradient strip solutions at physical depth R, using cutoffs with j-th derivative $C_j$ $R^{-j}$. The small graph boundary is straightened by In formulas: (q,p) -> (q,p-epsilon df_i(q)).

on its own source boundary arc; no single chart simultaneously straightens all different boundary arcs. Correct the logarithmic boundary values in these charts before cutting. On a vertex neck the zero-mode Taylor discrepancy is at most C ${ϵ}^2$ $R^2$, its first derivative at most C ${ϵ}^2$ R, and the complementary discrepancy is C $ϵ$ $\exp (-cR)$. These inequalities follow by Taylor's integral formula for $d{f}_i(q_v+ϵRe{w}_v)$, using $\mathrm{\mid }{w}_v\mathrm{\mid }\le C$ R, and by the gradient ODE Taylor formula. The graph flattening changes its dbar coefficients by $O(ϵ)$ with bounded differentiated coefficients; their derivatives with respect to the section variable are likewise $O(ϵ)$. The same bounds hold for the compactly supported time-dependent edge data, since a physical s derivative of their tau coefficient supplies another $ϵ$.

Build an approximate inverse by the ZERO-MODE inverse of the The actual slow graph inverse construction and the FAST inverses of the The two explicit local inverses construction. Glue only the complementary remainders with cutoffs. The slow endpoint constants are matched by the finite graph matrix, rather than by cutting a section of norm ${ϵ}^{-1}$ and claiming its cutoff error small. Differentiating a fast cutoff contributes C/R. The coefficient and zero-mode Taylor comparison across a vertex interval contributes at most C $ϵ$ $R^2$. Thus, in the adapted source/target norms, In formulas: ||D_app Q_app-I|| <= C(R^(-1)+epsilon^(1/4)+epsilon R^2),; ||F(v_app)|| <= C(epsilon R^2+exp(-cR)). (A).

To track the normalization in the second estimate, a physical averaged residual ${ϵ}^2$ R is divided by $ϵ$, giving $ϵ$ R; integrating it to an endpoint constant costs at most another R. Complementary residuals use their physical norm; their R-volume factor $R^{1\mathrm{/}p}$ is smaller than the retained $R^2$ bound. Every derivative up to the chosen finite order has the same conservative bound. Constants depend on the fixed regular tree set, never on $ϵ$. The matching inverse and end Fourier kernels are bounded in the adapted norms. Hence Neumann inversion of the first line gives an actual uniformly bounded right inverse $Q_{ϵ}$ for all sufficiently small $ϵ$. The same cut-and-match argument on the kernel complement proves injectivity there; otherwise breaking such a section gives zero slow graph data and zero fast data, and the error in (A) absorbs its norm.

Taylor's remainder after the boundary flattening satisfies In formulas: ||F(v_app+xi)-F(v_app+eta)-D_app(xi-eta)||; <= C(||xi||+||eta||)||xi-eta||. (B).

The constant is uniform. In the average equation its apparent ${ϵ}^{-1}$ is canceled by the $O(ϵ)$ derivative of the boundary/edge coefficients. In fast equations the physical coefficients are bounded and their nonlinear derivatives are $O(ϵ)$. Products are controlled on unit strip charts and on tau intervals by the fixed Sobolev embedding; the vertex constant evaluations control their common modes. Thus (B) is an actual uniform nonlinear estimate in the stated norms. Since $ϵ$ $R^2={ϵ}^{1\mathrm{/}2}$ and $R^{-1}={ϵ}^{1\mathrm{/}4}$, contraction gives one solution on each fixed kernel-coordinate slice, with correction at most C ${ϵ}^{1\mathrm{/}2}$. Its actual linearized operator is onto by a further Neumann inversion, so these rigid polygons are regular. The same kernel-complement argument gives uniqueness near the preglued tree. This supplies regularity, not just an assignment of a curve to a tree.

For rigid tree moduli every internal finite edge length is bounded away from zero and infinity. A zero-length stratum would be a regular negative-dimensional four-valent stratum; an infinite-length break would have a negative-dimensional regular factor by the index sum. Both are empty. Their compact set is therefore covered by the above fixed finite gluing constructions. This is why the delicate transition between two resolutions of a four-valent tree is not needed for the rigid bijection asserted here. One-dimensional polygon regularity and its collars are obtained separately by supported forcing and the actual generic polygon supplier.

Exhaustion: there are no other rigid polygons. Stokes gives energy at most C $ϵ$ for each fixed generator word. For the edge Hamiltonian data the curvature term has the same bound: its physical coefficient is $ϵ$ $h(ϵs,q)$, so its s derivative is $O({ϵ}^2)$ and its strip integral is $O(ϵ)$. The tropical barrier confines the image to a fixed compact interior region. With no perturbation, each $p_a$ is harmonic and its boundary values are bounded by C $ϵ$, hence $\mathrm{\mid }p\mathrm{\mid }\le C$ $ϵ$. With the supported edge data the same conclusion follows from the reflected strip equation. Its imaginary part has Dirichlet boundary values of size $O(ϵ)$, and its forcing is an $O(ϵ)$ term plus $O(ϵ)$ times its first derivatives. The Dirichlet Laplacian on a unit-width strip has transverse gap ${\pi }^2$; its Green kernel and its first derivative have uniformly bounded integrals, independently of strip length. The finite vertex cores give the same bound after reflection. Thus the resulting $O(ϵ)-coefficient$ integral operator is absorbed by Neumann inversion, proving the claimed uniform bound also for the inhomogeneous edge equation. In either case local reflection/bootstrap gives, away from puncture cores, complementary-mode norm $O(ϵ)$, including every fixed derivative on unit sub-strips.

Take any sequence of such polygons as $ϵ$ tends to zero. Use the actual ordered real disk-domain compactification and its strip fixture. Along a strip, let $q_0(s)={\int }_0^1$ q(s,t)dt. Integrating $q_s=p_t$ gives EXACTLY In formulas: (q_0)_s=epsilon[grad f_j(q(s,1))-grad f_i(q(s,0))]; +the prescribed edge forcing..

The complementary bound makes this In formulas: (q_0)_tau=grad(f_j-f_i)(q_0)+edge field+O(epsilon). (C).

Its differentiated local versions follow from the same strip equation. On a fixed core the real variation is $O(ϵ)$; on a core enlarged to physical depth R it is $O(ϵR)$, tending to zero. Thus each core has a limiting vertex value. The rescaled strip length $ϵ$ times its physical length either tends to a finite nonnegative number, or to infinity. Arzela--Ascoli applied to (C) and then the differentiated equation gives an actual gradient edge in the limit. Zero lengths identify vertices; infinite lengths break at critical points. The explicit critical-end ODE estimates and the finite positive drop between different critical actions give finitely many breaks. The collar barrier excludes every boundary limit. Hence the limit is an actual compactified gradient tree with the required critical inputs and output.

The index formula is |out|-sum|in|+k-2, from the actual polygon cap/domain construction, and is the identical flow-tree dimension. Coherent regularity of all the tree evaluation strata implies there is no negative-dimensional limiting tree. It also implies a rigid limit cannot have a collapsed internal edge or a break, since its corresponding stratum has smaller dimension. Consequently every rigid polygon sequence limits to one of the unbroken regular trees covered in the Pregluing estimates and the nonlinear solution construction. The same argument proves absence of polygons of negative expected dimension for sufficiently small $ϵ$.

To invoke the uniqueness just proved, the convergence must be in the adapted norm, not merely C0. On slow intervals (C) and ODE variation of constants give $W^{k,p}$ convergence in tau, with the endpoint translations fixed. On their fast complements the spectral-gap inverse gives convergence in physical $W^{k,p}$; on the fixed cores ordinary elliptic convergence gives the same, and on their expanding ends the exponential complementary tail and slow Taylor bounds used in (A) apply. Choose a fixed core cutoff first and then increase it; the weighted critical-end estimates handle the remaining tails. These give exactly the adapted convergence to the preglued solution. The uniqueness on the kernel complement therefore identifies every sufficiently small rigid polygon with the constructed one. Compactness of the rigid tree set makes $ϵ$ uniform for the chosen finite collection.

Signs, regular parameter spaces, and continuations. The block elimination in Sections1--3 gives an actual determinant-line isomorphism between the polygon derivative and its slow graph derivative. The fast block is invertible; its determinant line has the canonical positive generator, fixed by its Fourier/Cauchy model and continued without a zero crossing. The slow block is the gradient matching operator with its ordered vertex and critical orientation lines. Thus rigid signs agree. At a broken edge the same elimination is a product of the component eliminations and the real gluing-length direction; their determinant contractions agree. This constructs coherent orientations for these graph objects using their standard graph spin/gradings, rather than assuming a general orientation theorem. Character lifts do not change the Hessians or these determinant lines.

Choose pairwise Morse--Smale primitives so that arity-one edge perturbations are zero. Keep Hamiltonian one-forms ZERO on arity-one strips and their continuation strips; their energy remains the exact fixed action difference. Regularity of nonconstant strips and strip continuations can instead be arranged by source-dependent almost-complex variations: where du is nonzero an anti-complex-linear variation of J can send its nonzero tangent vector to any prescribed vector. A source bump distinguishes coincident image points, and adjoint unique continuation gives the same cokernel-spanning argument. Constants are already regular by the transverse linear energy identity. For higher arities at fixed positive $ϵ$, regularity of the other zero/one-dimensional or parameterized spaces is constructed by interior source-dependent Hamiltonian forcing. At a solution, a Hamiltonian linear in a target coordinate near one point produces an arbitrary forcing vector there; use a source bump to distinguish coincident image points. The actual bordered evaluation supplier says such supported forcings span the cokernel. Finitely many do so on each compact piece of the solution set. Finite-dimensional Sard and a convergent exhaustion give regular data, exactly as for the flow trees. Data on already fixed product boundary faces are retained, since the actual regular polygon collar theorem supplies transversality near those faces. Every change can be kept off the tropical barrier and arbitrarily small. The compact rigid solutions of the Pregluing estimates and the nonlinear solution construction are already onto and remain uniquely nearby under such changes; their signed counts and the absence of negative solutions persist. Compactness/collars from the actual exact polygon theorem now give the A-infinity identities and every parameterized functor/homotopy identity. No general virtual polygon theorem is used.

For variation of the rescaling parameter, choose it to change slowly: in a strip |d $ϵ(s)\mathrm{/}ds\mathrm{\mid }\le C$ $ϵ(s{)}^2$, with analogous differentiated bounds. Any two positive endpoint values can be joined this way by allowing a sufficiently long transition interval. On the slow coordinate $\tau =\int$ $ϵ(s)ds$ the zero equation is again the SAME gradient equation; the extra differentiated rescaling terms are $O(\max ϵ)$. For a variable $ϵ$, use $p_{prime}=ϵ(s)p$ on each strip. Its equation differs from the constant-parameter standard equation by $({ϵ}_{prime}\mathrm{/}ϵ)p_{prime}$, an $O(\max ϵ)$ fast coefficient under the displayed slow-variation bound. Its real average still has the same equation in tau; the original unscaled strip action identity bounds the transformed energy by C max $ϵ$. This retains the scale derivative instead of dropping it. The fast modes keep their $\pi$ gap. The estimates (A)--(C) therefore extend to a fixed painted/continuation tree, uniformly in the endpoint ratio, with smallness controlled by the largest rescaling parameter. The vertex parameter space is the disk-domain space with a paint level; its boundary has painted-tree compositions and the two endpoint faces. It is constructed directly by the same real cross-ratios plus the ordered paint-length coordinates, compactified by exponentials; zero/infinite lengths give precisely those faces. Its local core inverse is the same real-constant/Cauchy inverse with these finite extra domain parameters.

When the Morse functions and edge data are unchanged, pull them back independently of the paint. Forgetting paint then leaves a gradient-tree problem whose dimension for a higher continuation component is minus one when that painted component is rigid. Such an unpainted regular tree does not exist. In arity1 only the unstable constant critical solution survives, with sign+1. The same exhaustion and gluing estimates therefore show that the actual slow continuation functor has linear part the identity and higher parts zero on every word in the fixed finite collection, once its largest parameter is sufficiently small for that finite collection. This is a finite-collection assertion; its uniformity does not extend to arbitrary functions without new bounds.

Build the genuine all-A-infinity pre-functor. Choose a countable collection of the divisor sections and their collar copies which contains a representative of every divisor class and, for every object and finite positive word, copies before and after that word with rational collar slopes. Small interior perturbations can be included inductively. Enumerate it by nested finite sets $S_i$. Positivity is strict ordering of collar directions, so a positive word in $S_i$ cannot repeat an object and has length at most $\mathrm{\mid }{S}_i\mathrm{\mid }$. Thus there are FINITELY many words, arities and relevant character generators in $S_i$. This elementary observation is what permits a finite-collection adiabatic threshold; no false single-small-parameter assertion for all arities is needed.

Fix coherent original polygon data on this object set, extending finite stages by the supported-forcing construction. On each $S_i$ choose ${ϵ}_i$ decreasing to zero, below the rigid and painted thresholds for all its words. The rigid comparison is a strict identification $I_i$ from the ${ϵ}_i$ polygon counts to the Morse counts on $S_i$. Counts of dimensions zero agree exactly; negative counts are absent. The one-dimensional perturbations preserve these counts and give the same A-infinity identities.

Construct continuation pre-functors $C_i$ from the original counts on $S_i$ to the ${ϵ}_i$ counts. Their operations are counted by regular painted disks as above. Their linear term is an upper triangular matrix with diagonal1 in the action basis: variable complex structure does not add curvature to Stokes, so every nonconstant strip strictly lowers action; a constant strip contributes1. There are finitely many generators, so this matrix is an invertible chain map. At a constant solution its linearized kernel is zero by the linear energy identity with transverse real endpoint planes, and its index is zero, hence it is onto. Thus the diagonal term is not an assumed regular constant. A functor with invertible linear part has an actual bar inverse: solve its inverse component in increasing word length, subtracting the finite sum of proper-partition compositions and applying the inverse linear part. Therefore these are quasi-equivalences on their finite object scope.

Arrange strict compatibility rather than assuming independent continuation maps agree. Extend the already chosen data for $C_i$ to the new words of $S_{i+1}$, retaining them on $S_i$ and on product faces. Then concatenate with the slow rescaling continuation from ${ϵ}_i$ to ${ϵ}_{i+1}$. Define the concatenated functor by cascades: its k-th component is the finite sum over consecutive ordered partitions of the compositions of the two component counts. Its bar functor equation is the product of their two bar equations. On $S_i$ the last slow continuation is EXACTLY the identity by the Signs, regular parameter spaces, and continuations construction, so the concatenated $C_{i+1}$ restricts to $C_i$ after the strict identifications $I_i,I_{i+1}$. Original-to-${ϵ}_i$ data on new words are chosen by the same relative forcing extension. This constructs compatible finite-stage functors, rather than relying on an unproved equality for independently chosen homotopy maps.

For each fixed finite word, its component now stabilizes once that word belongs to $S_i$. Define F on that word to be the stabilized component $I_i$ $C_i$. Every A-infinity functor equation involves only finitely many words and partitions and hence holds at one common finite stage. This proves the FULL equation on the algebraic tensor bar module. Its linear component is a quasi-isomorphism on every morphism complex. The object map is the identity on the chosen divisor/function representatives, and every selected section is included in the countable full sub-pre-category; the positive copies supply its unital extension property. Combining with the partition-corrected shrub quasi-equivalence and the strict toric relative-Cech comparison gives exactly the line-bundle subcategory quasi-equivalence. It neither identifies nor generates the rest of the relative Fukaya category.

The algebraic construction identifies the specified toric line-bundle dg category with the relative, character-graded cellular dg model by actual monomial modules and a strict same-index refinement. The shrub construction gives a quasi-equivalence from that model to the boundary Morse pre-category with all partition-corrected higher components. The tropical construction supplies every divisor section and the two actual boundary barriers, and the adiabatic construction gives the signed rigid graph-polygon comparison and the compatible all-arity pre-functor from the selected full relative Fukaya sub-pre-category to the Morse model. Every morphism map is a quasi-isomorphism and every line-bundle object is represented. Composing this finite zigzag gives the claimed A-infinity pre-category quasi-equivalence.

The selected objects and all their transverse morphism complexes are the full ones in the declared relative pre-category, with the coherent positive-sequence operations and unit copies just constructed. No object outside this source section class is required or generated. Thus the conclusion is exactly the original line-bundle subcategory Statement for all smooth projective toric varieties in the declared source datum, without a Fano assumption or an equivalence assertion for the whole Fukaya category.