Analytic foundations for disk and polygon counts

The admissibility inputs are actual domain bounds, not a formal assumption that the desired moduli are compact. For a full single-diagram hat module, [F1] supplies a zero-Chern sector independently of whether it has generators; the required all-sector test therefore tests EVERY normalized periodic domain and gives the global weak pair condition. For several systems the multiperiodic condition in the Statement is imposed on the actual chosen auxiliary multidiagram; pairwise conditions alone are not substituted for it. Its nonnegative zero-mark domains are finite before any index or regularity choice, as proved in [F3]. For the ordinary weak specified-sector branch, fix the finitely many required corner tuples, arities and count dimensions zero and one. If a source has $d$ inputs and $p$ allowed family parameters, its fixed-map index is its count dimension minus $d-2+p$; the finite strata of a compact parameter family therefore require only a finite list of actual indices. Strips and anchored profiles have their stated respective index shifts. For each nonempty same-corner, same-Spin-c slice choose one actual smooth reference and identical transverse real-plane caps for it and every comparison map. The protected nearly symmetric comparison homotopy in [F7] preserves those boundary planes and caps. Consequently the actual fixed-map index difference is $\nu (D(u)-D(u_0))=⟨c_1(\mathfrak{s}),H(D(u)-D(u_0))⟩$ at zero entire mark vector. Thus each fixed actual index gives a fixed relative-index slice after subtracting the reference index. The common conformal and external parameter columns cancel; their finite dimension shifts have already been included. Apply [F2] to these relative-index slices to obtain finitely many nonnegative domains, independently of coefficients. Include the actual continuous concatenations representing the total classes of the required boundary identities: their total index is the same prescribed finite list by actual cap additivity, their marks are zero, and their total Spin-c extensions are the specified ones. Thus their domains belong to those same affine slices, not to a hypothetical finite class list. There are finitely many slices, so selecting their references uses only finite choice, and their union is finite. The domain-energy estimate of [F6] supplies a uniform energy bound $E_0$ before the coefficient choices, on a small family with uniformly bounded geometric constants. Enlarge the finite list for the finitely many separated compositions and relative homotopies required, and choose a strict energy budget $E>E_0$. In the strict branch [F2] separately supplies its stronger finite upper-index domains and lower strip index. No lower index bound is claimed for ordinary weak data.

Treat rank zero by [F3]'s direct model. Its map operator is the zero operator between zero spaces and is onto of index zero. A triangle has one matching constant map, no conformal parameter and no continuous automorphism. A higher stable polygon has its actual positive source dimension and no rigid contribution. The four-marked source is an interval whose two ends are the two constant triangle products, and its parity is zero over the coefficient field. A matching one-parameter triangle family is its parameter interval; other assigned labels or nonzero marked vectors give empty sets. There are no nonconstant strips or bubbles. This proves the point-target count identities without excluding an allowed genus-zero input.

For positive rank apply [F2], protecting the basepoints and a witness in each old region. Pair generators persist without births, and the swept-annulus integral lattice transport preserves old sign witnesses. The actual collar extensions carry the compression disks, capped classes and Spin-c lifts. Consequently the prescribed admissibility and labels hold on the chosen representative. No constant stable polygon can occur: its at least three distinct boundary labels would give a common point of three product tori. This curve choice is explicit and necessary; the coefficient variation at a constant map is zero because its derivative is zero, so a J-only genericity assertion on the original arbitrary fixed triple would be false. Nothing in this step asserts count invariance under the curve change.

Reconstruct the reverse divisor bound needed for ordinary weak components. For each chosen region point $a$, its actual divisor bundle has the canonical nonzero section $\sigma$ on a neighborhood of the compact union of the fixed boundary tori. If a permitted compact family of boundary tracks is used, take its whole union and require the chosen divisor to avoid it; here the analytic coefficient families keep the chosen attaching curves fixed. Choose $U_1⋐U_0$ about that union, disjoint from the divisor, and use the flat connection in frame $\sigma$ on $U_0$. Cover the remaining compact target by finitely many line frames with flat local connections; a finite smooth partition, whose $U_0$ coefficient is one on $U_1$, patches them to a connection $\mathrm{\nabla }$. In a frame write $\mathrm{\nabla }e=Ae$. The transformation $A^{\mathrm{\prime }}=A+g^{-1}dg$ makes $F=dA$ a global smooth bounded two-form, zero on $U_1$. In a pulled-back disk frame write $\sigma =fe$ along the boundary. There $A=-f^{-1}df$, so Stokes gives $\int u^{\ast }F=-2\pi i\mathrm{wind}f=-2\pi i{n}_a(u)$, with the actual relative degree convention in [F6]. For a finite-ended polygon or anchored strip truncate every end in its common exact generator chart inside $U_1$, multiply its coordinate map by a real cutoff and cap at the generator. Scalar multiplication preserves both product boundary planes, with arbitrary specified angles. All modified pieces and caps have $F=0$, so the same integral formula holds on the uncapped source, independently of those choices. Pointwise $\mathrm{\mid }{u}^{\ast }F\mathrm{\mid }\le C\mathrm{\mid }du{\mathrm{\mid }}^2\mathrm{/}2$, hence $\mathrm{\mid }{n}_a(u)\mathrm{\mid }\le C_a{E}_g(u)$. This is not a reversal of energy bounded by domain area. Finitely many region points simultaneously bound EVERY component domain with energy at most $E$, regardless of its split Spin-c label. The same curvature estimate on omitted necks and the actual energy identity give integer degree additivity: core integrals converge, remaining integrals tend to zero, and convergence of the resulting integers is eventual equality. This applies at nodes on divisors and to contained compact bubbles, without a zero-set avoidance assumption.

Choose genuine end paths good on the energy budget without a lower-index assumption. Start in any prescribed smooth neighborhood of the compatible reference paths. Use the actual source-free, time-slice-injective universal argument in [F6]. A nonconstant strip has a free site outside the protected divisor and torus neighborhoods, since an entire image in the latter misses EVERY region divisor and has zero energy. A time-slice injective point and adjoint unique continuation localize a time-only variation there. The explicit countable Hilbert smoothing series with flat time-end jets and ACω Sard give auxiliary paths onto at EVERY nonconstant zero-mark strip, over ALL actual indices and component sectors. This uses no global finite-domain assertion. Cut the auxiliary paths to their fixed endpoint values in shrinking time collars. Flatness gives smooth convergence. If nononto zero-mark strips of energy at most $E$ existed for infinitely many late cuts, [F6]'s actual compactness would give a finite auxiliary limiting chain. Degree additivity and positivity exclude every zero-mark sphere and one-torus disk, because every one-system cut component contains a basepoint. All retained nonconstant levels are onto, with the same exact transverse end operators. Patch their actual right inverses and whole transverse neck inverses by squared cutoffs. Fixed-core operator differences tend to zero; each neck commutator is $O(1\mathrm{/}T)$ and exact-generator tail errors tend to zero. Taking component-core transitions long, then the subsequence late, makes the finite sum of errors smaller than one; Neumann correction gives a right inverse, a contradiction. No bound on the unknown indices of these strips was used. If the entire limit is constant, energy tends to zero; step2.1 makes every integer region degree eventually zero, and the domain-energy estimate forces the original strip to be constant. Its exact transverse operator is invertible and cannot be a counterexample. Thus a late genuine-collar path is onto at every zero-mark solution of energy at most $E$. The same compactness argument proves openness on this budget; starting the auxiliary choice in any specified neighborhood proves density. Choose these paths for the finitely many end pairs.

In the global branch use [F3]'s all-index end choices and coherent polygon data. Its finite face induction preserves any supplied compatible protected tensor and every supplied product angle, and its source-local Hilbert variations are positive at every finite neck length and flat at the conformal faces. Thus genericity is genuine even at large finite necks. In the strict branch use [F4]. Its finite lower-index band and unconditional bubble index budget first exclude every nonconstant strip of index at most zero; its index-one regularity then suffices for every limiting strip which can occur in a low-dimensional polygon. In the ordinary weak branch use step3.1 on the chosen budget and repeat the source-local Hilbert coefficient construction: neither universal surjectivity nor the finite coherent face induction uses global weakness or a lower index bound. It makes nonconstant polygon vertices regular at every actual index. Their relevant limiting strip levels inherit energy at most $E_0<E$ by the actual energy identity, so those strips are onto and have index at least one from their nonzero translation vector. No regularity outside that energy budget is presumed. All three branches fix genuine time-constant endpoint collars and include every allowed source parameter in the actual operator.

For a compact auxiliary parameter manifold $B$ with finitely many boundary charts, construct regularity jointly at its conformal faces; do not assign its dimension separately to different vertices. The autonomous end paths are fixed. On a prescribed planar face with vertices $v$, use the configuration space $B\times {\prod }_v{\mathcal{B}}_v$ and the direct-sum section $({\bar{\mathrm{\partial }}}_{b,v}{u}_v{)}_v$, with each strip phase treated as in [F6]. Its derivative is $((D_v{\xi }_v+C_v\dot{b}{)}_v)$, with ONE shared vector $\dot{b}\in T_{b}B$. At stable vertices the ordered boundary-label sublists of a finite ordered polygon face are distinct. Thus their source-local coefficient bumps are independent even at a common target value; these are variations of different ordered coefficient systems. At each nonconstant polygon vertex the universal bump argument of [F6] kills its obstruction, while every autonomous end level within the energy budget is already onto by step3.1. Hence the joint universal derivative is onto. Its parameter projection has Fredholm index ${\sum }_v{a}_v+\dim B$, where $a_v$ is its fixed-parameter component count index, including its conformal parameters and the actual autonomous strip phase. Apply the countable Hilbert Sard construction simultaneously over integer-index strata, corner tuples and finite face trees up to the fixed maximum arity. There are countably many such map charts and only finitely many label lists and tree types. Coefficient bumps are flat at conformal faces but positive in every vertex's interior; the finite arity induction therefore retains lower prescribed face data and achieves these joint conditions. On boundary faces of $B$ retain already jointly regular compatible data and induct over their finite face strata; source/parameter bumps vanish there, and interior parameter regularity is obtained as in the actual relative construction. This produces joint regularity, not regularity of every individual fixed-parameter polygon slice at a critical parameter.

The joint regular zero chart also supplies actual common-parameter gluing when an individual polygon vertex is not onto at fixed $b$. Take one bounded right inverse $Q$ of $(({\xi }_v),\dot{b})\mapsto (D_v{\xi }_v+C_v\dot{b}{)}_v$ on the finite limiting tree; its finite-dimensional parameter vector is shared, not copied. For a forcing on the glued source, use the squared cutoffs to split it into the vector of component forcings, apply this single $Q$, and cut and glue the resulting section components while retaining its ONE output $\dot{b}$. On component cores the resulting composition is the identity by the defining joint equation for $Q$. Generator-chart parameter derivatives vanish, since protected geometry is fixed, and the remaining parameter coefficients decay along the ends by the rapid coefficient bounds. Consequently the commutator and mismatch estimate is exactly the finite sum $C\mathrm{/}{T}_e+o(1)$ of [F6], with the extra parameter term bounded by its vanishing tail coefficient times the common finite energy norm. Neumann correction gives the augmented glued right inverse. The two cutoff maps on kernels are inverse for long necks by the same error estimate, now retaining $\dot{b}$ once; thus the local zero-chart dimensions and injectivity are the actual ones. Apply the proved Newton correction in the map/source charts with that shared parameter and the neck variables. Its derivative estimates, and the converse closeness of an actual convergent sequence to its pregluing, are unchanged because the parameter ranges in a compact finite-dimensional chart. This gives an exhaustive collar modeled on the joint component zero space times its neck coordinates. In particular a zero-dimensional joint face is one broken configuration with a fixed common parameter, even when its two polygon components are not separately rigid universal spaces.

Use [F6] with step1.1's uniform total energy bound in the ordinary weak branch, or the proved applications [F3] and [F4] in the other branches. Degrees add at every actual region divisor. A zero-basepoint sphere has its constant surface coefficient zero; a one-torus disk has constant coefficient on each one-system cut component, each of which contains a basepoint. Their domain-area inequalities force zero energy, so no nonconstant such bubble remains, including when it is entirely contained in a protected divisor or the diagonal. Constant compact ghosts are deleted only when unstable; no stable polygon vertex is deleted by that convention. The latter cannot be constant here by step1.3. At fixed coefficients the total count dimension is the sum of the regular vertex count dimensions plus the number $q$ of nodes, with one longitudinal phase removed only for an actual autonomous nonconstant strip. In a common-parameter family it is instead the JOINT component count dimension in step5.1 plus $q$, by actual cap and source-tree additivity. That joint dimension is nonnegative by regularity, even when an individual vertex has negative fixed-parameter index. Consequently dimension zero is unbroken; a non-interior dimension-one limit has exactly one node and a rigid joint two-component configuration. With no auxiliary parameter these are the two ordinary rigid components. This rules out every extra multi-level, multi-conformal or bubble escape in the dimensions used by the asserted identities.

Finiteness and collars are topological conclusions of the proved analytic construction, not consequences of a merely formal dimension count. An unbroken convergent sequence converges in the global map chart by the exact end cutoff inverse. A rigid total set is discrete and sequentially compact across all its literal classes, hence finite under ACω by selecting ordered finite distinct tuples and flattening them. Each matching fixed-data pair, or matching common-parameter joint configuration, has its actual nonlinear chart and unique exhaustive half-interval collar by step6.1. All components inherit the total energy bound, independently of their possible split Spin-c labels; step2.1 bounds their domains without reversing the area inequality. The rigid joint boundary sets themselves are finite by the same compactness and joint dimension argument: a further splitting would have another node and negative joint face dimension. Thus no separate assumption of finitely many component classes or separately rigid parameterized vertices is required. Distinct positive-energy cores and strip phases separate distinct collar endpoints. The resulting Hausdorff second-countable sequentially compact one-manifold is compact; its finite interval boundary has even parity. The actual class, total Spin-c label and degrees are locally constant in map charts and carried by explicit neck sewing, so imposing any such label selects the corresponding open-and-closed part and does not assume a class/domain bijection.

For a polygon with $d$ inputs the count dimension is the actual fixed-map index plus $d-2$ conformal parameters; a triangle has no such parameter and no translation quotient. The rigid operations are therefore the index-one strip differential and the polygon counts of index $2-d$. Enumerating the actual one-node ends in step8.1 gives precisely the finite A-infinity sum, including the output differential and every input differential. This proves differential square-zero, the triangle chain-map equation and the quadrilateral composition homotopy. For a compact auxiliary family use the actual joint construction in steps5.1–6.1. Its broken boundaries are the zero-dimensional common-parameter fiber configurations, not in general products of individually rigid parameterized polygon spaces; its prescribed parameter boundary strata are included as well. The parameter is counted once. Fixed-data A-infinity products remain precisely the ordinary products already enumerated. In particular universal index-minus-one triangles are rigid and universal index-zero triangles have their endpoint and strip-breaking boundaries. Counts of those boundaries give the asserted homotopy identities. Zero-basepoint boundary-degeneration terms are zero by step7.1, hence no omitted contribution remains to cancel. The analytic theorem supplies these identities for the actual admissible diagrams and inputs; computing named top cycles, constructing surgery multi-diagrams, identifying filling homology and proving move inverses require their own proofs and have not been assumed here.

For continuation retain the compact-transition source profile as part of the equation. The global branch is [F5]. For ordinary weak specified-sector data reconstruct its analytic proof on step1.1's energy budget rather than apply its global-admissibility Statement. The actual anchored compactness has one unshifted central component and autonomous escaping levels; zero marks exclude bubbles by step2.1's degree argument. Universal source-dependent bumps make the central augmented operator onto on all index strata. Fixed ends only need regularity within the energy budget, supplied in step3.1; constants in a protected generator chart have the exact invertible transverse operator. For a relative parameter family fix its regular endpoints and use parameter-interior source bumps; endpoint zero charts are already regular. The flat time and rapid source smoothing series can be cut to genuine time collars and compact source transitions: a nononto sequence of energy at most $E$ has an anchored auxiliary limit with onto augmented central operator and onto autonomous end levels. The squared-cutoff inverse retains the one central parameter output and has error $O(1\mathrm{/}T)+o(1)$; its parameter forcing is zero on generator necks and uniformly vanishing on escaping cores. Neumann correction contradicts nononto-ness. An entirely constant limit is handled by step2.1 and the zero-domain energy estimate, and its constant operator is already invertible. Thus a late genuine profile or family is regular at every relevant energy-bounded zero. The central rigid continuation operator has index zero with NO translation quotient. A count-one fixed profile has a rigid central zero and one autonomous rigid end strip; a relative one-parameter count has a fixed-map-index-minus-one central zero and the parameter endpoints. Actual anchored product collars, global tail convergence and endpoint parity prove the continuation equations exactly as in [F5], now with these bounded-energy hypotheses checked. For separated compositions retain the two unshifted regular profiles and send their separation to infinity. Every relevant total class was included in step1.1, so the same energy bound gives its actual two-central-component limit; the cutoff inverse and joint kernel matching give the actual glued products and exhaust them. Parameter comparison with a direct profile uses the same bounded-energy relative argument. The constant longitudinal profile has only constant index-zero solutions because every nonconstant autonomous solution within the budget is onto and has its nonzero translation vector. Its exact constants count the identity. This proves the required continuation chain maps, composition and inverse homotopies within the specified sector without importing a global index bound. Together with the polygon and direct point-target arguments, it proves the full analytic assertion; named top cycles, filling identification and Heegaard-move invariance remain independent operation duties.