Named Fermat compact relative and first-order mixed Fukaya construction

Fermat geometry and coefficients. Let $X_N=\{{\sum }_{i=0}^{N-1}{x}_i^N=0\}$ in $C{P}^{N-1}$, $N=4$ or5. It is smooth: all partial derivatives N $x_i^{N-1}$ can vanish only when all projective coordinates vanish. It has complex dimension $n=N-2$. The residue of the homogeneous numerator ${\sum }_i(-1{)}^i$ $x_i$ dx0 wedge … omit dxi … wedge $d{x}_{N-1}$ divided by sum $x_i^N$ is a nowhere-zero volume form Ω. In a chart where a partial derivative is nonzero it is the ordinary coordinate volume divided by that nonzero derivative; these expressions glue because numerator and denominator have the same homogeneous degree N.

Put $D_i=X_N\cap \{x_i=0\}$, $D={\sum }_i{D}_i$. Every $D_i$ and every nonempty intersection is transverse: the remaining partial derivatives cannot all vanish at a projective point. Thus D is SNC, with at most n components through a point, and its actual local complex equations are a coordinate product. $O(D)=O(N)\mathrm{\mid }{X}_N$. Write ${\omega }_{FS}$ for the integral Fubini–Study form representing c1(O(1)). The SOURCE-SCALED form is $\omega =N{\omega }_{FS}$, as explicitly defined in Nohara–Ueda section 3. The induced metric on O(N) has curvature ω. Hence [ω]=PD[D], with $\kappa =1$. If $s=\prod x_i$, the connection primitive obtained from $-d^c$ log||s||² on $X_N\setminus D$ has $d\lambda =\omega$ and integral−1 on a positively linked small circle around each branch, in that integral-curvature convention. The excised source boundary has the opposite orientation, hence contributes+1 to the disk energy. With the alternative unscaled ${\omega }_{FS}$ one would instead have $\kappa =1\mathrm{/}N$; these normalizations are deliberately distinguished. Its logarithmic norm is a proper strictly plurisubharmonic exhaustion toward D. These are explicit metric formulas, not an appeal to an unspecified arbitrary ample divisor.

Even coordinate permutations preserve the polynomial, the metric, λ and D. Their determinant is+1, so they preserve the residue numerator and Ω exactly. The determinant-one diagonal root-of-unity group modulo simultaneous scalar roots also preserves them. On the complement define $u_i=x_i^N\mathrm{/}({\prod }_j{x}_j)$. Then $\prod u_i=1$ and $\sum u_i=0$. The map to that quotient pencil fiber is a finite unramified cover: its deck transformations are diagonal ${\mu }_N$ phases with product one, modulo diagonal scalars, of order $N^{N-2}$. Thus the groups are Γ16 and Γ125 at $N=4$ and5. Even permutations normalize that group. This identifies the actual affine quotient pencil geometry used by the HMS source with this compact relative pair.

The function $W=\sum u_i$ on $\prod u_i=1$ has critical points $u_i=\zeta$, ${\zeta }^N=1$, with critical values Nζ. In logarithmic tangent coordinates $\sum v_i=0$ its Hessian is $\zeta \sum v_i^2$, nondegenerate on that hyperplane. $A_N$ fixes these points and values. The invariant Kähler symplectic connection carries the standard real vanishing sphere from each critical chart to the reference fiber along the chosen path. $A_N$ preserves the real sum-zero orientation because its determinant there is+1. Each lifted vanishing cycle is a compact $S^n$, since $n\ge 2$ and a finite cover of that simply connected sphere splits into sphere sheets. Its λ restriction is closed because it is Lagrangian and exact because $H1(S^n)=0$. The full lifted set is finite and invariant.

The phase map of each sphere has a grading lift because $H1(S^n;Z)=0$. At a fixed downstairs sphere the Ω-preserving finite group action changes a lift by an integer; this is a homomorphism from a finite stabilizer to Z and is zero. Choose lifts on orbit representatives and transport; the grades are therefore coherent on the full finite lifted set. The spin structure on $S^n$ is unique since $H1(S^n;F2)=0$. The orthogonal action on the critical real sphere has its finite preimage in Spin(n+1); transport it with the thimbles. Taking the combined finite lift on the finite orbit of spheres gives a finite spin/character extension with its actual group law. Central spin elements are retained; their underlying target diffeomorphism can be identity. This is why the free-label CHORD model is used instead of claiming every physical translated copy is distinct.

For another smooth degree-N hypersurface with the same source-scaled Fubini–Study class, choose a smooth coefficient path in the complement of the discriminant from the Fermat polynomial to it. This complement is path connected: a generic complex line/polygonal path meets the algebraic discriminant in finitely many points, which can be bypassed by small complex arcs. The total family over the compact path is proper and smooth. Its fiberwise symplectic connection has complete transport because the total family is compact; closure of the ambient Fubini–Study form makes that transport symplectic. Transport D,λ,the spheres and the finite group action together. The transported J is integrable near the transported SNC divisor, so the exact local hypotheses of the relative category construction remain valid even when that divisor is not holomorphic for the endpoint's originally displayed complex structure. The family residue forms identify the source grading covers continuously, and the finite spin lift transports by conjugation. Compact-target parametrized virtual correspondences give the auxiliary-data comparison to other admissible choices; they do not change the underlying symplectic Fukaya category. This retains the original smooth-quartic/quintic A-side scope instead of restricting it to a Fermat polynomial.

With total-divisor exponent $d=\beta \cdot D$, the SOURCE-SCALED relative symplectic energy is d plus the fixed primitive/action and compact curvature terms. Thus every fixed q-order has a genuine energy bound; the relative chain model is q-adically finite. Positive d polygons and bubbles remain in the COMPACT target, rather than being deleted by the affine exhaustion. The $d=0$ sector is exactly the controlled affine branch. This is exactly the source total-intersection exponent $q^d$ in Seidel section 8d. With unscaled integral FS the geometric area would be d/N; it is not silently used as the source q exponent. Base extension to the source Puiseux field uses its actual prequantum/action LINE convention: $q^{A(x)}$ generates the appropriate one-dimensional action-coset line, not necessarily an element of the scalar field. The actual action-line map in Seidel Proposition8.8 identifies these relative and absolute complexes without claiming a real-exponent scalar lies in the rational-exponent Puiseux field.

Free labels and actual compact correspondences. Let X be a compact Calabi–Yau Kähler manifold of complex dimension $n=2$ or3, with its nowhere-zero volume form Ω. Let D be a positive divisor with simple normal crossings, [ω]=κ PD[D], $\kappa >0$, with a G-invariant Hermitian metric on O(D) of curvature ω/κ and the resulting primitive λ of ω on $X\setminus D$, whose integral on a positively linked small circle is−κ along each component. Its actual local charts have $D=\{z_1\cdots z_r=0\}$; this hypothesis is part of the input, not inferred from an arbitrary singular divisor. Assume the chosen auxiliary complex structure is integrable in a fixed neighborhood of D. Let $L_i$, i∈I, be the actual finite collection of compact oriented graded spin Lagrangian spheres in $X\setminus D$, exact for λ, with their chosen grading lifts. A finite lifted group G preserves ω,D,λ,Ω and permutes the spheres with their specified grade/spin/brane-line action. It acts trivially on the formal variable q. The single formal exponent is the intersection with the TOTAL divisor. Separate multivariable nonnegative exponents would require the corresponding individual positivity hypotheses and are not inferred here.

Use labels $a=(i,g)$, g∈G, with h $a=(hi,hg)$. The action is free. The physical brane is $L_i$ with the transported grade/spin data; adding labels does not change it. Choose a representative of every ordered-pair orbit. A compactly supported Hamiltonian on $X\setminus D$ has generic endpoint image ${\mathrm{\Phi }}_H(L_a)$ transverse to $L_b$: the endpoint variation of its local Hamiltonian gradient can realize every tangent vector, so finite-dimensional local Sard parameters and a finite compact cover give transversality. Its intersection/chord set is finite by compactness. Extend the data to every pair by $H_{ha,hb}=H_{a,b}$ composed with $h^{-1}$, and likewise J by pushforward. There is no stabilizer equation to satisfy, even for a repeated pair or a spin-central element acting trivially on X. Thus one cannot replace this construction by an assertion that all static translated physical copies are distinct or transverse.

For a self pair, and for any two labels of the SAME physical sphere, choose the pair Hamiltonian more specifically. The actual Weinstein neighborhood identifies it with the zero section of T*Sn. Take $H=-\epsilon f(q)$, with the convention ${\iota }_X\omega =-dH$ and $\omega =d(pdq)$, and f a two-critical-point height function and a fiber cutoff equal to one along its entire small time-one trajectory. Its time-one image is graph(ε df), so its only chords are the two critical points. The actual scalar-plane capping calculation, with the Ω-grading, gives integer degrees n−Morse-index, namely0 and n. Cutoff regions create no additional chords because every trajectory starting on the zero section stays inside the region where that cutoff is one. Thus these self/duplicate morphism modules are R in degrees0 and n, with no degrees1 or (for $n=3$)2. This is an explicit endpoint/degree calculation, not an assumed Floer–Morse adiabatic correspondence. Choose this data on pair-orbit representatives and translate as before.

Choose the time-only strip J on each representative generically by the following actual argument. For a nonconstant finite-energy strip, $u_s$ solves its homogeneous linearized real CR system, so its nonzero locus is dense by unique continuation. F(s,t)=(t,u(s,t)) is an immersion there. Critical-sheet images have zero two-dimensional Hausdorff measure: on a small box about a rank-one point its derivative is close to a rank-one map; cover its image by $O({\epsilon }^{-1})$ balls of radius εr, giving total two-area O(εr²), then use the finite compact cover and let ε decrease to zero. End-chord images have the same empty-interior property. Countably many compact regular immersion sheets cover the remainder. If an open regular patch had no injective point, Baire would give a smaller open coincidence with one other compact sheet, with s' separated from s. The inverse immersion branch is smooth and preserves t, so write u(s,t)=u(phi(s,t),t). Differentiating and subtracting the two Floer equations gives $({\varphi }_s-1)u_s(\varphi ,t)+{\varphi }_t$ $J_t$ $u_s(\varphi ,t)=0$. The vectors $u_s$ and $J{u}_s$ are real independent, so $\varphi =s+c$, $c\mathrm{\ne }0$. Unique continuation extends this local translation equality globally. Periodicity and finite energy force $u_s=0$, contradiction. Near an injective point end convergence bounds all other candidate preimages in a compact s interval; compact separation and local immersion then make injective points open. This proves their openness and density rather than assumes it.

A nonconstant strip has a nonempty regular patch outside D+, since its boundary/end images are there and $u_s$ cannot vanish on an open patch without vanishing globally. A compactly supported $J_t$ variation near (t,u) at an injective point affects only that preimage. Tangent compatible variations span every target vector: write $Y=JK$, with K selfadjoint and anti-commuting with J. In unitary coordinates K is a complex symmetric matrix followed by conjugation, whose first column is arbitrary, so Yv is arbitrary for $v\mathrm{\ne }0$. The path J exp(εK) is compatible because its metric is g exp(εK) and its square is minus identity. Thus any adjoint annihilator of the universal strip derivative vanishes on that patch and everywhere by unique continuation. The actual weighted Fredholm/weak-adjoint suppliers give the finite-dimensional cokernel. At each compact cutoff take the supported finite obstruction spaces and relative rational transverse branches of F3 on a translation slice; this supplies the required strip counts with fixed end data and no additional infinite-dimensional genericity premise. Constant chord strips are the explicitly invertible transverse-end problem and are unstable after translation quotient. Translate the chosen data to all pair orbits.

For stable words a0,…,ak choose coherent domain-dependent Hamiltonian one-forms and tame structures on a representative of each orbit, relative to the already prescribed lower-face product data and fixed pair ends. Interior source-local Hamiltonian gradients give arbitrary universal forcing, including when $du-X_K$ vanishes identically. Supported finite-dimensional Sard parameters therefore give regularity on every prescribed finite slice where no bubbles intervene. Where bubbles or degenerations require a virtual procedure, choose the actual supported obstruction atlas and rational branch construction instead. In both cases boundary prescriptions are finite product lists, and the extension is relative to those products. Induct by intersection/energy and arity as in the actual disk-correspondence proof. For EVERY word define the remaining data by exact translation from its unique chosen orbit representative. Freeness ensures uniqueness, and translation of a boundary product is the product of translated factors. Thus all gluing, branch and higher-face data obey the group law EXACTLY, not up to a comparison homotopy.

End data and Hamiltonians are supported away from D. Near D the map equation is ordinary holomorphic. In a normal-crossing chart a component not contained in D has holomorphic normal coordinates $z_j$ composed with u; their zero orders are nonnegative and the total divisor multiplicity is their SUM, including at a crossing. A component entirely contained in D is necessarily a sphere, since all disk boundary labels lie in the complement. Its TOTAL intersection with D is positive because its ω-area equals κ times that intersection. This total statement does not pretend every component self-intersection is positive for an arbitrary SNC divisor. Hence every stable class has $d=\beta \cdot D\ge 0$, and d is additive over actual components. The topological relative tree-collapse proof preserves that class, including its homotopy annotation and integral homology image. Perturbations of the virtual problem may cease to be holomorphic in D-normal coordinates, but remain in a fixed original class; its d-label is unchanged. No positivity theorem for the perturbed section's accidental geometric zero set is asserted.

Excise small circles about the divisor intersections. Stokes for λ, together with the linking integral−κ and the reversed orientation of the excised circles and the exact boundary primitives on $L_i$, gives E(u)=κ d + sum(input actions)−output action + integral(curvature K).

The same formula on each stable component gives additivity at breaks. Coherent data have uniformly bounded curvature integrals on compact domain cores, zero on product ends. Thus for each generator word a bound on d gives an actual area bound. Since X is compact, there is no escape. Its disk/sphere compactness therefore applies; only finitely many component classes and tree types occur at fixed d cutoff. The virtual coefficient at every word/d is a finite rational count, and extending Q to C gives a power series in $R=C\mathrm{⟦}q\mathrm{⟧}$, weighted by $q^d$. No convergence of unrelated real periods or unjustified specialization of an arbitrary Novikov series is used.

The $d=0$ sector avoids D. A standard proper plurisubharmonic potential near D, with leading term −κ ${\mathrm{\Sigma }}_j$ $log\mathrm{\mid }{z}_j{\mathrm{\mid }}^2$ in a crossing chart, prevents a $d=0$ curve with boundary/end data in the fixed interior compact set from entering an arbitrarily small divisor neighborhood: the potential is subharmonic on that ordinary holomorphic part and its maximum would occur at the fixed inner level. This is the affine no-escape branch only. Exactness then excludes spheres and one-label disk bubbles there. At $d>0$ such curves DO meet D, and are retained in the compact-target moduli; neither this barrier nor exactness is used to delete their disks or spheres.

At each nondegenerate chord end use the actual transverse weighted inverse, with one weight smaller than the minimum of the finitely many end gaps. A label-changing puncture is REMOVED and has that strip end; it is never treated as a point of a single smooth boundary trace. Away from these disjoint end fixtures, every boundary chart sees one smooth Lagrangian arc. At a boundary bubble node that arc has one label $L_i$: retain its real shared constant and the supported evaluation/real-node construction of F2. At an interior sphere node retain its complex shared constant. At a broken chord there is only the discrete common chord label, not a real constant matching variable. Make the JOINT component equation and all real/complex evaluation differences onto by finitely many supported core forcings; the bounded evaluation lifts and finite cokernel complements are those of the bordered supplier. Take the sum of all these forcing spaces on a tuple overlap. Include all its finite stabilizer translates. On the central matched kernel retain the actual finite kernel, shared node values, obstruction coordinates and slice equations. Preglue the exponentially decaying transverse tails to their fixed chord and the smooth node tails to their shared value, using disjoint fixtures. The patched inverse has error bounded by a finite sum of decaying cutoff errors; its Neumann correction is an actual right inverse on the fixed core-coordinate complement. For the real-node factor use the separate real-mean construction of F2, not a transverse inverse for the equal-plane problem. The unit-window Sobolev product estimate gives $\mathrm{\parallel }N(\xi )-N(\eta )\mathrm{\parallel }\le C(\mathrm{\parallel }\xi \mathrm{\parallel }+\mathrm{\parallel }\eta \mathrm{\parallel })\mathrm{\parallel }\xi -\eta \mathrm{\parallel }$ on these charts. The residual tends exponentially to zero with the minimum neck length, so contraction solves the complement equation with the finite kernel and obstruction coordinates prescribed. For the converse, normalize the ordinary marked components first; core compactness and one-boundary-trace rescaling are the single-label local arguments, while a translated label-changing end is retained as an ordinary transverse Floer strip. Its nonconstant levels have positive action difference between the fixed finite chords, so only finitely many can occur at the energy bound. After extracting those levels every remaining translated neck limit is its constant chord; the local action/Poincaré estimate and the fixed transverse gap give exponential weighted tail convergence. Thus convergence enters the constructed contraction chart, whose uniqueness gives inverse gluing. Smooth real/complex bubble nodes use exactly F2's already proved converse. Differentiate on higher Sobolev orders before passing to the common double-exponential profile, as in F2; all fixed derivatives are polynomial factors times exponential tails and become flat at the nodal face. Tuple overlaps retain the same stabilized components and extra-forcing-zero equations, so their actual forgetful maps have the finite normal/obstruction isomorphisms and exact cocycles. This constructs the compact cornered mixed atlas, including its broken-chord product collars; it does not invoke the single-Lagrangian Statement of F3 for a mixed polygon.

Relative-spin pinching orients each disk component with its actual boundary brane frames. At a corner the chosen graded chord orientation line is the capping determinant, computed using the actual transverse end operator. Tensoring these caps converts the mixed-boundary operator into the ordinary real-boundary determinant already oriented; its real matching maps and domain-face Jacobians give the same suspended Koszul rule. Sphere factors remain complex oriented. The Ω-grading makes every operation have its full integer cohomological degree2−k by the actual Fredholm/capping index. In the Ω trivialization, each graded boundary phase has a real lift, hence zero winding on a single-label disk; the mixed-end caps account for the input and output chord degrees. Thus the disk supplier's Maslov coefficient e^(μ/2) is trivial on single-label components, and its energy-only specialization loses no integer grading in this Calabi–Yau construction. The specified G lift acts on each cap and brane line, and the tuple action carries these determinant maps exactly. Its action on output lines equals the tensor action on inputs under every operation. A spin-central element may act by object scalars; this is retained in the external line factors, not erased in a mod-two simplification.

Here is the finite rational perturbation extension on that MIXED atlas; F3's single-object theorem is not being applied with new hypotheses. Fix an energy/arity budget and its finite set of generator words, classes, stable trees and tuple charts. Boundary breaks are products of the FULLY CONSTRAINED lower problems over their finite common chord set. Bubble nodes with an $L_i$ evaluation are instead thickened and matched before choosing branches; a zero-length pearl edge is sewn to that same nodal thickening with the common smooth profile. Those disk/zero-edge seams are internal, with equal sections and opposite normals, exactly as in the explicit F3 construction. Order the finite problems first by total divisor degree, then arity, then constrained virtual dimension for the remaining unary breaks. A zero-input bubble has positive divisor degree: at degree zero the chosen single-label raw disk is exact and constant unstable disks are not counted. A degree-zero non-unary factor has arity at least two and lowers its companion's arity; a remaining nontrivial unary break lowers the companion's constrained dimension. Negative-dimensional transverse factors are empty. Thus every genuine boundary factor is already prescribed, including curvature factors whose companion has arity one larger. On each finite tuple reduction choose smaller supports with intersecting closures comparable by barycentric subdivision of its finite nerve. The extra obstruction coordinates give normal form $s_J(x,z)=(a(x,z),z)$ with $a(x,0)=s_I(x)$. Construct nested retractions and direct-summand transports with ${\pi }_{IK}={\pi }_{IJ}{\pi }_{JK}$ on their actual common neighborhoods: on a protected product collar use its product retraction; elsewhere extend its normal-quotient splitting by vertical vector fields and their finite-group averages, then take the normal flow and shrink. Intersecting active labels form a chain, so the shorter composite prescriptions agree. This is the retraction construction in F3's proof, whose only inputs are these finite normal charts and product collars, now supplied above. Extend a finite rational section DISTRIBUTION from I to J by direct-summand transport after ${\pi }_{IJ}$, with zero in the extra obstruction summand. Its zero equations are $z=0$ and $s_I={\nu }_I$; no false identity $a=s_I\pi$ is required. The prescribed broken-face distributions are tensor products over DISCRETE chord labels, hence their derivatives are block diagonal and onto. On corners their tensor products agree as distributions with the same determinant permutations. On the unprotected compact region add finitely many local cutoff/frame parameters, together with their whole finite group orbit distributions and rational product weights. Every branch has an independent parameter onto its obstruction fibre; independent supported evaluation lifts supply any remaining constraint normals. Finite-dimensional Sard, relative to the already transverse protected collars, therefore makes every branch and stratum transverse simultaneously. Smallness keeps all zeros in the controlled supports, and extra-forcing-zero normal cancellation gives the same branch masses and orientation lines on overlaps. An isolated zero is counted with its rational branch mass divided by its actual point stabilizer; a tuple cover multiplies the number of lifts by exactly the group-order ratio, so this count is independent of its chart. For a one-dimensional zero system the auxiliary scalar-height construction of F3 is unchanged: regular slices have locally constant weighted count, an interior critical point creates opposite signs with equal mass, and a boundary half-arc contributes its outward mass. The total boundary mass is consequently zero. All its genuine boundary points are the constructed chord products and single-label curvature insertions; internal seams cancel and sphere nodes have real codimension two. The capping/real matching determinant comparisons above identify their suspended signs with the nested curved operations. This proves the full curved A-infinity identities coefficientwise. Repeat relative to the previous finite budget; each fixed coefficient stops changing, giving a coherent completed system. Its finite chord basis, bounded integer grading, strict group translation and $q^d$ weights are those already constructed, not extra conclusions imported from F3.

Unmarked collapsed boundary. An unmarked positive-energy stable disk can lose its entire boundary to a point p in L. Its stable limit is a sphere tree with one distinguished smooth point mapped to p, as in the corrected prestable disk compactification. Add one temporary OUTPUT boundary mark. The collapsed configuration is now represented by a constant disk with that output and one interior node attached to the sphere tree. It is a genuine stable marked map. Its actual marked disk atlas has a complex interior fixture q and the ordinary sphere/kernel/obstruction coordinates. Forgetting the temporary output must quotient the disk rotations, not incorrectly keep its complex q as two unmarked smoothing directions.

At the constant ghost disk choose its interior node at zero, boundary the unit circle and real tangent $TL=R^n$ in a complex frame. Its real CR operator is onto, with kernel the constant real vectors. For every a in $R^n$ choose ξ_a(z)=i a η(|z|²), where $\eta =1$ near zero and $\eta =0$ near the boundary. Its boundary value is zero, its node evaluation is i a, and ∂bar ${\xi }_a$ is supported on a compact rotational annulus away from the node and boundary. These n forcings make the ghost CR/evaluation map onto in the missing imaginary node directions. They are invariant under disk rotation as INTRINSIC form-valued sections: ∂barη(|z|²) is a multiple of z dbar z, which is rotationally invariant. Their real coefficients a carry the TRIVIAL circle representation. Thus no nontrivial-weight equivariant obstruction Euler class is being hidden.

The full augmented ghost kernel consists of its real constants plus these radial imaginary sections, hence is circle fixed. Match its complex node value with the sphere evaluation and retain the ordinary sphere/core obstructions. Every central finite kernel coordinate and every obstruction coordinate is therefore fixed by this ghost rotation; only the complex fixture q rotates. Choose ghost metrics/cutoffs radial. The central linear right inverse can be made rotation equivariant by averaging its conjugates over the circle: its RIGHT-INVERSE equation is linear, so is preserved. Its kernel complement is detected by the invariant node values/obstruction coordinates and sphere core evaluations and is preserved as well. This Haar average is an ANALYTIC choice of bounded operators; no virtual count is divided by an infinite group and no real coefficient is claimed rational by integration.

The actual gluing equation, the nonlinear exponential chart and the annular forcing extension are natural under these domain rotations. The equivariant linear splitting and uniqueness of the uniform contraction consequently make its final obstruction section σ(w,κ) invariant under the circle action $w\mapsto e^{i\theta }w$, with κ and E fixed and E carrying the trivial circle action. Use the coarse quotient coordinate $t=\mathrm{\mid }w{\mathrm{\mid }}^2\ge 0$, not |w|. To verify smooth descent, express each component in a circle-invariant local frame. Its Taylor polynomial of degree 2m+1 is circle-invariant, so each homogeneous term is $c_j(w\bar{w}{)}^j$. Taylor's theorem gives remainder derivatives of order ℓ bounded by $O(\mathrm{\mid }w{\mathrm{\mid }}^{2m+2-\mathrm{\ell }})$. Restricting to $w=\sqrt{t}$ and applying the chain rule shows that the jth t-derivative of the remainder is $O(t^{m+1-j})$ for $0\le j\le m$. Thus the descended component is $C^m$ up to $t=0$; since m is arbitrary it is smooth. Conversely a smooth quotient component F(t) pulls back to F(|w|²), which is smooth and invariant. The same weighted Taylor argument for a smooth weight-one equivariant coordinate change gives w'=w a(|w|²,κ), with a smooth and a(0,κ)≠0 after the finite sphere-tuple chart is fixed. Hence t'=t|a(t,κ)|² is smooth with positive normal derivative. This is a genuine smooth corner chart with the same finite sphere-tuple group and the SAME finite obstruction space E. The value $t=0$ records the sphere-point incidence condition p in L. The marked complex smoothing supplied two real dimensions; its unmarked quotient supplies one, giving the correct unmarked virtual dimension n−3 and collapsed stratum n−4. No infinite-isotropy orbifold or arbitrary smooth structure on an unspecified quotient is asserted.

Every other unmarked chart is obtained by stabilizing its NONCONSTANT components with interior slices independently of a boundary output. Usual unstable constant components contract; two-boundary-node constant artifacts are replaced by the already constructed real-node chart, not retained with a translation stabilizer. The only whole-boundary exception is the one in steps 1.21–1.24, with a single constant disk and single interior node; a disk with two interior nodes is already stable. Add a temporary output to extract compact limits, then forget it using these charts. The actual map/graph topology and the explicit quotient above show the forgetful map is continuous and onto the compact unmarked stable-map space. Its nonexceptional smooth-domain fibers are boundary circles modulo finite map automorphisms; above the collapsed limit that circle shrinks to the quotient center. All ordinary tuple coordinate changes and sphere fixtures are unchanged. Thus the unmarked positive-energy moduli has the required finite collared virtual atlas, not merely a presumed proper quotient.

Finite target symmetries act on the radial ghost construction by their actual tangent/spin maps; a sphere tuple automorphism may rotate its node frame, which only rotates q and is removed in the radial quotient. Choose all target data on free label orbits and transport. Rational branch weights remain those of the finite tuple groups. Hence the S2 unobstructedness construction is natural for the required finite spin extension. It does not rely on a generic simple-disk decomposition theorem or on the claim that an arbitrary interpolation of nonvanishing sections remains nonvanishing.

Bounding objects, continuation and finite replacement. curvature and bounding data

The Ω-gradings give Maslov index zero to every single-sphere disk class; q has degree zero. For L=S3, the actual rational single-object construction followed by its curved transfer to the perfect Morse cohomology model H*(S3) has no degree-two target. Thus its transferred curvature is zero. The self-chord model from the two-critical-point self-chord construction has only degrees0 and3, so its curvature also has no degree-two target; no bounding correction is needed. In more general auxiliary chain models, the comparison's positive-energy zero-input component supplies the actual MC solution, by its length-zero equation. This proves existence without assuming that the untransferred curvature itself vanished.

For the explicit self-chord choices choose $b_a=0$ on every label, using the zero-curvature models constructed in this proof. For more general equivalent auxiliary models choose bounding data on the FREE label-orbit representatives by the same comparison constructions, then translate. Every label then has an actual MC solution $b_a$ of positive q valuation, and $b_{ha}=h{b}_a$ exactly. Twisting every operation by all b insertions gives an uncurved category. At fixed q order only finitely many positive-valuation insertions occur, so the series converges; the curved bar identity implies the twisted bar identity directly by insertion. The group action remains strict because both the original operations and the bounding data are translated exactly.

Continuation between auxiliary choices is constructed by the same actual parameterized virtual moduli, with a compact s-window and fixed ends. Its curvature bound adds only the integral of the source derivative of H. It gives an A∞ comparison, including its positive zero-input term, so sends MC solutions to MC solutions. For a vanishing sphere H1(L;C)=0. If two transported MC solutions first differ at $q^r$, the leading difference is d0-closed of degree one; it is d0 of a degree-zero cochain. The corresponding degree-zero bar gauge changes one solution by exactly that differential at $q^r$ and no smaller order. Repeat for $r=1,2,\dots$ . The product converges q-adically and yields an actual gauge equivalence. Thus translated choices on two labels of the SAME physical sphere are genuinely equivalent bounding objects, not merely presumed so because their supports coincide.

A candidate geometric unit is the oriented fundamental cycle of the sphere, of cohomological degree zero. Use its specified two-critical-point self model after the actual bounding twist: its only degrees are $0,n$, $n=2,3$, and $q$ has degree zero, so its degree-zero fundamental element is closed. Its transport to other chain coordinates is by the actual external-edge/cap comparison, not an asserted new fundamental-cycle count for arbitrary forced data. Perform the binary unit calculation on the actual finite Morse/chord model. For each input generator x and output generator y, impose the supplied stable/unstable endpoint incidences, or the fixed nondegenerate chord ends; these are the FULLY CONSTRAINED problems of the rational-correspondence supplier [F3, proof 5.1] and [F3, proofs 10.1–12.1]. Since the fundamental input has cohomological degree zero, it imposes no additional incidence condition at its marked point. A coefficient of multiplication by it with |y|=|x| has virtual dimension zero. The parameter-family count for |y|=|x|−1 defines a degree-minus-one matrix K; for |y|=|x| the same parameter family has virtual dimension one and its boundary gives the matrix homotopy equation. Thus only these fully constrained zero/one-dimensional spaces are used. Impose compatible transverse rational sections on every stratum of their finite atlases; no such dimension assertion is made for the unpaired chain-valued correspondence. A constant disk carrying the forgotten unit mark and two node flags is a real-codimension-two stratum before forgetting; any chain of such contractions has at least this codimension. Interior sphere nodes likewise have real codimension two. In the zero/one-dimensional unit and homotopy problems their zero sets have negative virtual dimension and are empty. Compactness of the finite reductions gives zero-free collars of these strata, also relative to the prescribed lower face products. On the remaining noncontracting component cores, forgetting the free unit mark preserves the node parameters and is a smooth projection with one-dimensional marked-point fibre. Pull back the regular base sections there. The unit input cycle imposes no incidence condition, so a stable rigid contribution would lie over a regular base of virtual dimension minus one and is empty. The remaining contracted one-node unit cap is the ordinary strip fixture. At a nondegenerate chord its normal operator is ${\mathrm{\partial }}_s+A$, with the same invertible self-adjoint end operator at both ends; positive/negative spectral exponential integrals give its inverse. The constant strip contributes the identity on the chord determinant line, with the suspended left/right unit sign. Nonconstant autonomous index-zero strips have a free translation parameter and a regular negative-dimensional quotient, so contribute nothing. Interpolating these data to the original data, relative to the zero-free collars and the allowed one-node products, gives compact one-dimensional unit/cap families. The orientation supplier [F3, proofs 6.1–9.1] uses cohomological Morse degree $q=n-r$, degree-minus-one suspension, the outward-normal-first real-face orientation, and insertion sign $(-1{)}^{{\sum }_{\text{preceding}}(q_j-1)}$. Its real matching/capping orientation identifies the constant cap with the identity on its determinant line. The boundary-normal-first convention and suspended tensor order give the usual right-unit Koszul sign from moving the unit past x; writing $L_e(x)=m2(e,x)$ and $R_e(x)=(-1{)}^{\mathrm{\mid }x\mathrm{\mid }+1}{m}_2(x,e)$ in that suspended convention $(\mathrm{\mid }x\mathrm{\mid }=q_x-1)$, the two boundary identities are $L_e-id=d{K}_L+K_{L}d$ and $R_e-id=d{K}_R+K_{R}d$. These are matrix identities on the finite model, not a geometric homotopy on every singular chain. The separated pair of continuation windows gives the inverse cap comparison by the same constant-strip endpoint and gluing calculation. Curvature faces remain in the compactification and cancel after the supplied bounding-cochain insertion. Choose each finite-budget perturbation relative to the already fixed smaller-budget sections and all constrained product faces, as in the rational supplier [F3, proof 5.1] and [F3, proofs 12.1–14.1]. Their coefficient identities therefore agree under cutoff restriction, and the energy-completed limit preserves them coefficientwise. These arguments prove binary cohomological units and continuation inverses on the finite model. For other singular-chain coordinates, the actual external-edge/filtered A-infinity comparisons in [F3] and [F4], applied to the supplied bounding objects, induce cohomology isomorphisms. Their length-two functor identity says that their first component preserves the binary product on closed classes: the discrepancy is the differential of their second component. Consequently the image unit acts as identity on every cohomology class, since every class has a preimage. This transports a cohomological unit; it constructs no high-dimensional universal chain homotopy. No smooth forgetful map through simultaneous node contractions or geometric higher-unit-forgetting construction is asserted. Higher strict normalization is the algebraic bar argument after the stated freeness input. All finite choices are made on free label orbits and transported, with the actual brane determinant lines retained.

An invertible continuation class $c_{ab}$ for two labels of the same physical bounding sphere is the fundamental class in their continuation/chord complex, transported by those cap maps and the gauge of the positive-valuation gauge construction. The compactified pair-of-caps with variable intervening length has one end the composite and the other the identity cap; its oriented boundary gives $[c_{bc}][c_{ab}]=[c_{ac}]$ and $[c_{ba}][c_{ab}]=[e_a]$. All group translates use the same parameterized data and determinant maps, so these equalities are G-natural. This proves genuine cohomological invertibility and essential surjectivity when one label per physical sphere is retained. It does not require those equalities to hold as literal equal chains.

Establish relative cohomology freeness from the actual CENTRAL grading pattern, rather than assuming it from chain freeness. Suppose the source central vanishing-cycle calculation identifies each pair of DISTINCT physical character labels with a vector space in ONE integer degree, and each self/duplicate-physical-sphere pair with degrees0 and n, $n=2$ or3. Amplification creates off-diagonal COPY pairs of the same physical sphere, so these two-degree exceptions are explicitly retained. The common exact condition is: no fixed Hom has two adjacent degrees. Contract the finite central chord complex over the coefficient field onto that known cohomology, then extend the contraction to R and perturb by the positive-q differential. The explicit q-adic geometric series converges and transfers the differential to the same graded cohomology module tensored with R. It has degree+1 and q has degree zero. There are no adjacent degrees in any fixed Hom, so that transferred differential is identically zero. Thus the relative cohomology is R-free. In the exterior smash-product central model the degree pattern can be checked without an unspecified flatness assertion: for the determinant-one diagonal ${\mu }_N$ group, subsets J,K have equal characters only when $1_J-1_K$ is constant modulo N, hence $J=K$ except the empty/full pair. Every nontrivial character slot is one exterior subset degree; the trivial slot gives the physical degrees0 and $n=N-2$. Object grading shifts change off-diagonal degrees but cannot create adjacency within a fixed slot. The actual central IDENTIFICATION and its physical gradings remain the named vanishing-cycle computation, not claimed from this elementary character check alone.

Now use that proved input before claiming a strictly unital small model. The geometric chain category just constructed is bounded finite-free over R, strictly G-equivariant, uncurved, and has the actual cohomological units constructed from the fundamental cycles. Once its cohomology is proved finite-free over R, the equivariant splitting is explicit: over the PID R the finite boundaries and cycles are free; split $C^j\to B^{j+1}$ and $Z^j\to H^j$ by right inverses, then average those TWO linear right inverses over G. Their right-inverse equations survive averaging. Thus $C=B\oplus i(H)\oplus S$ equivariantly, d:S→B is an isomorphism, and $h=d^{-1}$ on B and zero elsewhere satisfies $dh+hd=1-ip$ and $h^2=hi=ph=0$. All finite matrices are q-adically continuous. This gives the actual continuous equivariant strong deformation retract; merely averaging a previously chosen h would not prove these side identities. Rooted-tree transfer puts i at leaves, h on internal edges, the old operations at vertices and p at the root. At each fixed arity finitely many trees occur once the differential is absorbed. The edge identity $dh+hd=1-ip$ cancels contracted edges against old bar relations and cut edges against new compositions, proving the transferred identities and comparison maps. Every assigned map commutes with G, so the resulting uncurved minimal model has the actual linear strict G action. This is the stated finite-free transfer proof incorporated here explicitly, not an unstated extra premise. Its binary product has the unit classes as its strict algebra units. The preceding normalized-bar proof now gives an explicit strictly G-natural aritywise gauge, with first component identity and unchanged binary product, that makes every higher operation vanish on unit inputs. This is the strictly unital finite-free chain model. No finite strict-unital extension is asserted from an unconstructed geometric cone on a forgotten unit mark. The cohomology freeness is an exact required computational input, not inferred from free chain modules; the stated central cohomology hypothesis remains necessary for this conditional replacement.

Let P be the finite direct sum of all freely indexed objects. Its total geometric endomorphism complex is bounded finite-free over C⟦q⟧, with object labels, a strict finite-group chain action, strictly natural operations and actual cohomological units. After the proved freeness input, steps 1.33–1.44 give its strictly unital finite-free minimal replacement with the object idempotents. The present assertion about a finite-free minimal replacement uses the stated central degree hypothesis; no torsion-freeness is inferred merely from free chain modules.

Retaining one copy for each physical vanishing sphere gives a full subcategory. Every omitted copy is isomorphic in H0 by the continuation-class calculation (1.32), so inclusion is a genuine A∞ quasi-equivalence after the standard pretriangulated and idempotent completions. More concretely its representable modules are equivalent by multiplication with $c_{ab}$ and $c_{ba}$, with the explicit cap homotopies. The bimodules implementing the matrix-amplified comparison have their tensor-evaluation maps identified with these products, and those homotopies prove their inverses. This is the actual Morita input; a chosen small skeleton need not itself carry a strict G action. The amplified generator does, and its Hochschild/Morita action is the geometric finite action. Central object-scalar conjugations act inner, with the usual insertion homotopy on the bar complex. The stated finite-free equivariant contraction can now be applied once its cohomology hypothesis is actually proved, rather than supplied by wishful averaging.

Under the conditional no-adjacent-degree hypothesis already used for the finite-free replacement, put $H=H(C)$, let E be its object-idempotent ring, and use its transferred operations with $\mu 0=\mu 1=0$ and associative μ2. Equivariant strict units. The degenerate bar complex has an actual equivariant contraction

Choose an E-bimodule projection H→E restricting to the identity on E. Such a projection exists over a field. Over C⟦q⟧, the chosen units reduce to independent unit vectors at $q=0$, hence extend to a basis of the finite-free local module; Nakayama and a nonzero unit minor give the same splitting. Average the projection's conjugates over G. This is a LINEAR right-inverse equation, so its average is still a projection onto E, still the identity there, and is equivariant. Put $H=E\oplus \bar{H}$ using its kernel. No product is averaged.

Use the two-sided E-relative bar resolution $B_n=H{\otimes }_E{H}^{\otimes n}{\otimes }_{E}H$ with its ordinary signed multiplication differential. Let D be the subcomplex spanned by tensors with at least one INTERNAL factor in E. The complement in each degree consists of the words all of whose internal factors lie in $\bar{H}$. Thus the normalized quotient B/D is a free H-bimodule in every degree. Splitting words into their $E\mathrm{/}\bar{H}$ components is a finite algebraic operation at each degree and commutes with G.

For a degenerate word consider its first internal E factor. Duplicate that idempotent next to itself, with the bar/Koszul sign for which the surviving adjacent multiplication has coefficient +1. Call this degree-plus-one map h0. Filter degenerate words by the position of their first E factor. Expanding d h0+h0 d, all faces after that factor cancel in pairs by the bar signs, the two faces next to it give the identity, and all other uncancelled terms have their first E factor strictly EARLIER. The latter arise exactly when multiplying preceding $\bar{H}$ factors acquires an E component; they must not be discarded by pretending $\bar{H}$ is an augmentation ideal. Therefore d h0+h0 d=1−N, where N strictly lowers that finite filtration. N commutes with d, since d²=0. In degree n its power $N^n$ vanishes. Define $h_D=h_0{\sum }_{r\ge 0}{N}^r$, a FINITE sum on every bar degree. Then $d{h}_D+h_{D}d=(1-N)\sum N^r=1$ on D. The construction is H-bimodule linear, E-relative, strictly G-natural and q-adically continuous. No assertion $h_D^2=0$ is needed.

Assume μ3,…,μ_(n−1) already vanish on unit inputs. Let δ=[μ2,−] be the Hochschild differential. The arity n+1 A∞ equation says δμn is a sum of composites of the lower higher operations. Restrict it to a word containing a unit. In every composite that unit lies either in the inner higher operation or in the remaining inputs of the outer higher operation; in either case it vanishes. Hence α=μn|D is δ-closed in Hom_(H-bimod)(D,H).

The explicit bar contraction makes this degenerate cochain complex contractible: precompose α with $h_D$, with its standard Hom differential sign. Its coboundary is α because $\delta \alpha =0$ and $d{h}_D+h_{D}d=1$. Extend that primitive to a cochain ψ on B by zero on the chosen all-$\bar{H}$ complement. Restriction is surjective since the word splitting is H-bimodule linear. All steps are equivariant by the projection, word decomposition and bar contraction of steps 1.38–1.40, so ψ is strictly G-natural. Thus μn−δψ vanishes on D. ψ has precisely the degree and arity n−1 of an A∞ coordinate change.

Take the coalgebra coordinate map F with F1=identity, F_(n−1)=ψ (or its negative according to the conjugation convention), and other new components zero. Its inverse exists aritywise by the finite geometric recursion on a conilpotent word: each nonlinear substitution strictly lowers length. Conjugating the coderivation changes μn by −δψ and does not change μ2 or any already fixed smaller arity, because μ0=μ1=0. Direct insertion of μ2 and ψ verifies this first affected term; all other terms have larger arity. Therefore the new μn is normalized. F and its inverse commute with G and preserve object idempotents.

Repeat for $n=3,4,\dots$ . At each fixed arity only finitely many coordinate changes act, so the resulting operations and comparison map are actual aritywise R-linear series. No q substitution, scalar extension, or alteration of μ2 occurs; q-adic continuity is preserved. The A∞ identities remain true under each conjugation and hence in the aritywise limit. μ2 has its original unit actions and all higher μn vanish on units. This proves strict unitality with a STRICT finite group action.

Local mixed comparison and its explicit graph. Put $H=\{x\in {\mathbb{R}}^N:{\sum }_i{x}_i=0\}$ and $S^n=\{x\in H:{\sum }_i{x}_i^2=1\}$, with the induced round metric, $n=N-2$. Define the smooth odd, permutation-invariant function $f:S^n\to \mathbb{R}$ by $f(x)=-{\sum }_i{x}_i^3$. Writing $s_3={\sum }_i{x}_i^3$, orthogonal projection of its ambient derivative onto $H\cap x^{\perp }$ gives $(\mathrm{\nabla }f{)}_i=-3x_i^2+3\mathrm{/}N+3s_3{x}_i$. In particular $(\mathrm{\nabla }f{)}_i=3\mathrm{/}N>0$ on $\{x_i=0\}$. Let $V$ be a sufficiently small round cotangent disk bundle of this sphere, $Z$ its zero section, and ${\theta }_{(x,p)}(v)=p(d\mathrm{pr}(v))$ its tautological one-form, where $\mathrm{pr}:V\to S^n$. For this graph chart use ${\omega }_V=-d\theta$; fibre negation converts to the opposite cotangent convention used for the self-chord chart. Let $\pi :V\to U\subset \mathbb{P}(H\otimes \mathbb{C})$ be the symplectic cotangent double cover of a real Lagrangian neighborhood. Choose its differential along $Z$ to send a vertical metric covector to $J$ times the corresponding tangent vector, with the positive graph convention. For $Q_{ϵ}=\mathrm{\Gamma }(ϵdf)$, a local real coordinate representative of $\pi (Q_{ϵ})$ is $x+iϵ\mathrm{\nabla }f(x)+O({ϵ}^2)$, uniformly on the compact sphere. Near $x_i=0$ its $i$th imaginary coordinate is strictly positive for small $ϵ>0$; outside a fixed neighborhood of that equator its real coordinate stays nonzero. This proves $L_{ϵ}=\pi (Q_{ϵ})$ misses every $D_i=\{z_i=0\}$, rather than assuming it. The deck map is the cotangent lift of $x\mapsto -x$; oddness gives $a^{\ast }df=-df$, so it exchanges $Q_{ϵ}$ and $\mathrm{\Gamma }(-ϵdf)$. The projected graph is Lagrangian and exact on the divisor complement: its pulled-back primitive is closed, and $H^1(S^n;\mathbb{R})=0$ for $n=2,3$. Ordinary integral singular chains are used below; their relative-homology argument requires no analytic transversality.

A local Morse-sector mixed polygon means a finite-energy punctured disk whose image lies in U, with a lift to V such that every Lε boundary arc lifts to the SAME graph Qε, and every real-locus boundary arc lifts to Z. At an intersection corner on those two boundary types, the lifts coincide at (p,0), df(p)=0. The analogous definition for a local mixed pearl includes its attached real-locus Morse edges and their lifts, and requires the same global graph sheet. A separate sheet assignment at each corner is insufficient.

Complete every Z boundary arc by the fiber interpolation Fτ(q,0)=(q,τεdf(q)), 0≤τ≤1, with the sign that replaces its boundary by an arc in Qε. At intersection corners this interpolation is constant, so all added arcs match. For a pearly edge take the same interpolation on its lifted edge path and include the two side collars with their induced opposite orientations. The resulting completed relative two-chain C has ∂C supported on Qε and lies entirely in V. This is exactly the collar used to define the admissibility class when the lift/graph data above are used.

The graph inclusion Qε→V is a homotopy equivalence. An explicit deformation retraction is Rt(q,p)=(q,(1−t)p+tεdf(q)), 0≤t≤1.

For a round disk bundle it stays in the bundle by fiber convexity, fixes Qε, and retracts onto Qε. Consequently H2(V,Qε;Z)=0. In particular $C=\mathrm{\partial }B+A$ for a singular three-chain B in V and a two-chain A in Qε. Projecting gives $\pi \ast C=\mathrm{\partial }(\pi \ast B)+\pi \ast A$ in the pair $(C{P}^n,L\epsilon )$. Since D is disjoint from Lε, its divisor line bundle has a canonical nonvanishing section on Lε, hence defines a relative first Chern class c1(O(D),s|Lε) in $H2(C{P}^n,L\epsilon ;Z)$. Pairing with π*C is zero. This is precisely the topological divisor intersection used to define admissibility. The same argument works for each $D_j$ separately.

Thus EVERY such local Morse-sector mixed polygon or pearl has divisor intersection vector zero. This conclusion does not require a C0 approximation by a gradient trajectory, monotonicity of its coordinate functions, or a choice of a gradient trajectory with the same endpoints. It also covers $n=2$: $\pi 2(S^2)$ is nonzero, but the relative group H2(V,Qε) is zero. Merely claiming that π2 vanishes would give a false proof.

The relative divisor class is well-defined even if π(Qε) is immersed: it is a class for the pair with its image, which is disjoint from D. Alternatively one can retain the brane domain and use the mapping-cone relative-chain complex. The projected relative null-homology above proves the same pairing vanishing in that formulation.

For an admissible mixed strip with generator actions of size O(ε), zero Hamiltonian, and standard J, the action identity gives $E\le C\epsilon$. The constant C is uniform over its finite set of generators. Choose U0 with $R{P}^n\subset U0$ compactly contained in U; for small ε both boundary branes lie in U0. If a strip leaves U, it passes through a ball of a fixed radius $r>0$ disjoint from all boundary images. The interior monotonicity estimate for standard $C{P}^n$ gives $E\ge c{r}^2$, with c independent of ε. Thus $\epsilon <c{r}^2\mathrm{/}C$ forces every such strip into U. This is the actual application of the supplied monotonicity supplier; it is not an independent proof of monotonicity. The same argument applies to the specific mixed product used to test the unit, once its finite generator action bound is supplied. Unbounded source-dependent Hamiltonian curvature would invalidate this estimate and is not allowed.

A strip in U lifts after specifying one lifted boundary point, since its compactified domain is simply connected and π is an ordinary covering on U. Its boundary lifts remain on their continuous graph sheet. For the Morse sector that is the graph Qε fixed above. The lifted strip is exactly a strip for (Z,Qε), with the pulled-back Floer data and determinant lines. Conversely every such cotangent strip projects; the relative-homology lemma proves its admissibility. These are inverse identifications, including translation quotients, automorphisms, and orientations. No ε→0 limit is used. For the identity-product configuration the same statement requires the global same-graph lift, and then follows identically.

Inside $T\ast S^n$ the Hamiltonian isotopy Z→Qε has an actual compactly supported extension. Indeed the Hamiltonian ±εf(q), with the sign fixed by the chosen canonical-form convention and a fiber cutoff, has vertical flow tεdf along Z, entirely in the region where the cutoff is one. Its continuation maps and homotopies are those of the already constructed embedded exact cotangent Fukaya category. They give inverse cohomological unit classes in HF(Z,Qε) and HF(Qε,Z). The projection identification above carries these differential and identity-product calculations to the mixed category. This establishes the LOCAL analytic part of Sheridan's Lemmas 5.6–5.7 without using a strip-to-Morse theorem.

For an actual mixed A∞ category containing Lε and L′, inverse closed degree-zero morphisms imply quasi-isomorphism of their endomorphism algebras by the ordinary A∞ Yoneda construction. The mixed category's existence is a separate premise; the preceding paragraph alone does not construct it. In particular it does not identify the full relative compact-CY deformation.

Nondegenerate auxiliary cover metric. Let $\rho :Q\to C{P}^n$ be the smooth double cover branched over the smooth complex quadric B, $n=2,3$, and R⊂Q its ramification hypersurface. The map is locally (v,z)↦(v²,z). Let ω be a Kähler form on $C{P}^n$. Let K⊂Q be a compact set disjoint from R containing every lifted brane. In the named construction $B=\{\sum z_j^2=0\}$ in the hyperplane $\mathrm{\Sigma }{z}_j=0$, so B misses $R{P}^n$; for sufficiently small fixed ε it also misses Lε. Thus the lifted zero section and both graph sheets lie in such a K. Standard holomorphic structure is retained near B so that the pulled-back curve is genuinely holomorphic there.

The closed form α=ρ*ω is semipositive on Q, positive away from R, and at R has precisely the complex normal line as its kernel. The positivity along TR follows because ρ|R is a local holomorphic isomorphism onto B. Choose a Hermitian metric on the normal line bundle of R and a smooth nonnegative function r² on a tubular neighborhood, vanishing to second order at R, whose normal complex Hessian there is positive. Such a function can be obtained by a partition of unity from the squared moduli of local holomorphic defining functions; on R the zero-order and first-order terms vanish, so partition derivatives introduce no normal Hessian errors. It has zero tangential and mixed complex Hessian on R.

Choose a smooth cutoff χ supported in that neighborhood and equal to one closer to R; choose the neighborhood disjoint from K. Put φ=χr². For sufficiently small δ>0, ωδ=α+δ dd^cφ

is Kähler. Here $d{d}^c$ is normalized so $d{d}^c\mathrm{\mid }v{\mathrm{\mid }}^2$ is positive. At R the added Hessian is positive on the normal kernel and zero on TR. On a fixed sufficiently small inner tube the tangential block of α has a uniform positive lower bound, its normal block is comparable to |v|², and its mixed block is O(|v|); the added normal block is bounded below by δc, while its tangential block is O(δ|v|²) and mixed block O(δ|v|). The Schur complement is therefore positive after first choosing the tube small and then δ small; equivalently compare in the coordinates (v²,z) with the nonsingular positive downstairs metric, then use the additional δ normal term. On the remaining compact cutoff annulus α has a uniform positive lower bound, so its bounded δ-Hessian perturbation preserves positivity. Outside the support $\omega \delta =\alpha$. Thus ωδ is globally nondegenerate and equals ρω on a full neighborhood of K. In particular the lifted branes remain Lagrangian for ωδ. One must use this adjusted metric, rather than call the degenerate form ρω Kähler.

Let u be a smooth pearl downstairs and let πΣ:Σ̂→Σ be the degree-two pullback branched cover of its domain, with lift û:Σ̂→Q. Its boundary maps into K. Stokes gives ∫Σ̂ û*ωδ = ∫Σ̂ û*ρ*ω + δ∫∂Σ̂ û*d^cφ = 2∫Σ u*ω, because φ is identically zero near K. Domain branch points have measure zero and do not alter the degree-two integral. The identity holds separately on every smooth component, and for stable maps by summing components. For inhomogeneous pearls with uniformly bounded perturbation terms the corresponding uniform geometric-energy bound follows from this area bound and the ordinary energy/curvature identity. The energy/curvature identity applies to the lifted smooth equation; an arbitrary perturbation crossing B would not lift smoothly, so the following support condition is imposed. Choose perturbations zero near B and lift only away from B, where ρ is an ordinary cover.

Consequently a downstairs area bound A yields the actual lifted bound 2A for a fixed nondegenerate Kähler metric. The metric can be chosen once for a fixed ε and compact brane collection; no assertion of a uniform metric over ε→0 is required by the mixed-category route.

Actual covering-domain compactness and descent. Setup and exact hypotheses

Fix $\epsilon >0$ and a finite generator word of the mixed Lε/L′ category downstairs in $C{P}^n$, $n=N-2$, $N=4$ or 5, with completed divisor class $d=0$ or $e_j$. Hamiltonian perturbations vanish on a fixed neighborhood of the coordinate divisors AND of the auxiliary smooth branch quadric $B=\{\sum z_i^2=0\}$. The standard integrable J is fixed on those neighborhoods. The ordinary strip-end data are fixed; finite-core curvature terms have a uniform bound. On L′ segments retain the actual lift to $S^n$. Let $\rho :Q\to C{P}^n$ be the auxiliary double cover with ramification R. Its lifted branes Z,Γ(εdf),Γ(−εdf) are compact embedded spheres; every boundary segment carries its individual brane label. They lie a fixed positive distance from R.

The energy/action identity at the fixed word and fixed d gives an actual uniform area bound A. Use the Kähler metric constructed in the present construction, equal to ρω near all lifted branes. Lifted area is exactly twice downstairs area, so the corresponding geometric energy is uniformly bounded after the controlled inhomogeneous terms are included. This form tames the lifted J: on the modification neighborhood J is the standard integrable one; outside it the map is unramified and the form is ρω. Lifted Hamiltonians vanish near R, hence their apparent inverse-derivative singularity there never occurs.

The fixed-source compactness, boundary removable-singularity, energy-atom extraction, transverse-end decay/breaking, and disk/sphere small-image-neck estimates already supplied in the local analytic lane are used below. Their use on finite cores and fixtures is stated explicitly, rather than replacing a varying-domain proof by a fixed-source citation.

First complete every L′ boundary segment to Lε inside the cotangent neighborhood, which misses B. For the fixed external chords choose once and for all paths in the graph brane domain $S^n$ connecting the two domain endpoints at a chord and fixed small cap ribbons. The paths need not be unique: a different choice changes the eventual closed cap by a two-cycle coming from $S^n$, whose image in $H2(C{P}^n)$ is zero because the graph is Lagrangian. The same is true for the unperturbed real sphere. Here $H1(S^n)=0$ ensures that the completed boundary one-chain can be capped in the brane domain. Add the fixed cap chains to each completed curve to obtain a closed two-cycle z(u) in $C{P}^n$. For a fixed word the caps are fixed, and d_i(u)=⟨H,z(u)⟩+c_i(word), where H is the hyperplane class and $c_i$ is the relative divisor-trivialization correction of those fixed caps. Thus fixed d determines the integer $h=⟨H,z(u)⟩$. Since B has degree two and all moving interpolation collars miss B, b(u)=u·B=2h+c_B(word).

The right side is fixed. It is nonnegative by ordinary holomorphic normal positivity near B. No disk pearl is entirely contained in B, because its boundary lies on branes disjoint from B. Thus b bounds the number of normal zeros, counted with multiplicity. It in particular bounds the number of odd-order zeros over which the normalized pullback domain actually branches. This argument uses the full relative class and fixed chord caps; a bare genus bound or the statement “B is ample” would not establish the needed uniform b.

At a fixed word, stable pearl vertices are bounded by the planar tree flag identity. A chain of semistable L′/L′ pearls is bounded by its positive doubled projective degree (at least one for each nonconstant such pearl), which is fixed by the relative winding/word formula; at |d|≤1 the corresponding construction also shows that each consumes at least one unit of d. Semistable transverse mixed strips have fixed finite end-action spectra, and their genuine nonconstant breaks consume a positive action drop; constant strips are contracted. Autonomous disk/sphere bubbles have the ordinary positive energy quantum. Thus there are finitely many original stable pearl-domain types after the usual transverse map-stabilizing marks are added. No positive energy quantum is asserted for every arbitrary source-dependent stable pearl: their number is bounded by their flags instead. Morse edge breaking is considered separately below. The finite relative label and word data therefore give a uniform total b over the finite pieces under consideration.

On each smooth pearl mark every zero over B, repeating neither the position nor its parity but recording its positive multiplicity. After a subsequence their number, multiplicity partition and sheet-monodromy labels are fixed; their number is at most b. Add the existing boundary/corner/stabilizing marks. Double a disk across its boundary and add the conjugate zeros. The doubled base is a genus-zero marked curve with a uniformly bounded finite number of marks. The explicit cross-ratio/cluster extraction in the supplied real disk-domain construction therefore supplies a finite stable base tree and plumbing coordinates at all its nodes.

The normalized pullback cover of a smooth base component is determined by the parity of its branch orders. On a coordinate sphere chart it is explicitly the normalization of y²=∏_(odd marks a)(z−a), with the appropriate branch condition at infinity. A nonvanishing holomorphic unit in a local branch function can be removed by its holomorphic square root on a disk chart. Over the complement of the marked points, a degree-two cover is determined by its Z/2 monodromy, which sends each small loop to the parity of that point. The product relation fixes the parity at infinity. There are only finitely many remaining global sheet choices, retained as labels.

At a node of the stable base tree, declare its parity to be the sum modulo two of the branch multiplicities on either side. The two answers agree after the infinity flag/parity is included. On each base sphere the total parity of marked points and incident nodes is even, so the preceding polynomial construction defines its normalized component cover. Glue preimages of flags as follows.

For even monodromy there are two preimage nodes. If the base plumbing is $XY=t$, both sheets have local plumbing $x_{\pm }{y}_{\pm }=t$, up to nonzero coordinate units. * For odd monodromy there is one ramified preimage node. Put $X=x^2$, $Y=y^2$ and $xy=s$. The base parameter is $t=s^2$. Choose one of its two square roots after passing to a subsequence/local finite base change; its sign is a deck choice. These are the complete local models.

To verify convergence on cores, normalize three base flags, so their other marked coordinates converge to distinct limits on each extracted component. The polynomial coefficients then converge, and away from their distinct roots the square-root charts converge smoothly. At a simple odd root use y as coordinate; at an even root normalize its square factor first. These give smooth finite covers of every fixed core. At a node the two explicit local models give annuli with lengths equal to the base neck length or half of it, up to bounded coordinate-unit errors. Therefore every long cover neck tends to infinity and there is no unrecorded domain degeneration. Colliding roots have already become distinct marks on child base components, so this construction includes their resulting positive-genus double-cover components. The genus is encoded by these finite polynomial charts; no arbitrary-genus compactness theorem is being assumed.

The construction commutes with the conjugation of the doubled disk and with the target deck involution. Choose the original disk half on the base, and take its cover half. Boundary branching cannot occur at a smooth original boundary point: the target brane is disjoint from B. A branch-point cluster can converge to a boundary node of the extracted base; it is retained in the explicit node models until map convergence below determines whether that model is possible. Do not delete it in advance by assuming the quotient is unramified at every limiting node.

This constructs a compact subsequential nodal covering-domain family using only the finite marked genus-zero base extraction, polynomial roots and the displayed plumbing models. Unstable constant cover components can be retained temporarily by the branch/stabilizing marks. They are forgotten only after map extraction and descent.

On every fixed smooth cover core use the actual local interior/boundary epsilon estimate. The lifted boundary arcs lie on individual embedded Lagrangians, and the boundary data are fixed there. There are finitely many original corner flags; their preimages have fixed transverse chord ends, treated by the supplied weighted strip-end inverse and breaking/compactness theorem. Smooth core coefficient convergence follows from the explicit covering-domain coordinates of section 3, the fixed perturbation data, and the fact that the perturbation is zero on the ramified neighborhood. Thus the fixed-source theorem applies on each core chart after pullback to a fixed chart.

At derivative blowup extract the actual interior sphere or embedded-boundary disk bubble by rescaling. Each costs the uniform positive energy quantum from the compact-target estimate, so only finitely many are extracted. Add their stabilizing points to the base/domain extraction and repeat, using the integer energy-atom induction already established by the bubble-tree supplier. Constant stable components are bounded by the flag identity. This makes a finite map/domain tree.

On a long neck, if a definite amount of energy remains in a middle unit window, translate to that window and extract another atom or strip component. This either consumes a positive quantum or produces one of the specified chord/Morse breaks. After all such extractions the residual energy on each neck is below the small-image threshold. The local cylinder/real-reflection estimate then gives exponential derivative decay and vanishing middle energy and image diameter, using the same primitives and Fourier gap as the supplied nodal analysis. Hence its two endpoint values match. This is the step that upgrades core convergence to stable-map convergence on the varying cover domains; fixed-source compactness by itself would not do it.

All of this takes place in the fixed compact target Q with its NONDEGENERATE metric and fixed finite embedded boundary data. The higher genus of an intermediate cover component makes no difference to its finite disk/half-disk chart estimates or its finite annular fixtures. The domain compactness needed to produce precisely those cores and fixtures was proved in section 3.

A potential odd branch node lying on the physical boundary would be fixed by the deck involution, so its limiting map value would belong to R. On the other hand every boundary limit value belongs to the closed lifted boundary brane, which is disjoint from R. Thus that configuration is impossible for an actual map limit. Odd nodes in the interior, and pairs of even boundary nodes, remain and are treated by their displayed fixtures. This conclusion follows AFTER compact map convergence, not from an unsupported claim that source branch points stay a positive distance from the boundary.

Before extraction add complete deck orbits of stabilizing choices. The map identity $\widehat{u}\circ \tau =\sigma \circ \widehat{u}$ holds on each smooth cover, where τ is the domain deck action and σ the target deck action. The explicit core/node construction retains τ; on a node it acts by exchanging the even sheets or by the indicated sign in the odd square-root model. Passing to the limit gives the same identity on every component and node. Quotienting these explicit fixtures gives exactly $XY=t$ on the base, including the odd model with $t=s^2$. Consequently the quotient domain is the marked genus-zero base tree already extracted; it is not an arbitrary stable curve whose quotient genus was merely guessed.

The map descends continuously and smoothly on its component interiors because it is σ-equivariant. Its real boundary lifts are the limits of the SAME embedded-brane labels in Q. Since these brane domains are spheres and their projections are the fixed graph/real immersions, this retains the original $S^n$ lift on every boundary segment. The discrete antipodal sheet change at a flip is retained by its chord or Morse flag. No new lift jump may be inserted at an unmarked smooth boundary point: the lifted limit there has one continuous embedded-boundary trace. A new node comes with its explicit limiting two-side lift data and belongs to the ordinary disk/strip tree inventory.

Discard the auxiliary B-zero marks and contract only the resulting unstable constant components. The final downstairs maps and their original external/Morse flags are the desired stable mixed pearls. Positive B-degree of any nonconstant component has been retained. Coordinate-divisor pairings are locally constant under the actual collapse/smoothing maps, so the sum of component d-vectors is the original d. Positivity gives nonnegative entries on every component. At |d|≤1 the preceding sphere and disk index inventory removes the forbidden bubbling components, after coherent regular perturbation; ordinary disk/strip/Morse tree faces remain. The auxiliary branch ghosts do not create new operations: after forgetting their auxiliary marks they are exactly unstable constant components and are contracted.

Every Morse edge lies in the compact sphere, with fixed gradient data on its long portion. Its gradient energy identity and finite critical set give convergence to a finite broken gradient trajectory: between distinct critical levels the energy drop is fixed and positive, so there are finitely many nontrivial breaks, while a constant segment is absorbed into its nodal fixture. Finite-length edges converge by ordinary ODE continuous dependence. Domain-dependent end transitions occur in fixed compact intervals and carry uniformly bounded forcing. Their broken-end collars use the exponential-dichotomy inverse described in the first-order construction below. Endpoint evaluation matching is closed under the core convergence above.

This supplies compactness/exhaustiveness for the DIRECT DOWNSTAIRS charts. Those charts are still obtained from the original disk CR/evaluation problem and the local inverses already constructed, not from a “smooth moduli space of admissible ramified covers.” Auxiliary branch collisions need no new local operator or finite constraint. At a downstairs node the real/complex fixture and cap matching are the ordinary ones already specified; at a zero-length Morse seam use the two-sided collar in the corresponding construction. Therefore the explicit covering-domain extraction is compatible with the required local atlas without requiring a ramified gluing inverse.

Finite target symmetries preserving the auxiliary quadric act on every polynomial parity label, deck orbit, boundary lift and node fixture. If the requested finite symmetry fails to preserve this auxiliary quadric, do not infer it does: use the finite orbit of auxiliary quadrics only as separate compactness tests, and construct the actual downstairs charts on free word orbits. Since the compactness argument is a proof of existence of a convergent subsequence, its auxiliary cover need not be one of the category's equivariant choices. The actual category data and tuple coordinate changes, rather than the chosen compactness proof, obey the strict finite group law. Spin-central elements act on brane/cap lines as before.

First-order incidence and bubble inventory. The restricted category one actually needs

Use the downstairs compact pair $(P(H_C),D)$, $H_C=\{\sum z_i=0\}\subset {\mathbb{C}}^N$, $D_i=\{z_i=0\}$, with $n=N-2$. Its real double-cover object is $L^{\mathrm{\prime }}:S^n\to R{P}^n$. Its graph perturbation is Lε. For the central mixed category retain completed divisor intersection vector $d=0$. For deformation-class comparison retain only $\mathrm{\mid }d\mathrm{\mid }=\mathrm{\Sigma }{d}_i\le 1$, over the square-zero ring R1=C[r_1,...,r_N]/(r_i r_j: all i,j).

Words may have arbitrary finite arity. All moduli used in the operations and relations have expected dimension zero or one. This is a finite-order truncation of the ANALYTIC comparison, not a truncation or narrowing of the compact-CY q-category ultimately to be compared. That category still retains every positive-q class and every compact-target bubble.

Morse and flip data for the cubic. A critical point of $f$ has at most two coordinate values, since $-3x_i^2=\lambda +\mu x_i$; one value is impossible on $S^n$. For each nonempty proper subset $K\subset \{1,\dots ,N\}$ of size $k$, there is exactly one point $p_K$ with negative coordinates $b=-\sqrt{(N-k)\mathrm{/}(Nk)}$ on $K$ and positive coordinates $a=\sqrt{k\mathrm{/}(N(N-k))}$ on $K^c$. These solve both constraints. At $p_K$ the constrained Hessian is $3(a-b)$ on the sum-zero variations within $K$ and $-3(a-b)$ on those within $K^c$. These subspaces exhaust $T_{p_K}{S}^n$, so $f$ is Morse, of index $N-k-1$, and $p_{K^c}=-p_K$. The graph self-intersections are precisely these antipodal pairs: its two lifts meet when $df=0$, and their Hessian difference $2ϵ\mathrm{Hess}f$ is invertible. For completeness the round-gradient pair is Morse-Smale, meaning all its stable and unstable manifolds intersect transversely. Indeed its ascending flow satisfies ${\dot{x}}_i-{\dot{x}}_j=3(x_i-x_j)(s_3-x_i-x_j)$, so equalities and strict coordinate orders persist. Its stable manifold at $p_K$ consists of points whose coordinates on $K$ are the common strict minimum; its unstable manifold consists of points whose coordinates on $K^c$ are the common strict maximum. To verify these descriptions, every flow has a critical limit: $f$ increases, the integral of $\mathrm{\mid }\mathrm{\nabla }f{\mathrm{\mid }}^2$ is bounded, and compactness plus bounded derivative makes every accumulation point critical; the connected limit set in the finite critical set is one point. Near a limiting negative cluster the difference equation has positive coefficient $3(a-b)$, so two coordinates converging to that cluster must have been identical throughout; near the positive cluster the coefficient is negative and the reversed-time argument applies. This proves both descriptions and their converses. At an intersection the minimum and maximum blocks are disjoint. Their equality constraints have independent normals, supported on the two blocks, with coordinate sum zero and perpendicular to $x$; hence the tangent spaces are transverse. Choose also $h(x)=⟨v,x⟩$ for a unit $v\in H$; its only critical points are $\pm v$, its Hessians there are $\mp$ the round metric, and its meridian gradient flow is Morse-Smale. Choose $v$ with no zero coordinate and transport this choice on free label orbits. Flipping edges use $\mathrm{\nabla }f$ and exchange the two antipodal boundary lifts; nonflipping edges use $\mathrm{\nabla }h$ and keep the lift. Complete their boundary ribbons to $L_{ϵ}$ by the interpolation of steps 1.46--1.47, with opposite orientations on the two sides of an edge. This defines each $d_i$ as the relative divisor pairing with its actual lift data.

The positivity rule is local: an ordinary interior zero of the i-th normal holomorphic coordinate contributes its positive order; a negative crossing of an L′ boundary lift through $\{x_i=0\}$ contributes +1 after graph completion; a positive boundary crossing contributes zero; a flipping Morse edge crossing contributes +1, a nonflipping edge contributes zero. For a transverse real boundary crossing w(z)=az+O(z²), the completed graph displacement is +iεc, $c>0$. The zero of az+iεc lies in the upper half-domain exactly when $a<0$, proving the boundary rule. The two sides of a flipping strip give one positively oriented zero, whereas the nonflipping strip folds with zero relative intersection. The projected-gradient formula gives the actual cubic's normal component $(\mathrm{\nabla }f{)}_i=3\mathrm{/}N>0$ at $x_i=0$; its flipping flow therefore crosses only in the positive direction. Nontransverse crossings may be resolved by a small real movement of the hyperplane, still disjoint from Lε; the relative pairing is unchanged. Perturbations used to establish this rule must vanish near the moved divisors. Virtual chart sections subsequently retain the original class labels rather than claiming positivity for an arbitrary forced map's zeros.

This is an exact index/character check, useful for distinguishing ordinary bubble faces from the forbidden single-output components in the mark-forgetting comparison.

Let the diagonal cover label group be the determinant-one group $\tilde{\mathrm{\Gamma }}=\{\zeta \in {\mu }_N^N:\prod {\zeta }_i=1\}$. The central generator $p_K$ has character ${\chi }_K={\prod }_{i\in K}{\zeta }_i$. The relative parameter $r_j$ has character ${\chi }_j$; this convention is fixed by the geometric first-order coefficient ${\mu }^1(p_j)=\pm r_j\cdot 1$. Equivalently the Hochschild dual variable $u_j$ has inverse character. Pick integers ${\nu }_i$ with $\mathrm{\Sigma }{\nu }_i=N-1$. The grading of $p_K$ is ${\mathrm{\Sigma }}_{i\in K}(2{\nu }_i-1)$, while the divisor variable $r_j$ has degree $2{\nu }_j$. This is forced by the Maslov winding of the normal coordinate and by the preceding differential: $deg(p_j)+1=2{\nu }_j$.

A zero-input contribution $r_j$ $p_K$ must have trivial total character and total degree 2. Trivial character forces $1_K+e_j$ to be constant modulo N. Since $N\ge 4$ and its entries belong to {0,1,2}, it must be the all-ones vector. Thus K is the complement of {j}. Its total degree is deg(r_j)+deg(p_{j^c})=2ν_j+[2(N−1−ν_j)−(N−1)]=N−1.

For $N=4,5$ this is 3 or 4, never 2. Hence there is no rigid first-order curvature coefficient. With the usual convention that a zero-input moduli with output total degree t has dimension t−2, the corresponding expected dimension is $(N-1)-2=N-3=n-1>0$. Thus these moduli are not asserted empty; they cannot contribute a rigid coefficient. Their possible appearance as bubbles requires the free-node matching calculation below. This calculation is independent of the chosen integer weights. At $d=0$ curvature is absent by the exact affine embedded-lift argument; the S² compact-CY curvature construction is a different issue and remains as previously supplied. No vanishing of arbitrary higher-order downstairs curvature is asserted here.

The character calculation alone concerns rigid zero-input moduli. For a one-dimensional operation's compactification one must additionally impose the actual matching/cap index at its bubble node, or use the free-node calculation in section 4. An empty zero-dimensional curvature count alone is not a compactness theorem.

This part concerns geometric disks with ALL boundary labels Lε (or their embedded finite lifts), which are disjoint from D, rather than L′ boundary segments.

Fix an unmarked solution u of the actual disk equation, with topological intersection vector $d=e_j$. Assume the perturbation vanishes near D and J is the fixed integrable structure there. Assume that its stable compactification in the zero/one-dimensional regime has no nonconstant spheres or single-boundary-component disk bubbles. Then every intersection is ordinary holomorphic, and positivity gives exactly one point z(u) with $u(z(u))\in D_j$, of multiplicity one. It lies in the smooth part of $D_j$: an SNC crossing would also give an intersection with another $D_i$, contradicting $d_i=0$. There are no intersections with the other divisors.

Write $D_j=\{w=0\}$ near that point and use a source coordinate z. The function F(u,z)=w(u(z)) is holomorphic in z near its zero, with ${\mathrm{\partial }}_{zF}\mathrm{\ne }0$. For a finite-dimensional thickened solution chart with the equation unchanged in this contact neighborhood, the ordinary real implicit-function theorem gives a unique smooth z(u). Its local graph identifies the unmarked chart with the marked incidence locus $F=0$. The derivative in the z variable is complex linear and invertible. Therefore its real determinant is $\mathrm{\mid }{\mathrm{\partial }}_{zF}{\mathrm{\mid }}^2>0$, so this identification preserves the determinant orientation after the marked-point complex orientation and the divisor-normal complex constraint cancel. No factorial, deck multiplicity or sign is introduced.

For strips quotient by translation first, or equivalently use the mark to fix the real translation gauge. Translating u translates its unique contact point. Thus the construction descends and gives precisely one marked-strip class per unmarked-strip class. It does not multiply a count by the length of the translation orbit.

For a family of solutions converging to an ordinary broken disk tree, the contact lies on the unique component carrying $d=e_j$. It cannot converge to a real node, because that node lies on a boundary brane at positive distance from D for fixed ε. It cannot converge to an interior node without a sphere component; those have been excluded in this regime. Consequently its position remains in a smooth core on that distinguished component. The previous local identification is compatible with the actual product collars and determinant maps there. Hence the marked and unmarked zero/one-dimensional virtual correspondences coincide for these pulled-back data. Higher-order contacts require the separate confluent jet module; this simple-point argument does not replace that construction.

A nonconstant sphere in $C{P}^n$ has positive integer hyperplane degree h and therefore intersection h with EVERY $D_i$. Nonnegative component intersections and |d|≤1 exclude it, since $N\ge 4$. This includes a sphere contained in one coordinate hyperplane: its intersection with a different hyperplane is still $h>0$.

For a single-boundary-component bubble with label L′, retain its attaching point as a FREE real node. After graph completion, the exact sheet/coordinate bookkeeping gives d=q(1,...,1)+1_K, where K is the outgoing flip label, or K=∅ for a nonflipping node; its doubled-real-curve degree is 2q+1 in the flipping case and 2q in the nonflipping case. This formula is obtained by traversing the lifted boundary: each sign change in a coordinate contributes one half-turn, the extra full turns contribute the same integer q to all coordinates, and doubling the half-disk gives total degree twice those full turns plus the one sheet switch. The nonflipping closed lift contributes no half-turn. These are integer relative winding numbers, not a conclusion from symplectic area alone.

If the node is nonflipping, |d|≤1 and $d_i\ge 0$ force $q=0$, hence zero doubled degree and a constant bubble. If it is flipping, $0\le d$ and |d|≤1 force $q=0$ and $K=\{j\}$, so $d=e_j$ and doubled degree one. The bubble is a half-line, with Maslov index $\mu =n+1$: doubling its tangent bundle gives the pullback of $TC{P}^n$ along a degree-one real sphere, whose first Chern number is n+1, which equals the disk Maslov index by the determinant doubling definition. Its expected dimension with one free boundary node is n+μ−3+1=2n−1.

At a real-node stratum the matching loss is n and the gluing normal contributes one. Thus a total one-dimensional moduli (or a lower-dimensional one) would have complementary expected dimension at most 1−(2n−1)+n−1=1−n<0.

Coherent relative regularity on the complementary stable word makes it empty. This is the actual reason the forbidden single-boundary-component face is absent, rather than the incorrect assertion that all positive-intersection disks disappear by affine exactness.

For an Lε boundary bubble there are two cases, which must not be conflated. With no branch jump its boundary lifts continuously to $S^n$. The relative winding formula has no proper-subset term, so $d=q(1,\mathrm{.}\mathrm{.}\mathrm{.},1)$; at |d|≤1 it is zero and exactness excludes a nonconstant bubble. With a branch jump use the actual transverse chord/capping determinant, so the node matching is DISCRETE, not an n-dimensional evaluation match. At $d=e_j$ the character calculation of section 2 forces the output chord $x=p_{j^c}$. The full capping index formula gives the bubble moduli dimension deg(r_j)+deg(x)−2=N−3=n−1.

Here the index formula is obtained by gluing the cap for x to the punctured bordered operator: the resulting closed boundary problem has the relative Maslov contribution $deg(r_j)$; removing that cap subtracts its graded chord contribution with the output convention, and the one-output disk domain contributes −2. Equivalently this is the ordinary operation formula $\dim =\mathrm{deg}(\text{output with its coefficient})-\sum \mathrm{deg}(\text{inputs})+k-2$ at $k=0$. The cap is the same actual graded transverse-end cap used to define the degree of $p_K$; no limiting Morse correspondence is invoked.

At such a discrete chord break the bubble and complementary component have index sum V−1, where V is the total operation dimension. Thus the complementary moduli has dimension at most $1-1-(n-1)=1-n<0$. It is empty for the chosen regular data. This proves the required exclusion for Lε as well. Adding two auxiliary-data colors does not change any of these indices, so the exclusion applies to the two-copy comparison. At a bubble with multiple independent nodes there is at least one further real degeneration parameter; the same index sum or an iterated one-node analysis excludes it in the zero/one-dimensional regime. This calculation explicitly distinguishes the free L′ real-node match from the discrete Lε chord match.

A standard-J pearl with boundary on $R{P}^n$ doubles by complex conjugation to a holomorphic sphere $v:C{P}^1\to C{P}^n$ of degree $h\ge 1$. Pull back the Euler sequence: 0→O→O(h)^(n+1)→v*TCP^n→0.

Since H¹(O(h))=0 and H²(O)=0, $H^1(v\ast TC{P}^n)=0$. Hence the sphere Dolbeault derivative is onto. For an arbitrary real-invariant forcing, choose a preimage and average it with its conjugate; this is still a preimage and is fixed by the real involution. The fixed real problem is exactly the doubled real-boundary disk problem. Thus the pearl CR derivative is onto, including multiply covered maps. For at most two prescribed real evaluations, twist by their degree-two divisor: H¹(O(h−2))=0 for $h\ge 1$, so the same exact-sequence argument proves $H^1(v\ast TC{P}^n(-2))=0$ and the joint derivative at those marks is onto before any additional flag-incidence restrictions. Domain automorphisms and their translation direction are then quotiented by the actual marked slices. This gives the automatic-regularity ingredient in Sheridan section 6.4 directly.

Two adjacent semistable L′ pearls at a zero-length seam do not arise at |d|≤1: a nonconstant semistable pearl with $d=0$ would have exactly two flip flags. The winding identity gives $q=-1$ and complementary flag labels, hence doubled degree $2q+2=0$, contradiction. Therefore each such pearl consumes at least one unit of d. Their pair consumes at least two. This statement uses the legal flag labels and does not assert that every nonconstant stable central pearl has positive d.

The auxiliary quadric cover is a device for compactness and for retaining boundary sheets. It should NOT be used as the coordinate space of the pearl atlas. At a smooth downstairs map, passage through the auxiliary branch quadric is an ordinary smooth interior event, with no incidence condition on that quadric imposed. At a tangency to it, no downstairs differential operator or domain fixture changes. Thus no gluing theorem for a changing family of branched covers is required to define local downstairs charts.

A chart uses the actual CR operator with the chosen local boundary lifts, Morse edge operators, and node/flag evaluation constraints. Its real and complex disk nodes use the supplied matched inverses in the present construction section section 3–4; transverse corner ends use the supplied weighted transverse-end inverse. A finite Morse edge has the smooth time-dependent ODE flow and its ordinary variation-of-constants derivative. At a broken infinite Morse edge the limiting Hessian is real symmetric and nonsingular; decompose into its positive/negative eigenspaces. The explicit right inverses are the integrals of exp(−|A|(s−t)) over the appropriate past/future half-lines. With weight smaller than the least absolute Hessian eigenvalue they are bounded. Coefficient errors decay exponentially along the two convergent gradient ends; cutoff, then Neumann inversion, gives the same long-neck right-inverse construction. Its shared finite asymptotic coordinate is matched with the pearl evaluations. The ordinary Sobolev product estimate and ODE quadratic estimate give the contraction, and inverse-gluing follows from exponential convergence on the Morse neck and the supplied disk necks.

At a zero-length Morse edge the source perturbation is identically zero for all sufficiently short edges. Its equation is therefore exactly “both endpoints equal,” independent of the length, not an approximation to it. On the negative side of this signed seam use the real-node smoothing chart; on the positive side use this identical evaluation matching and a nonnegative short-edge length. The common double-exponential disk profile makes the negative-side core correction flat at zero, so its finite-dimensional obstruction section joins smoothly to the constant positive-side section. The domain gluing normal on this seam is internal: its two sides cancel in an oriented one-chain. It is not an A∞ boundary term. Long Morse breaking, ordinary disk breaking and chord matching are the actual external faces.

Choose supported obstruction and auxiliary slice tuples and retain every evaluation constraint jointly, as in the supplied atlas. Stable pearls allow arbitrary source-local Hamiltonian forcing; semistable real pearls use section 5 instead. Data are chosen on free ordered-word orbits of the lifted finite symmetry group and transported, so no invariant nonlinear transversality assertion is needed. Finite stabilizers of source tuples act by their actual permutations and cap maps. Determinant tensoring uses the existing relative-spin/capping convention and complex orientations of all divisor constraints. This gives a LOCAL collar construction from the already supplied operator estimates. Compactness/exhaustiveness of its boundary inventory remains necessary to turn these local charts into the global count.

Coherent comparison neighborhoods. Fix all central data before any positive-order data

Use $R1=C[r_1,\mathrm{.}\mathrm{.}\mathrm{.},r_N]\mathrm{/}(r{)}^2$ and $n=N-2\ge 2$. Construct the central $d=0$ mixed category on the two object types Lε,L′, and also its finite colored copies used to compare auxiliary data. Choose its coherent regular/virtual data in ALL arities first. The free-label prescription keeps the finite spin/character symmetry exact. Ordinary $q=0$ sphere and one-label disk bubbles are absent by the central positivity/exactness inventory. Stable splitting faces therefore involve strictly smaller arities and permit the usual relative extension on finite associahedral charts, with the zero-length Morse seam treated as an internal two-sided seam.

Only after that choose $d=e_j$ data, arity by arity. An allowed $d=e_j$ disk-tree splitting has one $d=e_j$ component of smaller arity and central components. Its face data are already prescribed. A forbidden one-boundary-component positive-order bubble has a CENTRAL complementary component, possibly of HIGHER arity. That is not an induction problem, because every central arity was fixed first. This ordering is necessary: inducting only on arity would leave precisely that complementary moduli unspecified.

For a zero/one-dimensional total word the preceding index calculation makes the complementary central problem negative dimensional. Its chosen regular zero set is empty. The statement concerns the central problem with the extra node/chord flag and its actual matching convention: a free L′ real-node loss is n, while a transverse Lε chord match is discrete. Those must be handled as in the first-order inventory, rather than using a uniform fictitious n-dimensional matching loss.

Fix a finite arity, word, generator tuple and one-divisor class. Its stabilized domain compactification has finitely many boundary types. At a forbidden face choose any compact finite-dimensional family of bubble-side perturbation data needed to interpolate between pulled-back and valid relative data. Hold the complementary $d=0$ data FIXED. Then no limit map can lie on that face: its central factor would be a solution of the fixed negative-dimensional regular problem, whose zero set is empty. Parameter dimensions on the BUBBLE data do not change this argument, since they do not alter that central problem.

The mixed compactness construction applies uniformly to this compact family: coefficients and curvature bounds are uniform; all target/divisor neighborhoods are fixed; the auxiliary-cover intersection and area bounds are unchanged. If solutions existed in every sufficiently small source collar of that forbidden face for some parameter choices, take a sequence of parameters in the compact family and a sequence approaching the face. Compactness gives a limiting map at the face, contradicting the empty central factor. Hence some whole collar contains no solutions for the ENTIRE compact family. At multiple forbidden faces use their finite product parameter family and the same fixed empty central factor; shrink a finite system of collars. This proves a genuine no-solution neighborhood, not just emptiness of the boundary stratum itself.

All statements are local at a finite compact slice; no single collar width uniform in arbitrary arity is required. Countably many words are handled by the same diagonal generic-data construction as the existing category, preserving the finite lists of already prescribed face data.

For pure words of one auxiliary-data color, use valid relative marked perturbation data. For pure words of the other color, pull back the unmarked topological-intersection data along forgetting its unique simple divisor mark. This pullback agrees with allowed splitting faces: the one contact lies on exactly one component, and forgetting it commutes with the remaining stable disk/Morse decomposition. Its failure of full domain coherence is confined to the forbidden face where the contact component is a disk with a single boundary component.

On that forbidden face choose valid relative bubble/end data. Extend them over its no-solution collar, and interpolate to the forgetful pullback before exiting the collar. Do this relative to all allowed face products. The interpolation uses the compact bubble-side family of section 2 and keeps the complementary central data fixed. Therefore it creates no solutions in the modified region. Outside the collar the data are unchanged. The marked/unmarked oriented local diffeomorphism established in the corresponding construction identifies every actual zero/one-dimensional solution and its allowed collars, so their counts are unchanged exactly.

This yields globally coherent valid data with the same active pure-word operations as the forgetful model. It is stronger and more precise than calling the uncorrected pullback globally valid. The finite obstruction-atlas version is identical: retain the prescribed obstruction spaces and sections on a neighborhood of the compact active solution set; extend relative to the empty-collar and allowed product faces using the existing finite normal-product construction. Since there are no zeros in the changed collar, it contributes neither an operation nor a boundary term.

Now choose mixed-color word data relative to both pure prescriptions and to the lower allowed splitting products. Domain-local forcing and finite virtual obstruction tuples provide the ordinary relative extension. Central data were fixed in section 1. The same empty-collar argument applies to mixed words, so the compactified zero/one-dimensional moduli have only the allowed A∞ product faces. Their signed boundaries give the first-order A∞ relations, with strict units normalized afterward by the existing finite equivariant unitalization. The two colored copies are centrally equivalent by their ordinary continuation classes. This constructs the second two-copy category required in Sheridan Lemma 6.5.3, rather than assuming the naïve forgetful pullback already satisfies its hypotheses.

Forcing layers, exact caps and normalized Hochschild restriction. Raw divisor-preserving equations and finite virtual sections are different layers

Use two layers and do not exchange their conclusions.

RAW LAYER. Choose the actual geometric Hamiltonian one-forms zero in fixed neighborhoods of D and the auxiliary quadric B, and J standard integrable there. These data need not make every original map regular. Their stable moduli carry the completed relative d-vector and the auxiliary B-intersection number. Holomorphic normal positivity applies to THESE original maps and their homogeneous bubbles. At the fixed d and word their compactness is the explicit cover-domain argument in the corresponding construction. Components contained in a coordinate divisor are not discarded without argument; they are included among the original stable maps. The auxiliary cover still lifts the equation smoothly because it is homogeneous near B. There is no requirement that every original stable moduli be a regular manifold before a virtual atlas is constructed.

FINITE-ATLAS LAYER. Around this compact original zero set construct the finite supported-obstruction/tuple atlas using the actual local matched inverses. Finite obstruction forcing or a transverse multisection need not preserve the coordinate divisors. A perturbed chart solution may have accidental positive and negative geometric intersections with D. Its weight is its ORIGINAL topological d-label, locally constant on the thickening and on every gluing/tuple map. We never infer positivity from that perturbed solution's zero set. Compactness of the perturbed virtual chain is obtained from finite relatively compact charts and their proper product collars, not by reapplying an unproved divisor-positivity compactness argument to arbitrary forced maps. Choose nested chart cores covering the original compact zero set, perturb sufficiently small relative to their positive boundary-distance margins, and use the finite normal-product extension; this is exactly the previously constructed rational virtual-correspondence procedure.

Obstructions can be supported away from B: every disk component has an open boundary collar outside B because its boundary branes are a fixed positive distance from B. At such a collar source-local forcing spans the CR cokernel by the actual adjoint unique-continuation argument. The forcing is supported in the INTERIOR of the source collar, not on its physical boundary, so its Hamiltonian boundary restriction remains valid. Constant disk components have the ordinary onto real CR operator, and the node-evaluation deficits are spanned by the explicit annular sections already constructed. Their images lie away from B as well. In the present unmarked or singly-interior-marked genus-zero regime, no stable constant sphere tree with image in B remains: there are no nonconstant spheres at |d|≤1, and a tree of constant spheres with only one attachment and at most one actual interior mark has an unstable leaf. Auxiliary B-zero marks used to extract compact limits are NOT atlas constraints and are forgotten before this assertion. Thus no obstruction supported at an unavoidable B-valued sphere ghost is being suppressed.

At a genuine geometric simple-contact mark on an all-Lε word, keep forcing zero on its fixed contact neighborhood as well. There is an open boundary collar outside BOTH B and D, since Lε misses both. The joint CR/normal-jet adjoint argument of the existing jet supplier makes the finite thickening derivative onto using source-local forcing in that collar. Its polynomial test sections near the contact are allowed section VARIATIONS, not forbidden forcing parameters. Hence the contact equation remains holomorphic, its unique zero remains simple, and mark forgetting is the actual oriented local diffeomorphism in the earlier packet. If a contact also lies on B, the same argument still works: the supporting collar is disjoint from both, while local section variations at the contact need not be.

For L′ pearly words, the coefficient $d=e_j$ is a TOPOLOGICAL completed-intersection label, not an imposed geometric simple contact. A negative real boundary crossing can carry it. Such charts require no interior contact mark and may use coordinate-D-crossing virtual sections. The regular rigid real-line/central-curve configurations used in the explicit count are kept with zero obstruction perturbation on a small isolating neighborhood. Relative rational extension handles the rest. This distinction is essential to the first-order class calculation.

The forbidden-face exclusion is therefore an assertion about the PRODUCT virtual sections on the raw compactification, as detailed in section 4 below. It need not assert that an unperturbed negative-index central moduli has no solutions.

Here is the cap argument with internal flags accounted for. The outer generator word is fixed. Complete the entire mixed pearly tree to the graph immersion inside the cotangent neighborhood, including the two sides of every Morse edge with their opposite orientations. At INTERNAL flags these collars/cap paths cancel pairwise; the tree has no cycle, so no free internal cap loop remains. Only the fixed external chords require chosen paths between their two endpoints in the graph brane domain $S^n$. Choose these finitely many external paths once. The resulting completed outer boundary is a closed singular one-cycle in $S^n$, and $H1(S^n;Z)=0$ supplies a filling two-chain there. Project it to $C{P}^n$ and combine it with the completed map to obtain a closed two-cycle z(u).

Different fillings differ by a two-cycle from $S^n$. Its image in $H2(C{P}^n;Z)=Z$ is zero: integrate the Kähler form over it, whose pullback to the Lagrangian immersion is zero, and use the torsion-free generator of $H2(C{P}^n)$. Consequently its hyperplane degree h is independent of this filling. Differences between two maps with the same external word cancel their external cap corrections. For each coordinate divisor $D_i$ of degree one, d_i(u)−d_i(v)=h(u)−h(v).

Thus fixed d fixes h. The same calculation for B of degree two gives u·B−v·B=2[h(u)−h(v)]=0.

The cotangent interpolation collars miss B, so this is the sum of the actual nonnegative B-zero multiplicities on ALL pearls. Individual pearl branch degrees need not be fixed while internal evaluations move, and that stronger assertion is not used. Their total is fixed and nonnegative, hence bounds every individual number and the total number of marked branch zeros. This supplies the exact finite bound needed by the explicit covering-domain extraction.

The construction uses outer relative homology, with chord paths in the brane DOMAIN. Treating an immersed brane image as though every path lifted uniquely would be incorrect. Likewise π2(S²) is not zero; the proof instead uses its zero image in ambient H2, as just shown.

After marking the bounded zeros and stabilizing the doubled genus-zero base, every component has distinct marked coordinates. A normalized double cover over a punctured sphere is classified by its homomorphism of fundamental groups to Z/2. Its values on small loops are the zero multiplicities modulo two, and their product relation fixes infinity. A cover with all monodromies zero consists of two sphere sheets; otherwise it is the unique normalized curve y²=∏(z−a) over the odd marks, with the infinity convention. This proves the component classification and includes disconnected covers.

At a base tree edge the monodromy is the parity sum in either subtree. In the even case there are two preimage nodes; there are finitely many sheet pairings, so choose a constant pairing after a subsequence. In the odd case there is one node, with local equations $X=x^2$, $Y=y^2$, $XY=t$, $xy=s$, $t=s^2$. Coordinate units admit square roots after shrinking a simply connected fixture chart; changing the root is the retained deck sign. With several nodes choose a root at each, a finite product of two-point choices. These are ordinary finite parameter covers, not divisions by an infinite isotropy group.

Core coefficient convergence follows from the explicit polynomials after base normalization. A cluster of zeros is not silently deleted: it produces its own base sphere component and corresponding polynomial cover. On annuli the above equations show that the lifted neck length is either the base neck length or one half of it, up to bounded coordinate-unit terms. Therefore the same local energy-atom and small-image-neck extraction applies to every lifted fixture. No arbitrary varying-genus Deligne–Mumford compactness input is needed.

Choose full deck orbits of the auxiliary map-stabilizing marks. The domain action is retained by the sheet pairing/root sign in each fixture. The relation $\widehat{u}\tau =\sigma \widehat{u}$ passes to every core and node. The quotient node model is exactly $XY=t$ in both cases. Its map descends and has the original genus-zero base tree. Boundary traces lie on their individually labeled EMBEDDED graph/zero-section branes in Q; hence their limits give continuous $S^n$ lifts along each downstairs boundary segment. An odd boundary node would be fixed by the deck action and map into R, while its boundary value lies on a brane disjoint from R. It is thus excluded only after actual map convergence, rather than by assuming source branch points stay away from the boundary. Constant ghosts introduced solely by the added branch marks are contracted after descent and do not become operation vertices.

This supplies subsequential compactness and the sheet inventory. Local downstairs gluing does not use the root s as its coordinate: it uses the original ordinary disk fixture t and original CR operator. Consequently the two root choices do not double a downstairs count. The auxiliary cover is not a parametrization of the downstairs moduli. This also explains why collisions of branch points impose no extra constraint or orbifold weight on the downstairs atlas.

Fix all central $d=0$ data AND their transverse rational virtual sections in every arity before selecting first-order sections. On a forbidden $d=e_j$ bubble face the complementary central chart has virtual dimension at most $1-n<0$ after the correct free-node/discrete-chord matching. Its chosen transverse section has empty zero set. The face section is the PRODUCT of that central section with the bubble section. Hence it is empty for EVERY bubble-side section, without any genericity assertion about an added interpolation parameter.

Take finite relatively compact face-chart cores and a compact family of the bubble-side data/sections to interpolate. On the finite central cores the norm of the central section has a positive lower bound, since it has no zero. Product collars converge to the exact face-product section uniformly in that compact family. Shrink their collars until the central component differs from its face value by less than half the lower bound. The resulting entire collar has no zeros for the entire family. This proves the uniform collar assertion in the VIRTUAL atlas, even when the original negative-index central moduli was nonempty. It does not rely on applying raw divisor positivity to a forced chart map.

At a multiple forbidden face choose one of its fixed empty central factors and the finite product family. The same lower-bound argument applies; take the minimum over finitely many chart cores. Allowed face sections remain the already prescribed products. Thus the valid relative data near the forbidden face can be joined to the naïve forgetful pullback outside the empty collar without changing any zero/one-dimensional count. The higher central arity appearing after bubble removal causes no circularity because all central arities were fixed first. This is the precise relative extension used in the colored two-copy comparison, including its virtual version.

First use the actual objectwise contraction to pass the CENTRAL two-object category to its minimal category. Its m1 is zero and m2 is associative. The inverse cohomology classes f,g are literal m2 inverses. Set $u_X=v_X=e_X$, $u_Y=f$, $v_Y=g$. For a morphism a:Z→W define its A-coordinate by bar(a)=v_W a u_Z ∈A=Hom(X,X).

Associativity and $v_Z{u}_Z=e_X$ give bar(ba)=bar(b)bar(a). This identifies the binary category with the two-object matrix category over A; it is an actual algebra identification, including all graded Hom spaces. The maps $u_Z,v_Z$ have degree zero.

For a normalized Hochschild cochain φ on A define the binary-category extension on every composable word by Eφ(a_m,...,a_1)=u_(Z_m) φ(bar(a_m),...,bar(a_1)) v_(Z_0).

Restriction R to all-X words satisfies RE=1. In the associative Hochschild differential, multiplication of adjacent inputs becomes multiplication of their A-coordinates; left and right coefficient actions do likewise. Therefore E and R are cochain maps.

The cochain homotopy between ER and identity is the usual finite prism obtained by replacing the word vertices successively by X. At the j-th vertex insert $v_{Z_j}:Z_j\to X$, convert every vertex to its left to X using bar(a), apply the cochain, and multiply its output by $u_{Z_m}$. Sum these m+1 terms with the alternating suspended-bar signs. Computing dH+Hd pairs each multiplication strictly to the left/right of the inserted v with its neighboring prism term; the remaining internal terms cancel by $v_{W}a=\bar{a}{v}_Z$. The two endpoints are the unchanged original word and the fully converted word, whose output is E R of the cochain. This verifies the homotopy on each finite word. On all-X words every inserted v is a unit and a normalized cochain vanishes, so $RH=0$. Likewise $HE=0$ because an inserted v becomes the unit in A-coordinates. To obtain the strong side condition H²=0 without an unproved normalization assertion, write $P=1-ER$, first replace H by PHP, and then put $h=HdH$. We have $dH+Hd=P$, $PH=HP=H$, and dH²=H²d; the latter follows by expanding both sides using $dH=P-Hd$. Consequently $dh+hd=(dH{)}^2+(Hd{)}^2=P$, while h²=H d H² d $H=H^3{d}^{2}H=0$. Also $Rh=hE=0$. Since d for this minimal associative category raises cochain arity by one and H lowers it by one, h still lowers arity by one. Use h as the H in the next paragraph. Degenerate-unit tensors are removed by the explicit first-unit insertion contraction before these formulas, so the normalized quotient is preserved. This gives the actual associative Morita strong contraction without changing R,E or the filtration.

Now restore central operations of arity at least three. Their Hochschild perturbation Δ=[m≥3,−] raises cochain arity by at least two. H lowers arity by one, so HΔ and ΔH raise it by at least one. The geometric perturbation series therefore converge in the product/arity topology, coefficientwise with only finitely many terms at any specified arity. Restriction still obeys RΔ=Δ_A R because A is a full one-object subcategory. Since RH=0, RΔ(HΔ)^k E=0 for k≥1, RΔE=Δ_A.

Thus the transferred differential is EXACTLY the Hochschild differential of the actual full A∞ algebra A, not a new unrelated perturbation of it. The corrected maps prove that the actual Hochschild restriction is a quasi-isomorphism. Repeat at Y. Applying both restrictions to the same first-order category cocycle identifies the two first-order classes. The initial central minimalization is itself a tensor-length-finite bar perturbation, and it commutes with restricting objectwise contractions, so the conclusion transports back to the original chain category.

The cochains may include arity zero when comparing full square-zero deformations. The normalized convention removes unit INPUTS, not curvature coefficients. The arity estimates above remain valid. Finite group naturality follows by the orbit/linear primitive choices already specified. No claim that first-order inverse morphisms lift is used.

Bounded regular continuation geometry, without a critical degeneration. Use the Hamiltonian convention ${\iota }_{X_H}\omega =-dH$. On a punctured disk $\mathrm{\Sigma }$, write the Hamiltonian one-form $K=H_{s}ds+H_{t}dt$ and put ${\mathrm{\Omega }}_K=\omega -dK$ on $\mathrm{\Sigma }\times X$, where $d$ is the total differential. Its horizontal lifts are ${\mathrm{\partial }}_s+X_{H_s}$ and ${\mathrm{\partial }}_t+X_{H_t}$. For $\{F,G\}=\omega (X_F,X_G)$ set $R_K={\mathrm{\partial }}_s{H}_t-{\mathrm{\partial }}_t{H}_s+\{H_s,H_t\}$. The horizontal pairing is $-R_K$. Define the total almost-complex structure by the compatible vertical $J$ and the horizontal lift of the source structure. Its section equation is exactly ${\mathrm{\partial }}_{s}u-X_{H_s}+J({\mathrm{\partial }}_{t}u-X_{H_t})=0$. Writing its vertical derivatives as $v_s,v_t$ gives $v_t=J{v}_s$, hence $E_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}=\int U^{\ast }{\mathrm{\Omega }}_K+\int R_{K}dsdt$. On a compact core ${\mathrm{\Omega }}_K+Cds\wedge dt$ tames this structure whenever $C>\sup R_K$. These are actual equations and an energy bound; their zero sets are not thereby transverse.

One common contact-preserving atlas for the two graph-data colors. In this paragraph every physical boundary label is the graph $L_{ϵ}$, with its specified lifts, although its auxiliary-data color can vary. Fix a word, a divisor label $d=e_j$, an energy cutoff and a compact core of its original stable solution set. No disk component is contained in $D$: its boundary lies on $L_{ϵ}$ at positive distance from $D$. Nonconstant spheres are excluded by the zero intersection with any $D_i$, $i\mathrm{\ne }j$, as in the first-order inventory. A constant sphere tree carrying only the unique divisor mark and one attachment has an unstable leaf and disappears on stabilization. Thus each nonconstant component has an open interior source window with image outside $D\cup B$, and every actual contact has a source disk contained in the fixed integrable target chart. Keep these contact disks disjoint from the chosen forcing windows.

Actual transition maps on the collapsed-boundary quotient. At a collapsed boundary with one attached sphere tree choose the disk node at $0$ and a temporary output at $1$. The rotations fixing $0$ change only the node fixture $q$; the ghost kernel coordinates are its real constant value and the radial imaginary sections of the ghost construction. Both are invariant, and the sphere core parameters do not depend on the temporary output. Choose the marked right inverse by Haar averaging its LINEAR right-inverse equation and use the invariant kernel complement. All nonlinear local equations and exponential charts are natural under rotation. Uniqueness of the local contraction therefore makes its obstruction section invariant with values in the trivial ghost obstruction representation. The resulting chart action is $(q,\kappa )\mapsto (e^{i\theta }q,\kappa )$, with the ordinary finite sphere-tuple action retained separately.

Regular arbitrary-data comparison domains. In the upper half-plane put the output at infinity, $k$ ordered inputs on the real line and a seam $\mathrm{Im}z=h>0$, compactifying to a circle tangent to the output. Quotient by positive affine transformations. For $k\ge 2$ normalize the outer inputs to $0,1$, giving dimension $k-1$; for $k=1$ normalize its input to $0$ and $h=1$. Compactify by nested ordered collisions and the ratios $h\mathrm{/}\delta$ to every flag-cluster scale. A finite positive ratio retains the seam on that component, while zero/infinity moves it to descendants/ancestors. Every input-output path crosses one seam component; distinct branches may have different seam components. Recursive normalized-cluster extraction terminates on finitely many flags, reconstructs by the component scale equations and gives compact Hausdorff corner charts. Its two real face types are a source polygon feeding a seam component and several seam components feeding a target polygon; their intersections are the same painted trees.

Stable zero-input painted comparisons. The ordinary curved comparison coefficient $F^0$ has one output and a seam tangent there; normalize the output to infinity and seam to height $1$ in the upper half-plane. Real translation remains, so this is not a stable point. Each positive-energy stable map has a nonconstant disk/sphere component. Put labeled regular interior marks there with transverse codimension-two target slices, taking full finite map-automorphism orbits and supported joint-evaluation augmentations. If the painted principal component is nonconstant a marked point fixes translation by $\mathrm{Re}z=0$; if it is a constant ghost, use its first attachment toward an energy-bearing component. A ghost with no such attachment is the excluded unstable zero-energy artifact. No nontrivial real translation fixes a nonconstant finite-energy map or the marked attachment configuration. Thus these are genuine local slices.

Its real codimension-one faces are ordinary two-disk real-node splittings and the just-constructed collapsed-boundary sphere faces. At total virtual dimension−1 their transverse zero sets are empty. Interior sphere nodes have codimension two and are also empty transversely. Choose the rational finite-branch multisections ON THIS unmarked atlas, inductively over energy and faces, with all lower-face products prescribed. Every chart has negative virtual dimension, so its transverse perturbed zero set is empty; relative extension exists by the actual finite reduction/normal product proof. Near a collapsed chart use the smooth quotient coordinate $t=\mathrm{\mid }w{\mathrm{\mid }}^2$ from step 1.24 and choose quotient branches as smooth functions of t and the fixed obstruction coordinates. Since the circle acts trivially on E, their pullbacks are smooth invariant branches on the marked chart, including at $w=0$. No equivariant transversality in a nontrivial circle representation is needed.

Relative extension of compatible vertical structures is explicit. On a smaller protected face collar convert its prescribed compatible $J$ to the positive metric $g(v,w)=\omega (v,Jw)$. Extend this metric by a partition of unity with an arbitrary positive interior metric. Define $A$ by $g(Av,w)=\omega (v,w)$ and set $J=A(-A^2{)}^{-1\mathrm{/}2}$. Skew-adjointness and nondegeneracy of $A$ make the square root positive and smooth; this $J$ is compatible and equals the prescribed one on the protected collar. Intersecting face prescriptions must already agree there. One may average metrics to retain an actual finite stabilizer symmetry, but not average a transverse section. With a fixed critical fibration keep its standard critical chart protected and apply this construction on its regular complement; a large base form tames the resulting structure on each compact regular core. This constructs data on that fixed fibration, not a family in which a critical disk collides with a boundary arc.

Let ${\varphi }_t$ be a Hamiltonian isotopy generated by $G_t$, stationary at its endpoints and supported away from $D$. To move selected boundary labels by ${\varphi }_1$, choose a smooth $\tau :\mathrm{\Sigma }\to [0,1]$, equal to $1$ on their boundary arcs and $0$ on the others. On each puncture's strip prescribe the same fixed interpolation $\tau (t)$ between its two boundary values. The puncture is not filled in, so these end prescriptions are smooth. Extend them over the compact core by a partition of unity. Set $F(z,x)=(z,{\varphi }_{\tau (z)}x)$, ${\mathrm{\Omega }}^{\mathrm{\prime }}=(F^{-1}{)}^{\ast }{\mathrm{\Omega }}_K$ and ${\tilde{J}}^{\mathrm{\prime }}=F_{\ast }\tilde{J}$. The horizontal Hamiltonian is $K_e^{\mathrm{\prime }}=K_e\circ {\varphi }_{\tau }^{-1}+d\tau (e)G_{\tau }$, up to a source scalar; the defining form ${\mathrm{\Omega }}^{\mathrm{\prime }}$ fixes that scalar convention. Differentiating $F$ proves this formula and intertwines the complete section equations. On a boundary arc $d\tau$ is zero tangentially, so the transformed boundary condition is exactly the moved brane. On an end every chord is transported to ${\varphi }_{\tau (t)}x(t)$ with the transformed nondegenerate end operator.

The CR cokernel is spanned by forcings in those windows. Indeed a cokernel distribution annihilating every window-supported smooth forcing vanishes there; away from the finite contact points it solves the homogeneous adjoint CR equation. Elliptic weak regularity and unique continuation therefore make it zero on the punctured connected component. There are no point constraints in the unmarked problem. For a marked normal jet constraint, test independently against holomorphic normal polynomials with arbitrary jets, cut off in an annulus outside the contact point; its finite jet covector is zero, after which distributional elliptic regularity removes a possible point-supported remainder. This is the actual joint CR/jet surjectivity argument. Choose finitely many such window-supported forcings spanning the finite cokernel and retain every node-evaluation matching constraint. At constant components the ordinary real constant kernel and the annular node-evaluation sections give the missing matching directions, at image points outside $D\cup B$. These augmentations leave the equation literally holomorphic on the contact disks.

On an overlap choose the sphere stabilizing slices independently of the temporary output, and take their complete finite tuples. Their actual coordinate map commutes with these rotations. With the same oriented complex node convention its fixture component has the form $q^{\mathrm{\prime }}=qc(\mathrm{\mid }q{\mathrm{\mid }}^2,\kappa )$, where $c$ is smooth and nonzero on a smaller overlap, and its other component is ${\kappa }^{\mathrm{\prime }}=\psi (\mathrm{\mid }q{\mathrm{\mid }}^2,\kappa )$. To check smoothness without dividing at $q=0$, expand a smooth equivariant function in its Taylor polynomial in $(q,\bar{q})$ and use the weight-one equation: only monomials $q^{b+1}{\bar{q}}^b$ occur; the integral Taylor remainder satisfies the same equation and has the same divisibility after one derivative is factored. Equivalently restrict to the real axis and use the even/odd smooth-function division formula. The invariance equation gives the corresponding even function for $\psi$. Thus in the smooth quotient coordinate $t=\mathrm{\mid }q{\mathrm{\mid }}^2$ the overlap map is $t^{\mathrm{\prime }}=t\mathrm{\mid }c(t,\kappa ){\mathrm{\mid }}^2$, ${\kappa }^{\mathrm{\prime }}=\psi (t,\kappa )$, smooth with strictly positive inward normal derivative. The inverse overlap has the same form. This is the smooth corner atlas induced by invariant functions, with its actual cocycle coordinate changes.

On these actual domains extend Hamiltonian forms relative to prescribed source/target end data and lower product faces by their affine structure. Extend compatible structures by convex positive metrics and polar decomposition. Retain the fixed integrable structure and zero forcing near $D$. The finite core parameter family has bounded curvature, so its compact mixed-boundary rational atlas gives parameterized chains wherever that atlas and the painted stabilization have been constructed. Their boundary products are exactly the two face types above and curvature insertions, giving the full comparison equations after actual MC insertion. The zero-input principal domain is unstable: its map-stabilizing charts and transitions must be included separately, and are not proved by the $k\ge 1$ configuration formula. Until that curved stabilization compatibility is supplied this paragraph does not claim an arbitrary-model $F^0$ construction.

Retain all their mark positions, seam-height ratios and node scales in the painted cluster extraction. A colliding normalization mark gives its stable bubble; a different regular mark/attachment supplies the overlapping slice. The common chart uses the simultaneous tuple of both stabilizations and its finite target-slice normal constraints. At a node use their complete product tuples. At each positive-energy cutoff the ordinary homogeneous zero-input vertex energy quantum and stable tree flag bound give finitely many types. Compactness, matched inverses, finite reduction and relative rational sections therefore construct actual zero-input parameterized moduli, including constant painted ghosts attached to nonconstant trees. Count their rigid parts to define $F^0$, rather than inserting a formal zero.