Named Fermat graph algebra, compact grading, and geometric Fermat class

Explicit cubic geometry, embedded lifts and rigid central counts. A repaired explicit Morse model

Put $H_R=\{x\in R^N:\sum x_j=0\},S=\{x\in H_R:\sum x_j^2=1\}$, and $f(x)=-\sum x_j^3$. Use the round metric. Before radial projection its gradient in $H_R$ has coordinates $F_j=3\mathrm{/}N-3x_j^2$. Thus its normal component on $x_j=0$ is exactly $3\mathrm{/}N>0$. It is odd and invariant under every coordinate permutation.

For $x_j>x_k\ge 0$, radial projection cancels in the derivative of $x_k\mathrm{/}{x}_j$, giving

$\mathrm{\nabla }f(x_k\mathrm{/}{x}_j)=3(x_j-x_k)(1\mathrm{/}N+x_j{x}_k)\mathrm{/}{x}_j^2>0$.

For $x_j<x_k\le 0$ the same expression is negative. These are exact identities; no inference from the order of two mean-subtracted gradient coordinates is used. At a critical point, $-3x_j^2=\lambda +2\mu x_j$ for all j. This quadratic has at most two real roots. There cannot be just one root because sum $x_j=0$ and norm1. There are therefore two nonzero levels, one negative a on a nonempty proper K, the other positive b on $K^c$. The constraints give

a=−sqrt((N−r)/(Nr)), b=sqrt(r/(N(N−r))), r=|K|.

Conversely the permutation symmetry preserves this isolated two-block locus, so its two points $p_K,p_{K^c}$ are critical. On the tangent space the two block sums vanish separately. The constrained Hessian is +3(b−a) on the r−1 directions in K and −3(b−a) on the N−r−1 directions in $K^c$. Hence it is Morse with index N−r−1. These are ALL critical points, and $p_{K^c}=-p_K$.

For positive-gradient flow, equalities and strict inequalities between coordinates are preserved by uniqueness and permutation symmetry. Its stable set at $p_K$ is precisely the locus whose minimum-coordinate index set is K; its unstable set is the locus whose maximum-coordinate index set is $K^c$. To prove the first assertion, every trajectory converges to a critical point: an omega limit point is critical by the gradient energy identity and compactness; the omega limit set is connected and the critical set finite, so it is one point. The initial minimum set is contained in the limit minimum set. If an additional coordinate approached that minimum, it would eventually be negative, and the negative-coordinate ratio identity above would keep its ratio with an original minimum bounded away from1. This contradicts convergence. The unstable assertion is the time-reversed argument. These descriptions give the actual linear flag stable/unstable cells used in the counts, including preservation of each equal-coordinate linear subspace under flow.

This cubic replaces an unnecessary saturated scalar g. The printed saturated-g ratio argument cannot justify its ratio inequality merely by ordering the mean-subtracted coordinates. Our cubic proves all the required inequalities directly. If positive scalar g′ is desired, write $g(t)=Ct-t^3$ with $C>3$; its restriction sum g equals f on $H_R$ and changes none of these calculations.

Identify the double cover $S\to RP(H_R)$ with the zero section of T*S. A symplectic cotangent neighborhood exists by the following direct relative-Moser construction. Choose the normal map given by the complex structure along $RP(H_R)$. Its pullback symplectic form and the standard cotangent form agree to first order along the zero section. Radial integration of their difference gives a primitive vanishing there; contraction with their interpolation gives a vector field vanishing to second order there. On a sufficiently small compact neighborhood its flow is defined and gives the desired symplectic chart. Average the initial metric/map and primitive over the finite coordinate group and real conjugation; the interpolation and unique vector field inherit their actions. Thus conjugation is fibre negation. Pull back to S.

The exact graph εdf maps to a Lagrangian immersion $L_{\epsilon }$. It avoids every coordinate divisor: at an intersection of the zero section with $x_j=0$, its first imaginary normal displacement is $\epsilon \mathrm{\nabla }{f}_j=3\epsilon \mathrm{/}N$, while on compact sets away from that divisor the unperturbed coordinate is bounded away from zero. The local expansion is uniform; reducing the neighborhood and ε gives nonvanishing everywhere. In simultaneous divisor charts the same statement holds in each coordinate. Its two sheets are graphs εdf and −εdf, meeting exactly at the Morse points above and transversely by their nondegenerate Hessians. The sheet involution is antipodal; conjugation preserves the image and exchanges its sheets.

The immersion is exact for the Liouville primitive on the affine hyperplane complement. Indeed its primitive is closed on S, which is simply connected for $n=2,3$, so is exact. The grading lifts for the same reason; use the volume-form convention below. Each of its N lifts in the cyclic affine cover is embedded: at the double point associated to K, the two sheet lifts differ by |K| in Z/N, which is nonzero for $0<\mathrm{\mid }K\mathrm{\mid }<N$. To calculate this deck change, deform the connecting path to the positive normal crossings of the coordinate hyperplanes. Each changed sign contributes one to the cyclic root character; the common diagonal changes it by N and is immaterial. Thus the two branches never belong to the same lifted sphere. Additional finite diagonal lifts preserve compactness, exactness, grading and the unique sphere spin structure.

The lifted objects have two Morse generators for self morphisms and one ordered sheet generator $p_K$ for each allowed distinct-object deck label. No affine immersion compactness theorem is assumed: all disk operations may be defined on these embedded lifts using the actual finite-group multi-object analytic construction. The actual mixed-category comparison in F1 identifies these geometric operations with the oriented real-pearl operations used below.

The real pearls live in $P(H_C)\cong P^n$. For a partition into three nonempty blocks $K_1,K_2,K_3$, the block-constant subspace in $H_C$ is two-dimensional, hence a projective line. It contains exactly the three prescribed block points $[p_{K_i}]$. Conversely a line meeting their three unstable flag loci must be this line: after fixing two evaluations, their span lies in the subspace whose remaining block is constant, and the third flag intersection is the isolated complementary block point. The intersection is transverse because the corresponding linear equalities on the three blocks are independent modulo sum $x=0$. Its two hemispheres are distinguished by the sphere boundary lift; precisely one has the prescribed sheet switches and positive coordinate-normal crossings. Thus the binary coefficient for disjoint input subsets is ±1. Repeated-index terms vanish by the character/degree condition. Units and the top pairing are constant-pearl limits, with their determinant cap orientations.

Here is the full rational interpolation calculation for the central higher count. In $H_C$ put $v_j=N{e}_j-1$, so sum $v_j=0$ and this is their only linear relation. The singleton projective points are $[v_j]$. Fix a generic unit-lift point $w\in H_R$, with all $w_j$ nonzero and pairwise distinct. Choose real distinct ${\nu }_j$ with

${\nu }_j^{-1}=c-w_j\mathrm{/}N$.

Then

$U(z)={\sum }_j{v}_j\mathrm{/}(z-{\nu }_j)$

has $[U({\nu }_j)]=[v_j]$ and $[U(0)]=[w]$. Multiplying by $\prod (z-{\nu }_j)$ gives homogeneous coordinate polynomials of degree at most N−2: the degree N−1 terms cancel because sum $v_j=0$. There is no common zero. At ${\nu }_j$ its nonzero value is a multiple of $v_j$. At any other finite point, $U(z)=0$ would force all $1\mathrm{/}(z-{\nu }_j)$ equal, impossible. At infinity its leading vector is sum ${\nu }_j{v}_j$, nonzero for distinct ${\nu }_j$. The image spans $H_C$ because it contains the spanning singleton points. A nondegenerate degree N−2 map to $P^{N-2}$ is the complete Veronese linear system, so is an embedded rational normal curve.

It is unique. Any degree N−2 curve through those N singleton points spans $P^{N-2}$, so its normalization carries the complete degree N−2 linear system. Fix the last marked point w at0, choose no singleton point at infinity, and use their real preimages ${\nu }_j$. Partial fractions of its coordinate polynomials divided by $\prod (z-{\nu }_j)$ give sum $c_j{v}_j\mathrm{/}(z-{\nu }_j)$. Its missing degree N−1 coefficient gives sum $c_j{v}_j=0$, forcing all $c_j$ equal. Its value at0 then gives ${\nu }_j^{-1}=c-a{w}_j\mathrm{/}N$ for a common nonzero a. Scaling z changes a and a fractional-linear transformation fixing0 changes c. Hence all such parametrizations are reparametrizations of the displayed U. This also proves regularity: the reciprocal-parameter equations solve smoothly and uniquely for a nearby generic tuple of N+1 marked target points after the three source reparametrization freedoms are removed. Equivalently, the evaluation derivative on the degree-N−2 map space modulo $PG{L}_2$ is an isomorphism.

Admissibility has an exact sign check. The j-th coordinate is

$U_j(z)=(N-1)\mathrm{/}(z-{\nu }_j)-{\sum }_{l\mathrm{\ne }j}1\mathrm{/}(z-{\nu }_l)$.

At a real zero its derivative is

−(N−1)/(z−ν_j)²+sum_(l≠j)1/(z−ν_l)²>0.

Indeed the zero equation says the sum of the latter N−1 reciprocals is (N−1)/(z−ν_j); Cauchy–Schwarz gives the weak inequality and equality would force all ν_l equal. At each pole ν_j the real boundary lift changes sheet. Thus one half-curve has the required positive crossings; the opposite half has the antipodal unit lift. Its cyclic order is fixed by the order of the ν_j, so exactly one ordered permutation of the N singleton inputs contributes, with coefficient ±1.

Consider a terminal nonconstant pearl in a multiple-pearl tree, with $k+1$ singleton input flags and one internal flag. Write $e_K$ for a subset indicator and $\mathbf{1}$ for the all-one vector. With $s$ flipping flags and doubled degree $d$, the positive boundary-crossing rule gives ${\sum }_{a=1}^s{e}_{K_a}=(s-d)\mathbf{1}\mathrm{/}2$: the boundary segment after a flip starts with its complementary label, so summing the lost coordinate labels around the boundary gives $s\mathbf{1}-2{\sum }_a{e}_{K_a}=d\mathbf{1}$. This conservation has two cases. If the internal flag is nonflipping, the singleton-indicator sum is constant in all coordinates, which forces $k=n+1$, so this terminal pearl already consumes all $N=n+2$ singleton inputs and cannot belong to a multiple-nonconstant-pearl contribution. If it is flipping, its extra label must be exactly the complement of the $k+1$ singleton indices; the constant is one and $s=k+2$, so the doubled pearl has projective degree $k$. A degree-$k$ genus-zero map spans at most a $k$-plane: its coordinate sections lie in the $(k+1)$-dimensional space of sections of $\mathcal{O}(k)$. The $k+1$ distinct singleton points are independent, so they force exactly their projective span, the equal-coordinate subspace on the complementary indices. If $k=n$, no positive degree is left for another nonconstant pearl; otherwise $k<n$. In that case the internal-flag evaluation ranges over an open subset of this $k$-plane, by the same rational interpolation with $k+1$ singleton flags and one moving flag. Its attached cubic-gradient vector lies in the same evaluation tangent space because the complementary equalities are flow invariant. Consequently varying the positive Morse-edge length adds no evaluation direction: the multiple-pearl unit-evaluation map has rank at least one below its expected rank. A generic unit point misses these finitely many fixed-degree tree images. On their stable compactifications the same argument applies to each nonconstant terminal subtree; constant unstable components are contracted. Choose the perturbation relative to those empty compact strata. This excludes multiple-pearl contributions without changing the unique regular single-pearl count.

Characters and the actual exterior signs. Characters, auxiliary grading, and relative cohomology freeness

Let $tilde\mathrm{\Gamma }=\{({\zeta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\zeta }_N)\in {\mu }_N^N:\prod {\zeta }_j=1\}$, of order $N^{N-1}$. The N sphere labels times the diagonal-cover group of order $N^{N-2}$ give precisely this number of primitive character idempotents. The character of $p_K$ is the restriction of ${\prod }_{j\in K}{\zeta }_j$. Two subsets have the same character exactly when their indicator difference is constant modulo N. Since every entry of that difference is −1,0,1 and $N\ge 4$, this means either identical subsets or the empty/full pair.

For the DOWNSTAIRS meromorphic-volume grading choose integral weights ν with sum ν_j=N−1. Its auxiliary degree is

$i(p_K)={\sum }_{j\in K}(2{\nu }_j-1)$.

This comes from the Morse index N−1−|K| plus the phase-lift difference of the two antipodal sheets; crossing the j-th coordinate cut changes that sheet lift by $-{\nu }_j$. In particular empty has degree0, full has degree $N-2=n$. The parity is |K|. Each off-diagonal primitive-idempotent Hom has one central generator, hence after the actual object-dependent upstairs grading it is concentrated in a SINGLE integer degree; diagonal Homs have degrees0,n only. This concentration assertion uses the character multiplicities and the actual brane grading, rather than identifying the displayed auxiliary formula unchanged with the compact grading. The central operation on the N DISTINCT singleton indices has degree 2−N and sends them to the unit, as the formula requires. This integer-ν convention is not the unchanged compact Calabi–Yau grading upstairs. Repeated identical-singleton operations in the Fermat first-order class require the branched-cover and object-dependent phase regrading, or the symmetric rational downstairs degree n/N with its actual lifted brane grading. The auxiliary $r_j$ has degree $2{\nu }_j$; it must not be replaced directly by the degree-zero compact parameter q while retaining all these fixed integer singleton degrees. The actual mixed-pearl grading binding is supplied by the present construction.

Once the actual affine comparison proves this central complex, relative HF-freeness is immediate and does not need a new positivity theorem. Contract its finite chord complex over C to the displayed central cohomology. Extend the contraction over $R=C\mathrm{⟦}q\mathrm{⟧}$ and apply the explicit q-adic perturbation identity $h(1+\delta h{)}^{-1}$, where δ is q-positive. Each inverse converges coefficientwise q-adically. The transferred differential has degree+1 and preserves the primitive idempotent pair. There are no such adjacent degrees in ANY pair, so it is identically zero. Hence all relative Hom cohomology is R-free. This is an argument from the proved central complex and the actual relative analytic category, not from chain freeness alone.

The ±1 counts alone would not prove the EXTERIOR algebra rather than a commuting square-zero algebra. The required conjugation orientation is the determinant-cap action on each $p_K$. For volume weights ν it is $(-1{)}^{1+{\sum }_K{\nu }_j}$. One obtains it by first shifting the second sheet by $i_K=N-1-2{\sum }_K{\nu }_j$, making its cap the short Morse cap. The shift reversal contributes $(-1{)}^{i_K(i_K-1)\mathrm{/}2}$. The antipodal Pin lift squares to $(-1{)}^{n(n+1)\mathrm{/}2}$, computed by the product of the n+1 orthogonal unit vectors in the positive Clifford algebra. The conjugation cap contributes one additional minus sign. Multiplying these three signs gives the displayed factor. For $n=2,3$ the sphere spin structure is unique. Thus conjugation gives the opposite binary product with exactly the Koszul sign; the ordinary associative product $a\cdot b=(-1{)}^{\mathrm{\mid }b\mathrm{\mid }}{\mu }^2(a,b)$ is supercommutative. Each singleton squares to0 by the disjointness condition. The nonzero disjoint coefficients make the singleton generators span all $p_K$ by products. The resulting exterior-algebra map is surjective between vector spaces of the same dimension $2^N$ and hence an isomorphism. This is compatible with the diagonal characters and their primitive-idempotent splitting.

This cap calculation must be bound to the actual determinant orientations of the comparison moduli, rather than copied as an unoriented count convention. The geometric relative-spin provider supplies the determinant gluing framework; the sheet/cap shift normalization is the explicit one fixed here.

First-order real line. The first-order downstairs pearl coefficient

Use one relative parameter $r_j$ for each hyperplane $D_j$ in $P(H_C)$, before taking the branched cover. The coefficient of $r_j$ in ${\mu }^1(p_j)$ paired with the unit is a first-order strip-pearl problem. Its two external flowlines are constant: the singleton is an extremal critical point of f in the required direction and the chosen unit point is the extremum of its auxiliary perfect Morse function. Its doubled pearl has degree1, so is the unique real projective line through $[p_j]$ and [w]. On S its boundary lift is the unique half great circle from $p_j$ to $-p_j$ passing through w. Its oriented sheet change at $p_j$ fixes which half of the complex line fills it. At every coordinate $i\mathrm{\ne }j$ it crosses the real coordinate hyperplane positively; the j-th coordinate has its single negative crossing. A half complex line has no interior intersection with any real coordinate hyperplane, since its unique complex intersection lies on the real boundary. Pushing the boundary by the positive normal εdf therefore creates exactly one interior intersection with $D_j$ and none with the other $D_i$. This sign assertion is a local calculation: at a transverse real boundary crossing the normal map is a z in the upper half-plane; after the graph displacement it is a z+i $\epsilon c,c>0$. Its zero is below the domain when $a>0$ and inside when $a<0$. The error is $O(\epsilon \mathrm{\mid }z\mathrm{\mid }+\mathrm{\mid }z{\mathrm{\mid }}^2+{\epsilon }^2)$; a small boundary half-disk and the argument principle preserve that zero count.

There is exactly one such regular pearl, so this coefficient is ±1. Any additional nonconstant pearl would have positive half-integer projective degree and leave no degree for the required line; the degree1 problem has only one nonconstant component. Unstable constant components are contracted and add no multiplicity. Its regularity is the invertibility of the two-point line evaluation derivative; the cap lines fix the same orientation normalization as the central count. Thus the first-order downstairs Hochschild class is ${\sum }_j\pm r_j{u}_j$, where $u_j$ is the polynomial Hochschild generator dual to the singleton arrow. This is a cochain calculation, not an inference from existence of the relative category. The actual finite first-order mixed comparison in F1 identifies this with the geometric deformation class. The contact-power construction and corrected grading below pass through N-fold branching to the class $u_j^N$ upstairs; no positive-contact curve is incorrectly confined to the null-relative-homology small cotangent sector.

Actual confluent jet and contact-power correspondence. Let $(M,D={\sum }_i{D}_i)$ be the compact Kähler pair used downstairs, with at least two divisors, every $[D_i]$ a positive multiple of the Kähler class. Let D+ be the specified divisor neighborhood, with integrable complex coordinates there; the domain-dependent Hamiltonian perturbations vanish there and the almost-complex structure is the fixed integrable one there. Boundary branes lie outside D+. For a fixed j and $b\ge 1$, M1(p,j,b) has one interior mark of EXACT normal intersection multiplicity b with $D_j$, total intersection $b{y}_j$, and no other divisor intersections. M2(p,j,b) has marks q1,q2 of multiplicities b and1, total $(b+1)y_j$, and the source's domain slice: a unit-disk normalization sends output to−i and q1,q2 to−t,+t for $t\in (0,1)$. Its natural closure retains $t=0,1$ and ordinary disk-tree faces. These are the actual source definitions, where “tangent to order b−1” means normal vanishing order b.

In a contact coordinate chart write $D_j=\{w=0\}$ and $f(z)=w(u(z))$. Near the marks f is holomorphic because BOTH Hamiltonian perturbation and virtual forcing there are zero and J is integrable. Multiplicity at least b is $f(0)=\dots =f^{b-1}(0)=0$. These are b COMPLEX equations, not one ordinary evaluation equation. Their coordinate changes are multiplication by a holomorphic unit and an invertible change of source coordinate, hence invertible complex triangular transformations of the finite jet module. They give its intrinsic complex orientation.

For distinct or colliding marks p1,p2 in one small disk set $P(z)=(z-p1{)}^b(z-p2),m=b+1$. The quotient C[z]/P is FREE of rank m with basis $1,z,\dots ,z^{m-1}$, including $p1=p2$. The remainder of the normal function is the polynomial r_f(z)=(1/(2πi)) integral_|ζ|=r f(ζ) [P(ζ)−P(z)]/[(ζ−z)P(ζ)] dζ.

Choose the outer contour fixed and both marks in $\mathrm{\mid }z\mathrm{\mid }<r\mathrm{/}3$. Then P(ζ) is uniformly nonzero there, and this formula is smooth in every real chart/kernel/domain parameter and in p1,p2. The equation $r_f=0$ is equivalent to divisibility by P. At $p1=p2$ it is EXACTLY the multiplicity-m jet condition, without any singular Vandermonde denominator. The same construction with one factor gives M1. Taylor coefficients use $f^l\mathrm{/}l$!, so no hidden factorial multiplicity is introduced.

The local section is taken on the CONTRACTED parent, not on the constant ghost. This assertion has a direct closure-and-inverse argument. Forget the two additional interior marks and stabilize the remaining disk. In a chart around their limiting interior point the parent coordinate is fixed by three retained flags. For positive collision parameter the two marks have coordinates $-t,+t$ in that chart. A raw map is holomorphic on a disk containing them, so its values on a surrounding fixed contour determine the whole normal function. If a sequence converges on that contour, the Cauchy formula gives convergence of every jet on a smaller disk. Since each $f_t$ is divisible by $(z+t{)}^b(z-t)$, the remainder formula of1.40 gives $f_0$ divisible by $z^{b+1}$. Thus no jet condition is lost on the ghost. Conversely, at a parent solution with exact order $b+1$, the combined CR/remainder derivative of1.43, 1.44 and 1.45 is onto. The implicit-function construction with the polynomial coefficients and the REAL parameter $t\ge 0$ therefore gives precisely the deformations satisfying the separated-root equations, with the prescribed parent/exterior data. The unique constant ghost has three special points and no automorphisms after those flags are fixed. Its constant CR kernel is eliminated by the node evaluation; its surjective operator contributes no residual obstruction. This proves that the collision end is a single parent-contact half-collar, not an extra component declared equivalent by forgetting its marks.

Work at Sobolev order high enough that jets through m−1 are bounded. For the combined linear map (CR,normal jets), a possible cokernel element is a distribution η in the dual target Sobolev space and a finite jet covector λ. Suppose it annihilates every admissible perturbation parameter. There is a nonempty interior source patch mapping outside D+: the boundary branes lie in that complement, and a small interior collar therefore does as well. Arbitrary source-local Hamiltonian gradient variations there realize arbitrary smooth compactly supported CR forcing along u. Consequently η=0 on that patch.

Away from the finitely many contact points, η solves the homogeneous adjoint real CR system. Actual interior weak regularity and unique continuation make it zero on that entire punctured connected source. It can therefore only be a distribution supported at the contact points. Test the annihilation identity on a local holomorphic normal polynomial section with any prescribed jets, cut off in a surrounding annulus. Its CR image vanishes near every marked point, so η evaluates it to zero regardless of its possible distribution order. The prescribed polynomial jets are arbitrary, hence λ=0. The remaining η solves the homogeneous elliptic adjoint distribution equation. The actual Fourier weak-regularity bootstrap makes it smooth, and a smooth distribution supported on finitely many points is zero. This proves the universal derivative onto.

The same proof works at confluent collision, using the free remainder module and one arbitrary polynomial of degree $<m$. Thus finitely many admissible Hamiltonian/source parameters supported OUTSIDE D+ span the combined finite cokernel, uniformly on a compact slice. One must not instead use forbidden perturbations inside D+ and then claim the ramified-cover pullback still holds. Use the finite supported augmentation and relative rational transverse-branch construction of F1, retaining every prescribed face product. The argument also gives finite virtual augmentations if needed. All normal jet equations have their full complex codimension2m and canonical complex orientation.

At fixed generator word and intersection vector the actual energy formula gives a bound. No map meets any $D_i,i\mathrm{\ne }j$. A nonconstant sphere, including a sphere contained in D, has positive intersection with EVERY $D_i$ by the assumed positive proportional classes. Disk components with boundary in the complement have nonnegative intersections, by their holomorphic normal functions. Thus a sphere would contradict the total zero intersection with $i\mathrm{\ne }j$. Every sphere bubble is consequently constant. A nonconstant unmarked disk bubble would, by exactness in the complement, have to meet a divisor; that intersection persists positively and carries one of the marked contacts with it. Hence it is included in the marked disk-tree alternatives rather than lost as an unstable unrecorded disk.

Fixed total intersection also excludes higher-than-prescribed contact and SNC crossings at a mark: either would add positive intersection beyond the total vector. In particular the contact neighborhoods really lie in the SMOOTH part of $D_j$ and the local single-normal-coordinate proof above applies. The fixed-contour jet construction records the entire contact order when the two marks collide. There are no components contained in D left to carry an extra normal-decoration stratum in this particular moduli problem; this exclusion depends on the zero intersection with at least one OTHER ample divisor and is not asserted for arbitrary contact moduli.

The actual compact genus-zero marked-domain construction gives the source slice endpoints. At $t=0$ its marks−t,+t collide with their fixed real direction; the unique constant three-flag sphere ghost carries the two marks and parent node. The remainder section limits to multiplicity b+1 on the parent, giving M1(p,j,b+1). The collision scale is a positive multiple of t, so it contributes a single real boundary normal and no extra degree or branching factor.

At $t=1$ the marks approach the DISTINCT boundary points−1,+1. Their two marked disk bubbles carry respectively multiplicity b and1. The remaining parent disk tree avoids every divisor, and the b branch is left of the1 branch in the source planar order. The two real node scales have a fixed positive ratio determined by the normalized domain slice; the ordinary two-neck product gluing restricted to that diagonal gives ONE half-collar parameter. Thus this endpoint is the ordered cup-product stratum, not two unrelated boundary ends.

The remaining source faces are ordinary broken disk trees with the M2 configuration on one distinguished component. They produce the Hochschild differential of its count. Additional independent node breaks have their actual additional real codimensions and are absent in a transverse one-dimensional space unless accounted for as these faces. Constant sphere corners with extra nodes have complex codimension at least two; there are no nonconstant spheres by the preceding sphere-exclusion argument. These are precisely the three alternatives, not a compactness assertion without an exhaustive boundary inventory.

The additional jet constraints are included BEFORE constructing the matched inverse. More explicitly, let $D_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}$ be the already augmented CR/evaluation operator on the fixed contour cores and necks, and let $J_P$ be the remainder-coefficient derivative. Its finite-codimensional restriction has a bounded right inverse because1.43, 1.44 and 1.45 provide finitely many allowed supported sections whose $J_P$ values span its residual cokernel. Choose their finite right-inverse matrix on a compact chart slice; its smallest positive singular value has a positive lower bound after a finite chart refinement. The ordinary pregluing error on the surrounding contours tends to zero in every fixed core Sobolev norm. The Cauchy formula for $J_P$ converts this into a uniformly small error in the FULL jet module, even when the roots coincide, because $\mathrm{\mid }P(\zeta )\mathrm{\mid }$ is bounded below on the fixed contour. The same formula differentiated twice gives the quadratic remainder bound. Splicing the matched core/neck inverses, including these finite supported jet directions, therefore gives an approximate inverse whose error is less than1/2; the Neumann series gives the exact inverse. Contraction gives the constrained solution, and core convergence followed by the same local uniqueness gives inverse gluing. At $t=1$ apply this to the two distinct node charts and restrict their positive gluing scales to the actual normalized one-dimensional diagonal; at $t=0$ use the parent-contact scale just established. All these inverses preserve the perturbation-zero neighborhood of $D$, since every additional section is supported outside it. This is the missing constrained gluing argument; an unconstrained evaluation inverse would not establish it.

All contact equations are the coefficients of the holomorphic remainder in the ordered basis $1,z,\dots ,z^{m-1}$. Their normal determinant is a COMPLEX determinant line and hence has its canonical real orientation. For separated roots, the evaluation/derivative coordinates and these remainder coordinates are related by a complex-linear isomorphism; as roots meet one uses the remainder basis itself, not the singular evaluation Vandermonde matrix. Taylor coefficients are divided by their factorials, which are nonzero complex scalars and do not change degree or orientation. At the collision the single ordered three-flag ghost has trivial automorphism group and the gluing coordinate $t$ covers its half-collar ONCE. At the product end the two marks are ordered and their two real node coordinates are restricted to the given positive diagonal ONCE. There is no quotient by a permutation group and consequently no factor $b!$ or $(b+1)!$. With determinant caps glued in the fixed suspended convention, the outward source normals are $-{\mathrm{\partial }}_t$ and $+{\mathrm{\partial }}_t$ at the two ends. The other faces have exactly the Hochschild broken-tree orientation. Counting their signed boundary gives ${\beta }_j(b+1)=\pm {\beta }_j(b)⌣{\beta }_j(1)\pm \mathrm{\partial }{H}_j(b)$, as an actual cochain identity in that convention.

The single-mark compactification likewise gives ∂β_j(b)=0. No unknown positive integer multiplicity appears: the confluent polynomial basis, the unique three-flag ghost and the positive real collision scale all have degree one. Overall signs depend on the chosen source normalizations, as in its stated theorem. Thus the induced class is the indicated signed power. Its nonvanishing requires the ACTUAL algebraic power to be nonzero; a nonzero class alone does not imply its powers are nonzero. The rigid pair-of-pants coefficients are computed in the first phase, and F3 supplies the pure-power computation applied below.

The ramified lift is GLOBAL over the disk after one boundary sheet is fixed. Consider first one normal equation $w=v^a$, with a downstairs map holomorphic in the divisor neighborhood and all its zeros there of order divisible by$a$. On the punctured source the possible analytic continuations of a chosen root have monodromy $\exp (2\pi im\mathrm{/}a)$ around a zero of order$m$. All these monodromies are1; a punctured disk has its fundamental group generated by these small loops. A root selected at one boundary point therefore continues uniquely, and removable-singularity extension gives the holomorphic lift over every zero. Equivalently near a single contact write $w(z)=(z-p{)}^{a}g(z)$, $g(p)\mathrm{\ne }0$. A chosen branch of the holomorphic unit root $g^{1\mathrm{/}a}$ gives $v(z)=(z-p)g(z{)}^{1\mathrm{/}a}$; the other$a$ choices are the other deck sheets, not extra lifts with the same anchored boundary label. For the actual finite cover use the same continuation argument for its full deck group: the monodromy of a small loop around a contact is the specified branch generator raised to a multiple of its order, so it is trivial. Thus a fixed anchored lift has exactly one global lift, including the simultaneous coordinate relations of the Fermat cover; independent coordinate roots are NOT counted without those relations. In families the Cauchy remainder description gives the smooth factor $g$ and the chosen unit root varies smoothly after fixing its value at one reference point. Its derivative on the unit factor is multiplication by $a{\eta }^{a-1}$ with ${\eta }^a=g$, an invertible complex multiplier. One must not use multiplication by $a{v}^{a-1}$ on an unconstrained Sobolev space at the ramification point: that multiplier vanishes there. On the ACTUAL constrained holomorphic-contact chart the variables are the contact point$p$ and the nonvanishing unit$g$, and the displayed root map is a local diffeomorphism preserving the complex jet-normal orientation. Outside the contact charts the cover is a local symplectic diffeomorphism; pull back the supported reductions there. The resulting finite reduced kernels, equations and determinant caps identify, with the same real capping lines and the complex-oriented jet factors. Thus the anchored first-order moduli carry the same signed virtual weight. Free deck orbits transport the data without averaging a nonlinear operation. This establishes the one-divisor first-order ramified comparison, not an all-orders relative orbifold theorem.

Compact CY grading and nonzero Fermat tangent class. The two downstairs gradings must not be confused

Write $N=4$ or 5 and $n=N-2$. On the pair of pants in $P(H_C)$, choose meromorphic-volume weights ${\nu }_i$ with $\sum {\nu }_i=N-1$. With INTEGER weights the auxiliary grading is |p_K|=Σ_(i∈K)(2ν_i−1), |r_j|=2ν_j.

This is the correct grading for the first-order downstream bubble check: $\mathrm{\mid }{r}_j{p}_{j^c}\mathrm{\mid }=N-1$, so its zero-input moduli dimension is $N-3=n-1$. It is NOT the compact CY Ω-grading on every lifted sphere. Keeping this integer formula unchanged upstairs would make N repetitions of a fixed singleton incompatible with a degree-(2−N) operation to the unit when q has degree zero.

For the branched CY comparison use instead the symmetric RATIONAL weights ${\nu }_i=(N-1)\mathrm{/}N$. Then each singleton has degree |v_j|=2(N−1)/N−1=n/N, and $\mathrm{\mid }{r}_j\mathrm{\mid }=2(N-1)\mathrm{/}N$ downstairs. These rational degrees are converted to integer Hom degrees by the explicit lifted-object shifts below.

Let $X_N=\{\sum x_i^N=0\}\subset C{P}^{N-1}$, and let $F:X_N\to P(H_C)$ be $[x_i]\mapsto [z_i]=[x_i^N]$. Its ramification along $E_i=\{x_i=0\}$ has order N−1. In the affine chart $x_N=1$, use $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$ as independent variables and solve for $x_{N-1}$. The actual holomorphic residue form is Ω_X=dx_1∧...∧dx_n/(N x_(N−1)^(N−1)).

On the corresponding base chart $z_N=1,z_{N-1}=-1-{\sum }_{i=1}^n{z}_i$, take the multivalued fractional volume η_sym=dz_1∧...∧dz_n /(z_1...z_(N−1))^((N−1)/N).

Choose its local roots according to the given covering sheet. Since $d{z}_i=N{x}_i^{N-1}d{x}_i$, direct cancellation gives F*η_sym=N^(n+1) Ω_X.

The positive real constant does not change its phase. Thus this fractional volume is not an arbitrary grading convention: its pullback extends to the actual nowhere-zero compact CY form, including the ramification locus. The same conclusion follows from $K_X=F\ast K_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}+(N-1)\sum E_i$, whose ramification term cancels the pulled-back fractional pole divisor. Integer auxiliary meromorphic volumes generally have a nonconstant ratio to ${\mathrm{\Omega }}_X$ after pullback and cannot be substituted here.

The primitive character labels for the exterior smash-product model are C=(Z/N)^N /⟨(1,...,1)⟩.

Let ${\epsilon }_j$ be the image of the j-th coordinate vector. A singleton arrow has source χ and target $\chi +{\epsilon }_j$; this follows directly from $g{v}_j={\chi }_j(g)v_{j}g$ and the character idempotents. For a representative $m=(m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_N)$ of χ, put $s(\chi )=\sum m_{i}modN$. This is well-defined: changing one representative coordinate by N or adding the diagonal vector changes the sum by a multiple of N. Define the actual fractional part of the object grading by θ_χ=frac(2s(χ)/N)=frac(−n s(χ)/N), 0≤θ_χ<1.

The two fractions agree since $n=N-2$. This is the squared-volume phase shift: the diagonal deck phase indexed by m multiplies ${\mathrm{\Omega }}_X$ by exp(2πi s/N), so its Lagrangian grading changes by 2s/N modulo Z. Choose the integer parts of the brane grading lifts compatibly with these ${\theta }_{\chi }$; other choices merely shift the corresponding objects by integers.

For the arrow $p_K:\chi \to \chi +{\sum }_{j\in K}{\epsilon }_j$ its compact CY integer degree is |p_K|_CY=|K| n/N + θ_(χ+Σ ε_j)−θ_χ.

The right side is an integer, because its fractional part is |K|(n+2)/N=|K|. For a singleton it is 0 or 1 in these particular shifts. For an arbitrary composable higher operation, all θ shifts telescope; hence the operation retains the full integer degree 2−k. This is an explicit integer regrading, not an appeal to a parity-only index theorem.

Empty and full subsets have the same character. Their diagonal degrees are 0 and N(n/N)=n. Every other fixed ordered pair has at most one subset character, since two subset indicators differ by a constant modulo N only when they are equal or are empty/full. Thus each nontrivial pair has one degree, and the diagonal pair has degrees 0,n. Duplicate physical labels inherit the same degrees via their chosen continuation shifts. This proves the no-adjacent-degrees pattern used by the relative HF-freeness argument without assigning one fixed integer degree to a singleton across every object.

Even coordinate permutations preserve s(χ) and ${\theta }_{\chi }$. The determinant-one diagonal deck subgroup has $s=0$ and therefore also preserves this grading. Their finite spin lift can consequently act with actual degree-zero maps as required by the strict free-label construction; a larger deck symmetry multiplying Ω by a nontrivial phase is not silently declared grading-preserving.

For N repetitions of $v_j$, the object labels close because $N{\epsilon }_j=0$. Summing their integer degrees telescopes to $N(n\mathrm{/}N)=n$. The arity-N operation has degree $2-N=-n$ and can therefore map them to the degree-zero unit with q degree zero. For the N DISTINCT singleton arrows the labels likewise close because $\sum {\epsilon }_j=0$, and the same computation gives n. Both the central product monomial and the Fermat pure powers are now degree-correct in the same compact CY model.

Use the constructed central/first-order mixed category and the repaired two-copy marked comparison. The unique real half-line coefficient on the real-pearl side is $\pm r_j{u}_j$ downstairs. The simple-contact forgetful map has complex determinant $\mathrm{\mid }{\mathrm{\partial }}_z(w\circ u){\mathrm{\mid }}^2>0$, so it introduces neither multiplicity nor an extra orientation sign. The actual central bar/Yoneda restriction identifies this with the geometric relative deformation class of the explicit graph lifts, including its object characters and higher operations.

Apply the ordered contact cobordism repeatedly from$b=1$ through$b=N-1$. The endpoint normalizations of1.52 give a signed Yoneda power, with multiplicity one at each step, so $[{\beta }_j(N)]=\pm [{\beta }_j(1){]}^{⌣N}$. The underlying marked maps are those of the fixed compact relative construction because the allowed perturbations and reductions still vanish near the divisor and agree with the chosen mixed first-order model on its faces. The global lift and its constrained determinant identification in1.54 identify order-$N$ downstairs contact with a simple zero of the upstairs divisor for the uniquely anchored deck lift. In particular one does not sum over all$N$ root phases into one output sheet, and neither a deck factor$N$ nor a factorial is introduced. Combining this with the ACTUAL unique downstairs line count gives ${\kappa }_{\mathrm{u}\mathrm{p}}={\sum }_j{c}_j{q}_j[u_j^N]$ with every $c_j\in \{+1,-1\}$. The sign is the transported determinant-cap sign of that line, multiplied by the ordered contact-cobordism signs; the holomorphic root/jet comparison adds no further real sign.

in the source's first-order coefficient system. The variables $q_j$ have degree zero after the compact CY regrading, as the explicit arity computation above confirms. The signs use the same transported determinant caps as the central exterior-algebra count.

Nonvanishing is supplied by the actual pure-power slots of the Hochschild/Koszul calculation of F3, not by the implication “$u_j$ nonzero implies its power nonzero.” In that calculation the identity-sector arity-N classes consist of the product monomial w and the N pure powers $u_j^N$. The first higher differential sends each Euler class to w, killing w, and leaves the N pure powers independent. The explicit degree/character positions for $N=4,5$ exclude any later incoming or outgoing differential on those pure-power total-degree-two slots. Consequently every $[u_j^N]$ is nonzero and their displayed combination, with every $c_j$ nonzero, is nonzero. F3 contains that finite Koszul complex and invariant calculation; its geometric hypotheses are the actual exterior product and nonzero N-fold count established here.

For the actual even-coordinate action, the geometric deformation is invariant. Its action on the pure-power subspace is the coordinate permutation action after the same cap/character normalization. Transitivity then makes all $c_j$ equal in that normalization, so the invariant class is a nonzero scalar multiple of the Fermat sum $\sum u_j^N$. If one changes individual generator orientations, the displayed coefficients change together with the representative action; nonvanishing and the invariant line do not change.

Lower arities, primary class and finite-free conclusion. For completeness, character preservation gives the lower-arity vanishing without assuming a central classification theorem. In symmetric rational degree, a central operation with k inputs has $\mathrm{\mid }\text{output}\mathrm{\mid }-\sum \mathrm{\mid }\text{inputs}\mathrm{\mid }=2-k$. In any character-preserving coefficient the sum of the subset indicator exponents minus the output indicator is coordinatewise constant modulo N; its TOTAL is therefore a multiple of N. Multiplying that total by n/N shows k−2 is a multiple of n. For $2<k<N=n+2$ this is impossible. Thus those operations vanish in the central minimal model. At arity N the pure-power unit coefficient on N identical singleton inputs would require completed intersection $d=q(1,\mathrm{.}\mathrm{.}\mathrm{.},1)+N{e}_j=0$. The other coordinates force $q=0$, and the jth then cannot vanish. Hence every such CENTRAL pure-power slot is zero. For N distinct singleton inputs $d=-(1,\mathrm{.}\mathrm{.}\mathrm{.},1)+(1,\mathrm{.}\mathrm{.}\mathrm{.},1)=0$, and the unique rational-normal-curve calculation gives the nonzero w coefficient. The supplied Koszul calculation has no other primary arity-N total-degree-two slots, so the central primary Hochschild class is exactly the stated nonzero product monomial, not an unspecified mixture with pure powers.

Apply the no-adjacent-degree conclusion to the actual finite chord differential. Choose a contraction of its central complex onto the computed Hom cohomology. Extend it over R and perturb by the q-positive part. The geometric series is q-adically convergent, and the transferred differential has degree one with q of degree zero. Every fixed Hom has only one degree or degrees0,n; it has no adjacent degrees, so that differential is zero. This proves finite freeness rather than inferring it from finite free chains. Split finite cycles/boundaries equivariantly over the PID R by averaging linear right inverses; rooted-tree transfer gives the finite equivariant minimal model. Finally normalize the unit-input degeneracies using the explicit first-unit insertion contraction and aritywise coordinate changes described in the compact relative supplier. No nonlinear operation is averaged. These constructions prove the full relative model assertions, while all actual higher operations remain the geometric ones up to those explicit A-infinity comparisons.

Bind the marked, confluent and branched operations to the same geometric data. Use the mixed supplier's contact-preserving tuple construction for pure graph words and for both auxiliary-data colors. Every forcing window stays outside the fixed divisor neighborhood, and every contact is described by the literal holomorphic normal function on a protected source disk. The simple-contact correspondence is the graph of its unique zero, with the same obstruction space, branch section, orientation and ordinary disk-face collars. Its empty forbidden collars are corrected before forming the colored category. Thus the first-order class used in the line coefficient and the simple-mark class used in the contact-power construction are restrictions of the same actual first-order colored cocycle; the comparison is not obtained by matching two unrelated virtual counts after perturbing through the divisor.

Fix the polynomial Hochschild generator $u_j$ by evaluation one on the singleton $v_j$. Its powers here mean the ordered Hochschild cup powers. This also fixes the symmetric Koszul convention: if the one-variable resolution is written with $d(z^s)=sp{z}^{s-1}$ as in F3, its map to the normalized bar resolution sends $z^s$ to $s![v_j\mathrm{\mid }\cdots \mathrm{\mid }{v}_j]$. Indeed the bar differential on that repeated word is its left-right difference times the word of length $s-1$, while multiplication of the latter by $s!$ gives precisely $s$ times the comparison in length $s-1$; all internal products vanish since $v_j^2=0$. Thus the polynomial cochain $u_j^s$ evaluates as $s!$ on $z^s$, or as one on the divided-power basis $z^s\mathrm{/}s!$. The ordered bar diagonal then gives $u_j^b⌣u_j=u_j^{b+1}$ with coefficient one. A cochain defined instead as the plain dual of $z^s$ would differ by $s!$; it is not the polynomial normalization used here. This explicit convention prevents a symmetric-resolution basis change from being misread as an enumerative factorial.

For the contact-power family include the entire remainder module in the constrained derivative before choosing its supported augmentation. On every finite chart its contour denominator is bounded away from zero through collision. Therefore the contact constraints, finite right inverse and rational branch sections extend with the same protected lower-face products. The collision's ghost is contracted before taking its parent remainder; its normal factor is complex oriented. At the ordered product end keep the two real node scales on the fixed positive diagonal, and use precisely the same component reductions as the colored simple-mark comparison. These product choices agree on intersecting disk faces, since on each distinguished component the normal remainder is intrinsic and the other component data are unchanged. The signed one-dimensional boundary consequently compares this particular geometric first-order class with its actual ordered cup power.

The regularity claim for the central rational curve can be checked without a genericity assertion about an unspecified curve space. The $N=n+2$ singleton points form a projective frame in ${\mathbb{P}}^n$: every $n+1$ are independent and their unique relation has all coefficients nonzero. A nearby ordered frame is therefore carried to this one by a unique projective transformation, depending smoothly on the frame. After this transformation, the remaining point varies freely in the open set where its coordinates are distinct and nonzero. The reciprocal equations in steps 1.16 and 1.20 solve uniquely modulo the two reparametrizations fixing its preimage $0$; the common residue scale is projectively immaterial. Thus the evaluation map on degree-$n$ maps with $n+3$ marks, modulo ${\mathrm{PGL}}_2$, has a smooth local inverse. Its derivative is an isomorphism, proving the claimed transverse isolated central coefficient. For the binary count the three complementary nonempty blocks give a two-dimensional vector space in $H_{\mathbb{C}}$; the three independent flag incidences in step 1.14 isolate its projective line. The prescribed lifted boundary fixes one hemisphere in both counts. Both are signed single points, so their absolute coefficients are one.

At the order-$N$ endpoint, select one anchored deck sheet, and replace the normal function $(z-p{)}^{N}g(z)$ by the lift $(z-p)g(z{)}^{1\mathrm{/}N}$ using the local nonvanishing-unit logarithm. On the holomorphic-contact chart this is a smooth local bijection with inverse $h\mapsto h^N$, and its unit-factor derivative is the nonzero complex multiplier $N{h}^{N-1}$. The CR residual in the contact disk has first been eliminated by the Cauchy-Green splitting; no derivative of the branched map is inverted on unrestricted variations at a zero. Outside the contact disk the covering is unramified and the reductions are literal pullbacks. Hence the constrained finite equations, kernel spaces and matching evaluations identify, with the same rational branch weights. The complex jet/root factors preserve real orientation, and the exterior determinant caps are the same ones used in the line count.

Here is the finite determinant normalization through collision. On the basis $1,z,\dots ,z^b$ of $\mathbb{C}[z]\mathrm{/}((z-p_1{)}^b(z-p_2))$, the separated constraints are the ordered Taylor coefficients $f^{(r)}(p_1)\mathrm{/}r!$, $0\le r<b$, followed by $f(p_2)$. Translate the basis to powers of $z-p_1$, an upper triangular change of determinant one. The first $b$ rows are the first $b$ coordinate projections and the last remaining entry is $(p_2-p_1{)}^b$. Thus the complex determinant is $(p_2-p_1{)}^b$ and its real determinant is $\mathrm{\mid }{p}_2-p_1{\mathrm{\mid }}^{2b}>0$. This separated matrix is never inverted at collision: the remainder basis extends there as a free complex module, with its fixed complex orientation. Multiplication of a normal equation by a holomorphic unit and a change of source coordinate give invertible complex triangular matrices on this quotient. Hence they preserve real orientation as well. Together with the mixed supplier's joint CR/remainder augmentation and protected product tuples, this calculation fixes the constrained determinant orientation through collision, independently of any singular Vandermonde inverse.

On every ordinary broken disk tree root anchoring fixes the lift on the root component and then on every adjacent component. All node values lie outside the divisor. A loop around a contact has trivial cover monodromy because its multiplicity is divisible by $N$, and the disk's punctured fundamental group is generated by those loops. A tree has no further sheet-cycle condition. Both orders of smoothing two nodes produce the same anchored lift, so the local root identifications commute with every retained face and higher corner. An automorphism of the anchored downstairs object has exactly one anchored lift, and its inverse descends; finite automorphism weights agree. Thus neither a deck factor $N$ nor a factorial appears. This is the coherent endpoint comparison required for the compact first-order coefficient, not merely a pointwise choice of normal roots.

More precisely let ${\sigma }_j\in \{\pm 1\}$ be the signed single-line coefficient in step 1.37 and let ${ϵ}_{j,b}\in \{\pm 1\}$ be the determinant-cap/source-normal sign comparing the order-$(b+1)$ collision end with the ordered $(b,1)$ end in step 1.52. Then the signed boundary identity gives $[{\beta }_j(N)]={\sigma }_j^N({\prod }_{b=1}^{N-1}{ϵ}_{j,b})[u_j^N]$. On a finite unit-jet space the root map $h\mapsto h^N$ has triangular complex derivative with diagonal $Nh(0{)}^{N-1}\mathrm{\ne }0$; its real determinant is positive. The contact-position coordinate is unchanged. The anchored root correspondence therefore adds sign $+1$, and the disk-tree and stabilizer identifications of steps 3.1 and 4.1 add multiplicity one. Consequently the actual upstairs coefficient is exactly $c_j={\sigma }_j^N{\prod }_{b=1}^{N-1}{ϵ}_{j,b}\in \{\pm 1\}$ in these transported caps and the ordered cup normalization. A universal numerical choice of $+$ is unnecessary and would require additional cap choices; the Statement requires exactly the displayed signed unit.

The downstairs contact-power bordism is used entirely DOWNSTAIRS. Its separated order-$b$ and order-one contacts for $0<b<N$ generally do not lift to an unbranched source disk upstairs, and the argument does not claim they do. First compute the downstairs identity $[{\beta }_j(N)]=\pm [{\beta }_j(1){]}^{⌣N}$ with the coherent remainder charts; then use the anchored order-$N$ endpoint comparison in steps 3.1 and 4.1. It maps the unique simple upstairs divisor mark to exactly that downstairs endpoint, preserving the compact brane grading and the chosen boundary sheet. Applying the signed line coefficient therefore gives the same $c_j=\pm 1$ in the actual upstairs class ${\sum }_j{c}_j{q}_j[u_j^N]$, with all ordinary central disk splittings included.

The contact-preserving common charts, colored first-order comparison and anchored ramified endpoint bind the signed first-order coefficients to the repaired actual mixed-category construction. Together with the preceding central graph count, contact-power boundary identity, integer grading and finite-free transfer, they give the unchanged Statement in its specified regimes. The contact bordism is used downstairs before its order-N endpoint is lifted, so no unsupported family of ordinary lifts is invoked. These authored constructions are submitted for current native assessment and confer no proof-acceptance stamp.