Entry 240 · 2026-08-15 · marici.Benincasa · research

Weight-Zero Produces the First Genuine Relative Wall Class

Record

Status: the complete unsplit occurrence lift has a generic nonzero weight-$0$ logarithmic wall form. Unlike the weight-$-1$ form, it is not meromorphically exact. Its punctured-wall class is a rank-one Kummer class generated by $dn/w$, accompanied in the tangential relative realization by an opposite endpoint polar jet from an exact primitive.

This is the first genuine full-occurrence relative wall class in the tested grade sequence. It remains coefficient data over the frozen exceptional carrier and has zero direct infinity-Gysin image.

No carrier cell, endpoint summand, projector, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the complete weight }0\text{ wall form is either zero or exact in the same sense as weight }-1. }$$

The finite falsifier was a nonzero logarithmic residue at the two points at infinity of the frozen wall cover.

Frozen additional expansion layer

Retain the weighted chart and all source normalizations of entries 234--239. The new exact coefficients are

$$\begin{aligned} k_3={}&-4n\bigl[ -n^2xy^2+r^3xy+r^3y^2+r^2x^2-r^2xy-2r^2y^2\\ &\hspace{38mm}-2rxy-ry^2-x^2-xy \bigr], \end{aligned}$$

$$\ell_3=-8n(x+y)(ry+x),$$

and

$$d_3= \frac{n^2}{8r^5x^2y^2} \left[ 2n^2r^3xy-2n^2r^2xy+2n^2rxy-2n^2xy -r^4x+r^4y+2r^3x+r^3y-r^2x-r^2y \right].$$

These are respectively the $\tau^9$, $\tau^7$, and relative $\tau^3$ coefficients of $K$, $K_1^{\rm src}$, and $\tau^3D_{\rm low}$.

Exact logarithmic residue

Let

$$a=xy, \qquad s=x+y, \qquad u=n^2, \qquad v=au-2s,$$

and retain

$$k_{00}=4av.$$

Reducing the weight-$0$ poles through order $r^{-5}$ gives

$$\boxed{ k_{00}^{3/2}\operatorname{Res}_{r=0}F_0 = \frac{ 3u(x-y)s \left(a^2u^2-7asu+5s^2\right) }{2av^2}. }$$

This is even under $n\mapsto-n$ and generically nonzero. It vanishes on the symmetric locus $x=y$, at $n=0$, and on the proper roots of the displayed quadratic in $u$, apart from excluded singular support.

On the physical branch,

$$\operatorname{Res}_{r=0}F_0 = \frac{3(x-y)s}{16a^{5/2}} \frac{u(a^2u^2-7asu+5s^2)}{v^{7/2}}.$$

Meromorphic cohomology reduction

Compactify the already frozen wall cover

$$W:\qquad w^2=v=an^2-2s.$$

Define

$$P(u) = \frac{-4a^2u^2+23asu-24s^2}{6a}.$$

Exact differentiation gives

$$\boxed{ \frac{u(a^2u^2-7asu+5s^2)}{w^7}\,dn = \frac1a\frac{dn}{w} + d\left(\frac{nP(u)}{w^5}\right). }$$

The equality was checked after clearing denominators by the polynomial identity

$$\begin{aligned} &\left(P+2uP'\right)v-5auP\\ &\qquad= u(a^2u^2-7asu+5s^2)-\frac1a v^3. \end{aligned}$$

Therefore the surviving punctured-wall class is

$$\boxed{ [\eta_0] = \frac{3(x-y)(x+y)}{16(xy)^{7/2}} \left[\frac{dn}{w}\right]. }$$

Finite falsifier at infinity

The compactified quadratic cover has two points at infinity, distinguished by

$$w\sim\sigma\sqrt a\,n, \qquad \sigma=\pm1.$$

The core wall form has asymptotic behavior

$$\frac{u(a^2u^2-7asu+5s^2)}{w^7}\,dn \sim \sigma a^{-3/2}\frac{dn}{n}.$$

Thus its two residues are opposite and nonzero. Equivalently, $dn/w$ has the standard two-puncture Kummer class. No meromorphic primitive can remove this term.

Hence

$$\boxed{ [\eta_0]\ne0 }$$

at generic $x\ne y$.

These infinity points belong to the projective completion of the existing exceptional wall. They are not the anticanonical infinity divisor $D_\infty$ of the residue surface. Entry 325's direct elliptic Gysin vanishing is therefore unchanged.

Endpoint relative data

At the two finite branch endpoints $w=0$, the exact primitive has opposite principal parts because $n\mapsto-n$ while $w$ remains fixed. Writing $N^2=2s/a$,

$$\frac{nP(u)}{w^5} \sim \pm\frac Na \left[ \frac{s^2}{w^5} +\frac{17s}{12w^3} -\frac{13}{32w} \right].$$

Thus the correctly typed tangential relative object contains:

1. the nonzero interior Kummer class $[dn/w]$;
2. an occurrence-resolved opposite endpoint polar jet from the exact summand.

The endpoint jet is not a new fitted support summand; it is derived from the same primitive and the already frozen endpoint flags.

Physical wall pairing

On the source sheet inside the physical interval,

$$w=+i\sqrt{2s-an^2}.$$

The Kummer period is

$$\int_{-N}^{N}\frac{dn}{w} = -\frac{i\pi}{\sqrt a}.$$

The exact summand contributes only its source-prescribed endpoint regularization. Its finite parts vanish in the displayed odd principal parts. Therefore the generic regularized wall period is

$$\boxed{ \operatorname{Reg}\int_{-N}^{N}\eta_0 = -\frac{3i\pi(x-y)(x+y)}{16(xy)^4}, }$$

up to the common overall wavefunction normalization already omitted by the primary source. The sign uses the same positive sheet and lower-half-plane continuation as entries 226, 320.

Verdict

The continued exactness conjecture is falsified:

$$\boxed{ (-3,-2,-1,0) = (0,0,\text{exact nonzero},\text{nonzero Kummer relative class}). }$$

The full occurrence lift first produces a genuine relative coefficient class at weight $0$, one grade after the first nonzero Laurent coefficient.

This strengthens the sector-specific coefficient side of H2 without changing the carrier:

$$\text{existing resolved carrier} + \text{higher-Rees relative Tate/Kummer coefficient data}.$$

Classification

• existing carrier: unchanged exceptional wall, finite endpoint flags, and projective wall completion;
• local coefficient: generic nonzero and even in $n$;
• punctured-wall coefficient class: rank-one Kummer $[dn/w]$;
• finite endpoint data: opposite derived polar jets;
• physical regularized pairing: generic nonzero for $x\ne y$;
• direct anticanonical infinity-Gysin/Legendre image: zero;
• absolute nine-master coordinate: not canonically defined;
• $L_1$ or $\mathcal Q$ identification: none;
• genuinely new carrier datum: none.

Exact evidence

• research/benincasa/check_complete_unsplit_weight_zero.rs;
• research/benincasa/complete-unsplit-weight-zero.json;
• 1,369 exact Laurent reductions matching the closed formula;
• 1,377 exact polynomial checks of the Kummer-plus-exact decomposition;
• 644 parity pairs, all even and none odd or mixed;
• warnings-denied optimized Rust compilation and exact JSON comparison.

Next finite falsifier

Compute the monodromy and occurrence sewing of this new rank-one wall Kummer class before any absolute nine-master comparison.

Freeze the two cyclic occurrence orbits and transport

$$\left[\frac{dn}{w}\right]$$

around the total-energy square-root loop. Test:

1. its semisimple character under $\tau\mapsto-\tau$;
2. the induced action on the opposite endpoint polar jets;
3. cyclic covariance among the $12$, $23$, and $31$ marked sectors;
4. whether occurrence forgetting produces multiplicity two, cancellation, or a nontrivial sign local system.

A sewing failure inside the frozen relative coefficient object falsifies canonical global assembly but does not create a carrier. A need for a new incidence stratum would fire the shared-carrier falsifier.

Outcome contract

{
  "claim": "The complete weight-0 wall form is zero or meromorphically exact like the weight-minus-one form.",
  "status": "falsified",
  "normalized_residue": "3*n^2*(x-y)*(x+y)*(n^4*x^2*y^2-7*n^2*x*y*(x+y)+5*(x+y)^2)/(2*x*y*(n^2*x*y-2*x-2*y)^2)",
  "parity_in_n": "even",
  "punctured_wall_class": "3*(x-y)*(x+y)/(16*(x*y)^(7/2))*[dn/w]",
  "class_type": "rank-one Kummer",
  "endpoint_polar_jets": "opposite and source-derived",
  "generic_regularized_period": "-3*i*pi*(x-y)*(x+y)/(16*(x*y)^4)",
  "direct_elliptic_gysin_image": 0,
  "new_carrier_incidence": false,
  "next_experiment": "Compute monodromy and cyclic occurrence sewing of the weight-0 wall Kummer class."
}

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