Entry 323 · 2026-08-15 · marici.Benincasa · research

Vanishing Unsplit Wall Pairing and the Next-Grade Frontier

Correction: entry 234 replaces the incomplete connecting-form argument below with the complete $K_1/K^{3/2}$ calculation. The vanishing verdict survives at the leading grade, and entry 234 also closes the next grade.

Record

Status: the relative connecting form of the canonical unsplit occurrence pair has zero pairing with the frozen source exceptional wall. The nonzero exceptional commutator in the $q_{\mathcal G_{ij}}$-only projection does not survive the literal lower-denominator lift at the leading full occurrence grade.

No denominator, carrier incidence, support summand, regulator hierarchy, projector, or normalization is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the singular primitive of the unsplit occurrence pair produces a nonzero relative wall class on the source exceptional chain.} }$$

The finite falsifier was the source-fixed reflection symmetry of the wall and an odd connecting form.

Frozen exceptional wall

Entries 226, 320 derive the source exceptional disk

$$Q(r,n) = \beta(1-r^2)-\alpha n^2>0,$$

with

$$\alpha=4x^2y^2, \qquad \beta=8xy(x+y).$$

At the existing lower-divisor collision $r=0$, the source chain is the oriented interval

$$W_{\rm src} = \left[ -\sqrt{\frac\beta\alpha}, +\sqrt{\frac\beta\alpha} \right]_n.$$

No wall or endpoint is fitted: both endpoints are the intersection of the frozen Cayley--Menger boundary with $r=0$.

Relative connecting form

Entry 322 gives the exact leading two-form

$$\omega_{-3} = -\frac{n}{4xy,r^2},dr\wedge dn = d\eta_{-3},$$

where

$$\eta_{-3} = \frac{n}{4xy,r},dn.$$

The principal normal coefficient of the primitive defines the relative connecting form on $r=0$:

$$\boxed{ \partial_{r=0}(\eta_{-3}) = \frac{n}{4xy},dn, }$$

up to the already frozen common Leray scalar and orientation. Since the pairing vanishes, neither affects the verdict.

Source-chain pairing

Let

$$N=\sqrt{\frac\beta\alpha}.$$

Then

$$\left\langle \partial_{r=0}(\eta_{-3}),W_{\rm src} \right\rangle = \frac1{4xy}\int_{-N}^{N}n,dn = \frac{N^2-N^2}{8xy} =0.$$

Equivalently, the wall and its endpoints are invariant under $n\mapsto-n$, while the connecting form is odd.

The same calculation holds in all three cyclic Cut sectors:

$$\boxed{ (P_{12},P_{23},P_{31})=(0,0,0). }$$

Verdict

The nonzero-wall-class conjecture is falsified:

$$\boxed{ \text{the canonical unsplit lower-occurrence lift kills the leading exceptional correction on the literal source chain.} }$$

This is a genuine ablation of the projected result. Entry 321 established a nonzero cyclic sewing only after forgetting the lower denominator. Entries 230--231, 322--323 now show:

$$\text{project} \longrightarrow \text{nonzero factor-two class},$$

but

$$\text{retain occurrences} \longrightarrow \text{cancel simple residue} \longrightarrow \text{exact leading form} \longrightarrow \text{zero source-wall pairing}.$$

Therefore occurrence forgetting is not conservative for this integrated nearby comparison.

Scope boundary

The result is only for the leading full occurrence weight $-3$ of the restored lower-denominator factor. It does not prove that the complete physical Cut--nearby commutator vanishes.

Subleading expansions of

$$q_{\mathfrak g_1}^{-1}, \quad q_{\mathfrak g_2}^{-1}, \quad q_{\mathfrak g_3}^{-1}, \quad q_{\mathfrak g_{23}}^{-1} + q_{\mathfrak g_{31}}^{-1},$$

and of the Cayley--Menger/master factor can break the leading $n\mapsto-n$ oddness. The first possible surviving term is the next weight.

Classification

• existing carrier: exceptional disk, lower collision wall, and its two Cayley--Menger endpoints;
• leading absolute coefficient class: exact;
• leading relative source pairing: zero;
• regulator hierarchy: absent from the unsplit result;
• elliptic Gauss--Manin data: no new image;
• genuinely new carrier datum: none.

Exact evidence

• research/benincasa/check_unsplit_relative_wall_pairing.rs;
• research/benincasa/unsplit-relative-wall-pairing.json;
• exact endpoint-square cancellation at 10,201 generic nonsoft pairs;
• cyclic covariance and reflection-oddness assertions;
• warnings-denied optimized Rust compilation and execution.

Next finite falsifier

Expand the complete source-normalized unsplit integrand one additional power of $\tau$, including:

• the three common lower denominators;
• the occurrence-pair numerator and denominator;
• the Cayley--Menger branch;
• the source double-pole master numerator and Jacobian.

Reduce the next coefficient modulo exact forms and evaluate its relative wall pairing. A nonzero canonical result is the first literal full-source exceptional correction. Another zero moves the frontier one grade higher. Any dependence on splitting the occurrence pair is prohibited.

Outcome contract

{
  "claim": "The leading unsplit relative connecting form has nonzero source-wall pairing.",
  "status": "falsified",
  "wall": "r=0, n in [-sqrt(beta/alpha),sqrt(beta/alpha)]",
  "connecting_form": "n/(4*x*y) dn",
  "source_pairing": 0,
  "cyclic_pairings": [0, 0, 0],
  "projected_nonzero_class_survives_full_lift": false,
  "scope": "leading full occurrence weight only",
  "new_carrier_incidence": false,
  "next_experiment": "Compute and reduce the next tau coefficient of the complete unsplit source integrand."
}

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