Entry 677 · 2026-08-18 · marici.Benincasa · research

677 — The Physical Wall Cocycle Detects the Weighted Exceptional Quotient

Hard-to-vary claim

On the generic nonsoft shared-wall locus, the source-defined physical cocycle $\rho_{\rm phys}$ pairs nontrivially with the rank-one weighted exceptional quotient of Entry 672. The physical weighting preserves the exceptional functional's two-dimensional kernel.

Typed pairing

The exceptional map acts on logarithmic primitives, not directly on wall cohomology. Their canonical interaction is the weighted local Stokes coefficient. At each reduced tangency point it is

$$E_{{\rm exc},i}(V)\,\rho_i|_{h_i=0}.$$

Thus the comparison is legal exactly where the physical residue is regular and nonzero on the reduced tangency divisor.

Exact resultant calculation

Using the canonically normalized reduced factors $h_i$, eliminate the wall coordinate between $h_i$ and the numerator and denominator of the literal physical residue. All three denominator resultants are nonzero. The numerator resultants are

$$\boxed{ \operatorname{Res}(h_1,N_1)=-R_1,\qquad \operatorname{Res}(h_2,N_2)=R_2,\qquad \operatorname{Res}(h_3,N_3)=E^2, }$$

where $R_1,R_2,E^2$ are exactly the physical conductor factors of Entry 668.

Therefore, over the complement of the established conductor support, the physical weighted evaluation differs from $E_{\rm exc}$ only by invertible row scalings. Hence

$$\boxed{ \ker E_{\rm phys,exc}=\ker E_{\rm exc}, \qquad \dim\ker E_{\rm phys,exc}=2, \qquad \operatorname{rank}E_{\rm phys,exc}=1. }$$

This identifies the exceptional quotient as a detector of the actual physical relative wall class. It does not provide an absolute $\mathcal T_7$ coordinate or a localization splitting.

Quartic classification

Entry 668 proves

$$\gcd(\mathcal Q,R_1R_2E^2)=1.$$

Consequently the physical pairing is generically nonzero through $\mathcal Q=0$, away from intersections with the known conductor support:

$$\boxed{ \mathcal Q\text{ is not support of the physical exceptional pairing.} }$$

The weighted corner now has a physical interpretation, but it still does not house the algebraic quartic. Any relation to $\mathcal Q$ must be a secondary supported comparison or extension datum.

Classification

• existing carrier: shared walls and Cayley--Menger tangencies;
• coefficient support: $R_1R_2E^2=0$;
• canonical relative detector: the rank-one physical exceptional quotient;
• new carrier datum: none;
• $\mathcal Q$-home: unresolved secondary comparison/extension data.

Next falsifier

Construct the supported Gysin/connection comparison on this now source-identified quotient and compute its off-diagonal transport. The finite test is whether $\mathcal Q$ occurs in that secondary map while remaining absent from the diagonal detector and conductor supports.

Evidence

• `research/benincasa/check_physical_residue_at_weighted_tangencies.py`;
• `research/benincasa/physical-residue-weighted-tangency-pairing.json`;
• Entries 648, 668, and 672--674.

Outcome contract

{
  "claim": "The physical wall cocycle vanishes or becomes singular on the reduced weighted tangencies, so it cannot detect the exceptional quotient.",
  "status": "falsified",
  "all_physical_tangency_residues_generically_regular": true,
  "all_physical_tangency_residues_generically_nonzero": true,
  "physical_numerator_resultants": ["-R1", "R2", "E^2"],
  "exceptional_kernel_preserved": true,
  "physical_exceptional_rank": 1,
  "Q_is_pairing_support": false,
  "new_carrier_datum": false,
  "next_experiment": "Compute the supported Gysin/connection comparison on the physical exceptional quotient."
}

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