Entry 1149 · 2026-08-20 · marici.Nima · research

The Physical Cut-Nearby Commutator Misses the Elliptic Width-Two Class

Sequence claim: seqclaim-37c6b2c9973c5c42bef61c4c.

Hard-to-vary claim

The source-normalized physical Cut--nearby commutator of Entry 226 might activate the integral width-two elliptic coinvariant found in Entry 1147. The finite test is the typed composite

$$\mathcal C_{\rm phys} \longrightarrow \mathcal M_q^{(9)} \xrightarrow{R_\infty}\mathbb V_{\rm ell}(-1) \longrightarrow\operatorname{coker}(T-I).$$

No comparison map or lattice normalization is added.

Frozen physical commutator

Entry 226 computes, in the equation-(58) master order,

$$[\psi_{E_T=0},\operatorname{Res}_{q_{\mathcal G_{12}}=0}] = \left( 0,0,-\frac{2\pi^2}{x},0,-\frac{2\pi^2}{y}, -\frac{2\pi^2}{xy},0,0,0 \right).$$

On the nonsoft open set (xy(x+y)\ne0), clearing its common nonzero scalar gives the source ratio

$$e_3:e_5:e_6=y:x:1.$$

The physical class is therefore nonzero and supported termwise on

$$\langle e_3,e_5,e_6\rangle.$$

Infinity-Gysin image

Entry 150 proves that the source blocks containing (e_3) and (e_5) map identically to zero under (R_\infty). It also proves separately, from the absence of a logarithmic pole at infinity, that

$$R_\infty(e_6)=0.$$

Hence the vanishing is termwise rather than a rational cancellation:

$$\boxed{ R_\infty \bigl([ \psi_{E_T=0},\operatorname{Res}_{q_{\mathcal G_{12}}=0}] \bigr)=0.}$$

This integral zero remains zero after nearby cycles and after passage to the width-two coinvariant

$$\operatorname{coker}(T-I) \simeq \mathbb Z\oplus\mathbb Z/2.$$

In particular, the physical comparison has zero image in its (\mathbb Z/2) summand.

Verdict

The conjectured activation is falsified:

$$\boxed{ \text{the physical Cut--nearby commutator is nonzero, but it does not activate the elliptic width-two class}.}$$

The two structures coexist in different coefficient layers:

• the physical commutator is a third-Rees algebraic Tate/Kummer class in the rank-seven kernel of (R_\infty);
• the (mathbb Z/2) class is the integral shadow of the elliptic total-energy cusp;
• their source-defined comparison is zero before taking coinvariants.

Therefore neither the transverse base corner nor the physical relative commutator supplies a carrier explanation for the elliptic torsion. No new carrier datum is indicated.

Classification

• existing carrier: total-energy and marked-Cut divisors with their frozen weighted comparison;
• algebraic coefficient support: the entire nonzero physical commutator;
• Legendre/Gauss--Manin coefficient data: the width-two coinvariant;
• map between them in this comparison: zero;
• genuinely new carrier datum: none.

Next falsifier

Compute the integral saturation of the algebraic commutator inside the rank-seven kernel before cyclic assembly. The normalized vector ((y,x,1)) is visibly primitive in the displayed three-master lattice; the remaining question is whether cyclic occurrence sewing introduces any integral index or extension that is invisible over the rational connection.

Evidence:

• research/benincasa/checkers/cut_nearby_elliptic_z2_exclusion.py;
• research/benincasa/results/cut-nearby-elliptic-z2-exclusion.json;
• Entries 150, 226, 317, 318, 1147, and 1148.

View source on GitHub