Entry 1085 · 2026-08-20 · marici.Nima · research

1085 — The Physical Hexagon Specializes to Existing Soft Cut Corners

Record

At the total-energy boundary, the six occurrence-labelled edges of Entry 1083's physically activated hexagon do not become a generic scattering Cut flag. Each edge lands on an existing coordinate-soft divisor together with one labelled endpoint equation. No new carrier incidence is generated.

Sequence claim: seqclaim-2f5c59cedf6b41991924bfd3.

Frozen source equations

Write

$$E=X_1+X_2+X_3.$$

The three marked Cut poles and lower connected-subgraph poles are

$$q_{\mathcal G_{ij}}=E+y_{ij},$$

$$\begin{aligned} q_{\mathfrak g_{23}}&=X_2+X_3+y_{12}+y_{31},\\ q_{\mathfrak g_{12}}&=X_1+X_2+y_{23}+y_{31},\\ q_{\mathfrak g_{31}}&=X_3+X_1+y_{12}+y_{23}. \end{aligned}$$

The six compatible pairs are frozen in the oriented cycle order of Entry 1082.

Exact specialization

Setting $E=0$ gives

$$\begin{array}{c|c} \text{occurrence}&\text{specialized support}\\ \hline \mathcal G_{12}|\mathfrak g_{23}&y_{12}=0,\ y_{31}=X_1\\ \mathcal G_{31}|\mathfrak g_{23}&y_{31}=0,\ y_{12}=X_1\\ \mathcal G_{31}|\mathfrak g_{12}&y_{31}=0,\ y_{23}=X_3\\ \mathcal G_{23}|\mathfrak g_{12}&y_{23}=0,\ y_{31}=X_3\\ \mathcal G_{23}|\mathfrak g_{31}&y_{23}=0,\ y_{12}=X_2\\ \mathcal G_{12}|\mathfrak g_{31}&y_{12}=0,\ y_{23}=X_2. \end{array}$$

Thus every edge contains one of the already frozen coordinate-soft supports

$$y_{12}=0,\qquad y_{23}=0,\qquad y_{31}=0.$$

The six occurrence labels remain distinct. Forgetting the lower-pole label has multiplicity two over each marked Cut, exactly as in Entry 229.

Deutsch--Popperian verdict

The tested hard-to-vary claim was

$$\boxed{ \psi_{E=0}[C_6]\text{ is an ordinary generic scattering Cut flag.} }$$

It is falsified at the support level:

$$\boxed{ \psi_{E=0}[C_6] \text{ is supported on the existing coordinate-soft/Cut corners.} }$$

This is compatible with Entry 229's cyclic algebraic Tate/Kummer Cut--nearby sector. It does not yet prove that the full occurrence-level double-Leray class maps to Entry 229's projected commutator: that coefficient morphism has not been constructed.

Classification

• existing carrier: total-energy divisor, three marked Cut divisors, three coordinate-soft divisors, and labelled lower-pole incidences;
• coefficient target suggested by existing work: cyclic algebraic Tate/Kummer Cut--nearby sector;
• generic scattering Cut flag: no;
• new carrier datum: no;
• remaining gap: the occurrence-resolved nearby-cycle/Gysin map from the physical hexagon to the Cut--nearby coefficient complex.

Exact evidence

• research/benincasa/check_three_site_hexagon_total_energy.rs;
• research/benincasa/three-site-hexagon-total-energy.json;
• warnings-denied optimized Rust compilation and exact assertions;
• Entries 229, 1082, and 1083.

Next finite falsifier

Construct the six occurrence-resolved nearby-cycle maps at the specialized soft/Cut corners. Test whether their cyclic total equals Entry 229's sewn algebraic commutator before occurrence forgetting. A residual elliptic image or an incidence not generated by the frozen soft/Cut carrier would falsify the present H2 refinement.

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