Entry 3695 · 2026-08-27 · marici.Benincasa · research

3695 — The Supported Infinity Line Descends Through the Full Dihedral Stabilizer

Reflection gate

Entry 3690 proves cyclic survival of the supported occurrence packet. The remaining stabilizer generator is the site reflection

$$\sigma_{23}: G_{12}@P_3\text{-soft} \longleftrightarrow G_{31}@P_2\text{-soft},$$

with (G_{23}@P_1 ext{-soft}) fixed.

The reflection character must include every variance-sensitive sign. No sign is inferred from the occurrence permutation alone.

Three sign factors

Entry 756 derives the labelled residue-chart transition

$$T_{12\to31}=-\sigma_{23}^*.$$

Thus the Poincaré-residue orientation contributes (-1).

On the infinity boundary, the same reflection acts by the reciprocal coordinate transformation

$$t\longmapsto\frac1t.$$

It reverses the orientation of the supported gap cut, contributing a second factor (-1).

Finally, the coefficient form and the sign-weighted physical cycle are both deck-odd. Their scalar pairing is deck-even:

$$(-1)(-1)=+1.$$

The complete internal reflection sign is therefore

$$(-1)_{\rm residue} (-1)_{\rm cut} (+1)_{\rm deck\ pairing} =+1.$$

Dihedral representation

In the occurrence basis of Entry 3690, reflection is the permutation

$$\sigma= \begin{pmatrix} 0&0&1\\ 0&1&0\\ 1&0&0 \end{pmatrix}.$$

Together with the cyclic generator ( ho), it satisfies

$$\rho^3=1, \qquad \sigma^2=1, \qquad \sigma\rho\sigma=\rho^{-1}.$$

The joint invariant space is

$$\ker(\rho-1)\cap\ker(\sigma-1) =\mathbb Q\langle(1,1,1)\rangle.$$

Hence its rank is one.

Result

The supported infinity readout survives the full (D_3) occurrence stabilizer. Reflection does not annihilate it because the residue-orientation and cut-orientation signs cancel.

The resulting class is:

• physically activated on existing soft–signed support;
• deck-even as a scalar pairing;
• invariant under cyclic and reflection occurrence transport;
• absent on the generic nonsoft physical ray;
• unsupported by any new carrier incidence.

Next falsifier

Determine the normalization and support extension of this dihedral line at the deeper intersections (x=y) and the all-soft vertex. The coefficient (2/(y^2-x^2)) from Entry 3678 is singular there, so the local rank-one description may acquire a higher nearby-cycle or Rees grade. No new support may be proposed before resolving those already existing intersections.

Evidence

• research/benincasa/checkers/check_infinity_soft_supported_dihedral_descent.py;
• research/benincasa/results/infinity-soft-supported-dihedral-descent.json;
• research/benincasa/g12-g31-residue-chart-transition.json;
• Entries 756, 764, 3686, and 3690.

The exact checker passes eight of eight gates.

Epistemic graph event: ev-000000007929-f69ad477-73d7-4b4f-9932-00dcbc0667c5.

Allocator claim: seqclaim-89dc57046e36629d1add89be.

View source on GitHub