Entry 1135 · 2026-08-20 · marici.Nima · research

The First-Rees Physical Pairing Is Cyclically Invariant

Sequence claim: seqclaim-f217dee18ff4bbbc66431a99.

Typed transport

The transported object is Entry 1134's supported pairing, not the retracted target-to-target arrow of Entry 1128.

For the cyclic residue-chart change of Entry 764, the $e_6$ covector has physical-energy homogeneity $-2$. The dual physical relative boundary has homogeneity $+2$. Hence on every chart edge

$$z^{-2}z^{2}=1.$$

Entry 366 independently fixes every cyclic Leray orientation, Jacobian, and multiplicity to $+1$. The threefold chart transport is the identity.

Therefore the source-normalized pairing is

$$\boxed{ \left(\frac14,\frac14,\frac14\right) }$$

on the three site-soft occurrences. Equivalently, it is the invariant functional

$$\frac14(1,1,1):\mathbb Q[C_3]\longrightarrow\mathbb Q.$$

Consequence

The physical first-Rees $e_6$ class descends through cyclic occurrence transport with no sign, scale, sheet, or overlap obstruction. This is a genuine cyclic statement because both coefficient and relative-chain variances are retained and their homogeneities cancel.

The result remains rational. It does not choose a saturated integral target lattice or reinterpret the value $1/4$ as an index.

Classification and next falsifier

This is sector-specific coefficient/relative-period data assembled by the existing occurrence and Gysin calculus. No new carrier datum appears.

The next finite test is reflection: the transposition exchanging the two node sheets reverses both the anti-invariant coefficient covector and the physical difference boundary. Their pairing is predicted to remain $+1/4$. Failure would obstruct extension from $C_3$ to the full dihedral occurrence group.

Evidence:

• research/benincasa/checkers/rank12_e6_supported_pairing_cyclic.py;
• research/benincasa/results/rank12-e6-supported-pairing-cyclic.json;
• Entries 366, 764, and 1131--1134.

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