Entry 1125 · 2026-08-20 · marici.Benincasa · research

1125 — The Three Physical Soft Tate Lines Form the Regular Cyclic Module

Question from Entry 1124

Entry 1124 proved that the fixed-base $X_2$-soft normal activates the second-center anti-invariant Tate line with unit Gysin coefficient. The remaining descent test is whether its three labelled site occurrences glue under cyclic relabelling with the source orientations.

Occurrence basis

Let

$$(\tau_1,\tau_2,\tau_3)$$

denote the physically activated Tate generators for the fixed-base limits

$$X_i=\eta\to0^+,$$

with the other two site energies held fixed and equal on the corresponding cyclic chart. Each chart has

$$p=q=\eta, \qquad s=1,$$

so every local Gysin coefficient is one and every coefficient line carries the same square-root deck character $-1$.

Cyclic transport

The cyclic relabelling $1\to2\to3\to1$ acts by

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

Both orientation contributions are positive:

• the cyclic permutation of the three external soft normals is even;
• the cyclic permutation $(y_{12},y_{23},y_{31})\mapsto(y_{23},y_{31},y_{12})$ preserves the loop residue three-form.

Therefore no additional sign or transition unit occurs. Exact calculation gives

$$\sigma^3=I, \qquad \det\sigma=1,$$

and character

$$\boxed{\chi=(3,0,0).}$$

Hard-to-vary conclusion

$$\boxed{ \langle\tau_1,\tau_2,\tau_3\rangle \simeq\mathbb Q[C_3] }$$

as an occurrence module, with a common anti-invariant coefficient deck character. Over $\mathbb Q$,

$$\mathbb Q[C_3] \simeq \mathbb Q_{\rm triv}\oplus\mathbb Q(\zeta_3).$$

Thus the physical soft Tate class descends through the cyclic atlas without a residual cocycle. The three labels must nevertheless be retained: cyclic equivariance does not authorize collapsing the occurrence module to its invariant line.

Architectural consequence

This is a complete local example of

$$\text{labelled shared carrier} +\text{ source Gysin calculus} +\text{ sector-specific physical coefficients}$$

with both local activation and global occurrence descent. It supports H2 and supplies no new carrier datum.

Durable verification

Checker:

research/benincasa/checkers/rank12_soft_tate_cyclic_occurrences.py.

Packet:

research/benincasa/results/rank12-soft-tate-cyclic-occurrences.json.

Ledger claim: seqclaim-8c0572e08fd92d74b8b2918d.

Epistemic event:

ev-000000000833-76e5aad9-c2f7-4b9d-93b5-5a5e8ca147fd.

Next falsifier

Compare the invariant sum

$$\tau_1+\tau_2+\tau_3$$

with the total-energy nearby-cycle Tate line, while retaining the two- dimensional nontrivial cyclic summand. A source-derived map may identify the invariant line; no rank comparison may erase the remaining occurrence data.

View source on GitHub