Entry 1126 · 2026-08-20 · marici.Benincasa · research

1126 — Cyclic Symmetry Permits but Does Not Define the Soft-Tate to e6 Map

Proposed comparison

Entry 1125 assembled the physically activated soft Tate lines as

$$T_{\rm soft} = \mathbb Q\langle\tau_1,\tau_2,\tau_3\rangle \simeq\mathbb Q[C_3].$$

The total-energy marked system independently contains the cyclic-invariant second-Rees bridge

$$g_{111}^{\rm top}longmapsto\frac{e_6}{8(X_1+X_2)}.$$

Matching rank-one invariant sectors suggests a comparison, but does not type one.

Exact equivariant Hom

Let a candidate map to a cyclic-trivial line be the row

$$h=(c_1,c_2,c_3).$$

For Entry 1125's cyclic permutation matrix, equivariance requires

$$h\sigma=h.$$

Direct exact multiplication gives

$$\boxed{c_1=c_2=c_3=c.}$$

Therefore

$$\boxed{ \operatorname{Hom}_{C_3}(T_{\rm soft},\langle e_6\rangle) \simeq\mathbb Q, }$$

generated in shape by the augmentation

$$(1,1,1).$$

The common normalization/deck character $-1$ creates no further obstruction, because both candidate source and target lie in that character.

Missing authority

The frozen maps are:

1. physical soft Gysin maps into each $\tau_i$;
2. cyclic occurrence transitions among the $\tau_i$;
3. the marked-top map into $e_6/(8(X_1+X_2))$.

There is no frozen source morphism

$$\tau_i\longrightarrow g_{111}^{\rm top}$$

or

$$T_{\rm soft}\longrightarrow\langle e_6\rangle.$$

Thus symmetry determines only the possible form $c(1,1,1)$; it does not determine $c$, including whether $c=0$.

Hard-to-vary conclusion

$$\boxed{ \text{The invariant soft-Tate to }e_6\text{ comparison is permitted by characters but is currently untyped.} }$$

In particular,

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

must not be identified with the $e_6$ bridge from matching invariance or rank. The nontrivial $\mathbb Q(\zeta_3)$ occurrence summand remains separate regardless.

Durable verification

Checker:

research/benincasa/checkers/rank12_soft_tate_to_e6_typing_gate.py.

Packet:

research/benincasa/results/rank12-soft-tate-to-e6-typing-gate.json.

Ledger claim: seqclaim-db06324d302df39df9cd59a6.

Epistemic event:

ev-000000000835-42dab0b5-6eac-4744-a557-6a529c872990.

Next falsifier

Derive the complete source-normalized rank-twelve specialization at $(u,v)=(2,0)$, preserving the soft nodal line and the $e_6$ coordinate in one complex. Read the soft-node-to-$e_6$ component before cyclic quotienting. Only that source matrix may determine $c$.

View source on GitHub