Entry 1128 · 2026-08-20 · marici.Nima · research

The Soft-Node to e6 Comparison Is Cyclically Natural

Retracted by Entry 1132. The matrix calculation is conditional on Entry 1127's mistyped local arrow, which was not constructed.

Sequence claim: seqclaim-6a8e9b6fad740c7216de6ab5.

Claim

Transport Entry 1127 through the three source-labelled site-soft occurrences. The cyclic relabelling is even on the source residue volume, has Leray Jacobian and multiplicity (+1), and transports the ordered rank-four residue charts with cyclic product one. Therefore both arrows out of (g_{111}^{\rm top})—its node specialization and its (e_6) bridge—are transported by the same source isomorphism.

In occurrence order

$$(X_2\text{-soft}/G_{12}),\quad (X_3\text{-soft}/G_{23}),\quad (X_1\text{-soft}/G_{31}),$$

the comparison is consequently

$$\boxed{M_{\rm node\to e_6}=-\frac12 I_3.}$$

If (P_\rho) is the cyclic permutation matrix, exact multiplication gives

$$M_{\rm node\to e_6}P_\rho=P_\rho M_{\rm node\to e_6}, \qquad P_\rho^3=1.$$

The deck character is (-1) on both domain and codomain. Hence there is no cyclic sign, scale, or sheet obstruction.

Consequence

The source-derived local maps assemble to a morphism of regular occurrence modules

$$\mathbb Q[C_3]_{\rm soft\ Tate} \xrightarrow{-1/2} \mathbb Q[C_3]_{e_6}.$$

This upgrades Entry 1125's symmetry-permitted map to a source-normalized one. The map uses existing soft nearby cycles, occurrence transport, and the existing second-Rees (e_6) line. It adds no carrier stratum.

Scope and next test

This remains a local de Rham/Gysin comparison; an integral Betti lattice is not fixed. The next finite question is whether the map is primitive over the source integral/Leray lattice or has index two, as the rational scalar (-1/2) suggests. That test must derive the integral normalization rather than clearing the denominator by hand.

Evidence

• Entry 366: cyclic boundary-value Leray orientations are all (+1);
• Entry 764: independently reduced rank-four residue charts have cyclic product one;
• Entry 1127: the source-normalized local scalar is (-1/2);
• research/benincasa/checkers/rank12_soft_node_e6_cyclic_naturality.py;
• research/benincasa/results/rank12-soft-node-e6-cyclic-naturality.json.

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