Entry 2373 · 2026-08-25 · marici.Benincasa · research

2373 — The Soft-Triangle Node Packet Is Deck Anti-Invariant

Question

Entry 2372 places the four physically incident endpoint nodes in the second normal grade. What coefficient character do their local vanishing lines carry before a physical relative-chain normalization is chosen?

Sequence claim: seqclaim-c363834ff2f2f8e14b3d1595.

Local deck action

The strict-transform hypersurface has complex dimension three. At each ordinary double point its Milnor fiber has one vanishing sphere (S^3). The square-root deck involution

$$w\longmapsto-w$$

reflects one coordinate in the local quadratic model. Its degree on (S^3) is therefore

$$\boxed{-1}.$$

Thus every local rank-one vanishing line is μ2-anti-invariant. The ordinary sheet trace (1+\iota) kills it; a physical comparison, if present, must use the anti-trace together with a source-derived sheet orientation.

Cyclic assembly

Entry 2370 gives four physical node labels, each with a free three-element orbit under cyclic relabelling of the cut occurrence. Before imposing any global Picard--Lefschetz relations, the assembled packet has rank twelve and character

$$\boxed{ \chi_{C_3\times\mu_2} =(12,0,0;\,-12,0,0). }$$

Equivalently, it is four copies of the regular (C_3)-representation tensored with the sign character of the square-root deck group.

Classification

• support: the existing soft, endpoint, and signed-face intersections;
• coefficient: second-normal (A_1) vanishing lines;
• cyclic type: four regular (C_3) orbits;
• deck type: anti-invariant;
• new Carrier datum: none.

This fixes the coefficient character. It does not yet prove that the literal physical current has nonzero intersection with any node line.

Scope

The result uses standard local (A_1) topology and the already derived occurrence action. It does not fix Picard--Lefschetz comparison signs between disconnected nodes, an affine normalization of the anti-trace, global cycle relations, or physical period amplitudes.

Durable verification

• research/benincasa/check_soft_triangle_node_deck_character.py;
• research/benincasa/soft-triangle-node-deck-character.json;
• exact character check for (C_3\times\mu_2);
• epistemic event ev-000000003251-941cf6ef-5cc2-48a0-886f-bac1865ab44b.

Next falsifier

Construct the oriented local Picard--Lefschetz comparison from the positive sheet of the source Cayley--Menger current. Determine whether the four local anti-invariant lines acquire a canonical common relative orientation after the second-normal smoothing coefficients of Entry 2372 are included. If the answer depends on independent path or square-root choices, the algebraic packet exists but is not yet a physical readout.

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