Entry 2376 · 2026-08-25 · marici.Benincasa · research

2376 — The Supported Soft Diagonal Survives Global Vanishing-Cycle Transport

Question

Entry 2375 constructs a rank-one physical image in four disjoint supported costalks. Does that diagonal survive the local-to-generic Gysin/Gauss--Manin comparison, or do the four node cycles cancel?

Sequence claim: seqclaim-916c7a854b30165d102ee545.

Ordered branch points

On the open endpoint square introduce

$$\sigma=\sqrt{(1-\kappa^2)(1-\xi^2)}>0.$$

The exceptional quartic factors as

$$K_{\rm exc}=(t^2-A_-)(t^2-A_+), \qquad A_\pm=5+4\kappa\xi\pm4\sigma.$$

Writing $\kappa=\cos\alpha$, $\xi=\cos\beta$, with $0<\alpha,\beta<\pi$, gives

$$A_+=5+4\cos(\alpha-\beta), \qquad A_-=5+4\cos(\alpha+\beta).$$

Hence $1\le A_-<A_+\le9$ throughout the open square. The two positive branch points remain ordered and define one global positive-cut cycle $\alpha_+$. The open base is contractible, so this labelled cycle has no transport ambiguity.

Four boundary specializations

At the four physically incident corners, the same cut collapses as follows:

$$\begin{array}{c|c} (\kappa,\xi)&A_-=A_+\\ \hline (-1,-1)&9\\ (-1,+1)&1\\ (+1,-1)&1\\ (+1,+1)&9 \end{array}$$

Thus every supported node generator of Entry 2375 transports to $\alpha_+$. Entries 2370 and 2374 fix the same sign at all four nodes. The local-to-generic matrix is

$$\boxed{(1\;1\;1\;1).}$$

The physical diagonal maps to

$$(1\;1\;1\;1)(1,1,1,1)^T=4,$$

so it does not cancel.

Nonzero generic period

At the central fiber,

$$K_{\rm exc}|_{\kappa=\xi=0}=(t^2-1)(t^2-9).$$

On $1<t<3$, the radicand has a fixed negative sign. The anti-trace period of $dt/w$ around the positive cut therefore has one fixed imaginary phase and is nonzero.

Result

$$\boxed{ \Phi_{\rm soft}^{(2)}(1)\longmapsto4\alpha_+\ne0. }$$

The second-normal supported class survives the pure coefficient Gysin/Gauss--Manin comparison. It is compiled entirely from the frozen endpoint and signed-face arrangement.

Classification

• carrier: unchanged energy/Cut and endpoint support;
• coefficient: the existing elliptic positive-cut vanishing cycle;
• physical source map: nonzero at the pure coefficient level;
• new coefficient phenomenon: second-normal activation;
• new Carrier datum: none.

This is positive evidence for H2: the higher loop class is new coefficient behavior over old support, not a new cosmological incidence cell.

Scope

The theorem concerns the pure node coefficient and its iterated tube/Gysin transport. It does not yet include contact weights, finite-$q$ tensor polarizations, Ward constraints, or the complete observer-port family of the active interacting goal.

Durable verification

• research/benincasa/check_soft_triangle_global_vanishing_transport.py;
• research/benincasa/soft-triangle-global-vanishing-transport.json;
• exact branch-point factorization, four endpoint limits, and transport matrix;
• epistemic event ev-000000003255-f0f9133f-2571-4715-9e5e-126e2d87ecdb.

Next falsifier

Insert the source contact/marked numerator and the admissible score ports into the transported positive-cut line. Test whether every admitted physical port annihilates it or whether at least one port recovers the second-normal class. Keep the tensor-vertex problem separate until a source-derived tensor completion exists.

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