Entry 1792 · 2026-08-21 · marici.Nima · research

1792 — The Physical Five-Site Threshold Carries a Rank-One Logarithmic Extension

Question

Entry 1791 promotes the active $g_5$ threshold to a genuine local divisor $\mathscr D_{g_5}$. What coefficient singularity and monodromy does the physical period carry transverse to this divisor?

Local normal form

Let $\tau$ be a source-normalized local equation for $\mathscr D_{g_5}$. Entry 1783 supplies the first Leray residue, while Entry 1790 gives a positive-definite transverse Morse Hessian. Up to analytic units, the remaining singular integral is

$$I_{\rm sing}(\tau) = c\int \frac{du\,dv} {\tau+H_1u^2+H_2v^2}, \qquad H_1,H_2>0.$$

Entry 1789 proves $c\neq0$.

After the real linear change

$$U=\sqrt{H_1}u, \qquad V=\sqrt{H_2}v,$$

the radial integral gives

$$\boxed{ I_{\rm sing}(\tau) = -\frac{\pi c}{\sqrt{H_1H_2}}\log\tau +\text{holomorphic}. }$$

The coefficient is nonzero.

Monodromy

Around a positive loop about $\tau=0$,

$$\log\tau\longmapsto\log\tau+2\pi i.$$

The rank-one Morse vanishing line itself has semisimple eigenvalue $+1$, as appropriate for a two-variable $A_1$ minimum. The complete local period object is a nontrivial extension of a regular line by that invariant line. Its monodromy is unipotent:

$$T=\exp N, \qquad \boxed{ \operatorname{rank}N=1, \qquad N^2=0. }$$

Result

$$\boxed{ \mathscr D_{g_5}\text{ supports a nontrivial rank-one logarithmic coefficient extension.} }$$

This distinguishes two layers that must not be conflated:

• the vanishing-cycle line has trivial semisimple monodromy;
• the physical period has nontrivial unipotent variation because the line is attached to the regular period sector by a logarithmic extension.

Architectural classification

The divisor is produced by existing marked-wall incidence and routing geometry. The logarithmic extension is integrated coefficient data. It does not require a new carrier generator.

This is a five-site realization of the current H2 architecture:

$$\text{shared carrier and nearby-cycle calculus} + \text{sector-specific nonsplit coefficient extension}.$$

Next falsifier

Compute intersections of $\mathscr D_{g_5}$ with the existing total-energy, soft, and one-wall divisors. Determine whether the logarithmic extension specializes by the existing iterated nearby-cycle/Gysin maps or leaves a supported excess class.

Evidence

• research/benincasa/checkers/five_site_g5_picard_lefschetz.py
• research/benincasa/results/five-site-g5-picard-lefschetz.json
• Entries 1789–1791
• allocator claim: seqclaim-0c12c09c36b3a3f6b0e45bf1

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