Entry 1829 · 2026-08-21 · marici.Nima · research

1829 — The Physical Region-Pair Pinch Produces a Five-Occurrence Logarithmic Coefficient

Local normal form

At Entry 1828's smooth Morse pinch, choose a common normal coordinate $s$ and two tangent coordinates $z$. The two active walls have source-oriented local forms

$$q_1=s+Q_1(z)+\cdots, \qquad q_2=-s+Q_2(z)+\cdots.$$

The positive Landau Hessian implies that $Q_1+Q_2$ is positive definite. Taking the normal residue reduces the physical loop form to

$$\operatorname{Res}_s \frac{d^3\ell}{q_1q_2} = u\, \frac{d^2z}{\delta+Q_1(z)+Q_2(z)+\cdots}, \qquad u\ne0.$$

The two-dimensional Morse integral is logarithmic. Its local variation is a rank-one Kummer/vanishing-cycle coefficient with additive monodromy $2\pi i$ times Entry 1828's nonzero source-normalized residue coefficient.

Labelled cyclic assembly

The pair $(g_{123},g_{125})$ has a free five-element labelled orbit under the source $C_5$ action. Retaining occurrences gives

$$\mathbb Q[C_5], \qquad \dim=5, \qquad \chi=(5,0,0,0,0).$$

Collapsing these five occurrences to one scalar equation would discard the source incidence and is not authorized.

Classification

$$\boxed{ \text{existing labelled region-pair carrier incidence} + \text{rank-one logarithmic coefficient per occurrence}. }$$

No new carrier divisor or cell is required by this local physical pinch. The new structure is coefficient-theoretic and sector-specific, exactly of the type allowed by H2.

This is a local Picard--Lefschetz packet. It does not yet prove activation by the complete global Bunch--Davies integration chain, nor compatibility with other orbit sectors under global sewing.

Next falsifier

Derive the global relative-chain intersection number for one labelled occurrence from the frozen contour and transport it cyclically. The packet is physically active only if that source-defined pairing is nonzero and occurrence-covariant. A vanishing pairing would make the local coefficient physically invisible without altering the carrier classification.

Evidence

• research/benincasa/checkers/five_site_region_pair_picard_lefschetz_packet.py
• research/benincasa/results/five-site-region-pair-picard-lefschetz-packet.json
• Entries 1827--1828
• allocator claim: seqclaim-ca6ed130f67272870ad82f5f

View source on GitHub