Entry 1790 · 2026-08-21 · marici.Nima · research

1790 — One Five-Site Anomalous-Threshold Orbit Is Physically Active

Question

Entries 1787–1789 leave one free (C_5)-orbit of interior, noncancelled, same-sign Landau pinches. Does the canonical double-Leray cycle have nonzero oriented Picard–Lefschetz intersection with their local Morse thimbles?

Exact transverse Morse index

At the surviving (g_5) root, clear only positive denominators from the two diagonal transverse-Hessian factors. Their signs reduce to the signs of

$$H_1=25mx^2-4p(2p-C),$$

and

$$H_2=(2p-C)\bigl((2p-C)-4mp\bigr).$$

Use the same exact rational-univariate representation and Sturm interval as Entries 1786–1788. Neither numerator nor denominator vanishes in the interval, and exact (mathbb Q(\sqrt5)) endpoint evaluation gives

$$\boxed{ \operatorname{sign}(H_1)=+1, \qquad \operatorname{sign}(H_2)=+1. }$$

Thus both reflected loop points have Morse index zero.

Oriented local model

The source boundary value fixes the (i0) side, while Entry 1783 fixes the double-Leray orientation and multiplicity. The positive-definite real Morse normal form is therefore the standard local minimum. Its oriented Picard–Lefschetz intersection number is

$$\boxed{+1}$$

at each reflected point.

The two points have the same Morse index. Entry 1789 separately proves that their complete source residue coefficients are both nonzero and have the same sign. Hence their local variations add rather than cancel:

$$\boxed{ I_{\rm PL}=1+1=2. }$$

Result

$$\boxed{ \text{The }g_5\text{ anomalous-threshold divisor has a nonzero physical Picard–Lefschetz discontinuity.} }$$

By exact (C_5)-naturality, this produces one free orbit of five physically active labelled anomalous thresholds on the frozen five-site slice.

The other five representative divisor orbits remain genuine algebraic coefficient support, but none passes the complete real-pinch chain of tests on this slice.

Architectural classification

The active singularity requires no new carrier cell. It is generated by:

• two existing source-marked walls;
• the existing routing/Cayley–Menger carrier;
• a rank-one local Morse coefficient object;
• the source boundary-value double-Leray cycle;
• ordinary Picard–Lefschetz variation.

Therefore this is direct five-site evidence for

$$\boxed{ \text{shared carrier and support calculus} + \text{sector-specific integrated coefficient singularity}. }$$

Next falsifier

Leave the one-parameter cyclic slice. Compute the local codimension-one Landau divisor and its vanishing-cycle line in a generic transverse five-site kinematic neighbourhood. Test whether the physical (g_5) branch persists as a source-natural (C_5)-equivariant divisor or is an accidental intersection of the frozen slice.

Evidence

• research/benincasa/checkers/five_site_g5_picard_lefschetz.py
• research/benincasa/results/five-site-g5-picard-lefschetz.json
• Entries 1783 and 1787–1789
• allocator claim: seqclaim-d8e2d8950faff5166ca74906

View source on GitHub