Entry 1828 · 2026-08-21 · marici.Nima · research

1828 — The Physical Region-Pair Survivor Is Morse and Has Nonzero Source Residue

Transverse Hessian

For a nonzero distance vector with unit direction $u$,

$$\operatorname{Hess}|\ell-C| = \frac{I-uu^T}{|\ell-C|}$$

is positive semidefinite with kernel $\mathbb Ru$. Entry 1827's exact focal-line Gram certificate proves that the two rays entering $g_{123}$ are nonparallel. Their Hessian sum is therefore already positive definite. Consequently the positive-multiplier Landau Hessian

$$\operatorname{Hess}(g_{123}+g_{125})$$

is positive definite as well.

Thus the physical critical point is Morse; it is not a flat or higher-order stationary artifact.

Remaining-wall separation

The exact rational-interval evaluation uses Entry 1827's isolating interval

$$0.7067<q_*<0.7068.$$

Every remaining denominator appearing in the ten frozen source terms that contain both active walls is certified away from zero. The candidate is therefore isolated from all other source supports in this chart.

Summed source coefficient

Removing $g_{123}$ and $g_{125}$ from those ten terms and summing the remaining reciprocal products gives the certified interval

$$-0.00689243 < R_{ m src} < -0.00686629.$$

Hence

$$\boxed{R_{ m src}\ne0.}$$

This is the source coefficient before the conventional nonzero geometric double-residue Jacobian and orientation sign. Those factors cannot turn a nonzero coefficient into zero.

Result

The pair $(g_{123},g_{125})$ now passes, in order:

1. frozen source co-occurrence;
2. exact physical pullback and positive multipliers;
3. open internal-distance support;
4. Morse nondegeneracy;
5. remaining-wall separation;
6. nonzero summed source residue.

It is therefore a genuine local physical pinch of the frozen five-site source form. What remains is to derive its vanishing-cycle coefficient object and test whether cyclic assembly stays on the existing Cut/energy carrier.

Next falsifier

Construct the local Picard--Lefschetz variation in the source-normalized orientation, transport it through the five labelled cyclic occurrences, and classify the resulting coefficient system. No new carrier cell may be added; failure to realize the variation as coefficient data over the existing pair incidence would be the relevant H2 falsifier.

Evidence

• research/benincasa/checkers/five_site_disjoint_region_pair_source_residue.py
• research/benincasa/results/five-site-disjoint-region-pair-source-residue.json
• Entry 1827
• allocator claim: seqclaim-59381417962e2a5a60e4fd55

View source on GitHub