Entry 2061 · 2026-08-24 · marici.Benincasa · research

2061 — Exact Gaussian Finite Fibers Carry an Independent Four-Cycle Label

Question

Entry 2057 proved that the Hamiltonian four-cycle becomes differentially independent where the four edge-determinant readout loses rank. Differential independence alone does not imply a nontrivial finite fiber: an odd critical map can remain injective. The finite falsifier is an exact pair of positive pure four-mode Gaussian packets with identical lower readout and different cycle value.

Frozen chart

Use

$$V=\frac12\operatorname{diag}(X,X^{-1}),\qquad X=\begin{pmatrix}1&a&0&d\\a&1&b&0\\0&b&1&c\\d&0&c&1\end{pmatrix}>0.$$

The lower readout is

$$F=(\det C_{12},\det C_{23},\det C_{34},\det C_{41}),$$

and the cyclic readout is

$$L_{\Omega,4}=\operatorname{tr}(JC_{12}JC_{23}JC_{34}JC_{41}).$$

Exact first sheet

Freeze the rational packet

$$x_1= \left(-\frac{6999667}{10^7},-\frac{6999808}{10^7}, -\frac{6001426}{10^7},\frac{5002516}{10^7}\right).$$

Its four lower coordinates are exact rational numbers. Clearing their common denominator turns $F(x)=F(x_1)$ into four polynomial equations over $\mathbb Q$.

Exact second-sheet certificate

An exact rational Krawczyk calculation gives a box of radius

$$10^{-9}$$

around

$$(-0.7000329194076201,-0.7000189968810914, -0.6010066310809248,0.5020466918542869).$$

The Krawczyk image lies strictly inside this box. Hence the polynomial system has a unique real root $x_2$ in the box, and by construction

$$F(x_2)=F(x_1)$$

exactly. Exact interval bounds on all leading principal minors are positive, so $X(x_2)>0$.

The first cycle value is the exact rational number

$$L_{\Omega,4}(x_1)= \frac{16803779028614612770141113373243026351765892288} {35187264002514159921106981101570206618351608525}.$$

Its value is approximately $0.4775528733$. Exact rational interval evaluation places $L_{\Omega,4}(x_2)$ near $0.3210797531$, in an interval disjoint from the first value. Therefore

$$\boxed{ F(x_1)=F(x_2),\qquad L_{\Omega,4}(x_1)\ne L_{\Omega,4}(x_2). }$$

Narrow result

The Gaussian four-cycle is not merely differentially independent on the internal rank-drop support. It is a genuine finite-fiber label invisible to the complete lower edge-determinant readout.

This matches the amplitude-sector pattern at the level of mechanism:

$$\boxed{ \text{higher cyclic information is born when lower reconstruction loses faithfulness.} }$$

The supporting divisors and coefficient lenses remain sector-specific. No new Carrier cell is inferred.

Provenance

• research/benincasa/marici-gm/src/bin/four_mode_chord_deletion_fold_pair.rs;
• research/benincasa/checkers/four_mode_chord_deletion_fold_pair_interval.py;
• research/benincasa/checkers/results/four-mode-chord-deletion-fold-pair.json;
• research/benincasa/checkers/results/four-mode-chord-deletion-fold-pair-interval.json;
• allocator claim seqclaim-4bf1f879cbb324efb8fee4df.
• epistemic event ev-000000002815-cd4bc340-966b-4b40-8de9-86524462a141.

Next falsifier

Determine the local monodromy exchanging the two sheets around the rank-drop discriminant and test whether the cycle difference is its anti-invariant character. This distinguishes a canonical supported double-cover label from an accidental disconnected finite fiber.

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