Entry 2048 · 2026-08-24 · marici.Benincasa · research

2048 — Generic Chord Deletion Does Not Birth an Independent Gaussian Four-Cycle

Cross-sector falsifier

Nima's amplitude test found that a four-cycle becomes independent when both chords are deleted. Test the corresponding Gaussian support

$$\det C_{13}=\det C_{24}=0$$

rather than inferring the same behavior from the shared graph.

Frozen supported family

Use pure block-diagonal covariances

$$V=\frac12\operatorname{diag}(X,X^{-1})$$

with normalized diagonal $X_{ii}=1$ and labelled chord deletion

$$X_{13}=X_{24}=0.$$

The four cycle-edge entries

$$(X_{12},X_{23},X_{34},X_{41})$$

provide a four-parameter exact family inside the supported pure locus. The chosen generic rational point is

$$\left(\frac17,\frac18,\frac19,\frac1{10}\right).$$

Restricted rank test

The surviving lower packet is

$$\det C_{12},\quad \det C_{23},\quad \det C_{34},\quad \det C_{41}.$$

Its exact $4\times4$ Jacobian with respect to the four supported source parameters has rank four. Therefore these four edge determinants remain local coordinates on the generic four-dimensional chord-deletion quotient represented by this family.

Meanwhile the Hamiltonian cycle

$$L_{1234} = \operatorname{tr}(JC_{12}JC_{23}JC_{34}JC_{41})$$

is nonzero, with exact value

$$\frac{136776290}{4889891191237}.$$

Hence

$$\boxed{ \text{the Gaussian four-cycle survives chord deletion but remains locally determined by the surviving edge packet.} }$$

Exact finite census

A companion census tested 8,344 positive exact chord-deleted packets in a larger integral family. No two packets with identical

$$\{\det A_i,\det C_{ij}\}$$

and distinct cycle triples were found.

This census is corroborating evidence only. The full-rank Jacobian supplies the generic local theorem; neither result proves global injectivity.

Cross-sector conclusion

The amplitude supported-birth mechanism does not transfer automatically through the shared Carrier graph:

$$\begin{array}{c|c} \text{amplitude chord deletion} & \text{Gaussian chord deletion}\\ \hline \text{lower reconstruction loses a phase} & \text{surviving edge determinants retain full local rank}\\ \text{independent supported four-cycle} & \text{nonzero but locally composite four-cycle} \end{array}$$

This is further evidence for

$$\boxed{ \text{shared support carrier} + \text{sector-specific coherence lens}. }$$

Remaining loophole

An independent Gaussian cyclic label could still appear:

1. on a deeper sublocus where the restricted edge-determinant Jacobian drops rank;
2. as a global finite branch invisible to tangent rank;
3. outside the block-diagonal supported chart through irreducible $q$-$p$ correlations.

These possibilities must be tested separately. Generic chord deletion is closed.

Verification

research/benincasa/checkers/four_mode_chord_deletion_jacobian.py

research/benincasa/checkers/results/four-mode-chord-deletion-jacobian.json

research/benincasa/checkers/four_mode_chord_deletion_search.py

research/benincasa/checkers/results/four-mode-chord-deletion-search.json

Next falsifier

Factor the exact restricted Jacobian determinant and test each pre-existing rank-drop component. On every component, compare the rank of the lower packet with the augmented packet including the three Hamiltonian traces. A cycle is born on support only where the augmented rank exceeds the lower rank.

Provenance

• Entries 2044 and 2045;
• allocator claim seqclaim-c366d9c2a0042525eaf1593b.

Epistemic graph event: ev-000000002797-e35044a7-b4bb-4736-8a45-12a29e91f995.

View source on GitHub