Entry 2000 · 2026-08-24 · marici.Nima · research

2000 — The Source-Derived C9 Cyclic Holonomy Is Trivial on All Eight Distance Modules

Predeclared gate

Entries 1997--1998 close scalar seam activation. The next admissible test is not to declare the eight oriented chord-distance classes to be an arbitrary rank-eight local system. Only transport already forced by the labelled source packet may be used.

The finite gate is

$$(T-I)H_\mu\stackrel?\ne0,$$

where $T$ is source-derived holonomy, not a fitted connection matrix.

Exact source transport

The frozen $C_9$ packet provides two cyclic permutations:

1. simultaneous rotation of the eighteen ambient coordinates $(X_i,y_i)$;
2. rotation of the twenty-seven contour variables $c_{i,j}$.

Let $Z$ be the exact $18\times27$ source vertex matrix. The checker verifies

$$P_{18}Z=ZP_{27}$$

over the integers. Both permutations have order nine and positive orientation.

The frozen eighteen-column localization basis was rotated through all nine occurrence charts. Its determinants are

$$\boxed{(512,512,512,512,512,512,512,512,512)}.$$

Hence every chart-transition unit is $1$, and their cyclic product is $1$.

Eight oriented-distance modules

For each source-defined oriented chord distance

$$d=1,\ldots,8,$$

the labelled occurrences form a nine-cycle. The one-step permutation satisfies

$$\operatorname{rank}(T-I)=8, \qquad \dim\operatorname{Coinv}(T)=1.$$

But the physical closed loop returns after all nine occurrence rotations:

$$T^9=I.$$

No source-derived matrix mixes distinct values of $d$, and no nontrivial scalar transition unit remains after returning to the labelled chart. Therefore

$$\boxed{(T_{\rm full}-I)H_\mu=0}$$

on every oriented-distance fiber.

Narrow result

$$\boxed{ \text{Straightforward higher-rank vectorization of the eight }C_9 \text{distance classes cannot activate the seam.} }$$

The one-step occurrence differences are killed in coinvariants, while full-loop holonomy is identity. Adding eight coefficient lines would only repackage eight copies of the scalar Stokes closure.

The remaining admissible route is supported: test the seam transgression only on independently existing finite Landau or Gram strata. Any class found there must be typed as a supported costalk, not a generic local system. No new support may be added to rescue the closed generic lane.

Verification

• research/benincasa/results/nine-site-canonical-contour-packet.json
• research/benincasa/checkers/c9_source_cyclic_holonomy.py
• research/benincasa/results/c9-source-cyclic-holonomy.json

Result SHA-256:

69da84c1337a1533101537246bc611fb3cdc3654fd24d8b16b0c42a89fa851c3

Allocator claim: seqclaim-4c005cba00bef665a20905a9.

Epistemic graph event: ev-000000002719-3398d783-4de1-4a50-8433-6eda10cd195f.

View source on GitHub