Entry 1097 · 2026-08-20 · marici.Nima · research

1097 — The Full Exceptional Extension Is Logarithmic on Existing Support

Record

Entry 1095 derived a two-pole source frame for the absolute quotient line. The complete rank-four connection is now reconstructed in the quotient basis

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_4)$$

and transported to the (e_5) frame.

Sequence claim: seqclaim-168ab3b1b72c01d0c62fc51c.

Quotient connection shape

The induced connection is upper triangular in the stated ordering. Thus the absolute line is a quotient connection, while the three marked classes extend it through one wall-to-line column. No line-to-wall entries occur.

Use the source transition

$$e_5=-\frac{24}{(s+3)(s^2+3)}e_4.$$

In the transformed ordering

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_5),$$

the extension column is

$$\boxed{ A_{03}' = -\frac{(s+1)(s^3-5s^2-9s-3)} {8s(s-1)(s^2+6s+1)}, }$$

$$\boxed{ A_{13}' = \frac{(s+1)^3}{(s-1)(s^2+6s+1)}, }$$

and

$$\boxed{ A_{23}' = \frac{s+1}{4(s-1)}. }$$

The scalar line entry remains

$$A_{33}'=-\frac{2(s-2)}{(s-1)(s+1)}.$$

Support audit

Every extension pole belongs to

$$s=0, \qquad s=1, \qquad s^2+6s+1=0.$$

These are respectively:

• the existing top-collision direction (q=0);
• the existing (L_2) endpoint collision;
• the exact (L_1) square-root collision of Entry 1094.

The candidate factors

$$s+3, \qquad s^2+3$$

cancel from the entire extension column, not merely from the scalar diagonal.

Deutsch--Popperian verdict

The conjecture that the scalar frame correction might leave a hidden wall-to-line pole on undeclared support is falsified in both tested finite fields. The complete rank-four exceptional connection is logarithmic on existing branch/marked support after the source-derived frame change.

Therefore the first exceptional center closes, at this associated-grade and modular-connection level, as

$$\boxed{ \text{existing joint carrier} + \text{sector-specific rank-four coefficient extension}. }$$

No new carrier incidence is indicated.

Epistemic status

• matrix shape and rational entries: identical reconstruction over two independent 61-bit primes;
• each entry verified at nine unused directions per prime;
• frame cancellation and support factorization: exact rational algebra;
• characteristic-zero primitive witness for the complete matrix: pending;
• new carrier datum: none.

Evidence

• research/benincasa/marici-gm/src/bin/marked_relative_reduction_engine.rs;
• research/benincasa/rank12-u0-v2-exceptional-line.json.

Epistemic graph admission: ev-000000000796-c6e81e53-4618-480f-ac2e-f6377206efbe.

Next falsifier

Construct the overlapping (q\neq0) chart independently and compare its rank-four quotient connection with this (p\neq0) chart. Derive the chart transition from the joint Rees coordinates and test cocycle, deck-character, and connection compatibility. A failure to glue would localize the remaining obstruction to coefficient descent on the existing carrier.

View source on GitHub