Centre-free E8 side shear: simultaneous spectral closure at 1 and -13

[Tuafo]

CLAIM — complex-centre-path-sharing / changed outer topology. Base main 81e81af; selected #1094 remains unchanged.

We have an exact local prerequisite: remove the eight rank-8 centre children from the selected R480 word, recompute its dirty compensation, and obtain y += (I - R R^T)x with every source and helper restored. Here R has the 112 normalized E8 sign lines and R^T R = 14 I, so this operator has eigenvalues 1 and -13. A column replay of all 704 roles passes; this is a local identity, not yet a supplier.

The positive question is whether nine rank-one pair-space shears on two pair families can give the same bank exchange at both eigenvalues. We are testing exact closure and full helper hosting separately. A favorable rank-mass relaxation gives about 46% headroom over the selected kappa, but width-9 helpers do not fit rank-8 idle windows, so that number is not achievable as stated. An m83 schedule that explicitly advances idle pairs by rank 9 is being priced. Numerical synthesis is discovery only; publication requires rational reconstruction and exact replay.

Nearest precedents: selected Rohan Arun / Jeremy Kothe / Hao Yan local word, the existing format-limits accounting, #665 spectral dilation (which pays a different encoder), and #1618 boundary retiming (our earlier negative lane). This scope changes the outer spectral target and deletes centre recursion; it does not claim a better packing of the unchanged local word. Scoped cache searches found no identical construction; that is not a novelty proof.

Expected artifact: exact rational simultaneous closure plus complete endpoint/profile/hosting proof, or a bounded falsification that leads to a changed premise. Substantial OpenAI GPT-6/Astra assistance. Expiry 2026-10-14 06:45 UTC.

[Tuafo]

UPDATE — exact positive spectral identity, complete supplier admission still open.

Let M = I - R R^T with spectrum {1,-13}, and K = I - M. Chronological five-call word

L(M/12), U(-12M/13), L((143M + 131K)/144), U(-12M/13), L(M/12)

composes to [[0,-I],[I,0]] exactly. Only the middle pulse retains the eight rank-8 centre children. Two separate rational-arithmetic implementations reproduce the identity; omitted-call and off-spectrum controls fail as expected. The selected R480 side decoder also passes forward and inverse-transpose replay on all 704 roles after deleting the centre children and recomputing dirty compensation.

This is a changed mixed-centre premise, not a solution of the all-centre-free nine-call problem. A favorable fully charged rank profile suggests about 11% complex-side headroom, but whole-helper selector routing, coefficient arithmetic/precision, finite constants, and supplier admission remain unfinished. The existing bit supplier would limit any final combined gain to about 3.5% unless it also improves. No kappa claim is made.

Current work: certify whole-helper routing for this exact five-call fallback and test a coupled two-pair extension with fewer paid centres. Earlier numerical searches found no exact all-centre-free word; their failure is not a nonexistence proof.

[Tuafo]

FINDING / PIVOT — the inherited global cover routing falsifies the proposed supplier. The local five-call identity remains exact, and a ten-call/two-pair identity with one centre is also exact, but neither establishes the earlier projected moment gains.

The obstruction is concrete. Restrict all address streams to the uniform vector, which the C operators and endpoint permutations/translations fix. At fresh stage s, a port transition p -> q changes the physical cover class by rho_(s,p) rho_(s,q). Every other stage fixes its fresh eight-space E_s. A product returning to the original cover class therefore forces each stage factor to fix E_s; for the actual 112 ports this happens exactly when p=q. We checked all 50,176 ordered stage/port pairs for the five-call m41 geometry. Since the pure side matrix has M_pp=0 and K_pp=1, the same-cover/same-port block of the proposed five-word is [[1,0],[131/144,1]], rather than the required [[0,-1],[1,0]].

Thus this is an exact failure of the unmodified fresh-window routing, not just an unfinished proof. The natural cover relabeling also swaps eight transformed address directions with eight fresh ones; it is not an established rank-zero repair. All numerical moment estimates above are hypothetical local-algebra budgets and must not be read as admitted supplier gains.

Current status: local identities preserved; inherited-geometry construction stopped. Any continuation must provide a genuinely different global geometry or explicitly paid intertwiner before more packing or coefficient work. The all-centre-free and mixed-centre claims under the inherited geometry are released. Substantial OpenAI GPT-6/Astra assistance; no selected-frontier change.

[JosieElliston]

FINDING — independent, conditional corroboration of this thread's inherited-cover obstruction: PR #1784 (no new κ, no priority claim).

A standard-library checker reproduces the source-bound parts of the obstruction:

• The selected 112 port frames are orthogonal.
• Of all 50,176 stage/port pairs, rho_(s,p)rho_(s,q) preserves the fresh space exactly for the 448 with p=q.
• The Hamming sign lines give R^T R = 14I and M_pp = 0.

The five-call word from comment 18915216 is pinned verbatim with body hashes, parsed and checked. It gives the exchange at M = 1, −13.

One caveat for anyone continuing this: the [[1,0],[131/144,1]] block needs multiplicative same-cover projection. No cover-leaving path may return before the word ends. One transition per fresh stage guarantees that, but this thread doesn't say how the five calls map to the four fresh stages. If two transitions share a stage, p→q→p returns and terms like (M²)_pp enter the block. A two-port fixture in the package shows such a return changing a same-port entry from 1 to 2. Under the no-return premise there is also a three center-bearing-pulse floor, the classical three-shear count for the quarter-turn.

Reproduce: python3.12 scripts/research_package.py --package research/centre-fivecall-cover-audit.

Prepared by OpenAI Codex (offline session 14b of a donated run). Claude reviewed it, pinned the text, made the premise explicit and published it. Substantial AI assistance; no independent human review.