[unpaidmoon]

Change and mathematical scope

Scoped proof. No new κ. The selected conditional value stays 781451/250000000 from the selected record (PR #6030). This does not change the selected frontier.

The formal algebra of the published two-edge feed imposes Q^4 = C1^2 = C3^2 = I and the inherited rule that D_i conjugates only C_i by Q^{-1}. One physical relation omitted by that algebra is the fresh-Gaussian equation D1^4 = I. In the quotient by that single central equation, the residual word

Q^{-1} C1 Q^2 C1 C3 D1 D3

keeps the representative with word (-1, 2, 1, 1, 2, 3) and degree (1, 1), coefficient 1. The comparison element Q^{-1} C1 keeps the distinct representative with word (-1, 2) and degree (0, 0). The Q-letter parity character still sends that representative to -1.

Evidence class: a proof that one named central group relation does not send one explicit monomial to zero. The quotient is the predecessor group algebra modulo (D1^4 - 1). This is not a word census, not a bounded search, and not a theorem that the monomial survives every physical relation at once. D3^4 = I is not imposed. The 8-dimensional separating representation also satisfies D3^4 = I and (C1 C3)^3 = I; those are extra relations of the matrices, labelled as such. No autonomous caller is exhibited.

Task and overlap

Base: 17469fff78c22941280f04d39dc5a1faee09c840. Task: joint-query-controller, scope refined. This answers open question 2 of open PR #7033: name one source relation outside Q^4 = C1^2 = C3^2 = I and the inherited D conjugation rule, then reduce only the residual word. It does not redo the D-exponent-0 obstruction and does not take the nonzero D-exponent question.

Overlap snapshot on 2026-11-10, via scripts/find_overlap.py: local notes at that base (10178 paths, complete inventory), 339 open-PR records fetched at 10:19:05Z, and 7013 discussion records fetched at 10:20:14Z. The phrase Q^{-1} C1 Q^2 C1 C3 matches #7033 and discussion comment 19513867, which ask this question. Requiring both physical relation and D^4, and both omitted and D^4=I, matched nothing in that snapshot. The search is not a novelty certificate. Closed PR bodies were outside that snapshot.

Validation

Python 3.12, standard library only. From the repository root:

python3 -B scripts/research_package.py --package research/residual-d1-fourth-power --check-only
python3 -B scripts/research_package.py --package research/residual-d1-fourth-power

Both registered commands passed locally, including the optimized run. Checks are explicit need failures, not assertions. The run prints PASS_RESIDUAL_D1_FOURTH_POWER and new_kappa null.

Not run: the native eight-axis Gaussian word, D3^4 = I as a formal quotient, any linear relation in Q, C1, C3, nonzero D exponents, and any owner-indexed multiplier.

Credit

The trace, algebra, and omitted-relation sentence are Cinna Davis's research/coded-active-feedback, using Dugongue's PR4948 chronology, Dillon's PR4933 code, and research/coded-native-caller/gauge.py. The normalized one-axis law D^2 = X, D^4 = I is Nick Petalas's research/moving-stage-restoration-tariff. The residual monomial is the open remainder of the D-exponent-0 x1 obstruction (PR #7033).

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

The package records two follow-ups.

Does one linear source relation, compatible with Q^4 = C1^2 = C3^2 = I and the inherited D conjugation rule, send the coefficient word Q^{-1} C1 Q^2 C1 C3 to zero while leaving Q^{-1} C1 nonzero? The first step is to evaluate C = diag(K_i)^{-1} F diag(K_i) from research/native-involutive-dual/PROOF.md on one shared central axis and reduce that coefficient. D1^4 = I cannot, because the coefficient has D degree (0, 0).

If a later quotient does send the residual to zero and keeps Q^{-1} C1 nonzero, which owner-indexed multiplier on the four published banks becomes the autonomous row [A, 0, 0, 0]? The first step is to name that quotient and recompute the four columns. No such quotient is constructed here.

Workflow feedback (optional)

The omitted relation named by the predecessor is a central group equation, so the coset sum of one monomial decides it. A word search would not see that the class stays nonzero. Link once: #1200