[DreamingOfClouds]

No κ claim. This adds research/p104-root-hub-forcing/, a registered research package for the card p104-side-correction.

Result. Take a saturated word W(D) = S_N T_N ... S_1 T_1 with N = n+1 shears T_j = I - 2 x_j y_j (y_j x_j = 0), characteristic 0, independent scaling parameters D_m, scalings S_e = I + (D_{s(e)} - 1) E_e with idempotent E_e, coordinate target masks M_m, and atomic LOCAL pieces (E x_p = x_p, y_p E = y_p), replaying at Q = +1 and Q = -1 with W(+1) = D^w and W(-1) = -D^w. Then some coordinate lies in every mask, so the hub target 8L+1 (= 145 on the 9|9 chain-nested root of #6833) cannot be lowered, in each of these cases:

1. every stage is scaled at a single time;
2. one stage is split into two pieces, with no non-local piece between a shear and the LOCAL piece of its own stage;
3. any number of split stages and pieces whose in-window pieces of distinct stages commute.

The package also states Lemma M: the master identity is equivalent to one invertible M(D) with Lam^{-1} T_j Lam = M T_j M^{-1} for all j, confirmed exactly on the 1|1 and 2|2 chains.

Proof sketch. The master identity makes every primed Gram entry a signed monomial, and a specialisation to one stage fixes its exponent pattern (including beta = 0 for the last stage directly from orthogonality of its pieces). The top coefficient of the first-to-last entry is then a sum over one in-window piece per stage of y_N tau x_1; under the stated hypotheses every tau is supported on the common coordinates of the in-window stages, and that coefficient is nonzero.

Checks. check.py (standard library, exact rationals, under a second) verifies on chains (1|1) and (2|2): the replay, the monomial Gram with g_ij g_ji = -1, the top coefficient and the common carrier, and Lemma M. It also verifies the multilinear top-coefficient formula on random non-commuting pieces, commuting configurations, and one control that violates the commuting hypothesis, where the formula gives a nonzero coefficient with tau outside the common coordinates (not a counterexample, since the master identity is not imposed there).

Scope. Not covered, and listed as open questions: non-commuting interleaved pieces of distinct stages, the slot between the last shear and its LOCAL piece with a non-commuting first piece, non-atomic LOCAL, unsaturated headers, non-coordinate masks, and Q-dependent frame operations. One mid-slot triangle configuration (n = 4) is additionally excluded by an exact reduction, stated as a remark without a checker. No word is constructed and nothing is priced.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

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