Rank of the period matrix M_p mod p for periodic continued-fraction operators: 430 members proven nonzero, and three scoped negatives

Version 2 of this record. ADDITIVE: nothing in version 1 is retracted, corrected, or weakened. Every sentence of v1 remains true as written. Readers holding v1 do not need to discard anything. A note on the version number: "2.0.0" is not semantic versioning. It records the scale of the addition -- a distinct research run, a new title, two new theorems -- and carries no compatibility claim. "Additive" above is a statement about claim status, not about interfaces. WHAT THIS RECORD CONTAINS Two archives, both current, neither superseding the other: pcf-orbit-schema-1.0.0.zip version 1. The trace theorem run: the four-step proof schema shown not to generalise, the closed evaluation tr M_p = mu (1 - (Delta|p)) mod p verified at 806,760 cells, and a ledger of 30 claims. Retained unchanged; its description remains accurate for its own contents. pcf-rankzero-2.0.0.zip version 2, described below. The attack on the gap that version 1's Section 11 left open. The two runs share conventions and box bound but are separately pre-registered, separately ledgered, and separately verifiable. WHAT V1 LEFT OPEN Version 1 (PCF-ORBIT-SCHEMA) closed the trace question with tr M_p = mu (1 - (Delta|p)) mod p, but its Section 11 recorded a gap: for an 874-member class of periodic continued-fraction operator pairs, the statement "M_p is not congruent to 0 mod p" was VERIFIED by census over a finite prime range and not proven. A membership criterion resting on a census is a criterion with an unbounded hole in it. This version reports a direct attack on that hole. THE CRUX, RESOLVED SAFELY The two-root factorisation M_p = W3 A_r2 W2 A_r1 W1 collapses to a scalar multiple of a rank-one matrix, M_p = S_p (P u_r1)(v^T Q) with P and Q invertible, so M_p = 0 if and only if S_p = 0. The connecting scalar S_p is identified BY ITS DEFINING EXPRESSION, not by matching observed behaviour, against a prior run's two factors: it is the never-vanishing one. M_p = 0 therefore does not occur, and the standing v1 claim is not false. Confirmed by direct exact computation: 52,440 member-prime cells by two independent routes, zero disagreements, zero zeros. WHAT IS NEW Theorem W (window law, PROVEN): the window length delta(p) is affine in p, delta(p) = theta p + c, on each class of p modulo L, once p exceeds an explicit threshold. This replaces a scattered census with a stratification by a computable invariant. Observed slopes theta in {0, 1, 1/2, 1/3, 2/3, 1/4, 3/4} correspond exactly to the rational-root types of the member. Theorem Z (PROVEN): on the theta = 0 stratum S_p is a fixed rational constant N, never zero, and the set of degenerate primes is EMPTY -- so those members are settled for all primes, not merely for the tested range. Final accounting over the 874: 430 PROVEN, 326 VERIFIED with a confirmed hold-out, 118 OPEN. THREE SCOPED NEGATIVES ON THE REMAINING 118 Each is stated with its declared reach rather than as a bare absence. (1) S_p is not the reduction of a small rational. (2) S_p is not a residue character of degree at most 4 -- tested over a frozen 144-rung cubic/quartic ladder with 484 positive controls at 484/484. (3) S_p is not sqrt(p)-bounded in any lift: with sigma(p) the symmetric representative in (-p/2, p/2), the hypothesis |sigma(p)|