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)| <= C sqrt(p) is REJECTED 0 of 118 over 41,654 cells on a held-out prime band, every member exceeding 16 sqrt(p) at between 165 and 211 of its 353 primes. SECTION 11 IS NOT RETIRED The deposit gate refuses to retire it. 430 proven is not 874. The v2 replaces Section 11's text with a sharper statement of exactly what remains open and what has been excluded, and leaves the gap standing. FAILED INSTRUMENTS ARE DEPOSITED WITH THEIR REFUTATIONS Four pre-registrations are included, all sealed and hash-verified, including prereg3 -- whose gate contained a FALSE MATHEMATICAL CLAIM. It declared 484 control members must fail an O(sqrt(p)) bound when 206 of them provably must pass it, a consequence of the deposit's own Theorem Z. The gate detected this and halted before reading its hold-out; the artifact that would have recorded that reading was never written, which is the mechanical witness that the hold-out survived. A refuted pre-registration that vanishes from the record is indistinguishable from one that was never written, so it is deposited intact and its refutation is a ledger entry of class PROVEN. REPRODUCIBILITY Exact integer and finite-field arithmetic throughout; zero floats anywhere, including intermediates, enforced by an automated gate. Append-only ledger of 28 entries written under file locking with O_APPEND and fsync, claim numbers contiguous, every entry carrying its guards, its falsifier, and the hash of the pre-registration in force when it was written. Run code/verify_deposit.py to check all of this against the bytes you received. OPEN ITEM, NARROWED Version 1 closed with: "Open item: the second nilpotency clause (M_p not equal to 0) is reduced to a single scalar congruence but not closed." That sentence still stands and is not retracted. It is now understated. The scalar is named by the expression that computes it and is proven never to vanish for 430 of the 874 members; closed forms of the trace theorem's own shape cover a further 326 and are confirmed on a declared hold-out band; the remaining 118 are census only, but now carry three scoped negatives rather than no characterisation. Section 11 is therefore NOT retired -- 430 of 874 is not 874, and this version says so.