v10.23.0 — sharper N=6 obstruction (Room 11)

N=6's documented obstruction is sharpened from a proof-technique limitation to a verified, algebraically exact structural fact: the N=4-template contour construction's saddle-pinned heights coincide EXACTLY with two of N=6's five resonance loci along their entire length (not approximately, and not at isolated points), confirmed via an exact algebraic identity and calibrated against N=4 (whose own saddle heights miss its resonances by a clear margin, confirming this is N=6-specific). This rules out the direct N=4-template transplant specifically, for a precise reason -- it does not prove no technique whatsoever could work at N=6, and two escape routes remain explicitly open. N=6 remains OPEN. Reviewed adversarially (Reviewer AV).

v10.22.0 — sharper N=6 diagnosis (P1 follow-up)

The N=6 case (a/N=1/6,5/6) remains OPEN, but the reason is now precise: the q=i reflected-contour argument that closes N=4 despite tie height T<0 relies on a ray lemma needing sin(theta)!=0 at the tie-ray angle. N=6's certified tie-ray angle is exactly theta*=0, degenerating that lemma's proof exactly where N=6 needs it. Two alternative hypotheses (a 3-way tie, a composite-divisor obstruction) were checked and ruled out. This is a diagnosis, not an impossibility proof: a modulus-only rescue at theta=0 was not checked and remains conceivable. Reviewed by Reviewer AK (CORRECT, one wording fix).

v10.21.0 — beta2 arithmetic status (P6); research program P1-P6 complete

Final item of the research program. beta2 (the exponential growth rate of u_n, v_n, cor:beta) is proved not to be a root of any integer polynomial of degree <=64 under stated height caps (rigorous LLL certificate with an explicit unimodularity check); a further exclusion to degree <=128 is recorded at a weaker VERIFIED-only tier, since it predates that check. Irrationality and transcendence of beta2 remain OPEN. Five independent rational-approximation constructions all land at a decay/denominator-growth ratio rho=1/2, a factor of two below the threshold needed by known q-irrationality criteria; this is documented as a HEURISTIC, inductive obstruction, not a proof that no route exists. Reviewed by Reviewer AJ (one precision fix applied). This release closes the research program items P1 through P6: each is now either proved and released, or precisely documented as open with its exact obstruction.

v10.20.0 — sharpened W_T non-D-finiteness gap (P5)

W_T's D-finiteness status remains OPEN (conj:WT-nonDfinite). This release sharpens the problem statement: non-D-finiteness follows immediately, by the same mechanism used for U and V, once a meromorphic continuation of F_T with infinitely many poles is exhibited. Three precise missing ingredients are named: no known continuation of F_T; the natural transfer variable is non-contracting, unlike the rational-base numeration systems the analogous toolkit was built for; no carry-catalytic transfer equation exists. No new theorem is claimed. Reviewed by Reviewer AI (CORRECT, no fixes required).

v10.19.0 — D-algebraicity strategy redirect (P4)

D-algebraicity of U remains OPEN. This release documents why the natural strategy (extending the pole-density argument that proves non-D-finiteness to the nonlinear case) does not work: algebraic ODEs can have movable (Painleve-type) singularities dense on a curve, so pole density alone is not an obstruction the way it is for linear ODEs. The correct theorem family is identified (Malmquist/Steinmetz/Bank-Kaufman growth theorems), with the disc-analog version flagged as unconfirmed in citable form. An initial draft's flagship counterexample was found broken on adversarial review and is disclosed and retracted in the note; no false claim survives. No new theorem is claimed.

v10.18.0 — no Q=-1 q-difference relation for U,V (P3)

U and V satisfy no linear q-difference relation A0(x)f(x)+A1(x)f(-x)=b(x) with Q=-1 (paper2a cor:Qminusone). PROVED, conditional on cor:arcUV (hence on thm:model and thm:RJ). The proof uses a 2-adic valuation argument at odd-order arc points; the delicate steps were independently reverified in PARI/GP exact algebraic number theory by Reviewer AG, who found no false step. Three minor exposition/script fixes applied.

v10.17.0 — near-q=1 zeros of B (P2)

Three explicit zeros of B are certified (Arb, rigorous winding number) near e(1/13) and e(1/17), where the arc theorem's saddle mechanism does not apply. An oddness lemma is proved: the saddle heights near these roots of unity pair as +-V, with both members surviving for odd N and only one for even N. Whether the zeros actually accumulate at these points remains HEURISTIC and open; it needs an amplitude non-vanishing result and a uniform error bound, neither proved. No density result is claimed on this range. Reviewed by Reviewer AF (CORRECT, no fixes required).

v10.16.0 — small-N pole certificates (P1i)

Direct per-case certificates for 52 reduced a/N with 3<=N<=16 (excluding N=4, covered elsewhere, and N=6, structurally open). Adds one new U/V pole point outside the previously-stated arc: a/N=2/13. The N=6 case is diagnosed as structurally unclosable by this method (tie height T<0 against a head term of rate 0), not merely unproved. Reviewed by Reviewer AE (CORRECT, one wording fix).

v10.15.0 — wider arcs (P1h)

Zeros of B are dense on arg q in [0.286pi,1.714pi]. The poles of U and V are dense on arg x in [pi/6,5pi/6] and [7pi/6,11pi/6], assuming thm:model and thm:RJ. Both results are computer-assisted and were adversarially reviewed (Reviewer AD). The method cannot pass the threshold l~0.484. U/V density on [0.143,1/6) is open.

v10.14.0 — U/V poles dense on the middle arcs

The poles of U and V accumulate at every point of arg x in [pi/5,4pi/5] and [6pi/5,9pi/5] (paper2a cor:arcUV). This has the same standing as the other U/V pole results: it assumes thm:model and thm:RJ, and it is computer-assisted, using P1e, P1f and P1g, each adversarially reviewed. The release also fixes a covering-radius bug in P1e_caseA.py; rerunning the certificate gave identical output.