Iteration 23 — refereeing (a′) and attacking the crux

Status: The (a′) computation (integrator's own, prep commit af47a43) submitted to a binding adversarial referee; crux (b′) attacked through the two structural lenses the iteration-22 analysis sharpened (G1 energy-dependent Schur monotonicity; G2 Krein-space definitizability); binding assembling referee. All machine checks independent (1e−15 identity over 2000 random (c,s); independent 4×4 BC-determinant check at the roots to 1e−13). Outcome: Verdict UNCHANGED — 22nd consecutive confirmation; hedge HELD at R7. No proof assembles. The iteration's yield: (a′) ESTABLISHED; a fourth methods-no-go (global non-definitizability); the open channel's Krein type ESTABLISHED; one honest retraction (G1's z-slot monotonicity, internally contradicted and struck by the assembler); and the residual compressed to two explicitly named checkable lemmas. Last updated: 2026-07-11 Iteration: 23

1. (a′) — ESTABLISHED, with the framework honestly re-opened

The computation stands [ESTABLISHED]: cos²(πs) = (z+2+z⁻¹)/4 in z = e^{2πis}, so c − cos²(πs) = −(z² − 2(2c−1)z + 1)/(4z); discriminant 16c(c−1) > 0 for all c > 1; roots z± = (√c ± √(c−1))² = e^{±2πτ₀} (cosh(πτ₀) = √c); the transverse spectrum is two shifted simple uniformly-separated lattices (ℤ + iτ₀) ∪ (ℤ − iτ₀). Referee corrections: regularity degenerates as c → 1⁺ (the FG-1989 boundary; any Riesz bound is non-uniform there); and — decisive — "distinct roots ⟹ Riesz basis" has NO applicable off-the-shelf theorem for this pencil: the 4×4 first-order reduction has repeated characteristic roots (both components carry the same symbol, coupled only through the boundary conditions), outside the scalar chain (Mikhailov 1962; Kesel'man 1964; Dunford–Schwartz XIX.4; Shkalikov 1983/86 — live-verified) and the systems chain (Lunyov–Malamud, live-verified); Keldysh linearization cannot help (the problem is already linear in λ = s²).

The chain that does apply (one lemma open): (i) [ESTABLISHED] Δ(s) ∼ c − cos²(πs) is a sine-type entire function of exponential type 2π with simple separated zeros; (ii) Levin (1961)/Golovin (1964): such zeros generate an exponential Riesz basis of L²(0, 2π) (canonical: Avdonin–Ivanov); (iii) LEM-A1‴-T [OPEN] — the explicit map carrying e^{is_n·} to the normalized pencil eigenvectors (U_n, V_n) (explicit trig combinations) is bounded with bounded inverse: a concrete Gram-matrix bound on explicitly known functions, cold-checkable and numerically probeable first; bounds necessarily non-uniform as c → 1⁺.

2. The fourth methods-no-go: global definitizability fails `[INFERENCE, high]`

The transverse eigenvalues are ν_n = n² − τ₀² ± 2inτ₀: exactly one real branch (n = 0) and infinitely many non-real conjugate pairs — so σ_ess of the strip pair contains infinitely many non-real rays, exceeding Langer's finiteness bound for definitizable operators (Langer LNM 948; live-verified restatement). The strip pair is NOT globally definitizable in the Krein metric for any c > 1 — the same corner exponents n ± iτ₀ that define the gate kill definitizability. (Ray-completeness rests on the mode decomposition; unconditional once LEM-A1‴-T lands.)

What survives locally [INFERENCE, medium — conditional]: over the strip Ω = {|Im z| < 2τ₀} (the non-real branches sit at distance ≥ 2τ₀), local definitizability holds conditional on LEM-A1‴-T + relative compactness of Q in the Krein resolvent sense; then 0 is a spectral point of type π₋ (finite-index Pontryagin spectral subspace — Behrndt–Jonas; Azizov–Jonas–Trunk, live-verified) — a genuine nonlocal almost-definite structure that sidesteps the iteration-21 pointwise no-go. And [ESTABLISHED — machine-verified]: the n = 0 open channel w has constant negative Krein density ⟨J_c w, w⟩ = −(c−1) = −1/(κ−1).

3. The honest retraction (the discipline working)

G1's step (1) — the z-slot Herglotz/Loewner monotonicity of the Schur complement, initially submitted [ESTABLISHED] — was refuted in assembly by internal contradiction with G2's machine-verified non-real closed channels (the Loewner order has no meaning in the everywhere-indefinite strip metric) and is downgraded to [OPEN]. What survives of G1: the exact Feshbach–Schur reduction as a scheme; the BC-slot Hellmann–Feynman boundary identity with strict sign via the iteration-20 nowhere-locally-zero rigidity [INFERENCE, medium] (its compression through the Krein metric open); and the (★) reduction target. Recorded per house rule: submitted grades are provisional until assembly; a struck claim is documented, not erased.

4. The residual after iteration 23

LEM-A1‴-T [OPEN] (upstream bottleneck): the transfer/Gram bound converting the Levin–Golovin exponential Riesz basis into a Riesz basis of pencil eigenvectors. Feeds every live route (the Feshbach machinery, the local definitizability, and H1's mode decomposition). LEM-A1⁗ [OPEN] (the crux, renamed): Pontryagin-channel on-shell transversality — the on-shell nondegeneracy argument executed in the type-π₋ local spectral structure. Plus the standing (H1) [INFERENCE, medium — resolvent-probe supported, three falsifiers named] and (H2) [OPEN].

Designated next move: the numerical Gram-matrix probe of LEM-A1‴-T (condition numbers of the explicit eigenvector Gram matrices vs truncation and c) — cheap, decisive-in-expectation, and it feeds directly into the analytic bound.

5. Ledger

Hedge HELD at R7 (no combination assembles; one grade moved backward by retraction — honest bookkeeping). Verdict unchanged — 22nd consecutive confirmation; count FIVE; no numeric ID consumed. Methods-no-go tally now four (BKT family-level; Gaussian-BIC decay-class; and now global definitizability; plus the localized/naturality closures of iterations 16–20 on the operator-algebra side).