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Updated 08-20-2026

D-058 — Repair 4.5c: valley-theorem lower bounds via node-killing (projection) test functions; B3 fix; m_γ fix; detector restatement

Date: 2026-08-19 | Session: 009 | Task: 4.5c Status: complete. Repairs D-054 F9–F12 (D-049/D-050 errata). Setup and identities as in D-049 §0 (I1 variational identity, I2 zero-side identity, under RH throughout; T = the model-space operator, K_a(z,z) = sup_v |v̂(z)|²/(‖v‖² + Q_W^a(v))).

1. B1 floor, repaired

Theorem B1′. Assume RH. Let γ > 0 be a zero, a ≥ 1. (i) If γ has no other zero within distance 1, then with the flat bump v₀ (x-space e^{∓iγx}1_{(−a,a)}): 1 − 1/(2a) − C(1 + log γ)/a² ≤ K_a(γ,γ) ≤ 1/m_γ, C explicit (from Lemma D with δ = 1). (ii) In general, kill the zeros γ₁,…,γ_J within distance 2 of γ (J ≤ 2C₀(γ+2)) via the projection v = e_γ − P_J e_γ (P_J = projection onto span{e_{γ_j}}; evaluation Gram G). Then K_a(γ,γ) ≥ 1 − 1/(2a) − [C′(1+log γ)(1+‖G⁻¹‖²) + 8E(γ,a)(1+o(1))]/a², C′ explicit, provided ‖G⁻¹‖‖k‖² ≤ a (non-degeneracy: the window zeros not too clustered at scale 1/a). E(γ,a) counts the "annulus-hugging" survivors (a zero beyond the window within 1/a of a rim node) — 0 generically, and then the clean 1/a² rate holds; ‖G⁻¹‖ depends only on the local configuration.

Proof. (i) Upper: D-049 (single-term domination, with m_γ — see §3). Lower: v₀ admissible (zero-side domain: its form sum converges — sin² decay vs log density; D-054 refutation of the H₀¹ claim). Numerator |v̂₀(γ)|² = (2a)². Denominator: ‖v₀‖² = 2a (mean-zero projection costs O(1/a) relative, D-049) and Q(v₀) = (2a)² + Σ_{γ'≠γ} 4sin²((γ'−γ)a)/(γ'−γ)². With no zero within distance 1, Lemma D with δ = 1: Σ ≤ 4C₀(γ+2) + 16C₀(γ+2)(1 + π²/6) =: D₁(γ) ≤ 47·C₀(γ+2) = O(1 + log γ). Hence K ≥ (2a)²/[(2a)² + 2a + D₁] = 1/(1 + 1/(2a) + D₁/(4a²)) ≥ 1 − 1/(2a) − D₁/(4a²). ∎(i) The D-049 error (F9): the hypothesis g_γ ≥ 4/a allows neighbors at distance ~1/a, whose terms are Θ(a²) (sin² phase can be ~1); the 1/g²-term of Lemma D was dropped. Isolation (distance ≥ 1) is what makes the rate work; the sin² phases of neighbors at distance ≥ 1 contribute ≤ 4 each, and unit-interval counting gives D₁.

(ii) Kill window W = 2: let γ₁,…,γ_J be the zeros within distance 2 of γ, G their evaluation Gram matrix, v = e_γ − P_J e_γ. Then v̂(γ_j) = 0, v̂(γ) = ‖v‖² (residual identity), and with k_j = ⟨e_γ, e_{γ_j}⟩ = 2a·sinc((γ−γ_j)a): since |γ−γ_j| ≥ 1/a can be assumed WLOG (closer pairs are near-multiples; treat a cluster within 1/a as ONE node — Hermite/derivative conditions, same bounds with the Hermite Gram), |k_j| ≤ 2a·(1/(a·δ_j)) ≤ 2/δ_j, so ‖k‖² ≤ 4Σ1/δ_j² ≤ 4J·max(1, a²g_min^{-2}…)— configuration-dependent; in the generic case δ_j ∈ [1/a, 2]: ‖k‖² ≤ 4J·a²·… — we carry ‖k‖² symbolically. ‖v‖² = 2a − k*G⁻¹k ≥ 2a − ‖G⁻¹‖‖k‖². Q(v): γ-term = ‖v‖⁴; nodes vanish. Survivors γ' (|γ'−γ| > 2): |v̂(γ')| ≤ |K(γ',γ)| + Σ_j|α_j||K(γ',γ_j)|, |α| ≤ ‖G⁻¹‖‖k‖ = O(1) in a (generic case). For survivors with min_j|γ'−γ_j| =: ε_j(γ') ≥ 1/a: |v̂(γ')| ≤ 2/(|γ'−γ|) + 2Σ|α_j|/ε_j(γ')·… ≤ O(1)/distance — summable by Lemma D to O((1+log γ)(1+‖G⁻¹‖²‖k‖²)). For survivors within 1/a of a node — possible only in the boundary annulus |γ'−γ| ∈ (2, 2+1/a] with the node on the outer rim — |v̂(γ')| ≤ ε·sup|v̂′| ≤ (1/a)·a·sup|v̂| ≤ sup|v̂| ≤ ‖v‖₁ ≤ √(2a)‖v‖₂ ≤ 2a√2, so each contributes ≤ 8a²·(1+o(1)) to Σ_rest. Write E(γ,a) := #{survivors within 1/a of a node} (usually 0; requires a zero in the thin outer annulus hugging a rim node). Assemble (non-degeneracy condition ‖G⁻¹‖‖k‖² ≤ a): K ≥ ‖v‖⁴/[‖v‖² + ‖v‖⁴ + Σ_rest] ≥ 1 − 1/(2a) − [C'(1+log γ)(1+‖G⁻¹‖²) + 8E(γ,a)·(1+o(1))]/a². So: the clean 1/a² rate holds whenever E(γ,a) = 0 (generic — in particular whenever no zero sits in the outer annulus (2, 2+1/a] near a rim node); in general the correction is the explicit constant 8E(γ,a). ∎(ii)

Remarks. (1) The rate 1 − 1/(2a) − O((1+log γ)/a²) is confirmed for isolated zeros; the measured floors ≈ 1.00–1.04 (E-022 family) sit in regime (i) at typical heights (mean spacing ≫ 1 up to γ ~ 10³). (2) Regime (ii)'s correction has two explicit pieces: the configuration constant ‖G⁻¹‖ (small unless the window zeros cluster at scale 1/a) and the annulus count E(γ,a) (zero unless a survivor hugs a rim node — a measure-zero-type event needing a zero in a 1/a-wide annulus adjacent to another zero in the rim). For clustered zeros the valleys themselves merge below resolution ~1/a and no per-zero floor is expected — the statement is tight in this sense. (3) This repair uses the SAME variational identity (I1); no new inputs.

2. B3 profile, corrected argument

The γ-term of Q for the bump at z is |v̂₀(γ)|² = 4sin²((z−γ)a)/(z−γ)² = (2a)²·sinc²((z−γ)a), sinc u = sin u/u. Theorem B3′. Assume RH, γ isolated as in B1′(i), |z−γ| ≤ 1/2: K_a(z,z) ≥ 1/(sinc²((z−γ)a) + 1/(2a) + D₁/(4a²)). (D-049's printed sinc²((z−γ)a/2) was a factor-2 error — the envelope below is the correct, weaker one; the first zero of the envelope sits at |z−γ| = π/a, not 2π/a.) Proof. As D-049 with the corrected argument; the isolation hypothesis disposes of the F9 issue identically. ∎

3. m_γ fix (upper bounds)

Under RH alone the zero-side identity gives Q(v) ≥ m_γ|v̂(γ)|², so K_a(γ,γ) ≤ 1/m_γ (exact). "= 1" holds for γ ≤ T₀ = 3×10¹² (Platt–Trudgian simplicity, computational) and unconditionally for the first ~10¹³ zeros. Statement-level fix; the B1/B2 proofs are otherwise unchanged.

4. Detector restatement (folds E-027/D-055)

The three-level measured picture (E-027) with proof status: (L1) Bare form: for any off-line pair (δ > 0), Q_W^a is indefinite — PROVABLE: the pair's rank-2 form u v† + v u† on the evaluation functionals e_{γ∓iδ} has det = |⟨u,v⟩|² − ‖u‖²‖v‖² < 0 (strict Schwarz — distinct points in a RKHS), and the background is PSD, so Q has a negative eigen-direction whenever the pair direction is not covered... precisely: in the 2D span{u, v} the pair form has a negative eigenvalue Re⟨u,v⟩ − |⟨u,v⟩| < 0, while the background contributes ≥ 0; indefiniteness of the TOTAL form on that 2D subspace follows when the background compressed to it does not cover the negative eigenvalue — measured to never happen (E-027: onset immediate, ~1e13 × floor). The clean universal statement "any off-line pair ⟹ Q_W^a not PSD for every a" is the Weil-positivity theorem itself (support (−a,a) suffices for large a; the E-027 content is that NO threshold appears even at fixed small a). (L2) Pencil T_a = I + B_a: loses positivity at a magnitude-set onset δ_pencil(a) ≪ π/(2a) (measured 0.11 @ a=4, 0.038 @ a=8). (L3) Flat-carrier probe*: negative quotient iff δ > π/(2a) — PROVED (D-050 exact computation, one test function, both signs settled for THAT direction) and measured to ≤ 0.9% (E-027 V2c). [ERRATUM 2026-08-19 (session 010, D-060): "≤ 0.9%" should read "within 0.02%–2.3% of π/(2a) — three of four cases within 0.5%, worst case 2.3% (a=4, γ≈51.4)". See D-055 erratum.] The D-050 sentence "produces negative form directions only if δ > π/(2a)" is hereby scoped to (L3).

5. Updated theorem tags (D-052 T-3 line)