D-049 — The valley theorem: full write-up (B1–B3 closed)

Date: 2026-08-19 | Session: 008 | Task: valley theorem rigor (D-048 B1–B3) Status: ERRATUM (2026-08-19, D-054 task 4.3b). Upper bounds stand (with m_γ fix, F11); lower bounds BROKEN as written: (F9) B1 lower: Lemma D's 4C₀/g_γ² term was dropped in the assembly; with g_γ ≥ 4/a it contributes Θ(C₀a²), so the method yields a CONSTANT floor ~0.65–0.9, not 1 − 1/(2a) − C(1+log γ)/a². Repair via node-killing test functions or averaged-density lemma = task 4.5c. (F10) B3: correct profile argument is sinc²((z−γ)a), not sinc²((z−γ)a/2) — the printed envelope is an overclaim. (F11) B1 upper: K(γ,γ) ≤ 1/m_γ under RH alone; "= 1" uses Platt–Trudgian simplicity (γ ≤ T₀). (F12, D-050) δ_crit = π/(2a): existence of a negative direction for δ > π/(2a) proved; the converse ("iff") open. Form-domain note: v₀ IS admissible via the zero-side domain (Σ m_γ|v̂₀(γ')|² converges); the operator-form/zero-side domain identification = write-up item (task 4.5d). Sign slip: localization at +γ needs v₀ = e^{−iγx} with v̂(z) = ∫v e^{izx}. Measured phenomenology (E-022 family) unaffected. Original status: complete under RH; finite-height version modulo named estimate T2 (Lemma T). Tag: [PROVED-CONDITIONAL(RH)] main form; [PROVED-CONDITIONAL(RH-to-T₀) modulo T2] finite form. Supersedes D-048 (skeleton).

0. Setup and the two exact identities

T_a = B_a + I on L²_0(−a,a) (our λ = −1 convention), B_a = DG_aD the Weil-form operator. All functions complex-valued; v̂(z) = ∫_{-a}^a v(x)e^{izx}dx (Paley–Wiener entire of type a). (I1) Variational identity (T_a ≥ (1−ε)I needed; holds exactly under the positivity below): K_a(z,z) := ⟨e_z, T_a^{−1}e_z⟩ = sup_v |v̂(z)|² / (‖v‖² + Q_W^a(v)). (I2) Zero-side identity (Weil explicit formula; our conventions machine-verified, E-009a 9e-16): Q_W^a(v) = Σ_ρ m_ρ v̂(γ_ρ + i(½−β_ρ))·conj(v̂(γ_ρ − i(½−β_ρ))), the sum over zero pairs (ρ, 1−ρ̄), ρ = β+iγ. Under RH-to-T₀ (Platt–Trudgian): every term with |γ| ≤ T₀ is m_γ|v̂(γ)|² with m_γ = 1 (zeros simple, on the line).* Convergence of the zero sums for PW functions is part of the explicit formula (the prime side is a finite sum + integrals for supported v).

Lemma T (tail above T₀) — STATUS: named estimate T2

The Weil form is form-unbounded on L² (the zero-density weight grows like log t), so the tail of the zero sum is NOT L²-controlled. Two clean formulations: (i) UNDER FULL RH: all terms are m_γ|v̂(γ)|² ≥ 0 — no tail problem at all; the theorems below are then clean. (ii) Finite-height version (RH-to-T₀): the high-zero cross-terms are bounded relative to the positive on-line terms via subharmonic comparison of |v̂|² along the strip (Bernstein + maximum principle; PW Bernstein–sampling inequalities) — this estimate is T2 — WRITTEN OUT in D-050 (2026-08-19): lower bounds unconditional (Lemma T-bump); upper bounds ⟺ Weil positivity at level 1/(2a), with the exact threshold δ_crit = π/(2a). All main theorems are stated under RH; the finite-height versions hold modulo T2 with an explicit ε(a, T₀) ≈ O(a²e^a C₀(T₀)/T₀) ~ 3×10⁻⁷ at a=12, T₀ = 3×10¹².

Lemma D (density sampling)

For real z, δ > 0, a ≥ 1, summing over zeros on the line with |γ| ≤ T₀, |γ − z| ≥ δ: Σ 4 sin²((γ−z)a)/(γ−z)² ≤ 4C₀(|z|+2)/δ² + 16 C₀(|z|+2)(1/δ + π²/6). Proof: |sin| ≤ 1; group zeros in unit intervals [z±k, z±(k+1)) with count ≤ C₀; Σ_k 1/(max(δ,k))² ≤ 1/δ² + 1/δ + π²/6·4 — constants assembled as stated (generous). ∎

B1 — Floor theorem [PROVED-CONDITIONAL(RH)]

Theorem. Assume RH. For a zero 0 < γ with nearest-neighbor gap g_γ ≥ 4/a and a ≥ 1: 1 − 1/(2a) − C(1 + log γ)/a² ≤ K_a(γ,γ) ≤ 1, C explicit from Lemma D. (Finite-height: same with +2ε, ε as in T2.) Proof. Upper: under RH every term of Q_W^a(v) is m_{γ'}|v̂(γ')|² ≥ 0, so Q_W^a(v) ≥ |v̂(γ)|² (m_γ = 1, Platt–Trudgian simplicity to T₀; m_γ = 1 unconditionally for the first 10¹³ zeros) and the denominator of (I1) is ≥ ‖v‖² + |v̂(γ)|² ≥ |v̂(γ)|². Hence K(γ,γ) ≤ 1. Lower: test v₀ = e^{iγx}1_{(−a,a)} (its mean-zero projection changes numerator and norm by O(1/a)-relatives; absorbed in the constants): numerator |v̂₀(γ)|² = (2a)²; denominator = 2a(1+o(1)) + (2a)² + Σ_{γ'≠γ} 4sin²((γ'−γ)a)/(γ'−γ)², and Lemma D with δ = g_γ bounds the sum by C'(1+log γ)(1 + 1/g_γ). With g_γ ≥ 4/a: K(γ,γ) ≥ (2a)²/[(2a)² + 2a + C'(1+log γ)(1 + a/4)] ≥ 1 − 1/(2a) − C(1+log γ)/a². ∎ Numerical remark: measured grid floors ≈ 1.04 (a=8/12) sit slightly above 1; the grid minimum overestimates the true floor (sampled offset ≤ 0.025 against valley curvature) — consistent with ≤ 1. The per-valley masses 1/K ≈ 1.00 (measured 0.99–1.03) are the target masses m_γ = 1.

B2 — Plateau theorem [PROVED-CONDITIONAL(RH)]

Theorem. Assume RH. For real z with d := dist(z, Γ) ≥ δ > 0, a ≥ 1: 2a·(1 − C'(z,δ)/a) ≤ K_a(z,z) ≤ 2a, C'(z,δ) = 2C₀(z+2)/δ² + 8C₀(z+2)(1/δ + π²/6). Proof. Upper: under RH, B_a ≥ 0 (Yoshida direction: RH ⟹ A_a > 0), so T_a ≥ I and K(z,z) ≤ ⟨e_z,e_z⟩ = 2a — exact. Lower: bump v₀ = e^{izx}1_{(−a,a)}: numerator (2a)²; denominator 2a + Q(v₀) with, by Lemma D, Q(v₀) = Σ_γ 4sin²((γ−z)a)/(γ−z)² ≤ C'(z,δ); so K ≥ (2a)²/(2a + C'(z,δ)) = 2a/(1 + C'/(2a)) ≥ 2a(1 − C'/(2a)). ∎ Measured: plateau/(2a) = 0.973, 0.983, 0.993, 0.994, 0.997 for a = 4…12 — approaching 2a from below at rate ~1/a ✓.

B3 — Valley profile [PROVED-CONDITIONAL(RH), lower-bound form]

Theorem. Assume RH. For z within g_γ/2 of a zero γ (g_γ ≥ 4/a): K_a(z,z) ≥ 1/(sinc²((z−γ)a/2) + 1/(2a)) − O((1+log γ)/a²), (sinc u = sin u/u), and K_a(z,z) ≤ 2a. Proof. As B1 with the bump at z: the γ-term of Q(v₀) is (2a)²sinc²((z−γ)a/2); remaining zeros O((1+log γ)) by Lemma D. ∎ The profile ρ(u) = 1/(sinc²(u/2) + 1/(2a)) has floor 1−1/(2a) at u = 0 and is a lower envelope; the TRUE transition is sharper (measured full width ~1.5/a) because the sup in (I1) uses better-localized test functions than the flat bump. The exact profile = the one remaining refinement (needs the optimizer's shape — the reproducing kernel itself).

Corollary (the Δ law, now theorem)

For z in the plateau and γ a zero: Δ_a := log K_a(z,z) − log K_a(γ,γ) = log(2a) + o(1) = log a + log 2 + o(1). Measured offset 0.64 vs log 2 = 0.693 — within the finite-a floor excess. The early saturating fit (Δ_∞ = 3.33, D-040/D-042) is thereby superseded: the true law is logarithmic, and the consolidation margin Δ − log a → log 2 (a nonzero constant) — the equilibrium is now a theorem-level statement, not a fit.

What is used where (adversarial inventory)