Message #475 · Riemann Hypothesis Research

← Session 9 — Phase 4 synthesis assault + paper draft cleanup

#475 assistant 2026-08-19 20:50:02.046 UTC · 914→7443 tok open ⤢

thinking (18293 chars)

The task: B1–B3 write-up for the valley theorem (D-048). Let me think carefully about what needs proving and do it properly. Setup: T_a = B_a + I on L²_0(−a,a) (λ=−1 convention), Q_W^a(v) = ⟨B_a v, v⟩ the Weil form on the interval. The variational identity: K_a(z,z) = sup_{v∈L²_0} |v̂(z)|² / (‖v‖² + Q_W^a(v)). The three bookkeeping items: - **B1** (one-zero removal defect): Q_W^a(v) ≥ |v̂(γ)|² − ε(v) with controlled ε. - **B2** (comb–smooth cancellation on bumps): |Q_W^a(v₀)| small for plateau bump v₀ = e^{izx}1_{(−a,a)}. - **B3** (valley shape): the profile of K near γ. The right framework for all three is the Weil explicit formula in the form we machine-verified (NT-1). Let me recall what the exact identity is for Q_W^a(v). From E-009a (the Gaussian test) and D-019: the Weil form on the interval: Q_W^a(v) = ⟨G_a Dv, Dv⟩ where G_a = convolution by g on the interval. And Suzuki's key identity (his Prop 3.1 / eq 8.3): ⟨G_a Dv, Dv⟩ = Q_W(v) for v ∈ C∞_c(−a,a) — the Weil quadratic form. The Weil explicit formula for the form: with the convention from our NT-1 (validated): for v smooth compactly supported, Q_W^a(v) = Σ_ρ m_ρ |v̂(ρ̃)|²? — actually the precise statement needs the archimedean term. The Weil explicit formula: Σ_ρ m_ρ F(ρ) = (prime side) + (archimedean side) where F(ρ) = v̂(...)... For the QUADRATIC form Q_W(v) = Σ_ρ m_ρ |v̂(i(½−ρ̄))|² — is this exact or does it have an archimedean integral remainder? Let me think about what's true. The Weil form (Bombieri's functional): Q_W(v) := Σ_ρ m_ρ |∫v(x)e^{i...}|² — Bombieri defines the form BY the zero side and the theorem is the explicit formula equating it to the prime+arch side. In Suzuki's setup: A_a is defined via the prime-side g, and the equality with the zero side is a theorem (under RH-ish conditions...). Actually, from D-019 and the E-009a validation: "Galerkin A_a Galerkin = validated Weil-form operator (ledger #1; E-009a/b, D-019). Weil formula conventions machine-verified (Gaussian test, 9e-16)". The Gaussian test: Q_W^a(v) computed via the prime-side g vs the zero-side sum — they matched to 9e-16 for a Gaussian test function. So the identity Q_W^a(v) = Σ_ρ m_ρ|v̂(γ-thing)|² (zero side, with the first 10⁴ zeros numerically) + small tail — holds numerically. The exact identity: Q_W^a(v) = Σ_ρ m_ρ |v̂(i(½−ρ))|² — wait, for real v... our form is on real functions? v real-valued: v̂(−z) = conj(v̂(z)). The zeros pair ρ, 1−ρ̄ = ρ̄'... The zero-side pairing: |v̂(i(½−ρ))|² terms pair up nicely. Under RH: i(½−ρ) = i(½−½−iγ) = γ — real: |v̂(γ)|² ✓. So the exact identity (under RH): Q_W^a(v) = Σ_γ m_γ |v̂(γ)|² — with NO background term — this is Bombieri's theorem essentially (the Weil form IS the comb sum). Hmm — but then the plateau analysis: Q_W^a(v₀) for the bump v₀ = e^{izx}1: Σ_γ 4sin²((γ−z)a)/(γ−z)² ≈ a·log(z/2π) — LARGE (order a·log) — so Q_W^a(v₀) ≈ a log z, and K(z,z) ≥ (2a)²/(2a + a log z) = 4a/(2+log z)·... — contradicts the measured plateau 2a! Hmm wait — maybe I have the variational formula's denominator wrong: T_a = B_a + I, so the form is ‖v‖² + Q_W^a(v) — with v real? Our space: L²_0 real or complex? The Galerkin uses real hats but complex loads b_z... The e_z = e^{−izx} is complex — so the space is complex L². Then v̂(z) with v complex... fine. Let me recompute the plateau more carefully. K(z,z) = sup_v |v̂(z)|²/(‖v‖² + Q_W^a(v)). Take v₀ = e^{izx}·w(x) where w is a smooth cutoff ≈ 1 on (−a,a) (smooth to keep v̂ localized): v̂₀(ξ) = ŵ(ξ−z): a bump of height ~2a, width ~1/a (for sharp cutoff) around ξ=z. Denominator: ‖v₀‖² ≈ 2a + Σ_γ m_γ |ŵ(γ−z)|². The sum Σ_γ |ŵ(γ−z)|²: ŵ(0) ≈ 2a (height); |ŵ(u)|² ≈ 4a²·sinc²(au)-ish (width 1/a). The sum over γ within |γ−z| ≲ 1/a: number of zeros in that window: ~ (1/a)·(1/2π)log(z/2π) ≈ 0.3/a — LESS THAN ONE for a ≥ 1! So typically NO zero within the bump's width: the sum is dominated by the tails: Σ_γ 4a²sinc²((γ−z)a) — with zeros every ~2: Σ ≈ 4a²·(1/2)Σ_γ 1/((γ−z)²a²)·(sin oscillation)... sinc²(au) = sin²(au)/(au)²: Σ_γ 4 sin²((γ−z)a)/(γ−z)²: with the average of sin² = 1/2 and zeros at spacing d ~ 2: Σ ≈ 2·∫ ρ(t) dt/(t−z)² = 2ρ̄·(π²/6·2)·... ~ O(ρ̄) = O(log z). So Q_W^a(v₀) ≈ O(log z) — SMALL vs ‖v‖² = 2a!! Wait — I made an arithmetic error before: 4sin²((γ−z)a)/(γ−z)² — at the NEAREST zero (distance ~1): 4·sin²(a)/1 — oscillates 0..4 — O(1). Sum over all: Σ 4sin²((γ−z)a)/(γ−z)²: split near (|γ−z| ≤ 1): O(few); far: Σ 2/(γ−z)² ≈ 2ρ̄·π²/6·... ~ O(log z) — TOTAL O(log z), NOT O(a·log z). I previously wrote "Σ ≈ 2a·ρ̄·π" — that was WRONG: I incorrectly used ∫sin²(au)/u² du = πa/2 as if the zeros sample the integral densely — but zeros are spaced ~2 apart while the kernel oscillates at scale 1/a ≪ 2 — the zeros sample the kernel SPARSELY: the sum ≈ (1/2)·Σ 4/(γ−z)²·(avg sin²) = 2Σ1/(γ−z)² = O(log z). The dense-sampling regime would need zeros per unit ≫ a — i.e., z with (1/2π)log(z/2π) ≫ a — z ≫ e^{2πa} — far outside our measured range. So the plateau: Q_W^a(v₀) = O(log z) ⟹ K ≥ (2a)²/(2a + O(log z)) = 2a(1 − O(log z /a)) ✓ plateau 2a confirmed with the correct estimate. And this ALSO explains the plateau prefactor drift 0.973 → 0.997: the correction is O(log z / a) = O(1/a) at fixed z — 1−c/a with c ~ log z·const/(2): at a=4: 0.973 ⟹ c ≈ 0.11·... hmm 1−0.973 = 0.027 ~ c/4: c ≈ 0.11; at a=12: 1−0.997 = 0.003 ~ 0.11/12 = 0.009 — eh, order right. And the floor: at z = γ₀: the bump v₀ = e^{iγ₀x}w: Q = Σ_γ |v̂₀(γ)|² = |v̂₀(γ₀)|² + Σ_{γ≠γ₀}|v̂₀(γ)|² = (2a)²(1+o(1)) + O(log γ₀). Denominator = 2a + (2a)² + O(log γ₀). K ≥ (2a)²/((2a)² + 2a + O(log γ₀)) = 1 − O(1/a) ✓. And the valley WIDTH: K(z,z) for z near γ₀: the numerator test function v₀ = e^{izx}w: the γ₀-term: |v̂₀(γ₀)|² = 4a²sinc²((z−γ₀)a)-ish = (2a)²·(1 − (z−γ₀)²a²/3·...): dips quadratically near z=γ₀ over width 1/a ✓. So the valley profile: K(z,z) ≈ (2a)²·sinc²((z−γ₀)a)... hmm more precisely the numerator/denominator interplay: K(z,z) ≈ sup over test functions... the achieved K(z,z) ≈ (2a)²sinc²((z−γ₀)a)/(2a + (2a)²sinc²((z−γ₀)a) + O(log)) — for |z−γ₀| ≳ 1/a: the sinc² term ≪ 2a: K ≈ 2a; for |z−γ₀| ≲ 1/a: transitions to 1. Width ~1/a ✓. Profile: Lorentzian-like in sinc² — matches the measured smooth dip ✓. So B2's correct statement: Q_W^a(v₀) = O(log z) for bumps — the key estimate is Σ_γ 4sin²((γ−z)a)/(γ−z)² = O(log z) (sparse sampling, NOT dense). And B1 is actually trivial in this framing: Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² ≥ m_γ₀|v̂(γ₀)|² directly (all terms positive, under RH) — no defect needed!! The upper bound K(γ,γ) ≤ 1/m_γ = 1 is immediate. Under RH the whole thing becomes nearly clean! So the B1-B3 write-up reduces to: - B1: trivial from positivity (under RH): Q ≥ single term. ✓ - B2: the estimate Σ_γ 4sin²((γ−z)a)/(γ−z)² = O(log z) — needs the zero-density bound (N(t) — Trudgian, proved) — clean lemma. ✓ - B3: valley profile from the sinc² kernel: K(z,z) for |z−γ| ≤ c/a: asymptotic formula. ✓ And the identity Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² — the zero-side of the explicit formula — needs: (i) its validity (Bombieri/Weil theorem — under RH for the positive comb; unconditionally the indefinite version), (ii) convergence of the sum for PW functions (v̂ decays... v ∈ L²(−a,a): v̂ entire of type a with ∫|v̂|² < ∞; the sum Σ|v̂(γ)|² converges since... v̂(γ) ~ values of an L² bandlimited function at density-log points: Σ|v̂(γ)|² ≤ C‖v̂‖² — sampling theorem for PW (Plancherel–Pólya: Σ|v̂(γ_k)|² ≤ C‖v̂‖² for separated sequences — but γ's are NOT separated (gaps → 0)! The sum Σ_γ|v̂(γ)|² can diverge?! For PW functions, sampling at a sequence with density > 1/(2a)·... the zeros have density ~ log T → ∞ — ANY PW_a function is sampled above its Nyquist density eventually — the sum Σ|v̂(γ)|² converges iff... hmm, actually for v̂ ∈ PW_a ∩ L²: |v̂(γ)|² summable over a sequence with density going to infinity — NOT guaranteed by L². But v̂ is entire of exp type with L² on ℝ — such functions satisfy Σ|v̂(λ_n)|² < ∞ for any sequence with |λ_n − λ_m| ≥ δ... for non-separated it can diverge logarithmically... BUT the explicit formula says the sum equals Q_W^a(v) which is finite (the prime side is a finite sum + integrals). So convergence is fine — the explicit formula forces it. OK. Also the sup in the variational identity: K = sup |v̂(z)|²/(‖v‖²+Q) — under RH, the map v ↦ v̂ restricted to PW_a with the norm ‖v‖² + Σ_γ|v̂(γ)|² — the sup is the evaluation functional's norm at z in the space with norm² = L² + comb. So the valley theorem, conditional on RH (or unconditional with the indefinite caveat), is now a clean theorem: **THEOREM (valley law).** Assume RH (or work to height T₀ with the correspondingly restricted statements). For the Suzuki Weil-form operator T_a = B_a + I on L²_0(−a,a), the evaluation-kernel diagonal K_a(z,z) = ⟨e_z, T_a^{−1}e_z⟩ satisfies, for a ≥ a₀ and |z| ≤ Z with a ≥ C log Z: (i) [plateau] |K_a(z,z) − 2a| ≤ C log(|z|+2) for dist(z, Γ) ≥ 1 (say); (ii) [floor] K_a(γ,γ) = 1 − O((1+log γ)/a) for each simple zero γ; (iii) [profile] the transition has width ~1/a with the sinc² shape. Proof: (i): K = sup |v̂(z)|²/(‖v‖²+Q): upper bound: K(z,z) ≤ ‖e_z‖²/λ_min(T)·... hmm need K ≤ 2a(1+o(1)): from K = sup_v ... ≥ the bump test gives lower bound 2a(1−O(log z/a)). Upper bound: K(z,z) ≤ ⟨e_z, e_z⟩/μ where... K = ⟨e_z, T^{-1}e_z⟩ ≤ ⟨e_z,e_z⟩ = 2a since T ≥ I (T = B + I, B ≥ 0 under RH!) — SO K ≤ 2a EXACTLY (T ≥ I) — and the prefactor approaching 2 from below measured ✓✓. CLEAN. (ii): floor: K(γ,γ) ≤ 1/m_γ from Q ≥ m_γ|v̂(γ)|² term: denominator ≥ ‖v‖² + m_γ|v̂(γ)|² ≥ m_γ|v̂(γ)|² ⟹ K ≤ 1. Lower bound via bump test: K(γ,γ) ≥ (2a)²/(2a + (2a)² + O(log γ))·(1+...) = 1 − O((1+log γ)/a) — wait the numerator for the lower bound test: |v̂₀(γ)|² = (2a)² exactly (v̂₀(γ) = ∫e^{iγx}e^{−iγx} = 2a ✓); denominator ‖v₀‖² + Q(v₀) = 2a + Σ_{γ'}|v̂₀(γ')|² = 2a + (2a)² + Σ_{γ'≠γ}4sin²((γ'−γ)a)/(γ'−γ)² = 2a + (2a)² + O(log γ). So K ≥ (2a)²/(2a + (2a)² + c log γ) = 1/(1 + (2a + c log γ)/(2a)²) = 1 − (2a + c log γ)/(2a)² + ... = 1 − 1/(2a) − O(log γ/a²) — so K(γ,γ) ≥ 1 − 1/(2a) − O(log γ/a²). Measured floor 1.04 — hmm, ≥ 1 − 1/(2a) = 0.96 at a=8; measured 1.04 — slightly ABOVE 1: the upper bound K ≤ 1 is violated?! K(γ,γ) = 1.04 > 1?? Hmm. The variational upper bound: K(γ,γ) = sup |v̂(γ)|²/(‖v‖²+Q(v)): since Q(v) = Σ m_γ'|v̂(γ')|² ≥ |v̂(γ)|² (m=1): denominator ≥ ‖v‖² + |v̂(γ)|² ≥ |v̂(γ)|² ⟹ ratio ≤ 1 ⟹ K ≤ 1. But measured 1.04?! Contradiction — unless: (a) the measured K is the GALERKIN one (discretization), (b) the λ=−1 convention: our T = Q_Gal + M corresponds to B + I — Q(v) ≥ 0 requires RH — the measured operator uses the TRUNCATED... no — established: the interval form sees all zeros exactly. Hmm — (c) m_γ = 1 assumed simple — true for first zeros ✓. (d) the variational identity for the COMPLEX e_z: K = ⟨e_z, T^{-1}e_z⟩ = sup_v |⟨e_z,v⟩|²/⟨Tv,v⟩ ✓ for any PSD T. ⟨e_z,v⟩ = ∫e^{-izx}v̄?? conventions — |⟨e_z,v⟩| = |v̂(±z)| fine. So K(γ,γ) ≤ 1 should hold EXACTLY in the continuum and approximately in Galerkin... measured 1.04 at a=8 (floor mean 0.038 → e^0.038 = 1.039). 4% above 1. Is the Galerkin discretization responsible (4% at h=0.01?) — the refinement test E-022 V2: K stable to 1.1e-3 at a=4 — so 4% is NOT discretization. Hmm. OR the upper bound is K ≤ 1/m_γ + something... let me recheck: K = sup_v |v̂(γ)|²/(‖v‖² + Q(v)): Q(v) = Σ m|v̂(γ')|² ≥ |v̂(γ)|²: so denominator ≥ ‖v‖² + |v̂(γ)|²: ratio ≤ |v̂(γ)|²/(‖v‖²+|v̂(γ)|²) ≤ 1 ✓ rigorous. Measured 1.039 > 1 — so either the measured K isn't the continuum K (Galerkin/truncation effects) or my identity Q = Σ|v̂(γ)|² is missing a term (a NEGATIVE background: if Q = Σ|v̂(γ')|² − (background integral), then Q could be < the single term and K > 1 possible!). The measured floor 1.04 — slightly above 1 — actually SUGGESTS a small negative background. Interesting — the archimedean contribution to the form might be negative-ish. Whatever: at 4% level. For the theorem write-up: floor → 1 as a → ∞ with measured 4% excess at a=8–12; the excess direction (above 1) indicates the background sign — worth a remark. OK — so the B1–B3 write-up is genuinely closable: the three estimates are now clean lemmas with the zero-density input. Let me write the full proof carefully. Key lemmas: L1 (zero-density sampling): for any real z and a ≥ 1: Σ_γ 4 sin²((γ−z)a)/(γ−z)² ≤ C(1 + log(|z|+2)) — proof via N(t) bounds (Trudgian): split |γ−z| ≤ 1 (≤ C₀(z)·4·min(a²,1)-bounded... zeros within 1 of z: ≤ C₀ ≈ 0.433 log(z+2)+2.5: each term ≤ 4 min(a², 1/(γ−z)²)·... hmm if a zero is VERY close to z (within 1/a): term ≈ 4a² — big! For the plateau statement we need dist(z,Γ) ≥ δ: then near zeros contribute ≤ C₀·4/δ². Far: Σ_{|γ−z|>1} 4/(γ−z)² ≤ 8∫_1^∞ C₀(z+u)/u² du·... = O(log z). So Σ ≤ 4C₀(z)/δ² + C'log(z+2). Fine — plateau needs dist ≥ δ. For the FLOOR test-function denominator, the sum is Σ_{γ'≠γ}4sin²((γ'−γ)a)/(γ'−γ)² — same bound with z=γ: the nearest other zero at distance ≥ the actual gap (could be tiny — zeros can be close!) — but sin² kills close ones: 4sin²((γ'−γ)a)/(γ'−γ)² ≤ 4a²·(a(γ'−γ))²/((γ'−γ)²a²)... for |γ'−γ| ≤ 1/a: sin(x)/x ≈ 1: term ≈ 4a²·... wait: 4sin²((γ'−γ)a)/(γ'−γ)² with |γ'−γ| ≤ 1/a: sin((γ'−γ)a) ≈ (γ'−γ)a: term ≈ 4a² — HUGE?! If two zeros are within 1/a, the bump at γ also captures γ' fully: |v̂₀(γ')|² ≈ (2a)² — so the denominator gets 2(2a)² and the floor drops to ~1/2 — CORRECT physics: valleys merge when zeros are closer than 1/a (unresolved doublet). For the first ~135 zeros, min gap ~ 0.6-ish? At a=12: 1/a = 0.083: gaps ≫ 1/a for low zeros ✓ fine — the floor law holds for zeros with gap ≳ 1/a. State it. So B1: under RH, K(γ,γ) ≤ 1 EXACTLY (from PSD + single-term domination); note the measured 4% excess → remark on background sign. B2: plateau: K(z,z) ≤ 2a exactly (T ≥ I); lower bound via bump: K(z,z) ≥ (2a)²/(2a + Q(v₀)) with Q(v₀) ≤ 4C₀(z)/δ² + C' log(z+2) for dist(z,Γ) ≥ δ ⟹ K ≥ 2a(1 − (C₀(z)/δ² + C''log z)/a·...). B3: profile: K(z,z) for z near γ: using the two-sided bounds with the sinc² dependence: K(z,z) = (2a)²s̃²/((2a)²s̃² + 2a + O(log))·... approximately — the Lorentzian-sinc profile: K ≈ 2a·[1/(1 + (2a/(2a))·(...))]... more precisely: write u = (z−γ)a: K(z,z) ≈ (2a)²sinc²(u/2)²/(2a + (2a)²sinc²(u/2)²) = 2a·sinc²/(1 + 2a·sinc²/(2a))... let me just present the shape: K(z,z) ≈ 2a·σ(u)/(1+σ(u))·(1/...)... I'll write the profile function ρ(u) = (2a s²)/(2a + 2a s²... Hmm — let me simplify: with s = sinc((z−γ)a/2) (the bump's value at γ when centered at z): numerator (2as)² — wait no. Let me redo: test function centered at z (the evaluation point): v₀ = e^{izx}1: numerator |v̂₀(z)|² = (2a)² ALWAYS (bump at its own center). Denominator: 2a + Σ_γ' 4sin²((γ'−z)a)/(γ'−z)². For z near γ (within 1/a): the γ-term ≈ 4a²sinc²((z−γ)a/2)·4/4... 4sin²(Δa)/Δ² with Δ = z−γ: = 4a²·(sin(Δa)/(Δa))² = 4a² sinc²(Δa/2) (numpy sinc convention aside). So denominator ≈ 2a + 4a²sinc²(Δa/2) + O(log): K(z,z) ≥ (2a)²/(2a + 4a²sinc² + ...) = 2/(1 + 2a·sinc²(Δa/2))·... = (2a)/(1 + 2a·sinc²(Δa/2))·... hmm: (2a)²/(4a²sinc² + 2a) = (2a)²/(4a²(sinc² + 1/(2a))) = (1/sinc²)·1/(1+1/(2a·sinc²)) = 1/(sinc² + 1/(2a)). So K(z,z) ≈ 1/(sinc²((z−γ)a/2) + 1/(2a)) — at Δ=0: 1/(1+1/(2a)) = 1−1/(2a) ✓ floor; for |Δ| ≫ 1/a: sinc² → small oscillating: K ≈ 2a/(1+2a·sinc²) → oscillates between 2a/(1+2a·...)... hmm at sinc²=0 exactly: K = 2a·... 1/(0 + 1/2a) = 2a ✓ plateau. Between: Lorentzian-ish in sinc². The profile: K ≈ 1/(sinc²(u/2) + 1/(2a)) with u = (z−γ)a — a clean universal shape! Width: sinc²(u/2) = 1/(2a) at u/2 ≈ ... sinc(u/2) ≈ 1−u²/24: sinc² ≈ 1−u²/12 = 1/(2a) ⟹ u² ≈ 12/(2a) = 6/a ⟹ u ≈ 2.45/√a?? That gives width ~ a^{-1/2} not a^{-1} — hmm but that's only the near-center behavior; for the full profile, K reaches half-plateau when sinc² ~ small: sinc²(u/2) ≤ 1/(2a)·... at |u| ~ √(a/2)·2.4... The measured width was ~1.5/a (where logK < plateau−1). The profile 1/(sinc²+1/2a): logK = −log(sinc²+1/2a): plateau−1 ⟹ sinc² + 1/2a = e·... plateau logK = log 2a: condition logK < log 2a − 1 ⟹ 1/(sinc²+1/2a) < 2a/e ⟹ sinc² + 1/2a > e/(2a) ⟹ sinc² > (e−1)/(2a) ⟹ |sinc| > √((e−1)/2a) ≈ 0.93/√a: sinc(u/2) ≈ 2/u for large u: 4/u² > 1.71/(2a)·... u² < 8a/1.71·... hmm: 2/|u| > 0.93/√a ⟹ |u| < 2.15√a ⟹ |z−γ| < 2.15/√a — width ~ a^{−1/2}?! But measured width ~1.5/a. Discrepancy — because the lower-bound profile is only a lower bound (the true K transitions faster). The true profile: the sup over v can use better-concentrated test functions (not just the flat bump) — the achieved K is higher than the bump bound in the transition zone. The exact profile requires optimizing over test-function shapes — the true optimizer is the reproducing kernel itself... The measured width 1.5/a: the transition from floor to plateau over ~1.5/a — faster than the bump's sinc² transition. Fine — B3's full profile is an optimization; state the two-sided bounds and the measured width. Let me now write the D-049 (B1–B3 full write-up). Structure: - Setup and variational identity (from D-048). - Lemma B0 (zero-side identity under RH; unconditional indefinite version) + convergence via explicit formula. - B1 (floor): theorem + proof + the remark on the 4% excess (background sign). - B2 (plateau): theorem + proof (T ≥ I upper bound EXACT; lower bound via bump + density sampling lemma). - B3 (profile/width): two-sided analysis, universal profile lower bound, measured width note. - The Δ = log 2a law now theorem. - Conditional status: under RH (for PSD); unconditionally indefinite — and the detector remark. - What this yields: the K-valley law is essentially a THEOREM now (modulo the B0 identity's standing — which is the explicit formula = [PROVED literature] under RH). Also this SHARPENS D-048 and should be reflected: maybe write D-049 as the completed version. Also update the valley theorem claims in D-045 (plateau → 2a now has the exact upper bound T ≥ I). One more consideration: the identity Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² — is this EXACTLY right (no extra terms)? Bombieri's Weil functional: Q(v) = Σ_ρ m_ρ Φ(ρ)Φ(1−ρ̄)̄... with Φ(ρ) = ∫v... For v real on the interval and the form ⟨G_aDv, Dv⟩: from E-009a's Gaussian test — the zero-side sum over the first 10⁴ zeros matched the prime side to 9e-16 — with what exactly? Let me check E-009a's formulation quickly to get the exact identity we validated. Let me grep the E-009 script.

Starting the B1–B3 write-up. First, let me pin down the exact zero-side identity we validated in E-009a (conventions matter for the proof).

tool: Bash ok

input

{
  "command": "cd /d/Python/Quantum && grep -n \"zero.side\\|zeros\\|Q_W\\|explicit\" research/numerics/scripts/E-009-suzuki-galerkin.py | head -12 && sed -n '20,42p' research/numerics/scripts/E-009-suzuki-galerkin.py",
  "description": "Check E-009's validated identity conventions"
}

result 2026-08-19 20:50:02Z → 2026-08-26 21:53:30Z

=== Johansson @ 10091 ===
s arecursive machinery, of intricacy blowing up with n; [23, ﬁg. 6] moreover reports a loss of precision of ≈ 0.2 decimal place per step n (when working ex nihilo - i.e., using no Riemann zeros in input); only λn-values up to n ≈ 4000 were accessed that way until recently, with n = 10 5 attained by Johansson [18, § 4.2] (who states a loss of 1 bit ≈ 0.3 decimal place per step n). Still, the range (18) needed for new tests of RH stays far out of reach . All that motivates quests for simpler realizations of the Keiper–Li idea. But ﬁrst, as the asymptotic sensitivity to RH is the main property we will pro

=== Johansson @ 64443 ===
ction, and zeros computation at very large height , preprint (Oct. 2004), http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf 37 [17] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Prod- ucts, 5th ed. edited by A. Jeﬀrey, Academic Press (1994). [18] F. Johansson, Rigorous high-precision computation of the Hurwitz zeta function and its derivatives , Numer. Algor. 69 (2015) 253–270. [19] J.B. Keiper, Power series expansions of Riemann ’s ξ function, Math. Comput. 58 (1992) 765–773. [20] J.C. Lagarias, Li coeﬃcients for automorphic L-functions, Ann. Inst. Fou

=== Keiper @ 54 ===
arXiv:1703.02844v3  [math.NT]  3 Apr 2022 Discretized Keiper/Li approach to the Riemann Hypothesis Andr´ e Voros Institut de physique th´ eorique, Universit´ e Paris-Saclay, CEA, CNRS F-91191 Gif-sur-Yvette Cedex (France) E-mail: andre.voros@ipht.fr April 5, 2022 Abstract The Keiper–Li sequence {λn} is most sensitive to the Riemann Hypothesis asymptotically 

=== Keiper @ 277 ===
arXiv:1703.02844v3  [math.NT]  3 Apr 2022 Discretized Keiper/Li approach to the Riemann Hypothesis Andr´ e Voros Institut de physique th´ eorique, Universit´ e Paris-Saclay, CEA, CNRS F-91191 Gif-sur-Yvette Cedex (France) E-mail: andre.voros@ipht.fr April 5, 2022 Abstract The Keiper–Li sequence {λn} is most sensitive to the Riemann Hypothesis asymptotically ( n → ∞ ), but highly elusive both ana- lytically and numerically. We deform it to fully explicit se quences, simpler to analyze and to compute (up to n = 5 · 105 by G. Misguich). This also works on the Davenport–Heilbronn 

=== Keiper @ 1641 ===
he highest T up to which RH is veriﬁed (at a given time), that roughly sets T0 ≈ 2.4 · 1012 currently (as of 2004) . (1) 1 We will examine how novel sequential criteria might probe RH further, i.e., sense any zeros ρ′ = 1 2 +t + iT with t> 0 (violating RH) and T >T0. First (§ 1) we review the usual Keiper–Li constants {λn}n≥1. Their (real- valued) sequence is RH-sensitive and could detect such ρ′ forn /greaterorsimilarT 2/t (cur- rently: above ≈ 1025). But the λn involve derivatives of log ζ to any orders, which are most elusive both analytically and numerically (in which case th e diﬃculty steeply 

=== Keiper @ 2039 ===
′ forn /greaterorsimilarT 2/t (cur- rently: above ≈ 1025). But the λn involve derivatives of log ζ to any orders, which are most elusive both analytically and numerically (in which case th e diﬃculty steeply rises with n : only recently was n = 10 5 reached). We propose instead ( § 2) to modify the Keiper–Li deﬁnition, by discretiz- ing the above derivatives to ﬁnite diﬀerences while retaining RH-sensitivity. We thus obtain a variant sequence {Λ n} in the elementary explicit form (36), also more expedient to encode numerically (G. Misguich reached n = 5 · 105). In § 3 we conﬁrm analytically that the

=== Keiper @ 3738 ===
omputational stumbling block, and a purely logistic one (arbitrary-precision programs are fully ready, only computing power has to rise to the task). Hence overall, {Λ n} is easier to calculate, has a lower uncertainty-principle barrier, and its study only begins. This discre tized ver- sion of the Keiper–Li sequence thus ought to be more eﬃcient in the long run than the original {λn} for testing RH further. An Appendix describes a similar discretization for a variant (more sym - metrical) Keiper–Li sequence. 2 1 Main notations and background • 2ξ(·) : a completed zeta function, obeying Riemann’s Fu

=== Keiper @ 3933 ===
has a lower uncertainty-principle barrier, and its study only begins. This discre tized ver- sion of the Keiper–Li sequence thus ought to be more eﬃcient in the long run than the original {λn} for testing RH further. An Appendix describes a similar discretization for a variant (more sym - metrical) Keiper–Li sequence. 2 1 Main notations and background • 2ξ(·) : a completed zeta function, obeying Riemann’s Functional Equatio n, 2ξ(x) def = x(x − 1)π−x/2 Γ(x/2)ζ(x) ≡ 2ξ(1 −x) : (2) 2ξ is better normalized for us than Riemann’s ξ-function: 2ξ(0) = 2ξ(1) = 1. The symmetry in (2) makes us also denote x =

=== Keiper @ 5363 ===
 only, in which case k!! def = k(k − 2) · · ·1 for odd k >0, def = 2 (k+1)/2 Γ( 1 2k + 1)/√π for odd k ≷ 0 ( e.g., (−1)!! = 1, (−3)!! = −1). • (j,k,ℓ )! def = (j +k +ℓ)!/ (j!k!ℓ!) (3-index multinomial coeﬃcients). •B2m : Bernoulli numbers, B2m(·) : Bernoulli polynomials ( m = 0, 1, 2,... ). 1.1 The Keiper and Li coeﬃcients In 1992 Keiper [19] considered a real sequence {λn} of generating function ϕ(z) def = log 2ξ(M(z)) ≡ ∞∑ n=1 λK n zn, M (z) def = 1 1 −z, (5) 3 (we write λK n for Keiper’sλn), deduced that λK n ≡n−1 ∑ ρ [1 − (1 − 1/ρ)n], (6) and that RH ⇒ λK n > 0 ( ∀n), then wrote (without proof n

=== Keiper @ 5396 ===
k − 2) · · ·1 for odd k >0, def = 2 (k+1)/2 Γ( 1 2k + 1)/√π for odd k ≷ 0 ( e.g., (−1)!! = 1, (−3)!! = −1). • (j,k,ℓ )! def = (j +k +ℓ)!/ (j!k!ℓ!) (3-index multinomial coeﬃcients). •B2m : Bernoulli numbers, B2m(·) : Bernoulli polynomials ( m = 0, 1, 2,... ). 1.1 The Keiper and Li coeﬃcients In 1992 Keiper [19] considered a real sequence {λn} of generating function ϕ(z) def = log 2ξ(M(z)) ≡ ∞∑ n=1 λK n zn, M (z) def = 1 1 −z, (5) 3 (we write λK n for Keiper’sλn), deduced that λK n ≡n−1 ∑ ρ [1 − (1 − 1/ρ)n], (6) and that RH ⇒ λK n > 0 ( ∀n), then wrote (without proof nor elaboration): “In fact, if we 

=== Keiper @ 5550 ===
x multinomial coeﬃcients). •B2m : Bernoulli numbers, B2m(·) : Bernoulli polynomials ( m = 0, 1, 2,... ). 1.1 The Keiper and Li coeﬃcients In 1992 Keiper [19] considered a real sequence {λn} of generating function ϕ(z) def = log 2ξ(M(z)) ≡ ∞∑ n=1 λK n zn, M (z) def = 1 1 −z, (5) 3 (we write λK n for Keiper’sλn), deduced that λK n ≡n−1 ∑ ρ [1 − (1 − 1/ρ)n], (6) and that RH ⇒ λK n > 0 ( ∀n), then wrote (without proof nor elaboration): “In fact, if we assume the Riemann hypothesis, and further that t he zeros are very evenly distributed, we can show that λm ≈ logm 2 − log(2π) + 1 −γ 2 . ” [for λK m ; m 

=== Keiper @ 6600 ===
 proved the sharp equivalence (Li’s criterion) : RH ⇐ ⇒ λL n ≥ 0 for all n. (11) He also obtained λL n ≡ ∑ ρ [1 − (1 − 1/ρ)n], which in view of (6) entails λL n ≡nλ K n for all n = 1, 2,... ; (12) our superscripts K vs L will disambiguate λn whenever the factor n matters. 1.2 Probing RH through the Keiper–Li constants λn Li’s criterion (11) makes it clear that the Keiper–Li sequence is RH-sensitive: how eﬃciently can it then serve to test RH? 4 Known results actually imply that, beyond the present frontier (1) , the sequence {λn} may eﬀectively probe RH only in its tail n ≫ 1 and via its asymptotic 

=== Keiper @ 6667 ===
r all n. (11) He also obtained λL n ≡ ∑ ρ [1 − (1 − 1/ρ)n], which in view of (6) entails λL n ≡nλ K n for all n = 1, 2,... ; (12) our superscripts K vs L will disambiguate λn whenever the factor n matters. 1.2 Probing RH through the Keiper–Li constants λn Li’s criterion (11) makes it clear that the Keiper–Li sequence is RH-sensitive: how eﬃciently can it then serve to test RH? 4 Known results actually imply that, beyond the present frontier (1) , the sequence {λn} may eﬀectively probe RH only in its tail n ≫ 1 and via its asymptotic form for n → ∞. In 2000 Oesterl´ e [26, prop. 2] proved (but left u

=== Keiper @ 7192 ===
ts asymptotic form for n → ∞. In 2000 Oesterl´ e [26, prop. 2] proved (but left unpublished [6, § 2.3]) that Re ρ = 1 2 for all zeros with | Im ρ| ≤ T0 =⇒ λn ≥ 0 for all n ≤T 2 0, (13) and that under RH, [26, § 2] λL n =n( 1 2 logn +c) +o(n)n→∞ with c = 1 2 (γ − log 2π − 1), (14) which concurs with Keiper’s formula (7) but now assuming RH alone . In 2004 Ma´ slanka [22][23] computed a few thousandλL n-values numerically and inferred asymptotic conjectures on them for the case RH tru e. In 2004–2006, inspired by the latter (but unaware of [26]), we used the saddle-point method to draw an asymptotic c

=== Keiper @ 9497 ===
necessitates n /greaterorsimilar2T 2 0, currently implying n /greaterorsimilar1025. (18) It is then no wonder that published λn-plots (having n ≲ 7000) solely reﬂect the RH-true pattern (16) (already from n ≈ 30) [19][22]. So, whether one would take (11) or (15)–(16) to track violations o f RH, the Keiper–Li sequence {λn} only matters in its asymptotic tail {n ≫ 1}, where the alternative (15)–(16) rules (and enacts Li’s sign proper ty as well). At the same time, theλn are quite elusive analytically [8][10]. Numerically too (see Ma´ slanka [22][23] and Coﬀey [9]), their evaluation requires arecursive

=== Keiper @ 10306 ===
lues up to n ≈ 4000 were accessed that way until recently, with n = 10 5 attained by Johansson [18, § 4.2] (who states a loss of 1 bit ≈ 0.3 decimal place per step n). Still, the range (18) needed for new tests of RH stays far out of reach . All that motivates quests for simpler realizations of the Keiper–Li idea. But ﬁrst, as the asymptotic sensitivity to RH is the main property we will prove to generalize, we review that feature for {λn} itself. 1.3 Asymptotic analysis of {λn} (to be generalized) The following derivation of the large- n alternative (15)–(16) for λn readily settles the RH-false cas

=== Keiper @ 14608 ===
s. (3.721(1)), (4.421(1))]. So that all in all, λK n = 1 2 logn +c +o(1), c = 1 2 (γ − log 2π − 1) (under RH true) , (24) amounting to (14). □ 2 An explicit variant to the sequence {λK n } To alleviate the diﬃculties met with the original sequence {λn} (§ 1.2), we propose to deform it (speciﬁcally, Keiper’s form (5)) into a simpler one, still RH-sensitive but of elementary closed form . While the original speciﬁcation (5) for λn looks rigid, this is only due to an extraneous assumption implied on the mapping M: that the origin z = 0 has to map to x = 1 (the pole of ζ). Indeed, (5) at once built {λn}

=== Keiper @ 19856 ===
n = 1, 2,..., (36) 11 and the latter form is fully explicit : Anm are given by (32), and 2ξ(2m) = (−1)m+1B2m (2m − 3)!! (2π)m ≡ 2(−1)m+1B2m Γ(m − 1 2) πm+1/2, m = 0, 1,... (37) E.g., Λ 1 = 3 2 logπ 3, Λ 2 = 5 24 log [( 2 5 ) 7 311 π4 ] , Λ 3 = 21 80 log [ 525π8 23(32 · 7)11 ] . (38) So, we deformed Keiper’s {λK n } by discretizing the derivatives on log 2 ξ to ﬁnite diﬀerences anchored at locations 1, {2m} whereξ has known values, in a canonical way basically dictated by the preservation of RH-sensitivity .2 This gave {Λ n} the elementary closed form (36), which is moreover directly computable at an

=== Keiper @ 24143 ===
); hence this integral tends to 0 if R → ∞ while keeping far enough from ordinates of Riemann zeros in a classic fashion (i.e., so that ⏐ ⏐ ⏐1 ζ dζ dx ⏐ ⏐ ⏐(r + iR)<K log2R for all r ∈ [−1, +2], cf. [11, p. 108]), hence (44) results. □ 14 3 Resulting new sequential criterion for RH Like the elusive Keiper–Li sequence, the fully explicit one {Λ n} proves RH- sensitive (just slightly diﬀerently). The argument will use the funct ion Φ(X) def = X log ( 1 − 1 X 2 ) + logX + 1 X − 1 ≡ ∫ X ∞ log ( 1 − 1 Y 2 ) dY (46) ≡ (X + 1) log(X + 1) + (X − 1) log(X − 1) − 2X logX (47) made single-valued on C \ [−1, 1]

=== Keiper @ 38460 ===
values for Lχ (and ξχ) at all even, resp. odd, positive integers ac- cording to the even, resp. odd parity of χ. Moreover, Lχ(1) (hence ξχ(1)) is explicitly computable also for even parity [11, chap. 1][34, eq.(10.70)]. 22 • Generalized Riemann Hypothesis (GRH): all zerosρ ofξχ have Re ρ = 1 2. • A Keiper–Li sequence {λχ,n} exists for ξχ in full similarity to {λn} for ξ (applying [8, Cor. 1]). For GRH, the only change in our asymptotic alte rna- tive (15)–(16) aﬀects the constant c in (16) or (24) due to the replacement, in the asymptotic counting function N (T ) from (3) used in § 1.3.2, of the ter

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[{"type":"thinking","thinking":"The task: B1–B3 write-up for the valley theorem (D-048). Let me think carefully about what needs proving and do it properly.\n\nSetup: T_a = B_a + I on L²_0(−a,a) (λ=−1 convention), Q_W^a(v) = ⟨B_a v, v⟩ the Weil form on the interval. The variational identity:\nK_a(z,z) = sup_{v∈L²_0} |v̂(z)|² / (‖v‖² + Q_W^a(v)).\n\nThe three bookkeeping items:\n- **B1** (one-zero removal defect): Q_W^a(v) ≥ |v̂(γ)|² − ε(v) with controlled ε.\n- **B2** (comb–smooth cancellation on bumps): |Q_W^a(v₀)| small for plateau bump v₀ = e^{izx}1_{(−a,a)}.\n- **B3** (valley shape): the profile of K near γ.\n\nThe right framework for all three is the Weil explicit formula in the form we machine-verified (NT-1). Let me recall what the exact identity is for Q_W^a(v).\n\nFrom E-009a (the Gaussian test) and D-019: the Weil form on the interval:\nQ_W^a(v) = ⟨G_a Dv, Dv⟩ where G_a = convolution by g on the interval. And Suzuki's key identity (his Prop 3.1 / eq 8.3): ⟨G_a Dv, Dv⟩ = Q_W(v) for v ∈ C∞_c(−a,a) — the Weil quadratic form.\n\nThe Weil explicit formula for the form: with the convention from our NT-1 (validated): for v smooth compactly supported,\nQ_W^a(v) = Σ_ρ m_ρ |v̂(ρ̃)|²? — actually the precise statement needs the archimedean term. The Weil explicit formula: \nΣ_ρ m_ρ F(ρ) = (prime side) + (archimedean side)\nwhere F(ρ) = v̂(...)... For the QUADRATIC form Q_W(v) = Σ_ρ m_ρ |v̂(i(½−ρ̄))|² — is this exact or does it have an archimedean integral remainder?\n\nLet me think about what's true. The Weil form (Bombieri's functional): Q_W(v) := Σ_ρ m_ρ |∫v(x)e^{i...}|² — Bombieri defines the form BY the zero side and the theorem is the explicit formula equating it to the prime+arch side. In Suzuki's setup: A_a is defined via the prime-side g, and the equality with the zero side is a theorem (under RH-ish conditions...). \n\nActually, from D-019 and the E-009a validation: \"Galerkin A_a Galerkin = validated Weil-form operator (ledger #1; E-009a/b, D-019). Weil formula conventions machine-verified (Gaussian test, 9e-16)\". The Gaussian test: Q_W^a(v) computed via the prime-side g vs the zero-side sum — they matched to 9e-16 for a Gaussian test function. So the identity Q_W^a(v) = Σ_ρ m_ρ|v̂(γ-thing)|² (zero side, with the first 10⁴ zeros numerically) + small tail — holds numerically. The exact identity:\n\nQ_W^a(v) = Σ_ρ m_ρ |v̂(i(½−ρ))|² — wait, for real v... our form is on real functions? v real-valued: v̂(−z) = conj(v̂(z)). The zeros pair ρ, 1−ρ̄ = ρ̄'... The zero-side pairing: |v̂(i(½−ρ))|² terms pair up nicely. Under RH: i(½−ρ) = i(½−½−iγ) = γ — real: |v̂(γ)|² ✓.\n\nSo the exact identity (under RH): Q_W^a(v) = Σ_γ m_γ |v̂(γ)|² — with NO background term — this is Bombieri's theorem essentially (the Weil form IS the comb sum). Hmm — but then the plateau analysis: Q_W^a(v₀) for the bump v₀ = e^{izx}1: Σ_γ 4sin²((γ−z)a)/(γ−z)² ≈ a·log(z/2π) — LARGE (order a·log) — so Q_W^a(v₀) ≈ a log z, and K(z,z) ≥ (2a)²/(2a + a log z) = 4a/(2+log z)·... — contradicts the measured plateau 2a!\n\nHmm wait — maybe I have the variational formula's denominator wrong: T_a = B_a + I, so the form is ‖v‖² + Q_W^a(v) — with v real? Our space: L²_0 real or complex? The Galerkin uses real hats but complex loads b_z... The e_z = e^{−izx} is complex — so the space is complex L². Then v̂(z) with v complex... fine.\n\nLet me recompute the plateau more carefully. K(z,z) = sup_v |v̂(z)|²/(‖v‖² + Q_W^a(v)). Take v₀ = e^{izx}·w(x) where w is a smooth cutoff ≈ 1 on (−a,a) (smooth to keep v̂ localized): v̂₀(ξ) = ŵ(ξ−z): a bump of height ~2a, width ~1/a (for sharp cutoff) around ξ=z. Denominator: ‖v₀‖² ≈ 2a + Σ_γ m_γ |ŵ(γ−z)|².\n\nThe sum Σ_γ |ŵ(γ−z)|²: ŵ(0) ≈ 2a (height); |ŵ(u)|² ≈ 4a²·sinc²(au)-ish (width 1/a). The sum over γ within |γ−z| ≲ 1/a: number of zeros in that window: ~ (1/a)·(1/2π)log(z/2π) ≈ 0.3/a — LESS THAN ONE for a ≥ 1! So typically NO zero within the bump's width: the sum is dominated by the tails: Σ_γ 4a²sinc²((γ−z)a) — with zeros every ~2: Σ ≈ 4a²·(1/2)Σ_γ 1/((γ−z)²a²)·(sin oscillation)... sinc²(au) = sin²(au)/(au)²: Σ_γ 4 sin²((γ−z)a)/(γ−z)²: with the average of sin² = 1/2 and zeros at spacing d ~ 2: Σ ≈ 2·∫ ρ(t) dt/(t−z)² = 2ρ̄·(π²/6·2)·... ~ O(ρ̄) = O(log z). So Q_W^a(v₀) ≈ O(log z) — SMALL vs ‖v‖² = 2a!! \n\nWait — I made an arithmetic error before: 4sin²((γ−z)a)/(γ−z)² — at the NEAREST zero (distance ~1): 4·sin²(a)/1 — oscillates 0..4 — O(1). Sum over all: Σ 4sin²((γ−z)a)/(γ−z)²: split near (|γ−z| ≤ 1): O(few); far: Σ 2/(γ−z)² ≈ 2ρ̄·π²/6·... ~ O(log z) — TOTAL O(log z), NOT O(a·log z). I previously wrote \"Σ ≈ 2a·ρ̄·π\" — that was WRONG: I incorrectly used ∫sin²(au)/u² du = πa/2 as if the zeros sample the integral densely — but zeros are spaced ~2 apart while the kernel oscillates at scale 1/a ≪ 2 — the zeros sample the kernel SPARSELY: the sum ≈ (1/2)·Σ 4/(γ−z)²·(avg sin²) = 2Σ1/(γ−z)² = O(log z). The dense-sampling regime would need zeros per unit ≫ a — i.e., z with (1/2π)log(z/2π) ≫ a — z ≫ e^{2πa} — far outside our measured range. \n\nSo the plateau: Q_W^a(v₀) = O(log z) ⟹ K ≥ (2a)²/(2a + O(log z)) = 2a(1 − O(log z /a)) ✓ plateau 2a confirmed with the correct estimate. And this ALSO explains the plateau prefactor drift 0.973 → 0.997: the correction is O(log z / a) = O(1/a) at fixed z — 1−c/a with c ~ log z·const/(2): at a=4: 0.973 ⟹ c ≈ 0.11·... hmm 1−0.973 = 0.027 ~ c/4: c ≈ 0.11; at a=12: 1−0.997 = 0.003 ~ 0.11/12 = 0.009 — eh, order right.\n\nAnd the floor: at z = γ₀: the bump v₀ = e^{iγ₀x}w: Q = Σ_γ |v̂₀(γ)|² = |v̂₀(γ₀)|² + Σ_{γ≠γ₀}|v̂₀(γ)|² = (2a)²(1+o(1)) + O(log γ₀). Denominator = 2a + (2a)² + O(log γ₀). K ≥ (2a)²/((2a)² + 2a + O(log γ₀)) = 1 − O(1/a) ✓.\n\nAnd the valley WIDTH: K(z,z) for z near γ₀: the numerator test function v₀ = e^{izx}w: the γ₀-term: |v̂₀(γ₀)|² = 4a²sinc²((z−γ₀)a)-ish = (2a)²·(1 − (z−γ₀)²a²/3·...): dips quadratically near z=γ₀ over width 1/a ✓. So the valley profile: K(z,z) ≈ (2a)²·sinc²((z−γ₀)a)... hmm more precisely the numerator/denominator interplay: K(z,z) ≈ sup over test functions... the achieved K(z,z) ≈ (2a)²sinc²((z−γ₀)a)/(2a + (2a)²sinc²((z−γ₀)a) + O(log)) — for |z−γ₀| ≳ 1/a: the sinc² term ≪ 2a: K ≈ 2a; for |z−γ₀| ≲ 1/a: transitions to 1. Width ~1/a ✓. Profile: Lorentzian-like in sinc² — matches the measured smooth dip ✓.\n\nSo B2's correct statement: Q_W^a(v₀) = O(log z) for bumps — the key estimate is Σ_γ 4sin²((γ−z)a)/(γ−z)² = O(log z) (sparse sampling, NOT dense). And B1 is actually trivial in this framing: Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² ≥ m_γ₀|v̂(γ₀)|² directly (all terms positive, under RH) — no defect needed!! The upper bound K(γ,γ) ≤ 1/m_γ = 1 is immediate. Under RH the whole thing becomes nearly clean!\n\nSo the B1-B3 write-up reduces to:\n- B1: trivial from positivity (under RH): Q ≥ single term. ✓\n- B2: the estimate Σ_γ 4sin²((γ−z)a)/(γ−z)² = O(log z) — needs the zero-density bound (N(t) — Trudgian, proved) — clean lemma. ✓\n- B3: valley profile from the sinc² kernel: K(z,z) for |z−γ| ≤ c/a: asymptotic formula. ✓\n\nAnd the identity Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² — the zero-side of the explicit formula — needs: (i) its validity (Bombieri/Weil theorem — under RH for the positive comb; unconditionally the indefinite version), (ii) convergence of the sum for PW functions (v̂ decays... v ∈ L²(−a,a): v̂ entire of type a with ∫|v̂|² < ∞; the sum Σ|v̂(γ)|² converges since... v̂(γ) ~ values of an L² bandlimited function at density-log points: Σ|v̂(γ)|² ≤ C‖v̂‖² — sampling theorem for PW (Plancherel–Pólya: Σ|v̂(γ_k)|² ≤ C‖v̂‖² for separated sequences — but γ's are NOT separated (gaps → 0)! The sum Σ_γ|v̂(γ)|² can diverge?! For PW functions, sampling at a sequence with density > 1/(2a)·... the zeros have density ~ log T → ∞ — ANY PW_a function is sampled above its Nyquist density eventually — the sum Σ|v̂(γ)|² converges iff... hmm, actually for v̂ ∈ PW_a ∩ L²: |v̂(γ)|² summable over a sequence with density going to infinity — NOT guaranteed by L². But v̂ is entire of exp type with L² on ℝ — such functions satisfy Σ|v̂(λ_n)|² < ∞ for any sequence with |λ_n − λ_m| ≥ δ... for non-separated it can diverge logarithmically... BUT the explicit formula says the sum equals Q_W^a(v) which is finite (the prime side is a finite sum + integrals). So convergence is fine — the explicit formula forces it. OK.\n\nAlso the sup in the variational identity: K = sup |v̂(z)|²/(‖v‖²+Q) — under RH, the map v ↦ v̂ restricted to PW_a with the norm ‖v‖² + Σ_γ|v̂(γ)|² — the sup is the evaluation functional's norm at z in the space with norm² = L² + comb. \n\nSo the valley theorem, conditional on RH (or unconditional with the indefinite caveat), is now a clean theorem:\n\n**THEOREM (valley law).** Assume RH (or work to height T₀ with the correspondingly restricted statements). For the Suzuki Weil-form operator T_a = B_a + I on L²_0(−a,a), the evaluation-kernel diagonal K_a(z,z) = ⟨e_z, T_a^{−1}e_z⟩ satisfies, for a ≥ a₀ and |z| ≤ Z with a ≥ C log Z:\n(i) [plateau] |K_a(z,z) − 2a| ≤ C log(|z|+2) for dist(z, Γ) ≥ 1 (say);\n(ii) [floor] K_a(γ,γ) = 1 − O((1+log γ)/a) for each simple zero γ;\n(iii) [profile] the transition has width ~1/a with the sinc² shape.\n\nProof:\n(i): K = sup |v̂(z)|²/(‖v‖²+Q): upper bound: K(z,z) ≤ ‖e_z‖²/λ_min(T)·... hmm need K ≤ 2a(1+o(1)): from K = sup_v ... ≥ the bump test gives lower bound 2a(1−O(log z/a)). Upper bound: K(z,z) ≤ ⟨e_z, e_z⟩/μ where... K = ⟨e_z, T^{-1}e_z⟩ ≤ ⟨e_z,e_z⟩ = 2a since T ≥ I (T = B + I, B ≥ 0 under RH!) — SO K ≤ 2a EXACTLY (T ≥ I) — and the prefactor approaching 2 from below measured ✓✓. CLEAN.\n(ii): floor: K(γ,γ) ≤ 1/m_γ from Q ≥ m_γ|v̂(γ)|² term: denominator ≥ ‖v‖² + m_γ|v̂(γ)|² ≥ m_γ|v̂(γ)|² ⟹ K ≤ 1. Lower bound via bump test: K(γ,γ) ≥ (2a)²/(2a + (2a)² + O(log γ))·(1+...) = 1 − O((1+log γ)/a) — wait the numerator for the lower bound test: |v̂₀(γ)|² = (2a)² exactly (v̂₀(γ) = ∫e^{iγx}e^{−iγx} = 2a ✓); denominator ‖v₀‖² + Q(v₀) = 2a + Σ_{γ'}|v̂₀(γ')|² = 2a + (2a)² + Σ_{γ'≠γ}4sin²((γ'−γ)a)/(γ'−γ)² = 2a + (2a)² + O(log γ). So K ≥ (2a)²/(2a + (2a)² + c log γ) = 1/(1 + (2a + c log γ)/(2a)²) = 1 − (2a + c log γ)/(2a)² + ... = 1 − 1/(2a) − O(log γ/a²) — so K(γ,γ) ≥ 1 − 1/(2a) − O(log γ/a²). Measured floor 1.04 — hmm, ≥ 1 − 1/(2a) = 0.96 at a=8; measured 1.04 — slightly ABOVE 1: the upper bound K ≤ 1 is violated?! K(γ,γ) = 1.04 > 1?? Hmm. The variational upper bound: K(γ,γ) = sup |v̂(γ)|²/(‖v‖²+Q(v)): since Q(v) = Σ m_γ'|v̂(γ')|² ≥ |v̂(γ)|² (m=1): denominator ≥ ‖v‖² + |v̂(γ)|² ≥ |v̂(γ)|² ⟹ ratio ≤ 1 ⟹ K ≤ 1. But measured 1.04?! Contradiction — unless: (a) the measured K is the GALERKIN one (discretization), (b) the λ=−1 convention: our T = Q_Gal + M corresponds to B + I — Q(v) ≥ 0 requires RH — the measured operator uses the TRUNCATED... no — established: the interval form sees all zeros exactly. Hmm — (c) m_γ = 1 assumed simple — true for first zeros ✓. (d) the variational identity for the COMPLEX e_z: K = ⟨e_z, T^{-1}e_z⟩ = sup_v |⟨e_z,v⟩|²/⟨Tv,v⟩ ✓ for any PSD T. ⟨e_z,v⟩ = ∫e^{-izx}v̄?? conventions — |⟨e_z,v⟩| = |v̂(±z)| fine. So K(γ,γ) ≤ 1 should hold EXACTLY in the continuum and approximately in Galerkin... measured 1.04 at a=8 (floor mean 0.038 → e^0.038 = 1.039). 4% above 1. Is the Galerkin discretization responsible (4% at h=0.01?) — the refinement test E-022 V2: K stable to 1.1e-3 at a=4 — so 4% is NOT discretization. Hmm. OR the upper bound is K ≤ 1/m_γ + something... let me recheck: K = sup_v |v̂(γ)|²/(‖v‖² + Q(v)): Q(v) = Σ m|v̂(γ')|² ≥ |v̂(γ)|²: so denominator ≥ ‖v‖² + |v̂(γ)|²: ratio ≤ |v̂(γ)|²/(‖v‖²+|v̂(γ)|²) ≤ 1 ✓ rigorous. Measured 1.039 > 1 — so either the measured K isn't the continuum K (Galerkin/truncation effects) or my identity Q = Σ|v̂(γ)|² is missing a term (a NEGATIVE background: if Q = Σ|v̂(γ')|² − (background integral), then Q could be < the single term and K > 1 possible!). The measured floor 1.04 — slightly above 1 — actually SUGGESTS a small negative background. Interesting — the archimedean contribution to the form might be negative-ish. Whatever: at 4% level. For the theorem write-up: floor → 1 as a → ∞ with measured 4% excess at a=8–12; the excess direction (above 1) indicates the background sign — worth a remark.\n\nOK — so the B1–B3 write-up is genuinely closable: the three estimates are now clean lemmas with the zero-density input. Let me write the full proof carefully. Key lemmas:\n\nL1 (zero-density sampling): for any real z and a ≥ 1: Σ_γ 4 sin²((γ−z)a)/(γ−z)² ≤ C(1 + log(|z|+2)) — proof via N(t) bounds (Trudgian): split |γ−z| ≤ 1 (≤ C₀(z)·4·min(a²,1)-bounded... zeros within 1 of z: ≤ C₀ ≈ 0.433 log(z+2)+2.5: each term ≤ 4 min(a², 1/(γ−z)²)·... hmm if a zero is VERY close to z (within 1/a): term ≈ 4a² — big! For the plateau statement we need dist(z,Γ) ≥ δ: then near zeros contribute ≤ C₀·4/δ². Far: Σ_{|γ−z|>1} 4/(γ−z)² ≤ 8∫_1^∞ C₀(z+u)/u² du·... = O(log z). So Σ ≤ 4C₀(z)/δ² + C'log(z+2). Fine — plateau needs dist ≥ δ.\n\nFor the FLOOR test-function denominator, the sum is Σ_{γ'≠γ}4sin²((γ'−γ)a)/(γ'−γ)² — same bound with z=γ: the nearest other zero at distance ≥ the actual gap (could be tiny — zeros can be close!) — but sin² kills close ones: 4sin²((γ'−γ)a)/(γ'−γ)² ≤ 4a²·(a(γ'−γ))²/((γ'−γ)²a²)... for |γ'−γ| ≤ 1/a: sin(x)/x ≈ 1: term ≈ 4a²·... wait: 4sin²((γ'−γ)a)/(γ'−γ)² with |γ'−γ| ≤ 1/a: sin((γ'−γ)a) ≈ (γ'−γ)a: term ≈ 4a² — HUGE?! If two zeros are within 1/a, the bump at γ also captures γ' fully: |v̂₀(γ')|² ≈ (2a)² — so the denominator gets 2(2a)² and the floor drops to ~1/2 — CORRECT physics: valleys merge when zeros are closer than 1/a (unresolved doublet). For the first ~135 zeros, min gap ~ 0.6-ish? At a=12: 1/a = 0.083: gaps ≫ 1/a for low zeros ✓ fine — the floor law holds for zeros with gap ≳ 1/a. State it.\n\nSo B1: under RH, K(γ,γ) ≤ 1 EXACTLY (from PSD + single-term domination); note the measured 4% excess → remark on background sign.\nB2: plateau: K(z,z) ≤ 2a exactly (T ≥ I); lower bound via bump: K(z,z) ≥ (2a)²/(2a + Q(v₀)) with Q(v₀) ≤ 4C₀(z)/δ² + C' log(z+2) for dist(z,Γ) ≥ δ ⟹ K ≥ 2a(1 − (C₀(z)/δ² + C''log z)/a·...). \nB3: profile: K(z,z) for z near γ: using the two-sided bounds with the sinc² dependence: K(z,z) = (2a)²s̃²/((2a)²s̃² + 2a + O(log))·... approximately — the Lorentzian-sinc profile: K ≈ 2a·[1/(1 + (2a/(2a))·(...))]... more precisely: write u = (z−γ)a: K(z,z) ≈ (2a)²sinc²(u/2)²/(2a + (2a)²sinc²(u/2)²) = 2a·sinc²/(1 + 2a·sinc²/(2a))... let me just present the shape: K(z,z) ≈ 2a·σ(u)/(1+σ(u))·(1/...)... I'll write the profile function ρ(u) = (2a s²)/(2a + 2a s²... \n\nHmm — let me simplify: with s = sinc((z−γ)a/2) (the bump's value at γ when centered at z): numerator (2as)² — wait no. Let me redo: test function centered at z (the evaluation point): v₀ = e^{izx}1: numerator |v̂₀(z)|² = (2a)² ALWAYS (bump at its own center). Denominator: 2a + Σ_γ' 4sin²((γ'−z)a)/(γ'−z)². For z near γ (within 1/a): the γ-term ≈ 4a²sinc²((z−γ)a/2)·4/4... 4sin²(Δa)/Δ² with Δ = z−γ: = 4a²·(sin(Δa)/(Δa))² = 4a² sinc²(Δa/2) (numpy sinc convention aside). So denominator ≈ 2a + 4a²sinc²(Δa/2) + O(log): K(z,z) ≥ (2a)²/(2a + 4a²sinc² + ...) = 2/(1 + 2a·sinc²(Δa/2))·... = (2a)/(1 + 2a·sinc²(Δa/2))·... hmm: (2a)²/(4a²sinc² + 2a) = (2a)²/(4a²(sinc² + 1/(2a))) = (1/sinc²)·1/(1+1/(2a·sinc²)) = 1/(sinc² + 1/(2a)). So K(z,z) ≈ 1/(sinc²((z−γ)a/2) + 1/(2a)) — at Δ=0: 1/(1+1/(2a)) = 1−1/(2a) ✓ floor; for |Δ| ≫ 1/a: sinc² → small oscillating: K ≈ 2a/(1+2a·sinc²) → oscillates between 2a/(1+2a·...)... hmm at sinc²=0 exactly: K = 2a·... 1/(0 + 1/2a) = 2a ✓ plateau. Between: Lorentzian-ish in sinc². The profile: K ≈ 1/(sinc²(u/2) + 1/(2a)) with u = (z−γ)a — a clean universal shape! Width: sinc²(u/2) = 1/(2a) at u/2 ≈ ... sinc(u/2) ≈ 1−u²/24: sinc² ≈ 1−u²/12 = 1/(2a) ⟹ u² ≈ 12/(2a) = 6/a ⟹ u ≈ 2.45/√a?? That gives width ~ a^{-1/2} not a^{-1} — hmm but that's only the near-center behavior; for the full profile, K reaches half-plateau when sinc² ~ small: sinc²(u/2) ≤ 1/(2a)·... at |u| ~ √(a/2)·2.4... The measured width was ~1.5/a (where logK < plateau−1). The profile 1/(sinc²+1/2a): logK = −log(sinc²+1/2a): plateau−1 ⟹ sinc² + 1/2a = e·... plateau logK = log 2a: condition logK < log 2a − 1 ⟹ 1/(sinc²+1/2a) < 2a/e ⟹ sinc² + 1/2a > e/(2a) ⟹ sinc² > (e−1)/(2a) ⟹ |sinc| > √((e−1)/2a) ≈ 0.93/√a: sinc(u/2) ≈ 2/u for large u: 4/u² > 1.71/(2a)·... u² < 8a/1.71·... hmm: 2/|u| > 0.93/√a ⟹ |u| < 2.15√a ⟹ |z−γ| < 2.15/√a — width ~ a^{−1/2}?! But measured width ~1.5/a. Discrepancy — because the lower-bound profile is only a lower bound (the true K transitions faster). The true profile: the sup over v can use better-concentrated test functions (not just the flat bump) — the achieved K is higher than the bump bound in the transition zone. The exact profile requires optimizing over test-function shapes — the true optimizer is the reproducing kernel itself... The measured width 1.5/a: the transition from floor to plateau over ~1.5/a — faster than the bump's sinc² transition. Fine — B3's full profile is an optimization; state the two-sided bounds and the measured width.\n\nLet me now write the D-049 (B1–B3 full write-up). Structure:\n- Setup and variational identity (from D-048).\n- Lemma B0 (zero-side identity under RH; unconditional indefinite version) + convergence via explicit formula.\n- B1 (floor): theorem + proof + the remark on the 4% excess (background sign).\n- B2 (plateau): theorem + proof (T ≥ I upper bound EXACT; lower bound via bump + density sampling lemma).\n- B3 (profile/width): two-sided analysis, universal profile lower bound, measured width note.\n- The Δ = log 2a law now theorem.\n- Conditional status: under RH (for PSD); unconditionally indefinite — and the detector remark.\n- What this yields: the K-valley law is essentially a THEOREM now (modulo the B0 identity's standing — which is the explicit formula = [PROVED literature] under RH).\n\nAlso this SHARPENS D-048 and should be reflected: maybe write D-049 as the completed version. Also update the valley theorem claims in D-045 (plateau → 2a now has the exact upper bound T ≥ I).\n\nOne more consideration: the identity Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² — is this EXACTLY right (no extra terms)? Bombieri's Weil functional: Q(v) = Σ_ρ m_ρ Φ(ρ)Φ(1−ρ̄)̄... with Φ(ρ) = ∫v... For v real on the interval and the form ⟨G_aDv, Dv⟩: from E-009a's Gaussian test — the zero-side sum over the first 10⁴ zeros matched the prime side to 9e-16 — with what exactly? Let me check E-009a's formulation quickly to get the exact identity we validated. Let me grep the E-009 script.","signature":""},{"type":"text","text":"Starting the B1–B3 write-up. First, let me pin down the exact zero-side identity we validated in E-009a (conventions matter for the proof)."},{"type":"tool_use","id":"Bash_275","name":"Bash","input":{"command":"cd /d/Python/Quantum && grep -n \"zero.side\\|zeros\\|Q_W\\|explicit\" research/numerics/scripts/E-009-suzuki-galerkin.py | head -12 && sed -n '20,42p' research/numerics/scripts/E-009-suzuki-galerkin.py","description":"Check E-009's validated identity conventions"},"extra_content":null}]