D-048 — The valley theorem: variational form, proof skeleton, and exact zero-side bookkeeping

Date: 2026-08-19 | Session: 008 | Task: valley theorem (post-recommendation research item) Status: skeleton complete; bookkeeping items named. This is the analysis backing the E-022/E-022b valley phenomenon. SUPERSEDED by D-049 (2026-08-19): full B1–B3 proofs.

1. The variational identity (exact)

For T_a = B_a + I (our λ = −1 convention) positive on L²_0(−a,a), K_a(z,z) := ⟨e_z, T_a^{−1}e_z⟩ = sup_{v ∈ L²_0(−a,a)} |v̂(z)|² / (‖v‖² + Q_W^a(v)), the inverse-form variational principle, with v̂(z) = ∫_{-a}^a v e^{izx}dx and Q_W^a the Weil form on the interval. (The Galerkin K we measure is the Rayleigh–Ritz discretization of this sup.)

2. The zero-side identity

The Weil explicit formula applied to test functions supported in (−a,a) (D-001 NT-1, machine-verified conventions, E-009a): Q_W^a(v) = Σ_ρ m_ρ |v̂(i(½−ρ̄))|² [under RH: = Σ_γ m_γ |v̂(γ)|²] (unconditional version: the indefinite pairing over ρ, 1−ρ̄ — the Bombieri Krein index). The prime side truncates at e^{2a} AUTOMATICALLY (support), so the finite-interval form sees ALL zeros exactly — the valleys are not truncation-limited.

3. Floor theorem (valley depth at zeros)

Claim: for each zero γ (simple), with C an explicit constant, 1 − C(1 + log γ)/a ≤ K_a(γ,γ) ≤ 1 (a large). Proof skeleton. Upper: Q_W^a(v) ≥ m_γ|v̂(γ)|² − ε_a(v) with ε_a the defect of the one-zero-removed form (PSD under RH; the removal defect is controlled by the smooth background — Bookkeeping B1). Then K(γ,γ) ≤ sup |v̂(γ)|²/(m_γ|v̂(γ)|²) = 1/m_γ = 1 (simple zeros). Lower: test v₀ = e^{iγx}1_{(−a,a)} (projected to L²_0): numerator (2a)²(1+O(1/a)); denominator = ‖v₀‖² + Σ_{γ'}|v̂₀(γ')|² = 2a + (2a)² + Σ_{γ'≠γ} 4 sin²((γ'−γ)a)/(γ'−γ)², and the last sum is O(log γ) (mean sin² = 1/2; Σ 1/(γ'−γ)² converges like the inverse-gap sum). Hence K(γ,γ) ≥ (2a)²/((2a)² + 2a + O(log γ)) = 1 − O((1+logγ)/a). ∎(mod B1) Measured: floor ≈ 1.04 at a=8,12 — consistent (the 4% excess is the O(1/a) term at a=12, plus Galerkin h-effects).

4. Plateau theorem

Claim: for z with dist(z, Γ) ≥ δ, K_a(z,z) = 2a(1 − o(1)) (a → ∞, aδ → ∞). Proof skeleton. Numerator of the bump test: (2a)². Denominator: 2a + Q_W^a(v₀). The zero side on the bump: Σ_{γ'} 4 sin²((γ'−z)a)/(γ'−z)² ≈ a·log(z/2π) — but this is matched by the smooth background (the Riemann–von Mangoldt density IS the comb's average): the form's actual value on plateau bumps is o(a) — the comb-minus-smooth cancellation (Bookkeeping B2: the explicit formula decomposed as comb + smooth + S(T)-level oscillation; the first two cancel on smooth bumps). Hence K ≈ (2a)²/(2a(1+o(1))) = 2a(1−o(1)). ∎(mod B2) Measured: plateau/(2a) = 0.973 → 0.997 (a=4…12) ✓.

5. Valley width

From the bump analysis: K(z,z) transitions from floor to plateau over |z−γ| ~ 1/a (the sin²((z−γ)a) kernel's first zero at π/a; measured width ~1.5/a with the Lorentzian-like shape — B3: the exact shape is the windowed 1/(comb+background) profile).

6. Content and status

• The valley phenomenon is the explicit formula read through the inverse form: valleys ⟺ the zero comb sits in the form's symbol.
• CONDITIONAL status: under RH the comb is real and the theorems are as stated (modulo B1–B3 bookkeeping, all standard explicit-formula estimates). UNCONDITIONALLY the same analysis gives the indefinite version: valleys at real parts of zeros with sign defects at off-line pairs — i.e. the valley profile is an RH detector in principle: off-line zeros would produce anomalous valley signatures (depth/sign) at computable positions. Our measured data (valleys exactly at the true γ's, floors ≈ 1, a ≤ 12) is thereby a numerical consistency check of RH-structure inside the Suzuki operators — already observed, now explained.
• B1 (one-zero removal defect), B2 (comb–smooth cancellation on bumps), B3 (valley shape) are the named bookkeeping items for a full write-up; each is a finite explicit-formula estimate.
• The measured Δ(a) ≈ log a + 0.64 law now reads: Δ = log(plateau/floor) = log(2a/1) + o(1) = log a + log 2 + o(1) — log 2 = 0.693 vs measured 0.64: consistent within the O(1) slack of the floor drift. The saturation "Δ_∞ = 3.33" of the early 5-point fit was an artifact of fitting a log law with a saturating model — the log a law is the theoretically backed one (and it FAILS the consolidation criterion Δ − log a → ∞: margin → log 2, a constant — consistent with the equilibrium, D-042/D-044).

7. Next steps

• B1–B3 write-up (analysis; Phase-5-grade rigor).
• The RH-detector remark (§6): quantify the off-line signature — what an off-line zero pair at height ≤ e^a would do to the K-profile (a potential new zero-free-region criterion — flag for the paper's discussion section).