D-050 — T2 write-up: the finite-height tail, the phase threshold, and what positivity costs

Date: 2026-08-19 | Session: 008 | Task: T2 (D-049 finite-height tail) Status: complete (analysis) — with D-054 erratum (2026-08-19): the δ_crit statement is ONE-DIRECTION as proved — the tuned carrier exhibits a negative direction iff δ > π/(2a), but non-existence of negative directions for δ ≤ π/(2a) (other test functions) is unproved. Sharpened by E-027/D-055 (2026-08-19): as a statement about the bare form it is FALSE — the pair form is indefinite for ANY δ > 0 (strict Schwarz on the evaluation functionals; measured λ_min < 0 at the smallest scanned δ, ~1e13 × control floor). The correct statements: (i) tuned-carrier quotient threshold = π/(2a) EXACT (measured to ≤0.9% at a = 4, 8); (ii) bare-form onset immediate; (iii) T_a-pencil positivity loss magnitude-limited (δ ≈ 0.11 at a = 4, 0.038 at a = 8). Restatement = task 4.5c. Also note: the "lower bounds unconditional" conclusion below inherits D-049's F9/F10 errata (the Lemma-D assembly drops the 1/g² term; sinc² argument factor 2) — repair task 4.5c. Original conclusion: the valley theorem's LOWER bounds are unconditional (direct estimates); the UPPER bounds at finite height are exactly Weil-positivity-at-a statements (not provable from RH-to-T₀ alone; measured). Under full RH: everything clean (D-049 as written).

1. The pair-term phase analysis (exact computation)

For an off-line zero pair ρ = β+iγ, 1−ρ̄ (δ := β−½ ∈ (0,½)), the contribution to Q_W^a(v) is 2 m_ρ Re[ v̂(γ−iδ)·conj(v̂(γ+iδ)) ]. Take the carrier-shifted flat profile v(x) = e^{-i(γ+κ)x}1_{(−a,a)} (κ a free offset). Then v̂(γ±iδ) = 2 sinh(a(δ∓iκ))/(δ∓iκ)-type, and the pair term evaluates exactly to 2 Re[U(−δ)conj(U(δ))] = −8 cosh²(aδ)·(δ²−κ²)/(δ²+κ²)², which (for the tuned carrier κ = π/2a) is negative iff δ > π/(2a). Hence: an off-line zero pair at height γ produces negative form directions at scale a only if its deviation exceeds δ_crit = π/(2a); the negativity turns on continuously (∝ δ²−κ²). This is the quantitative content of finite resolution: the scale-a form is blind to off-line deviations below π/(2a). (The threshold matches the valley width scale 1.5/a up to the shape constant.)

2. High-frequency directions cannot corrupt low-z observables

For the variational quotient q_z(v) = |v̂(z)|²/(‖v‖² + Q(v)), |z| ≤ Z: a direction v whose v̂-mass sits at height γ₀ ≫ T₀ has |v̂(z)|² ≤ 4a²/(γ₀−z)²·(per-unit) — so q_z(v) ≤ 2a/(γ₀−z)²·η⁻¹ ≪ 2a whenever the denominator stays above η‖v‖²; and a denominator crossing zero makes q_z NEGATIVE (irrelevant for the sup). A mixed direction v = αv_hi + βv_lo has quotient denominator minimizing only through t = α/β = 0 unless the form's negative content reaches the threshold c₀ = 1/(2a) (§3). Hence: the upper-bound observables K(z,z) for z ≤ Z are sensitive only to the form's negativity at level ≥ 1/(2a) — small negativity, if any, moves nothing at our digits.

3. What the upper bounds cost (the honest inventory)

• B2's K(z,z) ≤ 2a and B1's K(γ,γ) ≤ 1 require T_a ≥ (1−ε)I with ε < 1/(2a): i.e. Weil positivity of the interval form at level 1/(2a). Under RH: free (Yoshida). Under RH-to-T₀ alone: NOT provable — an off-line pair above T₀ with δ > π/(2a) and the wrong phase could defeat it (§1), and nothing known rules that out.
• It IS measured: λ_a > 0 (E-009b…E-025, floor-caveated) and the valleys themselves (their existence at depth ≈ 1 IS the positivity signature at the γ-frequencies). So the measured valley picture is evidence of Weil positivity at scale a ≤ 12 — the detector remark (D-048 §6) made quantitative: an off-line pair at height γ with δ > π/(2a) would flatten/invert the valley at γ by a computable amount.
• The LOWER bounds (B1: K(γ,γ) ≥ 1 − 1/(2a) − …; B2: K ≥ 2a(1−C'/a); B3: the profile envelope) are UNCONDITIONAL given RH-to-T₀: they use only the bump test functions, whose tails are directly summable (|v̂₀(γ+iσ)| ≤ 2a e^{aσ}/|γ−z|; Σ_{γ>T₀} ≤ 4a²e^a·4C₀(T₀)/T₀ ≈ 3×10⁻⁷ at a=12 — Lemma T-bump, PROVED).

4. Final status of the valley theorem