← Session 9 — Phase 4 synthesis assault + paper draft cleanup
thinking (124 chars)
Committed. Report to user. The T2 write-up is done, with the exact threshold result. Summarize concisely and give the state.
T2 complete and committed (`685081c`). The valley theorem analysis is now fully closed out:
## D-050 — T2 results
**1. The exact negativity threshold.** For an off-line zero pair at height γ with deviation δ = β−½, the pair-term in the form evaluates exactly (carrier-tuned profile):
$$2\,\mathrm{Re}[U(-\delta)\overline{U(\delta)}] = -8\cosh^2(a\delta)\frac{\delta^2-\kappa^2}{(\delta^2+\kappa^2)^2},\quad \kappa=\pi/2a$$
— negative **iff δ > π/(2a)**. So the scale-a form is provably blind to off-line deviations below π/(2a); negativity turns on continuously past the threshold. (I caught and fixed a sign error in my first pass — the on-line limit must be positive, and the exact computation confirms the threshold rather than refuting it.)
**2. High-frequency directions can't corrupt low-z observables** — any direction living at height γ₀ has quotient numerator suppressed by 1/(γ₀−z)².
**3. The honest inventory**:
- Valley *lower* bounds: **unconditional** (bump tails directly summable; ~3×10⁻⁷ at a=12).
- *Upper* bounds (K(γ,γ) ≤ 1, K ≤ 2a): exactly **Weil positivity at level 1/(2a)** — free under RH, not provable from RH-to-T₀ alone, and *measured* (λ_a > 0 plus the valleys themselves).
**The payoff remark**: this makes the measured valley picture a *numerical verification of Weil positivity* in the ℋ(T_a) metric to scale 12 — independent of the λ_a cliff measurements, and quantitative: an off-line pair with δ > π/(2a) would flatten its valley by a computable amount.
Session state: clean, committed, everything resumable. Queue: paper valley-theorem section + compile test (your TeX), L3/L4 library assembly, or fresh direction.
raw content_json (database row)
[{"type":"thinking","thinking":"Committed. Report to user. The T2 write-up is done, with the exact threshold result. Summarize concisely and give the state.","signature":""},{"type":"text","text":"T2 complete and committed (`685081c`). The valley theorem analysis is now fully closed out:\n\n## D-050 — T2 results\n\n**1. The exact negativity threshold.** For an off-line zero pair at height γ with deviation δ = β−½, the pair-term in the form evaluates exactly (carrier-tuned profile): \n$$2\\,\\mathrm{Re}[U(-\\delta)\\overline{U(\\delta)}] = -8\\cosh^2(a\\delta)\\frac{\\delta^2-\\kappa^2}{(\\delta^2+\\kappa^2)^2},\\quad \\kappa=\\pi/2a$$\n— negative **iff δ > π/(2a)**. So the scale-a form is provably blind to off-line deviations below π/(2a); negativity turns on continuously past the threshold. (I caught and fixed a sign error in my first pass — the on-line limit must be positive, and the exact computation confirms the threshold rather than refuting it.)\n\n**2. High-frequency directions can't corrupt low-z observables** — any direction living at height γ₀ has quotient numerator suppressed by 1/(γ₀−z)².\n\n**3. The honest inventory**:\n- Valley *lower* bounds: **unconditional** (bump tails directly summable; ~3×10⁻⁷ at a=12).\n- *Upper* bounds (K(γ,γ) ≤ 1, K ≤ 2a): exactly **Weil positivity at level 1/(2a)** — free under RH, not provable from RH-to-T₀ alone, and *measured* (λ_a > 0 plus the valleys themselves).\n\n**The payoff remark**: this makes the measured valley picture a *numerical verification of Weil positivity* in the ℋ(T_a) metric to scale 12 — independent of the λ_a cliff measurements, and quantitative: an off-line pair with δ > π/(2a) would flatten its valley by a computable amount.\n\nSession state: clean, committed, everything resumable. Queue: paper valley-theorem section + compile test (your TeX), L3/L4 library assembly, or fresh direction."}]