RUNS

id: gauss_positivity-2026-10-03-main
hunt: gauss_positivity
started: 2026-10-03T09:00-05:00
finished: 2026-10-03T15:30-05:00
ran:
  - .venv/bin/python hunts/gauss_positivity/newton_battery.py --n-max 60 --d-max 10 --dps 60
  - .venv/bin/python hunts/gauss_positivity/turan_large_n.py --dps 60 --n-list 100,200,400,800,1600,3200
  - .venv/bin/python hunts/gauss_positivity/laplace_D.py --dps 80 --n-list 50,100,200,400,800,1600,3200
  - .venv/bin/python hunts/gauss_positivity/reproduce.py
outcome: the Turan inequality reduces exactly to a Cauchy-Schwarz deficit D_n <= 1/(n+1) on the moments of a positive weight; a saddle instrument reproduces D_n to 4-6 digits with 1/n error; the law n(r_n-1) -> 1 is measured to n=10^20; two counterexamples move the criterion from positivity to Schoenberg total positivity; the corrected Gate-3 rival is Davenport-Heilbronn, whose completed function has the complex t-zero 85.69934848537759 - 0.3085171824566374j, not W_a; Newton's inequalities are proved to follow from Turan, closing that family; and the higher-degree hyperbolicity route is closed from below by the measured de Bruijn-Newman smoothing t_eff(d) = |x0|/(8d), which makes low-degree scans unable to fail. No RH statement is claimed.
artifacts:
  - hunts/gauss_positivity/journal.md
  - hunts/gauss_positivity/RESULTS.md
  - hunts/gauss_positivity/reproduce.py
  - hunts/gauss_positivity/reproduce_results.json
  - hunts/gauss_positivity/turan_large_n.py
  - hunts/gauss_positivity/laplace_D.py
  - hunts/gauss_positivity/newton_battery.py
  - hunts/gauss_positivity/newton_results.json
  - hunts/gauss_positivity/modal_moments.py

Cost and compute notes

• No Modal run. The Modal token is absent on this machine (~/.modal.toml, MODAL_TOKEN_ID/MODAL_TOKEN_SECRET, ~/.config/modal*, the keychain and the shell rc files were all checked). modal_moments.py is written and ready but was never executed. Every number in RESULTS.md was computed locally from the committed zeta/li.py machinery.
• The expensive object a Modal run would have bought (b_n = ∫Φ u^{2n} at large n) was not needed: the exact relation b_n = (2n)!γ(n)/(64·4ⁿ·n!) was derived and confirmed (b_0 = Ξ(0)/8 = 0.0621400972735392637 to 30 digits), so the committed γ(n) table is the same data.
• Local wall time: newton_battery.py under 60 s; turan_large_n.py to n = 3200 under 5 min; laplace_D.py under 5 min; reproduce.py under 5 min. Nothing ran on a shared checkout and no long-lived process was left behind.

Known defects in this hunt's own artifacts

• modal_moments.py is untested against a live Modal endpoint. It is kept because it is the costed path for the next experiment, and it is labelled here as unexecuted rather than presented as working.
• newton_battery.py returns float margins; at d = 10, n = 50 the smallest is 1.2e-40, i.e. at the working precision. It was re-checked at dps 120 and is stable, so it is a genuine small margin rather than noise, but the file should carry interval arithmetic before any margin is quoted as decided.