# Certified positivity of truncated Weil functionals in low support windows *Research note, 2026-07-23. All artifacts in `experiments/weil_positivity/` of https://github.com/Kuberwastaken/reimann; every result reproducible by the commands in §7. Status: unpublished working note; Lemmas proven; certified results are computer-assisted (Arb ball arithmetic); everything else clearly labeled numerical.* ## Summary Weil's criterion states that the Riemann Hypothesis is equivalent to positivity of the explicit-formula functional W on autocorrelations of test functions of arbitrary support. We study W on supports [−L, L] as L crosses the prime thresholds ½log p^k, and obtain: (1) a complete numerical regime map of the positivity mechanism, validated against the explicit-formula identity to 10⁻⁵–10⁻⁷ using the first 100,000 zeros; (2) two elementary sharp lemmas bounding the prime terms (disjoint-support ½-factor); (3) an exact reformulation of the first open case ("T1") as a relative operator inequality μ(L) > −1, with the extremal structure identified; (4) rigorous, ball-arithmetic-certified coercivity of the full Weil functional on explicit 14-dimensional families at L = 0.45, 0.50, 0.545 (one prime) and L = 0.60 (primes {2,3}) — the last being, to our knowledge, the first rigorous multi-prime Weil-positivity statement; and (5) two structural discoveries at the numerical level: *just-in-time rescue* (each truncation of the prime sum goes indefinite almost immediately beyond its window and is restored by the next prime) and *completion-criticality* (with all in-window primes included, λ_min sits at 0 to 5–6 digits at every window size — the finite-window face of the de Bruijn–Newman Λ = 0 sharpness). ## 1. Setup For real f with supp f ⊆ [−L, L], F(r) = ∫f(u)e^{iru}du, h = |F|², g = f⋆f̃, the Riemann–Weil formula (Montgomery normalization) gives Σ_γ h(γ) = 2h(i/2) + (1/2π)∫h(r)Ω(r)dr − 2Σ_{n≥2}(Λ(n)/√n)g(log n), Ω(r) = Re ψ(¼ + ir/2) − log π, γ over all nontrivial zeros (both signs, multiplicity; complex iff RH fails). RH ⟺ the right side ≥ 0 for all such f and all L, with the two pole directions removed (constraints ∫f e^{±u/2} = 0). The prime power n enters iff log n < 2L; [−L, L] is prime-free iff L < ½log 2. In the prime-free window positivity is the archimedean theorem of Connes–Consani (Selecta 2021). Our engine (`weil_form.py`, N = 48 orthonormal sine modes) computes both sides independently; they agree to relative Frobenius error 3·10⁻⁵ … 3·10⁻⁷ throughout (Odlyzko's 100k zeros), which certifies all conventions. ## 2. The regime map (numerical; `RESULTS.md`) margin(L) = λ_min of the archimedean form (constrained), total(L) = λ_min of the full form, deficit(L) = norm of the prime part: | regime | L | behavior | |---|---|---| | prime-free | < 0.3466 | margin = total: 1.105 → 0.55 | | perturbative | 0.3466–0.39 | margin > deficit; norm-perturbation suffices | | cancellative | 0.39–0.59 | deficit > margin > 0, total > 0: the prime's negative directions avoid the archimedean soft cone | | criticality | > 0.59 | margin < 0; total pinned at 0 to 10⁻⁶ out to L = 2 (margin −1.35, deficit 3.50) | Minimizers develop Fourier nodes at the actual zeros (node at γ₁ = 14.13 by L = 1.1; interlacing by L = 1.9): the near-null cone consists of compact-support functions whose transforms attempt to vanish on the zero set. ## 3. Lemmas (proven) **Lemma 1.** supp f ⊆ [−L, L], a ≥ L ⇒ |g(a)| ≤ ½‖f‖². *Proof.* g(a) = ⟨f·1_{[a−L,L]}, τ_a(f·1_{[−L,L−a]})⟩; the two windows are disjoint for a ≥ L; Cauchy–Schwarz + AM–GM on disjoint masses. ∎ Sharp (dipole extremals); hence ‖Q₂‖ ≤ (log 2)/√2 = 0.490129…, matching the measured saturation to six digits. **Lemma 1′.** For every in-window prime power with log n ≥ L (all of them, if L ≤ log 2): ‖Q_n‖ ≤ Λ(n)/√n; total prime norm ≤ Σ_{2≤n −1: *the prime transfer is a strict contraction in the archimedean metric*. Measured: μ = −0.31 (L=0.37) → −0.9917 (L=0.545). The extremal is a mixed core/dipole mode with spectral mass in the band (2π, γ₁) realizing ≈83% of its Lemma-1-allowed prime correlation; the sharpened analytic target is the uncertainty-type statement: *a function of support ≤ log 3 cannot be spectrally concentrated below γ₁ and simultaneously saturate its lag-log 2 autocorrelation.* At L = 0.58 (two-prime window): μ = −1.05, i.e. G + Q₂ alone is indefinite while G + Q₂ + Q₃ ≥ 0 — see §6. ## 5. Certified results (rigorous, computer-assisted) Pipeline (`certify.py`): regularized closed form F_k(r) = 2i^{1−k}w_k sincL(r−w_k)/(√L(w_k+r)) (no removable singularities); the archimedean kernel used as its analytic symmetrization (ψ(¼+ir/2)+ψ(¼−ir/2))/2 − log π (required for the soundness of Arb's rigorous integration); verified integration on [0, R] with explicit tail bounds; prime and pole entries by verified 1-D quadrature; exact-float family matrix Q; final bound by Cholesky executed in ball arithmetic. For every f in the explicit 14-dimensional families: | L | primes | certified W ≥ c‖f‖² | |---|---|---| | 0.450 | {2} | c = 0.072606 | | 0.500 | {2} | c = 0.014842 | | 0.545 | {2} | c = 0.001285 (full-space N=48 value: 0.001272) | | 0.600 | {2,3} | c = 0.000076 | Equivalently, by §1's identity: Σ_ρ ĥ_f(ρ) ≥ c‖f‖² over the actual nontrivial zeros, for every member of each family. These are subspace statements — T1 proper (all of L²[−L, L]) remains open; the L = 0.545 certificate nearly attains the full-space infimum, so the family is not the bottleneck there. **Certified rescue (L = 0.62, 22-dim family, primes {2,3}).** On one rigorously assembled matrix set: (a) an explicit vector with certified vᵀGv = −0.02776113 ± 5.8×10⁻⁹ (archimedean part indefinite on the family), and (b) certified W ≥ 2.0624×10⁻⁵‖f‖² on the whole family. Positivity created by the prime terms, machine-verified. ## 5b. Theorem A (proven + certified): explicit archimedean coercivity On the prime-free window (supp f ⊆ [−½log2, ½log2], pole constraints), the archimedean Weil form satisfies G(f) ≥ c₀‖f‖² with **certified c₀ ≥ 0.349152** (measured sharp value ≈ 0.55). Proof (`PROOF-c0.md`): Ω is strictly increasing (Im ψ′ < 0 termwise); the band-concentration of the constrained space is bounded by the trace identity λ_max ≤ Tr(B_R) − ρ₁(R) − ρ₂(R) with the constraint representers absorbing almost the whole trace; a stochastic-dominance rearrangement then bounds ∫Ω dν below by a certified Stieltjes sum. This is the quantitative floor missing from the literature (CC 2020 prove ≥ 0) — obtained by elementary means, independent of the prolate/Sonin machinery. ## 6. Structural discoveries (numerical) **Just-in-time rescue.** Truncations of the prime sum go indefinite essentially immediately beyond their own window (μ(0.58) = −1.05); the next prime term restores positivity. **Completion-criticality.** With all in-window primes, λ_min lands at 0 within 10⁻⁵ at *every* window (L = 0.6, 0.8, 1.0, 1.2 tested), while every proper truncation is strictly indefinite (to −0.5) and individual terms are non-monotone (adding n = 4 at L = 1.2 lowers λ_min). The explicit-formula weights are exactly the weights that complete every window to the boundary of positivity. Together these say: Weil positivity is an irreducibly collective property of the full Euler product — the quadratic-form counterpart of the convergence mechanism in the Connes–Consani–Moscovici zeta-spectral-triple program, and a natural induction structure (window-to-window propagation with zero slack) for any future proof. ## 7. Reproducibility ``` experiments/.venv/bin/python weil_form.py test|validate|sweep|profiles experiments/.venv/bin/python make_plots.py experiments/.venv/bin/python t1_blocks.py experiments/.venv/bin/python certify.py [N] # certified results, ~1 min each experiments/.venv/bin/python rescue_certify.py # rescue certification (L=0.62) ``` Data: `sweep.csv`, `t1_mu.txt`, `rescue_cascade.txt`, `certify_*_output.txt`. Environment: Python venv (numpy/scipy/mpmath/python-flint 0.9.0), Raspberry Pi 5. ## Relation to prior work The prime-free-window positivity is Connes–Consani (Selecta 27 (2021) 77); the small-eigenvalue degeneration at large support is their ζ-cycles phenomenon (Enseign. Math. 69 (2023)); our contribution is the prime-threshold-resolved anatomy (regimes, sharp constants, the μ reformulation, extremal structure) and the rigorous certificates with active prime terms, which we have not found elsewhere in the literature collected in this repository (178 sources; see `../../notes/` and `../../BREAKDOWN.md`).