[{"type":"thinking","thinking":"Session 002 begins (after shutdown/resume). Actually — per protocol, this is a new session. I should note we're now in Session 002: start a new journal file at the end. Let me proceed with Track C, task 3.1.\n\nPlan for this sub-stage:\n1. Web research: Yoshida 1992 (Weil form positivity on small support) — what's the current record for support [−a, a]? Known facts I recall: Weil's form positivity on functions with Fourier support in [−1,1] (i.e. test function support log X ≤ ...) relates to Montgomery's theorem. Yoshida [Yos92] proved positivity of W on functions with support in (−1, 1)? Hmm — actually the known result: W(f,f) ≥ 0 for f with supp f̂ ⊂ [−1,1]-ish? No — Montgomery's theorem (unconditional form) gives positivity for certain test functions... Let me research.\n\nThe precise statements: Weil's explicit formula: W(f) := Σ_ρ f̂(γ_ρ)... positivity of the Weil form W(g·g̃) where support conditions... Bombieri: W positive on all C²_c ⟺ RH. Yoshida: W(f) ≥ 0 if supp f ⊂ [−1,1]·(some normalization) — \"On Hermitian forms attached to zeta functions\" — proved positivity for functions supported in [−1/2, 1/2]? I recall Yoshida's result: the Weil form is positive on test functions with support in [−log 2, log 2] trivially (no primes), and up to [−1,1]-type ranges via Montgomery. There's also the concept that positivity for support [−a,a] for a ≤ 1 follows from Montgomery's unconditional pair correlation. Beyond a = 1 needs support>1 form factors — the SAME wall as Track A! Interesting — W6 again. Hmm: Weil positivity on support [−a,a] relates to pair correlation with support a. So 3.1 may hit W6 too — but the cross-pollination angle (3.4) is whether 2/3-density HELPS positivity statements, not necessarily extending support.\n\nResearch plan:\n1. WebFetch: Yoshida 1992 result + current state of \"Weil positivity bounded support\".\n2. Also Li criterion and de Bruijn–Newman quick state-of-the-art (for 3.2/3.3 context) — keep focused on 3.1 first.\n3. Then design E-006: numerical positivity horizon — smallest eigenvalue of truncated Weil form matrices vs support.\n\nWeil form numerical setup: W(f,g) = Σ_ρ m_ρ f̂(γ_ρ)ḡ̂(γ_ρ). Finite truncation: take zeros |γ| ≤ H (from E-002), test functions from a finite-dimensional family with Fourier support in [−a,a]... Equivalent to our G̃ construction but with varying support (bandwidth). The smallest eigenvalue of the d×d matrix as function of bandwidth a: our E-003 already computed G̃ at \"bandwidth 1\" (L = log(T/2π)) — PSD ✔. The positivity horizon: how big can L (= bandwidth) be before the finite matrix develops negative eigenvalues... but on true zeros (all on-line for our 10⁴) W is PSD by construction (sum of |f̂(γ)|² over real γ!). Hmm — with zeros exactly on the line, W(f,f) = Σ|f̂(γ)|² ≥ 0 trivially. The interest is the PRIME SIDE: Weil positivity ⟺ Σρ f̂(γ)² ≥ 0 via the prime-side expression. The nontrivial content: express W via primes and ask if the prime-side quadratic form is PSD — its failure would signal off-line zeros. For our numerics: W computed via zeros is automatically PSD; the horizon question becomes: how large can support be while the PRIME-SIDE evaluation (using only primes ≤ X) stays PSD? That's exactly the bandwidth wall again — E-006 should compute the Weil form BOTH ways (zeros side vs prime side) and compare! The prime-side matrix uses Σ_{n≤X} Λ(n)/√n terms; the zero-side uses zeros. Positivity of the prime-side form with support log X — for X beyond T the prime side involves... hmm, the prime side is finite-dimensional approximation.\n\nActually the cleaner E-006 design (positivity horizon): the paper's G̃ at grid critical density with bandwidth parameter λ: the zero-side matrix is always PSD on our data; the question is whether the PRIME-SIDE evaluation matches. At λ ≤ 1 they match (Theorem 5.7 / Montgomery). At λ > 1, the prime-side evaluation is NOT available (off-diagonal fails) — but numerically we can still COMPUTE the prime-side matrix entries exactly as finite prime sums (no error control, but computable!) and check whether they match the zero-side matrix. The deviation measures the off-diagonal prime-pair contribution — numerically probing W6! That's a genuinely interesting experiment: numerically evaluate the support-(1+δ) form factor from primes alone (finite X) and compare with zeros. This could reveal the *rate* at which the off-diagonal cancellation happens — empirical Hardy–Littlewood in the relevant regime! That's a nice E-006.\n\nBut wait — task 3.1 is about Weil positivity, and I should stay on task. The positivity question on growing support: Yoshida proved W ≥ 0 on small support unconditionally (via trivial prime bounds: the prime contribution is small for small X). The record: support up to log 2 is trivial (no prime powers ≤ X=2... Λ terms only n ≥ 2); support [−log2, log2] trivially positive; beyond that, primes enter. Yoshida's result: positivity for supp ⊂ [−1,1]·... need research. Let me do the literature step first, then design E-006 accordingly.\n\nLet me do the arXiv/literature research on Yoshida and Weil positivity support records.","signature":""},{"type":"text","text":"Starting **Session 002 — Track C, task 3.1**. First sub-stage: the literature state of Weil positivity on bounded supports (per §4A.3), then the numerical design."},{"type":"tool_use","id":"WebFetch_245","name":"WebFetch","input":{"prompt":"List paper titles, authors, years, and abstract summaries. I'm researching the Weil explicit formula Hermitian form (Weil's quadratic functional / Weil form) positivity: RH is equivalent to positivity on all compactly supported test functions. I need to know: (1) what Yoshida proved about positivity of the Weil form on test functions with bounded Fourier support (\"On Hermitian forms attached to zeta functions\" 1992), and the exact support range; (2) any later results pushing the support range for unconditional positivity; (3) any results connecting zero-density or partial RH results to the inertia/positive index of truncated Weil forms.","url":"http://export.arxiv.org/api/query?search_query=all:%22Weil%22+AND+all:%22Hermitian+form%22+AND+all:%22explicit+formula%22+AND+all:%22positivity%22&start=0&max_results=12&sortBy=relevance"},"extra_content":null}]