darreal44/riemann-weil-phenomenology: v1.5 — The law: ℓ = 11·D_max

Abstract

Fifth tagged release. (A v1.4 tag exists on d965cf2, Grok's partial freeze of 5 September — 445 commits behind this one, no release; it is kept as history and superseded here.) The law. The depth of the well — the smallest eigenvalue of the windowed Weil form on the window $[0, L]$, $L = \log\mu$ — is a count: the eigenvalues of the zero Gram form a plunge ladder with near-universal rungs (16 → 5 nats, mean 11), and the number of exponentially small eigenvalues equals D_max = max_γ (γL/2π − N_Γ(γ)), the maximal lead of the window's Nyquist count over the zero count — a discrete Landau theorem, checked 10/10, 5/5, 3/3, 1/1 on ζ, χ₃, χ₅, χ₂₉. Hence ℓ ≈ 11.0 · D_max: within 3% on eight of twelve degree-1 windows (ζ 1.7%, χ₅ 0.3%, χ₂₉ 0%), 6% on two, 10% on three elliptic curves, with D_max read off the zero list and no fitted parameter but the mean rung. The marginal weight of each in-band zero is w(γ) ≈ 11 (1 − γ/γ_c), γ_c the Nyquist crossing (fourteen windows, server); zeros above the crossing are free. The two-term geometric formula, the τγ₁ line and the log-determinant are replaced by this count (notes/the-well.pdf v3, notebook §120–135). The well as one object. Desert (sampling floor), hyper-nullity and edge budget, spectral mass in the desert, super-exponential collapse of the ground state's autocorrelation at every lag — the "silence at the primes" of the quorum mechanism is not about primes — and the edge value: −ln λ₀ = 2(−ln|ψ(0)|) + O(1) (Grok), derived as the leakage of the edge jump onto the zeros beyond the band edge and verified on thirteen windows at 86–103% of the depth. Two independent chains (space side, zero side) agree. GL₂. The prime-side form of eight elliptic curves, built from a_p alone, agrees with each curve's zero Gram to the tail at three windows (five convention errors found by that judge); the four rank-1 curves require the central zero once on the constant mode and the rank-0 curves refuse it; depth desert-dominated, decreasing with conductor and rank; quorum complete on 11a1, 19a1, 32a1, 67a1 (to µ = 74) while 37a1 keeps a stable optional prime (the 3, +0.09 plateau to µ = 80 — a pre-registration killed, 39th): the degree-1 quorum theorem does not transfer verbatim; the quorum laws hold with a factor 2 for composite L-functions (notes/gl2-prime-side.pdf). Semi-local. Connes–Consani's archimedean operator rebuilt to every published digit; the compact-remainder mechanism does not cross the place 2, neither as is nor after renormalizing the units' log profile; the divergence is the sum of the 2-adic unit sub-shells, delocalized on the slice; the 2-adic peak sits at λ = 2^(±1) with exact shell masses derived by Grok (1/√2 vs √2 by convention); Q(µ = 3) certified positive definite without zeros (Arb). (notes/semilocal-step.pdf v4.) Bookkeeping. 283 tests, recomputing, green on two machines (tests/run_heavy.py runs the heavy files in parallel: 75 s on 32 cores); 27 notes; notebook report/le-milieu-des-premiers-v2.md 3191 lines, 138 sections, journal to 221, 39 pre-registered executions (thirteen of them the authors' own, dead), twenty artifact families (the last: a prime list hardcoded to 37 that made every window above µ = 41 an incomplete form — its non-positivity was the quorum). Task-level parallelism (--workers) is now standard. Status. Two theorems on the explicit forms (certified quorum, both halves; the 2×2 mechanism lemma), one under RH (the floor), one certificate without zeros beyond the prime-2 threshold. Everything else is measurement with stated numbers. The depth law is measured, not proved; its proof is a discrete Landau count and the universality of a ladder — pure harmonic analysis, no primes. Nothing here bears on RH. Contributors: D. Joubert, with the assistance of two language models (Claude, Grok). HEAD 1ab3477.