M1 — Mesoscopic-decay extremal function (working file)

Length

87,634 words

Claims

5 theorems

Status

Phase 11 WEEK 1 COMPLETE (2026-04-18, same session). Week 1 executed per Joker attack plan: D7 rigorously verified (Proposition J.D7, §47.1-§47.2) AND naive Idea #1 refuted (Theorem 47.4, §47.4). **(P2) is weakenable**: pointwise Re φ(1/2+iγ)≥0 for all γ is NOT needed — (P1) already gives it on |γ|≤T (indicator=0 at β=1/2), and the |γ|>T tail is handled by truncated Weil identity (Heath-Brown 1979; see §47.3 for Riemann-von-Mangoldt naive vs. truncated-Weil gap). **BUT naive Idea #1 fails**: for φ=V+αW with V Selberg-symmetric (V̂∈ℝ) and W Selberg-antisymmetric (Ŵ∈iℝ), the prime-sum decomposition |S_φ|²=|S_V|²+α²|S_W|² > |S_V|² (orthogonality of real and imaginary contributions in ℂ), so αW correction strictly WORSENS (P5) — which is the M1 bottleneck. Akhiezer-Krein escape (breaking Selberg symmetry) does NOT translate to (P5) escape. **Residual open problem — Idea #1-prime**: general complex-Fourier φ with φ̂=a+ib, a,b:[−L,L]→ℝ independent (NO symmetry constraint). SDP-scale optimization over (Re S_φ, Im S_φ) ∈ disk; ~10¹³ variables at T=10¹², requires physics-informed parameterization. **Revised Phase 11 Week 2+ priorities**: (1) TOP priority Idea #3 reverse-engineer φ_RH under RH (2-3 weeks, 3 informational scenarios), (2) SECONDARY Idea #2 localized-window M1 (~3-4 weeks, unaffected), (3) DEFERRED Idea #1-prime (SDP solver infrastructure, post-Phase-11), (4) UNCHANGED Ideas #4+#5 long-horizon. **Publishable Week 1 output**: Prop J.D7 + Thm 47.4 form a clean sharp-negative result (~8-page note): 'Pointwise (P2) in §15.2 can be weakened, but no antisymmetric-correction ansatz improves the M1 prime-sum bottleneck.' m1 §47 added (~270 lines); joker_phase11_20260418.md Week 1 section added (~130 lines). Phase 11 Week 2 begins with Idea #3.