[{"type":"thinking","thinking":"Committed. Report to the user. Summarize the valley theorem's final form, the adversarial catch, and what's left (T2). Keep it tight. Also suggest next steps.","signature":""},{"type":"text","text":"B1–B3 write-up complete and committed (`e507386`). The valley theorem is now proved (D-049):\n\n## The valley theorem (final form, under RH)\n\nThe key was confirming from E-009a's validated identity that the form is a **pure comb**: Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² — no background term. Then everything follows from the variational identity K = sup_v |v̂(z)|²/(‖v‖² + Q(v)):\n\n- **Floor (B1)**: K(γ,γ) ≤ 1 *exactly* (single-term domination: Q(v) ≥ |v̂(γ)|²), and ≥ 1 − 1/(2a) − C(1+log γ)/a² (bump test + density sampling). The measured 1.04 floors are explained as grid-minimum upward bias — consistent.\n- **Plateau (B2)**: K(z,z) ≤ 2a *exactly* (T_a ≥ I under RH, Yoshida direction) and ≥ 2a(1 − C′/a). The measured approach 0.973 → 0.997 matches the 1/a rate.\n- **Profile (B3)**: universal lower envelope 1/(sinc²((z−γ)a/2) + 1/(2a)).\n- **Δ law is now a theorem**: Δ = log 2a + o(1). The early saturating fit (Δ_∞ = 3.33) is formally superseded — it was a model artifact.\n\n## The adversarial catch (why I check)\nMid-write-up I found that the Weil form is **form-unbounded on L²** (the zero-density weight grows like log t), so the finite-height tail above T₀ is *not* L²-controlled as I'd first written. Resolution: the clean theorems are stated under RH (tail positive, no issue), and the finite-height version is tagged modulo a named estimate **T2** (PW sampling + subharmonic comparison — standard complex-analysis assembly, not yet written out).\n\n## Status\n- Valley theorem: proved under RH; finite-height version needs T2 (~a page of standard estimates).\n- This makes the valley/locking/equilibrium phenomenology a theorem-level picture, and it pins the exact meaning of the consolidation obstruction (the margin Δ − log a → log 2, a constant — equilibrium is structural).\n- Queue: T2 write-up; paper section for the valley theorem + compile test (your TeX); L3/L4 assembly. \n\nGood stopping point, or continue with T2?"}]