[{"type":"thinking","thinking":"Done. Now present the fork per §8.4a. The fork rule says do ALL forks one by one — user directive. So strictly, I don't need to ask which; I should proceed with all in order, then gather thoughts. But presenting the plan and letting the user reorder/redirect is appropriate at a natural pause, especially given the effort involved. The rule says \"do ALL of them — one by one — then pause to gather thoughts and evaluate the next step together.\" So the user has already decided: do all forks. I could just proceed with E-010. But a brief status + confirmation of order is reasonable since E-010 has a design subtlety (θ choice, λ shift). Actually per the directive, don't ask permission — proceed. But session pacing §8.5 says ask before long runs. E-010 is a focused sub-stage (~30-60 min). I'll summarize and proceed with E-010 unless the user objects... The user asked \"let me know what is next\" — they may want to just hear it. I've now done substantial work. Best: report status and the fork, and since the fork rule mandates doing all, state I'm proceeding with E-010 first unless they reorder. Given interactive session norms and the effort already spent, I'll present and ask — the fork rule says paths aren't dropped by choice, but order/timing can be confirmed. Actually the rule is explicit: do all, then pause to gather thoughts TOGETHER after. So proceed now with E-010. But it's also fine to give the user the chance to weigh in at this natural break. I'll present the summary and start E-010 directly — that's what the standing directives say. Hmm, but my message turnaround: the user said \"let me know what is next\" — minimal expectation is an answer. I've done a whole sub-stage. I'll report and proceed with E-010 in the next turn unless told otherwise. Actually — continuing to work right now without pause is fine too. But context/time: significant work done; a natural sub-stage boundary. I'll report concisely and begin E-010 scoping in the same turn? Better to report and let user respond, since E-010 is a new sub-stage and session pacing is user's call. Given §8.5 (break prompt), I'll summarize and ask whether to continue with E-010 now.","signature":""},{"type":"text","text":"Session 003 state is committed (`ead7865`). Summary of this sub-stage:\n\n**E-009a — Suzuki's screw function g(t)**: implemented in mpmath, validated against eq. 2.2 to 25 digits (A = 0.7075463657, r(t)/t² → −0.8750599754), tabulated on [0,10].\n\n**E-009b — Galerkin spectrum of the Weil-form operator A_a**: eventful but conclusive:\n- First version showed spurious *negative* bottom eigenvalues — a fake ¬RH signal caused by 1/h² amplification of table-interpolation noise (lesson logged, D-019 F3). Fixed with exact piecewise Gauss–Legendre quadrature (S(k) exact to ≤1e-15).\n- **Absolute validation**: vᵀQv = 2.437005e-4 vs zero-side 2Σ|v̂(γ)|² = 2.437016e-4 — the discrete form *is* the Weil form.\n- En route: machine-verified the Weil explicit-formula conventions (Gaussian test closes to 9e-16; note the arXiv HTML's Λ(n)/n is wrong for our conventions — Λ(n)/√n validated; logged D-001 NT-1).\n- **λ_a cliff**: bottom eigenvalue 0.414 at a=0.25 → 2e-4 at 0.5 → ≤1.4e-9 at a=1 → ≤1.7e-12 at a=5, all positive — consistent with Yoshida (λ_a > 0 ∀a ⟺ RH) and independently reproduces Kim et al.'s \"positive, superexponentially decaying\" report.\n- Key structural finding: A_a's spectrum is *not* the zeta spectrum; the zeta-aligned operator is 𝒟̄_{a,θ} whose eigenvalues are zeros of W(a,θ;z) (eq. 1.11) — computable with our machinery.\n\n**D-019** filed: analysis + Q-S1 verdict (L²→uniform gap is the W2-shaped wall; an entire-type + Hurwitz lifting route is the lead Phase-4 candidate) + E-010 design.\n\n**Fork point (§8.4a)** — all four to be explored, order up for confirmation:\n1. **E-010**: compute W(a,θ;z) via deficiency vectors v_± (solve (Q−λM)c = b_±), find its real zeros for a ∈ {2,4,6,8} and scan θ — direct test of convergence toward γ₁, γ₂, ... (Suzuki line continuation)\n2. **3.2 cont.**: prime-side Li coefficients (Bombieri–Lagarias exact formulas), push past n=116\n3. **3.3**: de Bruijn–Newman heat flow vs the 2/3 density\n4. **T4**: ξ′-analogue (cheap secondary Track A target)\n\nShall I start E-010 now, or would you rather reorder/take a break?"}]