[{"type":"text","text":"Linear-K analysis complete and committed (`afa55c4`). Summary of D-044:\n\n## The windowed-symbol law\n\n**K_a(z,z) ≈ a·(1/β ⊛ w_a)** — mode counting gives the linear prefactor; the window w_a has width ~1/a. Every measured feature falls out:\n\n| prediction | measured |\n|---|---|\n| plateau K ∝ a | log K = ln a + 0.64, 0.3% over 5 points |\n| valley floor O(1), a-stationary (windowed simple zero of 1/β) | floor logK ∈ [−0.11, +0.23] |\n| valley width ~ c/a | 0.35 (a=4) → 0.15 (a=12) |\n| valley mass ≈ 1/K(γ,γ) = 1 | 0.99–1.03 ✓ |\n| residual density (1/π)(1/ρ̄) | 1/6 → R → Z/6 ✓ |\n\nThe valleys are **simple zeros of 1/β at the true γ's** (symbol poles) — the operator's symbol carries the zeros. The equilibrium (margin C(a) → const) is now explained as *generic Szegő bulk behavior*, not a numerical accident.\n\n## The breaking mechanism — quantitative\n\nThe λ-shift leaves the symbol's poles unchanged, so **valley floors stay pinned (masses stay 1) while the plateau 1/(β̄−λ) diverges as λ → β_min — the residual dies**. That's precisely consolidation. E-025's measured response matches the direction. The blocker is purely machinery: the discretization floor beats λ_a for a ≳ 3, and dense refinement is infeasible — it needs a spectral/exponential-convergence Galerkin for the rough kernel.\n\n## Honestly open\n- **Closed form of β is unidentified** — the |Ξ/Ξ′| family was refuted numerically (corr ≤ 0.62); the flat plateau points to truncation-bulk dominance, and g's exponential growth means the classical symbol calculus needs distributional care.\n- The **6/π constant** is unexplained (curious: K_plateau ≈ 6 × the Paley–Wiener kernel diagonal a/π).\n\n**Route status**: the (★) route now has exactly one open theory item (symbol identification → exact equilibrium) and one open machinery item (spectral Galerkin → the λ-regime). Both are sharply scoped. Next per the queue: symbol identification analysis, or the spectral-Galerkin sizing study. Your call."}]