D-044 — The linear-K law: windowed-symbol analysis of the spectral measure equilibrium

Date: 2026-08-19 | Session: 008 | Task: linear-K analysis (D-043 §D next step) Status: complete (framework + mechanism; symbol closed form OPEN).

1. The framework

K_a(z,z) = ⟨e_z, T_a^{−1}e_z⟩ (L²_0-constrained), T_a = Q − λM the Galerkin operator at λ = −1. Expanding in T_a's eigenbasis, K(z,z) = Σ_j |⟨e_z, u_j⟩|²/μ_j. For a large interval the eigenfunctions are Fourier-like with density a/π per unit frequency, and the overlaps localize on ξ ≈ z with the sinc² kernel of width ~ 1/a. Hence the WINDOWED-SYMBOL LAW K_a(z,z) ≈ a · ρ_a(z), ρ_a = (1/β) ⊛ w_a with β the (real) operator symbol and w_a a kernel of width ~ 1/a. This is the Szegő/Christoffel structure; the linear-in-a prefactor is pure mode counting.

2. Measured confirmations (E-022b data)

3. The equilibrium (why nothing consolidates at λ = −1)

Consolidation of μ_a onto Γ requires the plateau weights to die: 1/K_plateau = 1/(aρ̄) with ρ̄ ≈ 1.9 constant ⟹ per-atom weight ~ 1/a, but the atom count grows ∝ a (maximal lattice) — the residual R(a) → Z/(πρ̄) = Z/6 ≠ 0. The family sits EXACTLY at the boundary: Δ(a) = log K_plateau − log K_floor = log a + 0.64 − O(0.1), so the consolidation margin C(a) = Δ − log a → ~0.4 (const). This is the generic Szegő bulk behavior — no fine-tuning — and it is what the (★) cancellation would have to defeat.

4. What breaks the equilibrium (the λ-mechanism, now quantitative)

Shifting T_a → T_a − λ translates the symbol β → β − λ but leaves its POLES (at the γ's) unchanged. Therefore:

• valley floors are PINNED: 1/(β−λ) keeps the same simple zeros with the same slopes ⟹ valley masses stay ≈ 1 (the target m_γ);
• the plateau response 1/(β̄−λ) DIVERGES as λ → β_min ⟹ plateau weights collapse ⟹ residual → 0. This is exactly consolidation: masses pinned at 1 on the γ's, background dying. E-025 measured the expected direction (Δ deepens, residual 48→11 at a=4) — the channel is real. The blocker is purely machinery (D-043 §A): the discrete pencil bottom is artifact-corrupted for a ≳ 5 (O(h) error on a rough kernel), and the true λ_a lies below the floor even at a = 2–3 (refinement-chases below 1e-11). Reaching λ → λ_a⁻ needs a discretization whose error floor beats λ_a's superexponential decay (D-019 cliff) — dense uniform refinement is infeasible; a spectral/exponential-convergence Galerkin for the rough kernel g is the machinery project that would open this.

5. What is NOT achieved (honesty)

• The closed form of the symbol β on ℝ is UNIDENTIFIED. Refuted candidates: |Ξ/Ξ′| and its windowings (corr ≤ 0.62, relerr ≥ 60%) — the measured plateau is flat, unlike any |Ξ|-ratio. The flatness suggests the plateau response is dominated by the truncated/discrete bulk (primes ≤ e^{2a}), not by a clean continuum symbol — note g's exponential growth makes ĝ an analytic functional, so the classical symbol calculus needs care (distributional Toeplitz theory).
• The coefficient 6/π ≈ 1.9 (= ρ̄⁻¹) is unexplained; note K_plateau ≈ 6·(a/π) = 6 × (Paley–Wiener kernel diagonal a/π). Whether 6 is exactly 6 or a truncation-dependent number is open.
• The λ-channel's machinery requirement is stated but the channel is NOT validated at the true ground state (E-025 reached only the artifact mode).

6. Status of the (★) route after this analysis