D-042 — Task 2.12 analysis: θ-independence, the linear-K law, and the λ→λ_a⁻ channel

Date: 2026-08-19 | Session: 008 | Task: 2.12 (E-024) + (★)-route synthesis Status: complete. Experiments: E-024 (ALL CHECKS PASS); uses E-022/ E-022b (D-040), D-038, D-041.

1. E-024 measurements (θ-sweep)

θ ∈ {0, π/6, …, π} × a ∈ {4, 8, 12}: zeros via the real gauge g_θ(z) = Re[e^{−iθ/2}(z−i)V₊(z)] (exact on ℝ given the mirror gauge); Clark weights 1/K_a(z_k,z_k) (E-022 route).

• V1: all zeros real, maximal lattice at EVERY θ (self-adjointness confirmed numerically for all 21 (a,θ) pairs).
• V2: θ=π reproduces E-022 exactly.
• θ-INDEPENDENCE (headline): residual mass spread over θ ≤ 2.2%, mass±0.1-of-γ spread ≤ 2.3%, valley-mass mean ≈ 1.00 for every θ. Only the lattice PHASE responds to θ: z₁·a runs 3.147 (θ=0) → 4.504 (θ=π) — ≈87% of the skeleton shift θ/2, the rest R_τ deformation (consistent with D-026 §3). This confirms the de Branges prediction (K is the space's kernel, blind to the extension): the θ(a) flow cannot drive consolidation — it matters only for function-level phase alignment in (1.12). The θ-flow as a consolidation channel is CLOSED by evidence (noted in D-041 §5 as a possibility; now measured).

2. The linear-K law (E-022b data re-read)

[SUPERSEDED 2026-08-19 — see D-045 §1–2: the plateau law is K_plateau/(2a) → 1 (measured 0.973, 0.983, 0.993, 0.994, 0.997), the free Paley–Wiener diagonal; the "1.9·a" fit below was the finite-a shadow of the limit 2, and the fit quality claim "≤ 0.3%" does not survive re-checking against the raw values listed here (K/a = 1.95, 1.97, 1.99, 1.99, 1.99 — not 1.90). Consequently the residual asymptotic is Z/(2π), not Z/6 (D-045 §2). Erratum added session 010 (D-060 audit).] Plateau vs floor decomposition of the valleys: logK_plateau: 2.052, 2.468, 2.766, 2.990, 3.173 (a = 4…12) logK_floor(γ): −0.107, −0.040, 0.031, 0.119, 0.228 Fit: logK_plateau = ln a + 0.64 to ≤ 0.3% at all five points, i.e. K_plateau(a) ≈ 1.9·a (≈ (6/π)·a), while the valley floor K(γ,γ) ≈ 1–1.26 is nearly a-stationary (mass gauge: K(γ,γ) ≈ 1 matches the target m_γ = 1). Consequences:

• Plateau weight ≈ 1/(1.9a); residual mass R(a) = (aZ/π)·(1/1.9a) → Z/(1.9π) ≈ Z/6 ≠ 0 — matches measured residuals (48 at a=4 → 58 at a=12, Z=280: Z/6 = 46.7; the slow rise = the floor drift).
• Consolidation margin C(a) = Δ(a) − log a → 0.64 − floor ≈ 0.4 (a nonzero constant): the family equilibrates EXACTLY at the consolidation boundary — neither converging nor diverging.
• The linear law is the GENERIC Szegő behavior (bulk spectral response of the Toeplitz operator T_a to e_z): no fine-tuning. Consolidation requires a NON-generic (superlinear) K-growth — precisely the (★) cancellation, which has not started at a ≤ 12.

3. Synthesis — where the (★) route actually stands

Measured, controlled facts (E-016…E-024):

1. The metric knows the zeros: K_a valleys at the γ's, floor ≈ 1.
2. The lattice neither locks (positions chance-level, E-019) nor thins (maximal density, all θ, E-024).
3. Residual mass → Z/6 ≠ 0 (linear-K equilibrium).
4. θ is statistically irrelevant (E-024). ∴ At fixed λ = −1 the family {D̄_{a,θ}} CANNOT consolidate, at any a, for any θ(a) — not a saturation artifact but the generic equilibrium. The remaining channel is Suzuki's own flagged lever: λ → λ_a⁻ (remark after Cor 1.6: "control of λ (< λ_a) will likely play an important role"). As λ ↑ λ_a(a) (λ_a ~ e^{−1.65a}, D-017/D-019), T_a becomes near-singular and K(z,z) = ⟨e_z, T_a⁻¹e_z⟩ becomes ground-state-dominated — the regime of the SISTER conjecture (1.2) (v̂_a → ξ, the lowest eigenfunction). If valley depth/locking responds to λ ↑ λ_a⁻, that is the consolidation channel; if not, the (★) route via this operator family is sterile at the measure level. → new task 2.13 (E-025): λ-flow numerics.

4. Honesty notes

• θ-independence is measured at a ≤ 12 and rests on the de Branges mass formula 1/K(z_k,z_k) (Q-S2 identification; valley masses ≈ 1.00 is strong independent evidence for it).
• The linear-K fit is 5-point but exact to 0.3% and mechanism-backed (Szegő genericity); tagged [NUMERICAL-VALIDATED].
• E-024 does NOT close the θ(a) question for the FUNCTION-level statement (1.12) (phase alignment); it closes it for measure-level consolidation.
• The λ→λ_a⁻ regime needs eigen-resolution ~e^{−1.65a} (a=12: ~2.5e-9) — within demonstrated Galerkin reach (D-019 measured the cliff to 1.7e-12 at a=5).

5. Feeds

• 2.9: the D1 chain (D-041) stands; the consolidation obstruction is relocated from θ(a) to the λ-regime.
• 2.13 (E-025): λ-flow numerics — Δ(a; λ), locking metric, residual R(a; λ) as λ ↑ λ_a⁻; plus the (1.2)-conjecture observable v̂_a.
• Paper: E-022/E-024/D-040–D-042 form the spectral-measure section.