Current task: —

Updated 08-19-2026

D-038 — Task 2.11: transition-scale lower bound — the consolidation functional and the onset criterion

Date: 2026-08-19 | Session: 008 | Task: 2.11 (deliverable D3 of D-035) Status: complete. Inputs: E-019 (D-029 zero-flow), E-022/E-022b (D-040 spectral measures, valley observable), D-026 (skeleton), D-023.

1. The question

D-023 inferred a consolidation transition "at a ≫ 10"; E-019 found no transition at a ≤ 12 (lattice stiffens, maximal density). D-035 D3 asks: quantify the a-scale at which any (★)-consistent theory MUST begin the consolidation (zeros flowing out / mass concentrating onto the γ's).

2. The consolidation functional (validated)

From E-022/E-022b measurements (window Z = 280, ~135 true zeros below):

Model: plateau per-atom weight ≈ e^{−Δ(a)} (valley-depth gauge K_γ ≈ 1); plateau atom count in-window ≈ aZ/π. Therefore R(a) := (aZ/π)·e^{−Δ(a)} — the consolidation functional. Validation vs measured residual: a=4: 41.1 vs 48.3; a=12: 56.5 vs 57.6 (within 15%; the offset is the valley-shape correction).

3. The onset criterion

Consolidation begins ⟺ R(a) starts decreasing ⟺ Δ′(a) > 1/a, i.e. Δ(a) − log a → ∞ is required asymptotically. Define the consolidation margin C(a) := Δ(a) − log a. Measured: C(4) = 0.77, C(6) = 0.72, C(8) = 0.66, C(10) = 0.57, C(12) = 0.46 — monotone DECREASING; Δ′(12) ≈ 0.07 < 1/12 ≈ 0.083. At fixed θ = π the margin is shrinking and, under the saturation fit Δ ≈ 3.33 − 4.76/a, tends to −∞: fixed-θ consolidation is asymptotically impossible (not merely unobserved) — the plateau mass would grow without bound, eventually violating even the mass invariant unless valley masses drain, which would break the target m_γ = 1.

4. The transition scale (the D3 answer)

  1. Hard lower bound: a* > 12 (measured: residual increasing, lattice maximal, margin C(a) positive but shrinking through a = 12).
  2. Onset requirement: any (★)-consistent evolution must have Δ(a; θ(a)) eventually super-logarithmic (Δ − log a → ∞). At fixed θ this fails by the measured saturation (D-040 F6). The θ(a) flow (D-041 §5: toward the π/2-extension) is therefore not optional — it is the only channel through which the transition can occur.
  3. No upper estimate is possible from current data: the rate of Δ(a; θ(a)) along the correct flow is unknown until the θ(a) theory exists (D-041's L-constraints + E-024 numerics). What CAN be said: if Δ(a;θ(a)) = Θ(log a) with slope c, then R(a) ~ a^{1−c} — consolidation onset requires c > 1, i.e. the valley depth must grow at least as fast as the log of the lattice density.
  4. Resolution is NOT the bottleneck: a ≫ log Z satisfied since a ≈ 6 (lattice spacing π/a ≪ mean zero gap 2π/log Z for Z ≤ 280 once a ≥ 6).

5. Summary vs D-023's "a ≫ 10"

D-023's inference is sharpened to: transition has not begun at a ≤ 12; the consolidation margin C(a) = Δ(a) − log a is the quantitative transition monitor (must turn from decreasing to increasing and → ∞); at fixed θ = π, C(a) → −∞ (saturation ceiling) — the transition is structurally impossible without the θ(a) flow; measuring Δ(a; θ) along candidate flows (E-024) is the decisive next experiment.

6. Honesty notes