D-028 — E-018 analysis: the (★) cancellation holds uniformly in the strip y₀ ∈ [0.1, 1]; the open content is the zero set of M, not its envelope

Date: 2026-08-18 | Session: 006 cont.3 | Task: 2.8 sub-stage (i) Status: complete. Descent-profile baseline established; transition signature identified as zero-flow-only; θ-preference found to be metric-dependent (tension recorded honestly).

1. What E-018 measured

|M(a,θ;z)|/|R₀(a,θ;z)| on z = x + iy₀, x ∈ [−20,20] (81 pts), y₀ ∈ {1, 0.3, 0.1}, a = 4,6,8,10, θ = π (θ = 0 control). Machinery: C1 off-axis W regression < 1e-10 vs mpmath dps-50; C2 complex-F vs flint acb 5.3e-10. ALL CHECKS PASS.

2. Findings [NUMERICAL-VALIDATED]

• F1. Envelope cancellation is strip-uniform. Median |M|/|R₀| ≈ 5–7e-3 at every y₀ ∈ {1, 0.3, 0.1}, both θ, all a. The (★) tail cancellation does NOT break down as y₀ → 0.1; E-017's y-dependence (0.2% → 4.7% toward y=0.5 on the imaginary axis) levels off — the leftover envelope is ~0.5–0.7% of the skeleton throughout the strip.
• F2. Near-stationary, slowly increasing in a. θ=π medians: 5.88e-3 (a=4) → 6.79e-3 (a=10) at y₀=0.1. Same slow creep as E-017. No transition signal in the envelope at a ≤ 10 (consistent with D-023's a ≫ 10 estimate).
• F3. θ-ordering is METRIC-DEPENDENT. On the imaginary axis at y=0.5 (E-017): π 4.7% < zero 6.8% (π preferred). Median over the strip lines (E-018): zero 4.6–6.2e-3 < π 5.9–6.8e-3 (zero preferred). The H4 θ=π preference is NOT robust to the choice of metric; the honest statement is that the x=0 line and the strip median order differently. Any θ(a) theory must explain both.

3. Interpretation [ANALYSIS]

The strip-uniform smallness of |M|/|R₀| means R_τ ≈ −R₀ + (0.5% envelope) is already the limiting off-axis behavior. What prevents e^{ϕ}M → 1 is purely M's ZERO SET: M = W/F vanishes at the W-lattice (dense, real) and blows up at F's zeros ({γ_j} ∪ {ξ′-zeros}, sparse, real). Both features live exactly on ℝ and are invisible to the envelope ratio (dips/spikes average out in medians). Hence: the entire remaining content of Cor 1.6 is the a-evolution of M's real zero/pole configuration — γ-locking + outflow (D-023) — and no off-axis envelope measurement can detect it. Consequence for the plan: task 2.8 sub-stage (ii) (backend push to a ~ 12–14) should focus on the ZERO FLOW: located-zero tracking vs a (π/a lattice inflow — does the inflow ever bounce?), nearest-zero distance to each γ_j (γ-locking metric), and winding counts in thin strips |Im z| < y₀. The prime-sum backend (blockwise Λ with checkpointed prefix sums) is required since a > 10 exceeds the e²⁰ sieve.

4. Cost note for sub-stage (ii)

Direct sieve to e^20 covers a ≤ 10 (~11 s, in-memory cumsums). a = 12 needs e^24 ≈ 2.6e10 (blockwise: feasible, ~10⁴ blocks of 2.6e6 or ~26 blocks of 1e9 — est. tens of minutes); a = 14 needs e^28 ≈ 1.45e12 (hours). Sub-stage (ii) targets a = 12 first, a = 13–14 only if the a=12 zero-flow data warrants it.

5. Honesty notes

• The median can hide zero/pole structure (spikes at skeleton zeros, dips at M zeros) — F1 is an ENVELOPE statement; the CSV keeps full grids for re-analysis.
• C1's first run failed on a sign bug in the mpmath reference (hat argument), not in the machinery — fixed, rerun, pass; logged per D-025 §8's lesson (independent anchors matter).
• λ = −1 numerics vs λ = 0 derivation: immaterial at this accuracy (D-026 §7, D-027 §4).