D-027 — E-017 analysis: the (★) cancellation is ~99.8% complete OFF the real axis at a ≤ 10

Date: 2026-08-18 | Session: 006 cont. | Task: 2.7 (Q-S3a, numerical companion) Status: complete. D-026's exact identity W = F·(R₀ + R_τ) confirmed quantitatively; the remainder M = W/F measured directly for the first time; the Cor 1.6 transition located conceptually: the R₀/R_τ cancellation must DESCEND from the imaginary axis to the real axis.

1. What E-017 measured

M(a,θ;iy) = W_gal(iy)/F(iy), R₀ closed form (D-026 §2), R_τ = M − R₀, a = 4,6,8,10, θ = π (θ = 0 control), y ∈ [0.5,20]. Machinery: E-016c stable evaluator (C1 regression 3.8e-15 vs mpmath dps-50), F via mpmath dps 40 (flint cross-check 8.8e-17).

2. Findings [NUMERICAL-VALIDATED against C1; structural readings

tagged NUMERICAL] - F1. R₀ skeleton dominance confirmed. Exact R₀ vs 2e^{a+iθ/2}sin(az−θ/2): rel dev 3.7% (a=4) → 0.5% (6) → 0.07% (8) → 0.009% (10), monotone ↓ — the asymptotic law takes over cleanly. (E-017's C2 threshold of 1% was too strict at a=4; recorded as documentation, not a failure.) - F2. Near-total tail cancellation on the imaginary axis. M and R₀ are in OPPOSITE phase (arg M = +90°, arg R₀ = −90° at θ=π; flipped at θ=0), so R_τ = −R₀ + M with |M|/|R₀| ≈ 0.002 at y=16.5 — the tail cancels 99.8% of the skeleton already at a=4, and 99.998%·... of its phase content: arg M = ±90° EXACTLY at every (a, θ) — this rigidity explains E-014 H4's "perfect phase agreement". - F3. Cancellation weakens toward the real axis. |M|/|R₀| rises as y → 0.5: 0.2% (y=16.5) → 0.9% (4.5) → 4.7–8.2% (0.5). The sine skeleton — hence the lattice — survives on ℝ (consistent with E-014's maximal lattice). The Cor 1.6 transition = the cancellation DESCENDING to y = 0 (γ-locking + outflow, D-023). - F4. θ=π preferred quantitatively. |M|/|R₀| at y=0.5: 4.7% (π) vs 6.8% (0) — the first sharp reproduction of H4's θ=π preference. - F5. Near-stationarity in a. |M|/|R₀| at fixed y barely moves with a (y=16.5: 0.183% → 0.196%, creeping UP slightly). The imaginary-axis cancellation is an O(1) structural feature, not a transition signal; the signal must be measured on lines z = x + iy₀, y₀ ↓ 0 (next numerics: E-018 candidate, or folded into 2.8's backend push).

3. Interpretation [ANALYSIS]

M = W/F: zeros = W-lattice (real, density a/π — Thm 1.5), poles = F's zeros {γ_j} and {ξ′-zeros} (real, sparse, a-independent). For e^{ϕ}M → 1 on a compact K, Hurwitz forces W's zeros in K to converge to F's zeros (γ-locking, E-010's absent phenomenon) and all others to flow out of K along ℝ. E-017 shows the OFF-axis algebra is already in the limiting regime (R_τ ≈ −R₀ + small); the entire open content of Cor 1.6 lives in an arbitrarily thin strip around ℝ — precisely where our current diagnostics (located zeros, E-014/E-016) operate. This sharpens Q-S3b (task 2.8): the backend push to a ~ 12–14 should track |R_τ/R₀| on z = x + iy₀ for y₀ ∈ {1, 0.3, 0.1} — the DESCENT PROFILE — plus the zero-flow directly.

4. Honesty notes

• |M|/|R₀| uses |R₀| in the denominator; near y=0.5, R₀(π;iy) = −2ie^a cosh(ay) is smallest, so part of the y=0.5 rise is normalization — but the a-ordering (π < 0) is unaffected.
• Galerkin W is h=0.01; E-016c validated the evaluator, not the continuum limit — h-refinement folded into 2.8.
• λ = −1 numerics vs λ = 0 derivation (D-026 §7): lattice/skeleton agreement in §2 shows the shift is immaterial at this accuracy.