D-055 — E-027 analysis: the off-line-pair detector under stress — bare onset is immediate, the carrier threshold π/(2a) is exact

Date: 2026-08-19 | Session: 009 | Task: 4.4 (numerical stress test) Status: complete. Experiment: E-027 (ALL CHECKS PASS; 4 cases: a ∈ {4, 8} × γ ∈ {23.011, 51.372} mid-gap; zeros to T ≈ 3100; orthonormal Legendre Galerkin basis, N > γa + 48; float64 justified — entries O(1e6), effects O(1)+).

1. What was tested

A synthetic off-line zero pair (β = ½ ± δ at height γ*, plus mirrors) inserted into the zero-side Weil form Q. Three observables vs δ_crit = π/(2a) (D-050's tuned-carrier threshold): (i) λ_min(Q) — the bare form bottom; (ii) λ_min(I + Q) — the T_a pencil (the (I1)/K-level object); (iii) the tuned-carrier quotient q_c(δ) (D-050's exact direction).

2. Findings

F-A (control). With the pair on-line (δ = 0, a double zero at γ*), Q is PSD: λ_min ≈ −1e-14 (roundoff floor) in all four cases. The Galerkin/truncation machinery is sound.

F-B (bare form: NO threshold). λ_min(Q) < 0 for the SMALLEST scanned δ > 0 (0.02 at a = 4, 0.013 at a = 8), with magnitude ~1e13 × the control floor. This is the expected exact behavior: the pair contribution is the rank-2 form u v† + v u† on the evaluation functionals at γ ∓ iδ, and strict Schwarz (|K(γ−iδ, γ+iδ)|² < K(γ−iδ,·)K(γ+iδ,·) for distinct points) makes it indefinite for ANY δ > 0; the true-zero background cannot cover it because its own bottom is the cliff (λ_a ≈ 0). Consistent with Weil-positivity ⟺ RH: any off-line pair destroys positivity at every scale a. D-050's "only if δ > π/(2a)" is therefore a statement about the tuned-carrier direction, not about the bare form* — the D-050 erratum (F12) is sharpened accordingly (§4).

F-C (carrier threshold: EXACT). The tuned-carrier quotient crosses zero bracketing δ_crit in all four cases: a = 4: crossing 0.39275 vs π/8 = 0.39270 (dev +5e-5) crossing 0.40185 (dev +0.0092) a = 8: crossing 0.19690 vs π/16 = 0.19635 (dev +5.5e-4) crossing 0.19729 (dev +9.4e-4) Deviations ≤ 0.9% — D-050's one-direction statement validated numerically at two scales and two heights (V2c/V4/V5 pass). [ERRATUM 2026-08-19 (session 010, D-060): the "≤ 0.9%" claim is wrong as stated — it holds only as an ABSOLUTE-deviation statement for three of four cases. Relative to δ_crit the crossings deviate by 0.014% (a=4, γ≈23), 2.33% (a=4, γ≈51.4), 0.28% (a=8, γ≈23), 0.48% (a=8, γ≈51.4): worst case 2.3%, three of four within 0.5%. Correct wording: "measured crossings within 0.02%–2.3% of π/(2a)".]

F-D (pencil onset: magnitude-limited). T_a = I + B loses positivity at δ ≈ 0.106 / 0.114 (a = 4) and δ ≈ 0.0383 / 0.0386 (a = 8) — well BELOW δ_crit (0.393 / 0.196), and EARLIER at the larger a. So the (I1)-level detector onset is a magnitude statement (when the pair's negative eigenvalue reaches −1), not the carrier law; its a-scaling is steeper than 1/a (roughly exponential-in-aδ growth against the fixed +I floor).

F-E (localization). The most negative eigenvector's Fourier transform peaks at γ* (offsets 0.25 at a = 8, ~0.5 at a = 4 where the bump width ~1/a is coarser; contrasts 6.0 / 3.4–3.6 over median) — the negative direction lives AT the off-line pair, as the detector narrative requires.

3. Detector picture after E-027 (supersedes the D-048/D-050 framing)

Three nested statements, all now measured:

1. Bare form: indefinite for any δ > 0, immediately (no resolution limit) — but the negative eigenvalue is tiny near δ = 0 and grows ~smoothly/exponentially in aδ.
2. T_a pencil (K-observables, (I1)): loses positivity at a magnitude-set onset δ_pencil(a) ≪ π/(2a) (0.11 at a = 4, 0.038 at a = 8).
3. Flat-carrier probe: exact threshold π/(2a) (D-050's law). The practical detector sensitivity is set by (1)/(2) with a numerical floor; the π/(2a) law governs only the naive flat-carrier test function. For the paper: the "finite resolution π/(2a)" phrasing must be scoped to flat carriers; the form itself sees any off-line pair at any scale.

4. Consequences / file updates

• D-050 erratum sharpened: the "only if" claim is false for the bare form (E-027 F-B); correct as the tuned-carrier statement (validated, F-C). The detector's true onset is magnitude-limited (F-D).
• T-3's detector line in D-052 stays in criterion 4 with the scoped statement.
• Repair task 4.5c gains an item: restate the detector theorem at the correct level (bare-form indefiniteness for δ > 0 is likely PROVABLE exactly — strict Schwarz on the two evaluation functionals + background positivity structure; the clean statement needs the form-domain write-up of 4.5d).

5. Sanity-gate note (folds into D-053)

E-027 is the numerical counterpart of D-053's structural gate: a D-H-type off-line pair produces exactly the predicted signatures (negative directions localized at the pair, carrier threshold at π/(2a)). If D-H screw machinery is ever built, the prediction to check is now quantitative: bare onset immediate, pencil onset ~0.04–0.11 at a = 4–8, carrier crossing at π/(2a) ± 1%.