{
  "content": "# D-055 \u2014 E-027 analysis: the off-line-pair detector under stress \u2014 bare onset is immediate, the carrier threshold \u03c0/(2a) is exact\nDate: 2026-08-19 | Session: 009 | Task: 4.4 (numerical stress test)\nStatus: complete. Experiment: E-027 (ALL CHECKS PASS; 4 cases:\na \u2208 {4, 8} \u00d7 \u03b3* \u2208 {23.011, 51.372} mid-gap; zeros to T \u2248 3100;\northonormal Legendre Galerkin basis, N > \u03b3*a + 48; float64 justified\n\u2014 entries O(1e6), effects O(1)+).\n\n## 1. What was tested\nA synthetic off-line zero pair (\u03b2 = \u00bd \u00b1 \u03b4 at height \u03b3*, plus mirrors)\ninserted into the zero-side Weil form Q. Three observables vs\n\u03b4_crit = \u03c0/(2a) (D-050's tuned-carrier threshold):\n (i) \u03bb_min(Q) \u2014 the bare form bottom;\n (ii) \u03bb_min(I + Q) \u2014 the T_a pencil (the (I1)/K-level object);\n (iii) the tuned-carrier quotient q_c(\u03b4) (D-050's exact direction).\n\n## 2. Findings\n**F-A (control).** With the pair on-line (\u03b4 = 0, a double zero at \u03b3*),\nQ is PSD: \u03bb_min \u2248 \u22121e-14 (roundoff floor) in all four cases. The\nGalerkin/truncation machinery is sound.\n\n**F-B (bare form: NO threshold).** \u03bb_min(Q) < 0 for the SMALLEST\nscanned \u03b4 > 0 (0.02 at a = 4, 0.013 at a = 8), with magnitude\n~1e13 \u00d7 the control floor. This is the expected exact behavior: the\npair contribution is the rank-2 form u v\u2020 + v u\u2020 on the evaluation\nfunctionals at \u03b3* \u2213 i\u03b4, and strict Schwarz\n(|K(\u03b3*\u2212i\u03b4, \u03b3*+i\u03b4)|\u00b2 < K(\u03b3*\u2212i\u03b4,\u00b7)K(\u03b3*+i\u03b4,\u00b7) for distinct points) makes\nit indefinite for ANY \u03b4 > 0; the true-zero background cannot cover it\nbecause its own bottom is the cliff (\u03bb_a \u2248 0). Consistent with\nWeil-positivity \u27fa RH: any off-line pair destroys positivity at every\nscale a. **D-050's \"only if \u03b4 > \u03c0/(2a)\" is therefore a statement\nabout the tuned-carrier direction, not about the bare form** \u2014 the\nD-050 erratum (F12) is sharpened accordingly (\u00a74).\n\n**F-C (carrier threshold: EXACT).** The tuned-carrier quotient crosses\nzero bracketing \u03b4_crit in all four cases:\n  a = 4:  crossing 0.39275 vs \u03c0/8 = 0.39270  (dev +5e-5)\n          crossing 0.40185                   (dev +0.0092)\n  a = 8:  crossing 0.19690 vs \u03c0/16 = 0.19635 (dev +5.5e-4)\n          crossing 0.19729                   (dev +9.4e-4)\nDeviations \u2264 0.9% \u2014 D-050's one-direction statement validated\nnumerically at two scales and two heights (V2c/V4/V5 pass).\n\n**F-D (pencil onset: magnitude-limited).** T_a = I + B loses\npositivity at \u03b4 \u2248 0.106 / 0.114 (a = 4) and \u03b4 \u2248 0.0383 / 0.0386\n(a = 8) \u2014 well BELOW \u03b4_crit (0.393 / 0.196), and EARLIER at the\nlarger a. So the (I1)-level detector onset is a magnitude statement\n(when the pair's negative eigenvalue reaches \u22121), not the carrier\nlaw; its a-scaling is steeper than 1/a (roughly exponential-in-a\u03b4\ngrowth against the fixed +I floor).\n\n**F-E (localization).** The most negative eigenvector's Fourier\ntransform peaks at \u03b3* (offsets 0.25 at a = 8, ~0.5 at a = 4 where\nthe bump width ~1/a is coarser; contrasts 6.0 / 3.4\u20133.6 over median)\n\u2014 the negative direction lives AT the off-line pair, as the detector\nnarrative requires.\n\n## 3. Detector picture after E-027 (supersedes the D-048/D-050 framing)\nThree nested statements, all now measured:\n1. **Bare form**: indefinite for any \u03b4 > 0, immediately (no\n   resolution limit) \u2014 but the negative eigenvalue is tiny near\n   \u03b4 = 0 and grows ~smoothly/exponentially in a\u03b4.\n2. **T_a pencil** (K-observables, (I1)): loses positivity at a\n   magnitude-set onset \u03b4_pencil(a) \u226a \u03c0/(2a) (0.11 at a = 4, 0.038 at\n   a = 8).\n3. **Flat-carrier probe**: exact threshold \u03c0/(2a) (D-050's law).\nThe practical detector sensitivity is set by (1)/(2) with a numerical\nfloor; the \u03c0/(2a) law governs only the naive flat-carrier test\nfunction. For the paper: the \"finite resolution \u03c0/(2a)\" phrasing\nmust be scoped to flat carriers; the form itself sees any off-line\npair at any scale.\n\n## 4. Consequences / file updates\n- D-050 erratum sharpened: the \"only if\" claim is false for the bare\n  form (E-027 F-B); correct as the tuned-carrier statement (validated,\n  F-C). The detector's true onset is magnitude-limited (F-D).\n- T-3's detector line in D-052 stays in criterion 4 with the scoped\n  statement.\n- Repair task 4.5c gains an item: restate the detector theorem at the\n  correct level (bare-form indefiniteness for \u03b4 > 0 is likely PROVABLE\n  exactly \u2014 strict Schwarz on the two evaluation functionals +\n  background positivity structure; the clean statement needs the\n  form-domain write-up of 4.5d).\n\n## 5. Sanity-gate note (folds into D-053)\nE-027 is the numerical counterpart of D-053's structural gate: a\nD-H-type off-line pair produces exactly the predicted signatures\n(negative directions localized at the pair, carrier threshold at\n\u03c0/(2a)). If D-H screw machinery is ever built, the prediction to\ncheck is now quantitative: bare onset immediate, pencil onset\n~0.04\u20130.11 at a = 4\u20138, carrier crossing at \u03c0/(2a) \u00b1 1%.\n",
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  "content": "# D-055 \u2014 E-027 analysis: the off-line-pair detector under stress \u2014 bare onset is immediate, the carrier threshold \u03c0/(2a) is exact\nDate: 2026-08-19 | Session: 009 | Task: 4.4 (numerical stress test)\nStatus: complete. Experiment: E-027 (ALL CHECKS PASS; 4 cases:\na \u2208 {4, 8} \u00d7 \u03b3* \u2208 {23.011, 51.372} mid-gap; zeros to T \u2248 3100;\northonormal Legendre Galerkin basis, N > \u03b3*a + 48; float64 justified\n\u2014 entries O(1e6), effects O(1)+).\n\n## 1. What was tested\nA synthetic off-line zero pair (\u03b2 = \u00bd \u00b1 \u03b4 at height \u03b3*, plus mirrors)\ninserted into the zero-side Weil form Q. Three observables vs\n\u03b4_crit = \u03c0/(2a) (D-050's tuned-carrier threshold):\n (i) \u03bb_min(Q) \u2014 the bare form bottom;\n (ii) \u03bb_min(I + Q) \u2014 the T_a pencil (the (I1)/K-level object);\n (iii) the tuned-carrier quotient q_c(\u03b4) (D-050's exact direction).\n\n## 2. Findings\n**F-A (control).** With the pair on-line (\u03b4 = 0, a double zero at \u03b3*),\nQ is PSD: \u03bb_min \u2248 \u22121e-14 (roundoff floor) in all four cases. The\nGalerkin/truncation machinery is sound.\n\n**F-B (bare form: NO threshold).** \u03bb_min(Q) < 0 for the SMALLEST\nscanned \u03b4 > 0 (0.02 at a = 4, 0.013 at a = 8), with magnitude\n~1e13 \u00d7 the control floor. This is the expected exact behavior: the\npair contribution is the rank-2 form u v\u2020 + v u\u2020 on the evaluation\nfunctionals at \u03b3* \u2213 i\u03b4, and strict Schwarz\n(|K(\u03b3*\u2212i\u03b4, \u03b3*+i\u03b4)|\u00b2 < K(\u03b3*\u2212i\u03b4,\u00b7)K(\u03b3*+i\u03b4,\u00b7) for distinct points) makes\nit indefinite for ANY \u03b4 > 0; the true-zero background cannot cover it\nbecause its own bottom is the cliff (\u03bb_a \u2248 0). Consistent with\nWeil-positivity \u27fa RH: any off-line pair destroys positivity at every\nscale a. **D-050's \"only if \u03b4 > \u03c0/(2a)\" is therefore a statement\nabout the tuned-carrier direction, not about the bare form** \u2014 the\nD-050 erratum (F12) is sharpened accordingly (\u00a74).\n\n**F-C (carrier threshold: EXACT).** The tuned-carrier quotient crosses\nzero bracketing \u03b4_crit in all four cases:\n  a = 4:  crossing 0.39275 vs \u03c0/8 = 0.39270  (dev +5e-5)\n          crossing 0.40185                   (dev +0.0092)\n  a = 8:  crossing 0.19690 vs \u03c0/16 = 0.19635 (dev +5.5e-4)\n          crossing 0.19729                   (dev +9.4e-4)\nDeviations \u2264 0.9% \u2014 D-050's one-direction statement validated\nnumerically at two scales and two heights (V2c/V4/V5 pass).\n\n**F-D (pencil onset: magnitude-limited).** T_a = I + B loses\npositivity at \u03b4 \u2248 0.106 / 0.114 (a = 4) and \u03b4 \u2248 0.0383 / 0.0386\n(a = 8) \u2014 well BELOW \u03b4_crit (0.393 / 0.196), and EARLIER at the\nlarger a. So the (I1)-level detector onset is a magnitude statement\n(when the pair's negative eigenvalue reaches \u22121), not the carrier\nlaw; its a-scaling is steeper than 1/a (roughly exponential-in-a\u03b4\ngrowth against the fixed +I floor).\n\n**F-E (localization).** The most negative eigenvector's Fourier\ntransform peaks at \u03b3* (offsets 0.25 at a = 8, ~0.5 at a = 4 where\nthe bump width ~1/a is coarser; contrasts 6.0 / 3.4\u20133.6 over median)\n\u2014 the negative direction lives AT the off-line pair, as the detector\nnarrative requires.\n\n## 3. Detector picture after E-027 (supersedes the D-048/D-050 framing)\nThree nested statements, all now measured:\n1. **Bare form**: indefinite for any \u03b4 > 0, immediately (no\n   resolution limit) \u2014 but the negative eigenvalue is tiny near\n   \u03b4 = 0 and grows ~smoothly/exponentially in a\u03b4.\n2. **T_a pencil** (K-observables, (I1)): loses positivity at a\n   magnitude-set onset \u03b4_pencil(a) \u226a \u03c0/(2a) (0.11 at a = 4, 0.038 at\n   a = 8).\n3. **Flat-carrier probe**: exact threshold \u03c0/(2a) (D-050's law).\nThe practical detector sensitivity is set by (1)/(2) with a numerical\nfloor; the \u03c0/(2a) law governs only the naive flat-carrier test\nfunction. For the paper: the \"finite resolution \u03c0/(2a)\" phrasing\nmust be scoped to flat carriers; the form itself sees any off-line\npair at any scale.\n\n## 4. Consequences / file updates\n- D-050 erratum sharpened: the \"only if\" claim is false for the bare\n  form (E-027 F-B); correct as the tuned-carrier statement (validated,\n  F-C). The detector's true onset is magnitude-limited (F-D).\n- T-3's detector line in D-052 stays in criterion 4 with the scoped\n  statement.\n- Repair task 4.5c gains an item: restate the detector theorem at the\n  correct level (bare-form indefiniteness for \u03b4 > 0 is likely PROVABLE\n  exactly \u2014 strict Schwarz on the two evaluation functionals +\n  background positivity structure; the clean statement needs the\n  form-domain write-up of 4.5d).\n\n## 5. Sanity-gate note (folds into D-053)\nE-027 is the numerical counterpart of D-053's structural gate: a\nD-H-type off-line pair produces exactly the predicted signatures\n(negative directions localized at the pair, carrier threshold at\n\u03c0/(2a)). If D-H screw machinery is ever built, the prediction to\ncheck is now quantitative: bare onset immediate, pencil onset\n~0.04\u20130.11 at a = 4\u20138, carrier crossing at \u03c0/(2a) \u00b1 1%.\n",
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