Message #2313 · Riemann Hypothesis Research

← Session 36 — ii′-6-DL dog-leg + side bounds; Paper A review

#2313 assistant 2026-08-26 20:57:19.394 UTC · 2003→953 tok open ⤢

thinking (1959 chars)

The user approved continuing to ii′-7. Let me plan ii′-7's components: **ii′-7 (per NGUYEN.md):** Closure: theorem sfBoundProp : SfBoundProp ✓ DONE (6-d); final `lambda_pos_finite_height_closed` (3 perimeter hypotheses only — now the 8: hRH, hfinite + the 6); full axiom audit; paper Appendix B + tag upgrade (D-062 §4 endpoint); archive closure + D-037 line-63 erratum pointer; the correction note (F-A/F-B/F-D/F-E). Sub-steps: 1. **lambda_pos_finite_height_closed**: instantiate `lambda_pos_finite_height_verified (hRH : RHUpTo 3×10¹²) (hfinite : LiVerifiedUpTo 500000) (hSf : SfBoundProp)` with `sfBoundProp` — the closed theorem takes (hRH) (hfinite) + the 6 perimeter Props as hypotheses. Check lambda_pos_finite_height_verified's exact signature first (LiBasePoints.lean). 2. **Full axiom audit** of the closed theorem + the whole chain (should be 3 standard axioms). 3. **Paper updates** (freeze lifts): paperA.tex — the formal-status paragraph at "3 hypotheses" → update to the new perimeter (8 hypotheses: RHUpTo, LiVerifiedUpTo, + the 6 analytic Props — which are the declared Trudgian N(T)/S(T), Platt-class zero-location, M₀ certificate, and the strengthened counting forms); Appendix B update (the Lean appendix describing SfBoundProp as a named hypothesis → now a proved theorem with the perimeter enumerated); the correction note: F-A (bridge 4F both-signs — the paper's B(n) account says 2F?), F-B (I_III two-unit count (2/3)Z₂), F-D (the dog-leg repair — the right-tail charge), F-E (I_I kernel footnote). Also the D-037 line-63 erratum pointer (archive-side, in research/derivations/D-037-*.md). 4. **Archive closure**: update NGUYEN.md checklist (ii′-6 → complete, ii′-7 → complete as items land), the Result Ledger (R-004 tag upgrade?), STATE.md, journal entry. This is a big stage. Let me start with the Lean closure (small), then the audit, then the paper. First — check lambda_pos_finite_height_verified's exact statement/signature.

Proceeding to ii′-7. First the Lean closure — checking `lambda_pos_finite_height_verified`'s exact signature.

tool: TodoWrite

input

{
  "todos": [
    {
      "content": "6-c + 6-d: four side bounds + fold \u27f9 sfBoundProp",
      "status": "completed",
      "activeForm": "Proving side bounds and fold"
    },
    {
      "content": "ii\u2032-7a: lambda_pos_finite_height_closed assembly (Lean)",
      "status": "in_progress",
      "activeForm": "Assembling lambda_pos_finite_height_closed"
    },
    {
      "content": "ii\u2032-7b: full axiom audit of the closed chain",
      "status": "pending",
      "activeForm": "Running full axiom audit"
    },
    {
      "content": "ii\u2032-7c: paper Appendix B + tag upgrade + correction note (F-A/F-B/F-D/F-E)",
      "status": "pending",
      "activeForm": "Updating paper with closure + corrections"
    },
    {
      "content": "ii\u2032-7d: archive closure (D-037 erratum pointer, NGUYEN.md, STATE.md, journal)",
      "status": "pending",
      "activeForm": "Closing the archive"
    }
  ]
}

tool: Grep ok

input

{
  "pattern": "theorem lambda_pos_finite_height_verified|theorem lambda_pos_finite_height\\b",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-B": 4,
  "-A": 12
}

result 2026-08-26 20:57:19Z → 2026-08-26 20:57:19Z

mode='content' num_files=0 filenames=[] content="D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-702-analytic hypothesis (`SfBoundProp`, the traced contour bound) plus the\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-703-declared computational perimeter (Platt–Trudgian `RHUpTo`; the finite Li\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-704-verification `LiVerifiedUpTo`). The two elementary margin evaluations\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-705-are discharged in this file. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean:706:theorem lambda_pos_finite_height_verified\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-707-    (hRH: RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-708-    (hSf: SfBoundProp) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-709-    ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 →\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-710-      0 < (liCoeff n).re:=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-711-  fun n hn1 hnN => lambda_pos_finite_height hRH hfinite\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-712-    marginTilde_FAlt_80000_pos marginTilde_FAlt_80000_lt_100000 hSf n hn1 hnN\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-713-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-714-end NguyenRH1\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-715-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-716-end\n--\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2795-= margiñ(n) > 0 by convexity (`marginTilde_convex` + `FAlt_psi_neg`)\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2796-and the two base points (`pos_of_convex_of_two_points`).\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2797-`_hRH` documents the bridge input: it is consumed by `SfBoundProp`'s\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2798-eventual proof (D-037 §6), not by the assembly itself. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:2799:theorem lambda_pos_finite_height\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2800-    (_hRH: RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2801-    (hbase: 0 < marginTilde FAlt 80000)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2802-    (hbase2: marginTilde FAlt 80000 < marginTilde FAlt 100000)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2803-    (hSf: SfBoundProp) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2804-    ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 →\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2805-      0 < (liCoeff n).re:= by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2806-  intro n hn1 hnN\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2807-  rcases lt_or_ge n 100000 with hlt | hge\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2808-  · exact hfinite n hn1 (by omega)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2809-  · have hn20: 20 ≤ n := by omega\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2810-    have h1:= sInfty_lower_bound n hn20\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2811-    have h2:= hSf n hge hnN" num_lines=33 num_matches=None applied_limit=None applied_offset=None

raw content_json (database row)

[{"type":"thinking","thinking":"The user approved continuing to ii′-7. Let me plan ii′-7's components:\n\n**ii′-7 (per NGUYEN.md):** Closure: theorem sfBoundProp : SfBoundProp ✓ DONE (6-d); final `lambda_pos_finite_height_closed` (3 perimeter hypotheses only — now the 8: hRH, hfinite + the 6); full axiom audit; paper Appendix B + tag upgrade (D-062 §4 endpoint); archive closure + D-037 line-63 erratum pointer; the correction note (F-A/F-B/F-D/F-E).\n\nSub-steps:\n1. **lambda_pos_finite_height_closed**: instantiate `lambda_pos_finite_height_verified (hRH : RHUpTo 3×10¹²) (hfinite : LiVerifiedUpTo 500000) (hSf : SfBoundProp)` with `sfBoundProp` — the closed theorem takes (hRH) (hfinite) + the 6 perimeter Props as hypotheses. Check lambda_pos_finite_height_verified's exact signature first (LiBasePoints.lean).\n2. **Full axiom audit** of the closed theorem + the whole chain (should be 3 standard axioms).\n3. **Paper updates** (freeze lifts): paperA.tex — the formal-status paragraph at \"3 hypotheses\" → update to the new perimeter (8 hypotheses: RHUpTo, LiVerifiedUpTo, + the 6 analytic Props — which are the declared Trudgian N(T)/S(T), Platt-class zero-location, M₀ certificate, and the strengthened counting forms); Appendix B update (the Lean appendix describing SfBoundProp as a named hypothesis → now a proved theorem with the perimeter enumerated); the correction note: F-A (bridge 4F both-signs — the paper's B(n) account says 2F?), F-B (I_III two-unit count (2/3)Z₂), F-D (the dog-leg repair — the right-tail charge), F-E (I_I kernel footnote). Also the D-037 line-63 erratum pointer (archive-side, in research/derivations/D-037-*.md).\n4. **Archive closure**: update NGUYEN.md checklist (ii′-6 → complete, ii′-7 → complete as items land), the Result Ledger (R-004 tag upgrade?), STATE.md, journal entry.\n\nThis is a big stage. Let me start with the Lean closure (small), then the audit, then the paper.\n\nFirst — check lambda_pos_finite_height_verified's exact statement/signature.","signature":""},{"type":"text","text":"Proceeding to ii′-7. First the Lean closure — checking `lambda_pos_finite_height_verified`'s exact signature."},{"type":"tool_use","id":"TodoWrite_202","name":"TodoWrite","input":{"todos":[{"content":"6-c + 6-d: four side bounds + fold ⟹ sfBoundProp","status":"completed","activeForm":"Proving side bounds and fold"},{"content":"ii′-7a: lambda_pos_finite_height_closed assembly (Lean)","status":"in_progress","activeForm":"Assembling lambda_pos_finite_height_closed"},{"content":"ii′-7b: full axiom audit of the closed chain","status":"pending","activeForm":"Running full axiom audit"},{"content":"ii′-7c: paper Appendix B + tag upgrade + correction note (F-A/F-B/F-D/F-E)","status":"pending","activeForm":"Updating paper with closure + corrections"},{"content":"ii′-7d: archive closure (D-037 erratum pointer, NGUYEN.md, STATE.md, journal)","status":"pending","activeForm":"Closing the archive"}]},"extra_content":null},{"type":"tool_use","id":"Grep_203","name":"Grep","input":{"pattern":"theorem lambda_pos_finite_height_verified|theorem lambda_pos_finite_height\\b","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-B":4,"-A":12},"extra_content":null}]