[{"type":"thinking","thinking":"Paper compiles clean, 12 pages. Now the journal, STATE.md, NGUYEN.md, db, and final commit. Also delete session-012-notes.txt? No — keep it (it's a record; STATE.md reference). Actually the git status showed session-012-notes.txt untracked — it should probably be committed as part of the record. Let me include it.\n\nWrite journal/session-012.md.","signature":""},{"type":"text","text":"Paper compiles clean (12 pp). Writing the session journal."},{"type":"tool_use","id":"Write_105","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-012.md","content":"# Session 012 — 2026-08-20 — Lean tier (ii) step 1: assembly modulo SfBoundProp\n\n## Directive\nsession-012-notes.txt (read at session start per the session-011 user\ndirective): pick option (a) Lean tier (ii), but SPLIT it first —\nassembly-first with the finite-part bound as a named hypothesis, then\ntriage. User confirmed option (a) at session start.\n\n## What happened\n\n### Step 1: assembly-first (COMPLETE, two commits)\nRead LiSkeleton.lean (tier-(i) state) + D-037/D-056/D-057 for the exact\nassembly chain: λ_n = S_∞ − S_f + 1 (BL Thm 2) ⟹ λ_n ≥ S(n) − B(n) + 1\n= margin(n), convexity + base point ⟹ margin > 0 on the range, finite\nverification below.\n\nKey design decisions (all in D-062):\n1. **Sf := SInfty n + 1 − (liCoeff n).re** — BL Thm 2 holds by\n   construction; the D-036 contour argument proves identity + bound in\n   one stroke, so no separate BL formalization is needed downstream.\n2. **FAlt majorant swap** — discovered during design: the true Trudgian\n   F + S_b has ψ > 0 on [5.47, ~16] (negative-weight −(log 2πe/2π)T\n   term against the negative-ψ T-block), which would FALSIFY the\n   marginTilde_convex hypothesis hψ for the explicit majorant. Dropped\n   the negative-weight term: FAlt ≥ F + S_b pointwise, all four blocks\n   provably ψ < 0 (D-057 rows #1/#3/#4), margin cost ≈ 0.9√n = 256 at\n   n = 8×10⁴ vs margin 15 764 — negligible. (Caught by hand-computing\n   ψ[F+S_b](5.47) = −0.0006 — barely negative — before attempting the\n   Lean instantiation.)\n3. **Two-point base (secant)** instead of the derivative base point:\n   pos_of_convex_of_two_points (ConvexOn.slope_mono_adjacent) — the\n   remaining numeric obligations are two PLAIN evaluations of the\n   explicit margin function, no deriv machinery. Caveat found: convex +\n   two points does NOT give positivity between the points (convex can\n   dip), so the regime split moved to n = 10⁵ (second point), with\n   finite verification covering below.\n4. Refactor: margin regularity ContDiff ℝ 2 F → ContDiffOn (Ioi 1)\n   (FAlt log-singularities at 0, 1); chain_rule_sqrt_shift_on\n   generalized to c ≤ 1; unused hmaj dropped from marginTilde_convex.\n\nResult: `lambda_pos_finite_height` PROVED from 5 named hypotheses\n(hRH perimeter, hfinite perimeter, hbase, hbase2, hSf : SfBoundProp).\nPrintAxioms = [propext, Classical.choice, Quot.sound] — NO sorryAx.\nLiSkeleton.lean is now sorry-free; sorryAx remains only on the 4 valley\nstatements + R003 skeleton (tiers iii/iv) by design.\n\nCommits: 741d2c5 (step 1), f39499a (two-point variant).\n\n### Step 2 design: triage + perimeter decision (COMPLETE, D-062)\nDecomposed SfBoundProp into layers L0–L7 with tractability verdicts:\n- Mathlib-tractable: kernel bounds, R-bound (mod N(T)), zero-avoidance.\n- Main lift: L0 rectangle-contour/residue machinery (thin in Mathlib).\n- DECISION (declared, per the notes' instruction to decide\n  deliberately): Trudgian 2014 explicit N(T)/S(T) → INSIDE the trusted\n  perimeter, same footing as Platt–Trudgian. Rationale: published\n  refereed infrastructure; formalizing it is Backlund–Rosser-class\n  (larger than everything done so far) with zero insight into R-004's\n  actual content. Paper appendix updated to declare exactly this.\n- M₀ → computational perimeter after E-023b (acb certificate).\n\n### Step 2 prep: base-point discharge plan VALIDATED (D-062 §5)\nFull rational-interval bound chain pre-verified at dps 40: log 80000 ∈\n[11.263, 11.3], log 100000 ≥ 11.5, log T ≤ 5.7/5.8, log log ≤ 1.75/1.77,\nlog 285 ≥ 5.6, log π ∈ [1.14, 1.1448] (upper link 0.007% — tightest),\ne⁴ ∈ [54.5978, 54.6015], IIMaj ≤ 4.56(s+1)·2. Certified outcome:\nmargin(8×10⁴) ∈ [9 960·slack, 63 989], margin(10⁵) ≥ 86 220 —\nhbase and hbase2 both comfortably provable (~400–600 lines of interval\ngrind along block_range_bounds/iimaj_bracket_bound patterns).\n\n### Paper\ndraft.tex Formal verification appendix: new paragraph (tier (ii)\nassembly form, five named hypotheses, Trudgian perimeter declaration,\nFAlt note) + remaining-tiers paragraph updated (ii → ii′ inside\nSfBoundProp). MiKTeX compile clean, 12 pp. Audit note (D-060 §5 rule):\nthe edited lines are exactly the new/updated paragraphs; claims match\nthe Lean state verbatim (hypothesis list, axiom profile).\n\n## Files\n- research/lean/NguyenRH1/NguyenRH1/LiSkeleton.lean (+~470 lines:\n  FAlt/Bn/Sf/SfBoundProp defs, marginTilde_eq_sub, ψ-blocks #1/#3/#4,\n  contDiffOn_FAlt, FAlt_psi_neg, pos_of_convex_of_base/two_points,\n  lambda_pos_finite_height; refactor of margin regularity)\n- research/lean/NguyenRH1/PrintAxioms.lean (new declarations added)\n- research/derivations/D-062-SfBound-triage.md (NEW)\n- D-061 §5 update (session-012 status)\n- research/paper/draft.tex (Formal verification appendix)\n- research/journal/session-012.md (this file)\n\n## State at stop\nTier (ii) step 1 COMPLETE and committed. Next: tier (ii) step 2 —\ndischarge hbase/hbase2 per the D-062 §5 plan (validated), then the long\npole: SfBoundProp layers L0–L3 (contour formalization), with L4\nperimeter decided. Optional hygiene: E-023b (M₀ acb certificate).\n"},"extra_content":null}]