← Session 17 — Lean tier (i): elementary proofs (after crash)
thinking (3156 chars)
Two documentation tasks:
1. Add a design note to the archive (D-061 update + journal) about the 11→8 block divergence: D-057's table has 11 rows, the Lean formalization uses 8 block types + separate IIMaj bound. Reason: B̃₄ (I_I term) is non-concave, handled by direct exponential bound instead of ψ-table. Note: actually D-057's table already handles I_I separately (§3) — the 11 blocks are for the OTHER terms. The Lean proof has 8 block types: id(#2), L1(#10), L2(#11), L1²(#7), L1·L2(#8), L2²(#9), T·L1(#5), T·L2(#6). Blocks #1, #3, #4 from D-057's table aren't needed because in the Lean formalization, F enters as a hypothesis-driven parameter (the 2F(T) term uses hψ directly, and there's no S_b(T) explicit majorant — blocks #1/#2 came from 2F(T) expansion, #3/#4 from 2S_b(T); in marginTilde as formalized, the F term is atomic). So the note should explain: D-057's 11 rows include blocks needed to expand the Trudgian F/S_b majorants; the Lean theorem is parameterized over F (with hψ as hypothesis), so only the 8 blocks from the closed-form Ĉ₀/Ĥ terms are formalized, and B̃₄ (non-concave) is handled by the separate analytic IIMaj″ bound. Thirty minutes of documentation.
Where to put it: D-057 (append a note), D-061 §5 update, journal session-011. The user says "the archive should say so explicitly" — add to D-057 (the source of the 11-block claim) a cross-reference note, and to journal/D-061.
2. Paper: add a line in the formalization appendix about E-029 upgrade: grid-measured 2n·|B̃₄″| max 0.189 → analytically proved bracket bound ≤ 1.316 < 2, presented as the measured-to-traced conversion worked example. Note the audit rule: any edit to draft.tex re-opens the audit for edited lines — keep the edit small and accurate. Let me look at the paper's formal-verification section to add appropriately.
Caution on accuracy: the Lean proof shows margiñ″ ≥ 1/(2n) − IIMaj″(n) and IIMaj″(n) < 1/(2n) via the bracket bound ψ̃″(s) − ψ̃′(s)/s < 2 for s ≥ √20 (value ≤ 1.316 < 2). E-029 grid-measured max 2n·|B̃₄″| = 0.189 over [20, 10⁴]. So the "≤ 1.316" is the bound on the bracket ψ̃″ − ψ̃′/s (which equals 2n·... hmm: IIMaj″(n) = [bracket]/(4n) and 1/(2n) threshold ⟺ bracket < 2. The grid measured 2n·|B̃₄″| max 0.189 → bracket = 4n·IIMaj″?? wait: B̃₄″(n) = [ψ̃″(s) − ψ̃′(s)/s]/(4n)·... let me recompute: IIMaj = B̃₄. IIMaj″(n) = [ψ̃″(s) − ψ̃′(s)/s]/(4n). So 2n·B̃₄″ = bracket/2 → grid max 0.189 ⟹ bracket ≈ 0.378. Our analytic bound: bracket ≤ 1.316 < 2. Consistent (bound ~3.5× loose vs measured peak). In the paper, phrase: "the grid-measured maximum 0.189 (E-029) is superseded by a fully analytic proof of the sufficient bound (bracket ≤ 1.316 < 2; iimaj_bracket_bound/IIMaj_deriv2_lt)".
Plan:
1. Edit D-057: append §6 "Lean formalization note (session 011)".
2. Edit D-061 §5 update: add the divergence note (already mentions 8 blocks — make the divergence explicit).
3. Journal session-011: add the design note line.
4. Paper draft.tex: add a paragraph in the formal verification appendix.
5. Rebuild paper (MiKTeX compile check), commit everything.
Let me first check the paper's formal verification section and D-057 tail.
Both are worth doing now. Let me look at the paper's formal-verification section and D-057's tail to place the notes.
tool: Bash ok
input
{
"command": "grep -n \"Formal verification\\|appendix\\|Appendix\" /d/Python/Quantum/research/paper/draft.tex | head -10; tail -8 /d/Python/Quantum/research/derivations/D-057-convexity-proof.md",
"description": "Locate paper formal verification section and D-057 tail"
}
result 2026-08-20 08:18:52Z → 2026-08-26 08:11:15Z
NguyenRH1/LiDogLeg.lean:815:11: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern (↑(2 * √↑n) + ↑t * I) / (↑(2 * √↑n) + ↑t * I - 1) in the target expression ‖((↑(2 * √↑n) + ↑t * I) / (↑(2 * √↑n - 1) + ↑t * I)) ^ n - 1‖ ≤ Real.exp (√↑n / 2) + 1 n : ℕ t : ℝ hn : 16 ≤ n ht : ↑n ≤ t ^ 2 NguyenRH1/LiDogLeg.lean:847:50: error: unsolved goals n : ℕ T U : ℝ hn : 16 ≤ n hT0 : 0 < T hTU : T ≤ U hTn : ↑n ≤ T ^ 2 hnorm : ‖VIntegral (Jlag n) (2 * √↑n) T U‖ = ‖∫ (t : ℝ) in T..U, Jlag n (↑(2 * √↑n) + ↑t * I)‖ t : ℝ ht : t ∈ Set.Ioc T U htT : T < t
raw content_json (database row)
[{"type":"thinking","thinking":"Two documentation tasks:\n\n1. Add a design note to the archive (D-061 update + journal) about the 11→8 block divergence: D-057's table has 11 rows, the Lean formalization uses 8 block types + separate IIMaj bound. Reason: B̃₄ (I_I term) is non-concave, handled by direct exponential bound instead of ψ-table. Note: actually D-057's table already handles I_I separately (§3) — the 11 blocks are for the OTHER terms. The Lean proof has 8 block types: id(#2), L1(#10), L2(#11), L1²(#7), L1·L2(#8), L2²(#9), T·L1(#5), T·L2(#6). Blocks #1, #3, #4 from D-057's table aren't needed because in the Lean formalization, F enters as a hypothesis-driven parameter (the 2F(T) term uses hψ directly, and there's no S_b(T) explicit majorant — blocks #1/#2 came from 2F(T) expansion, #3/#4 from 2S_b(T); in marginTilde as formalized, the F term is atomic). So the note should explain: D-057's 11 rows include blocks needed to expand the Trudgian F/S_b majorants; the Lean theorem is parameterized over F (with hψ as hypothesis), so only the 8 blocks from the closed-form Ĉ₀/Ĥ terms are formalized, and B̃₄ (non-concave) is handled by the separate analytic IIMaj″ bound. Thirty minutes of documentation.\n\nWhere to put it: D-057 (append a note), D-061 §5 update, journal session-011. The user says \"the archive should say so explicitly\" — add to D-057 (the source of the 11-block claim) a cross-reference note, and to journal/D-061.\n\n2. Paper: add a line in the formalization appendix about E-029 upgrade: grid-measured 2n·|B̃₄″| max 0.189 → analytically proved bracket bound ≤ 1.316 < 2, presented as the measured-to-traced conversion worked example. Note the audit rule: any edit to draft.tex re-opens the audit for edited lines — keep the edit small and accurate. Let me look at the paper's formal-verification section to add appropriately.\n\nCaution on accuracy: the Lean proof shows margiñ″ ≥ 1/(2n) − IIMaj″(n) and IIMaj″(n) < 1/(2n) via the bracket bound ψ̃″(s) − ψ̃′(s)/s < 2 for s ≥ √20 (value ≤ 1.316 < 2). E-029 grid-measured max 2n·|B̃₄″| = 0.189 over [20, 10⁴]. So the \"≤ 1.316\" is the bound on the bracket ψ̃″ − ψ̃′/s (which equals 2n·... hmm: IIMaj″(n) = [bracket]/(4n) and 1/(2n) threshold ⟺ bracket < 2. The grid measured 2n·|B̃₄″| max 0.189 → bracket = 4n·IIMaj″?? wait: B̃₄″(n) = [ψ̃″(s) − ψ̃′(s)/s]/(4n)·... let me recompute: IIMaj = B̃₄. IIMaj″(n) = [ψ̃″(s) − ψ̃′(s)/s]/(4n). So 2n·B̃₄″ = bracket/2 → grid max 0.189 ⟹ bracket ≈ 0.378. Our analytic bound: bracket ≤ 1.316 < 2. Consistent (bound ~3.5× loose vs measured peak). In the paper, phrase: \"the grid-measured maximum 0.189 (E-029) is superseded by a fully analytic proof of the sufficient bound (bracket ≤ 1.316 < 2; iimaj_bracket_bound/IIMaj_deriv2_lt)\".\n\nPlan:\n1. Edit D-057: append §6 \"Lean formalization note (session 011)\".\n2. Edit D-061 §5 update: add the divergence note (already mentions 8 blocks — make the divergence explicit).\n3. Journal session-011: add the design note line.\n4. Paper draft.tex: add a paragraph in the formal verification appendix.\n5. Rebuild paper (MiKTeX compile check), commit everything.\n\nLet me first check the paper's formal verification section and D-057 tail.","signature":""},{"type":"text","text":"Both are worth doing now. Let me look at the paper's formal-verification section and D-057's tail to place the notes."},{"type":"tool_use","id":"Bash_354","name":"Bash","input":{"command":"grep -n \"Formal verification\\|appendix\\|Appendix\" /d/Python/Quantum/research/paper/draft.tex | head -10; tail -8 /d/Python/Quantum/research/derivations/D-057-convexity-proof.md","description":"Locate paper formal verification section and D-057 tail"},"extra_content":null}]