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← Session 17 — Lean tier (i): elementary proofs (after crash)

#685 assistant 2026-08-20 07:20:43.316 UTC · 834→4208 tok open ⤢
thinking (8642 chars)
Now the C0hat² is expanded on both sides. Comparing LHS and RHS term by term (LHS shown first, truncated at top — I see only its tail: L2² terms, exp4·(−51591/50), IIMaj; RHS in full): RHS: - const: −969721/750 = −1292.96̄ - n·log n·(1/2) ✓ - n·lagariasC1 + nγ(−1/2) ✓ - √n·(−2996/15) ✓ id-block - √n·L1·(−16183/375) = −43.154̄ ✓ - √n·L2·(−886/375) ✓ - −F·2 ✓ - L1·(−2217251/7500) = −295.567 — L1-alone should be −252.4788! ✗ - L1·L2·(−4028199/125000) = −32.2255920 ✓ - L1·L2·exp4·(−4028199/125000) ✓ - L1·exp4·(−631197/2500) = −252.4788 ✓ - L1²·(−3937269/500000) = −7.8745380 ✓ - L1²·exp4·same ✓ - L2·(−818221/3750) = −218.1923̄ ✗ should be −215.8296 - L2·exp4·(−269787/1250) = −215.8296 ✓ - L2²·(−4121229/125000) = −32.9698320 ✓ - L2²·exp4 ✓ - exp4·(−51591/50) = −1031.82 ✓ - −IIMaj ✓ So RHS (MY expanded side) has L1-alone −295.567 (should −252.4788) and L2-alone −218.19 (should −215.8296). Differences: L1: 295.567 − 252.4788 = 43.088; L2: 218.1923 − 215.8296 = 2.3627. Hmm 43.088 ≈ 43.154̄ − 0.0667? −2217251/7500 = −295.56680. Expected −252.4788. Diff = 43.0880. And 43.088 = ? mTL1b's w = 43.1546̄. Not equal. L2 diff = 2.3626̄ = TL2's w exactly. Hmm — the L1-alone diff 43.0880 — what is it? 42·(2·0.433·2.9) + 147 = 105.4788 + 147 = 252.4788 expected L1 coefficient. −2217251/7500 = −295.56680. 295.56680 − 252.4788 = 43.0880. Hmm 43.088 = 42·1.0259...? or (8/3)·0.433·... no: 43.154̄ (TL1's w) − 0.0667. What's 43.088? = 42 + 1.088 = 42 + (8/3)·0.408?? Hmm 1.088 = (8/3)·0.408 — no. 43.088·375 = 16158 — vs √n·L1 coefficient −16183/375 ✓ TL1. So BOTH sides have separate √n·L1 (TL1) and L1-alone (mL1b) terms — but the COEFFICIENTS of the L1-alone differ: LHS? The LHS (def side) got truncated at top so I can't see its L1-alone. Both sides have all the SAME monomials — so the difference is purely in coefficients of L1-alone and L2-alone? But ring computes the difference — the remaining goal IS the difference (LHS − RHS = the unsolved thing). Wait no — the "unsolved goals" display shows `LHS = RHS` as a giant equation — ring FAILED to close it, i.e. the two sides differ. Since most coefficients match, compare the displayed LHS tail vs RHS: LHS ends: `Real.log (Real.log (3 + √n)) ^ 2 * (-4121229 / 125000) + Real.log (Real.log (3 + √n)) ^ 2 * Real.exp 4 * (-4121229 / 125000) + Real.exp 4 * (-51591 / 50) - IIMaj n` — SAME as RHS's corresponding ✓. The LHS's top (cut) contains its L1-alone/L2-alone/const — the MISMATCH must be there. RHS const: −969721/750 = −1292.9613̄; LHS const (from earlier display): −966721/750 = −1289.0̄ (earlier build) or −350903/375. Hmm. Let me compute the expected constants: DEF side const: −4 (S) − 40·1.3352 − 1031.82·(e⁴+1) − 1 + 1 = −4 − 53.408 − 1031.82·exp4 − 1031.82. So exp4·(−1031.82) and const-alone −(4+53.408+1031.82) = −1089.228. RHS (mine): exp4·(−51591/50) = −1031.82 ✓; const-alone: mK = −4 − 53.408 − 2(e⁴+1)·515.91 − 1 + 1: = −58.408 − 1031.82·exp4 − 1031.82 = −1090.228?? wait mK = −4 − 40·1.3352 − 2·(e⁴+1)·515.91 − 1 + 1: the −1+1 cancels! mK = −4 − 53.408 − 1031.82(e⁴+1) = −58.408 − 1031.82e⁴ − 1031.82 → const-alone: −58.408 − 1031.82 = −1090.228. DEF side: −4 − 53.408 − 1031.82(e⁴+1) + (−1+1 from B̃'s +1 and margin's +1: B̃ has +1, margin has +1: −(B̃) + 1 → −... − 1 (from B̃) + 1 (outer) = 0 ✓ so DEF const-alone: −4 − 53.408 − 1031.82 = −1089.228. MY mK: I wrote `... - 1 + 1` — mK := −4 − 40·1.3352 − 2(e⁴+1)(...) − 1 + 1 — that's −4 − 53.408 − 1031.82(e⁴+1) − 1 + 1 = same as DEF ✓ = −1089.228?? −4 −53.408 = −57.408; −57.408 −1031.82 = −1089.228 ✓ (my arithmetic above slipped: −58.408 was wrong; −4 − 53.408 = −57.408 ✓). And RHS display const −969721/750 = −1292.961̄ — that doesn't match −1089.228 either! −969721/750: −1292.96. Hmm — what's IN the RHS const: mK PLUS other constant contributions? My pieces: mIdb has (−((8/3)*2.9+192))·(√m+1) — the +1 contributes −199.73̄ to const! And DEF side's 4T·G's constant part: T·199.73̄ = √n·199.73̄ + 199.73̄ — both sides have it ✓ (the √n·(−2996/15) term ✓ and const +199.73̄·(−1) = −199.73̄). So RHS const should be −1089.228 − 199.733 = −1288.96 — vs displayed −1292.96: off by 4. And LHS const (first display): −350903/375 = −935.74 + ... no that was pre-simp. Current LHS top is cut. Hmm wait — actually where does mK's −4 (from S) and the S-piece's −4 collide? My mS piece: `fun m => (1/2)*m*Real.log m + c*m − 4` — INCLUDES −4 ✓. And mK := −4 − ... — I DOUBLE-COUNTED the −4!! The marginTilde def: margin = (S-part with −4) − B̃ + 1. My mS has −4 ✓ and mK also has −4 ✗. mK should be: −40·1.3352 − 2(e⁴+1)(7(3·2.9²+48)+3.3) − 1 + 1 = −53.408 − 1031.82(e⁴+1). So mK WITHOUT the leading −4. Then RHS const: −53.408 − 1031.82 − 199.733 = −1284.96. DEF: −4 − 53.408 − 1031.82 − 199.733 = −1288.96. Difference 4 = the double-count ✓ consistent with "off by 4". Now the L1-alone discrepancy: RHS −295.567 vs expected −252.4788: diff 43.088 — where from? mTL1b's composition: (fun t => t·log(t+2))(√m+1) = (√m+1)·L1 = √n·L1 + L1!! The TL1 piece contributes L1-alone too! DEF side's 4T·G: T·(43.154̄·L1) → √n·43.154̄·L1 + 43.154̄·L1. So BOTH sides have L1-alone +43.154̄·(−1) = −43.154̄ from the TL blocks ✓ — so L1-alone expected: −252.4788 − 43.154̄ = −295.633 vs RHS −295.567: off by 0.0667?? −2217251/7500 = −295.56680; expected −295.63347. Diff 0.06667 = 1/15. Hmm — √n·L1 coefficient −16183/375 = −43.15467 (TL1 w = 43.154̄ ✓ exact (8/3)·0.433+42 = 1.154̄+42 = 43.154̄ = 16183/375 ✓). L1-alone from TL1: same −16183/375 = −43.15467. mL1b: −252.4788·(e⁴+1) → L1-alone −252.4788. Total: −295.63347. RHS display: −2217251/7500 = −295.56680. Diff: 295.63347 − 295.56680 = 0.06667 = 1/15 = 0.066̄. Hmm — so mL1b's weight is off by 1/15?? My mL1b weight: (42*(2*0.433*2.9) + 147)·(e⁴+1): 2·0.433·2.9 = 2.5114; ×42 = 105.4788 ✓ +147 = 252.4788 ✓. Display RHS L1·exp4: −631197/2500 = −252.4788 ✓ correct! And L1-alone: −2217251/7500 = −295.56680 — but should be −(252.4788 + 43.15467) = −295.63347. Difference exactly 1/15. So the DEF side computes L1-alone as −295.56680? or the RHS — the display shows LHS = RHS with the RHS (my side) having −2217251/7500. Wait which side has −2217251/7500 — from the structure of the output (after `IIMaj n =`), it's the RHS = MY side. So my side's L1-alone = −295.5668, expected −295.6335 — my side is SHORT by 1/15. Components of my L1-alone: mL1b: −252.4788; mTL1b's (√m+1)-expansion: −43.15467·L1 − total −295.6335. Displayed −295.5668 → my mTL1b only contributed −43.0880?? = 43.15467 − 0.0667. Hmm 43.088 = 42 + 1.088 = 42 + (8/3)·0.408?? Odd. OR — mTL1b's weight: −((8/3)*0.433 + 42): 8·0.433 = 3.464; /3 = 1.154666...; +42 = 43.154666... ✓ = 16183/375?? 16183/375 = 43.154666... ✓ consistent. So where does 1/15 go missing? Wait — maybe the DEF side's TL1 coefficient is NOT (8/3)·0.433+42: 4T·G with G = (2/3)·C + 10.5·L1 + 48, C = 0.433·L1 + 0.886·L2 + 2.9: 4T·(2/3)·0.433·L1 = T·L1·(8/3)(0.433) ✓ and 4T·10.5·L1 = 42·T·L1 ✓ — TL1 w = (8/3)(0.433)+42 ✓. And mTL1b matches ✓. Hmm, actually — recheck: is it maybe the mL1b on the DEF side: 2(e⁴+1)(7H+3.3): 7H's L1 part: 7·(10.5·L1) = 73.5·L1 — WAIT: H = 3C² + 10.5·L1 + 48: 7H = 21C² + 73.5·L1 + 336: then 2(e⁴+1)·(7H+3.3) = (e⁴+1)·[42C² + 147·L1 + 672 + 6.6] ✓ 147·L1 ✓ (7·10.5 = 73.5, ×2 = 147 ✓) fine. And 42C²'s L1: 42·(2·0.433·2.9)·L1 = 42·2.5114 = 105.4788 ✓. So DEF-side L1-alone: −[105.4788 + 147]·(e⁴+1) − 43.15467(TL1's) = −252.4788·e⁴ − 252.4788 − 43.15467 — as normalized: L1-alone: −252.4788 − 43.15467 = −295.6335; L1·exp4: −252.4788. MY side matches this with mL1b −252.4788(e⁴+1) + mTL1b −43.15467·(√n+1)L1 → L1-alone −43.15467. Total my side: −295.6335 ✓ same! But display shows my side −295.5668... Let me recompute −2217251/7500 precisely: 2217251/7500 = 295.633466...!! 2217251/7500: 7500·295 = 2212500; remainder 4751; 4751/7500 = 0.6335; so 295.6335 ✓✓ — I MISDIVIDED earlier (295.567 was wrong arithmetic). So L1-alone MATCHES ✓. L2-alone: RHS −818221/3750 = 218.19226̄. Expected: −215.8296 (mL2b) − 2.362̄ (TL2's (√n+1) contribution: −886/375 = −2.3626̄) = −218.1923 ✓ MATCHES ✓. const: RHS −969721/750 = −1292.9613̄. Expected: −1089.228 − 199.73̄ (mIdb's +1) = −1288.961̄. Difference: 1292.9613 − 1288.9613 = 4.0 = the double-counted −4 in mK ✓ (as diagnosed). Also check LHS const — cut off, but if the only mismatch is the 4, ring would leave a residual −4 = 0-ish goal... the unsolved goal shows the FULL equation (ring failed on the whole thing? or displays the remaining difference). The remaining difference might be JUST the constant −4. Let me fix mK (remove leading −4) and see if ring closes. Fix: mK := -40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1.
Found it: `mK` double-counts the `−4` (already inside the S-piece). Fixing.
tool: Edit ok
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
  "old_string": "private def mK : \u211d := -4 - 40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1",
  "new_string": "private def mK : \u211d := -40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1"
}
result 2026-08-20 07:20:43Z → 2026-08-26 22:10:19Z
{
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiClosed.lean",
  "oldString": "/-- **THE CLOSED FINITE-HEIGHT LI THEOREM** (R-004, ii\u2032-7 endpoint):\n\u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074, from the eight\ndeclared hypotheses (two computational certificates + six analytic\nperimeter Props). The session-012 named analytic hypothesis\n`SfBoundProp` is discharged by ii\u2032-6's `sfBoundProp`. -/",
  "newString": "/-- **THE CLOSED FINITE-HEIGHT LI THEOREM** (R-004, ii\u2032-7 endpoint):\n\u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074, from the SEVEN\ndeclared hypotheses (two computational certificates + five analytic\nperimeter Props). The session-012 named analytic hypothesis\n`SfBoundProp` is discharged by ii\u2032-6's `sfBoundProp`. -/",
  "originalFile": "/-\nLiClosed.lean \u2014 tier ii\u2032-7: THE CLOSED finite-height Li theorem (R-004\nendpoint, D-062 \u00a74). `lambda_pos_finite_height_closed`: \u03bb_n > 0 for all\n1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 from the SEVEN declared hypotheses:\nthe two computational certificates (Platt\u2013Trudgian `RHUpTo`; the finite\nLi verification `LiVerifiedUpTo` \u2014 E-011/E-015 to 3000 + Johansson 2015\n\u00a7 4.2 to 10\u2075) and the five analytic perimeter Props (Trudgian N(T)/S(T):\n`ZeroCountProp` unit-window, `ZeroCountProp2` two-unit,\n`ZeroCountGlobalProp` global; the Platt-class zero-location certificate\n`NoLowZeros`; the M\u2080 interval certificate `MZeroProp`) \u2014 the session-012\nnamed analytic hypothesis `SfBoundProp` is now PROVED (tier ii\u2032-6,\nLiSideBounds).\n-/\nimport Mathlib\nimport NguyenRH1.LiBasePoints\nimport NguyenRH1.LiSideBounds\n\nopen Zeta23 Complex Filter Topology\n\nnamespace NguyenRH1\n\n/-- **THE CLOSED FINITE-HEIGHT LI THEOREM** (R-004, ii\u2032-7 endpoint):\n\u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074, from the eight\ndeclared hypotheses (two computational certificates + six analytic\nperimeter Props). The session-012 named analytic hypothesis\n`SfBoundProp` is discharged by ii\u2032-6's `sfBoundProp`. -/\ntheorem lambda_pos_finite_height_closed\n    (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (10 ^ 5))\n    (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp)\n    (hNLZ : NoLowZeros) (hMZ : MZeroProp) :\n    \u2200 n : \u2115, 1 \u2264 n \u2192 (n : \u211d) \u2264 (3 * 10 ^ 12 - 1 : \u211d) ^ 2 \u2192\n      0 < (liCoeff n).re :=\n  lambda_pos_finite_height_verified hRH hfinite\n    (sfBoundProp hRH hZC hZC2 hZCG hNLZ hMZ)\n\nend NguyenRH1\n",
  "structuredPatch": [
    {
      "oldStart": 20,
      "oldLines": 8,
      "newStart": 20,
      "newLines": 8,
      "lines": [
        " namespace NguyenRH1",
        " ",
        " /-- **THE CLOSED FINITE-HEIGHT LI THEOREM** (R-004, ii\u2032-7 endpoint):",
        "-\u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074, from the eight",
        "-declared hypotheses (two computational certificates + six analytic",
        "+\u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074, from the SEVEN",
        "+declared hypotheses (two computational certificates + five analytic",
        " perimeter Props). The session-012 named analytic hypothesis",
        " `SfBoundProp` is discharged by ii\u2032-6's `sfBoundProp`. -/",
        " theorem lambda_pos_finite_height_closed"
      ]
    }
  ],
  "userModified": false,
  "replaceAll": false
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raw content_json (database row) 
[{"type":"thinking","thinking":"Now the C0hat² is expanded on both sides. Comparing LHS and RHS term by term (LHS shown first, truncated at top — I see only its tail: L2² terms, exp4·(−51591/50), IIMaj; RHS in full):\n\nRHS:\n- const: −969721/750 = −1292.96̄\n- n·log n·(1/2) ✓\n- n·lagariasC1 + nγ(−1/2) ✓\n- √n·(−2996/15) ✓ id-block\n- √n·L1·(−16183/375) = −43.154̄ ✓\n- √n·L2·(−886/375) ✓\n- −F·2 ✓\n- L1·(−2217251/7500) = −295.567 — L1-alone should be −252.4788! ✗\n- L1·L2·(−4028199/125000) = −32.2255920 ✓\n- L1·L2·exp4·(−4028199/125000) ✓\n- L1·exp4·(−631197/2500) = −252.4788 ✓\n- L1²·(−3937269/500000) = −7.8745380 ✓\n- L1²·exp4·same ✓\n- L2·(−818221/3750) = −218.1923̄ ✗ should be −215.8296\n- L2·exp4·(−269787/1250) = −215.8296 ✓\n- L2²·(−4121229/125000) = −32.9698320 ✓\n- L2²·exp4 ✓\n- exp4·(−51591/50) = −1031.82 ✓\n- −IIMaj ✓\n\nSo RHS (MY expanded side) has L1-alone −295.567 (should −252.4788) and L2-alone −218.19 (should −215.8296). Differences: L1: 295.567 − 252.4788 = 43.088; L2: 218.1923 − 215.8296 = 2.3627. Hmm 43.088 ≈ 43.154̄ − 0.0667? −2217251/7500 = −295.56680. Expected −252.4788. Diff = 43.0880. And 43.088 = ? mTL1b's w = 43.1546̄. Not equal. L2 diff = 2.3626̄ = TL2's w exactly. Hmm — the L1-alone diff 43.0880 — what is it? 42·(2·0.433·2.9) + 147 = 105.4788 + 147 = 252.4788 expected L1 coefficient. −2217251/7500 = −295.56680. 295.56680 − 252.4788 = 43.0880. Hmm 43.088 = 42·1.0259...? or (8/3)·0.433·... no: 43.154̄ (TL1's w) − 0.0667. What's 43.088? = 42 + 1.088 = 42 + (8/3)·0.408?? Hmm 1.088 = (8/3)·0.408 — no. 43.088·375 = 16158 — vs √n·L1 coefficient −16183/375 ✓ TL1.\n\nSo BOTH sides have separate √n·L1 (TL1) and L1-alone (mL1b) terms — but the COEFFICIENTS of the L1-alone differ: LHS? The LHS (def side) got truncated at top so I can't see its L1-alone. Both sides have all the SAME monomials — so the difference is purely in coefficients of L1-alone and L2-alone? But ring computes the difference — the remaining goal IS the difference (LHS − RHS = the unsolved thing). Wait no — the \"unsolved goals\" display shows `LHS = RHS` as a giant equation — ring FAILED to close it, i.e. the two sides differ. Since most coefficients match, compare the displayed LHS tail vs RHS: LHS ends: `Real.log (Real.log (3 + √n)) ^ 2 * (-4121229 / 125000) + Real.log (Real.log (3 + √n)) ^ 2 * Real.exp 4 * (-4121229 / 125000) + Real.exp 4 * (-51591 / 50) - IIMaj n` — SAME as RHS's corresponding ✓. The LHS's top (cut) contains its L1-alone/L2-alone/const — the MISMATCH must be there. RHS const: −969721/750 = −1292.9613̄; LHS const (from earlier display): −966721/750 = −1289.0̄ (earlier build) or −350903/375. Hmm.\n\nLet me compute the expected constants:\nDEF side const: −4 (S) − 40·1.3352 − 1031.82·(e⁴+1) − 1 + 1 = −4 − 53.408 − 1031.82·exp4 − 1031.82. So exp4·(−1031.82) and const-alone −(4+53.408+1031.82) = −1089.228. RHS (mine): exp4·(−51591/50) = −1031.82 ✓; const-alone: mK = −4 − 53.408 − 2(e⁴+1)·515.91 − 1 + 1: = −58.408 − 1031.82·exp4 − 1031.82 = −1090.228?? wait mK = −4 − 40·1.3352 − 2·(e⁴+1)·515.91 − 1 + 1: the −1+1 cancels! mK = −4 − 53.408 − 1031.82(e⁴+1) = −58.408 − 1031.82e⁴ − 1031.82 → const-alone: −58.408 − 1031.82 = −1090.228. DEF side: −4 − 53.408 − 1031.82(e⁴+1) + (−1+1 from B̃'s +1 and margin's +1: B̃ has +1, margin has +1: −(B̃) + 1 → −... − 1 (from B̃) + 1 (outer) = 0 ✓ so DEF const-alone: −4 − 53.408 − 1031.82 = −1089.228. MY mK: I wrote `... - 1 + 1` — mK := −4 − 40·1.3352 − 2(e⁴+1)(...) − 1 + 1 — that's −4 − 53.408 − 1031.82(e⁴+1) − 1 + 1 = same as DEF ✓ = −1089.228?? −4 −53.408 = −57.408; −57.408 −1031.82 = −1089.228 ✓ (my arithmetic above slipped: −58.408 was wrong; −4 − 53.408 = −57.408 ✓). And RHS display const −969721/750 = −1292.961̄ — that doesn't match −1089.228 either! −969721/750: −1292.96. Hmm — what's IN the RHS const: mK PLUS other constant contributions? My pieces: mIdb has (−((8/3)*2.9+192))·(√m+1) — the +1 contributes −199.73̄ to const! And DEF side's 4T·G's constant part: T·199.73̄ = √n·199.73̄ + 199.73̄ — both sides have it ✓ (the √n·(−2996/15) term ✓ and const +199.73̄·(−1) = −199.73̄). So RHS const should be −1089.228 − 199.733 = −1288.96 — vs displayed −1292.96: off by 4. And LHS const (first display): −350903/375 = −935.74 + ... no that was pre-simp. Current LHS top is cut.\n\nHmm wait — actually where does mK's −4 (from S) and the S-piece's −4 collide? My mS piece: `fun m => (1/2)*m*Real.log m + c*m − 4` — INCLUDES −4 ✓. And mK := −4 − ... — I DOUBLE-COUNTED the −4!! The marginTilde def: margin = (S-part with −4) − B̃ + 1. My mS has −4 ✓ and mK also has −4 ✗. mK should be: −40·1.3352 − 2(e⁴+1)(7(3·2.9²+48)+3.3) − 1 + 1 = −53.408 − 1031.82(e⁴+1). So mK WITHOUT the leading −4. Then RHS const: −53.408 − 1031.82 − 199.733 = −1284.96. DEF: −4 − 53.408 − 1031.82 − 199.733 = −1288.96. Difference 4 = the double-count ✓ consistent with \"off by 4\".\n\nNow the L1-alone discrepancy: RHS −295.567 vs expected −252.4788: diff 43.088 — where from? mTL1b's composition: (fun t => t·log(t+2))(√m+1) = (√m+1)·L1 = √n·L1 + L1!! The TL1 piece contributes L1-alone too! DEF side's 4T·G: T·(43.154̄·L1) → √n·43.154̄·L1 + 43.154̄·L1. So BOTH sides have L1-alone +43.154̄·(−1) = −43.154̄ from the TL blocks ✓ — so L1-alone expected: −252.4788 − 43.154̄ = −295.633 vs RHS −295.567: off by 0.0667?? −2217251/7500 = −295.56680; expected −295.63347. Diff 0.06667 = 1/15. Hmm — √n·L1 coefficient −16183/375 = −43.15467 (TL1 w = 43.154̄ ✓ exact (8/3)·0.433+42 = 1.154̄+42 = 43.154̄ = 16183/375 ✓). L1-alone from TL1: same −16183/375 = −43.15467. mL1b: −252.4788·(e⁴+1) → L1-alone −252.4788. Total: −295.63347. RHS display: −2217251/7500 = −295.56680. Diff: 295.63347 − 295.56680 = 0.06667 = 1/15 = 0.066̄. Hmm — so mL1b's weight is off by 1/15?? My mL1b weight: (42*(2*0.433*2.9) + 147)·(e⁴+1): 2·0.433·2.9 = 2.5114; ×42 = 105.4788 ✓ +147 = 252.4788 ✓. Display RHS L1·exp4: −631197/2500 = −252.4788 ✓ correct! And L1-alone: −2217251/7500 = −295.56680 — but should be −(252.4788 + 43.15467) = −295.63347. Difference exactly 1/15. So the DEF side computes L1-alone as −295.56680? or the RHS — the display shows LHS = RHS with the RHS (my side) having −2217251/7500. Wait which side has −2217251/7500 — from the structure of the output (after `IIMaj n =`), it's the RHS = MY side. So my side's L1-alone = −295.5668, expected −295.6335 — my side is SHORT by 1/15. Components of my L1-alone: mL1b: −252.4788; mTL1b's (√m+1)-expansion: −43.15467·L1 − total −295.6335. Displayed −295.5668 → my mTL1b only contributed −43.0880?? = 43.15467 − 0.0667. Hmm 43.088 = 42 + 1.088 = 42 + (8/3)·0.408?? Odd. OR — mTL1b's weight: −((8/3)*0.433 + 42): 8·0.433 = 3.464; /3 = 1.154666...; +42 = 43.154666... ✓ = 16183/375?? 16183/375 = 43.154666... ✓ consistent. So where does 1/15 go missing?\n\nWait — maybe the DEF side's TL1 coefficient is NOT (8/3)·0.433+42: 4T·G with G = (2/3)·C + 10.5·L1 + 48, C = 0.433·L1 + 0.886·L2 + 2.9: 4T·(2/3)·0.433·L1 = T·L1·(8/3)(0.433) ✓ and 4T·10.5·L1 = 42·T·L1 ✓ — TL1 w = (8/3)(0.433)+42 ✓. And mTL1b matches ✓.\n\nHmm, actually — recheck: is it maybe the mL1b on the DEF side: 2(e⁴+1)(7H+3.3): 7H's L1 part: 7·(10.5·L1) = 73.5·L1 — WAIT: H = 3C² + 10.5·L1 + 48: 7H = 21C² + 73.5·L1 + 336: then 2(e⁴+1)·(7H+3.3) = (e⁴+1)·[42C² + 147·L1 + 672 + 6.6] ✓ 147·L1 ✓ (7·10.5 = 73.5, ×2 = 147 ✓) fine.\n\nAnd 42C²'s L1: 42·(2·0.433·2.9)·L1 = 42·2.5114 = 105.4788 ✓.\n\nSo DEF-side L1-alone: −[105.4788 + 147]·(e⁴+1) − 43.15467(TL1's) = −252.4788·e⁴ − 252.4788 − 43.15467 — as normalized: L1-alone: −252.4788 − 43.15467 = −295.6335; L1·exp4: −252.4788. MY side matches this with mL1b −252.4788(e⁴+1) + mTL1b −43.15467·(√n+1)L1 → L1-alone −43.15467. Total my side: −295.6335 ✓ same! But display shows my side −295.5668... Let me recompute −2217251/7500 precisely: 2217251/7500 = 295.633466...!! 2217251/7500: 7500·295 = 2212500; remainder 4751; 4751/7500 = 0.6335; so 295.6335 ✓✓ — I MISDIVIDED earlier (295.567 was wrong arithmetic). So L1-alone MATCHES ✓.\n\nL2-alone: RHS −818221/3750 = 218.19226̄. Expected: −215.8296 (mL2b) − 2.362̄ (TL2's (√n+1) contribution: −886/375 = −2.3626̄) = −218.1923 ✓ MATCHES ✓.\n\nconst: RHS −969721/750 = −1292.9613̄. Expected: −1089.228 − 199.73̄ (mIdb's +1) = −1288.961̄. Difference: 1292.9613 − 1288.9613 = 4.0 = the double-counted −4 in mK ✓ (as diagnosed). Also check LHS const — cut off, but if the only mismatch is the 4, ring would leave a residual −4 = 0-ish goal... the unsolved goal shows the FULL equation (ring failed on the whole thing? or displays the remaining difference). The remaining difference might be JUST the constant −4. Let me fix mK (remove leading −4) and see if ring closes.\n\nFix: mK := -40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1.","signature":""},{"type":"text","text":"Found it: `mK` double-counts the `−4` (already inside the S-piece). Fixing."},{"type":"tool_use","id":"Edit_294","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"private def mK : ℝ := -4 - 40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1","new_string":"private def mK : ℝ := -40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1"},"extra_content":null}]