← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (1384 chars)
`tendstoUniformlyOn_tsum_nat (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) : TendstoUniformlyOn (fun N => fun x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' n, f n x) atTop s` — the range-indexed M-test ✓ EXACTLY what I need for hC.
Let me now rewrite the whole Section 2 with all fixes:
1. h2's e0: match the beta-reduced form `1 / (x + ↑0)`.
2. hcle/hCle: membership-only application (hcle hx).
3. dpsiTermR_nonneg hx0 m (add m).
4. The n : ℕ annotations in hA's lambdas.
5. hC via tendstoLocallyUniformlyOn_iff_forall_isCompact + tendstoUniformlyOn_tsum_nat.
6. hdecomp/hlimit congr structure (that part was fine but the congr directions — let me double check: `TendstoLocallyUniformlyOn.congr (hf : TendstoLocallyUniformlyOn F f p s) (hg : ∀ n, s.EqOn (F n) (G n)) : TendstoLocallyUniformlyOn G f p s` — so hf is the PROVED convergence (of the decomposed sequence) and G := dpsiSeqR — hg n : EqOn (F n) (dpsiSeqR n) — F n := the decomposed lambda — EqOn (decomposed n) (dpsiSeqR n) means decomposed n x = dpsiSeqR n x for x ∈ s — that's (dpsiSeqR_eq n x).symm ✓ so `.congr (fun n x _ => (dpsiSeqR_eq n x).symm)` ✓ as I had.
And congr_right (hg : s.EqOn f g): f := decomposed-limit, g := DinfR — hg : ∀ x ∈ s, decomplimit x = DinfR x — my hlimit ✓.
Let me rewrite the whole Section 2 again with these fixes. Actually the fixes are localized; let me do targeted edits.
`tendstoUniformlyOn_tsum_nat` is the right form. Applying targeted fixes.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " rw [Finset.sum_congr rfl (fun i _ => e i)]\n have e0 : (fun m => 1 / (x + (m : \u211d))) 0 = 1 / x := by simp\n rw [e0]\n ring",
"new_string": " rw [Finset.sum_congr rfl (fun i _ => e i)]\n have e0 : 1 / (x + ((0 : \u2115) : \u211d)) = 1 / x := by simp\n rw [e0]\n ring"
}
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " by_cases hKe : K = \u2205\n \u00b7 subst hKe\n exact \u27e80, summable_zero, fun m x hx => by simp at hx\u27e9\n \u00b7 obtain \u27e8x0, hx0\u27e9 := Set.nonempty_iff_ne_empty.mpr hKe\n obtain \u27e8c, hcK, hcle\u27e9 := hKc.exists_isMinOn \u27e8x0, hx0\u27e9 (continuous_id.continuousOn)\n obtain \u27e8C, hCK, hCle\u27e9 := hKc.exists_isMaxOn \u27e8x0, hx0\u27e9 (continuous_id.continuousOn)\n have hc0 : 0 < c := hK hcK\n refine \u27e8fun m => C / ((m : \u211d) + 1) ^ 2, summable_div_add_one_sq C, fun m x hx => ?_\u27e9\n have hx0 : 0 < x := hK hx\n have hxC : x \u2264 C := hCle x hx\n rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0)]\n calc dpsiTermR x m \u2264 x / ((m : \u211d) + 1) ^ 2 := dpsiTermR_le hx0 m\n _ \u2264 C / ((m : \u211d) + 1) ^ 2 :=\n (div_le_div_iff_of_pos_right (by positivity : (0 : \u211d) < ((m : \u211d) + 1) ^ 2)).mpr hxC",
"new_string": " by_cases hKe : K = \u2205\n \u00b7 subst hKe\n exact \u27e80, summable_zero, fun m x hx => by simp at hx\u27e9\n \u00b7 obtain \u27e8x0, hx0\u27e9 := Set.nonempty_iff_ne_empty.mpr hKe\n obtain \u27e8C, hCK, hCle\u27e9 := hKc.exists_isMaxOn \u27e8x0, hx0\u27e9 (continuous_id.continuousOn)\n refine \u27e8fun m => C / ((m : \u211d) + 1) ^ 2, summable_div_add_one_sq C, fun m x hx => ?_\u27e9\n have hx0 : 0 < x := hK hx\n have hxC : x \u2264 C := hCle hx\n rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0 m)]\n calc dpsiTermR x m \u2264 x / ((m : \u211d) + 1) ^ 2 := dpsiTermR_le hx0 m\n _ \u2264 C / ((m : \u211d) + 1) ^ 2 :=\n (div_le_div_iff_of_pos_right (by positivity : (0 : \u211d) < ((m : \u211d) + 1) ^ 2)).mpr hxC"
}
result 2026-08-22 07:18:37Z → 2026-08-26 22:38:11Z
{
"filePath": "D:\\Python\\Quantum\\research\\results\\R-004-finite-height-li-theorem.md",
"oldString": "4. Bridge: RH to height \u221an+1 \u27f9 each zero contributes \u2264 2 \u27f9\n |\u03bb_n(T)| \u2264 2N(T); holds for n \u2264 (T\u2080\u22121)\u00b2.\n5. margin(n) := S_\u221e^lower \u2212 B + 1 is convex for n \u2265 20 (PROVED D-057;\n E-029), margin(8\u00d710\u2074) = 16 020 > 0 with positive derivative \u27f9\n margin(n) > 0 for all n \u2208 [8\u00d710\u2074, N\u2081] (E-028;\n margin(N\u2081) = 2.458\u00d710\u00b2\u2076).\n6. Finite verification: n \u2264 3000 (E-011/E-015, cross-validated,\n ledger #3) \u222a n \u2264 5\u00d710\u2075 (Misguich\u2013Voros) \u2014 overlaps N\u2080 = 8\u00d710\u2074\n by a factor 6.25.\n\n## Dependencies\nR-001 (finite verification), D-032/D-033 (design + wall), D-036 (contour\nconstants), D-037 (assembly), E-021 (exact S_\u221e/measured constants),\nE-023 (assembly numerics), Trudgian 2014 [PROVED lit], Platt\u2013Trudgian\n2021 [computational], Lagarias math/0404394, Bombieri\u2013Lagarias 1999,\nVoros 1703.02844 [Misguich numerics].",
"newString": "4. Bridge (CORRECTED ii\u2032-6, F-A): RH to height \u221an+1 \u27f9 each zero\n contributes \u2264 2 \u27f9 |\u03bb_n(T)| \u2264 4F(T) (both signs of \u03b3; F the global\n Trudgian majorant); holds for n \u2264 (T\u2080\u22121)\u00b2.\n5. margin(n) := S_\u221e^lower \u2212 B + 1 is convex for n \u2265 20 (PROVED D-057;\n E-029), margin(8\u00d710\u2074) = 16 020 > 0 with positive derivative \u27f9\n margin(n) > 0 for all n \u2208 [8\u00d710\u2074, N\u2081] (E-028;\n margin(N\u2081) = 2.458\u00d710\u00b2\u2076).\n6. Finite verification (CORRECTED session 042): n \u2264 3000 (E-011/E-015,\n cross-validated, ledger #3) \u222a n \u2264 10\u2075 (Johansson 2015 \u00a7 4.2,\n rigorous interval-arithmetic ordinary-\u03bb_n table, all positive) \u2014\n covers N\u2080 = 8\u00d710\u2074 by a factor 1.25 and meets the n = 10\u2075 split\n exactly. [Supersedes: \"n \u2264 5\u00d710\u2075 (Misguich\u2013Voros)\" \u2014 that source\n verifies a deformed sequence, not the ordinary \u03bb_n.]\n\n## Dependencies\nR-001 (finite verification), D-032/D-033 (design + wall), D-036 (contour\nconstants), D-037 (assembly), E-021 (exact S_\u221e/measured constants),\nE-023 (assembly numerics), Trudgian 2014 [PROVED lit], Platt\u2013Trudgian\n2021 [computational], Lagarias math/0404394, Bombieri\u2013Lagarias 1999,\nJohansson 2015 (Numer. Algor. 69:253\u2013270, \u00a7 4.2) [finite verification].",
"originalFile": "# R-004 \u2014 Explicit finite-height Li theorem: \u03bb_n > 0 for all n \u2264 (T\u2080\u22121)\u00b2 = 8.999\u2026\u00d710\u00b2\u2074\nDate: 2026-08-19 | Session: 008 | Track: C (task 3.8)\nTag: **[PROVED-CONDITIONAL(computational RH-to-height-T\u2080; computational\nfinite verification)]** \u2014 R-bound FULLY TRACED 2026-08-19 (D-056,\ntask 4.5a: reference-point method, A = 10.5, B\u2080 = 48 with the honest\nC\u2080 incl. log log term; digamma remainder 1.2/|w|\u00b2 proved; E-028\nverified, 26\u00d7 slack); N\u2080 = 80 000 (1.25\u00d7 overlap with\nJohansson's 10\u2075 table). M\u2080 remains grid-verified (E-023 S1; additive\n~53). Margin convexity: PROVED (D-057, task 4.5b; E-029 ALL CHECKS\nPASS). History: tag downgraded then restored same day (D-054 erratum:\nbroken D-039/D-036-\u00a71 proofs; repairs D-056\u2013D-059).\nLean closure (2026-08-26, session 041, tier ii\u2032 COMPLETE): the analytic\nchain is now FULLY MACHINE-PROVED \u2014 `LiSideBounds.sfBoundProp` (the\ncontour bound |S_f(n)| \u2264 Bn FAlt n on [10\u2075, N\u2081]) and\n`LiClosed.lambda_pos_finite_height_closed` (\u03bb_n > 0 for all\n1 \u2264 n \u2264 N\u2081) in Lean 4, from exactly SEVEN declared hypotheses:\nRHUpTo (Platt\u2013Trudgian), LiVerifiedUpTo(10\u2075) (finite Li verification:\nE-011/E-015 to 3000 + Johansson 2015 \u00a7 4.2, rigorous interval-arithmetic\nordinary-\u03bb_n table to 10\u2075), ZeroCountProp/ZeroCountProp2/\nZeroCountGlobalProp (Trudgian N(T)/S(T) forms), NoLowZeros (Platt-class\nzero-location), MZeroProp (M\u2080 interval certificate). Axiom audit on the\nfull chain: [propext, Classical.choice, Quot.sound] only, 0 sorryAx.\nThe B(n) account was corrected in four places during the closure\n(F-A/F-B/F-D/F-E \u2014 see the D-037 erratum block and Paper A \u00a73 Step-2\nfootnote); the theorem statement is unchanged.\nCertificate correction (2026-08-26, session 042, user review): the\nfinite-verification citation Misguich\u2013Voros (Voros 2020, Exp. Math.\n29(4)) was found to carry a DEFORMED sequence \u039b_n (Misguich, private\ncommunication), not the ordinary \u03bb_n \u2014 it has been replaced by\nJohansson 2015 \u00a7 4.2 (rigorous, published, ordinary Keiper\u2013Li\ncoefficients to n = 10\u2075, all positive) and the Lean perimeter\n`LiVerifiedUpTo` reduced from (5\u00b710\u2075) to (10\u2075) to match the checkable\nevidence (commit 79c96f2; the assembly's hfinite branch uses exactly\nn < 10\u2075, so nothing else changes).\n\n## Statement\nLet \u03bb_n = \u03a3\u2032_\u03c1 [1\u2212(1\u22121/\u03c1)\u207f] be the Li coefficients of the Riemann zeta\nfunction. Assume the Riemann hypothesis verified to height\nT\u2080 = 3\u00d710\u00b9\u00b2 (Platt\u2013Trudgian 2021, interval arithmetic). Then\n\n **\u03bb_n > 0 for every 1 \u2264 n \u2264 N\u2081 := (T\u2080\u22121)\u00b2\n = 8 999 999 999 994 000 000 000 001 \u2248 9.0\u00d710\u00b2\u2074.**\n\nConservative variant (using Lagarias's printed bridge window (1.16)\ninstead of the traced proof): N\u2081* = T\u2080\u00b2/(4 log\u00b2T\u2080) = 2.726\u00d710\u00b2\u00b9.\n\nThis is the first assembled EXPLICIT finite-height Li bound: the chain\n\"RH to height T\u2080 + fully traced constants \u27f9 \u03bb_n > 0 to n ~ T\u2080\u00b2\" existed\nonly with untraced O-constants (Lagarias math/0404394 Thm 1.1/(1.16)).\nBy Voros's sharpening, no statement of the form \"\u03bb_n > 0 \u2200n \u2265 N\u2080\" is\npossible short of RH \u2014 this finite-height theorem is the maximal content\nof the Li route (D-032/D-033).\n\n## Proof / Derivation\nFull assembly: **D-037** (derivation), support computation **E-023**\n(ALL CHECKS PASS). Skeleton:\n1. \u03bb_n = S_\u221e(n) \u2212 S_f(n) + 1 (Bombieri\u2013Lagarias Thm 2).\n2. Archimedean: S_\u221e(n) \u2265 \u00bdn log n + (C\u2081\u2212\u03b3/2)n \u2212 4 for n \u2265 20,\n C\u2081 = (\u03b3\u22121\u2212log 2\u03c0)/2 \u2014 elementary from Lagarias (4.11) via finite\n differences + explicit harmonic bounds; validated against E-021's\n exact S_\u221e to n = 5000 (E-023 V1, min slack 9.52).\n3. Finite: |S_f(n)| \u2264 B(n) (D-037 (4.1)) \u2014 Lagarias Thm 6.1 contour with\n all constants traced (Trudgian N(T)/S(T); kernel e\u2074+1; R-bound\n A = 10.5, B\u2080 = 48 traced (D-056, reference-point method) /\n 0.9, 1.0 measured; short-segment max\n M\u2080 = 1.3352 closing the D-036 |t|<10 gap).\n4. Bridge: RH to height \u221an+1 \u27f9 each zero contributes \u2264 2 \u27f9\n |\u03bb_n(T)| \u2264 2N(T); holds for n \u2264 (T\u2080\u22121)\u00b2.\n5. margin(n) := S_\u221e^lower \u2212 B + 1 is convex for n \u2265 20 (PROVED D-057;\n E-029), margin(8\u00d710\u2074) = 16 020 > 0 with positive derivative \u27f9\n margin(n) > 0 for all n \u2208 [8\u00d710\u2074, N\u2081] (E-028;\n margin(N\u2081) = 2.458\u00d710\u00b2\u2076).\n6. Finite verification: n \u2264 3000 (E-011/E-015, cross-validated,\n ledger #3) \u222a n \u2264 5\u00d710\u2075 (Misguich\u2013Voros) \u2014 overlaps N\u2080 = 8\u00d710\u2074\n by a factor 6.25.\n\n## Dependencies\nR-001 (finite verification), D-032/D-033 (design + wall), D-036 (contour\nconstants), D-037 (assembly), E-021 (exact S_\u221e/measured constants),\nE-023 (assembly numerics), Trudgian 2014 [PROVED lit], Platt\u2013Trudgian\n2021 [computational], Lagarias math/0404394, Bombieri\u2013Lagarias 1999,\nVoros 1703.02844 [Misguich numerics].\n\n## Adversarial review\nD-037 \u00a77: full input inventory P1\u2013P6 (proved/elementary) vs H1\u2013H5\n(computational/heuristic). Update (session 009 repairs): H2 (digamma\nsub-bound) is now PROVED (D-056: |r| \u2264 1.2/|w|\u00b2, Binet split +\nmax-modulus majorant); the only remaining grid-verified constant link\nis H3 (M\u2080 = 1.3352, E-023 S1; additive ~53.4). The theorem's\nsensitivity to M\u2080 is negligible \u2014 it enters a sub-dominant additive\nconstant; even 5\u00d7 worse moves N\u2080 by ~15 %, irrelevant against the\n6.25\u00d7 overlap margin. Upgrade path listed (E-023b: acb box cover).\nSanity gates: no claim beyond N\u2081 (Voros wall respected); no\nD-H/Epstein analogue claimed. Honest scope: this is NOT progress on\n\"\u03bb_n > 0 \u2200 large n\" (\u27fa RH).\n\n## Paper section\nNumerics/positivity appendix of draft.tex: \"Explicit finite-height\npositivity of the Keiper\u2013Li coefficients\"; constants table from D-037\n\u00a74; N\u2081 headline in the abstract-worthy results list.\n",
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{
"oldStart": 64,
"oldLines": 22,
"newStart": 64,
"newLines": 26,
"lines": [
" A = 10.5, B\u2080 = 48 traced (D-056, reference-point method) /",
" 0.9, 1.0 measured; short-segment max",
" M\u2080 = 1.3352 closing the D-036 |t|<10 gap).",
"-4. Bridge: RH to height \u221an+1 \u27f9 each zero contributes \u2264 2 \u27f9",
"- |\u03bb_n(T)| \u2264 2N(T); holds for n \u2264 (T\u2080\u22121)\u00b2.",
"+4. Bridge (CORRECTED ii\u2032-6, F-A): RH to height \u221an+1 \u27f9 each zero",
"+ contributes \u2264 2 \u27f9 |\u03bb_n(T)| \u2264 4F(T) (both signs of \u03b3; F the global",
"+ Trudgian majorant); holds for n \u2264 (T\u2080\u22121)\u00b2.",
" 5. margin(n) := S_\u221e^lower \u2212 B + 1 is convex for n \u2265 20 (PROVED D-057;",
" E-029), margin(8\u00d710\u2074) = 16 020 > 0 with positive derivative \u27f9",
" margin(n) > 0 for all n \u2208 [8\u00d710\u2074, N\u2081] (E-028;",
" margin(N\u2081) = 2.458\u00d710\u00b2\u2076).",
"-6. Finite verification: n \u2264 3000 (E-011/E-015, cross-validated,",
"- ledger #3) \u222a n \u2264 5\u00d710\u2075 (Misguich\u2013Voros) \u2014 overlaps N\u2080 = 8\u00d710\u2074",
"- by a factor 6.25.",
"+6. Finite verification (CORRECTED session 042): n \u2264 3000 (E-011/E-015,",
"+ cross-validated, ledger #3) \u222a n \u2264 10\u2075 (Johansson 2015 \u00a7 4.2,",
"+ rigorous interval-arithmetic ordinary-\u03bb_n table, all positive) \u2014",
"+ covers N\u2080 = 8\u00d710\u2074 by a factor 1.25 and meets the n = 10\u2075 split",
"+ exactly. [Supersedes: \"n \u2264 5\u00d710\u2075 (Misguich\u2013Voros)\" \u2014 that source",
"+ verifies a deformed sequence, not the ordinary \u03bb_n.]",
" ",
" ## Dependencies",
" R-001 (finite verification), D-032/D-033 (design + wall), D-036 (contour",
" constants), D-037 (assembly), E-021 (exact S_\u221e/measured constants),",
" E-023 (assembly numerics), Trudgian 2014 [PROVED lit], Platt\u2013Trudgian",
" 2021 [computational], Lagarias math/0404394, Bombieri\u2013Lagarias 1999,",
"-Voros 1703.02844 [Misguich numerics].",
"+Johansson 2015 (Numer. Algor. 69:253\u2013270, \u00a7 4.2) [finite verification].",
" ",
" ## Adversarial review",
" D-037 \u00a77: full input inventory P1\u2013P6 (proved/elementary) vs H1\u2013H5"
]
}
],
"userModified": false,
"replaceAll": false
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " have hA : TendstoLocallyUniformlyOn (fun n => fun _ : \u211d => Real.log n - (harmonic n : \u211d))\n (fun _ => -Real.eulerMascheroniConstant) atTop (Set.Ioi 0) := by\n have h := Real.tendsto_harmonic_sub_log\n have h2 : Tendsto (fun n : \u2115 => Real.log n - (harmonic n : \u211d)) atTop\n (\ud835\udcdd (-Real.eulerMascheroniConstant)) := by\n have h3 : (fun n : \u2115 => Real.log n - (harmonic n : \u211d)) =\n fun n => -((harmonic n : \u211d) - Real.log n) := by\n funext n; ring\n rw [h3]\n exact h.neg\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro \u03b5 h\u03b5\n rw [Metric.tendsto_atTop] at h2\n obtain \u27e8N, hN\u27e9 := h2 \u03b5 h\u03b5\n rw [Filter.eventually_atTop]\n exact \u27e8N, fun n hn x hx => by simpa using hN n hn\u27e9\n have hB : TendstoLocallyUniformlyOn (fun _ : \u2115 => fun x : \u211d => -1 / x) (fun x => -1 / x)\n atTop (Set.Ioi 0) := by\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro \u03b5 h\u03b5\n rw [Filter.eventually_atTop]\n exact \u27e80, fun n _ x _ => by simpa using h\u03b5\u27e9\n have hC : TendstoLocallyUniformlyOn (fun n => fun x : \u211d => \u2211 m \u2208 Finset.range n, dpsiTermR x m)\n (fun x => \u2211' m : \u2115, dpsiTermR x m) atTop (Set.Ioi 0) := by\n obtain \u27e8g, hg\u27e9 := dpsiTermR_summableLocallyUniformlyOn\n apply (hasSumLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn.mp hg).congr_right\n intro t ht\n exact (hg.tsum_eqOn ht).symm",
"new_string": " have hA : TendstoLocallyUniformlyOn (fun n : \u2115 => fun _ : \u211d => Real.log n - (harmonic n : \u211d))\n (fun _ => -Real.eulerMascheroniConstant) atTop (Set.Ioi 0) := by\n have h := Real.tendsto_harmonic_sub_log\n have h2 : Tendsto (fun n : \u2115 => Real.log n - (harmonic n : \u211d)) atTop\n (\ud835\udcdd (-Real.eulerMascheroniConstant)) := by\n have h3 : (fun n : \u2115 => Real.log n - (harmonic n : \u211d)) =\n fun n => -((harmonic n : \u211d) - Real.log n) := by\n funext n; ring\n rw [h3]\n exact h.neg\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro \u03b5 h\u03b5\n rw [Metric.tendsto_atTop] at h2\n obtain \u27e8N, hN\u27e9 := h2 \u03b5 h\u03b5\n rw [Filter.eventually_atTop]\n exact \u27e8N, fun n hn x hx => by simpa using hN n hn\u27e9\n have hB : TendstoLocallyUniformlyOn (fun _ : \u2115 => fun x : \u211d => -1 / x) (fun x => -1 / x)\n atTop (Set.Ioi 0) := by\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro \u03b5 h\u03b5\n rw [Filter.eventually_atTop]\n exact \u27e80, fun n _ x _ => by simpa using h\u03b5\u27e9\n have hC : TendstoLocallyUniformlyOn (fun n : \u2115 => fun x : \u211d => \u2211 m \u2208 Finset.range n, dpsiTermR x m)\n (fun x => \u2211' m : \u2115, dpsiTermR x m) atTop (Set.Ioi 0) := by\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n by_cases hKe : K = \u2205\n \u00b7 subst hKe\n rw [tendstoUniformlyOn_empty]\n intro \u03b5 h\u03b5\n exact Filter.eventually_of_forall fun n => by simp\n \u00b7 obtain \u27e8x0, hx0\u27e9 := Set.nonempty_iff_ne_empty.mpr hKe\n obtain \u27e8C, hCK, hCle\u27e9 := hKc.exists_isMaxOn \u27e8x0, hx0\u27e9 (continuous_id.continuousOn)\n apply tendstoUniformlyOn_tsum_nat (summable_div_add_one_sq C)\n intro m x hx\n have hx0 : 0 < x := hK hx\n have hxC : x \u2264 C := hCle hx\n rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0 m)]\n calc dpsiTermR x m \u2264 x / ((m : \u211d) + 1) ^ 2 := dpsiTermR_le hx0 m\n _ \u2264 C / ((m : \u211d) + 1) ^ 2 :=\n (div_le_div_iff_of_pos_right (by positivity : (0 : \u211d) < ((m : \u211d) + 1) ^ 2)).mpr hxC"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`tendstoUniformlyOn_tsum_nat (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) : TendstoUniformlyOn (fun N => fun x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' n, f n x) atTop s` — the range-indexed M-test ✓ EXACTLY what I need for hC.\n\nLet me now rewrite the whole Section 2 with all fixes:\n1. h2's e0: match the beta-reduced form `1 / (x + ↑0)`.\n2. hcle/hCle: membership-only application (hcle hx).\n3. dpsiTermR_nonneg hx0 m (add m).\n4. The n : ℕ annotations in hA's lambdas.\n5. hC via tendstoLocallyUniformlyOn_iff_forall_isCompact + tendstoUniformlyOn_tsum_nat.\n6. hdecomp/hlimit congr structure (that part was fine but the congr directions — let me double check: `TendstoLocallyUniformlyOn.congr (hf : TendstoLocallyUniformlyOn F f p s) (hg : ∀ n, s.EqOn (F n) (G n)) : TendstoLocallyUniformlyOn G f p s` — so hf is the PROVED convergence (of the decomposed sequence) and G := dpsiSeqR — hg n : EqOn (F n) (dpsiSeqR n) — F n := the decomposed lambda — EqOn (decomposed n) (dpsiSeqR n) means decomposed n x = dpsiSeqR n x for x ∈ s — that's (dpsiSeqR_eq n x).symm ✓ so `.congr (fun n x _ => (dpsiSeqR_eq n x).symm)` ✓ as I had.\n And congr_right (hg : s.EqOn f g): f := decomposed-limit, g := DinfR — hg : ∀ x ∈ s, decomplimit x = DinfR x — my hlimit ✓.\n\nLet me rewrite the whole Section 2 again with these fixes. Actually the fixes are localized; let me do targeted edits.","signature":""},{"type":"text","text":"`tendstoUniformlyOn_tsum_nat` is the right form. Applying targeted fixes."},{"type":"tool_use","id":"Edit_326","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" rw [Finset.sum_congr rfl (fun i _ => e i)]\n have e0 : (fun m => 1 / (x + (m : ℝ))) 0 = 1 / x := by simp\n rw [e0]\n ring","new_string":" rw [Finset.sum_congr rfl (fun i _ => e i)]\n have e0 : 1 / (x + ((0 : ℕ) : ℝ)) = 1 / x := by simp\n rw [e0]\n ring"},"extra_content":null},{"type":"tool_use","id":"Edit_327","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" by_cases hKe : K = ∅\n · subst hKe\n exact ⟨0, summable_zero, fun m x hx => by simp at hx⟩\n · obtain ⟨x0, hx0⟩ := Set.nonempty_iff_ne_empty.mpr hKe\n obtain ⟨c, hcK, hcle⟩ := hKc.exists_isMinOn ⟨x0, hx0⟩ (continuous_id.continuousOn)\n obtain ⟨C, hCK, hCle⟩ := hKc.exists_isMaxOn ⟨x0, hx0⟩ (continuous_id.continuousOn)\n have hc0 : 0 < c := hK hcK\n refine ⟨fun m => C / ((m : ℝ) + 1) ^ 2, summable_div_add_one_sq C, fun m x hx => ?_⟩\n have hx0 : 0 < x := hK hx\n have hxC : x ≤ C := hCle x hx\n rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0)]\n calc dpsiTermR x m ≤ x / ((m : ℝ) + 1) ^ 2 := dpsiTermR_le hx0 m\n _ ≤ C / ((m : ℝ) + 1) ^ 2 :=\n (div_le_div_iff_of_pos_right (by positivity : (0 : ℝ) < ((m : ℝ) + 1) ^ 2)).mpr hxC","new_string":" by_cases hKe : K = ∅\n · subst hKe\n exact ⟨0, summable_zero, fun m x hx => by simp at hx⟩\n · obtain ⟨x0, hx0⟩ := Set.nonempty_iff_ne_empty.mpr hKe\n obtain ⟨C, hCK, hCle⟩ := hKc.exists_isMaxOn ⟨x0, hx0⟩ (continuous_id.continuousOn)\n refine ⟨fun m => C / ((m : ℝ) + 1) ^ 2, summable_div_add_one_sq C, fun m x hx => ?_⟩\n have hx0 : 0 < x := hK hx\n have hxC : x ≤ C := hCle hx\n rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0 m)]\n calc dpsiTermR x m ≤ x / ((m : ℝ) + 1) ^ 2 := dpsiTermR_le hx0 m\n _ ≤ C / ((m : ℝ) + 1) ^ 2 :=\n (div_le_div_iff_of_pos_right (by positivity : (0 : ℝ) < ((m : ℝ) + 1) ^ 2)).mpr hxC"},"extra_content":null},{"type":"tool_use","id":"Edit_328","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" have hA : TendstoLocallyUniformlyOn (fun n => fun _ : ℝ => Real.log n - (harmonic n : ℝ))\n (fun _ => -Real.eulerMascheroniConstant) atTop (Set.Ioi 0) := by\n have h := Real.tendsto_harmonic_sub_log\n have h2 : Tendsto (fun n : ℕ => Real.log n - (harmonic n : ℝ)) atTop\n (𝓝 (-Real.eulerMascheroniConstant)) := by\n have h3 : (fun n : ℕ => Real.log n - (harmonic n : ℝ)) =\n fun n => -((harmonic n : ℝ) - Real.log n) := by\n funext n; ring\n rw [h3]\n exact h.neg\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro ε hε\n rw [Metric.tendsto_atTop] at h2\n obtain ⟨N, hN⟩ := h2 ε hε\n rw [Filter.eventually_atTop]\n exact ⟨N, fun n hn x hx => by simpa using hN n hn⟩\n have hB : TendstoLocallyUniformlyOn (fun _ : ℕ => fun x : ℝ => -1 / x) (fun x => -1 / x)\n atTop (Set.Ioi 0) := by\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro ε hε\n rw [Filter.eventually_atTop]\n exact ⟨0, fun n _ x _ => by simpa using hε⟩\n have hC : TendstoLocallyUniformlyOn (fun n => fun x : ℝ => ∑ m ∈ Finset.range n, dpsiTermR x m)\n (fun x => ∑' m : ℕ, dpsiTermR x m) atTop (Set.Ioi 0) := by\n obtain ⟨g, hg⟩ := dpsiTermR_summableLocallyUniformlyOn\n apply (hasSumLocallyUniformlyOn_iff_tendstoLocallyUniformlyOn.mp hg).congr_right\n intro t ht\n exact (hg.tsum_eqOn ht).symm","new_string":" have hA : TendstoLocallyUniformlyOn (fun n : ℕ => fun _ : ℝ => Real.log n - (harmonic n : ℝ))\n (fun _ => -Real.eulerMascheroniConstant) atTop (Set.Ioi 0) := by\n have h := Real.tendsto_harmonic_sub_log\n have h2 : Tendsto (fun n : ℕ => Real.log n - (harmonic n : ℝ)) atTop\n (𝓝 (-Real.eulerMascheroniConstant)) := by\n have h3 : (fun n : ℕ => Real.log n - (harmonic n : ℝ)) =\n fun n => -((harmonic n : ℝ) - Real.log n) := by\n funext n; ring\n rw [h3]\n exact h.neg\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro ε hε\n rw [Metric.tendsto_atTop] at h2\n obtain ⟨N, hN⟩ := h2 ε hε\n rw [Filter.eventually_atTop]\n exact ⟨N, fun n hn x hx => by simpa using hN n hn⟩\n have hB : TendstoLocallyUniformlyOn (fun _ : ℕ => fun x : ℝ => -1 / x) (fun x => -1 / x)\n atTop (Set.Ioi 0) := by\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n rw [Metric.tendstoUniformlyOn_iff]\n intro ε hε\n rw [Filter.eventually_atTop]\n exact ⟨0, fun n _ x _ => by simpa using hε⟩\n have hC : TendstoLocallyUniformlyOn (fun n : ℕ => fun x : ℝ => ∑ m ∈ Finset.range n, dpsiTermR x m)\n (fun x => ∑' m : ℕ, dpsiTermR x m) atTop (Set.Ioi 0) := by\n rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_Ioi]\n intro K hK hKc\n by_cases hKe : K = ∅\n · subst hKe\n rw [tendstoUniformlyOn_empty]\n intro ε hε\n exact Filter.eventually_of_forall fun n => by simp\n · obtain ⟨x0, hx0⟩ := Set.nonempty_iff_ne_empty.mpr hKe\n obtain ⟨C, hCK, hCle⟩ := hKc.exists_isMaxOn ⟨x0, hx0⟩ (continuous_id.continuousOn)\n apply tendstoUniformlyOn_tsum_nat (summable_div_add_one_sq C)\n intro m x hx\n have hx0 : 0 < x := hK hx\n have hxC : x ≤ C := hCle hx\n rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0 m)]\n calc dpsiTermR x m ≤ x / ((m : ℝ) + 1) ^ 2 := dpsiTermR_le hx0 m\n _ ≤ C / ((m : ℝ) + 1) ^ 2 :=\n (div_le_div_iff_of_pos_right (by positivity : (0 : ℝ) < ((m : ℝ) + 1) ^ 2)).mpr hxC"},"extra_content":null}]