← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (1980 chars)
1. Line 740: `add_le_add_left hrec _` — produced ‖1/(s−1)‖ + ? ≤ ... — the ORDER: add_le_add_left (h : b ≤ c) (a) : a + b ≤ a + c. The result shows ‖1/(s−1)‖ + ? — so it unified b := ‖1/(s−1)‖, a := ? — hmm the result has b + a on the left — meaning `add_le_add_left hrec ?` gives ‖1/(s−1)‖ + ?m ≤ 1/‖s−1‖ + ?m — but I need 1/2 + ‖1/(s−1)‖ — i.e. a + b with a = 1/2: the result form is a + b ≤ a + c — so `add_le_add_left hrec (1/2)` should give (1/2) + ‖1/(s−1)‖ ≤ (1/2) + 1/‖s−1‖. But the error shows it gave ‖1/(s−1)‖ + ?m ≤ ... — so the argument got placed as b?? add_le_add_left (h : b ≤ c) (a) : a + b ≤ a + c — hrec : ‖1/(s−1)‖ ≤ 1/‖s−1‖ — so b = ‖1/(s−1)‖, c = 1/‖s−1‖, a = the `_` — the RESULT is a + b ≤ a + c = ?m + ‖1/(s−1)‖ ≤ ... — the displayed result `‖1 / (s - 1)‖ + ?m.2176 ≤ ...` has the ORDER FLIPPED — meaning add_le_add_left in this Mathlib is (h : b ≤ c) (a) : b + a ≤ c + a?! No wait — the error's "full type" from before: `∀ {b c}, b ≤ c → ∀ (a : α), b + a ≤ c + a` — IT'S b + a ≤ c + a (right addition, confusingly named "left"). So the constant goes on the RIGHT. My goal: 1/2 + ‖..‖ ≤ 1/2 + 1/‖..‖ — constant on LEFT — need `add_le_add_right`! In this Mathlib: add_le_add_left (h : b ≤ c) (a) : b + a ≤ c + a; add_le_add_right (h : b ≤ c) (a) : a + b ≤ a + c. So use add_le_add_right. Let me fix both spots: `add_le_add_right hrec _` and `add_le_add_right (...) _`.
2. Line 745: `(one_div_le_one_div_of_le ...).trans_eq (by ring)` — the ring left `2/15 = ?m` — the trans_eq's equation goal: 1/(15/2) = 2/15: `by ring` failed? It shows `2 / 15 = ?m.2221` unsolved — the metavariable in trans_eq — trans_eq (h : a = b)? `LE.le.trans_eq (h₁ : a ≤ b) (h₂ : b = c) : a ≤ c`? so the by ring proves 1/(15/2) = 2/15 — the ?m is c — from my `.trans_eq (by ring)` the ring goal `1/(15/2) = 2/15` should close... the error shows `2/15 = ?m.2221` — ring turned it into `2/15 = ?`?? weird. Provide explicitly: `.trans_eq (show (1 : ℝ) / (15/2) = 2/15 by norm_num)`.
`add_le_add_left` is the right-addition form in this Mathlib — need `add_le_add_right`. And the trans_eq norm.
tool: Edit ok
input
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"old_string": " _ \u2264 1 / 2 + 1 / \u2016s - 1\u2016 := by\n have hrec : \u20161 / (s - 1)\u2016 \u2264 1 / \u2016s - 1\u2016 := by\n rw [norm_div, norm_one]\n exact add_le_add_left hrec _\n \u00b7 rw [norm_mul, norm_mul]\n _ \u2264 (1 / 2 + 2 / 15) + (|t| + 9 / 2) * (|t| + 11 / 2) * (1 / 4) := by\n apply add_le_add\n \u00b7 exact add_le_add_left\n ((one_div_le_one_div_of_le (by norm_num) hsm1norm).trans_eq (by ring)) _",
"new_string": " _ \u2264 1 / 2 + 1 / \u2016s - 1\u2016 := by\n have hrec : \u20161 / (s - 1)\u2016 \u2264 1 / \u2016s - 1\u2016 := by\n rw [norm_div, norm_one]\n exact add_le_add_right hrec _\n \u00b7 rw [norm_mul, norm_mul]\n _ \u2264 (1 / 2 + 2 / 15) + (|t| + 9 / 2) * (|t| + 11 / 2) * (1 / 4) := by\n apply add_le_add\n \u00b7 exact add_le_add_right\n ((one_div_le_one_div_of_le (by norm_num) hsm1norm).trans_eq\n (show (1 : \u211d) / (15 / 2) = 2 / 15 by norm_num)) _"
}
result 2026-08-22 18:56:15Z → 2026-08-25 06:55:54Z
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"oldString": " NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/",
"newString": " NguyenRH1.LiAssembly \u2014 ii\u2032-6-b (session 038, D-071 \u00a74): the\n perimeter Props (ZeroCountGlobalProp global\n Trudgian N(T) \u2264 FAlt; MZeroProp M\u2080 \u2264 1.3352)\n + the wide zero-avoiding ordinate\n (exists_zero_avoiding_T_wide, the Finding-2\n 3-unit repair) + the sides identity\n (sf_abs_le_rectInt_add_zeroSum: |Sf n| \u2264\n \u2016RectInt\u2016 + \u2016\u03a3\u2016 on the variant-B rectangle)\n + the bridge (zeroSum_norm_le_four_FAlt_add_one:\n \u2016\u03a3\u2016 \u2264 4\u00b7FAlt T + 1 via RHUpTo + conj symmetry\n + ZeroCountGlobalProp)\n NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/",
"originalFile": "/-\nNguyenRH1.lean \u2014 root of the NGUYEN-RH-1 Phase-5 skeleton library.\n\nStatement-level Lean 4 skeletons for the project's new results\n(paper draft.tex \u00a73), built on zeta-23-lean (tag v1.0) conventions.\nEvery proof is `sorry` at this stage BY DESIGN: this is task 5.1's\nfirst pass \u2014 get the *statements* machine-checked against Mathlib +\nZeta23 types. `#print axioms` on any declaration here shows `sorryAx`\nuntil the corresponding proof is formalized; that marker is the\nhonesty tag, mirrored by the [SKELETON] comments.\n\nContents:\n NguyenRH1.LiSkeleton \u2014 R-004 finite-height Li theorem + the\n archimedean lower bound (draft \u00a73.1);\n assembly proved modulo SfBoundProp\n NguyenRH1.LiBasePoints \u2014 tier (ii) step 2: certified interval-bound\n discharge of the two margin evaluations;\n lambda_pos_finite_height_verified (perimeter\n + SfBoundProp only)\n NguyenRH1.LiResidue \u2014 tier ii\u2032-1b(i): the w = 1 residue of the\n Lagarias-kernel integrand (ResW1_eq_sum)\n NguyenRH1.LiPrincipal \u2014 tier ii\u2032-1b(ii): the principal-part integral\n (rectInt'_ppart)\n NguyenRH1.LiSides \u2014 tier ii\u2032-1b(iii): rectangle residue theorem\n for the integrand (rectInt'_Jlag)\n NguyenRH1.LiSwaps \u2014 tier ii\u2032-2: the elementary swaps\n (liCoeff_eq_sum_sigma, sInfty_eq_sum)\n NguyenRH1.LiZetaCoeff \u2014 tier ii\u2032-3a piece (B): the \u03b6-piece Taylor\n coefficients (iteratedDeriv_hLiExt)\n NguyenRH1.LiDigamma \u2014 tier ii\u2032-3a piece (A) core: the digamma\n series + polygamma values at 1/2\n NguyenRH1.LiArchimedean \u2014 tier ii\u2032-3a piece (A): the archimedean\n coefficients (iteratedDeriv_logDeriv_Gamma\u211d)\n NguyenRH1.LiContour \u2014 tier ii\u2032-4: L3 zero-avoiding T (pigeonhole)\n + L1 kernel bounds + zeroMult(\u22122) = 1\n NguyenRH1.LiRBound \u2014 tier ii\u2032-5b: the level-2 Euler\u2013Maclaurin \u03b6\n representation (re > \u22121) + the polynomial\n disc bound \u2016\u03b6\u2016 \u2264 (3/4)|t|\u00b2\n NguyenRH1.LiDisc \u2014 tier ii\u2032-5c: the loose Davenport-disc\n R-bound \u2016\u03b6\u2032/\u03b6 \u2212 \u03a3_near m/(s\u2212\u03c1)\u2016 \u2264\n 100000\u00b7(log|t|)\u00b2 on \u03c3 \u2208 [1/2, 7/2], |t| \u2265 10\n (Blaschke quotient + log + Borel\u2013Carath\u00e9odory)\n NguyenRH1.LiReflect \u2014 tier ii\u2032-5d: the FE-reflection of the loose\n R-bound to \u03c3 \u2208 [\u22125/2, 1/2] (exact zero-sum\n cancellation under \u03c1 \u21a6 1\u2212\u03c1; \u0393R-bracket via\n the +4 shift + digamma_stirling) + the\n left-plane uniform digamma bound\n \u2016\u03c8(w) \u2212 log w\u2016 \u2264 3 for |im w| \u2265 2\n NguyenRH1.LiTrivialZeros \u2014 ii\u2032-5e prep: zeroMult(\u22122n) = 1 for ALL\n n \u2265 1 (product-form \u0393 recurrence\n gamma_mul_prod + the HfunT cofactor;\n generalizes ii\u2032-4c's zeroMult_neg_two)\n NguyenRH1.LiGapCircles \u2014 ii\u2032-5e-i: the \u03b3-gap circle radii\n (exists_gapCircle_radius): R \u2208 [4k+5/2,\n 4k+3] with dist(R/2, \u2124) \u2265 1/4 and every\n zero ordinate either \u2265 1 away or at\n clearance \u2265 1/(16\u00b7C0hat(4k+9/2)) \u2014 the\n \u00d72-rescaled midpoint pigeonhole\n NguyenRH1.LiDigammaLeft \u2014 ii\u2032-5e-ii: the telescoping tail + the\n right-half-plane digamma bound\n (norm_digamma_le_of_two_le_re) + the\n separated reciprocal sum bound\n (digamma_recip_sum_le) + the near-axis\n bound (norm_digamma_le_of_dist_nat) + the\n general logDeriv \u0393R identity\n (logDeriv_Gamma\u211d_eq)\n NguyenRH1.LiSeries \u2014 ii\u2032-5e-iii (D-068 \u00a73): the zero-side\n partial-fraction series seriesFun \u2014\n summable everywhere (seriesSummable),\n analytic off the nontrivial zeros\n (seriesFun_analyticOnNhd), isolation +\n the reduced series analytic at each zero\n (seriesFunSub_analyticAt), the polar\n subtraction analytic at each zero\n (seriesFun_sub_polar_analyticAt)\n NguyenRH1.LiFEntire \u2014 ii\u2032-5e-iv (D-068 \u00a74): F entire \u2014 the\n punctured-limit extension Gentire of the\n partial-fraction difference Fexpr is\n everywhere analytic\n (gentire_analyticAt_and_eq): the\n principal-part splits at the nontrivial\n zeros (NF + 5e-iii polar subtraction),\n at 1 (logDeriv zetaA), at 0 and at \u22122n\n (the digamma-series splits) + the zero\n classification + analyticAt_digamma\n NguyenRH1.LiCircleBound \u2014 ii\u2032-5e-v (sessions 030\u2013033): the count\n bridges + the Dirichlet bound + the far-tail\n bound + the series circle bound\n (seriesFun_norm_le_gapCircle) + the Fexpr\n circle bound (Fexpr_circle_le): \u2016Fexpr\u2016 \u2264\n 2\u00b7lowZeroConst + dirichletC2 +\n 400000\u00b7A\u2080\u00b7\u0108\u2080(R+2)\u00b7(log R)\u00b2 on the \u03b3-gap\n circles, k \u2265 4000\n NguyenRH1.LiSqueeze \u2014 ii\u2032-5e-v-d (session 034): the Schwarz\n squeeze \u2014 max-modulus + the ball-form\n Schwarz lemma on the \u03b3-gap circle sequence\n with the (log R)\u00b3/R \u2192 0 estimate\n (gentire_const: Gentire \u2261 Gentire 0 under\n ZeroCountProp) \u27f9 the capstone\n partialFraction_identity: \u2203 B, \u03b6\u2032/\u03b6(s) =\n B + seriesFun s \u2212 1/s \u2212 1/(s\u22121) +\n \u00bdlog \u03c0 \u2212 \u00bd\u03c8(s/2) off the excluded set\n NguyenRH1.LiTightR \u2014 ii\u2032-5f (sessions 035\u2013036): the tight\n R-bound from the partial-fraction identity \u2014\n the six-piece reference-point decomposition\n + the far-sum interval machine + the pinch\n lemma \u27f9 the capstone\n logDeriv_riemannZeta_sub_nearZeros_le_tight\n (hZC) (hZC2) (hNLZ): \u2016\u03b6\u2032/\u03b6(s) \u2212 \u03a3_near\n m/(s\u2212\u03c1)\u2016 \u2264 10.5\u00b7log(t+2) + 48\n NguyenRH1.LiMoments \u2014 ii\u2032-3b (sessions 037\u2013038, D-070 route):\n the contour moments from the partial-fraction\n identity by pure algebra + series calculus \u2014\n xiL + (G) gFun = B + seriesFun + the k = 1\n reindex + the termwise iterated derivation\n at 0 \u27f9 the capstone\n sigmaP_eq_neg_iteratedDeriv_gFun_div (hZC):\n sigmaP k = \u2212gFun^{(k\u22121)}(0)/(k\u22121)!, all\n k \u2265 1 (perimeter: ZeroCountProp only)\n NguyenRH1.LiCrux \u2014 ii\u2032-3c (session 038, D-066 \u00a72): the\n (4.10-a) assembly \u2014 FE derivative relation\n (gFun_one_sub_fun + iteratedDeriv_gFun_zero),\n the shift expansion (gFun_one_add_punctured/\n gFun_one_add_eventuallyEq), the coefficient\n extraction (iteratedDeriv_gFun_one) \u27f9\n **lagarias_4_10_a** (hZC) \u27f9 THE CRUX\n **sf_eq_resW1** (hZC): (ResW1 n).re = Sf n\n for n \u2265 1 \u2014 Sf(n) = Res\u2081(n) PROVED\n (perimeter: ZeroCountProp only)\n NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/\nimport NguyenRH1.LiSkeleton\nimport NguyenRH1.LiBasePoints\nimport NguyenRH1.LiResidue\nimport NguyenRH1.LiPrincipal\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiSwaps\nimport NguyenRH1.LiZetaCoeff\nimport NguyenRH1.LiDigamma\nimport NguyenRH1.LiArchimedean\nimport NguyenRH1.LiContour\nimport NguyenRH1.LiRBound\nimport NguyenRH1.LiDisc\nimport NguyenRH1.LiReflect\nimport NguyenRH1.LiTrivialZeros\nimport NguyenRH1.LiGapCircles\nimport NguyenRH1.LiDigammaLeft\nimport NguyenRH1.LiSeries\nimport NguyenRH1.LiFEntire\nimport NguyenRH1.LiCircleBound\nimport NguyenRH1.LiSqueeze\nimport NguyenRH1.LiTightR\nimport NguyenRH1.LiMoments\nimport NguyenRH1.LiCrux\nimport NguyenRH1.SuzukiSkeleton\nimport NguyenRH1.ValleySkeleton\n",
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" for n \u2265 1 \u2014 Sf(n) = Res\u2081(n) PROVED",
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"+ NguyenRH1.LiAssembly \u2014 ii\u2032-6-b (session 038, D-071 \u00a74): the",
"+ perimeter Props (ZeroCountGlobalProp global",
"+ Trudgian N(T) \u2264 FAlt; MZeroProp M\u2080 \u2264 1.3352)",
"+ + the wide zero-avoiding ordinate",
"+ (exists_zero_avoiding_T_wide, the Finding-2",
"+ 3-unit repair) + the sides identity",
"+ (sf_abs_le_rectInt_add_zeroSum: |Sf n| \u2264",
"+ \u2016RectInt\u2016 + \u2016\u03a3\u2016 on the variant-B rectangle)",
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"+ \u2016\u03a3\u2016 \u2264 4\u00b7FAlt T + 1 via RHUpTo + conj symmetry",
"+ + ZeroCountGlobalProp)",
" NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)",
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[{"type":"thinking","thinking":"1. Line 740: `add_le_add_left hrec _` — produced ‖1/(s−1)‖ + ? ≤ ... — the ORDER: add_le_add_left (h : b ≤ c) (a) : a + b ≤ a + c. The result shows ‖1/(s−1)‖ + ? — so it unified b := ‖1/(s−1)‖, a := ? — hmm the result has b + a on the left — meaning `add_le_add_left hrec ?` gives ‖1/(s−1)‖ + ?m ≤ 1/‖s−1‖ + ?m — but I need 1/2 + ‖1/(s−1)‖ — i.e. a + b with a = 1/2: the result form is a + b ≤ a + c — so `add_le_add_left hrec (1/2)` should give (1/2) + ‖1/(s−1)‖ ≤ (1/2) + 1/‖s−1‖. But the error shows it gave ‖1/(s−1)‖ + ?m ≤ ... — so the argument got placed as b?? add_le_add_left (h : b ≤ c) (a) : a + b ≤ a + c — hrec : ‖1/(s−1)‖ ≤ 1/‖s−1‖ — so b = ‖1/(s−1)‖, c = 1/‖s−1‖, a = the `_` — the RESULT is a + b ≤ a + c = ?m + ‖1/(s−1)‖ ≤ ... — the displayed result `‖1 / (s - 1)‖ + ?m.2176 ≤ ...` has the ORDER FLIPPED — meaning add_le_add_left in this Mathlib is (h : b ≤ c) (a) : b + a ≤ c + a?! No wait — the error's \"full type\" from before: `∀ {b c}, b ≤ c → ∀ (a : α), b + a ≤ c + a` — IT'S b + a ≤ c + a (right addition, confusingly named \"left\"). So the constant goes on the RIGHT. My goal: 1/2 + ‖..‖ ≤ 1/2 + 1/‖..‖ — constant on LEFT — need `add_le_add_right`! In this Mathlib: add_le_add_left (h : b ≤ c) (a) : b + a ≤ c + a; add_le_add_right (h : b ≤ c) (a) : a + b ≤ a + c. So use add_le_add_right. Let me fix both spots: `add_le_add_right hrec _` and `add_le_add_right (...) _`.\n\n2. Line 745: `(one_div_le_one_div_of_le ...).trans_eq (by ring)` — the ring left `2/15 = ?m` — the trans_eq's equation goal: 1/(15/2) = 2/15: `by ring` failed? It shows `2 / 15 = ?m.2221` unsolved — the metavariable in trans_eq — trans_eq (h : a = b)? `LE.le.trans_eq (h₁ : a ≤ b) (h₂ : b = c) : a ≤ c`? so the by ring proves 1/(15/2) = 2/15 — the ?m is c — from my `.trans_eq (by ring)` the ring goal `1/(15/2) = 2/15` should close... the error shows `2/15 = ?m.2221` — ring turned it into `2/15 = ?`?? weird. Provide explicitly: `.trans_eq (show (1 : ℝ) / (15/2) = 2/15 by norm_num)`.","signature":""},{"type":"text","text":"`add_le_add_left` is the right-addition form in this Mathlib — need `add_le_add_right`. And the trans_eq norm."},{"type":"tool_use","id":"Edit_313","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":" _ ≤ 1 / 2 + 1 / ‖s - 1‖ := by\n have hrec : ‖1 / (s - 1)‖ ≤ 1 / ‖s - 1‖ := by\n rw [norm_div, norm_one]\n exact add_le_add_left hrec _\n · rw [norm_mul, norm_mul]\n _ ≤ (1 / 2 + 2 / 15) + (|t| + 9 / 2) * (|t| + 11 / 2) * (1 / 4) := by\n apply add_le_add\n · exact add_le_add_left\n ((one_div_le_one_div_of_le (by norm_num) hsm1norm).trans_eq (by ring)) _","new_string":" _ ≤ 1 / 2 + 1 / ‖s - 1‖ := by\n have hrec : ‖1 / (s - 1)‖ ≤ 1 / ‖s - 1‖ := by\n rw [norm_div, norm_one]\n exact add_le_add_right hrec _\n · rw [norm_mul, norm_mul]\n _ ≤ (1 / 2 + 2 / 15) + (|t| + 9 / 2) * (|t| + 11 / 2) * (1 / 4) := by\n apply add_le_add\n · exact add_le_add_right\n ((one_div_le_one_div_of_le (by norm_num) hsm1norm).trans_eq\n (show (1 : ℝ) / (15 / 2) = 2 / 15 by norm_num)) _"},"extra_content":null}]