← Session 31 — ii′-5f: tight R-bound 10.5/48 (piece 3 + pinch)
thinking (317 chars)
ii′-5f is COMPLETE. Per §8.4 session end protocol (mandatory before shutdown):
1. Write journal/session-036.md.
2. Update STATE.md with NEXT ACTION.
3. Update NGUYEN.md checklist (ii′-5f → complete) + session log.
4. Register in research.db.
5. Final commit with session number.
Let me write the journal entry first.
ii′-5f is complete. Executing the session-end protocol (§8.4): journal, STATE, NGUYEN.md, db registration.
tool: Write
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-036.md",
"content": "# Session 036 \u2014 2026-08-24 \u2014 ii\u2032-5f COMPLETE: piece 3 + 5f-iii capstone\n\n## Directive\nContinue ii\u2032-5f from the session-035 state: piece 3 (the far-sum interval\nmachine) then 5f-iii (assembly + pinch + capstone). User: \"Let's do it\nnow\" / \"let's continue to 5f-iii now\".\n\n## Outcome part 1: PIECE 3 COMPLETE (commits 79c506b..52bb2cb)\nLiTightR.lean sections 9\u201319 (~2700 new lines), the far difference series\nbounded by `far_series_le : \u2016far\u2016 \u2264 12.1284\u00b7C0hat(t+1) + 7.2425`:\n- **\u00a79 certified numerics**: the integral-comparison log machines\n (log_one_add_ge, neg_log_one_sub_le), all certified log/exp/sqrt bounds\n (log 10 both sides, log 11/14 below, log 13/2.3/(23/13)/(17/14) above,\n 1/e \u2264 0.3679, log u \u2264 u/e).\n- **\u00a710 the weight farW + S\u2083**: farW k = 1/(k\u221a(1+k\u00b2)); S\u2083 = \u03a3_{k\u22651} farW k\n \u2264 1.3476 (4 exact sqrt-rational terms: \u221a2 \u2265 1.414, \u221a5 \u2265 2.23,\n \u221a10 \u2265 3.16, \u221a17 \u2265 4.123; telescope \u03a3_{k\u22655} 1/(k(k\u22121)) = 1/4).\n- **\u00a711/12 the six tail bricks** (antitone-integral machine):\n RW\u2080 0.1, RW\u2081 0.3303, RW\u2082 0.1216 (loglog \u2264 (1/e)\u00b7log trick);\n LA\u2080 1/23, LA\u2081 0.1687 (the F = \u2212log x/(x+10) + (1/10)log(x/(x+10))\n antiderivative), LA\u2082 0.0621.\n- **\u00a713 C0hat bridges**: C0hat(2k+1) \u2264 0.433(log k+0.833)+0.886(loglog k\n +0.3475)+2.9 (k \u2265 11), the j+1 form (j \u2265 14: 0.1942/0.0736), and\n **C0hat(2t+1) \u2264 C0hat(t+1) + 0.55** (t \u2265 10).\n- **\u00a714 conj machinery**: hasDerivAt_conj_of_conj_eq (the conj-derivative\n lemma via the slope characterization) \u2192 iteratedDeriv_riemannZeta_conj\n \u2192 analyticOrderAt_riemannZeta_conj \u2192 **zeroMult_conj** (the left-tail\n count bridge).\n- **\u00a715 farG + norm reduction**: farG (4.5m/(x\u221a(1+x\u00b2))), farG_summable\n (dominated by 18(1+t)\u00b2\u00b7m/(1+\u2016\u03b3\u2016\u00b2) via the normSq comparison),\n seriesDiffTerm_norm_eq, the per-term bound (\u2016s\u2080\u2212s\u2016 \u2264 4.5, \u2016s\u2212\u03c1\u2016 \u2265 x,\n \u2016s\u2080\u2212\u03c1\u2016 \u2265 \u221a(1+x\u00b2)), far_series_norm_le_farG_tsum.\n- **\u00a716 the main bucket machine**: unit_family_le (dyadic_family_le\n generalized), mainFinset + bucket range, farG_le_farW_mul (the monotone\n weight chain), main_bucket_count (the two-window perimeter bridge:\n right via hZC + C0hat_two_t_add_one_le, left via hZC or NoLowZeros\n emptiness), farG_main_le \u2264 4.5(2C\u2080+0.55)\u00b71.3476.\n- **\u00a717/18 the two tail machines** (ENNReal fiberwise, cloning the\n LiCircleBound fiber_tail_le/zero_tail_inv_sq_le pattern):\n farGLeft_le \u2264 4.5\u00b70.260558 = 1.1729 (left, j \u2265 14, conj + zeroMult_conj);\n farGRight_le \u2264 4.5\u00b70.607614 = 2.7343 (right, k \u2265 \u230at\u230b\u208a+1, C0hat(2k+1)).\n- **\u00a719 the tail split + capstone**: farG_tail_eq (outside mainFinset,\n farG = farGRight + farGLeft \u2014 the main finset has no \u03b3<0 zeros by\n NoLowZeros) and **far_series_le** (main + right + left:\n 4.5(2C\u2080+0.55)\u00b71.3476 + 2.7343 + 1.1729 = 12.1284\u00b7C\u2080 + 7.2425).\n\n## Outcome part 2: 5f-iii COMPLETE (commit e6d57e1) \u2014 **ii\u2032-5f DONE**\nLiTightR.lean sections 20\u201321 (~550 lines):\n- **the pinch lemma pinch_tight_R**: 11.6318\u00b7log L \u2212 4.6484\u00b7L \u2264 0.397094\n for L \u2265 log 12 \u2014 the concave tangent at the certified critical point\n 2.5024: f\u2032 = 11.6318/L \u2212 4.6484 antitone on Ici (log 12), FTC in both\n directions, f(2.5024) \u2264 \u22120.96231 certified (log 2.5024 \u2264 0.9173 via\n exp partial sums; 4.6484\u00b72.5024 = 11.63215616 \u2265 11.6318). Correction\n side \u2264 0.001. Supporting: log 12 = 4 log 2 \u2212 log(4/3) \u2265 2.4848 (the\n 4-term neg_log_one_sub_le_four machine), log 12 \u2264 2.5024, log 3 \u2264 1.0987.\n- **the C0hat L/LL conversion**: C0hat(t+1) \u2264 0.433(L+1/12) +\n 0.886(LL+0.03354)+2.9.\n- **THE CAPSTONE `logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC)\n (hZC2) (hNLZ)`**: for \u03c3 \u2208 [\u22125/2, 3], t \u2265 10, \u03b6 s \u2260 0:\n \u2016logDeriv \u03b6 s \u2212 \u03a3_{near} m/(s\u2212\u03c1)\u2016 \u2264 10.5\u00b7log(t+2) + 48 \u2014\n the six-piece assembly: piece 1 dirichletC2 \u2264 1, piece 2 the honest\n two-unit near count 0.6L+0.886LL+2.7, piece 3 far_series_le, pieces\n 4/5 0.045 each, piece 6 digamma_diff_le \u2264 6/t \u2264 0.6; fold:\n 5.8516L + 11.6318LL + 47.6029 \u2264 10.5L + 48 via the pinch\n (margin \u2248 1.36 at the worst point L*).\n\n**ii\u2032-5f is COMPLETE**: the tight R-bound (D-056's Theorem R, A = 10.5,\nB\u2080 = 48) is now Lean-proved modulo ONLY the declared perimeter\n(ZeroCountProp, ZeroCountProp2, NoLowZeros).\n\n## Axiom audits\n- audit-session036-axioms.txt (piece 3, 22 decls): 3 standard axioms only.\n- audit-session036b-axioms.txt (5f-iii, 11 decls): 3 standard axioms only.\nFull build GREEN (8762 jobs).\n\n## Banked gotchas (session 036)\n- **`a * b / c` parses as `(a\u00b7b)/c`** \u2014 with mul_left-coefficient proofs\n the goal must be written `c * (b / d)` with explicit parens, else\n isDefEq explodes (200k+ heartbeats) proving `(a\u00b7b)/c = a\u00b7(b/c)` by\n unfolding. THE key Lean-4 gotcha of the session (reproduced in\n isolation).\n- **Decimal literals can default to Float** in literal-only statements \u2014\n pin `(x : \u211d)` on the first literal when the statement's type isn't\n otherwise forced.\n- **`rw [lemma]` rewrites only the FIRST unifier's instances** \u2014 a\n two-sided rewrite (e.g. div_eq_mul_inv on both sides) needs two calls.\n- **x\u207b\u00b9 vs 1/x are different terms** (Inv.inv vs HDiv); normalize via\n one_div/div_eq_mul_inv with care about which form each lemma produces.\n- **`div_le_div_of_nonneg_left (ha : 0\u2264a) (hc : 0<c) (h : c\u2264b) : a/b \u2264 a/c`** \u2014\n the 2nd arg is 0 < (SECOND denominator); **`div_le_iff\u2080 (hc : 0<c) :\n b/c \u2264 a \u2194 b \u2264 a\u00b7c`**; **`le_div_iff\u2080 : a \u2264 b/c \u2194 a\u00b7c \u2264 b`** (directions\n swapped vs naive expectation \u2014 banked twice).\n- **mul_le_mul (h\u2081 : a\u2264b) (h\u2082 : c\u2264d) (hc : 0\u2264c) (hb : 0\u2264b) : a\u00b7c \u2264 b\u00b7d**\n (arg order: (left-ineq)(right-ineq)(0\u2264right-left)(0\u2264left-left)).\n- **`Real.exp_one_gt_d9`/`lt_d9` trans direction**: `(by norm_num : rounder\n \u2264 d9-const).trans X_gt_d9.le` for lower bounds; `X_lt_d9.le.trans\n (by norm_num)` for upper bounds \u2014 getting these backwards produces\n `Real.log 2 \u2264 ?m` metavariable goals.\n- **`t ^ 3` needs `pow_nonneg ht.1 3`** (odd powers aren't positivity-provable);\n t\u00b2 is fine.\n- **Summable.mul_left/add produce lambdas with \u00b7 inside** (not Pi.mul/add) \u2014\n exact at the term level works; `apply Summable.add` with function args\n fails (it takes proofs).\n- **tsum_add as a rewrite needs beta-rfl bridges**: state the goal with\n `from rfl`-provable explicit (f x + g x) forms before rw [e1, e2].\n- **Nat.floor needs exact_mod_cast for mixed \u2115/\u211d conditions**; and\n \u230at\u230b\u208a+1 vs (\u230at\u230b\u208a:\u211d)+1 bridge via push_cast.\n- **The farGLeft tail's left interval count needs NoLowZeros for the\n main-farGLeft emptiness** (any \u03b3<0 zero has |\u03b3| \u2265 14 \u27f9 x \u2265 t+14 \u27f9\n never in mainFinset).\n- **`two_le_C0hat (ht : 1 \u2264 t)`** is the C0hat nonnegativity source\n (positivity can't see through C0hat).\n- **The pinch's coefficient must match the assembly exactly** \u2014 a\n rounded-up LL-coefficient with a rounded-up L-coefficient is safe\n (both LL, L \u2265 0), but a mixed-sign rounding (LL up, L-coeff down via\n 10.5\u2212L) breaks the pinch for large L (the f\u2212f_actual difference has\n mixed sign). Use exact coefficients 11.6318/4.6484 with explicit\n rounding lemmas 5.8515972 \u2264 5.8516, 10.7457624 \u2264 11.6318.\n\n## Status / next\n**ii\u2032-5f COMPLETE.** Remaining map (per D-064 \u00a76 + session-035 notes):\n**ii\u2032-3b** (the contour moments \u03c3_k = \u2212g^{(k\u22121)}(0)/(k\u22121)! \u2014 the R-bound\nis now proved, unblocking the moment estimates) \u2192 **ii\u2032-3c** (assembly of\nLagarias (4.10): \u03c4_j = c_j + (\u22121)\u02b2\u03c3_{j+1} \u2212 (\u22121)\u02b2 \u2014 the crux identity\nSf(n) = Res\u2081(n)) \u2192 **ii\u2032-6** (L0 assembly: sides identity + bridge\n|\u03bb_n(T)| \u2264 2\u00b7FAlt(T) (hRH) + |Sf n| \u2264 Bn FAlt n on [10\u2075, N\u2081]; D-069\nFinding 2's H(T) near-sum repair map lives here) \u2192 **ii\u2032-7** (closure:\ntheorem sfBoundProp : SfBoundProp; final lambda_pos_finite_height_closed\nwith the 3 perimeter hypotheses only; full axiom audit; paper Appendix B\n+ tag upgrade; archive closure + D-037 line-63 erratum pointer).\n**NEXT ACTION**: Session 037 \u2014 ii\u2032-3b (the contour moments).\n"
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raw content_json (database row)
[{"type":"thinking","thinking":"ii′-5f is COMPLETE. Per §8.4 session end protocol (mandatory before shutdown):\n1. Write journal/session-036.md.\n2. Update STATE.md with NEXT ACTION.\n3. Update NGUYEN.md checklist (ii′-5f → complete) + session log.\n4. Register in research.db.\n5. Final commit with session number.\n\nLet me write the journal entry first.","signature":""},{"type":"text","text":"ii′-5f is complete. Executing the session-end protocol (§8.4): journal, STATE, NGUYEN.md, db registration."},{"type":"tool_use","id":"Write_328","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-036.md","content":"# Session 036 — 2026-08-24 — ii′-5f COMPLETE: piece 3 + 5f-iii capstone\n\n## Directive\nContinue ii′-5f from the session-035 state: piece 3 (the far-sum interval\nmachine) then 5f-iii (assembly + pinch + capstone). User: \"Let's do it\nnow\" / \"let's continue to 5f-iii now\".\n\n## Outcome part 1: PIECE 3 COMPLETE (commits 79c506b..52bb2cb)\nLiTightR.lean sections 9–19 (~2700 new lines), the far difference series\nbounded by `far_series_le : ‖far‖ ≤ 12.1284·C0hat(t+1) + 7.2425`:\n- **§9 certified numerics**: the integral-comparison log machines\n (log_one_add_ge, neg_log_one_sub_le), all certified log/exp/sqrt bounds\n (log 10 both sides, log 11/14 below, log 13/2.3/(23/13)/(17/14) above,\n 1/e ≤ 0.3679, log u ≤ u/e).\n- **§10 the weight farW + S₃**: farW k = 1/(k√(1+k²)); S₃ = Σ_{k≥1} farW k\n ≤ 1.3476 (4 exact sqrt-rational terms: √2 ≥ 1.414, √5 ≥ 2.23,\n √10 ≥ 3.16, √17 ≥ 4.123; telescope Σ_{k≥5} 1/(k(k−1)) = 1/4).\n- **§11/12 the six tail bricks** (antitone-integral machine):\n RW₀ 0.1, RW₁ 0.3303, RW₂ 0.1216 (loglog ≤ (1/e)·log trick);\n LA₀ 1/23, LA₁ 0.1687 (the F = −log x/(x+10) + (1/10)log(x/(x+10))\n antiderivative), LA₂ 0.0621.\n- **§13 C0hat bridges**: C0hat(2k+1) ≤ 0.433(log k+0.833)+0.886(loglog k\n +0.3475)+2.9 (k ≥ 11), the j+1 form (j ≥ 14: 0.1942/0.0736), and\n **C0hat(2t+1) ≤ C0hat(t+1) + 0.55** (t ≥ 10).\n- **§14 conj machinery**: hasDerivAt_conj_of_conj_eq (the conj-derivative\n lemma via the slope characterization) → iteratedDeriv_riemannZeta_conj\n → analyticOrderAt_riemannZeta_conj → **zeroMult_conj** (the left-tail\n count bridge).\n- **§15 farG + norm reduction**: farG (4.5m/(x√(1+x²))), farG_summable\n (dominated by 18(1+t)²·m/(1+‖γ‖²) via the normSq comparison),\n seriesDiffTerm_norm_eq, the per-term bound (‖s₀−s‖ ≤ 4.5, ‖s−ρ‖ ≥ x,\n ‖s₀−ρ‖ ≥ √(1+x²)), far_series_norm_le_farG_tsum.\n- **§16 the main bucket machine**: unit_family_le (dyadic_family_le\n generalized), mainFinset + bucket range, farG_le_farW_mul (the monotone\n weight chain), main_bucket_count (the two-window perimeter bridge:\n right via hZC + C0hat_two_t_add_one_le, left via hZC or NoLowZeros\n emptiness), farG_main_le ≤ 4.5(2C₀+0.55)·1.3476.\n- **§17/18 the two tail machines** (ENNReal fiberwise, cloning the\n LiCircleBound fiber_tail_le/zero_tail_inv_sq_le pattern):\n farGLeft_le ≤ 4.5·0.260558 = 1.1729 (left, j ≥ 14, conj + zeroMult_conj);\n farGRight_le ≤ 4.5·0.607614 = 2.7343 (right, k ≥ ⌊t⌋₊+1, C0hat(2k+1)).\n- **§19 the tail split + capstone**: farG_tail_eq (outside mainFinset,\n farG = farGRight + farGLeft — the main finset has no γ<0 zeros by\n NoLowZeros) and **far_series_le** (main + right + left:\n 4.5(2C₀+0.55)·1.3476 + 2.7343 + 1.1729 = 12.1284·C₀ + 7.2425).\n\n## Outcome part 2: 5f-iii COMPLETE (commit e6d57e1) — **ii′-5f DONE**\nLiTightR.lean sections 20–21 (~550 lines):\n- **the pinch lemma pinch_tight_R**: 11.6318·log L − 4.6484·L ≤ 0.397094\n for L ≥ log 12 — the concave tangent at the certified critical point\n 2.5024: f′ = 11.6318/L − 4.6484 antitone on Ici (log 12), FTC in both\n directions, f(2.5024) ≤ −0.96231 certified (log 2.5024 ≤ 0.9173 via\n exp partial sums; 4.6484·2.5024 = 11.63215616 ≥ 11.6318). Correction\n side ≤ 0.001. Supporting: log 12 = 4 log 2 − log(4/3) ≥ 2.4848 (the\n 4-term neg_log_one_sub_le_four machine), log 12 ≤ 2.5024, log 3 ≤ 1.0987.\n- **the C0hat L/LL conversion**: C0hat(t+1) ≤ 0.433(L+1/12) +\n 0.886(LL+0.03354)+2.9.\n- **THE CAPSTONE `logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC)\n (hZC2) (hNLZ)`**: for σ ∈ [−5/2, 3], t ≥ 10, ζ s ≠ 0:\n ‖logDeriv ζ s − Σ_{near} m/(s−ρ)‖ ≤ 10.5·log(t+2) + 48 —\n the six-piece assembly: piece 1 dirichletC2 ≤ 1, piece 2 the honest\n two-unit near count 0.6L+0.886LL+2.7, piece 3 far_series_le, pieces\n 4/5 0.045 each, piece 6 digamma_diff_le ≤ 6/t ≤ 0.6; fold:\n 5.8516L + 11.6318LL + 47.6029 ≤ 10.5L + 48 via the pinch\n (margin ≈ 1.36 at the worst point L*).\n\n**ii′-5f is COMPLETE**: the tight R-bound (D-056's Theorem R, A = 10.5,\nB₀ = 48) is now Lean-proved modulo ONLY the declared perimeter\n(ZeroCountProp, ZeroCountProp2, NoLowZeros).\n\n## Axiom audits\n- audit-session036-axioms.txt (piece 3, 22 decls): 3 standard axioms only.\n- audit-session036b-axioms.txt (5f-iii, 11 decls): 3 standard axioms only.\nFull build GREEN (8762 jobs).\n\n## Banked gotchas (session 036)\n- **`a * b / c` parses as `(a·b)/c`** — with mul_left-coefficient proofs\n the goal must be written `c * (b / d)` with explicit parens, else\n isDefEq explodes (200k+ heartbeats) proving `(a·b)/c = a·(b/c)` by\n unfolding. THE key Lean-4 gotcha of the session (reproduced in\n isolation).\n- **Decimal literals can default to Float** in literal-only statements —\n pin `(x : ℝ)` on the first literal when the statement's type isn't\n otherwise forced.\n- **`rw [lemma]` rewrites only the FIRST unifier's instances** — a\n two-sided rewrite (e.g. div_eq_mul_inv on both sides) needs two calls.\n- **x⁻¹ vs 1/x are different terms** (Inv.inv vs HDiv); normalize via\n one_div/div_eq_mul_inv with care about which form each lemma produces.\n- **`div_le_div_of_nonneg_left (ha : 0≤a) (hc : 0<c) (h : c≤b) : a/b ≤ a/c`** —\n the 2nd arg is 0 < (SECOND denominator); **`div_le_iff₀ (hc : 0<c) :\n b/c ≤ a ↔ b ≤ a·c`**; **`le_div_iff₀ : a ≤ b/c ↔ a·c ≤ b`** (directions\n swapped vs naive expectation — banked twice).\n- **mul_le_mul (h₁ : a≤b) (h₂ : c≤d) (hc : 0≤c) (hb : 0≤b) : a·c ≤ b·d**\n (arg order: (left-ineq)(right-ineq)(0≤right-left)(0≤left-left)).\n- **`Real.exp_one_gt_d9`/`lt_d9` trans direction**: `(by norm_num : rounder\n ≤ d9-const).trans X_gt_d9.le` for lower bounds; `X_lt_d9.le.trans\n (by norm_num)` for upper bounds — getting these backwards produces\n `Real.log 2 ≤ ?m` metavariable goals.\n- **`t ^ 3` needs `pow_nonneg ht.1 3`** (odd powers aren't positivity-provable);\n t² is fine.\n- **Summable.mul_left/add produce lambdas with · inside** (not Pi.mul/add) —\n exact at the term level works; `apply Summable.add` with function args\n fails (it takes proofs).\n- **tsum_add as a rewrite needs beta-rfl bridges**: state the goal with\n `from rfl`-provable explicit (f x + g x) forms before rw [e1, e2].\n- **Nat.floor needs exact_mod_cast for mixed ℕ/ℝ conditions**; and\n ⌊t⌋₊+1 vs (⌊t⌋₊:ℝ)+1 bridge via push_cast.\n- **The farGLeft tail's left interval count needs NoLowZeros for the\n main-farGLeft emptiness** (any γ<0 zero has |γ| ≥ 14 ⟹ x ≥ t+14 ⟹\n never in mainFinset).\n- **`two_le_C0hat (ht : 1 ≤ t)`** is the C0hat nonnegativity source\n (positivity can't see through C0hat).\n- **The pinch's coefficient must match the assembly exactly** — a\n rounded-up LL-coefficient with a rounded-up L-coefficient is safe\n (both LL, L ≥ 0), but a mixed-sign rounding (LL up, L-coeff down via\n 10.5−L) breaks the pinch for large L (the f−f_actual difference has\n mixed sign). Use exact coefficients 11.6318/4.6484 with explicit\n rounding lemmas 5.8515972 ≤ 5.8516, 10.7457624 ≤ 11.6318.\n\n## Status / next\n**ii′-5f COMPLETE.** Remaining map (per D-064 §6 + session-035 notes):\n**ii′-3b** (the contour moments σ_k = −g^{(k−1)}(0)/(k−1)! — the R-bound\nis now proved, unblocking the moment estimates) → **ii′-3c** (assembly of\nLagarias (4.10): τ_j = c_j + (−1)ʲσ_{j+1} − (−1)ʲ — the crux identity\nSf(n) = Res₁(n)) → **ii′-6** (L0 assembly: sides identity + bridge\n|λ_n(T)| ≤ 2·FAlt(T) (hRH) + |Sf n| ≤ Bn FAlt n on [10⁵, N₁]; D-069\nFinding 2's H(T) near-sum repair map lives here) → **ii′-7** (closure:\ntheorem sfBoundProp : SfBoundProp; final lambda_pos_finite_height_closed\nwith the 3 perimeter hypotheses only; full axiom audit; paper Appendix B\n+ tag upgrade; archive closure + D-037 line-63 erratum pointer).\n**NEXT ACTION**: Session 037 — ii′-3b (the contour moments).\n"},"extra_content":null}]