← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (1998 chars)
The simpa also failed to close hsplit — the goal is unchanged after it. That means `h.symm`'s simp-normal form doesn't match. The key question: what IS h's form? Let me get Lean to print it — use `trace_state` or just #check via a scratch. Quicker: run lean with a #check on the term form. Actually — I bet the issue: `Summable.sum_add_tsum_nat_add` might not be the name (unknown identifier would error though — no unknown-identifier error shown, so the name resolved). Hmm — but if `h := hker_summ.sum_add_tsum_nat_add N` resolved and h's type is something unexpected (e.g., the PRIMED version with a different shape), then simpa wouldn't close. Let me write a tiny probe file: #check @NguyenRH1... no — probe: `example := hker_summ.sum_add_tsum_nat_add N` with a #check? Simplest: add `set_option pp.all true`? Too heavy. Use a probe in the Scratch... Actually — let me think about what `Summable.sum_add_tsum_nat_add` could be: maybe it's `Summable.sum_add_tsum_nat_add (hf : Summable f) (k : ℕ) : ∑' i, f i = ∑ i ∈ range k, f i + ∑' i, f (i + k)` — h then k. If so, `hker_summ.sum_add_tsum_nat_add N` — dot notation: hker_summ fills the FIRST explicit argument of type Summable — if the signature is (hf) (k), then hf := hker_summ and N fills k ✓ same result. The statement's direction: ∑' f = ∑ + ∑' — then h.symm is the split form ✓ and simpa should close... UNLESS the issue is that my goal's tail cast `↑(n + N)` differs from h's `↑(n + N)`...
Hold on — maybe the problem is entirely different: maybe `simpa using h.symm` FAILED because simpa's default simp set TRANSFORMED my goal (e.g., simp might rewrite the ⁻¹ forms or the casts) making it not match — and then simpa reports "unsolved goals"? no — simpa failing reports "simp made no progress" or a mismatch... it reported "unsolved goals" with the ORIGINAL goal — meaning simpa transformed h.symm to something and the goal unchanged, unclosed.
Enough guessing — probe the actual h. Let me create a tiny probe file that #checks it.
Let me probe the actual statement form instead of guessing.
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Probe.lean",
"content": "/- probe: the exact form of Summable.sum_add_tsum_nat_add -/\nimport Mathlib\n\nexample : True := by\n trivial\n\n#check (fun (f : \u2115 \u2192 \u211d) (k : \u2115) (hf : Summable f) => hf.sum_add_tsum_nat_add k)\n#check (fun (f : \u2115 \u2192 \u211d) (k : \u2115) (hf : Summable f) => hf.sum_add_tsum_nat_add' k)\n#check @Summable.sum_add_tsum_nat_add\n"
}
result 2026-08-23 09:55:53Z → 2026-08-24 22:25:21Z
{
"type": "create",
"filePath": "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":"The simpa also failed to close hsplit — the goal is unchanged after it. That means `h.symm`'s simp-normal form doesn't match. The key question: what IS h's form? Let me get Lean to print it — use `trace_state` or just #check via a scratch. Quicker: run lean with a #check on the term form. Actually — I bet the issue: `Summable.sum_add_tsum_nat_add` might not be the name (unknown identifier would error though — no unknown-identifier error shown, so the name resolved). Hmm — but if `h := hker_summ.sum_add_tsum_nat_add N` resolved and h's type is something unexpected (e.g., the PRIMED version with a different shape), then simpa wouldn't close. Let me write a tiny probe file: #check @NguyenRH1... no — probe: `example := hker_summ.sum_add_tsum_nat_add N` with a #check? Simplest: add `set_option pp.all true`? Too heavy. Use a probe in the Scratch... Actually — let me think about what `Summable.sum_add_tsum_nat_add` could be: maybe it's `Summable.sum_add_tsum_nat_add (hf : Summable f) (k : ℕ) : ∑' i, f i = ∑ i ∈ range k, f i + ∑' i, f (i + k)` — h then k. If so, `hker_summ.sum_add_tsum_nat_add N` — dot notation: hker_summ fills the FIRST explicit argument of type Summable — if the signature is (hf) (k), then hf := hker_summ and N fills k ✓ same result. The statement's direction: ∑' f = ∑ + ∑' — then h.symm is the split form ✓ and simpa should close... UNLESS the issue is that my goal's tail cast `↑(n + N)` differs from h's `↑(n + N)`... \n\nHold on — maybe the problem is entirely different: maybe `simpa using h.symm` FAILED because simpa's default simp set TRANSFORMED my goal (e.g., simp might rewrite the ⁻¹ forms or the casts) making it not match — and then simpa reports \"unsolved goals\"? no — simpa failing reports \"simp made no progress\" or a mismatch... it reported \"unsolved goals\" with the ORIGINAL goal — meaning simpa transformed h.symm to something and the goal unchanged, unclosed.\n\nEnough guessing — probe the actual h. Let me create a tiny probe file that #checks it.","signature":""},{"type":"text","text":"Let me probe the actual statement form instead of guessing."},{"type":"tool_use","id":"Write_328","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Probe.lean","content":"/- probe: the exact form of Summable.sum_add_tsum_nat_add -/\nimport Mathlib\n\nexample : True := by\n trivial\n\n#check (fun (f : ℕ → ℝ) (k : ℕ) (hf : Summable f) => hf.sum_add_tsum_nat_add k)\n#check (fun (f : ℕ → ℝ) (k : ℕ) (hf : Summable f) => hf.sum_add_tsum_nat_add' k)\n#check @Summable.sum_add_tsum_nat_add\n"},"extra_content":null}]