← Session 21 — Tier ii′ planning + launch: close SfBoundProp in Lean (D-064)
thinking (1704 chars)
Committed. Now — session 016 has delivered ii′-1b(i) and (ii). The remaining piece of ii′-1b is (iii): the full rectangle residue theorem for J. Given the session has been very long (many Lean iterations), let me check the time/pacing. The user is on telegram and said "let's go" — I should keep working as long as productive, but this is a natural milestone. Let me assess what (iii) needs and decide whether to continue or wrap with a clean status update.
ii′-1b(iii) plan (from the design):
- R := J − P: holomorphic on Rectangle \ (zeros ∪ {−2}) with simple poles at zeros (residues mult·k_n(ρ−1) = mult·(1−(ρ/(ρ−1))ⁿ)) and at −2 (residue 1−(2/3)ⁿ).
- The order-at-1 computation (R = G near 1, where G = [F − Tayl_n F]/(w−1)^{n+1} analytic).
- residueTheorem_finset (vendored Zeta23.Analytic.RectangleLogDeriv) for R.
- Assemble: sides = ∮J = ∮P + ∮R = ResW1 + Σ_residues.
That's another substantial file (300+ lines, with the order computation being the trickiest part). Given how long this session has run, and per the pacing rules (§8.5 BREAK PROMPT — approaching long work should pause and check with the user), this is a good point to close session 016 cleanly with the milestone committed, give the user the status, and let them decide to continue.
Actually — the user said "let's go" from telegram and I should keep momentum but also be honest about session boundaries. The session protocol (§8.4) wants clean end-states. I have a clean committed milestone (ii′-1b(i)+(ii)). Let me wrap with the journal + STATE update for session 016, then report.
Let me do the session-016 closure: journal, STATE, NGUYEN checklist update (ii′-1b partially done — mark (i)/(ii)), db registration, commit.
A natural milestone: ii′-1b(i) + (ii) complete and committed. Let me close session 016 cleanly with the archive updates.
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-016.md",
"content": "# Session 016 \u2014 2026-08-21 \u2014 Tier ii\u2032 sub-stage ii\u2032-1b (i)+(ii) COMPLETE\n\n## Directive\nContinue tier ii\u2032 per D-064: sub-stage ii\u2032-1b (R1 residue machinery).\n\n## Work done\n**ii\u2032-1b(i) COMPLETE (commit 1c35d31)** \u2014 LiResidue.lean:\n- `zetaA = (w\u22121)\u00b7\u03b6(w)` extended by 1 at 1: analytic (removable\n singularity via Mathlib `riemannZeta_residue_one` +\n RemovableSingularity.lean's `differentiableOn_update_limUnder_of_bddAbove`).\n- `hLi = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)`: analytic at 1, value 1, agrees with\n (w\u22121)(\u2212\u03b6\u2032/\u03b6) on a punctured neighborhood of 1 (`hLi_eq_near_one`,\n via riemannZeta_residue_one.eventually_ne).\n- `lcoeff k := iteratedDeriv (k+1) hLi 1 / (k+1)!` \u2014 operational\n Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1.\n- `A_iter`: iterated derivatives of A(w) = w\u207f\u2212(w\u22121)\u207f at 1\n (descFactorial for i < n, 0 for i = n; via iteratedDeriv_pow +\n iteratedDeriv_comp_sub_const).\n- **`ResW1_eq_sum`**: the w=1 residue of the Lagarias-kernel integrand\n equals \u03a3_{j=1}^n C(n,j)\u00b7lcoeff(j\u22121) \u2014 via iteratedDeriv_mul\n (Leibniz), as designed (D-064 R1) and verified numerically (session 014).\n\n**ii\u2032-1b(ii) COMPLETE (commit 973fe66)** \u2014 LiPrincipal.lean:\n- `rectInt_deriv_eq_zero`: RectangleIntegral (deriv G) = 0 when G is\n differentiable on the border \u2014 per-side FTC with the mixed-field chain\n rule (HasDerivAt.comp in the \u211d\u2192\u2102\u2192\u2102 algebra tower), border MapsTo from\n vendored Rectangle.lean.\n- `RectangleIntegral'_add` linearity.\n- `Acoeff/ppart/Gprim`; `deriv_Gprim` (termwise: hasDerivAt_zpow + comp);\n `tail_continuousAt` (Finset induction); `ppart_eq_deriv_add`.\n- **`rectInt'_ppart`**: RectangleIntegral' (ppart n) z w = ResW1 n on\n any rectangle with 1 interior \u2014 A_1 term by the vendored\n `ResidueTheoremInRectangle`, i \u2265 2 tail vanishes by the primitive FTC.\n\nBoth axiom-audited: 3 standard axioms only.\n\n## Lean notes (hard-won this session, additions to the handoff list)\n- Factorial `n !` is SCOPED notation \u2014 needs `open scoped Nat`.\n- `conj` is scope-locked (ComplexConjugate) \u2014 use `starRingEnd \u2102`.\n- Parametrized lambdas over \u211d into \u2102 need explicit `(x : \u211d)` annotations\n (else x elaborates at \u2102 and every downstream unify fails).\n- `HasDerivAt.comp` handles mixed fields via NormedAlgebra/IsScalarTower.\n- `ContinuousLinearMap.hasDerivAt : HasDerivAt e (e 1) x` (ofRealCLM).\n- `intervalIntegral.integral_eq_sub_of_hasDerivAt` (FTC-2) with explicit\n (f := ...) (f' := ...) args \u2014 unification is brittle otherwise.\n- `ContinuousAt` of expressions containing the point variable: write the\n `fun w =>` lambda EXPLICITLY (Lean's eta-unification misfires).\n- Finset.sum_apply / funext for sum-of-functions \u2194 lambda-of-sums.\n- Finset.sum_bij arg order: (i, hi, i_inj, i_surj, h).\n- differentiableOn_update_limUnder_of_bddAbove (removable singularity);\n Tendsto.eventually_ne; DifferentiableOn.deriv (CauchyIntegral.lean).\n\n## State at stop\nii\u2032-1b(i)+(ii) complete. Remaining for ii\u2032-1b: (iii) the full rectangle\nresidue theorem for J \u2014 R = J \u2212 P holomorphic off {zeros \u222a {\u22122}}\n(order \u2265 0 at 1 via Taylor-remainder), simple poles at zeros (residue\nmult\u00b7(1\u2212(\u03c1/(\u03c1\u22121))\u207f)) and at \u22122 (residue 1\u2212(2/3)\u207f), via vendored\n`residueTheorem_finset` (Zeta23.Analytic.RectangleLogDeriv), then\nassemble sides = ResW1 + \u03a3 + triv.\nNext session: ii\u2032-1b(iii).\n"
}
result 2026-08-21 22:28:49Z → 2026-08-22 06:36:48Z
{
"type": "update",
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\PrintAxioms.lean",
"content": "/- Session 020 axiom audit: ii\u2032-3a piece (A) part 1 new declarations. -/\nimport Mathlib\nimport NguyenRH1.LiArchimedean\n\nopen NguyenRH1\n\n#print axioms logDeriv_Gamma\u211d_one\n#print axioms logDeriv_Gamma\u211d_one_add\n#print axioms pi_cpow_neg_half_eq_exp\n#print axioms log_four_pi\n",
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"-/- Session 019 axiom audit: ii\u2032-3a piece (B) new declarations. -/",
"+/- Session 020 axiom audit: ii\u2032-3a piece (A) part 1 new declarations. -/",
" import Mathlib",
"-import NguyenRH1.LiZetaCoeff",
"+import NguyenRH1.LiArchimedean",
" ",
" open NguyenRH1",
" ",
"-#print axioms iteratedDeriv_hLiExt",
"-#print axioms hLiExt_analyticAt_zero",
"-#print axioms hLiExt_mul_id",
"-#print axioms hLiExt_tendsto",
"-#print axioms iteratedDeriv_hLi_one_add",
"-#print axioms hLiExt_eq_near_zero",
"+#print axioms logDeriv_Gamma\u211d_one",
"+#print axioms logDeriv_Gamma\u211d_one_add",
"+#print axioms pi_cpow_neg_half_eq_exp",
"+#print axioms log_four_pi",
" "
]
}
],
"originalFile": "/- Session 019 axiom audit: ii\u2032-3a piece (B) new declarations. -/\nimport Mathlib\nimport NguyenRH1.LiZetaCoeff\n\nopen NguyenRH1\n\n#print axioms iteratedDeriv_hLiExt\n#print axioms hLiExt_analyticAt_zero\n#print axioms hLiExt_mul_id\n#print axioms hLiExt_tendsto\n#print axioms iteratedDeriv_hLi_one_add\n#print axioms hLiExt_eq_near_zero\n",
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raw content_json (database row)
[{"type":"thinking","thinking":"Committed. Now — session 016 has delivered ii′-1b(i) and (ii). The remaining piece of ii′-1b is (iii): the full rectangle residue theorem for J. Given the session has been very long (many Lean iterations), let me check the time/pacing. The user is on telegram and said \"let's go\" — I should keep working as long as productive, but this is a natural milestone. Let me assess what (iii) needs and decide whether to continue or wrap with a clean status update.\n\nii′-1b(iii) plan (from the design):\n- R := J − P: holomorphic on Rectangle \\ (zeros ∪ {−2}) with simple poles at zeros (residues mult·k_n(ρ−1) = mult·(1−(ρ/(ρ−1))ⁿ)) and at −2 (residue 1−(2/3)ⁿ).\n- The order-at-1 computation (R = G near 1, where G = [F − Tayl_n F]/(w−1)^{n+1} analytic).\n- residueTheorem_finset (vendored Zeta23.Analytic.RectangleLogDeriv) for R.\n- Assemble: sides = ∮J = ∮P + ∮R = ResW1 + Σ_residues.\n\nThat's another substantial file (300+ lines, with the order computation being the trickiest part). Given how long this session has run, and per the pacing rules (§8.5 BREAK PROMPT — approaching long work should pause and check with the user), this is a good point to close session 016 cleanly with the milestone committed, give the user the status, and let them decide to continue.\n\nActually — the user said \"let's go\" from telegram and I should keep momentum but also be honest about session boundaries. The session protocol (§8.4) wants clean end-states. I have a clean committed milestone (ii′-1b(i)+(ii)). Let me wrap with the journal + STATE update for session 016, then report.\n\nLet me do the session-016 closure: journal, STATE, NGUYEN checklist update (ii′-1b partially done — mark (i)/(ii)), db registration, commit.","signature":""},{"type":"text","text":"A natural milestone: ii′-1b(i) + (ii) complete and committed. Let me close session 016 cleanly with the archive updates."},{"type":"tool_use","id":"Write_286","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-016.md","content":"# Session 016 — 2026-08-21 — Tier ii′ sub-stage ii′-1b (i)+(ii) COMPLETE\n\n## Directive\nContinue tier ii′ per D-064: sub-stage ii′-1b (R1 residue machinery).\n\n## Work done\n**ii′-1b(i) COMPLETE (commit 1c35d31)** — LiResidue.lean:\n- `zetaA = (w−1)·ζ(w)` extended by 1 at 1: analytic (removable\n singularity via Mathlib `riemannZeta_residue_one` +\n RemovableSingularity.lean's `differentiableOn_update_limUnder_of_bddAbove`).\n- `hLi = 1 − (w−1)·(zetaA′/zetaA)`: analytic at 1, value 1, agrees with\n (w−1)(−ζ′/ζ) on a punctured neighborhood of 1 (`hLi_eq_near_one`,\n via riemannZeta_residue_one.eventually_ne).\n- `lcoeff k := iteratedDeriv (k+1) hLi 1 / (k+1)!` — operational\n Laurent coefficients of −ζ′/ζ at 1.\n- `A_iter`: iterated derivatives of A(w) = wⁿ−(w−1)ⁿ at 1\n (descFactorial for i < n, 0 for i = n; via iteratedDeriv_pow +\n iteratedDeriv_comp_sub_const).\n- **`ResW1_eq_sum`**: the w=1 residue of the Lagarias-kernel integrand\n equals Σ_{j=1}^n C(n,j)·lcoeff(j−1) — via iteratedDeriv_mul\n (Leibniz), as designed (D-064 R1) and verified numerically (session 014).\n\n**ii′-1b(ii) COMPLETE (commit 973fe66)** — LiPrincipal.lean:\n- `rectInt_deriv_eq_zero`: RectangleIntegral (deriv G) = 0 when G is\n differentiable on the border — per-side FTC with the mixed-field chain\n rule (HasDerivAt.comp in the ℝ→ℂ→ℂ algebra tower), border MapsTo from\n vendored Rectangle.lean.\n- `RectangleIntegral'_add` linearity.\n- `Acoeff/ppart/Gprim`; `deriv_Gprim` (termwise: hasDerivAt_zpow + comp);\n `tail_continuousAt` (Finset induction); `ppart_eq_deriv_add`.\n- **`rectInt'_ppart`**: RectangleIntegral' (ppart n) z w = ResW1 n on\n any rectangle with 1 interior — A_1 term by the vendored\n `ResidueTheoremInRectangle`, i ≥ 2 tail vanishes by the primitive FTC.\n\nBoth axiom-audited: 3 standard axioms only.\n\n## Lean notes (hard-won this session, additions to the handoff list)\n- Factorial `n !` is SCOPED notation — needs `open scoped Nat`.\n- `conj` is scope-locked (ComplexConjugate) — use `starRingEnd ℂ`.\n- Parametrized lambdas over ℝ into ℂ need explicit `(x : ℝ)` annotations\n (else x elaborates at ℂ and every downstream unify fails).\n- `HasDerivAt.comp` handles mixed fields via NormedAlgebra/IsScalarTower.\n- `ContinuousLinearMap.hasDerivAt : HasDerivAt e (e 1) x` (ofRealCLM).\n- `intervalIntegral.integral_eq_sub_of_hasDerivAt` (FTC-2) with explicit\n (f := ...) (f' := ...) args — unification is brittle otherwise.\n- `ContinuousAt` of expressions containing the point variable: write the\n `fun w =>` lambda EXPLICITLY (Lean's eta-unification misfires).\n- Finset.sum_apply / funext for sum-of-functions ↔ lambda-of-sums.\n- Finset.sum_bij arg order: (i, hi, i_inj, i_surj, h).\n- differentiableOn_update_limUnder_of_bddAbove (removable singularity);\n Tendsto.eventually_ne; DifferentiableOn.deriv (CauchyIntegral.lean).\n\n## State at stop\nii′-1b(i)+(ii) complete. Remaining for ii′-1b: (iii) the full rectangle\nresidue theorem for J — R = J − P holomorphic off {zeros ∪ {−2}}\n(order ≥ 0 at 1 via Taylor-remainder), simple poles at zeros (residue\nmult·(1−(ρ/(ρ−1))ⁿ)) and at −2 (residue 1−(2/3)ⁿ), via vendored\n`residueTheorem_finset` (Zeta23.Analytic.RectangleLogDeriv), then\nassemble sides = ResW1 + Σ + triv.\nNext session: ii′-1b(iii).\n"},"extra_content":null}]