← Session 30 — ii′-5e squeeze capstone + 5f perimeter repair
thinking (812 chars)
Milestone committed. Now the session-end artifacts:
1. journal/session-034.md
2. STATE.md update (NEXT ACTION → ii′-5f)
3. NGUYEN.md §5 checklist (ii′-5e entry) + session log
4. db registration (session + derivation? — the Lean sub-stages get journal + STATE; check how previous sessions registered. From handoff: "Session registration via direct sqlite3 insert (journal_file column)".)
Let me check how session 033 did the db registration to mirror. Actually the commits show "journal + STATE + NGUYEN + db". Let me look at the db.py CLI quickly and the session table pattern. From handoff notes: `db.py: init, log-experiment, add-result, add-derivation, add-graveyard, list`. Session registration via direct sqlite3 insert.
Let me write the journal first, then STATE, then NGUYEN, then db, then final commit.
Milestone committed. Now the session-end archive artifacts (journal, STATE, NGUYEN, db).
tool: Write
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-034.md",
"content": "# Session 034 \u2014 2026-08-23 \u2014 ii\u2032-5e-v-d COMPLETE: the Schwarz squeeze \u27f9\n# the partial-fraction capstone\n\n## Directive\nContinue ii\u2032-5e-v per STATE.md: chunk 5e-v-d \u2014 the Schwarz squeeze \u27f9\nthe \u2203 B capstone `partialFraction_identity` (LiCircleBound.lean \u00a713 or a\nnew LiSqueeze.lean). User: \"Let's start ii\u2032-5e-v-d\".\n\n## Outcome: 5e-v-d COMPLETE (commit 5a262a1; new file LiSqueeze.lean,\n~330 lines, clean elaboration, zero warnings; LiCircleBound + LiSqueeze\nroot-imported; full build GREEN; audit-session034-axioms.txt: 5 decls,\n3 standard axioms only)\n\nNew file **LiSqueeze.lean** (imports LiCircleBound; LiCircleBound built\nfor the first time \u2014 51 s \u2014 and both now root-imported):\n\n- **\u00a71 `C0hat_add_two_le`**: \u0108\u2080(R+2) \u2264 5\u00b7log R for R \u2265 16002. Via\n log(R+4) \u2264 2\u00b7log R (R+4 \u2264 R\u00b2, Real.log_pow), log log(R+4) \u2264\n log(2\u00b7log R) = log 2 + log log R \u2264 1 + log R \u2264 2\u00b7log R\n (Real.log_le_sub_one_of_pos twice), 2.9 \u2264 (1/3)\u00b7log R; total\n 0.433\u00b72 + 0.886\u00b72 + 1/3 \u2248 2.97 \u2264 5.\n- **\u00a72 `not_badPoint_on_gapCircle`**: the four-way badness split \u2014\n not nontrivial (session-032's `not_isNontrivialZero_on_gapCircle`),\n \u2260 0/1 (norms: \u2016s\u2016 = R \u2265 16002), \u2260 \u22122n (else R/2 = n \u2208 \u2124 against\n hgapInt with m := n; `Int.cast_natCast` bridge, `Complex.norm_natCast`).\n- **\u00a73 `gentire_circle_le`**: Gentire s = Fexpr s on the circle (\u00a72 +\n 5e-iv's `gentire_eq_Fexpr`) \u27f9 the Fexpr_circle_le bound transfers.\n- **\u00a74 `gentire_const (hZC : ZeroCountProp)` : Gentire z\u2080 = Gentire 0**.\n The squeeze: for \u03b5 > 0, set C\u2080' := 2\u00b7lowZeroConst + dirichletC2 +\n \u2016Gentire 0\u2016, C\u2081 := 2\u00b710\u2076\u00b7A\u2080 ( = 400000\u00b7A\u2080\u00b75), c := 2 +\n (128\u00b7C\u2081 + 2\u00b7C\u2080')\u00b7\u2016z\u2080\u2016/\u03b5, M := max (c\u2074) (\u2016z\u2080\u2016+1), k := max 4000 \u2308M/4\u2309\u208a;\n the perimeter gap circle R \u2208 [4k+5/2, 4k+3] (exists_gapCircle_radius)\n has R \u2265 M. Max-modulus (Zeta23\n `AnalyticOn.norm_le_of_norm_le_on_sphere`, r = R, analytic from\n `gentire_analyticAt_and_eq.1`) lifts the sphere bound to the ball;\n MapsTo into closedBall (Gentire 0) (RB + \u2016Gentire 0\u2016) by norm_sub_le;\n the ball-form Schwarz lemma\n (`Complex.dist_le_div_mul_dist_of_mapsTo_ball`) gives\n dist \u2264 (RB + \u2016Gentire 0\u2016)/R\u00b7\u2016z\u2080\u2016. Growth: RB \u2264 C\u2080' + 64\u00b7C\u2081\u00b7Q\u00b3 with\n Q := R^{1/4} (Real.log_le_rpow_div at \u03b5 = 1/4 \u27f9 log R \u2264 4Q;\n pow_le_pow_left\u2080 \u27f9 (log R)\u00b3 \u2264 64Q\u00b3; \u0108\u2080(R+2) \u2264 5\u00b7log R by \u00a71) and\n R = Q\u2074 (Real.rpow_inv_natCast_pow); Q \u2265 c \u27f9 the two branches\n 64\u00b7C\u2081\u00b7Q\u00b3\u00b7\u2016z\u2080\u2016 \u2264 (\u03b5/2)\u00b7Q\u2074 (from 128\u00b7C\u2081\u00b7\u2016z\u2080\u2016 \u2264 \u03b5\u00b7Q) and C\u2080'\u00b7\u2016z\u2080\u2016 \u2264\n (\u03b5/2)\u00b7Q\u2074 (from 2\u00b7C\u2080'\u00b7\u2016z\u2080\u2016 \u2264 \u03b5\u00b7Q and \u03b5\u00b7Q \u2264 (\u03b5/2)\u00b7Q\u2074 via Q\u00b3 \u2265 8);\n sum \u27f9 (RB + \u2016Gentire 0\u2016)\u00b7\u2016z\u2080\u2016 \u2264 \u03b5\u00b7Q\u2074 = \u03b5\u00b7R. Then dist \u2264 \u03b5 for all\n \u03b5 \u27f9 dist = 0 (le_or_gt rcases + the \u03b5 := dist/2 contradiction) \u27f9\n Gentire z\u2080 = Gentire 0.\n- **\u00a75 `partialFraction_identity (hZC : ZeroCountProp)`**: \u2203 B : \u2102,\n \u2200 s, riemannZeta s \u2260 0 \u2192 s \u2260 0 \u2192 s \u2260 1 \u2192 (\u2200 n \u2265 1, s \u2260 \u22122n) \u2192\n logDeriv riemannZeta s = B + seriesFun s \u2212 1/s \u2212 1/(s\u22121) +\n (Real.log Real.pi : \u2102)/2 \u2212 Complex.digamma (s/2)/2. Proof: \u03b6 s \u2260 0\n kills the nontrivial-zero disjunct (IsNontrivialZero.1) \u27f9 \u00ac badPoint s\n \u27f9 Gentire s = Fexpr s (5e-iv) and Gentire s = Gentire 0 (\u00a74) \u27f9\n Fexpr s = Gentire 0; unfold Fexpr; linear_combination. B := Gentire 0\n (no evaluation needed \u2014 D-068 \u00a70: B cancels in 5f's two-point\n analysis).\n\n**This is the first use of the perimeter in the 5e chain** (the gap\ncircles need ZeroCountProp; 5c/5d/5e-i..iv were Jensen/unconditional).\nThe capstone is exactly the D-068 \u00a75 target statement.\n\n## Inventory verification (all live-checked before writing)\n- `Complex.dist_le_div_mul_dist_of_mapsTo_ball` (Mathlib Schwarz.lean):\n (hd : DifferentiableOn \u2102 f (ball c R\u2081)) (MapsTo f (ball c R\u2081)\n (closedBall (f c) R\u2082)) (z \u2208 ball) \u27f9 dist (f z) (f c) \u2264 R\u2082/R\u2081\u00b7dist z c.\n- `AnalyticOn.norm_le_of_norm_le_on_sphere` (Zeta23 StrongPNTPrefix):\n r \u2264 R, AnalyticOn (closedBall 0 R), sphere-r bound, w \u2208 closedBall 0 r\n \u27f9 \u2016f w\u2016 \u2264 C.\n- `Real.log_le_rpow_div (0 \u2264 x) (0 < \u03b5) : log x \u2264 x^\u03b5/\u03b5` (Pow/Real).\n- `Real.pow_rpow_inv_natCast_pow` / `Real.rpow_inv_natCast_pow` (the\n (c\u2074)^{1/4} = c / (R^{1/4})\u2074 = R directions).\n- `pow_le_pow_left\u2080` (the \u2080-suffixed form in this Mathlib;\n pow_le_pow_left is the preorder one WITHOUT the 0 \u2264 a).\n\n## Banked gotchas (session 034)\n- **`le_or_lt` does NOT exist** (recurrence of the session-030/032\n gotcha \u2014 now banked here too): use `le_or_gt (a b) : a \u2264 b \u2228 a > b`.\n The dist-squeeze close: rcases le_or_gt d 0; the d > 0 branch dies\n by key (d/2).\n- `set Q := R ^ ((4:\u211d)\u207b\u00b9) with hQ` + `rw [\u2190 hQ] at h` after push_cast\n is the clean (R^{1/4})\u2074 = R bridge; write haves with Q in the\n statement and give R^{1/4} proofs \u2014 the set-let defeq makes `exact`\n accept them.\n- `Real.log_le_rpow_div` wants the rpow EXPONENT as the \u03b5 hypothesis:\n (4:\u211d)\u207b\u00b9 with positivity; the x^\u03b5/\u03b5 \u2192 4\u00b7x^\u03b5 conversion is\n div_eq_mul_inv + inv_inv + ring.\n- push_cast turns rpow_inv_natCast_pow's ((4:\u2115):\u211d)\u207b\u00b9 exponent into\n (4:\u211d)\u207b\u00b9 BEFORE the rw [\u2190 hQ] match (cast-mismatch rw failure\n otherwise).\n- linarith closes every \u2264-chain once the products are pre-formed\n (div_mul_cancel\u2080 for the /\u03b5\u00b7\u03b5 round trips; mul_le_mul_of_nonneg_right\n for the single-factor lifts); the banked \"no add_le_add_left/right\"\n rule kept.\n- lake build of a NON-root-imported file's importer fails with a stale\n \"object file does not exist\" \u2014 `lake build NguyenRH1.LiSqueeze`\n builds the dependency chain (LiCircleBound 51 s first time).\n\n## Status / next\n**ii\u2032-5e COMPLETE** (5e-0, i, ii, iii, iv, v-a/b/c/d all done). The\npartial-fraction identity is now Lean-proved modulo ONLY the declared\nperimeter (ZeroCountProp \u2014 Trudgian N(T)/S(T), D-062 \u00a74). Remaining\nmap: **ii\u2032-5f** (D-056's tight constants A = 10.5, B\u2080 = 48 from the\nidentity \u2014 the partial-fraction R-bound, ~1 session) \u2192 ii\u2032-3b (moments)\n\u2192 ii\u2032-3c (crux assembly Sf = Res\u2081) \u2192 ii\u2032-6 (L0) \u2192 ii\u2032-7 (closure +\npaper Appendix B + tag upgrade).\n"
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[{"type":"thinking","thinking":"Milestone committed. Now the session-end artifacts:\n1. journal/session-034.md\n2. STATE.md update (NEXT ACTION → ii′-5f)\n3. NGUYEN.md §5 checklist (ii′-5e entry) + session log\n4. db registration (session + derivation? — the Lean sub-stages get journal + STATE; check how previous sessions registered. From handoff: \"Session registration via direct sqlite3 insert (journal_file column)\".)\n\nLet me check how session 033 did the db registration to mirror. Actually the commits show \"journal + STATE + NGUYEN + db\". Let me look at the db.py CLI quickly and the session table pattern. From handoff notes: `db.py: init, log-experiment, add-result, add-derivation, add-graveyard, list`. Session registration via direct sqlite3 insert.\n\nLet me write the journal first, then STATE, then NGUYEN, then db, then final commit.","signature":""},{"type":"text","text":"Milestone committed. Now the session-end archive artifacts (journal, STATE, NGUYEN, db)."},{"type":"tool_use","id":"Write_45","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-034.md","content":"# Session 034 — 2026-08-23 — ii′-5e-v-d COMPLETE: the Schwarz squeeze ⟹\n# the partial-fraction capstone\n\n## Directive\nContinue ii′-5e-v per STATE.md: chunk 5e-v-d — the Schwarz squeeze ⟹\nthe ∃ B capstone `partialFraction_identity` (LiCircleBound.lean §13 or a\nnew LiSqueeze.lean). User: \"Let's start ii′-5e-v-d\".\n\n## Outcome: 5e-v-d COMPLETE (commit 5a262a1; new file LiSqueeze.lean,\n~330 lines, clean elaboration, zero warnings; LiCircleBound + LiSqueeze\nroot-imported; full build GREEN; audit-session034-axioms.txt: 5 decls,\n3 standard axioms only)\n\nNew file **LiSqueeze.lean** (imports LiCircleBound; LiCircleBound built\nfor the first time — 51 s — and both now root-imported):\n\n- **§1 `C0hat_add_two_le`**: Ĉ₀(R+2) ≤ 5·log R for R ≥ 16002. Via\n log(R+4) ≤ 2·log R (R+4 ≤ R², Real.log_pow), log log(R+4) ≤\n log(2·log R) = log 2 + log log R ≤ 1 + log R ≤ 2·log R\n (Real.log_le_sub_one_of_pos twice), 2.9 ≤ (1/3)·log R; total\n 0.433·2 + 0.886·2 + 1/3 ≈ 2.97 ≤ 5.\n- **§2 `not_badPoint_on_gapCircle`**: the four-way badness split —\n not nontrivial (session-032's `not_isNontrivialZero_on_gapCircle`),\n ≠ 0/1 (norms: ‖s‖ = R ≥ 16002), ≠ −2n (else R/2 = n ∈ ℤ against\n hgapInt with m := n; `Int.cast_natCast` bridge, `Complex.norm_natCast`).\n- **§3 `gentire_circle_le`**: Gentire s = Fexpr s on the circle (§2 +\n 5e-iv's `gentire_eq_Fexpr`) ⟹ the Fexpr_circle_le bound transfers.\n- **§4 `gentire_const (hZC : ZeroCountProp)` : Gentire z₀ = Gentire 0**.\n The squeeze: for ε > 0, set C₀' := 2·lowZeroConst + dirichletC2 +\n ‖Gentire 0‖, C₁ := 2·10⁶·A₀ ( = 400000·A₀·5), c := 2 +\n (128·C₁ + 2·C₀')·‖z₀‖/ε, M := max (c⁴) (‖z₀‖+1), k := max 4000 ⌈M/4⌉₊;\n the perimeter gap circle R ∈ [4k+5/2, 4k+3] (exists_gapCircle_radius)\n has R ≥ M. Max-modulus (Zeta23\n `AnalyticOn.norm_le_of_norm_le_on_sphere`, r = R, analytic from\n `gentire_analyticAt_and_eq.1`) lifts the sphere bound to the ball;\n MapsTo into closedBall (Gentire 0) (RB + ‖Gentire 0‖) by norm_sub_le;\n the ball-form Schwarz lemma\n (`Complex.dist_le_div_mul_dist_of_mapsTo_ball`) gives\n dist ≤ (RB + ‖Gentire 0‖)/R·‖z₀‖. Growth: RB ≤ C₀' + 64·C₁·Q³ with\n Q := R^{1/4} (Real.log_le_rpow_div at ε = 1/4 ⟹ log R ≤ 4Q;\n pow_le_pow_left₀ ⟹ (log R)³ ≤ 64Q³; Ĉ₀(R+2) ≤ 5·log R by §1) and\n R = Q⁴ (Real.rpow_inv_natCast_pow); Q ≥ c ⟹ the two branches\n 64·C₁·Q³·‖z₀‖ ≤ (ε/2)·Q⁴ (from 128·C₁·‖z₀‖ ≤ ε·Q) and C₀'·‖z₀‖ ≤\n (ε/2)·Q⁴ (from 2·C₀'·‖z₀‖ ≤ ε·Q and ε·Q ≤ (ε/2)·Q⁴ via Q³ ≥ 8);\n sum ⟹ (RB + ‖Gentire 0‖)·‖z₀‖ ≤ ε·Q⁴ = ε·R. Then dist ≤ ε for all\n ε ⟹ dist = 0 (le_or_gt rcases + the ε := dist/2 contradiction) ⟹\n Gentire z₀ = Gentire 0.\n- **§5 `partialFraction_identity (hZC : ZeroCountProp)`**: ∃ B : ℂ,\n ∀ s, riemannZeta s ≠ 0 → s ≠ 0 → s ≠ 1 → (∀ n ≥ 1, s ≠ −2n) →\n logDeriv riemannZeta s = B + seriesFun s − 1/s − 1/(s−1) +\n (Real.log Real.pi : ℂ)/2 − Complex.digamma (s/2)/2. Proof: ζ s ≠ 0\n kills the nontrivial-zero disjunct (IsNontrivialZero.1) ⟹ ¬ badPoint s\n ⟹ Gentire s = Fexpr s (5e-iv) and Gentire s = Gentire 0 (§4) ⟹\n Fexpr s = Gentire 0; unfold Fexpr; linear_combination. B := Gentire 0\n (no evaluation needed — D-068 §0: B cancels in 5f's two-point\n analysis).\n\n**This is the first use of the perimeter in the 5e chain** (the gap\ncircles need ZeroCountProp; 5c/5d/5e-i..iv were Jensen/unconditional).\nThe capstone is exactly the D-068 §5 target statement.\n\n## Inventory verification (all live-checked before writing)\n- `Complex.dist_le_div_mul_dist_of_mapsTo_ball` (Mathlib Schwarz.lean):\n (hd : DifferentiableOn ℂ f (ball c R₁)) (MapsTo f (ball c R₁)\n (closedBall (f c) R₂)) (z ∈ ball) ⟹ dist (f z) (f c) ≤ R₂/R₁·dist z c.\n- `AnalyticOn.norm_le_of_norm_le_on_sphere` (Zeta23 StrongPNTPrefix):\n r ≤ R, AnalyticOn (closedBall 0 R), sphere-r bound, w ∈ closedBall 0 r\n ⟹ ‖f w‖ ≤ C.\n- `Real.log_le_rpow_div (0 ≤ x) (0 < ε) : log x ≤ x^ε/ε` (Pow/Real).\n- `Real.pow_rpow_inv_natCast_pow` / `Real.rpow_inv_natCast_pow` (the\n (c⁴)^{1/4} = c / (R^{1/4})⁴ = R directions).\n- `pow_le_pow_left₀` (the ₀-suffixed form in this Mathlib;\n pow_le_pow_left is the preorder one WITHOUT the 0 ≤ a).\n\n## Banked gotchas (session 034)\n- **`le_or_lt` does NOT exist** (recurrence of the session-030/032\n gotcha — now banked here too): use `le_or_gt (a b) : a ≤ b ∨ a > b`.\n The dist-squeeze close: rcases le_or_gt d 0; the d > 0 branch dies\n by key (d/2).\n- `set Q := R ^ ((4:ℝ)⁻¹) with hQ` + `rw [← hQ] at h` after push_cast\n is the clean (R^{1/4})⁴ = R bridge; write haves with Q in the\n statement and give R^{1/4} proofs — the set-let defeq makes `exact`\n accept them.\n- `Real.log_le_rpow_div` wants the rpow EXPONENT as the ε hypothesis:\n (4:ℝ)⁻¹ with positivity; the x^ε/ε → 4·x^ε conversion is\n div_eq_mul_inv + inv_inv + ring.\n- push_cast turns rpow_inv_natCast_pow's ((4:ℕ):ℝ)⁻¹ exponent into\n (4:ℝ)⁻¹ BEFORE the rw [← hQ] match (cast-mismatch rw failure\n otherwise).\n- linarith closes every ≤-chain once the products are pre-formed\n (div_mul_cancel₀ for the /ε·ε round trips; mul_le_mul_of_nonneg_right\n for the single-factor lifts); the banked \"no add_le_add_left/right\"\n rule kept.\n- lake build of a NON-root-imported file's importer fails with a stale\n \"object file does not exist\" — `lake build NguyenRH1.LiSqueeze`\n builds the dependency chain (LiCircleBound 51 s first time).\n\n## Status / next\n**ii′-5e COMPLETE** (5e-0, i, ii, iii, iv, v-a/b/c/d all done). The\npartial-fraction identity is now Lean-proved modulo ONLY the declared\nperimeter (ZeroCountProp — Trudgian N(T)/S(T), D-062 §4). Remaining\nmap: **ii′-5f** (D-056's tight constants A = 10.5, B₀ = 48 from the\nidentity — the partial-fraction R-bound, ~1 session) → ii′-3b (moments)\n→ ii′-3c (crux assembly Sf = Res₁) → ii′-6 (L0) → ii′-7 (closure +\npaper Appendix B + tag upgrade).\n"},"extra_content":null}]