← Session 25 — ii′-5 R-bound: design + level-2 EM disc bound
thinking (392 chars)
Committed. Now per the session-end protocol: journal entry, STATE.md update, NGUYEN.md checklist update, final commit. The sub-stage ii′-5c is fully complete. The user said "let's do it" for the session — the work is done. Per §8.4 I should write the journal + update STATE + NGUYEN and commit. Let me do that now (the session reached a clean boundary).
Let me write the journal entry first.
ii′-5c committed (c7cbae5). Now the session-end protocol: journal + STATE + NGUYEN updates.
tool: Write
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-024.md",
"content": "# Session 024 \u2014 2026-08-22 \u2014 ii\u2032-5c COMPLETE: the loose Davenport-disc R-bound\n\n## Directive\nContinue tier ii\u2032-5 per STATE.md: the ii\u2032-5c back half (R-gap, g, BC\nassembly). User: \"I am ready, let's do it.\"\n\n## Outcome: ii\u2032-5c COMPLETE (commit c7cbae5)\n`logDeriv_riemannZeta_sub_nearZeros_le` PROVED: for s with s.im = t,\nre s \u2208 [1/2, 7/2], |t| \u2265 10, \u03b6 s \u2260 0:\n \u2016logDeriv \u03b6 s \u2212 \u03a3_{\u03c1 \u2208 nearZeros t} zeroMult \u03c1/(s\u2212\u03c1)\u2016 \u2264 100000\u00b7(log|t|)\u00b2.\nConstant deliberately loose (D-067 \u00a76 two-stage plan: this is the GROWTH\nINPUT for the ii\u2032-5e partial-fraction identity; the tight 10.5/48 come\nfrom the identity's analysis in ii\u2032-5f). No perimeter hypothesis needed \u2014\nthe count input is the session-023 Jensen bound, not ZeroCountProp.\n\n## Build path (~1000 new lines in LiDisc.lean)\n1. `norm_scaledZeta_le` \u2014 the (9/4)|t|\u00b2 bound extracted from the Jensen\n proof (standalone, for the outer-circle estimate).\n2. Order helpers: `scaledZeta_order_ne_top` (via\n AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected, F(0) = 1),\n `one_le_analyticOrderNatAt_scaledZeta`.\n3. `scaledZeroFinset t r (hr : r \u2264 1)` (subset-of-unit-disc construction,\n avoiding the dite-proof finsets of Zeta23's finiteSetOfZeros_mono) +\n `scaledZeroFinset_24_25_eq` (toFinset_inj.mpr rfl) bridging to the\n Jensen finset; `scaledZeroFinset_sum_order_le` (monotonicity + Jensen),\n `scaledZeroFinset_card_le_sum` (orders \u2265 1).\n4. R-gap: `exists_collection_radius` \u2014 the \u00d710 RESCALE TRICK: apply\n ii\u2032-4's unit-window `exists_avoiding_midpoint` at a = 23 to the \u00d710\n image of the (5/2)-scaled norm finset, divide by 10 \u27f9 R_c \u2208 [2.3, 2.4]\n with clearance 1/(20(k+1)) from every zero-norm \u2014 ZERO new pigeonhole\n code (STATE.md's suggested route). `collRadius`/`rC` bundle via\n Classical.choose; `rC_gap` (gapD t := 1/(50(k+1)) in the w-scale);\n `collZeros_norm_le` (collected zeros satisfy \u2016w\u2016 \u2264 r_c \u2212 gapD).\n5. The Blaschke quotient g := Cf r_c F entirely from Zeta23's Cf API:\n `Cf_eq_div_collProd` / `Cf_eq_zeroFactor_div` (the dite-unfold pair),\n `Cf_collZeros_analytic` (CfAnalytic at R = (1+r_c)/2),\n `Cf_collZeros_ne_zero` (ZeroFactorization at zeros; else-branch off),\n `Cf_collZeros_log_norm_zero_nonneg` (g(0) = 1/\u220f(\u2212\u03c1)^m, \u2016\u03c1\u2016 < 1),\n `Cf_collZeros_norm_le_sphere` (sphere zero-free by the gap; each\n \u2016w\u2212\u03c1\u2016 \u2265 gapD \u27f9 \u2016g\u2016 \u2264 (9/4)|t|\u00b2\u00b7gapD^{\u2212M_c}),\n `Cf_collZeros_norm_le_ball` (AnalyticOn.norm_le_of_norm_le_on_sphere).\n6. Constant fold (all real-analysis, no new input): `log_9_4_le_one`\n (9/4 \u2264 e), `log_33_32_ge` (log x \u2265 1\u22121/x at 33/32 \u27f9 \u2265 1/33),\n `two_le_log_abs_t` (e\u00b2 \u2264 2.7183\u00b2 < 10), `jensen_bound_elementary`\n (\u2264 33+66L), `card96_le`, `collMultSum_le`, `log_50_card_le`\n (50(k+1) \u2264 4150L, log 4150 \u2264 9 via e > 2.71, log L \u2264 L\u22121 \u27f9 \u2264 L+8),\n `Mbound_le` (\u2264 420L\u00b2 via (L\u22122)(354L+131) \u2265 0).\n7. Main assembly: w\u2080 = (2/5)(s\u2212s\u2080) (\u2016w\u2080\u2016 \u2264 3/5 from s.im = t,\n \u03c3 \u2208 [1/2, 7/2]); B =\u1da0[\ud835\udcdd w\u2080] F/P (zero set closed = finset carrier);\n logDeriv chain (logDeriv_div/prod/comp + per-factor\n logDeriv_fun_pow \u27f9 m/(w\u2080\u2212\u03c1)); LogOfAnalyticFunction (r = 3/5,\n R = r_c) + BorelCaratheodoryDeriv (16\u00b7M\u00b7r_c\u00b2/(r_c\u22123/5)\u00b3 \u2264 450M);\n deriv J_B w\u2080 = (5/2)logDeriv \u03b6 s \u2212 \u03a3 m/(w\u2080\u2212\u03c1); the (2/5)-rescaled\n sum pushes forward through the affine image (Finset.sum_image +\n zeroMult_affMap) and splits near/extras (Finset.sum_sdiff); extras\n \u2016s\u2212\u03c1\u2016 \u2265 |t\u2212\u03b3| \u2265 1 \u27f9 \u2264 \u03a3m \u2264 Jensen. Final: 180\u00b7420L\u00b2 + 42L\u00b2 = 75642L\u00b2\n \u2264 100000L\u00b2.\n\n## Banked gotchas (session 024 additions)\n- `Set.Finite.toFinset_inj.mpr rfl` equates toFinsets from different\n finiteness proofs of the same set \u2014 the clean way to handle Cf's\n anonymous dite-bound proof.\n- `div_le_div_iff\u2080 (hb : 0 < b) (hd : 0 < d) : a/b \u2264 c/d \u2194 a*d \u2264 c*b`\n (GroupWithZero/Basic); the 4-arg `div_le_div` does NOT exist in this\n Mathlib. `div_le_iff\u2080 (hc : 0 < c) : a/c \u2264 b \u2194 a \u2264 b*c`.\n- `EventuallyEq.deriv` is the EVENTUALLY version (deriv f =\u1da0 deriv g);\n the pointwise one is `Filter.EventuallyEq.deriv_eq`.\n- `Filter.EventuallyEq.eq_of_nhds` gives pointwise value equality.\n- `Finset.sum_sdiff` exists via to_additive (prod_sdiff) \u2014 grep for\n \"sum_sdiff\" fails since it's generated; same for Finset.sum_image\n (takes Set.InjOn in this Mathlib).\n- `positivity` maxRecDepth-loops on `set`-bound finsets' card casts \u2014\n use explicit `mul_pos`/`add_pos_of_nonneg_of_pos (Nat.cast_nonneg _)`\n terms; and `Nat.cast_nonneg _` inside compound terms needs an\n ascribed target type (IsOrderedRing metavariable stuck otherwise).\n- `Real.log_le_log (0 < x) (x \u2264 y)`, `Real.one_sub_inv_le_log_of_pos`,\n `Real.log_le_iff_le_exp`, `Real.exp_one_gt_d9`/`_lt_d9`,\n `Real.exp_nat_mul : exp (n*x) = (exp x)^n` \u2014 all as expected.\n- `norm_natCast` is `RCLike.norm_natCast` (not Complex.*).\n- `mem_sphere_iff_norm` is root-namespace (not Metric.*).\n- rw [\u2190 mul_one]/[\u2190 div_one] rewrites ALL occurrences of the pattern \u2014\n for one-sided division comparisons use div_le_iff\u2080 + explicit\n mul_nonneg hints to nlinarith instead.\n- `field_simp` sometimes closes the goal entirely (trailing `ring` then\n errors \"no goals\") \u2014 check before appending ring.\n- deriv-computation chain that worked: funext-reorder to (f + const)\n form \u2192 `deriv_add_const` \u2192 `show` with `id w` (defeq) \u2192\n `deriv_const_mul _ differentiableAt_id` \u2192 `deriv_id'` \u2192 `simp`.\n (`differentiableAt_id'`/`deriv_id''` pointwise forms do NOT exist as\n stated; deriv_id' : deriv id = fun _ => 1, deriv_id'' for (fun x=>x).)\n\n## Status / next\nii\u2032-5c COMPLETE. Next per D-067 \u00a76: **ii\u2032-5d** (FE-reflection to\n\u03c3 \u2208 [\u22125/2, 1/2] + the left-plane uniform digamma bound |\u03c8(w) \u2212 log w| \u2264 C\naway from poles), then ii\u2032-5e (partial-fraction identity; needs\nzeroMult(\u22122n) = 1 for ALL n \u2014 generalize ii\u2032-4c's zeroMult_neg_two),\nii\u2032-5f (tight analysis \u2192 10.5/48), ii\u2032-3b (moments), ii\u2032-3c (crux),\nii\u2032-6, ii\u2032-7.\n"
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raw content_json (database row)
[{"type":"thinking","thinking":"Committed. Now per the session-end protocol: journal entry, STATE.md update, NGUYEN.md checklist update, final commit. The sub-stage ii′-5c is fully complete. The user said \"let's do it\" for the session — the work is done. Per §8.4 I should write the journal + update STATE + NGUYEN and commit. Let me do that now (the session reached a clean boundary).\n\nLet me write the journal entry first.","signature":""},{"type":"text","text":"ii′-5c committed (c7cbae5). Now the session-end protocol: journal + STATE + NGUYEN updates."},{"type":"tool_use","id":"Write_122","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-024.md","content":"# Session 024 — 2026-08-22 — ii′-5c COMPLETE: the loose Davenport-disc R-bound\n\n## Directive\nContinue tier ii′-5 per STATE.md: the ii′-5c back half (R-gap, g, BC\nassembly). User: \"I am ready, let's do it.\"\n\n## Outcome: ii′-5c COMPLETE (commit c7cbae5)\n`logDeriv_riemannZeta_sub_nearZeros_le` PROVED: for s with s.im = t,\nre s ∈ [1/2, 7/2], |t| ≥ 10, ζ s ≠ 0:\n ‖logDeriv ζ s − Σ_{ρ ∈ nearZeros t} zeroMult ρ/(s−ρ)‖ ≤ 100000·(log|t|)².\nConstant deliberately loose (D-067 §6 two-stage plan: this is the GROWTH\nINPUT for the ii′-5e partial-fraction identity; the tight 10.5/48 come\nfrom the identity's analysis in ii′-5f). No perimeter hypothesis needed —\nthe count input is the session-023 Jensen bound, not ZeroCountProp.\n\n## Build path (~1000 new lines in LiDisc.lean)\n1. `norm_scaledZeta_le` — the (9/4)|t|² bound extracted from the Jensen\n proof (standalone, for the outer-circle estimate).\n2. Order helpers: `scaledZeta_order_ne_top` (via\n AnalyticOnNhd.analyticOrderAt_ne_top_of_isPreconnected, F(0) = 1),\n `one_le_analyticOrderNatAt_scaledZeta`.\n3. `scaledZeroFinset t r (hr : r ≤ 1)` (subset-of-unit-disc construction,\n avoiding the dite-proof finsets of Zeta23's finiteSetOfZeros_mono) +\n `scaledZeroFinset_24_25_eq` (toFinset_inj.mpr rfl) bridging to the\n Jensen finset; `scaledZeroFinset_sum_order_le` (monotonicity + Jensen),\n `scaledZeroFinset_card_le_sum` (orders ≥ 1).\n4. R-gap: `exists_collection_radius` — the ×10 RESCALE TRICK: apply\n ii′-4's unit-window `exists_avoiding_midpoint` at a = 23 to the ×10\n image of the (5/2)-scaled norm finset, divide by 10 ⟹ R_c ∈ [2.3, 2.4]\n with clearance 1/(20(k+1)) from every zero-norm — ZERO new pigeonhole\n code (STATE.md's suggested route). `collRadius`/`rC` bundle via\n Classical.choose; `rC_gap` (gapD t := 1/(50(k+1)) in the w-scale);\n `collZeros_norm_le` (collected zeros satisfy ‖w‖ ≤ r_c − gapD).\n5. The Blaschke quotient g := Cf r_c F entirely from Zeta23's Cf API:\n `Cf_eq_div_collProd` / `Cf_eq_zeroFactor_div` (the dite-unfold pair),\n `Cf_collZeros_analytic` (CfAnalytic at R = (1+r_c)/2),\n `Cf_collZeros_ne_zero` (ZeroFactorization at zeros; else-branch off),\n `Cf_collZeros_log_norm_zero_nonneg` (g(0) = 1/∏(−ρ)^m, ‖ρ‖ < 1),\n `Cf_collZeros_norm_le_sphere` (sphere zero-free by the gap; each\n ‖w−ρ‖ ≥ gapD ⟹ ‖g‖ ≤ (9/4)|t|²·gapD^{−M_c}),\n `Cf_collZeros_norm_le_ball` (AnalyticOn.norm_le_of_norm_le_on_sphere).\n6. Constant fold (all real-analysis, no new input): `log_9_4_le_one`\n (9/4 ≤ e), `log_33_32_ge` (log x ≥ 1−1/x at 33/32 ⟹ ≥ 1/33),\n `two_le_log_abs_t` (e² ≤ 2.7183² < 10), `jensen_bound_elementary`\n (≤ 33+66L), `card96_le`, `collMultSum_le`, `log_50_card_le`\n (50(k+1) ≤ 4150L, log 4150 ≤ 9 via e > 2.71, log L ≤ L−1 ⟹ ≤ L+8),\n `Mbound_le` (≤ 420L² via (L−2)(354L+131) ≥ 0).\n7. Main assembly: w₀ = (2/5)(s−s₀) (‖w₀‖ ≤ 3/5 from s.im = t,\n σ ∈ [1/2, 7/2]); B =ᶠ[𝓝 w₀] F/P (zero set closed = finset carrier);\n logDeriv chain (logDeriv_div/prod/comp + per-factor\n logDeriv_fun_pow ⟹ m/(w₀−ρ)); LogOfAnalyticFunction (r = 3/5,\n R = r_c) + BorelCaratheodoryDeriv (16·M·r_c²/(r_c−3/5)³ ≤ 450M);\n deriv J_B w₀ = (5/2)logDeriv ζ s − Σ m/(w₀−ρ); the (2/5)-rescaled\n sum pushes forward through the affine image (Finset.sum_image +\n zeroMult_affMap) and splits near/extras (Finset.sum_sdiff); extras\n ‖s−ρ‖ ≥ |t−γ| ≥ 1 ⟹ ≤ Σm ≤ Jensen. Final: 180·420L² + 42L² = 75642L²\n ≤ 100000L².\n\n## Banked gotchas (session 024 additions)\n- `Set.Finite.toFinset_inj.mpr rfl` equates toFinsets from different\n finiteness proofs of the same set — the clean way to handle Cf's\n anonymous dite-bound proof.\n- `div_le_div_iff₀ (hb : 0 < b) (hd : 0 < d) : a/b ≤ c/d ↔ a*d ≤ c*b`\n (GroupWithZero/Basic); the 4-arg `div_le_div` does NOT exist in this\n Mathlib. `div_le_iff₀ (hc : 0 < c) : a/c ≤ b ↔ a ≤ b*c`.\n- `EventuallyEq.deriv` is the EVENTUALLY version (deriv f =ᶠ deriv g);\n the pointwise one is `Filter.EventuallyEq.deriv_eq`.\n- `Filter.EventuallyEq.eq_of_nhds` gives pointwise value equality.\n- `Finset.sum_sdiff` exists via to_additive (prod_sdiff) — grep for\n \"sum_sdiff\" fails since it's generated; same for Finset.sum_image\n (takes Set.InjOn in this Mathlib).\n- `positivity` maxRecDepth-loops on `set`-bound finsets' card casts —\n use explicit `mul_pos`/`add_pos_of_nonneg_of_pos (Nat.cast_nonneg _)`\n terms; and `Nat.cast_nonneg _` inside compound terms needs an\n ascribed target type (IsOrderedRing metavariable stuck otherwise).\n- `Real.log_le_log (0 < x) (x ≤ y)`, `Real.one_sub_inv_le_log_of_pos`,\n `Real.log_le_iff_le_exp`, `Real.exp_one_gt_d9`/`_lt_d9`,\n `Real.exp_nat_mul : exp (n*x) = (exp x)^n` — all as expected.\n- `norm_natCast` is `RCLike.norm_natCast` (not Complex.*).\n- `mem_sphere_iff_norm` is root-namespace (not Metric.*).\n- rw [← mul_one]/[← div_one] rewrites ALL occurrences of the pattern —\n for one-sided division comparisons use div_le_iff₀ + explicit\n mul_nonneg hints to nlinarith instead.\n- `field_simp` sometimes closes the goal entirely (trailing `ring` then\n errors \"no goals\") — check before appending ring.\n- deriv-computation chain that worked: funext-reorder to (f + const)\n form → `deriv_add_const` → `show` with `id w` (defeq) →\n `deriv_const_mul _ differentiableAt_id` → `deriv_id'` → `simp`.\n (`differentiableAt_id'`/`deriv_id''` pointwise forms do NOT exist as\n stated; deriv_id' : deriv id = fun _ => 1, deriv_id'' for (fun x=>x).)\n\n## Status / next\nii′-5c COMPLETE. Next per D-067 §6: **ii′-5d** (FE-reflection to\nσ ∈ [−5/2, 1/2] + the left-plane uniform digamma bound |ψ(w) − log w| ≤ C\naway from poles), then ii′-5e (partial-fraction identity; needs\nzeroMult(−2n) = 1 for ALL n — generalize ii′-4c's zeroMult_neg_two),\nii′-5f (tight analysis → 10.5/48), ii′-3b (moments), ii′-3c (crux),\nii′-6, ii′-7.\n"},"extra_content":null}]