[{"type":"thinking","thinking":"6-c COMPLETE (committed 5d3dc0a). Now 6-d — the fold ⟹ sfBoundProp. The recipe:\n\n**6-d RECIPE (the fold):** sides identity\nsf_abs_le_rectInt_add_zeroSum hZC (T from exists_zero_avoiding_T_wide\nhZC at a := √n, √n ≥ 316 ≥ 3) + bridge\nzeroSum_norm_le_four_FAlt_add_one hRH hZCG hNLZ (T ≤ 3×10¹²−1 from\nn ≤ N₁; 2 ≤ T) + ‖RectInt'‖ = (1/(2π))·‖RectInt‖ ≤ (1/2π)·(sides)\n(RectangleIntegral' def, norm_smul, ‖1/(2πi)‖ = 1/(2π)) ⟹\n|Sf n| ≤ (1/2π)(I_III + Horiz + e^{1/3}IIMaj) + (4F+1) ≤\n(1/2π)(I_III-charge + Horiz-charge) + IIMaj + 4F + 1 (e^{1/3} ≤ 1.4 ≤\n2π — the F-E absorption, one linarith; e^{1/3} ≤ 1.4 via\n(exp(1/3))³ = e ≤ 2.7183 ≤ 2.744 = 1.4³ + le_of_pow_le_pow_left₀) ≤\nBn FAlt n — with the HORIZONTAL PINCH as the only real-analytic\ncomparison: NI + 5.5(10.5 log+48) ≤ 7·Hhat T (pinch-lemma pattern\nfrom 5f-iii: certified endpoint at n = 10⁵ + growth direction; E-030\nV3 witness ratio ≤ 0.40; suffices 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤\n21·C0hat(T)² since 5.5 ≤ 7 on the (10.5 log+48) part — elementary\nL/LL comparisons with log x ≤ x/e (Real.log_le_sub_one_of_pos at\nx/e), certified numerics for the endpoint constants).\n\nComponents needed:\n1. `sf_abs_le_rectInt_add_zeroSum` (ii′-6-b, LiAssembly) — signature?\n2. `exists_zero_avoiding_T_wide` (LiContour/LiAssembly) — signature (gives the T with hTwide).\n3. `zeroSum_norm_le_four_FAlt_add_one` (LiAssembly) — signature (hRH hZCG hNLZ + T conditions).\n4. RectangleIntegral' def — the 1/(2πi) factor.\n5. `Bn`/`FAlt`/`SfBoundProp`/`Hhat`/`Z2hat`/`C0hat` defs and the relationship: Bn FAlt n's horizontal share: 2(e⁴+1)(7·Hhat T) + 2·DogLegRight (from the ii′-6-a2 redefinition) — need Bn's exact def to do the final ≤ Bn comparison.\n6. The horizontal pinch: 2·Z2hat(T)·log(148.5·C0hat(√n+2)) + 5.5·(10.5·log(T+2)+48) ≤ 7·Hhat T where Hhat T = 3·C0hat(T)² + 10.5·log(T+2) + 48 (LiSkeleton:749).\n7. e^{1/3} ≤ 1.4 via (e^{1/3})³ = e ≤ 2.7183... wait — need e^{1/3} ≤ 1.4 ⟺ e ≤ 1.4³ = 2.744: `le_of_pow_le_pow_left₀`? — x³ ≤ y³ with nonneg ⟹ x ≤ y — or Real.rpow nat route: e^{1/3} ≤ 1.4 ⟺ e ≤ 1.4³: (e^{1/3})³ = e^1 = e ≤ 2.744: need (e^{1/3})³ = e — `Real.exp_nat_mul`? or `Real.exp_mul`? — (e^{1/3})³ = e^{3·(1/3)} = e^1 — Real.exp_nat_mul (1/3) 3 : exp(3·(1/3)) = (exp(1/3))^3 — hmm — exp_nat_mul (x) (n) : exp(↑n * x) = (exp x)^n ✓ — so (exp(1/3))^3 = exp(3·(1/3)) = exp 1 — 3·(1/3) = 1 by ring. Then le_of_pow_le_pow_left₀ (n := 3) (exp(1/3) ≥ 0? hmm — le_of_pow_le_pow_left₀ (hn : n ≠ 0) (hx : 0 ≤ x) (h : x^n ≤ y^n) : x ≤ y? — signature: le_of_pow_le_pow_left₀ (hn : n ≠ 0) (ha : 0 ≤ a) (h : a ^ n ≤ b ^ n) : a ≤ b ✓ — with a := e^{1/3}, b := 1.4: a³ = e ≤ 2.7183 ≤ 2.744 = 1.4³ — need e ≤ 2.7183 — `Real.exp_one_lt_d9 : exp 1 < 2.7182818286` ✓ so e ≤ 2.7183 < 2.744 ✓ and 1.4³ = 2.744 by norm_num ✓.\n\n8. The sides sum: RectInt' (Jlag n) z w = (1/(2πi))·RectInt — from Zeta23: VIntegral'/HIntegral' have the 1/(2πi) — RectangleIntegral' := HIntegral' ... — let me check the Zeta23 RectangleIntegral' def. And the ii′-6-b identity sf_abs_le_rectInt_add_zeroSum : |Sf n| ≤ ‖RectangleIntegral (Jlag n) z w‖ + ‖Σ‖?? or RectangleIntegral'? Let me check.\n\nThe recipe says: \"sides identity sf_abs_le_rectInt_add_zeroSum hZC + ‖RectInt'‖ = (1/(2π))·‖RectInt‖ ≤ (1/2π)·(sides) (RectangleIntegral' def, norm_smul, ‖1/(2πi)‖ = 1/(2π))\" — so sf_abs_le_rectInt_add_zeroSum gives |Sf n| ≤ ‖RectangleIntegral' (Jlag n) z w‖ + ‖Σ‖ — where RectangleIntegral' has the 1/(2πi) factor — then ‖RectangleIntegral'‖ = (1/(2π))‖RectangleIntegral‖ ≤ (1/(2π))(‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖) via rectangleIntegral_Jlag_eq + norm triangulation (norm_sub_le).\n\nLet me look at: sf_abs_le_rectInt_add_zeroSum, zeroSum_norm_le_four_FAlt_add_one, exists_zero_avoiding_T_wide, RectangleIntegral'/RectangleIntegral defs, Bn's def, FAlt's def, SfBoundProp's exact form.","signature":""},{"type":"text","text":"6-c COMPLETE (`5d3dc0a`). Now 6-d — the fold. Signature recon: the 6-b identity, bridge, wide pigeonhole, `RectangleIntegral'`, `Bn`, `Hhat`."},{"type":"tool_use","id":"TodoWrite_157","name":"TodoWrite","input":{"todos":[{"content":"6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le","status":"completed","activeForm":"Proving 6-c-ii I_III left-vertical bound"},{"content":"6-c-iii: I_I right-vertical bound (F-E form)","status":"completed","activeForm":"Proving 6-c-iii I_I bound"},{"content":"6-c-iv: mid-horizontal segment machine","status":"completed","activeForm":"Proving 6-c-iv mid-horizontal machine"},{"content":"6-c-v: horizontal assembly (mid + DL-5 tails)","status":"completed","activeForm":"Assembling 6-c-v horizontal bound"},{"content":"6-d: fold ⟹ sfBoundProp (then commit, update archive)","status":"in_progress","activeForm":"Proving 6-d fold to sfBoundProp"}]},"extra_content":null},{"type":"tool_use","id":"Grep_158","name":"Grep","input":{"pattern":"theorem sf_abs_le_rectInt_add_zeroSum|theorem zeroSum_norm_le_four_FAlt_add_one|theorem exists_zero_avoiding_T_wide|def Bn\\b|def FAlt|noncomputable def RectangleIntegral|def SfBoundProp|def Sf\\b","path":"D:\\Python\\Quantum","output_mode":"content","-A":12,"glob":"*.lean","head_limit":120},"extra_content":null}]