[{"type":"thinking","thinking":"Piece 6 committed (`c9c1362`). It's ~2:50 AM. Only piece 3 (the far-sum interval machine — the biggest piece, 1–1.5 sessions) and 5f-iii (assembly + pinch lemma) remain for 5f.\n\nThe user answered \"Continue with piece 6\" — that's done. Given the time and that piece 3 is the biggest sub-build, this is the right point to checkpoint the session properly (journal + STATE) and report. Let me update the journal + STATE and do a final session-035 state, then report to the user.\n\nActually — the session is still going (user keeps saying continue). But it's 2:50 AM. Let me update journal/STATE (crash-safe) and report the milestone, asking about piece 3 vs log off.","signature":""},{"type":"text","text":"Piece 6 committed (`c9c1362`). Updating journal + STATE for the checkpoint."},{"type":"tool_use","id":"Edit_272","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-035.md","old_string":"## Status / next\n**5f-i COMPLETE.** Remaining: 5f-ii — piece 6 (the sharp digamma\nremainder |r| ≤ 1.2/|w|² via the level-2 EM, LiRBound b2Fun machinery;\nthen the ψ-difference with the crude log bound |log(1+u)| ≤ 2|u| —\nbudget ½|ψ-diff| ≤ 2.1, generous) and piece 3 (the far-sum interval\nmachine: per-term 4.5/(x√(1+x²)) ≤ 4.5/(k√(1+k²)); per-interval C0hat\ncounts from the perimeter + conj for γ < 0 + NoLowZeros below 14; the\nsum bound S₃ ≤ 1.3703 via k ≤ 3 exact + the telescope\n1/(k√(1+k²)) ≤ 1/(k(k−1)) for k ≥ 4, Σ = 1/3). Then 5f-iii: the\nassembly + the pinch lemma (11.813·LL − 4.560·L ≤ 1.315 for L ≥ log 12,\ninterior max at L* = 2.590, value ≈ −0.57 — decreasing throughout, so\nthe endpoint evaluation suffices) + the capstone\n`logDeriv_riemannZeta_sub_nearZeros_le_tight`.","new_string":"## Outcome part 4: 5f-ii piece 6 COMPLETE (commit c9c1362)\nLiTightR.lean §8 (~290 lines): `log_diff_le` (the log difference along\nthe constant-im segment, ≤ 5/t — the segment FTC with\nComplex.hasDerivAt_log.comp + comp_ofReal +\nintervalIntegral.norm_integral_le_of_norm_le_const) and\n**`digamma_diff_le`** (piece 6): ‖(ψ(s/2) − ψ(s₀/2))/2‖ ≤ 6/t — via\nthe +2 shift (re(w+2) ≥ 3/4 ⟹ StirlingVert.digamma_stirling applies,\nremainder ≤ 3/(t/2)² each), the recurrence (digamma_add_natCast), and\nlog_diff_le. NOTE: the sharp Lemma-D remainder (|r| ≤ 1.2/|w|²) turned\nout UNNECESSARY — digamma_stirling's 3/im² form suffices, and the\ncrude |log(1+u)| ≤ 2|u| log bound sufficed because piece 3's sharper\ninterval sum (S₃ ≤ 1.3703 via the k≥4 telescope, not the D-056\n√2-form) freed the constant budget: the full assembly is\n5.94·L + 11.813·LL + 46.7 ≤ 10.5·L + 48 with the pinch margin ≈ 1.9\n(the LL−L function is DECREASING on L ≥ log 12 — max at the boundary).\n\n## Banked gotchas (session 035, piece 6)\n- `(t/2)·I` elaborates with the division INSIDE ℂ (↑t·(1/2:ℂ)) —\n  write the real-cast-first form `((t/2 : ℝ) : ℂ)·I` for clean re/im.\n- re/im ext-goals: explicit per-component rw chains (Complex.add_re +\n  the value haves + ring) — constructor+simp is unpredictable (simp\n  sometimes closes, sometimes leaves arithmetic).\n- `field_simp [hx0]` can CLOSE the goal (no trailing ring) — twice\n  bitten (\"No goals to be solved\").\n- `linear_combination -ht1` when goal_diff = −ht1_diff.\n- `Set.uIoc` is the Ι-notation's real name.\n- `Real.norm_eq_abs` after `Complex.norm_real` for the |c| form\n  (norm_real gives ‖c‖ not |c|).\n- The segment-FTC pattern (banked from 5d): Complex.hasDerivAt_log\n  (slit) ∘ (affine via hasDerivAt_id'.mul_const/.const_add) then\n  .comp_ofReal; the comp's point-match needs the product-of-casts\n  segment form (z y = w₀ + ↑y·↑c, NOT ↑(y·c)).\n\n## Status / next\n**5f remaining: piece 3** (the far-sum interval machine, the big one)\n**+ 5f-iii** (assembly + pinch + capstone). Piece 3 map (journal-035\npiece-3 map + D-069 §3): the far difference series Σ_far m·(s₀−s)/\n((s−ρ)(s₀−ρ)) with per-term ≤ 4.5/(x√(1+x²)) ≤ 4.5/(k√(1+k²)) on\nk ≤ x = |t−γ| < k+1; per-interval C0hat counts (left ≤ C0hat(t+1) via\nthe perimeter; right ≤ C0hat(2t+1) ≤ C0hat(t+1) + 0.55); the negative-γ\nside via isNontrivialZero_conj; the height-< 14 intervals empty by\nNoLowZeros; S₃ := Σ_{k≥1} 1/(k√(1+k²)) ≤ 1.3703 via k ≤ 3 exact\n(√2 ≥ 1.414, √5 ≥ 2.23, √10 ≥ 3.16) + the telescope\n1/(k√(1+k²)) ≤ 1/(k(k−1)) for k ≥ 4 (Σ_{k≥4} 1/(k(k−1)) = 1/3 via\nFinset.sum_range_sub + the limit); the k > t tail ≤ 0.5828 (Σ log(3k)/k²\nand Σ 1/k² via the same telescopes). Total piece 3 ≤ 4.5·[2·S₃·C₀ +\n0.55·S₃ + 0.5828] = 12.333·C₀ + 6.014 with C₀ = C0hat(t+1) ≤ 0.433L +\n0.886LL + 2.9662 (the L' conversion).\n5f-iii: the assembly (six pieces into Rexpr_decomp's norm chain) + the\npinch lemma + the capstone\n`logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC) (hZC2) (hNLZ)`."},"extra_content":null}]