← Session 31 — ii′-5f: tight R-bound 10.5/48 (piece 3 + pinch)
thinking (479 chars)
Piece 3 is fully complete and committed. That's a major milestone for the session. Now the question is whether to continue to 5f-iii (assembly + pinch + capstone) in this session or report to the user. Given the session length (piece 3 alone was ~2700 lines of new Lean across sections 9–19), and the BREAK PROMPT directive (§8.5: pause and ask when work runs long), I should report the milestone and ask whether to continue with 5f-iii now.
Let me give a concise status report.
**Piece 3 is complete and committed** — that was the big one. Status:
## Session 036 so far — ii′-5f-ii COMPLETE (pieces 6 + 3)
**Piece 3 done** (`far_series_le`, commit 52bb2cb): the far difference series is bounded by **12.1284·C0hat(t+1) + 7.2425**. The full interval machine is in place in `LiTightR.lean` (~2700 new lines across sections 9–19, file now ~3900 lines):
- **§9** certified numerics (integral-comparison log machines; all log/exp/sqrt bounds)
- **§10** the weight `farW k = 1/(k√(1+k²))` with **S₃ ≤ 1.3476** (4 exact sqrt-rational terms + telescope ¼)
- **§11–12** the six tail-weight bricks via the antitone-integral machine (RW₀/1/2 = 0.1/0.3303/0.1216, LA₀/1/2 = 1/23/0.1687/0.0621)
- **§13** the C0hat bridges: `C0hat(2k+1) ≤ 0.433(log k+0.833)+0.886(loglog k+0.3475)+2.9` (k≥11), the j+1 form (j≥14), and **`C0hat(2t+1) ≤ C0hat(t+1)+0.55`**
- **§14** conj machinery: `zeroMult_conj` (conj-derivative lemma → iteratedDeriv conj-invariance → analytic order invariance)
- **§15** farG + summability + per-term bound (‖s₀−s‖≤4.5, ‖s−ρ‖≥x, ‖s₀−ρ‖≥√(1+x²)) + norm-tsum reduction
- **§16** main bucket machine: `unit_family_le`, `main_bucket_count` (two-window perimeter + NoLowZeros emptiness), `farG_main_le ≤ 4.5(2C₀+0.55)·1.3476`
- **§17/18** the two tail machines (ENNReal fiberwise, cloning the LiCircleBound pattern): left tail ≤ 1.1729, right tail ≤ 2.7343
- **§19** tail split + capstone assembly
**Axiom audit** (`audit-session036-axioms.txt`): all 22 key declarations show **3 standard axioms only**. Full build green (8762 jobs).
One notable banked gotcha: `a * b / c` parses as `(a·b)/c` — with `mul_left`-coefficient proofs the goal must be written `c * (b / d)` with explicit parens, or isDefEq explodes (200k+ heartbeats) trying to prove `(a·b)/c = a·(b/c)` by unfolding.
## Remaining in 5f
**5f-iii** (smaller): assemble the six pieces into `Rexpr_decomp`'s norm chain + the pinch lemma (11.813·LL − 4.56·L ≤ const, one-variable calculus + certified point evaluation) + the capstone `logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC) (hZC2) (hNLZ)`. All pieces now exist (piece 1 dirichletC2≤1, piece 2 near_sum_refPoint_le, piece 3 far_series_le, pieces 4/5 one_div_diff_le, piece 6 digamma_diff_le).
Continue to 5f-iii now, or pause here? (Piece 3 was the session's main objective; 5f-iii closes out the whole ii′-5f sub-stage.)
raw content_json (database row)
[{"type":"thinking","thinking":"Piece 3 is fully complete and committed. That's a major milestone for the session. Now the question is whether to continue to 5f-iii (assembly + pinch + capstone) in this session or report to the user. Given the session length (piece 3 alone was ~2700 lines of new Lean across sections 9–19), and the BREAK PROMPT directive (§8.5: pause and ask when work runs long), I should report the milestone and ask whether to continue with 5f-iii now.\n\nLet me give a concise status report.","signature":""},{"type":"text","text":"**Piece 3 is complete and committed** — that was the big one. Status:\n\n## Session 036 so far — ii′-5f-ii COMPLETE (pieces 6 + 3)\n\n**Piece 3 done** (`far_series_le`, commit 52bb2cb): the far difference series is bounded by **12.1284·C0hat(t+1) + 7.2425**. The full interval machine is in place in `LiTightR.lean` (~2700 new lines across sections 9–19, file now ~3900 lines):\n\n- **§9** certified numerics (integral-comparison log machines; all log/exp/sqrt bounds)\n- **§10** the weight `farW k = 1/(k√(1+k²))` with **S₃ ≤ 1.3476** (4 exact sqrt-rational terms + telescope ¼)\n- **§11–12** the six tail-weight bricks via the antitone-integral machine (RW₀/1/2 = 0.1/0.3303/0.1216, LA₀/1/2 = 1/23/0.1687/0.0621)\n- **§13** the C0hat bridges: `C0hat(2k+1) ≤ 0.433(log k+0.833)+0.886(loglog k+0.3475)+2.9` (k≥11), the j+1 form (j≥14), and **`C0hat(2t+1) ≤ C0hat(t+1)+0.55`**\n- **§14** conj machinery: `zeroMult_conj` (conj-derivative lemma → iteratedDeriv conj-invariance → analytic order invariance)\n- **§15** farG + summability + per-term bound (‖s₀−s‖≤4.5, ‖s−ρ‖≥x, ‖s₀−ρ‖≥√(1+x²)) + norm-tsum reduction\n- **§16** main bucket machine: `unit_family_le`, `main_bucket_count` (two-window perimeter + NoLowZeros emptiness), `farG_main_le ≤ 4.5(2C₀+0.55)·1.3476`\n- **§17/18** the two tail machines (ENNReal fiberwise, cloning the LiCircleBound pattern): left tail ≤ 1.1729, right tail ≤ 2.7343\n- **§19** tail split + capstone assembly\n\n**Axiom audit** (`audit-session036-axioms.txt`): all 22 key declarations show **3 standard axioms only**. Full build green (8762 jobs).\n\nOne notable banked gotcha: `a * b / c` parses as `(a·b)/c` — with `mul_left`-coefficient proofs the goal must be written `c * (b / d)` with explicit parens, or isDefEq explodes (200k+ heartbeats) trying to prove `(a·b)/c = a·(b/c)` by unfolding.\n\n## Remaining in 5f\n\n**5f-iii** (smaller): assemble the six pieces into `Rexpr_decomp`'s norm chain + the pinch lemma (11.813·LL − 4.56·L ≤ const, one-variable calculus + certified point evaluation) + the capstone `logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC) (hZC2) (hNLZ)`. All pieces now exist (piece 1 dirichletC2≤1, piece 2 near_sum_refPoint_le, piece 3 far_series_le, pieces 4/5 one_div_diff_le, piece 6 digamma_diff_le).\n\nContinue to 5f-iii now, or pause here? (Piece 3 was the session's main objective; 5f-iii closes out the whole ii′-5f sub-stage.)"}]