Session 041 (2026-08-26) — ii′-6-c..6-d + ii′-7: THE CLOSURE SESSION

Mission

Execute the rest of ii′-6-c (the four side bounds, from 6-c-ii per the STATE.md recipes), then 6-d (the fold ⟹ sfBoundProp), then ii′-7 (closure: lambda_pos_finite_height_closed, full axiom audit, Paper A Appendix B + correction note, archive closure). User green-lit ii′-7 at the ii′-6 milestone ("Continue to ii′-7 (Recommended)").

Headline results

1. ii′-6 COMPLETE — all four side bounds + the fold

• 6-c-ii (6e5a98f): vIntegral_Jlag_left_norm_le — the I_III bound, I_III ≤ 4T((2/3)Z₂(T) + 10.5 log(T+2) + 48) + 40·1.3352. The tails [10,T] go through the 5f tight R-bound capstone plus the near count ≤ (2/5)·Z₂(t) (‖s−ρ‖ ≥ 5/2 from re = −5/2); the middle |t| ≤ 10 uses the perimeter MZeroProp (cast bridge ↑(-5/2:ℝ) = (-5/2:ℂ)); the left tail equals the right tail by the even symmetry + integral_comp_neg.
• 6-c-iii (9bd8ba9): vIntegral_Jlag_right_le_exp_third_IIMaj — I_I ≤ e^{1/3}·IIMaj. The key step is the NEW F-E kernel lemma kFun_right_vertical_norm_le_exp_sqrt_add_third (uniform in t, n ≥ 4): |w/(w−1)| = |1+1/(w−1)| ≤ 1+1/‖w−1‖ ≤ 1+1/(2√n−1) by the plain triangle inequality plus ‖w−1‖ ≥ re(w−1) = 2√n−1; the worst case is t = 0, which repairs the off-by-one of the t² ≥ n / re ≥ 2√n+1 variants. The kernel power closes via (1+1/(2√n−1))ⁿ ≤ exp(n/(2√n−1)) ≤ exp(√n/2+1/3) ⟺ n ≥ 4.
• 6-c-iv (1fed1df): the mid-horizontal segment machine. The arcsinh-FTC integral_one_div_sqrt_sq_add_sq was built from scratch with the direct antiderivative F(σ) = log((σ−β)+√((σ−β)²+δ²)) — g > 0 since √h > |σ−β|, g′ = g/√h exactly, FTC via integral_eq_sub_of_hasDerivAt. The endpoints give F(3) ≤ log(6+δ) and −F(−5/2) ≤ log(7+δ)+2log(1/δ) via the rationalization −u+√(u²+δ²) = δ²/(u+√) ≥ δ²/(7+δ), so each zero costs ≤ 2·log(16.5/δ) by PURE log-mono ((6+δ)(7+δ) ≤ 56 ≤ 272.25 = 16.5² — no certified log numerics needed). The near-sum uses integral_finsetSum plus the two-unit count and the wide-pigeonhole clearance; the R-part is the 5f capstone (constant in σ), and the bottom side comes from the DL-5 conj trick.
• 6-c-v (5d3dc0a): hIntegral_Jlag_full_two_sides. Split at 3 (HIntegral_split), the middle by 6-c-iv, the tails by DL-5's dogLegUpper bounds; the −T side follows by the conj trick at the FULL level (intervalIntegral_conj is unconditional — no split needed).
• 6-d (774fec5): THE FOLD — sfBoundProp : SfBoundProp PROVED. |Sf n| ≤ (1/2π)(sides) + (4F+1) ≤ Bn FAlt n on [10⁵, N₁] from EXACTLY the six declared perimeter hypotheses. The pinch_horizontal step (NI + 5.5R ≤ 7·Ĥ) is a fully elementary chain: Ĉ₀(√n+2) ≤ √n+2 (C0hat_le_self, log x ≤ x/e with 1/e ≤ 0.36789 certified from Real.exp_one_gt_d9, loglog ≤ log−1), so 148.5·Ĉ₀(√n+2) ≤ (√n+2)² and its log ≤ 2·log(T+2); also Z₂ ≤ 2·Ĉ₀, Ĉ₀ ≥ 0.433·log with 8 ≤ 21·0.433 = 9.093, and 5.5 ≤ 7. The bridge was cloned at (hTle : T ≤ 3·10¹²) — the range the internal zeroSum_nontriv_norm_le actually uses; the public T ≤ 3·10¹²−1 hypothesis FAILS at the very top of the N₁ range (n = N₁ has √n = 3·10¹²−1 exactly, and the pigeonhole T can land in (3·10¹²−1, 3·10¹²]) — caught pre-instantiation by the hypothesis audit. The F-E absorption is e^{1/3} ≤ 1.4 ≤ 2π ((e^{1/3})³ = e ≤ 2.7183 ≤ 2.744 = 1.4³ + le_of_pow_le_pow_left₀).

2. ii′-7 COMPLETE — closure

• 7a+7b (a0943a6): lambda_pos_finite_height_closed PROVED (LiClosed.lean) — λ_n > 0 for ALL 1 ≤ n ≤ N₁ = (3×10¹²−1)² from EXACTLY SEVEN declared hypotheses (RHUpTo, LiVerifiedUpTo, ZeroCountProp, ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp). The axiom audit (AuditSession041) covers 29 decls — ALL [propext, Classical.choice, Quot.sound] only, 0 sorryAx.
• 7c (b64a839): Paper A closed — compiles clean, 9 pp, zero undefined refs. The abstract/intro moved to the seven-hypotheses headline; §3 Step 2 has the corrected B(n) display plus the F-A/F-B/F-D/F-E correction-note footnote; §3 Step 3 has the bridge 4F (both signs); Appendix B was rewritten (second tier COMPLETE, the six analytic inputs enumerated, and the full complex-analytic content list: rectInt'_Jlag, partialFraction_identity, logDeriv_riemannZeta_sub_nearZeros_le_tight, sf_eq_resW1, upper_right_horizontal_eq_dogleg, LiSideBounds, sfBoundProp); and the AI Contribution Statement got a factual refresh (sessions 001–041).
• 7d: archive closed — the D-037 erratum block (the four corrections), R-004 Lean-closure status, NGUYEN.md (Phase 5B → COMPLETE), STATE.md (rewritten at the new endpoint), and this journal.

Process notes (the error loops)

• 6-c-ii: 4 elaboration iterations — the ∫ x in a..b, body parse capture (the body grabs trailing + terms into the integrand — parenthesize EVERY integral summand; ≤ is NOT captured), the ContinuousAt.comp HO-decomposition (pin (g :=) (f :=) (x :=)), norm_le_insert arg-order (norm_le_insert' is the ‖a‖ ≤ ‖b‖+‖a−b‖ one), and the hσlo rw-auto-rfl -5/2 ≤ -5/2 (le_of_eq hre.symm is the deterministic form).
• 6-c-iii: show P by ring inside rw lists (no :=); Real.add_one_le_exp is x+1 ≤ e^x (bridge by linarith).
• 6-c-iv: the big one — the Σ-identifier bug: Σ (U+03A3) is NOT a Lean identifier character (lexes as the Sigma-type notation); the tell-tale is a phantom ?m.NNN metavariable goal reported at the DECLARATION HEAD position — always suspect a downstream parse break. Also: Continuous.div/.mul/.sub produce Pi-op forms — the const ?c leaks as an unsolved ?m unless the per-term type is ascribed (or the show-bridge to the Pi form); HasDerivAt.pow yields Pi.pow (simpa bridges); eq_div_iff for c = a/b; Finset.sum_mul for constant-RIGHT extraction; neg_le_abs : −a ≤ |a|.
• 6-c-v: ContinuousWithinAt.comp needs the same (g/f/x) pinning; ContinuousOn.intervalIntegrable takes the uIcc form.
• 6-d: linarith CANNOT multiply by atom-constants ((1/(2π)), (e⁴+1)) — every such step needs a pre-packaged mul_le_mul_of_nonneg_* (the hHhat' (2(e⁴+1)·7)·Ĥ(T) ≤ (2(e⁴+1)·7)·Ĥ(√n+1) miss was the final failure); in large contexts use linarith only/nlinarith only (the context scan on the giant hsf/hbridge types deterministically times out at isDefEq — three heartbeats-lost iterations); the bridge's public (hTle : T ≤ 3·10¹²−1) is over-strong vs its internal use (T ≤ 3·10¹²) — cloned the last mile with the honest range.
• le_of_pow_le_pow_left₀ takes the nonneg on the RIGHT (b) side.

Commits (this session, all on master)

6e5a98f (6-c-ii), 9bd8ba9 (6-c-iii), 1fed1df (6-c-iv), 5d3dc0a (6-c-v), 774fec5 (6-d fold — ii′-6 COMPLETE), a0943a6 (ii′-7a+7b — the closed theorem + audit), b64a839 (ii′-7c — Paper A closure), + this one (ii′-7d — archive closure).

Status at close

The finite-height Li theorem is FULLY MACHINE-PROVED with the ORIGINAL N₁ = (3×10¹²−1)² ≈ 9×10²⁴ PRESERVED (the D-074 dog-leg endpoint). Next: the user's read-through of paperA.pdf + the submission decision (cover-letter.md + referee-list.md from session 013, target Experimental Mathematics). Open research threads: Paper B Task 1 (R-003 tempered-FT bookkeeping), the parked Track-B remainders, the T = √n hidden-cancellation open question (D-073 §8 D4). The Lagarias F-D correspondence package in research/correspondence/ is the user's own work-in-progress (left uncommitted by design).