[{"type":"text","text":"Sub-stage ii′-0 complete and committed (e56cb66). Summary of what happened today:\n\n## Session 014 — tier ii′ launched, design complete, two discoveries\n\n**Discovery 1 — a real encoding defect (your audit instinct paid off).** `liCoeff` is defined as an *unordered* tsum whose family is provably **not Summable** for any n ≥ 1 (summand ~ n/|γ|, and Σ mult/|ρ| = ∞). In Lean that means `liCoeff n` evaluates to the junk value 0 — so as currently encoded, `LiVerifiedUpTo` is **false** and `SfBoundProp` is **false** (at n = 10⁵: |Sf| ≥ 432 501 > 347 000 ≥ Bn, by our own certified bounds). The session-012 theorems are vacuously true; the session-013 audit couldn't see this (it checks axioms, not hypothesis truth). Paper A's *mathematics* is untouched — this is purely a Lean encoding defect, and the fix is designed: the classical **paired symmetric sum** (absolutely summable via Zeta23's proved `zero_sum_inv_sq`). This alone justifies doing tier ii′ before submission.\n\n**Discovery 2 — the crux is simpler than feared.** The exact identity chain, settled against Lagarias's paper and verified numerically:\n- **Sf(n) = S_∞(n) + 1 − λ_n = Res₁(n)** — the w=1 residue Σⱼ C(n,j)c_{j−1} — *exactly* (via Lagarias (4.10) summed binomially). Verified: (4.10) at j=0 exact to 15 digits; the implied λ₂ = 0.09234573523 matches E-015's certified value to all digits.\n- Residue theorem `sides(T) = λ_n(T) + triv + Res₁(n)` verified to **2e-21** by direct contour integration.\n- The assembly is **tail-free**: |Sf| ≤ |λ_n(T)| + |sides| + |triv| ≤ 2N(T) + [side pieces] + 1 = B(n). No tail term — the D-062 \"long pole\" splits into two bounded pieces (a Leibniz-rule residue computation + the (4.10) Laurent relation).\n\n**Feasibility confirmed concretely**: rectangle residue theorem already vendored (needs a finite-order extension, bounded); Borel–Carathéodory/Jensen/digamma in Mathlib; Σ mult/(1+‖γ‖²) summability proved in Zeta23; all 8 planned dependencies axiom-checked live (clean).\n\n**Plan**: sub-stages ii′-1a…7 (~9–16 sessions), long pole = the (4.10) Laurent relation (EF-reuse first, Hadamard fallback). Full design in `research/derivations/D-064-tier-ii-prime-design.md`. Paper A untouched and submittable at every point (D-064 §8).\n\n**Next action (session 015)**: ii′-1a — the liCoeff pairing fix + summability proof. Ready when you are."}]