← Session 30 — ii′-5e squeeze capstone + 5f perimeter repair
**ii′-5e-v-d COMPLETE — and with it, ii′-5e is DONE.** Two commits: `5a262a1` (milestone) + `0b5d2bc` (closure).
## What was built — LiSqueeze.lean (~330 lines, 5 decls, all audited: 3 standard axioms only)
1. **`C0hat_add_two_le`** — Ĉ₀(R+2) ≤ 5·log R for R ≥ 16002 (the growth bound).
2. **`not_badPoint_on_gapCircle`** — every γ-gap-circle point is good (norm excludes 0/1; the ℤ-clearance dist(R/2,ℤ) ≥ 1/4 excludes −2n; session-032's lemma excludes nontrivial zeros).
3. **`gentire_circle_le`** — the Fexpr circle bound transfers to Gentire on the circles.
4. **`gentire_const`** — the squeeze: max-modulus (Zeta23's sphere lemma) lifts the circle bound to the ball; the ball-form Schwarz lemma gives `dist (Gentire z₀) (Gentire 0) ≤ (RB + ‖Gentire 0‖)/R · ‖z₀‖`; with Q := R^{1/4}, the bound is ≤ (C₀′ + 64·C₁·Q³)·‖z₀‖/Q⁴, and choosing the perimeter gap circle with Q ≥ 2 + (128·C₁ + 2·C₀′)·‖z₀‖/ε forces it ≤ ε — for all ε, so Gentire ≡ Gentire 0.
5. **THE CAPSTONE `partialFraction_identity (hZC : ZeroCountProp)`** — ∃ B : ℂ, for all s with ζ s ≠ 0, s ≠ 0, s ≠ 1, s ≠ −2n:
`ζ′/ζ(s) = B + Σ_ρ m_ρ(1/(s−ρ) + 1/ρ) − 1/s − 1/(s−1) + ½log π − ½ψ(s/2)`
The partial-fraction identity is now **Lean-proved modulo only the declared perimeter** (ZeroCountProp — Trudgian N(T)/S(T), D-062 §4). This is the first perimeter use in the 5e chain (the gap circles need it); everything else was unconditional.
Build: full green, LiCircleBound + LiSqueeze root-imported; sorryAx remains only on the 5 intended skeletons. One build hiccup: `le_or_lt` doesn't exist in this Mathlib (banked gotcha — `le_or_gt` + the dist/2 contradiction closes the squeeze).
**Next (session 035): ii′-5f** — the tight constants A = 10.5, B₀ = 48 from D-056's two-point analysis of the identity (B cancels — no evaluation needed), ~1 session. Then ii′-3b (moments) → ii′-3c (crux) → ii′-6 → ii′-7 closure.
raw content_json (database row)
[{"type":"text","text":"**ii′-5e-v-d COMPLETE — and with it, ii′-5e is DONE.** Two commits: `5a262a1` (milestone) + `0b5d2bc` (closure).\n\n## What was built — LiSqueeze.lean (~330 lines, 5 decls, all audited: 3 standard axioms only)\n\n1. **`C0hat_add_two_le`** — Ĉ₀(R+2) ≤ 5·log R for R ≥ 16002 (the growth bound).\n2. **`not_badPoint_on_gapCircle`** — every γ-gap-circle point is good (norm excludes 0/1; the ℤ-clearance dist(R/2,ℤ) ≥ 1/4 excludes −2n; session-032's lemma excludes nontrivial zeros).\n3. **`gentire_circle_le`** — the Fexpr circle bound transfers to Gentire on the circles.\n4. **`gentire_const`** — the squeeze: max-modulus (Zeta23's sphere lemma) lifts the circle bound to the ball; the ball-form Schwarz lemma gives `dist (Gentire z₀) (Gentire 0) ≤ (RB + ‖Gentire 0‖)/R · ‖z₀‖`; with Q := R^{1/4}, the bound is ≤ (C₀′ + 64·C₁·Q³)·‖z₀‖/Q⁴, and choosing the perimeter gap circle with Q ≥ 2 + (128·C₁ + 2·C₀′)·‖z₀‖/ε forces it ≤ ε — for all ε, so Gentire ≡ Gentire 0.\n5. **THE CAPSTONE `partialFraction_identity (hZC : ZeroCountProp)`** — ∃ B : ℂ, for all s with ζ s ≠ 0, s ≠ 0, s ≠ 1, s ≠ −2n:\n `ζ′/ζ(s) = B + Σ_ρ m_ρ(1/(s−ρ) + 1/ρ) − 1/s − 1/(s−1) + ½log π − ½ψ(s/2)`\n\nThe partial-fraction identity is now **Lean-proved modulo only the declared perimeter** (ZeroCountProp — Trudgian N(T)/S(T), D-062 §4). This is the first perimeter use in the 5e chain (the gap circles need it); everything else was unconditional.\n\nBuild: full green, LiCircleBound + LiSqueeze root-imported; sorryAx remains only on the 5 intended skeletons. One build hiccup: `le_or_lt` doesn't exist in this Mathlib (banked gotcha — `le_or_gt` + the dist/2 contradiction closes the squeeze).\n\n**Next (session 035): ii′-5f** — the tight constants A = 10.5, B₀ = 48 from D-056's two-point analysis of the identity (B cancels — no evaluation needed), ~1 session. Then ii′-3b (moments) → ii′-3c (crux) → ii′-6 → ii′-7 closure."}]