← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (45 chars)
Session 029 complete. Summarize for the user.
**ii′-5e-iv COMPLETE — F is entire** (commits `58e03e2`, `5213bb9`, `74f3c76`).
**LiFEntire.lean** (~1130 lines, 30 decls, all audited: 3 standard axioms only; build GREEN, 8766 jobs):
The difference function `Fexpr s = logDeriv ζ s − seriesFun s + 1/s + 1/(s−1) − (log π)/2 + ψ(s/2)/2` has only removable singularities, and Mathlib has no general removable-singularity theorem — so the extension is packaged as **`Gentire s := limUnder (𝓝[≠] s) Fexpr`**, with the junk point-values at excluded points absorbed by limUnder uniqueness. Capstone **`gentire_analyticAt_and_eq`**: `Gentire` is analytic at *every* point of ℂ and equals `Fexpr` at every good point.
The four principal-part-split witnesses, matching D-068 §0's residue bookkeeping:
- **nontrivial zeros**: the order-normal-form witness (`logDeriv ζ − m/(s−ρ₀) =ᶠ logDeriv h`) + 5e-iii's `seriesFun_sub_polar_analyticAt`;
- **s = 1**: `logDeriv ζ = logDeriv zetaA − 1/(s−1)` punctured;
- **s = 0**: `ψ(s/2)/2 + 1/s = −γ/2 + ½·digammaSeriesPart(s/2)` (M-test on ball(0,½));
- **s = −2n**: NF with `zeroMult(−2n) = 1` + the dropped-term digamma series `H_n` (eventually M-test on ball(−n,½)).
Supporting: **`zeta_zero_classification`** (every ζ zero is nontrivial, 0, or −2n — the LiDisc argument globalized) and **`analyticAt_digamma`**.
Archive updated (journal-029 with 8 new banked gotchas, STATE, NGUYEN, db).
**Next (session 030): ii′-5e-v** — the final 5e stage: bound `|Gentire| ≤ C(log R_k)³` on the γ-gap circles (5c/5d R-bounds + Dirichlet bound + the radial series split + 5e-ii digamma bounds), then the difference-quotient max-modulus squeeze ⟹ `Gentire ≡ const =: B` ⟹ the **partial-fraction identity capstone `∃ B, …`**. After that: 5f (tight constants) → 3b → 3c → 6 → 7.
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[{"type":"thinking","thinking":"Session 029 complete. Summarize for the user.","signature":""},{"type":"text","text":"**ii′-5e-iv COMPLETE — F is entire** (commits `58e03e2`, `5213bb9`, `74f3c76`).\n\n**LiFEntire.lean** (~1130 lines, 30 decls, all audited: 3 standard axioms only; build GREEN, 8766 jobs):\n\nThe difference function `Fexpr s = logDeriv ζ s − seriesFun s + 1/s + 1/(s−1) − (log π)/2 + ψ(s/2)/2` has only removable singularities, and Mathlib has no general removable-singularity theorem — so the extension is packaged as **`Gentire s := limUnder (𝓝[≠] s) Fexpr`**, with the junk point-values at excluded points absorbed by limUnder uniqueness. Capstone **`gentire_analyticAt_and_eq`**: `Gentire` is analytic at *every* point of ℂ and equals `Fexpr` at every good point.\n\nThe four principal-part-split witnesses, matching D-068 §0's residue bookkeeping:\n- **nontrivial zeros**: the order-normal-form witness (`logDeriv ζ − m/(s−ρ₀) =ᶠ logDeriv h`) + 5e-iii's `seriesFun_sub_polar_analyticAt`;\n- **s = 1**: `logDeriv ζ = logDeriv zetaA − 1/(s−1)` punctured;\n- **s = 0**: `ψ(s/2)/2 + 1/s = −γ/2 + ½·digammaSeriesPart(s/2)` (M-test on ball(0,½));\n- **s = −2n**: NF with `zeroMult(−2n) = 1` + the dropped-term digamma series `H_n` (eventually M-test on ball(−n,½)).\n\nSupporting: **`zeta_zero_classification`** (every ζ zero is nontrivial, 0, or −2n — the LiDisc argument globalized) and **`analyticAt_digamma`**.\n\nArchive updated (journal-029 with 8 new banked gotchas, STATE, NGUYEN, db). \n\n**Next (session 030): ii′-5e-v** — the final 5e stage: bound `|Gentire| ≤ C(log R_k)³` on the γ-gap circles (5c/5d R-bounds + Dirichlet bound + the radial series split + 5e-ii digamma bounds), then the difference-quotient max-modulus squeeze ⟹ `Gentire ≡ const =: B` ⟹ the **partial-fraction identity capstone `∃ B, …`**. After that: 5f (tight constants) → 3b → 3c → 6 → 7."}]