← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (323 chars)
All done. Summary for the user:
ii′-4 COMPLETE in one session (as D-064 estimated):
- ii′-4a: L3 — exists_zero_avoiding_T
- ii′-4b: L1 — kernel bounds
- ii′-4c: zeroMult(−2) = 1
- Build green 8751 jobs, 17 decls, 3 standard axioms only
- Archive updated, committed
Next: ii′-5 (R-bound), the long pole (2–3 sessions est).
**ii′-4 COMPLETE** — all in one session, as D-064 estimated. Four commits (3 sub-stages + closure).
## What was proved (new file `LiContour.lean`, ~400 lines)
**ii′-4a — L3, the zero-avoiding ordinate:**
- `ZeroCountProp` — the perimeter hypothesis (Trudgian N(T)/S(T)) in the exact form the assembly needs: ≤ Ĉ₀(t+1) zero ordinates in any unit window [t, t+1]
- `exists_avoiding_midpoint` — pure midpoint pigeonhole (k+1 midpoints; each point kills at most one since adjacent midpoints are 1/(k+1) apart)
- `exists_zero_avoiding_T` — T ∈ [a, a+1] with clearance ≥ 1/(3·Ĉ₀(a+1)) from every zero ordinate in the window
**ii′-4b — L1, kernel bounds on `kFun n s = (s/(s−1))ⁿ − 1`:**
- ‖kFun‖ ≤ 2 for re s ≤ 1/2; ≤ e⁴+1 on horizontals (re s ≤ 3, n ≤ im²); ≤ e^{√n/2}+1 on the right vertical (2√n ≤ re s) — all via the normSq core `norm_one_add_one_div_le` + `add_one_le_exp`
- `kFun n (−2) = (2/3)ⁿ − 1`, norm ≤ 1 (trivial-zero term)
**ii′-4c — the deferred simplicity result, `zeroMult (−2) = 1`:**
Local factorization ζ(s) = (s+2)·Λ(s)/Hfun(s) near −2 with Hfun = 4π^{−s/2}Γ(s/2+2)/s analytic, Hfun(−2) = −2π ≠ 0, and Λ(−2) = Λ(3) = ΓR(3)ζ(3) ≠ 0 (FE + `Gammaℝ_eq_zero_iff` + `riemannZeta_ne_zero_of_one_lt_re`), closed via `analyticOrderAt_eq_natCast`.
## Verification
- Full `lake build` GREEN (8751 jobs)
- Axiom sweep of all 17 new theorems: **3 standard axioms only** (propext, Classical.choice, Quot.sound), no `sorryAx`
- Archive updated: journal-021 (incl. ~10 new Lean gotchas banked), NGUYEN.md checklist, STATE.md, research.db; NEXT ACTION is now **ii′-5 — the R-bound** (Borel–Carathéodory + Jensen + Blaschke route; est. 2–3 sessions, the long pole of the remaining ii′ map)
One housekeeping note: the website fixes from earlier (`web/dashboard/research_data.py`, RH-WEBSITE.md) are still uncommitted in the working tree — they're excluded from all research commits per the §0 rule. Want me to commit them as a separate website commit?
raw content_json (database row)
[{"type":"thinking","thinking":"All done. Summary for the user:\n\nii′-4 COMPLETE in one session (as D-064 estimated):\n- ii′-4a: L3 — exists_zero_avoiding_T\n- ii′-4b: L1 — kernel bounds\n- ii′-4c: zeroMult(−2) = 1\n- Build green 8751 jobs, 17 decls, 3 standard axioms only\n- Archive updated, committed\n\nNext: ii′-5 (R-bound), the long pole (2–3 sessions est).","signature":""},{"type":"text","text":"**ii′-4 COMPLETE** — all in one session, as D-064 estimated. Four commits (3 sub-stages + closure).\n\n## What was proved (new file `LiContour.lean`, ~400 lines)\n\n**ii′-4a — L3, the zero-avoiding ordinate:**\n- `ZeroCountProp` — the perimeter hypothesis (Trudgian N(T)/S(T)) in the exact form the assembly needs: ≤ Ĉ₀(t+1) zero ordinates in any unit window [t, t+1]\n- `exists_avoiding_midpoint` — pure midpoint pigeonhole (k+1 midpoints; each point kills at most one since adjacent midpoints are 1/(k+1) apart)\n- `exists_zero_avoiding_T` — T ∈ [a, a+1] with clearance ≥ 1/(3·Ĉ₀(a+1)) from every zero ordinate in the window\n\n**ii′-4b — L1, kernel bounds on `kFun n s = (s/(s−1))ⁿ − 1`:**\n- ‖kFun‖ ≤ 2 for re s ≤ 1/2; ≤ e⁴+1 on horizontals (re s ≤ 3, n ≤ im²); ≤ e^{√n/2}+1 on the right vertical (2√n ≤ re s) — all via the normSq core `norm_one_add_one_div_le` + `add_one_le_exp`\n- `kFun n (−2) = (2/3)ⁿ − 1`, norm ≤ 1 (trivial-zero term)\n\n**ii′-4c — the deferred simplicity result, `zeroMult (−2) = 1`:**\nLocal factorization ζ(s) = (s+2)·Λ(s)/Hfun(s) near −2 with Hfun = 4π^{−s/2}Γ(s/2+2)/s analytic, Hfun(−2) = −2π ≠ 0, and Λ(−2) = Λ(3) = ΓR(3)ζ(3) ≠ 0 (FE + `Gammaℝ_eq_zero_iff` + `riemannZeta_ne_zero_of_one_lt_re`), closed via `analyticOrderAt_eq_natCast`.\n\n## Verification\n- Full `lake build` GREEN (8751 jobs)\n- Axiom sweep of all 17 new theorems: **3 standard axioms only** (propext, Classical.choice, Quot.sound), no `sorryAx`\n- Archive updated: journal-021 (incl. ~10 new Lean gotchas banked), NGUYEN.md checklist, STATE.md, research.db; NEXT ACTION is now **ii′-5 — the R-bound** (Borel–Carathéodory + Jensen + Blaschke route; est. 2–3 sessions, the long pole of the remaining ii′ map)\n\nOne housekeeping note: the website fixes from earlier (`web/dashboard/research_data.py`, RH-WEBSITE.md) are still uncommitted in the working tree — they're excluded from all research commits per the §0 rule. Want me to commit them as a separate website commit?"}]