← Session 25 — ii′-5 R-bound: design + level-2 EM disc bound
thinking (403 chars)
The user wants to know where we left off. STATE.md is comprehensive. Let me also skim the latest journal entry (session-023) briefly for any additional context, then give a concise summary.
Actually, STATE.md is quite complete. Let me just skim the journal-023 quickly to confirm gotchas, but STATE.md already summarizes them. I think I can answer directly from STATE.md. Let me give a concise summary.
Here's where we stand.
## Big picture
**Tier ii′** (Phase 5B) is in progress — closing `SfBoundProp` in Lean **before** Paper A submission (your directive; Paper A stays frozen/submittable throughout).
**Done so far:** ii′-0 (design D-064 + liCoeff defect found + crux identity `Sf = Res₁`), ii′-1a (defect repaired), ii′-1b (residue machinery, `rectInt'_Jlag`), ii′-2 (elementary swaps), ii′-3 design (D-066, route β4) + ii′-3a (pieces A+B proved), ii′-4 (L3 + L1 + `zeroMult_neg_two`), ii′-5a (D-067 design), ii′-5b (level-2 ζ representation, `‖ζ‖ ≤ (3/4)|t|²` on the disc), **ii′-5c PART 1** (LiDisc.lean: `zeroDiscFinset`, `scaledZeta`, F(0)=1, multiplicity invariance, `scaledZeta_jensen_count`).
## Next action — Session 024: ii′-5c back half (~1 session est.)
In LiDisc.lean:
1. **R-gap selection** — scale `exists_avoiding_midpoint` to window [2.3, 2.4] (ℓ-parameterized variant, reuse ii′-4's proof); k from the Jensen count.
2. **g := ζ/∏(s−ρₖ)^{mₖ}** analytic + nonvanishing on the closed ball (normal form removes each collected zero).
3. **Bounds** — |g| ≤ (3/4)|t|²·δ^{−M} on the outer circle; −log|g(s₀)| ≤ log 3 + M·log(2.4).
4. **BC assembly** — `LogOfAnalyticFunction` + `BorelCaratheodoryDeriv` ⟹ loose R-bound `‖ζ′/ζ − Σ m/(s−ρ)‖ ≤ C·(log|t|)²`, constants deliberately loose (per D-067 §6: the tight 10.5/48 come later from the partial-fraction identity in 5f, not from the disc).
**Read first:** D-067 §6 (route revision) + journal-023 gotchas (`analyticOrderNatAt = analyticOrderAt.toNat` by def; `Filter.EventuallyEq.xxx` terms not dot-notation on unfolded forms; `AnalyticAt.comp_of_eq` for beta-reduced points; `Nat.cast_analyticOrderNatAt`).
## Remaining map after 5c
ii′-5d (FE-reflection + left digamma bound) → ii′-5e (partial-fraction identity) → ii′-5f (tight analysis) → ii′-3b (moments) → ii′-3c (crux) → ii′-6 (L0 assembly) → ii′-7 (close + audit + paper Appendix B).
No blockers. Ready to start on ii′-5c back half whenever you are.
raw content_json (database row)
[{"type":"thinking","thinking":"The user wants to know where we left off. STATE.md is comprehensive. Let me also skim the latest journal entry (session-023) briefly for any additional context, then give a concise summary.\n\nActually, STATE.md is quite complete. Let me just skim the journal-023 quickly to confirm gotchas, but STATE.md already summarizes them. I think I can answer directly from STATE.md. Let me give a concise summary.","signature":""},{"type":"text","text":"Here's where we stand.\n\n## Big picture\n**Tier ii′** (Phase 5B) is in progress — closing `SfBoundProp` in Lean **before** Paper A submission (your directive; Paper A stays frozen/submittable throughout).\n\n**Done so far:** ii′-0 (design D-064 + liCoeff defect found + crux identity `Sf = Res₁`), ii′-1a (defect repaired), ii′-1b (residue machinery, `rectInt'_Jlag`), ii′-2 (elementary swaps), ii′-3 design (D-066, route β4) + ii′-3a (pieces A+B proved), ii′-4 (L3 + L1 + `zeroMult_neg_two`), ii′-5a (D-067 design), ii′-5b (level-2 ζ representation, `‖ζ‖ ≤ (3/4)|t|²` on the disc), **ii′-5c PART 1** (LiDisc.lean: `zeroDiscFinset`, `scaledZeta`, F(0)=1, multiplicity invariance, `scaledZeta_jensen_count`).\n\n## Next action — Session 024: ii′-5c back half (~1 session est.)\nIn LiDisc.lean:\n1. **R-gap selection** — scale `exists_avoiding_midpoint` to window [2.3, 2.4] (ℓ-parameterized variant, reuse ii′-4's proof); k from the Jensen count.\n2. **g := ζ/∏(s−ρₖ)^{mₖ}** analytic + nonvanishing on the closed ball (normal form removes each collected zero).\n3. **Bounds** — |g| ≤ (3/4)|t|²·δ^{−M} on the outer circle; −log|g(s₀)| ≤ log 3 + M·log(2.4).\n4. **BC assembly** — `LogOfAnalyticFunction` + `BorelCaratheodoryDeriv` ⟹ loose R-bound `‖ζ′/ζ − Σ m/(s−ρ)‖ ≤ C·(log|t|)²`, constants deliberately loose (per D-067 §6: the tight 10.5/48 come later from the partial-fraction identity in 5f, not from the disc).\n\n**Read first:** D-067 §6 (route revision) + journal-023 gotchas (`analyticOrderNatAt = analyticOrderAt.toNat` by def; `Filter.EventuallyEq.xxx` terms not dot-notation on unfolded forms; `AnalyticAt.comp_of_eq` for beta-reduced points; `Nat.cast_analyticOrderNatAt`).\n\n## Remaining map after 5c\nii′-5d (FE-reflection + left digamma bound) → ii′-5e (partial-fraction identity) → ii′-5f (tight analysis) → ii′-3b (moments) → ii′-3c (crux) → ii′-6 (L0 assembly) → ii′-7 (close + audit + paper Appendix B).\n\nNo blockers. Ready to start on ii′-5c back half whenever you are."}]