[{"type":"thinking","thinking":"The user chose option (a): do tier ii′ now, submit after. But with a hard requirement: Paper A and its information must remain intact for submission if we change course. So:\n\n1. Paper A must stay submission-ready at all times during tier ii′ work — no edits that destabilize it. Any changes to paperA.tex should be avoided until ii′ completes (or only made at the end in the closure sub-stage ii′-7, which is where the Appendix B update + tag upgrade lands).\n\n2. Start sub-stage ii′-0: design document D-064 + LiContour.lean skeleton.\n\nThis is session 014. Per protocol §8.3 I already read STATE.md. I should:\n- Register the session plan\n- Begin ii′-0\n\nFor safety of Paper A, concrete measures:\n- Do NOT touch paperA.tex during tier ii′ (all paper updates deferred to ii′-7, and even then only Appendix B/abstract sentence via a careful re-audit per D-060 §5 rule).\n- Work happens in research/lean/NguyenRH1/ (new file LiContour.lean) and research/derivations/D-064-*.md.\n- The existing Lean theorems stay untouched: lambda_pos_finite_height_verified keeps its current form with the hSf hypothesis; the new work PROVES SfBoundProp as a theorem, and only at ii′-7 do we swap. Even then, we can keep the old theorem (it's a corollary) — nothing breaks.\n- git: commit frequently per sub-stage so every state is recoverable. Current tree is clean except untracked LaTeX build artifacts — fine.\n\nLet me start session 014 now:\n1. Todo list for ii′-0.\n2. Set up the session: verify Lean environment works (quick check).\n3. Begin D-064 design doc — but first the capability inventory part of ii′-0: I need to survey:\n   - The vendored residue API in detail (ResidueCalcOnRectangles.lean — read fully)\n   - MeromorphicAt / meromorphicOrderAt API in Mathlib\n   - digamma in Mathlib\n   - Zeta23/Defs zero-counting infrastructure\n   - Mathlib zeta Laurent at 1 / trivial zeros / functional equation\n   - What liCoeff is in NguyenRH1 (L7)\n   - D-036 (the contour computation — the actual proof to formalize), D-056 (R-bound proof), D-057, D-039\n   - LiSkeleton.lean's SfBoundProp + Bn + FAlt + IIMaj + C0hat + Hhat definitions (what the bound actually is formally)\n\nThat's a lot of reading. ii′-0 is estimated 1–2 sessions. Let me be systematic.\n\nPlan for this session:\n1. TodoWrite with the ii′-0 breakdown (and note the Paper-A-freeze rule).\n2. First, the Paper A protection: nothing to do actively — just the rule. Maybe also verify paperA.pdf builds clean right now as a baseline? It exists (untracked). Quick pdflatex check might take a while but is a good baseline. Actually — the git status shows paperA.pdf untracked. Baseline: `pdflatex` compile check. That takes ~30s. Worth it as the \"intact baseline\". Actually simpler: the paper compiled clean at session 013 end (committed state), and we're not touching it. A baseline rebuild is cheap insurance though. Hmm — MiKTeX compile of 9pp takes seconds. Let me do it as part of session start (verify + note hash). Actually let me not over-engineer; the file is committed in git, that's the protection. I'll state the freeze rule and rely on git.\n\n3. Capability inventory (the bulk): read the key files. Use subagents? The Explore agent could survey Mathlib/Zeta23 APIs in parallel while I read D-036/D-056 and LiSkeleton myself. That's a good use of parallel agents:\n   - Agent 1 (Explore): full inventory of the vendored residue/meromorphic API: ResidueCalcOnRectangles.lean (all public decls + hypotheses of RectangleIntegral'_eq_sumResiduesIn), Rectangle.lean, ZetaBounds.lean (logDerivResidue family), Mathlib MeromorphicAt/meromorphicOrderAt API surface (files + key lemma names), digamma file.\n   - Agent 2 (Explore): Zeta23 zero-counting infrastructure (Defs/, WeilEF?, what N(T)-like objects exist, riemannZeta zeros API used), Mathlib riemannZeta API (Laurent at 1? trivial zeros? functional equation names), and what liCoeff refers to in NguyenRH1 (it's in LiSkeleton or imported?).\n   \n   Meanwhile I read: D-036 (contour computation), D-056 (R-bound), LiSkeleton.lean's tier-(ii) definitions (Bn, C0hat, Hhat, IIMaj, SInfty, sInftyLower, liCoeff).\n\n4. Then write D-064 (design) — the big deliverable: lemma dependency graph, route decisions, sorry skeleton design.\n5. Create LiContour.lean skeleton with named sorries, make it compile.\n6. Commit.\n\nGiven the sub-stage rule, ii′-0 might span this session and the next. Fine.\n\nOne important design consideration for the skeleton: how to structure SfBoundProp's proof. The final theorem must be `∀ n : ℕ, 100000 ≤ n → (n:ℝ) ≤ (3*10^12-1)^2 → |Sf n| ≤ Bn FAlt n`. The proof: fix n in range; set T := zero-avoiding in [√n, √n+1] (L3, needs existence from perimeter N count); identity (L0): Sf n = truncatedSum + sides + trivial; bound each; assemble ≤ Bn FAlt n.\n\nKey design decision: how to state the L0 identity as a Lean theorem with the exact RHS form that matches D-036's accounting (so that bounding is mechanical).\n\nAlso the route decisions:\n- w=1 identification: local Laurent recursion vs global. I need to read D-036's actual handling first.\n- L2: Hadamard vs local route; need to see what Zeta23/Mathlib has. Agent survey informs this.\n- Higher-order residue: design the general residue def. Actually — check: maybe current Mathlib HAS a general residue now? The vendored file says \"planned for Mathlib but not yet available\" — but that comment is from the zeta-23-lean port date; Mathlib v4.33.0-rc2 (current vendored) might have added residue API. Agent 1 should check Mathlib's own residue API (Mathlib/Analysis/Complex/... maybe Residue.lean exists?).\n\nLet me also remember the lean gotchas from STATE.md (handoff notes) when writing the skeleton.\n\nSandbox note: Lean commands — short ones in foreground OK per handoff (\"short PrintAxioms in foreground\"; iterate single-file with `lake env lean` from package dir). For the skeleton compile, that's a medium task; try foreground with timeout.\n\nLet me start. First: TodoWrite + session registration note + launch the two Explore agents in parallel + start reading D-036 myself.\n\nSession numbering: this is session 014. I'll note it. I won't create journal until end, but I should keep track.\n\nAlso: user directive — Paper A intact. I'll state the freeze rule explicitly:\n1. No edits to paperA.tex/paperB.tex/draft.tex/references.bib during ii′-0..ii′-6. All paper changes land in ii′-7 only, after the axiom audit passes, and even then Paper A's change is minimal (Appendix B paragraph + one abstract sentence + tag), re-audited per D-060 §5.\n2. Existing Lean theorems untouched: LiSkeleton.lean/LiBasePoints.lean are frozen; new work goes in NEW files (LiContour.lean etc.). SfBoundProp stays as-is until ii′-7, when we add `theorem sfBoundProp : SfBoundProp` (the def SfBoundProp is a Prop; we prove a theorem with that statement) and swap hypotheses — actually careful: SfBoundProp is a def; proving `theorem SfBoundProp.proof : SfBoundProp` then lambda_pos_finite_height_verified can be re-stated without hSf. But rather than editing LiBasePoints.lean (frozen), in ii′-7 we'd add a NEW final theorem `lambda_pos_finite_height_closed` in the new file that applies the old one. Even cleaner: zero edits to existing files, ever. The paper then cites the new final theorem. \n\nLet me kick off. First message to user: brief acknowledgment of the (a) decision + protection protocol + starting ii′-0. Then tools.\n\nActually — should I use EnterPlanMode here? The user already approved the plan (they said \"let's do (a)\"). The planning is done at the coarse level; ii′-0 IS the detailed design sub-stage. No need for plan mode; proceed with work. TodoWrite to track.\n\nLet me define todos:\n1. Paper-A freeze protocol (state it; verify git-clean baseline for paper files) — quick.\n2. Inventory: vendored residue/meromorphic API (Agent 1).\n3. Inventory: Zeta23 zero-counting + Mathlib zeta API + liCoeff (Agent 2).\n4. Read D-036 + D-056 + LiSkeleton tier-ii defs (myself).\n5. Write D-064 design doc (route decisions + lemma graph).\n6. Create LiContour.lean sorry skeleton; compiles.\n7. Session-014 journal + STATE.md update + commit.\n\nThat's the session. Let me go. I'll launch the two agents and read D-036 in parallel.","signature":""},{"type":"text","text":"Locked in: **option (a) — tier ii′ now, submit after, with Paper A frozen and submission-ready at all times.**\n\n## Protection protocol (in force until ii′-7)\n\n1. **Zero edits** to `paperA.tex`, `paperB.tex`, `draft.tex`, `references.bib` — they stay at the session-013 audited state, recoverable via git at any moment.\n2. **Zero edits to existing Lean proofs** — `LiSkeleton.lean` / `LiBasePoints.lean` are frozen. All new work goes in **new files** (`LiContour.lean`, …). Even at the end, we *add* a final closed-form theorem rather than rewrite the old ones — the current three-hypothesis `lambda_pos_finite_height_verified` remains in the tree permanently.\n3. Commit at every sub-stage boundary — every state recoverable.\n\nIf we change course at any point, nothing has moved: Paper A submits as-is.\n\n## Session 014 — starting sub-stage ii′-0 (design + skeleton, → D-064)"},{"type":"tool_use","id":"TodoWrite_23","name":"TodoWrite","input":{"todos":[{"content":"Verify paper/Lean baseline is clean and frozen","status":"in_progress","activeForm":"Verifying paper/Lean baseline"},{"content":"Inventory vendored residue/meromorphic API (agent)","status":"pending","activeForm":"Inventorying residue/meromorphic API"},{"content":"Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)","status":"pending","activeForm":"Inventorying Zeta23/Mathlib zeta API"},{"content":"Read D-036, D-056, LiSkeleton tier-(ii) defs","status":"pending","activeForm":"Reading D-036/D-056/LiSkeleton"},{"content":"Write D-064 design doc (routes + lemma graph)","status":"pending","activeForm":"Writing D-064 design doc"},{"content":"Create LiContour.lean sorry skeleton, compiles","status":"pending","activeForm":"Creating LiContour.lean skeleton"},{"content":"Session-014 closure (journal, STATE, commit)","status":"pending","activeForm":"Closing session 014"}]},"extra_content":null}]