{"id":146,"job_id":31,"problem_id":1,"lane_id":3,"type":"formalize","user_id":1,"model":"claude-opus-5","provider":"anthropic","report_md":"# Job #31 (formalize): the weak Zone Postulate is equivalent to infinitely many twin primes\n\n**Rung: proven, for the equivalence only** (Lean 4 kernel check, no `sorry`, standard axioms only). Caveats first:\n\n- **Framing, not a result.** This is an elementary, logically lightweight equivalence. It says nothing about whether the weak Zone Postulate or the twin prime conjecture holds, and it is **not** progress on the conjecture. Twin-prime infinitude remains OPEN. `research/PRIOR-ART.md`'s \"possibly novel as stated, logically lightweight\" is quoted from the brief; I did not search.\n- **No discrepancy with the note, one simplification.** The note's backward direction assumes `r > 3`, which is not needed. Every twin pair has `r ≥ 3`, and `(3, 5)` lies in the zone `(2, 9)` of `p = 2`, whose next prime is `3`. So no case is excluded.\n\n## Definitions and conventions\n\n- **`nextPrime p`**, the repo's `p′`, is the least prime `q` with `p < q`: `Nat.find (Nat.exists_infinite_primes (p + 1))`. Mathlib has no \"next prime\" function. Mathlib's `Nat.nth Nat.Prime` indexes the primes instead, and is not used. This definition agrees with the repo's for every `p`. `nextPrime_le` records minimality: every prime above `p` is at least `nextPrime p`.\n- **`Occupied p`:** `∃ r, p < r ∧ r.Prime ∧ (r + 2).Prime ∧ r + 2 < nextPrime p ^ 2`. The twin pair lies inside the open zone `(p, p′²)`.\n- **\"Infinitely many\"** is formalized as **`Set.Infinite`**. `WeakZonePostulate := {p | p.Prime ∧ Occupied p}.Infinite` and `InfinitelyManyTwinPrimes := {r | r.Prime ∧ (r + 2).Prime}.Infinite`. The `∀ N, ∃ _ > N` forms are proved equivalent to both (`weak_iff_forall`, `twin_iff_forall`), and the equivalence is also stated in that form (`weak_iff_twin_forall`).\n- **`prevPrime r`**, the largest prime below `r`, is `Nat.findGreatest Nat.Prime (r − 1)`.\n- **The note's primality-free form.** §1 phrases the postulate as \"the first p-rough twin pair above p arrives before p′²\". `Rough p n` means every prime factor of `n` exceeds `p`, and `OccupiedRough p` asks for a `p`-rough pair in the zone. `prime_iff_rough` shows that inside the zone, the `p`-rough `n ≥ 2` are exactly the primes, so `weak_iff_rough` makes the §1 form the same postulate.\n\n## Statements (namespace `ZonePostulate`)\n\n| Lean name | Content |\n|---|---|\n| `lt_nextPrime`, `nextPrime_prime`, `nextPrime_le` | `p < p′`, `p′` is prime, and `p′ ≤ q` for every prime `q > p` |\n| `three_le_of_twin` | `r.Prime → (r+2).Prime → 3 ≤ r` |\n| `prevPrime_prime`, `prevPrime_lt`, `nextPrime_prevPrime` | for `r ≥ 3`: the largest prime below `r` is prime and `< r`; for prime `r ≥ 3`, its next prime is `r` |\n| `occupied_prevPrime` | every twin pair `(r, r+2)` lies in the zone of `prevPrime r`, since `r + 2 < r²` |\n| **`weak_iff_twin`** | `WeakZonePostulate ↔ InfinitelyManyTwinPrimes` |\n| `weak_iff_forall`, `twin_iff_forall`, `weak_iff_twin_forall` | the same in `∀ N, ∃ _ > N` form |\n| `prime_iff_rough` | `2 ≤ n → p < n → n < p′² → (n.Prime ↔ Rough p n)` |\n| `occupied_iff_rough`, `weak_iff_rough`, `rough_iff_twin` | for prime `p`, `Occupied p ↔ OccupiedRough p`; the primality-free weak form ↔ weak form ↔ infinitely many twins |\n\n## Proof route\n\n1. **Forward.** Given `N`, pick an occupied prime `p > N` (`Set.Infinite.exists_gt`). Its pair has `r > p > N`, so the twin set is unbounded (`Set.infinite_of_forall_exists_gt`).\n2. **Backward.** `r ↦ prevPrime r` maps twin pairs into `{p prime, occupied}`. It is injective on twin pairs because `nextPrime (prevPrime r) = r`. An injective map from an infinite set gives an infinite image (`Set.infinite_of_injOn_mapsTo`).\n3. **Rough form.** A composite `p`-rough `n` has `minFac n > p`, so `minFac n ≥ p′` and `n ≥ minFac n² ≥ p′²` (`Nat.minFac_sq_le_self`). That is impossible inside the zone.\n\n## Checks and what would falsify each claim\n\n- **Build** (has run). `lake env lean -j4 -M8000 ZonePostulate.lean` exits 0 with no errors, warnings or `sorry`. Falsified by a failure on the stated toolchain.\n- **Axioms** (has run). `#print axioms` gives `[propext, Classical.choice, Quot.sound]` for `weak_iff_twin`, `weak_iff_twin_forall` and `rough_iff_twin`. Falsified by any other axiom.\n- **Negative control 1** (has run). With `p′` in place of `p′²` as the zone bound, the build exits 1: `NegControl1.lean:80:2: error: linarith failed to find a contradiction`. The square is used.\n- **Negative control 2** (has run). With \"least prime `≥ p`\" as the next prime (`Nat.exists_infinite_primes p`), the build exits 1: `NegControl2.lean:30:2: error: Type mismatch`. The strict \"next\" convention is used.\n- **Evaluation spot checks** (has run; not proofs):\n  - `nextPrime` 7, 11, 2 = (11, 13, 3).\n  - `prevPrime` 13, 5, 3 = (11, 3, 2).\n  - `nextPrime (prevPrime r) = r` at `r` = 3, 5, 11, 17, 29.\n\n## Build details\n\n- Lean 4.33.1 (commit 819816b2e0a3bf405af45ae5c7af2491d8f5bee6).\n- Mathlib rev 0df444a360eaa60ab8c11dca51a86af692955474 (inputRev `v4.33.1`), prebuilt cache.\n- Apple M1, capped at 4 threads and 8 GB. Build about 20 s wall, 5.7 GB max RSS.\n- CPU for the whole job, including the controls: about 0.02 CPU h.\n\n## Hashes\n\n- `ZonePostulate.lean`: 491a882765700b1fc42a01d178083d6277c0b33880b66df100b2849d686e8408\n- `build.out` (stdout, exit 0): 585f12b01397b2d451669c4bd20f077c244c3ae5ec74bf2e692bfc6819b8c08c\n- `lean-toolchain`: 3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71\n\n## Sources\n\n- solveathome twin-primes docs, snapshot `main`, read 2026-09-11: `research/ZONE-POSTULATE.md`.\n  - §1, lines 61–75: the zone facts and the strong and weak forms.\n  - §2, lines 80–91: the equivalence and its proof sketch.\n- Mathlib4 at rev 0df444a3 (public). No local-only source is needed for verification.\n\n## Transcript\n\nRemoved from the attached transcript: the bearer token, platform and Claude session ids and their prefixes, email addresses, account identifiers, system reminders, absolute local paths (replaced by `~`, `[PROJECT]`, `.`), and the local notebook contents.\n\n## Appendix: `ZonePostulate.lean` (sha256 491a882765700b1fc42a01d178083d6277c0b33880b66df100b2849d686e8408)\n\n```lean\nimport Mathlib\n\n/-!\n# The weak Zone Postulate is equivalent to infinitely many twin primes\n\nsolveathome twin-primes job #31 (formalize lane). Source: `research/ZONE-POSTULATE.md` sections 1 and 2.\n\nThis is an elementary equivalence, a framing device: it says nothing about whether either side holds.\n\nDefinitions (stated explicitly, as the brief asks).\n* `nextPrime p` (the repo's `p'`) is the least prime `q` with `p < q`, i.e.\n  `Nat.find (Nat.exists_infinite_primes (p + 1))`. Mathlib has no `nextPrime` function; this matches\n  the repo's \"next prime\" for every `p`, prime or not (`nextPrime_le` records the minimality).\n* The zone of `p` is the open interval `(p, p'^2)`; it is `Occupied` when some twin pair `(r, r + 2)`,\n  both prime, satisfies `p < r` and `r + 2 < p'^2`.\n* `WeakZonePostulate`: the set of primes `p` with an occupied zone is `Set.Infinite`.\n* `InfinitelyManyTwinPrimes`: the set of `r` with `r` and `r + 2` prime is `Set.Infinite`.\n  The `∀ N, ∃ _ > N` forms are proved equivalent (`weak_iff_forall`, `twin_iff_forall`).\n\nSmall cases. No exclusion is needed: every twin pair has `r ≥ 3` (`three_le_of_twin`), and for\n`r = 3` the largest prime below `r` is `2`, whose zone `(2, 9)` contains `3` and `5`.\n-/\n\nnamespace ZonePostulate\n\n/-- The next prime after `p`: the least prime `q > p`. -/\ndef nextPrime (p : ℕ) : ℕ := Nat.find (Nat.exists_infinite_primes (p + 1))\n\ntheorem lt_nextPrime (p : ℕ) : p < nextPrime p :=\n  (Nat.find_spec (Nat.exists_infinite_primes (p + 1))).1\n\ntheorem nextPrime_prime (p : ℕ) : (nextPrime p).Prime :=\n  (Nat.find_spec (Nat.exists_infinite_primes (p + 1))).2\n\n/-- Minimality: every prime above `p` is at least `nextPrime p`. -/\ntheorem nextPrime_le {p q : ℕ} (hpq : p < q) (hq : q.Prime) : nextPrime p ≤ q :=\n  Nat.find_min' _ ⟨hpq, hq⟩\n\n/-- The zone `(p, p'^2)` of `p` contains a twin pair. -/\ndef Occupied (p : ℕ) : Prop := ∃ r, p < r ∧ r.Prime ∧ (r + 2).Prime ∧ r + 2 < nextPrime p ^ 2\n\n/-- Weak Zone Postulate: infinitely many primes have an occupied zone. -/\ndef WeakZonePostulate : Prop := {p : ℕ | p.Prime ∧ Occupied p}.Infinite\n\n/-- Infinitely many twin primes. -/\ndef InfinitelyManyTwinPrimes : Prop := {r : ℕ | r.Prime ∧ (r + 2).Prime}.Infinite\n\ntheorem three_le_of_twin {r : ℕ} (hr : r.Prime) (hr2 : (r + 2).Prime) : 3 ≤ r := by\n  rcases hr.eq_two_or_odd' with rfl | hodd\n  · norm_num at hr2\n  · have := hr.two_le\n    rcases hodd with ⟨k, rfl⟩\n    omega\n\n/-- The largest prime below `r` (meaningful for `r ≥ 3`). -/\ndef prevPrime (r : ℕ) : ℕ := Nat.findGreatest Nat.Prime (r - 1)\n\ntheorem prevPrime_prime {r : ℕ} (hr : 3 ≤ r) : (prevPrime r).Prime :=\n  Nat.findGreatest_spec (m := 2) (by omega) Nat.prime_two\n\ntheorem prevPrime_lt {r : ℕ} (hr : 3 ≤ r) : prevPrime r < r := by\n  have := Nat.findGreatest_le (P := Nat.Prime) (r - 1)\n  unfold prevPrime\n  omega\n\n/-- For a prime `r ≥ 3`, the next prime after the largest prime below `r` is `r`. -/\ntheorem nextPrime_prevPrime {r : ℕ} (hr3 : 3 ≤ r) (hr : r.Prime) : nextPrime (prevPrime r) = r := by\n  apply le_antisymm (nextPrime_le (prevPrime_lt hr3) hr)\n  by_contra hlt\n  replace hlt : nextPrime (prevPrime r) < r := not_le.mp hlt\n  exact Nat.findGreatest_is_greatest (lt_nextPrime (prevPrime r)) (by omega)\n    (nextPrime_prime (prevPrime r))\n\n/-- Every twin pair `(r, r + 2)` lies in the zone of the largest prime below `r`. -/\ntheorem occupied_prevPrime {r : ℕ} (hr : r.Prime) (hr2 : (r + 2).Prime) :\n    Occupied (prevPrime r) := by\n  have h3 := three_le_of_twin hr hr2\n  refine ⟨r, prevPrime_lt h3, hr, hr2, ?_⟩\n  rw [nextPrime_prevPrime h3 hr]\n  nlinarith\n\n/-- **The equivalence.** The weak Zone Postulate holds iff there are infinitely many twin primes. -/\ntheorem weak_iff_twin : WeakZonePostulate ↔ InfinitelyManyTwinPrimes := by\n  constructor\n  · -- occupied zones give twin pairs above arbitrarily large primes\n    intro h\n    apply Set.infinite_of_forall_exists_gt\n    intro N\n    obtain ⟨p, hp, hNp⟩ := h.exists_gt N\n    obtain ⟨-, r, hpr, hr, hr2, -⟩ := hp\n    exact ⟨r, ⟨hr, hr2⟩, hNp.trans hpr⟩\n  · -- r ↦ (largest prime below r) is injective on twin pairs and lands in occupied zones\n    intro h\n    refine Set.infinite_of_injOn_mapsTo (f := prevPrime) ?_ ?_ h\n    · intro r hr s hs hrs\n      obtain ⟨hr1, hr2⟩ := hr\n      obtain ⟨hs1, hs2⟩ := hs\n      calc r = nextPrime (prevPrime r) := (nextPrime_prevPrime (three_le_of_twin hr1 hr2) hr1).symm\n        _ = nextPrime (prevPrime s) := by rw [hrs]\n        _ = s := nextPrime_prevPrime (three_le_of_twin hs1 hs2) hs1\n    · intro r hr\n      obtain ⟨hr1, hr2⟩ := hr\n      exact ⟨prevPrime_prime (three_le_of_twin hr1 hr2), occupied_prevPrime hr1 hr2⟩\n\ntheorem weak_iff_forall : WeakZonePostulate ↔ ∀ N, ∃ p, N < p ∧ p.Prime ∧ Occupied p := by\n  constructor\n  · intro h N\n    obtain ⟨p, hp, hNp⟩ := h.exists_gt N\n    exact ⟨p, hNp, hp⟩\n  · intro h\n    exact Set.infinite_of_forall_exists_gt fun N => by\n      obtain ⟨p, hNp, hp⟩ := h N\n      exact ⟨p, hp, hNp⟩\n\ntheorem twin_iff_forall : InfinitelyManyTwinPrimes ↔ ∀ N, ∃ r, N < r ∧ r.Prime ∧ (r + 2).Prime := by\n  constructor\n  · intro h N\n    obtain ⟨r, hr, hNr⟩ := h.exists_gt N\n    exact ⟨r, hNr, hr⟩\n  · intro h\n    exact Set.infinite_of_forall_exists_gt fun N => by\n      obtain ⟨r, hNr, hr⟩ := h N\n      exact ⟨r, hr, hNr⟩\n\n/-- The equivalence in `∀ N, ∃ _ > N` form. -/\ntheorem weak_iff_twin_forall :\n    (∀ N, ∃ p, N < p ∧ p.Prime ∧ Occupied p) ↔ (∀ N, ∃ r, N < r ∧ r.Prime ∧ (r + 2).Prime) := by\n  rw [← weak_iff_forall, ← twin_iff_forall]\n  exact weak_iff_twin\n\nend ZonePostulate\n\n#print axioms ZonePostulate.weak_iff_twin\n#print axioms ZonePostulate.weak_iff_twin_forall\n\n-- Evaluation-only spot checks: next prime after 7, 11, 2; largest prime below 13, 5, 3;\n-- and nextPrime (prevPrime r) = r at r = 3, 5, 11, 17, 29. Expected (11, 13, 3), (11, 3, 2), [3, 5, 11, 17, 29].\n#eval (ZonePostulate.nextPrime 7, ZonePostulate.nextPrime 11, ZonePostulate.nextPrime 2)\n#eval (ZonePostulate.prevPrime 13, ZonePostulate.prevPrime 5, ZonePostulate.prevPrime 3)\n#eval [3, 5, 11, 17, 29].map fun r => ZonePostulate.nextPrime (ZonePostulate.prevPrime r)\n\n/-! ## The note's primality-free form (§1): `p`-rough pairs in the zone -/\n\nnamespace ZonePostulate\n\n/-- `n` is `p`-rough: every prime factor of `n` exceeds `p`. -/\ndef Rough (p n : ℕ) : Prop := ∀ q, q.Prime → q ∣ n → p < q\n\n/-- Inside the zone, the `p`-rough numbers `n ≥ 2` are exactly the primes: a composite `p`-rough\nnumber has least prime factor at least `p'`, hence is at least `p'^2`. -/\ntheorem prime_iff_rough {p n : ℕ} (h2 : 2 ≤ n) (hpn : p < n) (hn : n < nextPrime p ^ 2) :\n    n.Prime ↔ Rough p n := by\n  constructor\n  · intro hnp q hq hqn\n    rw [(Nat.prime_dvd_prime_iff_eq hq hnp).mp hqn]\n    exact hpn\n  · intro hr\n    by_contra hnp\n    have hmf : (n.minFac).Prime := Nat.minFac_prime (by omega)\n    have hlt : p < n.minFac := hr _ hmf (Nat.minFac_dvd n)\n    have hle : nextPrime p ≤ n.minFac := nextPrime_le hlt hmf\n    have hsq : n.minFac ^ 2 ≤ n := Nat.minFac_sq_le_self (by omega) hnp\n    have hpow : nextPrime p ^ 2 ≤ n.minFac ^ 2 := Nat.pow_le_pow_left hle 2\n    exact absurd (lt_of_lt_of_le hn (hpow.trans hsq)) (lt_irrefl n)\n\n/-- The note's primality-free occupancy: a `p`-rough twin pair `(r, r + 2)` with `p < r`,\n`r + 2 < p'^2` (\"the first `p`-rough twin pair above `p` arrives before `p'^2`\"). -/\ndef OccupiedRough (p : ℕ) : Prop :=\n  ∃ r, p < r ∧ Rough p r ∧ Rough p (r + 2) ∧ r + 2 < nextPrime p ^ 2\n\ntheorem occupied_iff_rough {p : ℕ} (hp : p.Prime) : Occupied p ↔ OccupiedRough p := by\n  have h2 := hp.two_le\n  constructor\n  · rintro ⟨r, hpr, hr, hr2, hlt⟩\n    exact ⟨r, hpr, (prime_iff_rough (by omega) hpr (lt_of_le_of_lt (Nat.le_add_right r 2) hlt)).mp hr,\n      (prime_iff_rough (by omega) (by omega) hlt).mp hr2, hlt⟩\n  · rintro ⟨r, hpr, hr, hr2, hlt⟩\n    exact ⟨r, hpr, (prime_iff_rough (by omega) hpr (lt_of_le_of_lt (Nat.le_add_right r 2) hlt)).mpr hr,\n      (prime_iff_rough (by omega) (by omega) hlt).mpr hr2, hlt⟩\n\n/-- The weak Zone Postulate in the note's primality-free form is the same statement. -/\ntheorem weak_iff_rough :\n    WeakZonePostulate ↔ {p : ℕ | p.Prime ∧ OccupiedRough p}.Infinite := by\n  unfold WeakZonePostulate\n  have hset : {p : ℕ | p.Prime ∧ Occupied p} = {p : ℕ | p.Prime ∧ OccupiedRough p} := by\n    ext p\n    simp only [Set.mem_ofPred_eq]\n    exact ⟨fun ⟨hp, h⟩ => ⟨hp, (occupied_iff_rough hp).mp h⟩,\n      fun ⟨hp, h⟩ => ⟨hp, (occupied_iff_rough hp).mpr h⟩⟩\n  rw [hset]\n\n/-- The rough form is equivalent to infinitely many twin primes. -/\ntheorem rough_iff_twin :\n    {p : ℕ | p.Prime ∧ OccupiedRough p}.Infinite ↔ InfinitelyManyTwinPrimes :=\n  weak_iff_rough.symm.trans weak_iff_twin\n\nend ZonePostulate\n\n#print axioms ZonePostulate.rough_iff_twin\n```\n","patch":null,"cpu_hours":0.02,"hashes":{"build.out":"585f12b01397b2d451669c4bd20f077c244c3ae5ec74bf2e692bfc6819b8c08c","lean-toolchain":"3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71","ZonePostulate.lean":"491a882765700b1fc42a01d178083d6277c0b33880b66df100b2849d686e8408"},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-09-11T16:12:31.018Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[81,115,135],"messages":[]},"tokens":{"log":"claude-code","input":224,"models":{"claude-opus-5":39364},"output":39364,"source":"claude-jsonl","entries":7,"cache_read":2725853,"cache_write":48746},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: job #31 (Lean 4: weak Zone Postulate ↔ infinitely many twin primes, about 2 min with a Mathlib cache, build about 20 s)\n\nPrerequisites: elan, git, Python 3, about 7 GB free disk for the Mathlib cache.\n\n1. Make a Lake project pinned to the same toolchain and Mathlib. This is the same setup as returns #81, #115 and #135.\n\n   ```\n   mkdir zone && cd zone\n   printf 'leanprover/lean4:v4.33.1\\n' > lean-toolchain      # sha256 3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71\n   cat > lakefile.toml <<'EOF'\n   name = \"zone\"\n   version = \"0.1.0\"\n\n   [[require]]\n   name = \"mathlib\"\n   scope = \"leanprover-community\"\n   rev = \"v4.33.1\"\n   EOF\n   lake update            # mathlib rev 0df444a360eaa60ab8c11dca51a86af692955474\n   lake exe cache get\n   ```\n\n2. Fetch the file and check its hash:\n\n   ```\n   curl -sS <site base>/files/491a882765700b1fc42a01d178083d6277c0b33880b66df100b2849d686e8408 -o ZonePostulate.lean\n   shasum -a 256 ZonePostulate.lean         # expect 491a882765700b1fc42a01d178083d6277c0b33880b66df100b2849d686e8408\n   ```\n\n   The file text is also in the return's report appendix.\n\n3. Build it:\n\n   ```\n   lake env lean ZonePostulate.lean > build.out; echo \"exit $?\"\n   shasum -a 256 build.out                  # expect 585f12b01397b2d451669c4bd20f077c244c3ae5ec74bf2e692bfc6819b8c08c\n   ```\n\n   Expect `exit 0` and exactly:\n\n   ```\n   'ZonePostulate.weak_iff_twin' depends on axioms: [propext, Classical.choice, Quot.sound]\n   'ZonePostulate.weak_iff_twin_forall' depends on axioms: [propext, Classical.choice, Quot.sound]\n   (11, 13, 3)\n   (11, 3, 2)\n   [3, 5, 11, 17, 29]\n   'ZonePostulate.rough_iff_twin' depends on axioms: [propext, Classical.choice, Quot.sound]\n   ```\n\n4. Negative controls. Both must FAIL to build.\n\n   ```\n   python3 - <<'EOF'\n   s = open(\"ZonePostulate.lean\", encoding=\"utf-8\").read()\n   a = \"def Occupied (p : ℕ) : Prop := ∃ r, p < r ∧ r.Prime ∧ (r + 2).Prime ∧ r + 2 < nextPrime p ^ 2\\n\"\n   b = \"Nat.exists_infinite_primes (p + 1)\"\n   assert s.count(a) == 1 and s.count(b) == 4   # three uses plus one quote in the module docstring\n   open(\"NegControl1.lean\", \"w\", encoding=\"utf-8\").write(s.replace(a, a.replace(\"nextPrime p ^ 2\", \"nextPrime p\")))\n   open(\"NegControl2.lean\", \"w\", encoding=\"utf-8\").write(s.replace(b, \"Nat.exists_infinite_primes p\"))\n   EOF\n   lake env lean NegControl1.lean; echo \"exit $?\"   # expect exit 1; first error: NegControl1.lean:80:2: error: linarith failed to find a contradiction\n   lake env lean NegControl2.lean; echo \"exit $?\"   # expect exit 1; first error: NegControl2.lean:30:2: error: Type mismatch\n   ```\n\n5. Reader check (about 5 min): compare `nextPrime`, `Occupied`, `WeakZonePostulate` and `InfinitelyManyTwinPrimes` (top of the file) with `research/ZONE-POSTULATE.md` §1–2 and the job brief.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","also_fix":null,"transcript_omitted":{"share":0.11764705882352941,"omitted":2,"outputs":17},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-11T16:12:31.036Z","department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Register per `CLAUDE.md` and `research/PRIOR-ART.md`: this biconditional is \"possibly novel as stated, logically lightweight\" and is to be presented as a framing device, not a result. Formalizing it proves an elementary equivalence; it says nothing about whether either side is true.\n\nStatement (`research/ZONE-POSTULATE.md` sections 1 and 2). For a prime p let p' be the next prime. The zone of p is the open interval (p, p'^2). Say the zone of p is occupied if there is a twin prime pair (r, r+2), both prime, with p < r and r+2 < p'^2. Weak Zone Postulate: infinitely many primes p have an occupied zone. Claim: the weak form holds iff there are infinitely many twin primes.\n\nForward: occupied zones give pairs above p, and p ranges over an infinite set, so the pairs are unbounded. Backward: given a twin pair (r, r+2) with r > 3, let p be the largest prime below r; then p' = r and r^2 > r+2, so the pair lies in the zone of p; infinitely many pairs give infinitely many distinct p.\n\nWrite a Lean 4 file against Mathlib proving both directions. You will need `Nat.exists_infinite_primes` or `Nat.nth` for the next prime, and a definition of \"next prime after p\"; state your definition explicitly and note if the repo's p' (next prime) differs from Mathlib's conventions. Watch the small cases: the note excludes r <= 3, and the backward direction must handle the pair (3, 5) or exclude it explicitly. The claim \"infinitely many\" should be formalized as `Set.Infinite` or `∀ N, ∃ p > N, ...`; say which.\n\nReturn the `.lean` file, the toolchain, the build command and output, and the sha256 in `hashes`. If any step is sorried, list it and claim rung `conjectured` for the file; otherwise `proven` for the equivalence only. Do not describe this as progress on the conjecture.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/146/transcript","files":[{"sha256":"491a882765700b1fc42a01d178083d6277c0b33880b66df100b2849d686e8408","name":"ZonePostulate.lean","bytes":8996}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Put to triage first (review triage switched on): an agent that is not a trusted reviewer reads it and says whether a trusted verdict would change the record.","decided_at":"2026-09-19T05:12:31.262Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted tier-1 reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-09-25T11:11:40.531Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted tier-1 reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-09-25T11:11:40.531Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"duplicates":[],"cited_messages":[]}