{"id":79,"job_id":25,"problem_id":1,"lane_id":3,"type":"formalize","user_id":1,"model":"claude-opus-5","provider":"anthropic","report_md":"# Job #25 (formalize): ∑_{i=j₀}^{j} 2^i/i^K = (2 + o(1)) 2^j/j^K, and the (P) ⟺ (P') consumer equivalence, proved in Lean 4 against Mathlib\n\n**Caveat first.** This is a lemma about real sequences. It proves that two OPEN sufficient conditions are\nequivalent:\n- (P): a_j ≥ c·w_j infinitely often;\n- (P'): the partial sums are ≥ c'·w_j infinitely often.\n\nIt proves neither condition, and it changes nothing about the open gap in `research/consumer-comparison.md`\n§2 item 3 or `research/TWIN-REDUCTION.md` §2. The mathematics is elementary (a geometric-tail estimate).\nThe Lean file adds a proven rung to a step the notes already use, and nothing else. Conflict of interest:\nmy person owns the repository.\n\n**Rung: proven** for both statements, as stated below. The file is sorry-free and builds with no errors\nand no warnings. `#print axioms` on all seven declarations lists only `propext`, `Classical.choice` and\n`Quot.sound`. A negative control fails to build.\n\n## 0. The statements as formalized\n\n`w K i := (2:ℝ)^i / (i:ℝ)^K` with `K : ℝ`. The brief's hypotheses are kept: `0 ≤ K`, `1 ≤ j₀`, `∀ i, 0 ≤ a i`.\n\n| Lean name (namespace `ConsumerGrowth`) | statement |\n|---|---|\n| **`sum_div_w_tendsto`** (a) | `0 ≤ K → 1 ≤ j₀ → Tendsto (fun j => (∑ i ∈ Icc j₀ j, w K i) / w K j) atTop (𝓝 2)` |\n| **`consumer_equiv`** (b) | `0 ≤ K → (∀ i, 0 ≤ a i) → ((∃ c > 0, ∃ᶠ j in atTop, c * w K j ≤ a j) ↔ (∃ c > 0, ∃ᶠ j in atTop, c * w K j ≤ ∑ i ∈ range (j + 1), a i))` |\n| `w_pos` | `1 ≤ i → 0 < w K i` |\n| `w_ratio` | `m + 1 ≤ j → w K (j - m) / w K j = (1/2)^m * (j / (j - m))^K` |\n| `sum_reflect` | `j₀ ≤ j → ∑ i ∈ Icc j₀ j, w K i = ∑ m ∈ range (j - j₀ + 1), w K (j - m)` |\n| `succ_pow_le` | `(m + 1)^n ≤ 2^n * (m^n + 1)` for natural m, n |\n| `w_tendsto_atTop` | `Tendsto (w K) atTop atTop` for every real K; no sign condition is needed |\n\n**Constants.** Forward (P) ⟹ (P') keeps the same c, because a_j ≤ ∑_{i≤j} a_i when a ≥ 0. Backward,\n(P') with constant c' gives (P) with constant **c'/4**. The note instead argues with c/4 against c' > c/2.\nThe Lean proof runs as follows:\n1. Suppose a_i < (c'/4)·w_i for all i ≥ N. Put N₁ = max N 1.\n2. Part (a) with j₀ = N₁ gives ∑_{i∈[N₁,j]} w_i < 3·w_j eventually.\n3. w_j → ∞ gives A := ∑_{i<N₁} a_i < (c'/4)·w_j eventually.\n4. Hence ∑_{i≤j} a_i < (c'/4)·w_j + (c'/4)·3·w_j = c'·w_j eventually, which contradicts (P').\n\n**Index conventions.** The brief warns about j₀ = 1 versus i = 0 (w₀ is undefined for K > 0).\n- In Lean, `w K 0` is `1 / 0^K`. That is `1/0 = 0` for K ≠ 0 and 1 for K = 0. The value is never used.\n- (a) sums over `Icc j₀ j` with `j₀ ≥ 1`, so every term has i ≥ 1.\n- In (b) the partial sum is `∑ i ∈ range (j+1)`, i.e. it includes a₀, as in the note's ∑_{i≤j}. `w K` is\n  evaluated only at the indices j quantified by `∃ᶠ j in atTop`. A single index cannot change whether a\n  property holds frequently. The tail estimate starts at N₁ ≥ 1.\n\n**Route for (a).**\n1. Reflect i = j − m, so the ratio becomes ∑_{m ≤ j−j₀} (1/2)^m·(j/(j−m))^K.\n2. Term by term, (j/(j−m))^K → 1, by continuity of x ↦ x^K at 1.\n3. For m + j₀ ≤ j, j/(j−m) ≤ m + 1. With n = ⌈K⌉₊ the term is bounded by 2^n·(m^n + 1)·(1/2)^m, which is\n   summable (`summable_pow_mul_geometric_of_norm_lt_one` and `summable_geometric_two`).\n4. Tannery's theorem (`tendsto_tsum_of_dominated_convergence`) gives the limit ∑_m (1/2)^m = 2\n   (`tsum_geometric_two`).\n\nK ≥ 0 is used in step 3 (`Real.rpow_le_rpow`).\n\n**What would falsify this.** Any of the following:\n- a `sorryAx` or a nonstandard axiom in `#print axioms`;\n- the file failing to build at the pinned toolchain and Mathlib commit;\n- the Lean statements differing from the brief's (a) and (b).\n\nFor the last, compare the table above with the brief. The negative control shows that the limit value in\n(a) is checked.\n\n## 1. Build, checks, custody\n\n- **Toolchain.** `leanprover/lean4:v4.33.1` and Lake 5.0.0, with Mathlib `0df444a360eaa60ab8c11dca51a86af692955474`\n  (inputRev v4.33.1). This is the same project as returns #76 and #77.\n- **Build.** `lake env lean Growth.lean` exits 0 in 17.7 s wall on an Apple M1, with no output. The file is\n  `Growth.lean`, 228 lines, sha256 `3c66cec3e878b9add77df1f1b581f962331cfff9e3fb5a80b2500f64e5dbe90d`, and\n  `grep -c sorry` returns 0.\n- **Iterations.** The build took two iterations before it was clean:\n  1. three errors:\n     - `field_simp` left the goal `1 = 2^m * (1/2)^m` in `w_ratio`, closed by `rw [← mul_pow]; norm_num`;\n     - a composition-form type mismatch in the rpow limit, fixed by rewriting `Real.one_rpow` before `exact`;\n     - a redundant tactic after `congr 1` in the Ico/Icc split, which I replaced by an explicit\n       `Ico N₁ (j+1) = Icc N₁ j`;\n  2. clean. The unused hypothesis `0 ≤ K` was also dropped from `w_tendsto_atTop`.\n- **Axioms.** A copy with seven `#print axioms` lines exits 0. None of the declarations depends on `sorryAx`.\n- **Negative control.** A copy with the limit in (a) changed from `𝓝 2` to `𝓝 3` fails with\n  \"Type mismatch … `Tendsto ?m atTop (𝓝 2)` but is expected to have type … `(𝓝 3)`\" at `hT.congr'`.\n  The failure also leaves `consumer_equiv` unprovable, because `3 < 3` cannot be discharged, and the build\n  exits 1.\n\n**Files.** `Growth.lean` in full, the three logs, the toolchain line and the manifest pins are inline in\nthe recipe, with their sha256 values in `hashes`. `Growth.lean` and this report are also uploaded if the\nfile store accepts them; `files` lists what was accepted.\n\n## Sources\n\n- The job #25 brief (statements (a) and (b), and the note's constants c/4 and c' > c/2).\n  `research/consumer-comparison.md` §2 item 3 and `research/TWIN-REDUCTION.md` §2, snapshot `main`, as the\n  brief cites them for where the lemma is used. I formalized the brief's statement; checking the Lean\n  statements does not require those documents.\n- Mathlib at the commit above: `tendsto_tsum_of_dominated_convergence` (Mathlib/Analysis/Normed/Group/Tannery.lean),\n  `summable_pow_mul_geometric_of_norm_lt_one`, `tsum_geometric_two`, `tendsto_pow_const_div_const_pow_of_one_lt`,\n  `Filter.Tendsto.inv_tendsto_nhdsGT_zero`, `Real.continuousAt_rpow_const`, `Real.rpow_le_rpow`,\n  `Real.rpow_le_rpow_of_exponent_le`, `Finset.sum_nbij'`, `Finset.sum_range_add_sum_Ico`.\n\nNo local-only sources.\n\n**Transcript:** from the GET /start that received job #25 to this return. Removed: system reminders,\nthe bearer token, session ids, home, scratchpad and task-directory paths, and an e-mail address.\n","patch":null,"cpu_hours":0.05,"hashes":{"Growth.lean":"3c66cec3e878b9add77df1f1b581f962331cfff9e3fb5a80b2500f64e5dbe90d","lean-toolchain (inline in recipe)":"3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71","growth-build.log (inline in recipe)":"b4a259e24e80c867f4caea53026547c0209dac4fdd8dc5a928be3685fa93378c","growth-axioms.log (inline in recipe)":"58000fcd70bc3ae5602328c7c00a6ecfdeaef89deddc8c7bde18c82c9821647c","growth-negative.log (inline in recipe)":"2ab32760ec57514a4f48a46526e057d192e91fbccfd86a6d9c196d9d4b9c0fb0","lake-manifest.json (pins inline in recipe)":"713e3b169ede20be0a2767072005a84a3527c83b748a9420ab769b8e8cd8f278"},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-09-11T15:18:30.660Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"claude-code","input":352,"models":{"claude-opus-5":49799},"output":49799,"source":"claude-jsonl","entries":11,"cache_read":5916205,"cache_write":100337},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe, job #25 (Lean 4 + Mathlib cache; about 20 seconds to build once the cache is present)\n\n1. Toolchain and project, as in job #22 (return #76):\n   ```\n   curl -sSf https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh | sh -s -- -y --no-modify-path --default-toolchain none\n   export PATH=\"$HOME/.elan/bin:$PATH\"\n   lake +leanprover/lean4:stable new alt22 math && cd alt22\n   lake exe cache get\n   ```\n   Pins are below. `lean-toolchain` is `leanprover/lean4:v4.33.1` (sha256 `3aac669c7a910ec2389f4e4f921b605adf6ebf2d1e0c9b9cd0be4d33f3f5db71`), and Mathlib is at `0df444a360eaa60ab8c11dca51a86af692955474`.\n2. Save the file below as `Growth.lean` (sha256 `3c66cec3e878b9add77df1f1b581f962331cfff9e3fb5a80b2500f64e5dbe90d`) and run `/usr/bin/time -p lake env lean Growth.lean`.\n   Expected: exit 0, and no Lean output (only the three timing lines), about 18 s on an Apple M1. `grep -c sorry Growth.lean` prints 0.\n3. Axioms: append `#print axioms ConsumerGrowth.<name>` for `w_pos`, `w_ratio`, `sum_reflect`, `succ_pow_le`, `sum_div_w_tendsto`, `w_tendsto_atTop`, `consumer_equiv`, then build with\n   `lake env lean GrowthAxioms.lean`. Expected output below (the log's last line `exit 0` was appended by the shell); there is no `sorryAx`.\n4. Negative control: in a copy, change `(𝓝 2)` to `(𝓝 3)` in the statement of `sum_div_w_tendsto`. Expected: exit 1 with\n   \"Type mismatch ... Tendsto ?m atTop (𝓝 2) but is expected to have type ... (𝓝 3)\", plus an unsolved goal `False` in `consumer_equiv`.\n\nWhat reviewers should check by eye: that the statements of `sum_div_w_tendsto` and `consumer_equiv` (lines 82-83 and 183-185) say the brief's (a) and (b).\nConstants: forward keeps c; backward, c' gives c'/4. Indices: (a) sums over `Icc j₀ j` with `1 ≤ j₀`; (b) sums over `range (j+1)`.\n\n## Growth.lean (full text)\n\n```lean\nimport Mathlib\n\n/-!\n# `∑_{i=j₀}^{j} 2^i/i^K = (2 + o(1)) 2^j/j^K`, and the (P) ⟺ (P') consumer equivalence\n\nJob #25 (formalize). `research/consumer-comparison.md` §2 item 3 and `research/TWIN-REDUCTION.md` §2 use this\nlemma to pass between two OPEN sufficient conditions. Proving it proves neither condition: it is a\nstatement about real sequences.\n\n* `w K i = 2^i / i^K` for `i ≥ 1` and a fixed real `K ≥ 0`. The value at `i = 0` is never used: every\n  statement below either starts its sums at `j₀ ≥ 1` or only uses `w` at large indices.\n* (a) `sum_div_w_tendsto`: for every `j₀ ≥ 1`, `(∑_{i ∈ [j₀, j]} w K i) / w K j → 2` as `j → ∞`.\n* (b) `consumer_equiv`: for `a i ≥ 0`, `(∃ c > 0, ∃ᶠ j, c·w K j ≤ a j) ↔ (∃ c > 0, ∃ᶠ j, c·w K j ≤ ∑_{i ≤ j} a i)`.\n  Forward uses the same `c`. Backward: (P') with `c'` gives (P) with `c = c'/4`.\n-/\n\nopen Filter Topology Finset\n\nnamespace ConsumerGrowth\n\n/-- `w K i = 2^i / i^K`. -/\nnoncomputable def w (K : ℝ) (i : ℕ) : ℝ := (2 : ℝ) ^ i / (i : ℝ) ^ K\n\nlemma w_pos (K : ℝ) {i : ℕ} (hi : 1 ≤ i) : 0 < w K i := by\n  have h : (0 : ℝ) < (i : ℝ) := by exact_mod_cast hi\n  exact div_pos (pow_pos two_pos i) (Real.rpow_pos_of_pos h K)\n\n/-- The ratio of a shifted term to the last one. -/\nlemma w_ratio (K : ℝ) {j m : ℕ} (h : m + 1 ≤ j) :\n    w K (j - m) / w K j = (1 / 2 : ℝ) ^ m * ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K := by\n  obtain ⟨a, rfl⟩ : ∃ a, j = a + m := ⟨j - m, by omega⟩\n  have ha : (0 : ℝ) < (a : ℝ) := by exact_mod_cast (by omega : 0 < a)\n  have hj : (0 : ℝ) < ((a + m : ℕ) : ℝ) := by exact_mod_cast (by omega : 0 < a + m)\n  have hsub : a + m - m = a := by omega\n  rw [hsub]\n  unfold w\n  rw [Real.div_rpow hj.le ha.le]\n  have h1 : (0 : ℝ) < (a : ℝ) ^ K := Real.rpow_pos_of_pos ha K\n  have h2 : (0 : ℝ) < ((a + m : ℕ) : ℝ) ^ K := Real.rpow_pos_of_pos hj K\n  rw [pow_add]\n  field_simp\n  rw [← mul_pow]\n  norm_num\n\n/-- Reflect the window sum so that the last term comes first. -/\nlemma sum_reflect (K : ℝ) {j₀ j : ℕ} (h : j₀ ≤ j) :\n    ∑ i ∈ Icc j₀ j, w K i = ∑ m ∈ range (j - j₀ + 1), w K (j - m) := by\n  refine Finset.sum_nbij' (fun i => j - i) (fun m => j - m) ?_ ?_ ?_ ?_ ?_\n  · intro i hi\n    simp only [Finset.mem_Icc, Finset.mem_range] at hi ⊢\n    omega\n  · intro m hm\n    simp only [Finset.mem_Icc, Finset.mem_range] at hm ⊢\n    omega\n  · intro i hi\n    simp only [Finset.mem_Icc] at hi\n    omega\n  · intro m hm\n    simp only [Finset.mem_range] at hm\n    omega\n  · intro i hi\n    simp only [Finset.mem_Icc] at hi\n    congr 1\n    omega\n\n/-- `(m+1)^n ≤ 2^n (m^n + 1)` for natural `m`, `n`. -/\nlemma succ_pow_le (m n : ℕ) : ((m : ℝ) + 1) ^ n ≤ 2 ^ n * ((m : ℝ) ^ n + 1) := by\n  rcases Nat.eq_zero_or_pos m with rfl | hm\n  · simp only [Nat.cast_zero, zero_add, one_pow]\n    have : (1 : ℝ) ≤ 2 ^ n := one_le_pow₀ (by norm_num)\n    rcases Nat.eq_zero_or_pos n with rfl | hn\n    · norm_num\n    · rw [zero_pow (by omega)]; linarith\n  · have hm1 : (1 : ℝ) ≤ (m : ℝ) := by exact_mod_cast hm\n    calc ((m : ℝ) + 1) ^ n ≤ (2 * (m : ℝ)) ^ n := by gcongr; linarith\n      _ = 2 ^ n * (m : ℝ) ^ n := by rw [mul_pow]\n      _ ≤ 2 ^ n * ((m : ℝ) ^ n + 1) := by\n          have : (0 : ℝ) ≤ 2 ^ n := by positivity\n          nlinarith\n\n/-- (a): the window sum is `(2 + o(1))` times its last term. -/\ntheorem sum_div_w_tendsto (K : ℝ) (hK : 0 ≤ K) (j₀ : ℕ) (hj₀ : 1 ≤ j₀) :\n    Tendsto (fun j => (∑ i ∈ Icc j₀ j, w K i) / w K j) atTop (𝓝 2) := by\n  set n := ⌈K⌉₊\n  let f : ℕ → ℕ → ℝ := fun j m =>\n    if m + j₀ ≤ j then (1 / 2 : ℝ) ^ m * ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K else 0\n  let bound : ℕ → ℝ := fun m => 2 ^ n * ((m : ℝ) ^ n * (1 / 2 : ℝ) ^ m + (1 / 2 : ℝ) ^ m)\n  have h_sum : Summable bound :=\n    ((summable_pow_mul_geometric_of_norm_lt_one n (by norm_num : ‖(1 / 2 : ℝ)‖ < 1)).add\n      summable_geometric_two).mul_left _\n  have hab : ∀ m : ℕ, Tendsto (fun j => f j m) atTop (𝓝 ((1 / 2 : ℝ) ^ m)) := by\n    intro m\n    have hq : Tendsto (fun j : ℕ => (m : ℝ) / ((j - m : ℕ) : ℝ)) atTop (𝓝 0) :=\n      (tendsto_const_div_atTop_nhds_zero_nat (m : ℝ)).comp (tendsto_sub_atTop_nat m)\n    have hr : Tendsto (fun j : ℕ => (j : ℝ) / ((j - m : ℕ) : ℝ)) atTop (𝓝 1) := by\n      have h1 : Tendsto (fun j : ℕ => 1 + (m : ℝ) / ((j - m : ℕ) : ℝ)) atTop (𝓝 1) := by\n        simpa using tendsto_const_nhds.add hq\n      refine h1.congr' ?_\n      filter_upwards [eventually_ge_atTop (m + 1)] with j hj\n      have hpos : (0 : ℝ) < ((j - m : ℕ) : ℝ) := by exact_mod_cast (by omega : 0 < j - m)\n      have hcast : ((j - m : ℕ) : ℝ) = (j : ℝ) - m := by\n        rw [Nat.cast_sub (by omega : m ≤ j)]\n      field_simp\n      rw [hcast]\n      ring\n    have hrK : Tendsto (fun j : ℕ => ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K) atTop (𝓝 1) := by\n      have := (Real.continuousAt_rpow_const 1 K (Or.inl one_ne_zero)).tendsto.comp hr\n      rw [Real.one_rpow] at this\n      exact this\n    have hlim : Tendsto (fun j : ℕ => (1 / 2 : ℝ) ^ m * ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K) atTop\n        (𝓝 ((1 / 2 : ℝ) ^ m)) := by\n      simpa using tendsto_const_nhds.mul hrK\n    refine hlim.congr' ?_\n    filter_upwards [eventually_ge_atTop (m + j₀)] with j hj\n    simp only [f, if_pos hj]\n  have h_bound : ∀ᶠ j in atTop, ∀ m, ‖f j m‖ ≤ bound m := by\n    refine Eventually.of_forall fun j m => ?_\n    simp only [f]\n    split_ifs with hmj\n    · have hjm : 1 ≤ j - m := by omega\n      have ha : (0 : ℝ) < ((j - m : ℕ) : ℝ) := by exact_mod_cast (by omega : 0 < j - m)\n      have hx0 : (0 : ℝ) ≤ (j : ℝ) / ((j - m : ℕ) : ℝ) := by positivity\n      have hxle : (j : ℝ) / ((j - m : ℕ) : ℝ) ≤ (m : ℝ) + 1 := by\n        rw [div_le_iff₀ ha]\n        have hcast : ((j - m : ℕ) : ℝ) = (j : ℝ) - m := by rw [Nat.cast_sub (by omega : m ≤ j)]\n        have h1 : (1 : ℝ) ≤ ((j - m : ℕ) : ℝ) := by exact_mod_cast hjm\n        rw [hcast] at h1 ⊢\n        nlinarith\n      have hK1 : ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K ≤ ((m : ℝ) + 1) ^ K :=\n        Real.rpow_le_rpow hx0 hxle hK\n      have hK2 : ((m : ℝ) + 1) ^ K ≤ ((m : ℝ) + 1) ^ (n : ℝ) :=\n        Real.rpow_le_rpow_of_exponent_le (by linarith [(Nat.cast_nonneg m : (0 : ℝ) ≤ m)])\n          (Nat.le_ceil K)\n      rw [Real.rpow_natCast] at hK2\n      have hK3 := succ_pow_le m n\n      have hg : (0 : ℝ) ≤ (1 / 2 : ℝ) ^ m := by positivity\n      have hnonneg : 0 ≤ (1 / 2 : ℝ) ^ m * ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K := by positivity\n      rw [Real.norm_of_nonneg hnonneg]\n      calc (1 / 2 : ℝ) ^ m * ((j : ℝ) / ((j - m : ℕ) : ℝ)) ^ K\n          ≤ (1 / 2 : ℝ) ^ m * (2 ^ n * ((m : ℝ) ^ n + 1)) := by\n            apply mul_le_mul_of_nonneg_left _ hg\n            linarith\n        _ = bound m := by simp only [bound]; ring\n    · simp only [norm_zero, bound]\n      positivity\n  have hT := tendsto_tsum_of_dominated_convergence h_sum hab h_bound\n  rw [tsum_geometric_two] at hT\n  refine hT.congr' ?_\n  filter_upwards [eventually_ge_atTop j₀] with j hj\n  have hwj : 0 < w K j := w_pos K (le_trans hj₀ hj)\n  rw [tsum_eq_sum (s := range (j - j₀ + 1))]\n  · rw [sum_reflect K hj, Finset.sum_div]\n    apply Finset.sum_congr rfl\n    intro m hm\n    rw [Finset.mem_range] at hm\n    simp only [f, if_pos (by omega : m + j₀ ≤ j)]\n    rw [w_ratio K (by omega : m + 1 ≤ j)]\n  · intro m hm\n    rw [Finset.mem_range] at hm\n    simp only [f, if_neg (by omega : ¬ (m + j₀ ≤ j))]\n\n/-- `w K j → ∞`. -/\nlemma w_tendsto_atTop (K : ℝ) : Tendsto (w K) atTop atTop := by\n  set n := ⌈K⌉₊\n  have h0 : Tendsto (fun i : ℕ => (i : ℝ) ^ n / (2 : ℝ) ^ i) atTop (𝓝[>] 0) := by\n    rw [tendsto_nhdsWithin_iff]\n    refine ⟨tendsto_pow_const_div_const_pow_of_one_lt n one_lt_two, ?_⟩\n    filter_upwards [eventually_ge_atTop 1] with i hi\n    have : (0 : ℝ) < (i : ℝ) := by exact_mod_cast hi\n    exact Set.mem_Ioi.mpr (by positivity)\n  have hinv := h0.inv_tendsto_nhdsGT_zero\n  refine tendsto_atTop_mono' atTop ?_ hinv\n  filter_upwards [eventually_ge_atTop 1] with i hi\n  have hi' : (1 : ℝ) ≤ (i : ℝ) := by exact_mod_cast hi\n  have hipos : (0 : ℝ) < (i : ℝ) := by linarith\n  simp only [Pi.inv_apply, inv_div]\n  unfold w\n  have hle : (i : ℝ) ^ K ≤ (i : ℝ) ^ (n : ℝ) := Real.rpow_le_rpow_of_exponent_le hi' (Nat.le_ceil K)\n  rw [Real.rpow_natCast] at hle\n  exact div_le_div_of_nonneg_left (by positivity) (Real.rpow_pos_of_pos hipos K) hle\n\n/-- (b): the (P) ⟺ (P') consumer equivalence. Forward keeps `c`; backward (P') with `c'` gives (P) with `c'/4`. -/\ntheorem consumer_equiv (K : ℝ) (hK : 0 ≤ K) (a : ℕ → ℝ) (ha : ∀ i, 0 ≤ a i) :\n    (∃ c > 0, ∃ᶠ j in atTop, c * w K j ≤ a j) ↔\n      (∃ c > 0, ∃ᶠ j in atTop, c * w K j ≤ ∑ i ∈ range (j + 1), a i) := by\n  constructor\n  · rintro ⟨c, hc, hfreq⟩\n    refine ⟨c, hc, hfreq.mono fun j hj => hj.trans ?_⟩\n    exact Finset.single_le_sum (fun i _ => ha i) (Finset.self_mem_range_succ j)\n  · rintro ⟨c, hc, hfreq⟩\n    refine ⟨c / 4, by positivity, ?_⟩\n    by_contra hnot\n    rw [Filter.not_frequently] at hnot\n    obtain ⟨N, hN⟩ := Filter.eventually_atTop.mp hnot\n    set N₁ := max N 1 with hN₁\n    have hsum := sum_div_w_tendsto K hK N₁ (le_max_right _ _)\n    have h3 : ∀ᶠ j in atTop, (∑ i ∈ Icc N₁ j, w K i) / w K j < 3 :=\n      hsum.eventually (gt_mem_nhds (by norm_num))\n    set A := ∑ i ∈ range N₁, a i\n    have hbig : ∀ᶠ j in atTop, 4 * A / c < w K j := (w_tendsto_atTop K).eventually_gt_atTop _\n    have hev : ∀ᶠ j in atTop, ∑ i ∈ range (j + 1), a i < c * w K j := by\n      filter_upwards [h3, hbig, eventually_ge_atTop N₁] with j hj3 hjb hjN\n      have hwpos : 0 < w K j := w_pos K (le_trans (le_max_right N 1) hjN)\n      have hsplit : ∑ i ∈ range (j + 1), a i = A + ∑ i ∈ Icc N₁ j, a i := by\n        rw [← Finset.sum_range_add_sum_Ico _ (by omega : N₁ ≤ j + 1)]\n        have hI : Finset.Ico N₁ (j + 1) = Finset.Icc N₁ j := by\n          ext i\n          simp only [Finset.mem_Ico, Finset.mem_Icc]\n          omega\n        rw [hI]\n      have htail : ∑ i ∈ Icc N₁ j, a i ≤ ∑ i ∈ Icc N₁ j, c / 4 * w K i := by\n        apply Finset.sum_le_sum\n        intro i hi\n        have hiN : N ≤ i := le_trans (le_max_left N 1) (Finset.mem_Icc.mp hi).1\n        exact (not_le.mp (hN i hiN)).le\n      rw [← Finset.mul_sum] at htail\n      have hS : ∑ i ∈ Icc N₁ j, w K i < 3 * w K j := by rwa [div_lt_iff₀ hwpos] at hj3\n      have hA : A < c / 4 * w K j := by\n        rw [div_lt_iff₀ hc] at hjb\n        linarith\n      have hmul : c / 4 * ∑ i ∈ Icc N₁ j, w K i < c / 4 * (3 * w K j) :=\n        mul_lt_mul_of_pos_left hS (by positivity)\n      rw [hsplit]\n      linarith\n    obtain ⟨j, h1, h2⟩ := (hfreq.and_eventually hev).exists\n    exact absurd h1 (not_le.mpr h2)\n\nend ConsumerGrowth\n```\n\n## Build output, `/usr/bin/time -p lake env lean Growth.lean > growth-build.log 2>&1` (growth-build.log, sha256 `b4a259e24e80c867f4caea53026547c0209dac4fdd8dc5a928be3685fa93378c`)\n\n```\nreal 17.74\nuser 4.53\nsys 11.78\n```\n\n## Axioms (growth-axioms.log, sha256 `58000fcd70bc3ae5602328c7c00a6ecfdeaef89deddc8c7bde18c82c9821647c`)\n\n```\n'ConsumerGrowth.w_pos' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ConsumerGrowth.w_ratio' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ConsumerGrowth.sum_reflect' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ConsumerGrowth.succ_pow_le' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ConsumerGrowth.sum_div_w_tendsto' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ConsumerGrowth.w_tendsto_atTop' depends on axioms: [propext, Classical.choice, Quot.sound]\n'ConsumerGrowth.consumer_equiv' depends on axioms: [propext, Classical.choice, Quot.sound]\nexit 0\n```\n\n## Negative control, full log (growth-negative.log, sha256 `2ab32760ec57514a4f48a46526e057d192e91fbccfd86a6d9c196d9d4b9c0fb0`)\n\n```\nGrowthNeg.lean:148:2: error: Type mismatch\n  Tendsto.congr' ?m.986 hT\nhas type\n  Tendsto ?m.983 atTop (𝓝 2)\nbut is expected to have type\n  Tendsto (fun j => (∑ i ∈ Icc j₀ j, w K i) / w K j) atTop (𝓝 3)\nGrowthNeg.lean:198:36: error: unsolved goals\nK : ℝ\nhK : 0 ≤ K\na : ℕ → ℝ\nha : ∀ (i : ℕ), 0 ≤ a i\nc : ℝ\nhc : c > 0\nhfreq : ∃ᶠ (j : ℕ) in atTop, c * w K j ≤ ∑ i ∈ range (j + 1), a i\nhnot : ∀ᶠ (x : ℕ) in atTop, ¬c / 4 * w K x ≤ a x\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → ¬c / 4 * w K b ≤ a b\nN₁ : ℕ := max N 1\nhN₁ : N₁ = max N 1\nhsum : Tendsto (fun j => (∑ i ∈ Icc N₁ j, w K i) / w K j) atTop (𝓝 3)\n⊢ False\nexit 1\n```\n\n## lake-manifest.json pins (sha256 `713e3b169ede20be0a2767072005a84a3527c83b748a9420ab769b8e8cd8f278`)\n\n```\nmathlib 0df444a360eaa60ab8c11dca51a86af692955474 v4.33.1\nplausible b7eb3304aeae834b12dda98993a37f6a41f6f0bb main\nLeanSearchClient 5f4d51b81cbd3f6b32b156bfad9056621a040404 main\nimportGraph 16f02aa7642864af59f1ff0e384a015994db9118 main\nproofwidgets 4be2e3d5087eeb272cf5a8853b8f9dd025ef5957 main\naesop 3448c0bcc5ce01b2d1546e483ec3620e32df3d0e master\nQq 92c15be17b7caf78c2ad767ec40f89052d908d81 master\nbatteries 4488d40d070b9700d4d5a6aa342f0d40c31b2a2d main\nCli 6130a47896ce867c6a4a55373441e59e565bad0f v4.33.0\n```","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0.04,"omitted":1,"outputs":25},"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":"2026-09-11T15:18:30.687Z","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`. This is a lemma about sequences; it converts one OPEN sufficient consumer into another equivalent OPEN one (`research/consumer-comparison.md` section 2 item 3, `research/TWIN-REDUCTION.md` section 2). Formalizing it proves neither consumer.\n\nStatements. (a) For fixed K >= 0 and w_i = 2^i / i^K (i >= 1), sum_{i=j_0}^{j} w_i = (2 + o(1)) w_j as j tends to infinity, for any fixed j_0 >= 1. (b) Let a_i >= 0 be a sequence. The following are equivalent with existential positive constants: (P) there is c > 0 with a_j >= c w_j for infinitely many j; (P') there is c' > 0 with sum_{i <= j} a_i >= c' w_j for infinitely many j. Forward is immediate from a_i >= 0. Backward, per the note: if a_i < c w_i / 4 for all large i then sum_{i<=j} a_i <= (c/4)(2 + o(1)) w_j plus a fixed initial sum, which is below c' w_j for c' > c/2 and large j, contradicting (P').\n\nWrite a Lean 4 file against Mathlib proving (a) with `Filter.Tendsto` (ratio of the sum to w_j tends to 2) and (b) as an iff between `∃ c > 0, ∃ᶠ j in atTop, ...` statements. The note's constants (c/4, c/2) are one choice; any correct choice is acceptable but state it. Watch j_0 = 1 versus i = 0 (w_0 is undefined for K > 0).\n\nReturn the `.lean` file, the toolchain, build output and the file's sha256 in `hashes`. Describe the result as a proven equivalence of two open conditions and nothing more. Grade `conjectured` if any sorry remains.","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/79/transcript","files":[{"sha256":"f872e7b7efb26355568f53c00bf049e9b5e071bcf4fbc17b7d2ce0d95811fefd","name":"report25.md","bytes":6696},{"sha256":"3c66cec3e878b9add77df1f1b581f962331cfff9e3fb5a80b2500f64e5dbe90d","name":"Growth.lean","bytes":11065}],"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":"2026-09-25T11:05:01.634Z","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":"2026-09-25T11:05:01.634Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"duplicates":[],"cited_messages":[]}