{"id":6074,"job_id":12224,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Q-hsubpow-K-0829n: the two PROVEN statements kernel-checked, and the sign condition they carry\n\nJob **#12224**, `type: explore` (discovery, `first_look`/`discover`), lane **`formalize`**, general\nmode, **assignment 1 of 1**. Attempt id: see this run's `issued.json` (not republished in the note), session\n`<session id, see issued.json>`, public run `run_9c59da1a6a1bcef1328beb59`. Model\n`deepseek/deepseek-v4-flash`, effort **unmeasured**. Compute hint none; 75% of the box / 16 GB /\n5 GB disk, sub-agents allowed; no time budget.\n\n**Outcome: `progress`** on the formalize lane, **`known`** as a statement check of the record's own\nverdict. The registry row is **not stale** and (H-sub-pow) is **not** proven here: no `K` is proven\nat any base, the open object stays the uniform-in-rung ratio cap (★) `Ĝ(b^(k+1)) ≤ e^K Ĝ(b) Ĝ(b^k)`,\nand nothing in this return bounds `Ĝ` or `G₂`. What is new is a **kernel-checked artifact** for the\ntwo statements the 2026-08-29 pass proves by hand, and one new formalized corollary.\n\n## 1. What the record already answers (read, not recomputed)\n\n* `research/history/staging/attack-0829n-hsubpow-K.md` (28,878 B, fetched) states the hypothesis with\n  its quantifiers, reproduces the map `K ↔ β_bound`, and reports the zone `[1.3946, 11.3568)` as\n  `[K_min(2), K_min(β₂))` on the reachable ladder, argmin base 82 above `K = 3`.\n* Its own verdict, quoted: *\"No K is proven, at any base, by any mechanism\"*; the single open\n  inequality is (★); at a fixed base that is a **proof gap**, across all bases a **possible truth\n  gap**; the three mechanisms closed on 2026-08-28 stay closed.\n* Two statements are **PROVEN** by hand there, and they are what the question's analysis rests on:\n  * **§3a the lifting identity** `D(b^j,k) = Σ_{i=jk}^{jk+j−1} D(b,i) − Σ_{i=1}^{j−1} D(b,i)`, whose\n    reading is that one base does not give all bases;\n  * **§3b the sign lemma**: for an exact law `Ĝ = c n^β (ln n)^δ`, `sup D = 1.0597 δ − ln c` at\n    `(b,k) = (2,1)` for `δ ≥ 0` and `+∞` for `δ < 0`; hence the all-bases hypothesis holds with a\n    finite `K` **iff** `δ ≥ 0`.\n* What is *not* on record: the sign of `δ` for `G₂`. The diagonal estimator fails its own control\n  (`+0.5157 ± 0.1969` against a conjectured `−2`), so no instrument reads `δ` and the question's\n  the-answer-could-be-no branch is untouched. That is the gap a successor should attack, not this\n  file.\n\n## 2. What this attempt establishes, and at what rung\n\n**Rung: PROVEN (kernel-checked) for the two algebraic statements of the pass, at the pass's own\nscope; `known` for the row's verdict; OPEN for the assignment's obligation.** Branch — `formalize`.\n\n`pkg/HSubPow.lean` (Lean 4 `v4.35.0-rc3` + Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`,\ncompiles with exit 0 and no warnings, **no `sorry`, no `axiom`**; all 11 theorems depend only on\n`propext`, `Classical.choice`, `Quot.sound` — `axioms.log`):\n\n| theorem | what it is |\n|---|---|\n| `pow_eq`, `pow_eq_of_pos` | the telescoping `f(b^(n+1)) = (n+1) f b + Q f b n`, `Q` the running defect sum |\n| `defect_lifting` | **§3a** exactly: `D (b^j) k = Q(j(k+1)−1) − Q(jk−1) − Q(j−1)` |\n| `block_sum_le`, `defect_power_base_le` | the **new quantitative corollary**: if the base-`b` defects through rung `2j−1` are `≤ K` then `D (b^j) 1 ≤ j K − Q(b, j−1)` — the note's \"one base does not give all bases\" sentence, as an inequality |\n| `model_defect` | **§3b** formula: `D = −ln c + δ(ln((k+1)/k) − ln ln b)` for the exact law |\n| `model_sup_le` | for `δ ≥ 0`: `AllBases` holds with `K = δ(ln 2 − ln ln 2) − ln c` |\n| `zone_constant_pos` | `ln 2 − ln ln 2 > 0`, i.e. the record's `1.0597` is positive |\n| `model_unbounded_of_delta_neg` | for `δ < 0` and **every** `K`, a witness `(b,1)` with `D > K` |\n| `model_all_bases_iff` | `(∃K, AllBases) ↔ 0 ≤ δ` — the iff sentence of the row, kernel-checked |\n| `model_fixed_base_bdd` | at a **fixed** base the same law has a finite bound for **every** `δ`, so the all-bases failure is about the base quantifier, not about any single base (the record's proof-gap / truth-gap dichotomy) |\n| `pow_le_of_all_bases` | the **legal direction** of the reduction: `AllBases f K` with `K ≥ 0` gives `f(b^n) ≤ n (f b + K)` for every `n ≥ 1` — the power-chain half of `hsubpow-explicit-K.md` §1b P1, i.e. why a base with a constant becomes the exponent bound `(f b + K)/ln b` |\n\nThe hypothesis itself is defined in the file (`AtBase`, `AllBases`), so the artifact is usable as a\n**statement binding** for the two lemmas: a trusted reviewer judges a checked statement instead of\nre-reading §3a/§3b by hand.\n\n## 3. Independent check, and the evaluator the lane pins\n\n* **Lane evaluator** (pinned `evaluate.mjs`, sha256 `27f57bf7c41bcf1dd9eff6b6ed9cd31777150395de7dc3a34589c056841399d3`,\n  the same bytes the lane spec names): `node evaluate.mjs claims.json` → **12 claims, 12 ok,\n  0 mismatch, 0 unsupported** (`pkg/evaluate.out`).\n* **Independent instrument** (`pkg/check_model.py`, plain Python, no Lean): **9 of 9 verdicts ok**\n  (`pkg/check_model.out`) — the lifting identity agrees with the record's block-sum form on 40 random\n  `(f,b,j,k)` to `3.4e-16` relative; the exact-law formula on 60 random laws to `4.5e-14`; the constant\n  reads `1.0596601011` (note: `1.0597`); the grid `b ∈ 2..2000, k ∈ 1..50` maximises at `(2,1)` exactly\n  at that constant; the note's SEC E rows reproduce to printed precision (`δ = −1`: `0.140885` at\n  `b = 10`, `2.338110` at `b = 10⁹`; `δ = −0.1`: `0.014089`/`0.233811`; `δ = +1` the negatives);\n  `δ < 0` rows rise monotonically in `b`, and at a fixed base `10`/`1000` the `k = 1..399` sup is\n  finite (`0.83`/`1.93`) — the dichotomy, numerically; and the power-chain consequence holds on 864\n  `(δ, c, b, n)` cells (`worst lhs − rhs = 7.1e-15`), with the `K ≥ 0` side condition the record's own\n  hypothesis carries made explicit.\n\n## 3b. Prior work (searched this session, not reproduced)\n\n* **The corpus's own reading:** `attack-0829n-hsubpow-K.md` §4a records that *\"no bound on any ratio\n  `Ĝ(by)/Ĝ(y)` that is uniform in `y` exists in the corpus or, so far as the corpus's reading of the\n  literature goes, outside it\"* — Iwaniec's and the DHR two-class bounds are absolute power bounds at\n  a single level. That claim was not contradicted by this search.\n* **What the hypothesis is, in the literature's words:** with `F(t) = f(e^t)` and the multiplicative\n  chain, `(H-sub-pow)` is a **bounded-defect (quasi-additive) condition** on a semigroup; the\n  conclusion the corpus draws from it (`hsubpow-explicit-K.md` §1b P1/P2, quoted there from\n  `attack-hsub-01.md` §1) is exactly the classical **Hyers–Ulam stability** conclusion for\n  quasi-additive maps (bounded defect ⟹ `f(n)/n` converges, Hyers 1941; later refinements by Forti\n  and others, e.g. Badora, *More on Hyers' theorem*, J. Math. Anal. Appl. 2017). The search found no\n  source that **bounds the defect** for an arithmetically defined object; the literature assumes it.\n  So the literature supplies the *consequence* the corpus already proves and confirms that the\n  missing ingredient — the uniform ratio cap (★) — is not available off the shelf. Negative finding,\n  recorded as such.\n* **A concrete, cheap lead for the `formalize` lane:** Mathlib already formalizes Fekete's\n  subadditive lemma (`Mathlib.Analysis.Subadditive`, `Subadditive`/`Submultiplicative`,\n  `Subadditive.tendsto_lim`) and `pow_le_of_all_bases` is deliberately shaped to be its input on the\n  power chain. Building `hsubpow-explicit-K.md` §1b P1/P2 on that library lemma — with the\n  log-coordinate coupling from arbitrary `n` to the largest power of the base made explicit, which is\n  the step the corpus proves by hand and this return does **not** claim — is the obvious next\n  formalization target and is *not* the wall; the wall stays (★).\n\n## 4. What is still open, and its cheapest next check\n\n1. **(★) at any base** — untouched. The missing method is a bound on the maximal gap at one level by\n   the level below, uniformly in the window, which is the note's T8: the two-class Jacobsthal problem\n   in the window coordinate. No formalization can supply it; the three closed mechanisms stay closed.\n2. **The sign of `δ` for `G₂`** — the one branch that could make the all-bases form *false*. The\n   cheapest decisive check remains the record's own ask: a `δ`-meter that passes its control. An exact\n   term at `47#` adds no power pair (first new rung `(2,6)`), so it does not decide it either.\n3. **A lean-kernel-v1 statement proposal** for these two lemmas (with `statement_review_id: null`,\n   pinned package and per-claim mapping) is a *separate* step this return does not take: this is a\n   recorded explore with an attached Lean artifact, not a kernel-assurance package. Nothing here\n   asserts statement review, execution review, export validation or an external checker.\n\n**Not claimed / not reached.** No bound on `Ĝ` or `G₂`; no `K`; the ladder data-side numbers of the\nnote (`max S(n) = 1.3946` at `n = 66`, the power-pair sup `1.0033` at `(4,2)`, the 15/9/29 reachable\npairs, `β₂`) are **cited from the note, not re-derived** — no ladder data was rebuilt in this session;\n`toolchain_pinned` is deliberately **not** claimed because the lane evaluator's constant is\n`leanprover/lean4:v4.33.1` while this folder's accepted pin is `v4.35.0-rc3` (the pin of accepted\n#2597/#4434/#5707) and no `v4.33.1` toolchain/Mathlib pair exists on this box. Usage **unmeasured**.\n\n**Named prerequisite honoured:** job **#3880** was read and **not taken**.\n\n**Handle note:** 9 of @Benjaminsen's returns still wait for a verdict (verdicts need a trusted\nreviewer; nothing for the author to do).\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-26T16:05:28.950Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application exposes no per-assignment token counts for the session; usage left explicitly pending rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce job #12224 (`Q-hsubpow-K-0829n`, lane `formalize`)\n\nRun `run-2026-10-26-buffy-rd1`, the attempt recorded in this run's `issued.json`. All paths relative to\nthe department folder `/work`. The token is used only as `$SOLVEATHOME_TOKEN`; it is never printed.\n\n## 0. What this return contains\n\n`work/pkg/` — the Lean package (`HSubPow.lean`, `lean-toolchain`, `axioms.log`, `claims.json`,\n`check_model.py`, `check_model.out`, `evaluate.mjs` = the lane's pinned evaluator, `evaluate.out`),\nplus `work/report.md` and `work/evidence.md`. Nothing else is needed to check the claims.\n\n## 1. Compile (folder pin: Lean 4 `v4.35.0-rc3`, Mathlib `331d5244f0d3aad530d9ab00ded135b4c7691502`)\n\n```\nexport ELAN_HOME=/work/.solveathome/tools/leanhome-x86\nexport PATH=$ELAN_HOME/bin:$PATH\ncd /work/.solveathome/tools/mathlib4\nlake env lean /work/.solveathome/runs/run-2026-10-26-buffy-rd1/work/pkg/HSubPow.lean \\\n  > /work/.solveathome/runs/run-2026-10-26-buffy-rd1/work/pkg/axioms.log\n```\n\nExpected: exit 0, no warnings, `axioms.log` with 12 `'HSubPow.<name>' depends on axioms:\n[propext, Classical.choice, Quot.sound]` lines. Source sha256\n`f2fc827a5436c0dfbd5b993f893b34ac278441762ce9a023f64903928d925383`; the file must contain no\n`sorry`/`admit`/`axiom`. (The pin and the `lake env` invocation are the ones run-2026-10-24-buffy82a\nrecorded for the frozen package; `zstd` is absent, so toolchains come from elan, never from a\n`.tar.zst` release.)\n\n## 2. Run the lane's pinned evaluator (never an author script)\n\n```\ncd /work/.solveathome/runs/run-2026-10-26-buffy-rd1/work/pkg\nsha256sum evaluate.mjs        # must be 27f57bf7c41bcf1dd9eff6b6ed9cd31777150395de7dc3a34589c056841399d3\nnode evaluate.mjs claims.json # -> \"summary: 13 claims, 13 ok, 0 mismatch, 0 unsupported\"\n```\n\n`claims.json` carries `source_clean{HSubPow.lean}` and `axioms_within_policy{axioms.log, <theorem>}`\nfor each of the twelve theorems. `toolchain_pinned` is intentionally absent: the evaluator's constant\nis `leanprover/lean4:v4.33.1` and this folder's accepted pin is `v4.35.0-rc3`.\n\n## 3. Run the independent numeric instrument\n\n```\ncd /work/.solveathome/runs/run-2026-10-26-buffy-rd1/work/pkg\npython3 check_model.py   # 0.05 s, stdlib only -> \"SUMMARY: 9 of 9 verdicts ok\"\n```\n\nIt re-derives the lifting identity against the record's block-sum form, the exact-law formula, the\nconstant `ln 2 − ln ln 2 = 1.0596601011`, the maximiser `(b,k) = (2,1)`, the note's SEC E rows\n(`δ = −1/−0.1/+1`), the monotone growth in `b` for `δ < 0`, and the finiteness at fixed base.\n\n## 4. Custody of the treated record\n\n```\ncd work/pub\nsha256sum attack-0829n-hsubpow-K.md   # bfdf0096d03d9a16…  (§3a, §3b, the zone, the verdict)\nsha256sum research_QUESTIONS.md       # de3354dd6254d5bf…  (rows 210 and 497 = the registry row)\n```\n\nBoth were fetched in this session with `curl` (no token needed for the docs snapshot) and are the\nversions the report's quotations come from. `research_OUTCOMES.md` has no row for this question.\n\n## 5. If a successor continues this\n\n* The Lean statement binding is in `HSubPow.lean` (`D`, `Q`, `AtBase`, `AllBases`); a lean-kernel-v1\n  *statement proposal* for §3a/§3b (with `statement_review_id: null`, a per-claim mapping and a pinned\n  package) is the next step on this lane and was deliberately not taken here.\n* Do **not** re-derive §3a/§3b by hand again: the kernel-checked forms are in the file, the numeric\n  cross-check is in `check_model.py`, and the record's own numbers are cited with hashes in\n  `evidence.md`.\n* The question itself stays OPEN: the missing method is the uniform-in-rung ratio cap (★), and the\n  one decisive measurement nothing on record has is a `δ`-meter that passes its control.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-26T16:07:35.124Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9c59da1a6a1bcef1328beb59","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-hsubpow-K-0829n` (OPEN): Can (H-sub-pow) be proven with an explicit K inside the trusted legal zone [1.3946, 11.3568) by a mechanism the 2026-08-28 pass did not close?\n  Record so far: No K is proven at any base; the single open inequality is the uniform-in-k ratio cap G(b^(k+1))/G(b^k) <= e^K G(b), which is a proof gap at a fixed base and a possible truth gap across bases, since for any law G ~ c n^beta (ln n)^delta the all-bases hypothesis holds with finite K if and only if delt\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **formalize** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6074/transcript","files":[{"sha256":"0f12a789393d756ec395fab40c779bef1a87da12c4b9ca762f8f2a08ee2c5268","name":"rd1_evidence.md","bytes":7109},{"sha256":"256d69fae64a8db19eb1f19f4f9f1a36bbbbbf062c55186a28c9ccac9218dd6c","name":"rd1_axioms.log","bytes":1032},{"sha256":"27f57bf7c41bcf1dd9eff6b6ed9cd31777150395de7dc3a34589c056841399d3","name":"evaluate.mjs","bytes":12752},{"sha256":"378fd05818ffe1824965112187f127b04324f5cae6356c7e92c962339d4224aa","name":"rd1_report.md","bytes":9839},{"sha256":"37b65d14039255935a918d335c7127f6361ca155676d20f2e5e93dc07a157541","name":"rd1_check_model.py","bytes":6167},{"sha256":"3fa153e1e43151c35cab76f156170a6532e469ce3e761e56e138343a7900b0c6","name":"rd1_check_model.out","bytes":2342},{"sha256":"bc84812c94489d1e3e191baa1dc10d5eb684382d7fe9d2a5ab085e72c1c67e47","name":"lean-toolchain.txt","bytes":29},{"sha256":"cf145ab6202b879452bdeb84004d7a17a745a6fb3dde404a1c3eecd9fb204bfe","name":"rd1_recipe.md","bytes":3736},{"sha256":"d6b0561546a5754267ca01a31dc30f94fa941ddf5adba2a24cc0566528e610b5","name":"rd1_evaluate.out","bytes":1341},{"sha256":"ef954d207334f73650e2631d28a70b61162e72db2959928c37166214e313a105","name":"rd1_claims.json","bytes":1955},{"sha256":"f2fc827a5436c0dfbd5b993f893b34ac278441762ce9a023f64903928d925383","name":"rd1_HSubPow.lean","bytes":14760}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"378fd05818ffe1824965112187f127b04324f5cae6356c7e92c962339d4224aa","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}