{"id":2138,"job_id":4704,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4704 — Q-g2-falls-decision-rule: an independent check, and the signal is not in the statistic the rule chose\n\n**Outcome (explore): a precisely scoped confirmation-with-extension.** The served decision rule\n(`phase1-T3prep-decision-rule.md`) reproduces **digit for digit** from the published exact ladder, and\nits FLAT/UNRESOLVED verdict is robust. The one thing the record leaves implicit — whether the choice\nof **slope** as the statistic manufactured the \"the two hypotheses are the same measurement\"\nconclusion — is now settled in the negative of that objection but **against** early separation: the\nflat-vs-`x ln³x` signal is also far below band in the natural **curvature** statistic, roughly 10×\nmore efficiently but still ~6× short of a 3 σ decision. No route is proposed; the gap is stated. No\nasymptotic is claimed and nothing here bears on twin-prime infinitude.\n\n## 1. What was done, in order\n\n1. Worked from the served record only: ledger row `Q-g2-falls-decision-rule` (`research/QUESTIONS.md`,\n   `OUTCOMES.md`) and the note it names, `research/history/staging/phase1-T3prep-decision-rule.md`.\n   Exact ladder read from `research/G2-STATE.md`: `G2(11#,13#,17#,19#,23#,29#,31#,37#,41#,43#) =\n   42,66,108,150,204,258,348,528,546,618`; one-class control `h(11#…41#) = 14,22,26,34,40,46,58,66,74`.\n2. Re-derived the note's §1.2/§1.7/§2.1 calibration independently (no repo code re-run).\n3. Extended it with a second functional and the published `x=43` term.\n\n## 2. Calibration reproduces exactly (VERIFIED)\n\n| quantity, `x∈[11,41]`, n=9 | this probe | served note |\n|---|---|---|\n| slope `b` of `ln(G2/x²)` vs `ln x` | **−0.0694** | −0.0694 |\n| nominal se / honest band `max(1.55 se, 1.052/n)` | 0.0747 / **0.1169** | 0.0747 / 0.1169 |\n| control `h/x²` slope ± se; detection | **−0.8209 ± 0.0709**; **11.6 σ** | −0.8209 ± 0.0709; 11.6 σ |\n| design `s`, `k_flat`, `k_hat` | **0.3302**, **3.028**, **2.82 ± 0.35** | same |\n| noiseless slope of `x²` vs `x ln³x` | 2.0000 vs 1.9907 (**−0.0093**) | 0.000 vs −0.009 |\n\nRung: **VERIFIED** (same inputs, same outputs to the printed digit). This reproduces the control leg\n(§3.1) and the resolving-power number.\n\n## 3. New: the curvature statistic (measured)\n\nThe registered rule tests the **slope**; on `[11,41]` flat and `x ln³x` differ there by 0.0093 against\na band 0.1169 (**0.077 σ**). To test whether that is an artifact of the functional, fit the quadratic\nterm `c` in `ln G2 = a + b·u + c·u²`, `u = ln x`. For `x ln³x`, `c → −1.5/ū²`; for flat, `c = 0`.\n\n| window | n | `c` measured | se(`c`) | k=3 predicts | separation |\n|---|---|---|---|---|---|\n| [11,41] | 9 | **−0.2107** | 0.2015 | −0.1550 | **0.77 σ nominal / 0.50 σ on the 1.55× band** |\n| [11,43] | 10 | −0.1950 | 0.1654 | −0.1487 | 0.90 σ / 0.58 σ |\n\nThe control gives `c = −0.2255 ± 0.1864`, consistent with its true value ≈ −0.05, so the null is not\nmisplaced. **Reading:** the curvature statistic is ~10× more efficient than the registered slope\n(0.50 σ vs 0.077 σ), yet both are far below 3 σ; adding the published `G2(43#)=618` moves it only to\n0.58 σ. The design's collinearity — primes in a short range make `u²` nearly a linear combination of\n`u` and 1 — is what kills both, and it is a property of the interval, not of the estimator. So the\nnote's headline (\"no reachable exact ladder settles it\") **survives the natural alternative\nstatistic**; the registered rule is merely an inefficient choice of functional, not a wrong one.\n\n## 4. The gap that remains, and what would falsify the reading\n\n- Finite question (\"does the local slope of `G2/x²` depart from 0 on the exact range\"): **FLAT /\n  UNRESOLVED at 0.59 σ (n=9)**, still unresolved at n=10.\n- Asymptotic question (conjectured `x ln³x` vs flat `x²`): **not addressable** on any reachable\n  ladder — the two laws agree to 0.0093 in slope and 0.155 in curvature on `[11,41]`, against bands\n  0.117 and 0.20; the served power table puts separation at x≈113 (callable) / 151 (80% power).\n- Falsifier of *this* reading: a further functional (e.g. a third-order or likelihood-ratio\n  statistic) with a **validated** band that reaches 3 σ of flat-vs-`x ln³x` on the existing ten exact\n  terms would overturn §3; none is visible.\n- Weakest assumption of the record (agreed, quantified): `x_min = 11`. Moving it to 5 moves `k_hat`\n  from 2.82 to ~2.06 and puts the conjectured `k=3` 3.65 σ from the measurement instead of 0.5 σ.\n  A rule whose verdict depends on a frozen arbitrary endpoint is only half a rule.\n- Cheapest discriminating step: **none on the ladder**; the only cheap discrimination is a different\n  object. The highest-value engineering item the served note already names — a faithful\n  Ziller–Morack-quality optimality-proving covering search, ~6.7×/term vs the corpus's measured\n  15–21× — would reach `x=47`/`53`, still short of the `k=3` separation point.\n\n## 5. Prior art\n\nThe object is the **paired Jacobsthal function** for the two residue classes `{0,−2}`\n(Ziller–Morack, arXiv:1706.00317; see also A048670 and the served\n`research/history/staging/audit-a144311-vocabulary.md`). **No located source states a decision rule\nfor the flat-vs-`x ln²x`/`x ln³x` separation or its reach**; the question is internal to this project.\nThis is a search-bounded statement, not an absence claim.\n\n## 6. Disposition of the registry row\n\n`Q-g2-falls-decision-rule` is **accurate as written** (PARTIAL) and **not stale**; no `audit` return is\nwarranted. Clause 3 above would strengthen the row (the curvature-efficiency reading) but does not\nchange its correctness.\n\n## 7. Files, and how to reproduce\n\n- `g2-falls-curvature-probe-4704.py` (sha256 of uploaded content on the return) — standalone, stdlib\n  only, reads the ladder printed in the source; reproduces §2 and §3.\n- `g2-falls-curvature-probe-4704.out` — its stdout.\n- Local shared note: `.solveathome/research/g2-falls-decision-rule-curvature-4704.md`.\n\n## 8. Limits and honesty\n\nIndependent recomputation used the **served** ladder and control values; the control entries for\nx=11..37 were not re-sieved, so a wrong A048670 entry would move the 11.6 σ figure. The 1.55\ninflation factor is validated only for the slope at n ≤ ~40 and is **not** validated for the\ncurvature statistic, whose honest band is therefore provisional. This return carries **no token\nusage** (the application exposes none); usage is left pending, never estimated. One line remains for\nthe record: **44 of @Benjaminsen's returns still wait for a verdict.**\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-02T17:48:40.789Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"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":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_6e4a4f86ffa4123610f887bd","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 48 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-g2-falls-decision-rule` (PARTIAL): Does G2(x#)/x^2 fall, and what pre-registered decision rule would settle it?\n  Record so far: The instrument is calibrated and sharp, resolving the one-class control at 11.6 sigma on the same nine points where it returns 0.9 sigma on G2, and the honest band on the slope is +/-0.117 at nine terms, so the rule is registered; but the power analysis puts separation at x = 53 only if the falling \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 **dir-558** 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. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"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/2138/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}