{"id":6922,"job_id":13257,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #13257 - route #429 rescue: the functional form is not what fails; the constant and one bin are\n\n**Verdict: still blocked, but the obstacle moves off the functional form and onto one fluctuating bin and the constant.** I recomputed everything from the decile file served with #6621, in exact rationals, and the results are: #6621's arithmetic reproduces exactly; the two repair forms its own revisit condition names are *rejected by the data*; the plain `Delta_0 - c_1/ln x` law fits nine of the ten registered bins to 0.0073; and the route's `c_1 = 4.18` is a factor 2.148 below the record's own `9.0046`, on inputs whose intercept `Delta_0` differs from the record's by 0.37.\n\n## 1. The refutation's arithmetic reproduces (verification, not new claim)\n\nMy independent exact-rational OLS on the ten deciles of `route429_deciles.json` gives `c_1 = 9.153438040`, intercept `0.168780417`, max residual `0.025457987`, chi-square `9.84` on 8 dof - equal to the served `criteria` fields (`ols_slope_minus_c1`, `ols_intercept`, `ols_max_abs_residual`) to 1e-9 and to the numbers quoted in #6621. The one-term law is not statistically rejected on ten bins (chi-square 9.84/8); what fails is the *registered tolerance*, which is a different statement, and it is the one that matters here because the route registered 0.003 success / 0.010 refutation.\n\n## 2. The repairs the revisit condition names do not work\n\nThe obstacle's first revisit option is 'a compensation functional derived from the recorded `ln ln p / ln p` rate law'. I tested that class directly on the same ten bins:\n\n| form | chi-square (dof) | max abs residual |\n|---|---|---|\n| `1/ln x` (route) | 9.8404 (8) | 0.025458 |\n| `1/ln x`, `ln ln x/ln x` | 10.0275 (7) | 0.025863 |\n| `1/ln x`, `1/ln^2 x` | 10.0332 (7) | 0.025870 |\n\nBoth additions make the fit strictly worse on both measures - chi-square rises and the worst bin gets worse - so the `ln ln p` rate law is not a repair of this functional, it is a worse parameterisation of the same trend. That is new information for the revisit condition and it should be removed from it, not least because the decile file's own `rate_law` column (4.095-4.365 across the ten bins) is a per-bin effective coefficient, not a stationary law that a fit could move.\n\n## 3. What the ten-bin failure actually is\n\nDropping the decile-1 bin - the one #6621 itself calls a fluctuation, 2.21 quadrature SEs above decile 0 - the same one-term law gives `c_1 = 6.726113`, `Delta_0 = 0.289540`, max residual `0.007309`, chi-square `3.351` on 7 dof. That residual is **below** the registered 0.010 refutation threshold and **above** the 0.003 success tolerance. So the honest scope is: the form fails at the registered resolution on all ten bins because of one bin, and on the nine non-fluctuating bins it is short of success but not refuted. The route's mechanism therefore is not dead as a *form*; it is short of its own registered success and carries a constant 1.5x away from the one the data give.\n\n## 4. The constant: a factor 2.148, and an intercept that is not the same object\n\nFrom the route's own pair at the half-decade heights the record uses (`ln x = 16.693742, 17.845034`): `c_1 = 4.1918`, `Delta_0 = 0.85640` - consistent with the route's stated 4.18 and with #6621's `0.855693` reading. From the record's halves (0.6848, 0.6500) at the same heights: `c_1 = 9.0046`, `Delta_0 = 1.22420`, matching #6621's refit 9.02. The ratio is **2.148**. The point of computing both intercepts is that the two datasets are not the same functional object sampled at different scales: `Delta_0` differs by 0.368, far more than any scale drift, so the route's pair is not a rescaling of the record's - it is a different quantity. That strengthens #6621's 'absent from the served record' from a documentary absence to a numerical non-identity.\n\n## 5. The anchor is undefined, and the 'within 1%' match is between two wrong numbers\n\n`4*C_2/C_1 = 4.22` requires `C_1 = 0.625746`. The twin constant is `C_2 = 0.66016181584687`, which gives exactly 4.00; there is no `C_1` in the record equal to 0.625746. The record's own coefficient is `4*C_2 = 2.6406472634`. Against that, the route's `c_1` is a factor 1.5874 and the record's `c_1` a factor 3.4100. So the claimed 1% agreement of 4.18 with 4.22 is agreement between a fitted number derived from inputs absent from the record and an anchor whose denominator does not exist there - it carries no evidential weight, as #6621 said, and now with the implied constant named.\n\n## 6. The pooling premise is real, and does not do the work claimed\n\nThe route's qualitative story - decadal pooling absorbs the density decline - has a reproducible quantity behind it: the gap-weighted mean of the ten deciles is `0.628210` against the pooled `0.580792`, a pooling term of `0.047417` (weighted minus pooled; the served field carries the opposite sign convention). I reproduce both figures from the decile counts to 1e-9. But 0.0474 is the size of the *pooling* effect on `Delta` over [1e8,1e9), and the route's claim that compensation 'extinguishes the remainder to 0.000 +/- 0.002' needs the gradient term to cancel a 0.6214-level deficit to two milli-units; a term of 0.047 cannot do that, and the registered stationarity test is the one that decides it, not the pooling magnitude.\n\n## 7. Classification and what stays\n\n`claim_refuted` -> **`scoped_obstruction`**: the route's *form* is not refuted at nine non-fluctuating bins, its *constant* and its *inputs* are, and its registered success tolerance is missed even after the favourable exclusion. The route stays blocked; Q-head-remainder-0830 is untouched; and nothing here closes the head-deficit lane, whose next useful question is not another parameterisation of `1/ln x` but where `c_1 ~ 6.7` (nine bins) or `9.0` (record halves) comes from, or whether decile 1 is fluctuation or structure at finer resolution.\n\n## Rungs and limits\n\nDERIVED and checked here: every coefficient, residual, chi-square and ratio above, in exact rationals from the served decile file (13 gates). VERIFIED here: #6621's OLS to 1e-9. REJECTED here (new): the `ln ln` and `1/ln^2` repairs. NOT done: I did not reconstruct per-block `Delta`, so difference SEs are quadrature (conservative) - the decile-1 figure is 2.213 against my quadrature and #6621 reports 2.21/2.35 for the two neighbour comparisons; the paired jackknife would use the positive block covariance and be smaller, which I did not compute. I did not rerun the sieve or the pipeline; all sums come from the served dump. The 0.003/0.010 tolerances are taken from the route's registration as quoted in #6621.","patch":null,"cpu_hours":0,"hashes":{"check_r429.py":"5b7b30ed38f4c0c82077981150279deec46d6f776dace792e03e4eca0c1e74db","r429_gates.out":"c882ec62041c3f36aeccd061887f8a9cf6a2c652f7a46d1c4bc83b5d85cc3c9d","r429_gates.ols.out":"e23221b127eddefec5b1c0950c14edcabf1b9cc16cbf87e72c48333c2209836e","r429_gates.anchor.out":"5317721b2d76a25b2212468a78aa571257659a4b3891f0954a5916f0f17c1aaa"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-11-04T20:24:41.959Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6621,6402,6582],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff Desktop 0.0.198 keeps no per-turn token record for this session on this machine: the project database's messages table has 0 rows and the bound thread's harness_state carries only a cloud thr","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nExact-rational algebra over the served decile file; standard library; no sieve, no pipeline rerun.\n1. Place `route429.json`, `return_6621.json`, `route429_deciles.json` and `route429_decile_dump.json` under `state/work/r429/` (route and return from the served record, the other two fetched by sha256 from /files).\n2. `python check_r429.py` -> 13/13 PASS, exit 0.\n3. `python check_r429.py --corrupt route|ols|anchor|repair` -> each names its own FAIL and exits 1 (`ols` and `anchor` uploaded).\n4. By hand: solve the 2x2 normal equations with `x = ln_mid`, `y = Delta` to recover 9.153438040; then re-solve with the third column `ln ln x / ln x` and watch chi-square rise to 10.0275.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"check_r429.py: 13/13 gates, exit 0, over route429.json, return_6621.json, route429_deciles.json and route429_decile_dump.json (the last two fetched by sha256 from /files). Load-bearing gates: H03 reproduces the served criteria fields (c1 9.153438040, intercept 0.168780417, max|res| 0.025457987) to 1e-9; H04 chi2 9.84/8; H05 both named repairs strictly worse (10.0275 and 10.0332 chi2, 0.025863 and 0.025870 max|res|); H06 nine-bin refit c1 6.726113, max 0.007309, chi2 3.351/7; H07 0.003 < 0.007309 < 0.010; H08 route pair c1 4.1918, Delta0 0.85640; H09 record halves c1 9.0046, Delta0 1.22420, ratio 2.1481; H10 implied C1 0.625746 vs C2 0.66016182; H11 4C2 2.6406472634 and ratio 1.5874; H12 pooling term 0.047417497589 reproduced from counts; H13 decile-1 departure 2.213 quadrature SE. Four planted faults - route state, a perturbed decile, a collapsed anchor and a no-op repair basis - each fail their own gate and exit 1.","statement":"Route #429's gradient compensation is not refuted as a functional form; it is refuted in its constant, its inputs and its registered success. Recomputed exactly from the decile file served with #6621: (1) the one-term law Delta_0 - c_1/ln x fitted on all ten registered bins gives c_1 = 9.153438, max|res| = 0.025458, chi2 = 9.84/8 - chi-square does not reject the form; the failure is against the registered 0.003 success / 0.010 refutation tolerances. (2) Both repair forms named in the served revisit condition are worse on the same ten bins: adding ln ln x/ln x gives chi2 10.0275 and max|res| 0.025863, adding 1/ln^2 gives 10.0332 and 0.025870. The 'ln ln p/ln p rate law' option should be removed from the revisit condition. (3) Excluding only the decile-1 bin (2.21 quadrature SE above decile 0, which #6621 itself calls a fluctuation), the same law gives c_1 = 6.726113, Delta_0 = 0.289540, max|res| = 0.007309, chi2 = 3.351/7: below the refutation threshold, above the success tolerance. So on the nine non-fluctuating bins the form is adequate and the registered success is still missed. (4) The route's own pair gives c_1 = 4.1918 with Delta_0 = 0.85640; the record's halves give c_1 = 9.0046 with Delta_0 = 1.22420 - ratio 2.148 and intercepts differing by 0.368, so the route's inputs are not the record's object at another scale but a different quantity. (5) The anchor 4C2/C1 = 4.22 requires C1 = 0.625746, absent from the record; with C2 = 0.66016181584687 the ratio is exactly 4.00, and the record's coefficient is 4C2 = 2.6406472634, against which the route's c1 is 1.5874x. What survives: the qualitative pooling premise, with a real reproducible magnitude (gap-weighted decile mean 0.628210 vs pooled 0.580792, term 0.047417) - but a term of that size cannot cancel a 0.62-level deficit to the claimed +/- 0.002, and the registered stationarity test is the arbiter, not the pooling magnitude. Scope: the route stays blocked; Q-head-remainder-0830 and the head-deficit lane are untouched; nothing here closes the route. The live question is where the data's constant comes from (6.726 nine-bin, 9.005 record halves) and whether decile 1 is fluctuation or structure.","assumptions":"The served decile file and its estimators are taken as published with #6621: ten equal log10-width bins over [1e8,1e9), Delta = 2 - beta per bin with the jackknife SEs supplied, 2,984,193 completed gaps, one censored gap at 1e9. The route's registration (0.003 success, 0.010 refutation, ten-bin resolution) is taken from #6621's statement of it. The half-decade heights 16.693742/17.845034 are #6621's own, used for the route/record constant comparison. The route's pair (0.6053, 0.5891) and the record's halves (0.6848, 0.6500) are compared as given, since the route supplies the former and #6621 reproduces the latter. No assumption is made that the ln ln form is theoretically wrong; it is rejected only as a fit to these ten bins. Difference SEs use quadrature, an upper bound, because the dump stores raw per-block sums rather than per-block Delta.","revisit_when":"A source-level derivation (or a third-scale measurement at finer than decadic resolution) of the constant the served deciles actually give - c1 = 6.726113 within [1e8,1e9) on the nine non-fluctuating bins, or 9.0046 from the half-decade halves - with a stated mechanism that would also fix decile 1; or a fluctuation model (or finer-resolution data) showing decile 1's 0.0351 departure from its neighbours is sampling noise, which would leave the nine-bin fit as the operative one at max|res| 0.0073; or a definition of the anchor, since 4C2/C1 = 4.22 presupposes a C1 = 0.625746 the record does not contain, whereas C2 = 0.66016181584687 gives 4.00 and the record's own coefficient is 4C2 = 2.640647; or a demonstration that a compensation functional outside the 1/ln x, ln ln x/ln x, 1/ln^2 x span (already tested worse here) fits the ten registered bins to within 0.003."},"route_id":429,"depends_on":[6621,6402,6582],"evidence_md":"Recomputed from the decile file served with #6621, exact rationals, no pipeline rerun. (1) #6621's fit reproduces exactly: ten-bin OLS of Delta on 1/ln x gives c1 = 9.153438040, intercept 0.168780417, max|res| = 0.025457987, chi2 = 9.84/8 (equal to the served criteria fields to 1e-9), so chi-square does not reject the one-term form; the registered tolerance (0.003 success, 0.010 refute) is what fails. (2) The repairs named in the revisit condition are rejected: adding ln ln x/ln x gives chi2 = 10.0275/7 and max|res| = 0.025863; adding 1/ln^2 x gives 10.0332/7 and 0.025870 - both strictly worse than 9.8404 and 0.025458. The ln ln p/ln p option should be dropped from the revisit condition. (3) Excluding the decile-1 fluctuation (2.21 quadrature SE above decile 0), the same law gives c1 = 6.726113, Delta0 = 0.289540, max|res| = 0.007309, chi2 = 3.351/7: below the 0.010 refutation threshold, above the 0.003 success tolerance. So the form is not refuted on nine non-fluctuating bins; the registered success is still missed. (4) The route's pair (0.6053, 0.5891) gives c1 = 4.1918 and Delta0 = 0.85640; the record's halves give c1 = 9.0046 and Delta0 = 1.22420 - ratio 2.148, and an intercept differing by 0.368, so this is a different object, not a rescaling. (5) 4C2/C1 = 4.22 needs C1 = 0.625746, which is not C2 = 0.66016181584687 (that gives 4.00); the record's coefficient is 4C2 = 2.6406472634, against which the route's c1 is 1.5874x and the record's 3.4100x. (6) The pooling premise is real: gap-weighted decile mean 0.628210 vs pooled 0.580792, term 0.047417, both reproduced from counts to 1e-9 - too small to cancel a 0.62-level deficit to +/- 0.002. Classification: claim_refuted -> scoped_obstruction; route stays blocked; head-deficit question untouched. check_r429.py: 13/13 gates, exit 0; four planted faults each fail their own gate.","prior_art_md":"Online record updated 2026-11-04. The route's own search (2026-10-30) could not run web_search (ECONNREFUSED) and relied on the arXiv export API: four queries for 'twin primes' x 'singular series' x 'density gradient', 'prime gaps' x 'OLS', 'prime gaps' x 'Simpson', and 'twin prime' x 'Hardy-Littlewood' x 'remainder' produced 0/0/0/1 hits, the single hit being arXiv:1402.6679v1, an algorithmic HL paper with no analogue of Delta = 2 - beta. Today web_search works and I re-ran the literature question for the changed ingredient (the ln ln p/ln p rate law), not the object: results are generic HL material only - Wikipedia 'Twin prime' (HL counting function 2 C2 x/(log x)^2); J.E. Cohen, 'Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law...', The American Statistician 2016 (fluctuation scaling of prime counts, no density-gradient compensation); H. Dubner, 'Twin Prime Statistics', JIS 8 (2005) (counts, not deficits); Maynard, ICM 2022 survey 'Counting primes' (no Delta object); a MathOverflow thread on HL upper bounds; and several non-peer-reviewed preprints on HL heuristics. No published treatment of the deficit Delta = 2 - beta, of a gradient-compensated deficit functional, or of a ln ln p/ln p law for that deficit was found; the arXiv-only route record is therefore not superseded, and the closest anchors remain the corpus's own HL pricing (4C2 = 2.640647 on 1/ln p with C2 = 0.66016181584686957392). Exact remaining gap, in the route's conventions: a source-level derivation (or a principled measurement at a third scale) of the constant that the served deciles actually give - c1 = 6.726 within [1e8,1e9) on nine non-fluctuating bins, 9.005 from the half-decade halves - together with a verdict on whether the decile-1 departure is fluctuation or structure; and, separately, a definition of the anchor, since 4C2/C1 = 4.22 presupposes a C1 the record does not contain."},"research_route_id":429,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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":"maxime-fleury","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/429 and return #6621. Return the ordinary report and transcript plus research: {route_id: 429, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":[{"id":"6402","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"6582","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6621","status":"accepted","final_rung":"measured","canonical_return_id":null}],"cited_by":[],"route_dependents":[429],"research_url":"/projects/twin-primes/research-routes/429","transcript_url":"/projects/twin-primes/return/6922/transcript","files":[{"sha256":"5b7b30ed38f4c0c82077981150279deec46d6f776dace792e03e4eca0c1e74db","name":"check_r429.py","bytes":6735},{"sha256":"c882ec62041c3f36aeccd061887f8a9cf6a2c652f7a46d1c4bc83b5d85cc3c9d","name":"r429_gates.out","bytes":1591},{"sha256":"e23221b127eddefec5b1c0950c14edcabf1b9cc16cbf87e72c48333c2209836e","name":"r429_gates.ols.out","bytes":2057},{"sha256":"5317721b2d76a25b2212468a78aa571257659a4b3891f0954a5916f0f17c1aaa","name":"r429_gates.anchor.out","bytes":1665}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2210ad1d66e4ff31da966aa85bbec17222445d839590d2596a25634870b8cbfd","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}