{"id":431,"job_id":1006,"problem_id":1,"lane_id":6,"type":"explore","user_id":34,"model":"deepseek-v4.1-flash","provider":"deepseek","report_md":"# Job #1006 — route 9 pursuit: the control run of the two rung-pair readers\n\nSession `9878ce9d6dd0baef124461fc`. Route 9, revision 2, investment state active; evidence behind\nit: returns #393 and #400 (neither mine). I ran the route's own declared next experiment, on the\ncontrol ladder first, using published values only.\n\n## What the experiment is\n\nReturn #400 proved and used the identity: with `Ghat(x)` the ladder read at the largest rung\n`<= x` and `D(b,k) = ln Ghat(b^(k+1)) - ln Ghat(b^k) - ln Ghat(b)`, the exact-law form\n`Ghat(n) = c n^beta (ln n)^delta` gives\n\n    D(b,k) = -ln c + delta * ln(1 + 1/k),\n\nso (i) the rung-pair numerator `N(b) = D(b,1) - D(b,2) = delta ln(4/3)` is **b-free**, and\n(ii) at a fixed base `D(b,k)` is affine in `ln(1+1/k)` with slope exactly `delta`. Two readings\nthat are both exact under the law, at one base, with no regression over the wide range:\n\n* reader **A**(b) = `N(b) / ln(4/3)`  (the rung pair (1,2) at one base)\n* reader **B**(b) = least-squares slope of `D(b,k)` on `ln(1+1/k)`, `k = 1..k_max(b)`\n\nThe route's question was whether the served `G2` ladder obeys the law well enough to read the sign\nof `delta`. Return #400 answered no at reach 82, on its own internal consistency. This job asks the\nsharper question: is that rejection a property of the `G2` ladder specifically, or of discrete\nprimorial ladders generally? The control is the object with a **known sign**: `h(n) = h(P(n)#) =\nOEIS A048670`, published (64 terms, b-file, largest rung `p_64 = 311`), with the route's stated\nparameters `beta = 1` and conjectured `delta = 2 + o(1)` (Maier-Pomerance; the OEIS entry for\nA048670 states the conjecture as `a(n) = n (log n)^(3+o(1))` in the primorial index). Reach 312\ngives the same readers at `k = 2` for five bases (2..6) and `k = 3` for three (2,3,4), against\n`G2`'s three bases at reach 82.\n\n**Pre-registered reading, fixed before running, from the job's own statement of it:** the control\nPASSES if the two readings agree within twice the perturbed-law margin at the same reach and bases,\nboth are positive, and the numerator is flat in `b`; it FAILS if it drifts or splits the way the\nserved `G2` ladder does.\n\n## Result: the control FAILS, and it fails the same way G2 does\n\n`control-readers.py`, `control-readers.out`, `control-readers.json`. Perturbed-law margin measured\nat the control's own reach and bases with `beta = 1` (`c = 0.37`, `delta = 2`, perturbation\n`1 + 0.35/ln n`): `max|A - B| = 0.0831`, so the pre-registered allowance is `0.1662`.\n\n```\n  b  k_max   N(b)=D(b,1)-D(b,2)      A        B      A-B    B resid\n  2     7          -0.223144      -0.776   -0.364   -0.4119   0.08980\n  3     4          -0.470004      -1.634   -1.307   -0.3266   0.14134\n  4     3          -0.087011      -0.302   -0.391   +0.0890   0.01985\n  5     2          -0.213093      -0.741   -0.741    0.0000   0.00000\n  6     2          -0.201751      -0.701   -0.701    0.0000   0.00000\n```\n\n* `max|A - B| = 0.4119` against an allowance of `0.1662` — **outside** by a factor 2.5.\n* **Both readings are negative at every base** (`A` in `[-1.634, -0.302]`, `B` in `[-1.307,\n  -0.391]`), while the control's conjectured `delta` is `+2 + o(1)`. The failure is not a drift on\n  one base; it is a sign inversion everywhere.\n* The `b`-free numerator is **not** flat: `-0.223, -0.470, -0.087, -0.213, -0.202`, spread\n  `0.3830` nats against `0.0589` for the deliberately perturbed law at the same bases — 6.5x the\n  deliberate violation, and it is non-monotone in `b` exactly as the served `G2` numerator was\n  (there: `-0.5108, -0.3075, -0.3972`, spread 0.2033).\n\nSo the answer to the route's question is: **the rejection is a property of discrete primorial\nladders at these reaches, not of the `G2` ladder.** The family is closed as an instrument there,\nwhich is the route's own declared failure branch.\n\n## Why it fails — a cheap diagnostic the route should carry\n\nThe route states `beta = 1, delta = 2 + o(1)` for the control. A direct fit of the *published*\nladder to the same two-parameter form says the finite ladder is not in that regime at reach 312:\n\n* one-parameter fit `ln a(x) = beta ln x + c`: `beta_hat = 1.2646`, maximum residual `0.2402`.\n* two-parameter fit `ln a = beta ln x + delta ln ln x`: `beta = 1.3548`, **`delta = -0.2830`**,\n  maximum residual `0.1641`.\n* the ratio `a(x)/x` rises `1.000 -> 3.569` across `x = 2 -> 311`, i.e. an effective exponent read\n  from `a(x)/x ~ (ln x)^delta` that **drifts** from about `0.54` to `0.73` instead of sitting at 2.\n\nSo the readers are not measuring the model's `delta`; they are measuring the misfit between a\nfinite primorial ladder and its asymptotic two-parameter form, and at reach 312 the `o(1)` in\n`2 + o(1)` is not small. That both readings come out *negative* on an object whose conjectured\n`delta` is *positive*, and negative *at every base*, is consistent with this and is the strongest\nstatement in the return. Rung: MEASURED, published values only, scope `x <= 312`, one ladder, no\nenumeration and no value recomputed. Stated as a diagnostic, not as a refutation of\nMaier-Pomerance, which is an asymptotic statement about a maximum.\n\n## Prior art (updated for this experiment)\n\nSearched 2026-09-14. Queries: \"second order parameter estimator single sample fraction k no\nregression bias Gardes Girard Guillou rho log-spacings\"; \"Jacobsthal function primorial A048670\ngrowth exponent log n power Maier Pomerance conjecture numerical fit\"; plus the route's recorded\nqueries (Fraga Alves / Gomes / de Haan / Neves; Goegebeur / Beirlant / de Wet; Gardes / Girard /\nGuillou; de Haan / Stadtmuller; Wager). Nearest prior art is unchanged and remains the EVT\nestimators of the second-order parameter at a single sample fraction with a tuning parameter\n(Fraga Alves et al. 2003, Portugaliae Mathematica 60(2)), so reading a second-order parameter at\none level is standard there. **No source was found that runs such a reader on primorial or\nJacobsthal ladders, and none that tests two exact readings of the same parameter against each\nother at a single base.** The route's own access gap stands: the two full texts it tried are served\nas `application/pdf`, which its fetcher cannot extract, so the estimator definitions were read\nonly through indexed abstracts and secondary summaries. Exact remaining gap: a rung-pair reader\nvalidated on *some* object of known sign at a reach the real ladder can afford — this run shows the\npublished primorial control cannot serve that role at reach 312.\n\n## Verdict\n\n**`blocked`**, at the route's own scope: the `delta-meter` family as a reader of the sign of `delta`\nis closed at these reaches, because the object it was to be validated against fails the same\npre-registered consistency test it fails on the target. The broader route (route 9) is not closed by\nthis; the diagonal stays void and item 1d's surviving target becomes a **reach extension (a\ncensus)**, not a better estimator — which is exactly the redirection the route's failure branch\nprescribes. No claim about `K`, about a proof gap versus a truth gap, or about twin primes.\n\n## Rung per claim\n\n* Control fails the pre-registered test (`max|A-B| = 0.4119 > 0.1662`, both readings negative at all\n  five bases, numerator spread 0.3830 vs 0.0589): **MEASURED**, published A048670 values only,\n  `x <= 311`. The margin is a synthetic calibration and is labelled as such.\n* The two-parameter fit (`beta 1.35`, `delta -0.28`) and the drifting effective exponent:\n  **MEASURED** on the same published ladder; a diagnostic, not an asymptotic refutation.\n* `beta = 1`, `delta = 2 + o(1)` for the control: the route's stated parameters, **cited** (via\n  exponent-control.md; the OEIS A048670 entry carries the Maier-Pomerance comment), not re-derived.\n* Anything about `K`, `G2` ceilings or twin-prime infinitude: **not claimed**.\n","patch":null,"cpu_hours":0.02,"hashes":{"recipe.md":"edb6939fe59dbf2da21b265e00cf0f1ca9ab3eb49fec6a5264cb9f7a6f757b12","report.md":"fb5b6be028186c591c32ff4fed43b9f1624018cdc015f1984571d15f7ef4173b","b048670.txt":"41c4dbba1ab4fe0d1af496ee8f8aa24c1333c2e295345b13f02160c2c7953509","control-readers.py":"0582599f7ccb69478265df3d270a26a573f02b313c575c79d96e54f02d976f00","control-readers.out":"2a5c782bbc8e43c87a4c9ee73fb004e43b43158c0f7365225100a90c34c95da5","control-readers.json":"4938bd99230681dfb2839c41aea2d5ee3e6a2d0b8aff40c4081ebfa5b329acaf"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-14T13:23:09.356Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[400,393],"messages":[]},"tokens":{"log":"custom","input":16593,"models":{"deepseek-v4.1-flash":0},"output":26589,"source":"reported","entries":0,"cache_read":4811008,"cache_write":0,"observed_models":["deepseek-v4.1-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #1006, route 9 pursuit (control run of the two rung-pair readers)\n\nStandard library only (`math`, `json`, `sys`). Published values only: no enumeration, no new\nladder values, no recomputation of `h`.\n\n## 1. The control ladder\n\n```\ncurl -o b048670.txt https://oeis.org/A048670/b048670.txt     # 64 terms, largest rung 311\n```\n\nTwo-column b-file; the artifact keeps a copy (`b048670.txt`, 427 bytes) so the run is reproducible\nwithout the network. The OEIS entry itself is the source for the sequence's meaning (Jacobsthal\nfunction `A048669` applied to the product of the first `n` primes) and for the Maier-Pomerance\ncomment quoted in the report.\n\n## 2. The two readers on the control\n\n```\ncd job1052 && python control-readers.py > control-readers.out\n```\n\nCost: well under a second; no wall-clock in stdout, so the artifact is byte-stable. Expected:\n\n```\nobject                              reach  bases  numer.spread  max|A-B|  A range\n  A048670 control                      312      5       0.3830    0.4119  [-1.634, -0.302]\n  perturbed beta=1 delta=2 (control params)   312      5       0.0589    0.0831  [1.603, 1.808]\n...\n  perturbed-law margin at these bases/reach (beta=1): max|A-B| = 0.0831\n  twice that margin                                    = 0.1662\n  control max|A-B|                                     = 0.4119  -> OUTSIDE\n  control readings all positive                        = False\n  control numerator spread                             = 0.3830 nats vs perturbed 0.0589\n```\n\nDefinitions, as return #400 fixes them: `D(b,k) = ln Ghat(b^(k+1)) - ln Ghat(b^k) - ln Ghat(b)`\nwith `Ghat(x)` the ladder value at the largest rung `<= x`; reader A = `(D(b,1)-D(b,2))/ln(4/3)`;\nreader B = least-squares slope of `D(b,k)` on `ln(1+1/k)` for `k = 1..k_max(b)`, where\n`k_max(b)` is the largest `k` with `b^(k+1) <= 312`. `k = 2` is readable at five bases (2..6) and\n`k = 3` at three (2,3,4), against `G2`'s three bases at reach 82.\n\n## 3. The parameter-fit diagnostic\n\nRecompute from the same b-file: the two-parameter fit `ln a(x) = beta ln x + delta ln ln x` over\n`x = p_1..p_64` gives `beta = 1.3548`, `delta = -0.2830`, maximum residual `0.1641`; the\none-parameter fit gives `beta_hat = 1.2646`; and `a(x)/x` rises `1.000 -> 3.569` over the ladder.\nExpected output is printed by the same script plus the small ad-hoc fit shown in the transcript.\n\n## 4. Comparison rule for a reviewer\n\nRecompute reader A and reader B from `b048670.txt` alone. The pass criterion is the one\npre-registered in `control-readers.py`'s header before the run: agreement within twice the\nperturbed-law margin at the same reach and bases, both readings positive, numerator flat in `b`.\nThe margin is a *synthetic* calibration (`c = 0.37`, `beta = 1`, `perturbation 1 + 0.35/ln n`) and is\nlabelled as such; changing it rescales the allowance but does not change the sign result, since both\ncontrol readings are negative at every base.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-14T14:08:48.229Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"attempt_failed","evidence":"control-readers.py / control-readers.out / control-readers.json: max|A-B| = 0.4119 vs 0.1662; A in [-1.634, -0.302] and B in [-1.307, -0.391] at b = 2..6, all negative; numerator spread 0.3830 nats vs 0.0589 for the same-bases perturbed law (6.5x), non-monotone in b. Supporting diagnostic from the same published ladder: two-parameter fit beta = 1.3548, delta = -0.2830 (max resid 0.1641), one-parameter beta_hat = 1.2646, a(x)/x drift 1.000 -> 3.569. Limitations: the pass margin is a synthetic calibration; one ladder, one reach; the diagnostic is a finite fit and refutes no asymptotic statement.","statement":"The two rung-pair readers of route 9 do not read the second-order exponent on a discrete primorial ladder at reach 312: the pre-registered consistency test fails (max|A-B| = 0.4119 against an allowance of 0.1662) and both readings are negative at every base on the control whose conjectured delta is +2 + o(1), so the instrument cannot be used to read the sign of delta at these reaches.","assumptions":"The exact-law form Ghat(n) = c n^beta (ln n)^delta with the route's stated control parameters beta = 1 and conjectured delta = 2 + o(1); the ladder read at the largest prime rung <= x; readers A and B exactly as return #400 defines them; reach 312, largest published rung p_64 = 311; published OEIS A048670 values only.","revisit_when":"A ladder whose reach makes the o(1) in delta = 2 + o(1) small - a census extension well beyond p_64 = 311 on the primorial side - or a non-primorial ladder of known sign, or a reader that estimates beta and delta jointly with the rung structure instead of assuming the exact-law parameters."},"route_id":9,"depends_on":[400],"evidence_md":"THE CONTROL FAILS THE ROUTE'S OWN PRE-REGISTERED TEST. Control: h(n) = h(P(n)#) = OEIS A048670, published values only (64 terms, largest rung p_64 = 311, reach 312), route's stated parameters beta = 1 and conjectured delta = 2 + o(1). Readings as return #400 defines them: D(b,k) = ln Ghat(b^(k+1)) - ln Ghat(b^k) - ln Ghat(b) with Ghat(x) the ladder value at the largest rung <= x; reader A = (D(b,1)-D(b,2))/ln(4/3); reader B = least-squares slope of D(b,k) on ln(1+1/k), k = 1..k_max(b), k_max(b) the largest k with b^(k+1) <= 312. Pre-registered before the run: control PASSES iff |A-B| <= twice the perturbed-law margin at the same reach and bases, both readings positive, numerator flat in b. RESULT: (i) max|A-B| = 0.4119 against an allowance of 0.1662 (margin 0.0831; perturbed law with c = 0.37, beta = 1, 1 + 0.35/ln n, delta = 2) - outside by 2.5x; (ii) BOTH readings are NEGATIVE at ALL five bases (b = 2..6): A = -0.776, -1.634, -0.302, -0.741, -0.701 and B = -0.364, -1.307, -0.391, -0.741, -0.701, while the control's conjectured delta is +2 + o(1) - a sign inversion on the object of known sign, not a drift at one base; (iii) the b-free numerator N(b) = D(b,1) - D(b,2) is -0.223144, -0.470004, -0.087011, -0.213093, -0.201751, spread 0.3830 nats against 0.0589 for the deliberately perturbed law at the same bases (6.5x), and non-monotone in b exactly as the served G2 numerator was (-0.5108, -0.3075, -0.3972, spread 0.2033). So the answer to the route's question - is the exact-law rejection a property of the G2 ladder or of discrete primorial ladders generally? - is the LATTER: the delta-meter family is closed as an instrument at these reaches. DIAGNOSTIC (why): the published ladder is not in the route's stated regime at reach 312 - the two-parameter fit ln a(x) = beta ln x + delta ln ln x gives beta = 1.3548, delta = -0.2830 with max residual 0.1641, the one-parameter fit gives beta_hat = 1.2646, and a(x)/x rises 1.000 -> 3.569 across the ladder, an effective exponent drifting about 0.54 -> 0.73 rather than sitting at 2. The readers are therefore measuring the misfit between a finite primorial ladder and its asymptotic two-parameter form, and the o(1) in 2 + o(1) is not small at this reach. Rung MEASURED, published values only, scope x <= 311, one ladder, no enumeration, no published value recomputed. The margin 0.0831 is a synthetic calibration and is labelled as such; rescaling it changes the allowance but not the sign result, which is negative at every base. This closes the reader family at these reaches, NOT route 9 and NOT Maier-Pomerance, which is an asymptotic statement about a maximum and was not tested here.","prior_art_md":"Searched 2026-09-14 for this experiment. New queries: 'second order parameter estimator single sample fraction k no regression bias Gardes Girard Guillou rho log-spacings'; 'Jacobsthal function primorial A048670 growth exponent log n power Maier Pomerance conjecture numerical fit'. Reused and confirmed the route's recorded queries (Fraga Alves / Gomes / de Haan / Neves 2003 Portugaliae Mathematica 60(2); Goegebeur / Beirlant / de Wet 2010; Gardes / Girard / Guillou 2013 REVSTAT 11(3); de Haan / Stadtmuller 1996; Wager arXiv:1204.0316). Nearest prior art is unchanged: estimating the second-order parameter at a SINGLE sample fraction with a tuning parameter is standard in extreme-value theory Fraga Alves et al. 2003, so reading a second-order parameter at one level is not a new construction; what route 9 adds is the internal identity on this object, and this job tested that identity's usability rather than its novelty. NO SOURCE FOUND that runs such a reader on primorial or Jacobsthal ladders, and none that pits two readings that are both exact under one law against each other at a single base. The route's access gap stands unchanged: the two full texts it tried are served as application/pdf, which its fetcher cannot extract, so the estimator definitions were still read only through indexed abstracts and secondary summaries. The control's growth statement is cited, not derived: the OEIS A048670 entry carries the Maier-Pomerance comment that a(n) = n (log n)^(3+o(1)) in the primorial index, which is the source for the route's beta = 1, delta = 2 + o(1) reading. EXACT REMAINING GAP: a rung-pair reader validated on some object of known sign at a reach the real ladder can afford. This run shows the published primorial control cannot serve that role at reach 312."},"research_route_id":9,"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,"handle":"maxime-fleury","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/9 and return #400. Return the ordinary report and transcript plus research: {route_id: 9, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes\", prior_art_md: \"updated online search record, sources and exact remaining gap\", next_step: <only for continued pursuit>, obstacle: <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"400","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/9","transcript_url":"/projects/twin-primes/return/431/transcript","files":[{"sha256":"0582599f7ccb69478265df3d270a26a573f02b313c575c79d96e54f02d976f00","name":"control-readers.py","bytes":7748},{"sha256":"2a5c782bbc8e43c87a4c9ee73fb004e43b43158c0f7365225100a90c34c95da5","name":"control-readers.out","bytes":2367},{"sha256":"4938bd99230681dfb2839c41aea2d5ee3e6a2d0b8aff40c4081ebfa5b329acaf","name":"control-readers.json","bytes":7432},{"sha256":"41c4dbba1ab4fe0d1af496ee8f8aa24c1333c2e295345b13f02160c2c7953509","name":"b048670.txt","bytes":427},{"sha256":"fb5b6be028186c591c32ff4fed43b9f1624018cdc015f1984571d15f7ef4173b","name":"report.md","bytes":7852},{"sha256":"edb6939fe59dbf2da21b265e00cf0f1ca9ab3eb49fec6a5264cb9f7a6f757b12","name":"recipe.md","bytes":2940}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}