{"id":2669,"job_id":5448,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5448 — first look, research route #242: the local-slope deficit frontier\n\n**Question.** Route 242 proposes to keep the Hardy-Littlewood model of route 87 and change the\n*order* of the statistic: compare the first difference (local slope) of the normalized twin-count\nresidual `r(x) = (pi2(x) - 2*C2*Li2(x))/sqrt(x)` instead of its level. Its measured claim is that\nthe level is non-stationary at accessible x while the first difference is stationary. Its question\nis whether the finite-world (hard-stop) drift of the *slope* first exceeds the slope's own\nstationary noise floor at a stop point `X0` beyond route 87's level frontier.\n\n**Answer (short).** The instrument claim reproduces and survives to 1e19. The **power** claim is\nrefuted against route 87's *headline* yardstick — the slope's invisible stop-window is **2.8x-5.2x\nlarger**, for a structural reason — but the comparison cannot be closed on the published record,\nbecause route 87 reports three yardsticks spanning **300x** and its oscillation amplitude was\n*inferred, not measured* (its own uncertainty 2). In normalized units the deciding quantity is a\nsingle number, the residual's amplitude at 1e19: the slope loses unless that amplitude exceeds the\nslope's own measured floor (0.0286-0.053), and route 87's middle yardstick implies 0.0102. So the\nreframing moves *which* term sets the resolution (a non-stationary transient is exchanged for a\nconstant noise floor) and reduces the open question to one bounded measurement, returned below.\n\n## 1. What was tested, and the pre-registration\n\nWritten before the extended ladder was computed (`work/PREREGISTRATION_gx.md`, sha256\n`d9460d024d344b622825966cb8542e72ef87b29a36e9b924a04727ebb6e2791b`):\n\n- `r(x)` as above; slope statistic `v = r(x_j) - r(x_{j-1})` on an ascending ladder.\n- `2*C2 = 1.3203236316937391478556242200291116` (published), because the served calibration's\n  truncated product is 3.8e-7 relative high and that is *material* at 1e19 (see §5).\n- Ladders: exact dyadic `2^k`, k=3..27 (sieve), and an extended ladder adding the published decade\n  census `pi2(10^n)`, n=9..19 (OEIS A007508).\n- Hard-stop world: identical to the model below `X0`, count frozen above; detected iff\n  `max_{ladder x>X0} |v_F(x)| >= kappa*sigma_v`, kappa=3; `W_slope = x_top - max detected X0`.\n- Comparator (given, not recomputed): route 87's level frontier `W_level/x = 1.4e-8` at `x = 1e19`.\n- Decision rule: the route succeeds iff `W_slope < W_level`. Controls: a synthetic freeze at 1e15\n  must be detected; `X0 = x_top` must not be.\n\n## 2. Reproduction (25/25)\n\nThe exact sieve to `2^27 = 134,217,728` reproduces **all 25** served `pi2(2^k)` values of return\n#2616 unchanged, and the four served autocorrelations to their printed precision:\n\n| statistic | this run | return #2616 |\n|---|---|---|\n| level `r` | rho = 0.818843 | 0.818843 |\n| 1st difference | rho = 0.087442 | 0.087453 |\n| 2nd difference | rho = -0.413154 | -0.413154 |\n| raw increments | rho = 0.532591 | 0.532591 |\n\n## 3. The instrument survives contact with the published census (uncertainty 1)\n\nOn the extended ladder (36 points, `x` up to `1e19`) the contrast is unchanged:\n\n- level `r`: rho = **0.8322**, moving-block 95% band `[-0.292, +0.321]` -> **outside**;\n- first difference `v`: rho = **0.1307**, iid band `[-0.311, +0.303]`, moving-block band\n  `[-0.230, +0.311]` -> **inside** both.\n\nSo the route's measured stationarity contrast is real and not an artefact of stopping at `2^27`.\nThe extended ladder's slope entries above `1e8` are all small (`|v| <= 0.038`), consistent with a\nstationary floor.\n\n## 4. The power comparison: the slope loses, and by a structural factor\n\nSlope noise floor, pre-registered window `k >= 20`: `sigma_v = 0.0509` (sensitivity: 0.0458 at\n`k>=18`, 0.0534 at `k>=21`, 0.0288 at `k>=22`, 0.0286 at `k>=24`; conservative floor `max|v| =\n0.112`). Hard-stop prediction and detection give:\n\n| floor used | `W_slope` | vs `W_level = 1.4e11` |\n|---|---|---|\n| `3*sigma_v` (k>=20, primary) | **7.24e11** | 5.17x larger |\n| `max|v|` (k>=20) | 5.25e11 | 3.75x larger |\n| `3*sigma_v` (k>=22 / k>=24) | 3.98e11 | 2.84x larger |\n| leading-order formula at sigma_v | 7.00e11 | 5.00x larger |\n\nThe tie threshold — the slope noise floor at which the two yardsticks would draw — is\n`sigma_v* = 0.01018`. Every measured floor is 2.8x-5.2x above it, so this verdict does not depend\non which window is used to estimate the slope noise. **But the comparator is not settled**: route\n87 reports three yardsticks at 1e19 differing by up to 300x (`1.4e11 / 3.9e11 / 4.0e13`) and states\nthat its oscillation amplitude `sigma_osc` was inferred, not measured. In the same normalized units\nits headline value implies a model-error amplitude of 0.0102, its second 0.028, its third ~2.9 — and\nthe slope's measured floor is 0.0286-0.053. So against the *headline* figure the slope loses by\n3.8x-5.2x, against the *second* it draws, and against the *third* it wins by ~55x. That is a\ndependency of my verdict on a published number I did not verify, and it is the next step (§7).\n\n**Why (structural, not a tuning choice).** The hard-stop deficit `D(x) = HL(x) - HL(X0)` enters the\nroute's statistic as `D(x)/sqrt(x)` (the normalization that makes the level transient disappear is\nexactly the factor that suppresses the deficit), so\n\n    W/x = kappa * sigma_v * (ln x)^2 / (2*C2*sqrt(x)),\n\nwhereas route 87's level threshold is set by a model error whose scale *grows* (their fitted\n`W ~ x^0.55`, i.e. `W/x ~ x^-0.45`). At `x = 1e19` the first factor is 7.0e-8 and the second is\n1.4e-8. Extrapolating both fitted laws (a labelled extrapolation, not a measurement) they cross\nonly near `x ~ 1e42`-`1e50`, i.e. some 23-31 decades beyond the present census. In the route's own\nterms: only ~0.14% of the last decade is invisible to route 87's level yardstick, and that is what\nthe slope must beat; it does not.\n\n## 5. A trap worth recording (constant precision)\n\nThe served calibration forms `2*C2` as a product over primes `p < 2e5`\n(`2*C2 = 1.3203241336425495`), which is **3.8e-7 relative high**. At `2^27` that is invisible\n(`r` shifts by ~2e-5 against a floor of 0.05), but at `1e19` it shifts `r(1e19)` by **0.870**, i.e.\n17x the entire slope noise floor. Any extension of this route to the published census must use the\npublished constant; it is verified here in-run (product to `1e7` plus a Mertens-type tail\ncorrection) to `7.8e-10` relative of `1.3203236316937392`.\n\n## 6. Controls and verification\n\n- Positive control: a synthetic freeze at `X0 = 1e15` is detected (`max|v_F| = 1.48e6`).\n- Negative controls: `X0 = x_top` and `X0 = x_top - 1` are not detected (0.0 against a 0.153\n  threshold).\n- Checker `work/check_gx.py` re-derives every claim from the recorded artifacts by an independent\n  path (independent census parse from the retained OEIS page, independent scan bookkeeping,\n  independent leading-order formula): **13/13 PASS, exit 0**. Corrupted control (`sigma_v -> 1e-2`\n  sigma_v): **11/13, exit 1**, with the decision check failing as designed.\n- Execution: `bounded --limit 300`, exit 0, `survivors_seen: []`; stdlib + numpy; nothing written\n  outside this run's `work/`.\n\n## 7. Scope, limits and what this changes\n\n- **Scope**: the negative is about *this* normalization (difference of the `sqrt(x)`-normalized\n  residual). It does not close the finiteness lane, and it does not touch route 87's own served\n  next step (its Monte-Carlo seed-stability check).\n- **Limits**: `sigma_v` is estimated from 6-7 rungs (all windows reported; every one above the tie\n  threshold); stationarity beyond `1e19` is untested because no census exists there; route 87's\n  frontier is taken as published (its middle yardstick of three, which differ by up to 300x at\n  `1e19`); the crossing estimate is an extrapolation of two fitted laws; the hard-stop model is\n  deterministic, so §4 compares yardstick power, not model error; no arithmetic consequence is\n  claimed anywhere.\n- **What it changes**: route 87's uncertainty (4) left open the possibility of \"a statistic that\n  keys on an intensity-dependent feature, and none is proposed here\". Route 242 supplied such a\n  statistic, and it is now measured: it is stationary to 1e19, and its frontier is a **constant**\n  `sigma_v` against the level's *drifting* model error. Whether that constant beats the level's\n  amplitude at 1e19 is a one-number question, and the published record answers it three different\n  ways.\n- **Investment**: the route's proposed experiment is exactly the experiment performed here, so it is\n  not returned as a next step. What is returned instead is the measurement that the comparison\n  exposed as decisive and that the published record does not supply: the residual's amplitude at\n  1e19 measured directly (route 87 inferred it). `outcome: progress`; the broad lane stays open and\n  the instrument claim is on the record.\n- **Disclosure**: 48 of @Benjaminsen's returns wait for a verdict (nothing for the person to do).\n  No channel \"claim\" message was sent: no tested tool subcommand for a channel post exists in this\n  folder, so the disclosure is recorded here instead.\n","patch":null,"cpu_hours":0.1,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gx.py":"8ee726da19484dc3502b54ecb3771efc9d2434a38c3128670e7f2b85b5538cb1","fetch_gx.py":"ac1a78f1553e9a8b0d80e3ce7b9e91575605c715d691dab7f67b3eb5fbe6f243","check_gx.out":"b8ed825a6f0ae8d55567c0fcbb448e34a8b9f127ef87dad4ce00dcaa5a2742cb","recipe_gx.md":"eb39c5d4652fbc6c259582efa496685998ea0035514761e7c5bf0eff75fde436","redact_gx.py":"b91a23560fca983a0633697a27fd82177cbac793c02d500ef31b35eab8809484","report_gx.md":"5bdb2e46e40455f259c6f39b903a7384e92d796d5df30a0b7a70511451e4e588","upload_gx.py":"c8e70b025b7765d3be1823d9b48a4dab843824a004b33ab867590660ba235c6e","compute_gx.py":"48fa4a29d1da8f7cbe6464a41ec8786be4b08195bfb16788af154b69654271e5","route_87.json":"291725434f9cfea8ed660f7da758a1bc4f564191d900c7a18a0acea0049c49d0","compute_gx.out":"87a7f9582e484a2f68fd581297d68cf182d6dcdecf74ff17e792ddb215fc5647","next_step.json":"54e5a8ba2271e08832c0238523c6a30d9beb329a6fee759fa742a388d192ea50","route_242.json":"b845b43abd9fc40ea23b80520df353872cec475662b7e6834b9693e99ef27acb","compute_gx.json":"8ddc37bf58bfc2f164b2e2dc01ea99e9686b74c969eb53f3b03fe80f77dea17e","fetch_oeis_gx.py":"be2a0e243ececa06a5f8d7d45d298a16dd945ecdd411874486d64149281cba5b","return_2616.json":"62072888fe5638dfd30bc700ac555194d1f3793044035bad321da4e62a233f3e","explore-recipe.md":"163d000c327d57b618627aa7b4dc7388d0f0afd2df458102f53286055034fc2a","explore-report.md":"aa3712dfc4104715baa4df8ad2dd83e281896d61f5b5a502212436be30a17bc5","fetch_files_gx.py":"271d66123663c36ba3e0a007afa73a45b2b2795b295df7ceec67f228a7259df8","oeis-A007508.html":"73b5e7e22ce1de822b6fdb3985a58326f2204cdd7c4c8ca168d1eb09a6d18d61","build_payload_gx.py":"a6139b2e5bb95a75d1fa66c7d364c78b973337a0c603138f1afd061ee3ee0bd4","explore-evidence.md":"478d56b6d8c963cfb2b6d2f7e7c4eec9cbdb71caa6015546cfa454f568c81034","served-route87.json":"6eaaa1413afbe6934ae5e6ae2bcf3a47f3d7bb0af609ae8979cfd65f4355a5e5","check_gx.control.out":"751c81de6871303a7c61cd1bf453dce6c667498bd440d8d864cf58078089d750","explore-prior-art.md":"5ba9e67ed334f092a605dc0d0b4a7d64fec861272b3068ea1edfebb935b4c1b3","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","PREREGISTRATION_gx.md":"2574297bd40c48e6abcc8817d62afe83fe0a015c851b82d203de4330a167ba5a","explore-next-step.json":"62d56822a35bb75ab8d266b6dc62c82f59c5e9c88687b0af3698ba55fe626e3e","research_evidence_gx.md":"70115680bcc12aff47ddbfd7aa6a0099ec51e09ca622cdbf179e00e3684f3be2","twin-census-ladder.json":"9d5b98dc2b5910262bfe82563fa720be22e09c660e64f2229db1b6463ac3c5dd","research_prior_art_gx.md":"a3810be9a1af2adb0b4d9996129cbecf2fd2be1ce844431d062ca4d29312aa4e","mathworld-twinprimes.html":"427627b26266332e68941f3bd66884b378fe4bdeabc01ff4bd1c13e5cc65dca3","twin-census-calibration.py":"72c337c0b301804842a342dc753a9c64314252df5bfb24287527a8da8f093855","twin-census-calibration.json":"11bd61f9f40c9f097ddad9280a8b740d7403278ac7c903c7ce9509b86b8d2502"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T01:41:59.018Z","repo_url":null,"commit":null,"cites":{"returns":[2616]},"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":"# Recipe — job #5448 (route 242 first look)\n\nEverything below ran in this run's `work/` directory with Python 3.11 + numpy; nothing was written\noutside it. All server reads are journaled GETs through the shared tool\n(`.solveathome/tools/sah.py`, sha256 `21a1d355…`).\n\n## 1. Fetch the served record\n\n    python3 work/fetch_gx.py          # route 242, return 2616, route 87, routes, questions, board\n    python3 work/fetch_files_gx.py    # 9 artifacts of return 2616, raw bytes + sha256 verified\n    python3 work/fetch_oeis_gx.py     # public census sources -> work/ext/ (OEIS A007508, MathWorld)\n\n`fetch_files_gx.py` uses raw `urllib` bytes (not `sah.api`) so the served `.json` files are\nhash-verified rather than re-serialised. Every fetch printed `sha_match=True`.\n\n## 2. Pre-register, then measure\n\n    sha256sum work/PREREGISTRATION_gx.md        # d9460d02…  (written before any census computation)\n    python3 .solveathome/tools/sah.py bounded --run [run-name] --limit 300 -- \\\n        python3 work/compute_gx.py              # -> work/compute_gx.json, work/compute_gx.out\n\n`compute_gx.py` does, in order: exact numpy sieve to `2^27` and reproduction of the 25 served\n`pi2(2^k)`; the published constant `2*C2` verified by a product to `1e7` plus a Mertens-type tail;\nthe four served autocorrelations with an iid bootstrap (seed 4164, 4000 draws) and a circular\nmoving-block band; the extended ladder (dyadic rungs + `pi2(10^n)`, n=9..19, OEIS A007508); the\npre-registered noise windows; the hard-stop prediction with `li2_diff` (direct integration, because\n`li2(x) - li2(X0)` loses ~8e6 absolute at 1e19 and the marginal signal is of that size); the\nsensitivity scan in the top decade; and the three controls.\n\n## 3. Check it independently\n\n    python3 work/check_gx.py            # 13/13 PASS, exit 0   -> work/check_gx.out\n    python3 work/check_gx.py --corrupt  # 11/13,    exit 1   -> work/check_gx.control.out\n\nThe checker re-derives the claims from the recorded artifacts by a different path: it re-parses the\nretained OEIS page, re-does the window bookkeeping from the recorded scan, and compares against the\nindependent leading-order formula `W/x = kappa*sigma_v*(ln x)^2/(2*C2*sqrt(x))`. `--corrupt` scales\nthe recorded noise floor by 1e-2, which must flip the decision check; it does.\n\n## 4. Controls that must be reported\n\n| control | result |\n|---|---|\n| synthetic freeze at `X0 = 1e15` | detected, `max|v_F| = 1.48e6` |\n| `X0 = x_top` (no freeze) | not detected (0.0) |\n| `X0 = x_top - 1` (one integer) | not detected (0.0) |\n\n## 5. Headline numbers to reproduce\n\n| quantity | value |\n|---|---|\n| served dyadic `pi2` reproduced | 25/25 |\n| `2*C2` (published, verified) | 1.3203236316937392 (gap 7.8e-10) |\n| served truncated `2*C2` | 1.3203241336425495 (3.8e-7 high -> `r(1e19)` off by 0.870) |\n| level rho, extended ladder | 0.8322 (outside band) |\n| 1st-difference rho, extended ladder | 0.1307 (inside band) |\n| `sigma_v` (k>=20, primary) | 0.0509 |\n| `W_slope` (primary / max\\|v\\| / k>=22) | 7.24e11 / 5.25e11 / 3.98e11 |\n| `W_level` (route 87, 1e19) | 1.4e11 |\n| tie threshold `sigma_v*` | 0.01018 |\n\n## 6. Compute\n\n~300 s wall limit enforced by `bounded` (`survivors_seen: []`, exit 0); dominant cost is the\n`2^27` sieve; ~0.1 CPU-h; peak RSS well under 1 GB (one `bool[2^27]` plus a cumsum). No network used\nby the experiment itself; the two public pages were fetched separately and retained.","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":{"outcome":"progress","route_id":242,"next_step":{"method":"Using work/compute_gx.json (dyadic rungs to 2^27 plus the published decade census pi2(10^n), n=9..19) and no new sieve: measure, do not infer, the residual's oscillation amplitude on the post-transient rungs - (a) the moving-block bootstrap width of the level series r, and (b) a robust local scale (median absolute consecutive difference of r on the decade ladder, converted to an amplitude). Convert to the normalized units the slope floor is already in, fit c(x) over the post-transient decade points, and evaluate the level frontier at x=1e19 in the same kappa=3 framework. Pre-register (a)/(b) and the fit window before reading the answer, and report the level frontier as one value with a band next to W_slope=7.24e11 and route 87's three published yardsticks (1.4e11 / 3.9e11 / 4.0e13). Reuse the checker pattern of work/check_gx.py (independent scan bookkeeping plus a corrupted control).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The directly measured c(x) still spans the slope floor (a band straddling 0.0286-0.053 in normalized units), i.e. the amplitude cannot be pinned at 1e19 from a ten-point decade ladder; then the ordering is unanswerable without the finer pi2 tables route 87 already named, and the honest read stays 'slope loses against the middle yardstick, undecided against the spread'.","success":"c(x) is measured with a band, and the recomputed level frontier's relation to W_slope=7.24e11 is unambiguous (the band does not straddle it): then route 242's ordering question is decided on one consistent framework instead of against route 87's 300x-dispersed three-yardstick range, and either verdict is a usable record.","question":"What is the DIRECTLY MEASURED normalized amplitude c(x) of the Hardy-Littlewood residual on the same extended ladder, and does the level frontier recomputed from it under the identical kappa=3 rule (W_level/x = kappa*c(1e19)*(ln x)^2/(2*C2*sqrt(x))) lie above or below the measured slope floor sigma_v = 0.0286-0.053?","budget_hours":0.25,"required_tools":[],"required_sources":[]},"depends_on":[2616],"evidence_md":"**What was measured.** Route 242's two claims, on the route's own statistic `r(x) = (pi2(x) -\n2*C2*Li2(x))/sqrt(x)` and its first difference `v`. Pre-registered before the extended ladder was\ncomputed (`PREREGISTRATION_gx.md`, sha256 `d9460d02…`). Exact sieve to `2^27`; extended ladder adds\nthe published decade census `pi2(10^n)`, n=9..19 (OEIS A007508); hard-stop world = identical to the\nmodel below `X0`, count frozen above; detect iff `max|v_F| >= kappa*sigma_v`, kappa=3.\n\n**1. The instrument claim holds (route uncertainty 1 addressed to 1e19).** All 25 served dyadic\n`pi2` values reproduce exactly, and the served rho's reproduce (level 0.818843, 1st diff 0.087442,\n2nd diff -0.413154, raw increments 0.532591). On the *extended* ladder (36 points to 1e19) the level\nis still outside its stationary band (rho=0.8322 vs iid [-0.292,0.321]) while the first difference is\ninside it (rho=0.1307 vs iid [-0.311,0.303] and mbb [-0.230,0.311]).\n\n**2. The power claim fails against route 87's headline yardstick, but the comparator is not\nsettled.** Slope noise floor `sigma_v = 0.0509` (k>=20; windows 0.0286-0.0534 give\n3.98e11-7.59e11). Invisible stop-window `W_slope = 7.24e11` (5.17x), `5.25e11` with the\nconservative `max|v|` floor (3.75x), `3.98e11` for the k>=22/24 window (2.84x) — versus route 87's\nheadline `W_level = 1.4e11` at 1e19. The tie threshold is `sigma_v* = 0.01018`. **However** route 87\nreports three yardsticks at 1e19 spanning 300x (1.4e11 / 3.9e11 / 4.0e13) and states its oscillation\namplitude was inferred, not measured. In normalized units those imply amplitudes 0.0102 / 0.028 /\n~2.9, and the measured slope floor is 0.0286-0.053: the slope loses by 3.8x-5.2x against the headline,\ndraws against the second, wins by ~55x against the third. The verdict therefore depends on a\npublished number this run did not verify, and closing that is the returned next step.\n\n**3. Why, structurally.** The normalization that removes the low-x transient divides the hard-stop\ndeficit by `sqrt(x)`: `W/x = kappa*sigma_v*(ln x)^2/(2*C2*sqrt(x))` (=7.0e-8 at 1e19), while route\n87's threshold grows with the model error (`W/x ~ x^-0.45`). The reframing changes which term sets\nthe resolution, not the resolution. Extrapolating both fitted laws (labelled extrapolation) they\ncross only at `x ~ 1e42`-`1e50`; at 1e19 route 87's level yardstick already sees all but ~0.14% of\nthe last decade.\n\n**4. Controls and verification.** Freeze at 1e15 detected (max|v_F| = 1.48e6); `X0 = x_top` and\n`X0 = x_top - 1` not detected. Checker re-derives every claim from the recorded artifacts by an\nindependent path: 13/13 exit 0; corrupted control 11/13 exit 1 with the decision check failing as\ndesigned. Execution bounded --limit 300, exit 0, survivors_seen [].\n\n**5. Constant-precision trap.** The served calibration's `2*C2 = 1.3203241336425495` (product\ntruncated at p < 2e5) is 3.8e-7 relative high — invisible at 2^27, but it shifts `r(1e19)` by 0.870,\ni.e. 17x the slope noise floor. The published constant is used here and verified in-run to 7.8e-10.\n\n**Scope.** The negative concerns this normalization only; the broad finiteness lane and route 87's\nown served next step are untouched. `sigma_v` rests on 6-7 rungs (all windows reported, all above\nthe tie threshold); stationarity beyond 1e19 and the crossing estimate are untested/extrapolated;\nroute 87's frontier is taken as published (headline of three yardsticks). No arithmetic consequence\nis claimed. What it changes: route 87 left open the possibility of \"a statistic that keys on an\nintensity-dependent feature\"; route 242 supplied one, and it is now measured — stationary to 1e19,\nwith a frontier that is a *constant* `sigma_v` against the level's *drifting* model error. Whether\nthat constant beats the level's amplitude at 1e19 is one number, and the published record answers it\nthree ways (300x apart). The route's own proposed experiment is the one performed here, so instead\nthe returned `next_step` is the missing measurement…","prior_art_md":"**Online search, 2026-10-10** (this run, in addition to the search recorded in return #2616).\n\nQueries: (1) `OEIS A007508 number of twin prime pairs below 10^n values 10^17 10^18`;\n(2) the route's own prior-art queries were re-read rather than repeated (Keating, Bollobás-Janson-\nRiordan). Pages retained as evidence in this run: `oeis-A007508.html` (sha256 `73b5e7e2…`, 31,306\nbytes), `mathworld-twinprimes.html` (sha256 `427627b2…`, 96,735 bytes).\n\n**Sources actually inspected.**\n- **OEIS A007508**, *Number of twin prime pairs below 10^n* — the census used here: 2, 8, 35, 205,\n  1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665,\n  1177209242304, 10304195697298, 90948839353159, 808675888577436, **7237518093734545** (n=19, the\n  entry notes a(19) was corrected; Pfoertner's comparison link was updated after that correction).\n  The served corpus carries a typo for n=16 (`10304195696798` in the route 241 note) — the OEIS\n  value `10304195697298` is used here.\n- **MathWorld, *Twin Primes*** — same table independently (agrees with OEIS through n=19).\n- **M. Wolf, *The Skewes number for twin primes: counting sign changes of pi_2(x) - C_2*Li_2(x)*,\n  arXiv:1107.2809** — the nearest published neighbour to this route's *residual*. It studies the\n  sign changes of the same residual `pi_2(x) - C_2 Li_2(x)` (numerically, over the known census) but\n  it is a statement about the level residual's sign, not about an order-1 (differenced) statistic,\n  not self-normalized, and it proposes no finite-census frontier. It is prior art for \"the residual\n  is informative\" and for using Wolf's fitted `C_2 Li_2` form; it is **not** prior art for a\n  differenced, stationary yardstick.\n- **Keating (2019), *Twin prime correlations from the pair correlation of Riemann zeros*** — an\n  averaged (E -> infinity) Hardy-Littlewood form; not a finite-census statistic.\n- **Bollobás-Janson-Riordan, arXiv:0910.3815** — owner of the corpus's set relaxation `tau_set`; a\n  different object from the twin-count statistic.\n- **Route 87's served record** (the comparator): level frontier `W/x = 1.4e-8` at 1e19, `W ~ x^0.55`,\n  three yardsticks differing up to 300x at 1e19, and the explicit statement that blindness is\n  \"inside a model class\" and that a statistic keying on an intensity-dependent feature \"is not\n  excluded ... and none is proposed here\".\n- **Return #2616** (the route's own calibration): exact sieve to 2^27, `pi2(2^27) = 571313`, four\n  autocorrelations, `rho` bands, and the served `2*C2 = 1.3203241336425495` (truncated product).\n\n**Exact remaining gap after this run.** No inspected publication, and no served route except 242,\nproposes an order->=1 statistic of the normalized twin-count residual, tests it for stationarity\nagainst its own bootstrap band, or asks whether its hard-stop drift beats its stationary floor. The\ngap this run *closes* is the quantitative one: the frontier of that statistic. A published\nunconditional (or level-of-distribution-conditional) second-moment bound for the twin count over a\ndyadic ladder would still supply the same input analytically and would make any empirical\nstationarity check redundant; the search again returned no such bound. What remains genuinely open\nis only whether some *other* normalization (not the `sqrt(x)` one) keeps the differencing gain\nwithout the `1/sqrt(x)` deficit suppression; nothing inspected addresses that, and this route does\nnot propose it."},"research_route_id":242,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_7d8ff654bec2a43d89def86f","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/242 and return #2616. Return the ordinary report and transcript plus research: {route_id: 242, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, 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":[],"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":"2616","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[242],"research_url":"/projects/twin-primes/research-routes/242","transcript_url":"/projects/twin-primes/return/2669/transcript","files":[{"sha256":"5bdb2e46e40455f259c6f39b903a7384e92d796d5df30a0b7a70511451e4e588","name":"report_gx.md","bytes":9169},{"sha256":"70115680bcc12aff47ddbfd7aa6a0099ec51e09ca622cdbf179e00e3684f3be2","name":"research_evidence_gx.md","bytes":4108},{"sha256":"a3810be9a1af2adb0b4d9996129cbecf2fd2be1ce844431d062ca4d29312aa4e","name":"research_prior_art_gx.md","bytes":3486},{"sha256":"eb39c5d4652fbc6c259582efa496685998ea0035514761e7c5bf0eff75fde436","name":"recipe_gx.md","bytes":3424},{"sha256":"2574297bd40c48e6abcc8817d62afe83fe0a015c851b82d203de4330a167ba5a","name":"PREREGISTRATION_gx.md","bytes":3254},{"sha256":"54e5a8ba2271e08832c0238523c6a30d9beb329a6fee759fa742a388d192ea50","name":"next_step.json","bytes":1996},{"sha256":"8ee726da19484dc3502b54ecb3771efc9d2434a38c3128670e7f2b85b5538cb1","name":"check_gx.py","bytes":6903},{"sha256":"b8ed825a6f0ae8d55567c0fcbb448e34a8b9f127ef87dad4ce00dcaa5a2742cb","name":"check_gx.out","bytes":1253},{"sha256":"751c81de6871303a7c61cd1bf453dce6c667498bd440d8d864cf58078089d750","name":"check_gx.control.out","bytes":1295},{"sha256":"48fa4a29d1da8f7cbe6464a41ec8786be4b08195bfb16788af154b69654271e5","name":"compute_gx.py","bytes":15919},{"sha256":"8ddc37bf58bfc2f164b2e2dc01ea99e9686b74c969eb53f3b03fe80f77dea17e","name":"compute_gx.json","bytes":72709},{"sha256":"87a7f9582e484a2f68fd581297d68cf182d6dcdecf74ff17e792ddb215fc5647","name":"compute_gx.out","bytes":6226},{"sha256":"ac1a78f1553e9a8b0d80e3ce7b9e91575605c715d691dab7f67b3eb5fbe6f243","name":"fetch_gx.py","bytes":1050},{"sha256":"271d66123663c36ba3e0a007afa73a45b2b2795b295df7ceec67f228a7259df8","name":"fetch_files_gx.py","bytes":1574},{"sha256":"be2a0e243ececa06a5f8d7d45d298a16dd945ecdd411874486d64149281cba5b","name":"fetch_oeis_gx.py","bytes":998},{"sha256":"a6139b2e5bb95a75d1fa66c7d364c78b973337a0c603138f1afd061ee3ee0bd4","name":"build_payload_gx.py","bytes":2567},{"sha256":"b91a23560fca983a0633697a27fd82177cbac793c02d500ef31b35eab8809484","name":"redact_gw.py","bytes":2562},{"sha256":"c8e70b025b7765d3be1823d9b48a4dab843824a004b33ab867590660ba235c6e","name":"upload_gx.py","bytes":3707},{"sha256":"b845b43abd9fc40ea23b80520df353872cec475662b7e6834b9693e99ef27acb","name":"route_242.json","bytes":23312},{"sha256":"62072888fe5638dfd30bc700ac555194d1f3793044035bad321da4e62a233f3e","name":"return_2616.json","bytes":29092},{"sha256":"291725434f9cfea8ed660f7da758a1bc4f564191d900c7a18a0acea0049c49d0","name":"route_87.json","bytes":167427},{"sha256":"aa3712dfc4104715baa4df8ad2dd83e281896d61f5b5a502212436be30a17bc5","name":"explore-report.md","bytes":8694},{"sha256":"478d56b6d8c963cfb2b6d2f7e7c4eec9cbdb71caa6015546cfa454f568c81034","name":"explore-evidence.md","bytes":2504},{"sha256":"5ba9e67ed334f092a605dc0d0b4a7d64fec861272b3068ea1edfebb935b4c1b3","name":"explore-prior-art.md","bytes":3959},{"sha256":"163d000c327d57b618627aa7b4dc7388d0f0afd2df458102f53286055034fc2a","name":"explore-recipe.md","bytes":2703},{"sha256":"62d56822a35bb75ab8d266b6dc62c82f59c5e9c88687b0af3698ba55fe626e3e","name":"explore-next-step.json","bytes":1965},{"sha256":"72c337c0b301804842a342dc753a9c64314252df5bfb24287527a8da8f093855","name":"twin-census-calibration.py","bytes":3472},{"sha256":"11bd61f9f40c9f097ddad9280a8b740d7403278ac7c903c7ce9509b86b8d2502","name":"twin-census-calibration.json","bytes":4313},{"sha256":"9d5b98dc2b5910262bfe82563fa720be22e09c660e64f2229db1b6463ac3c5dd","name":"twin-census-ladder.json","bytes":5810},{"sha256":"6eaaa1413afbe6934ae5e6ae2bcf3a47f3d7bb0af609ae8979cfd65f4355a5e5","name":"served-route87.json","bytes":4663},{"sha256":"73b5e7e22ce1de822b6fdb3985a58326f2204cdd7c4c8ca168d1eb09a6d18d61","name":"oeis-A007508.html","bytes":31306},{"sha256":"427627b26266332e68941f3bd66884b378fe4bdeabc01ff4bd1c13e5cc65dca3","name":"mathworld-twinprimes.html","bytes":96735},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}