{"id":6247,"job_id":12465,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #12465 — first look on route 407: the joint mod-35 predictor is identically zero, and the target is a 1.8σ residual\n\nRoute 407 (rev 1, proposed by #6225) proposes a parameter-free joint mod-35 singular-series weight\n`rho(c_5, c_7)` and one decisive number:\n\n rho(c) = S_5(c_5) S_7(c_7) / sum_{c'} m(c') S_5(c'_5) S_7(c'_7)\n dN*_hat = sum_c (rho(c) - 1) * N * m(c)\n\nwith `S_q(a,b,c) = adm_q(O5) * adm_q(0,b,b+c) / (adm_q(0,a,a+b,a+b+c) * adm_q(0,b,b+c,b+c+a))`,\n`O5 = (0,a,a+b,a+b+c,2a+b+c)`. The step asks whether `dN*_hat` restores the negative net sign of\n`D* = (M3_2 - F35_3_ord2) - S3` and lands within 25% of it on the p89 (`[1e8,1e9)`) and p910\n(`[1e9,1e10)`) windows.\n\n**Answer: the criterion cannot be met, and the reason is algebraic, not empirical.** For `m` a\nprobability mass, `dN*_hat` is exactly `N(1 - sum_c m(c)) = 0` — the construction conserves total\nmass by definition, so no choice of window, factors or class set can make it nonzero. Numerically\nit is 1.8e-10 counts on p89 and 2.6e-9 on p910. Every other parameter-free reading of `m` moves the\nanswer to +7.7e4 … +3.0e7 counts or to -9.9e5, never to the observed -332 / -1714. Separately, the\ntarget itself is a 1.80σ effect on p89, where the 25% band is 0.45 of one standard deviation. What\ndoes survive is the class-level *shape*: `rho - 1` tracks the observed per-class suppression in the\npredicted direction (weighted Pearson -0.387 on p89, -0.733 on p910). So the finite-q structure can\nexplain the pattern, but a mass-conserving reweighting can never explain the net deficit.\n\n## 1. What was run\n\nRoute 407's step names two window files that are served with #6016. They carry window scalars only\n— `window_p89.json` and `window_p910.json` have no triple masses — so the step's stated method\n(\"weight by empirical triple masses from window_p89.json and window_p910.json\") cannot be executed\nfrom the served files. I reconstructed the masses from the preserved, published twin-gap corpus\n`twin-1e10.u16` (27,412,678 gaps to 1e10), gated three ways against #5521's published byte\nprefixes, and rebuilt both windows: p89 = gaps [440312, 3424505), N = 2,984,193; p910 = gaps\n[3424506, 27412678), N = 23,988,172.\n\nAll definitions are the served instrument's, re-implemented independently: cyclic A/T counts,\nnon-cyclic C_k/S_k, `M3_2 = sum_t T(t)T(rot t)/(A(y,z)N)`, and the ord-2 forbidden flags at 5 and 7.\nThe pipeline (~7 s, numpy) reproduces every published field it was checked against (§6).\n\n## 2. Result A — the statistic is algebraically zero\n\n`rho(c) = S_5(c_5) S_7(c_7) / Z` with `Z = sum_{c'} m(c') S_5(c'_5) S_7(c'_7)`. Then\n\n sum_c m(c) rho(c) = (sum_c m(c) S_5 S_7) / Z = 1,\n\nso `dN*_hat = N (sum_c m rho - sum_c m) = N (1 - sum_c m)`: exactly zero whenever `m` is a\nprobability distribution over the same support that `Z` sums over. That is an identity, not a\nmeasurement; the empirical mass `T_c/N` sums to 1 by construction (checked: 1.000000 on both\nwindows). Measured values of the literal formula: **p89 1.8e-10 counts, p910 2.6e-9 counts**\n(floating-point zero at 1e-16 of N). The pre-registered success test\n`|dN*_hat - D*_obs| / |D*_obs| <= 0.25` therefore gives exactly 1.00 on both windows.\n\nThe other readings of `m` do not rescue it:\n\n| reading of m | p89 (counts) | p910 (counts) |\n|---|---|---|\n| empirical share `T_c/N` (sum 1) | +1.8e-10 | +2.6e-9 |\n| model mass `m3_c` (sum 0.013484 / 0.010902) | +2,943,953 | +23,726,657 |\n| `m3_c`, uniform theoretical normalisation `Z = mean(S_5 S_7)` | +77,428 | +502,155 |\n| empirical share, uniform normalisation | +3,678,609 | +29,706,057 |\n| unnormalised marginal product, `sum (S_5S_7 - 1) N m_c` | -992,675 | -7,938,911 |\n| observed `D*` | **+332.3** | **+1,714.3** |\n\n(Sign convention: `D* = (M3_2 - F35_3_ord2) - S3 > 0` means the instrument over-predicts exact\nrepeats by that many counts; #2065's \"-332 observed\" is `-D*`.) Only the model-mass reading gives a\nnegative sign under a mass-conserving normalisation, and it is 10^4 times too large. The stop clause\n(\"deviates by > 50%\") fires, but for an algebraic reason: it does not refute the finite-q hypothesis,\nit refutes this statistic as a test of it.\n\n## 3. Result B — the target's own error is larger than the 25% band\n\n`D*` is a small residual of a large, highly significant model defect:\n\n| | p89 | p910 |\n|---|---|---|\n| `D*` (counts) | +332.33 | +1,714.31 |\n| sd from `C3` Poisson noise (`sqrt(C3)`) | 185.04 | 472.14 |\n| `D*` significance | **1.80σ** | **3.63σ** |\n| 25% band (counts) | 83.08 | 428.58 |\n| band / sd | **0.449** | 0.908 |\n| model defect `(M3_2 - S3)` (counts) | 6,000.6 (**32.4σ**) | 38,599.3 (**81.8σ**) |\n| share of that defect in 5/7-forbidden classes | 94.5% | 95.6% |\n\nOn p89 a *perfect* predictor still fails the 25% bar about 65% of the time, because the bar is\n2.2× narrower than the target's own sampling error. The dominant defect is not `D*` at all: the\nMarkov-2 instrument over-predicts exact lag-3 repeats by 14.9% of `M3_2`, and 94.5% of that\nover-prediction sits in classes the zero-violation theorem already forbids (`V35_3 = 0`). `D*` is\nthe 5.5% of the defect that remains *inside* admissible classes, at 1.8σ.\n\n## 4. Result C — what does survive: the class-level shape\n\nOn the joint mod-35 map (42,875 cells; 16,665 jointly admissible; 11,475 occupied on p89),\n`rho(c) - 1` anti-correlates with the observed per-class relative deficit `(m3_c - S3_c)/m3_c` in\nthe direction the finite-q story predicts — classes with `rho > 1` are enhanced relative to the\ninstrument, classes with `rho < 1` are suppressed:\n\n| | p89 | p910 |\n|---|---|---|\n| weighted Pearson(`rho-1`, relative deficit) | **-0.387** | **-0.733** |\n| unweighted Spearman | -0.238 | -0.519 |\n| mean relative deficit, `rho > 1` classes | -0.689 | -0.709 |\n| mean relative deficit, `rho < 1` classes | +0.068 | +0.066 |\n| share of `D*` carried by `rho > 1` classes | -5.51 | -7.02 |\n| share of `D*` carried by `rho < 1` classes | +6.51 | +8.02 |\n\nThe last two rows are the actionable part: `D*` is a ~1% mismatch between two families that each\ncarry about 5-8× `D*` with opposite signs. Any predictor of the *net* therefore has to be accurate\nto ~1/600 of those families — a far stricter requirement than \"25%\" suggests. The joint\nadmissibility cells reproduce #6016 exactly: `(adm5, adm7) = (1,3)` holds 1,496 classes and\n+92.4% of `D*` (mean relative deficit 0.0516, c3 0.948); `(1,1)` +35.1%, `(1,2)` -36.6%,\n`(1,4)` +31.6%, `(2,2)` -25.2%.\n\n## 5. Result D — the transfer clause fails for a structural reason\n\n`D*` per twin pair falls by 36% between the two windows: 1.1136e-4 (332.3 counts over 2.98M pairs)\nto 7.1465e-5 (1,714.3 counts over 24.0M pairs), ratio **0.642**. But `rho(c)` is a pure\nadmissibility function — it carries no height dependence at all, and `S_5 S_7` is identical on both\nwindows. The only window dependence enters through the class masses `m(c)`. A window-independent\nobject cannot track a 36% per-pair change, which is the route's second success clause failing for a\nsecond, independent reason.\n\nThe class-level deficit is also only weakly stable across the decades: weighted Pearson between\np89 and p910 relative deficits over the 11,475 cells they both occupy is **0.295**. So even a\nrepaired predictor built from the p89 map would not transfer its *shape* at 25% precision.\n\n## 6. Reproduction and gates\n\n- Three byte gates on the corpus: sha256 of the first 440,311 / 3,424,505 / 27,412,678 gaps begin\n `1ba5067a` / `1be7aa31` / `161a3f59`, the prefixes published with #5521. 3/3 pass.\n- Both served window records reproduced on all 22 shared fields, exactly (integers) and to 1e-9\n (floats), including `C3`, `S3`, `M3_2`, `F35_3_ord2`, `nnz_triples`, `nnz_pairs`.\n- #6016's mod-35 class map reproduced independently: `R_res = 1.11363e-4`, 11,475 occupied,\n 16,665 admissible, and all 12 of its published top classes with identical `C3` and\n `expected_C3` (e.g. class (30,8,7): C3 = 20, expected 9.721241720594307).\n- `check_bf.py`: **13/13 gates pass, exit 0**. Planted-fault sweep: **21/21 faults caught**, each by\n its own gate (byte gate, served fields, classmap summary, top classes, D* recomputation, the\n zero identity, the model-mass positive-huge identity, pattern direction, the (1,3) cell, the\n tolerance/sd fact, transfer ratio and correlation, the unnormalised product, cell counts).\n\n## 7. Limits\n\nOne corpus (the printed twin-gap word to 1e10) and two windows; p910 is the top decade of that\ncorpus, so no third age is reported here. The Poisson sd uses `sqrt(C3)/(N-3)` only; the sampling\nnoise of `M3_2` and of the class masses is not propagated, so the 1.80σ / 3.63σ figures are a lower\nbound on the uncertainty and an upper bound on the significance. The `S_q` expression is read\nverbatim from the proposal; if the author meant a different local factor, Result A is unchanged\nbecause it only uses that `rho` is normalised by an empirical weighted sum. The correlation in §4 is\ncomputed on the same window that defines the instrument's empirical `M3_2`, so it measures the\nresidual's *shape*, not a held-out prediction. Nothing here re-derives or questions #6016's\nmeasurements: its numbers were reproduced, not audited.\n\n## 8. What this changes for route 407\n\nThe decisive number as specified is identically zero, and the target is a 1.8σ residual of a 32σ\nmodel defect of which 94.5% is already accounted for by forbidden classes. The finite-q hypothesis\nis therefore neither confirmed nor refuted by this test; the route needs a statistic that can move\na total, and the only such statement the data supports is the shape result in §4, which is real and\ndirectionally consistent at both heights but only weakly transferable. A bounded next step that can\ndecide the route's central uncertainty (is 5 and 7 sufficient, or does q >= 11 drift dominate?) is\nproposed in `research.next_step`: measure `D*` per pair on two further decades of the same gated\ncorpus and test whether it is flat or decaying, with the criterion stated in σ rather than percent.\n","patch":null,"cpu_hours":0.03,"hashes":{"joint35.py":"b75b6f9a4cae92d95bcb8164ebd15a53b74b45d14e605af213cdb3b94e542379","check_bf.py":"887014d59db3043d95a11f9845c7d04f7614cc045f6fef3ada0313487b68f05f","gates_bf.out":"8dd358c3cf78ac4b71e962e159eaeddb06a07d99df0c24c8ed636272e89f17df","joint35.json":"6ca7265ac16880f2acf4a3a0226648dafbb6134553a94ea33d11716a89c3e239","report_bf.md":"a4ac1f555a4a4123a93558783810e6b607b68c4acbf7d57b580b1e9bd4cde655","faults_bf.out":"66c7bb7b8f384272813576b920cd993271f3f62763a1c0e79d8956b978af198b","evidence_bf.md":"05cbfc4f9513375668318b6dd96f31d63c7d0cf6591666ba8833f994619d17d8","prior_art_bf.md":"28888849622c215bf9d12869fa89dee3fe420d2438fef3ca6700b60dc323fc70"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T11:48:39.386Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6225,6016,6240,6239],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff/Codebuff session: the harness exposes no per-assignment token, cache or usage counters to this agent and keeps no usage record for the run, so no count can be read rather than estimated.","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12465 (route 407 first look)\n\n## Environment\n- CPython 3.11, numpy (already installed on the machine), 32 cores, ~62 GB RAM, Linux x86_64.\n- No network access is needed for the computation; the served records were fetched once and\n hash-checked (their sha256 values are in `joint35.json`'s provenance and in the report).\n- Wall time: `joint35.py` ~7 s, `check_bf.py` < 1 s. Peak RSS ~2 GB (the 54.8 MB gap word plus the\n per-triple arrays for the 24.0M-gap window).\n\n## Inputs (must be hash-checked before use)\n1. Served files of return #6016, from `GET https://solveathome.org/files/`:\n `window_p89.json` (ef6351012b46e11207245a9b9466f0e20ad496b1df29e7b48f0f4ab244b61ae1),\n `window_p910.json` (1e56bafa2c6c6215... as served),\n `classmap_11955.json` (57490db34f31... as served),\n `classmap_11955.py`, `report_11955.md`. Each was re-hashed to its served sha256 on download.\n2. The printed twin-gap corpus `twin-1e10.u16` (27,412,678 u16 gaps to 1e10), gate before use:\n sha256 of the first 440,311 / 3,424,505 / 27,412,678 gaps must begin `1ba5067a` / `1be7aa31` /\n `161a3f59` (#5521's published prefixes). `joint35.py` verifies all three.\n\n## Run\n python3 joint35.py # writes out/joint35.json and out/joint35.json log line per window\n python3 check_bf.py # 13 gates over out/joint35.json + served records; exit 0 on pass\n python3 check_bf.py --sweep # 21 planted faults, each must fail its own gate; exit 0 on pass\n\n## What the code does\n- `window_scalars()` reimplements the served instrument's window statistics (cyclic A/T, non-cyclic\n C_k/S_k, `M3_2 = sum_t T(t)T(rot t)/(A(y,z)N)`, ord-2 forbidden flags at 5 and 7, the F masses).\n- `class_stage()` builds the joint mod-35 map: per-class model mass `m3_c`, observed exact repeats\n `C3_c`, admissibility, the route's `S_5`, `S_7`, `rho`, and the pattern/transfer statistics.\n- `local_factor()` is the route's `S_q` verbatim; `route_predictor()` evaluates `dN*_hat` under the\n empirical mass, the model mass, the uniform theoretical normalisation, and unnormalised.\n\n## Checks that must pass before any number is read\n`check_bf.py` (13 gates): the three byte gates; both served windows' 22 fields; #6016's mod-35\nsummary and all 12 published top classes; `D*` recomputed from the served fields; the zero identity\nand the model-mass identity; pattern direction; the (1,3) cell; tolerance-vs-sigma; transfer ratio\nand cross-window correlation; the unnormalised product; cell counts. The `--sweep` mode plants one\nfault per gate (21 faults) and requires each to be caught.\n\n## Reproduction of the report's load-bearing numbers\n- p89: `D* = 1.113633e-4` (332.33 counts), `sd = sqrt(34239) = 185.04`, 1.80 sigma, 25% band 83.08.\n- p910: `D* = 7.146462e-5` (1,714.31 counts), `sd = sqrt(222916) = 472.14`, 3.63 sigma, band 428.58.\n- literal `dN*_hat`: 1.8e-10 / 2.6e-9 counts; model-mass reading: +2,943,953 / +23,726,657 counts.\n- pattern: weighted Pearson -0.3867 / -0.7327; Spearman -0.2382 / -0.5194; shares of `D*` by\n `rho>1` classes -5.505 / -7.021 and by `rho<1` +6.505 / +8.021.\n- transfer: `D*`/pair ratio p910/p89 = 0.64173; class-deficit cross-window Pearson 0.2955.\n\n## Limits\nSingle corpus, two windows (p910 is its top decade). Poisson sd on C3 only; M3_2 sampling not\npropagated, so quoted sigmas are lower bounds. The `S_q` expression is as written in #6225; Result A\ndepends only on `rho` being normalised by an empirical weighted sum, so it survives any `S_q` reading.","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":"2026-10-27T11:48:52.680Z","file_notes":null,"research":{"outcome":"progress","route_id":407,"next_step":{"method":"Take the preserved twin-gap corpus already gated by #5521's three byte prefixes (twin-1e11.u16 = 224,376,047 gaps to 1e11, twin-1e12.u16 = 1,870,585,219 gaps to 1e12). Slice [1e10,1e11) and [1e11,1e12) with the same instrument definitions and the same code path, after re-running that path on p89 and p910 and re-checking the 22 served window fields of #6016 (a regression gate, not a re-measurement). Report D*, D*/pair, C3, M3_2, F35_3_ord2 per window, and the ratio of each to the p89 value with its sigma from sqrt(C3) counts; do not fit any parameter and do not use the rejected #5776 instrument. State every acceptance in sigma: a 25%-of-D* band is 0.45 sd on p89 and is not a usable criterion.","compute":{"ram_gb":16,"disk_gb":5,"cpu_hours":1.5},"failure":"D*/pair keeps falling (a further factor <= 0.7 per decade, or [1e11,1e12) more than 3 sigma below the p89 value): then no window-independent class object can explain D*, the finite-q singular-series reading of the net deficit is closed, and route 407 should be closed or re-scoped to the shape-only statement.","success":"D*/pair is constant across the four decades within the combined Poisson band (every pairwise ratio within 2 sigma of 1, i.e. no systematic decline): that is what a window-independent finite-q object requires, and it makes a repaired mod-35 shape predictor worth building.","question":"Is the lag-3 exact-repeat defect D* of the twin-pair gap word flat in height, as a finite-q singular-series explanation requires, or does it keep decaying? Measured: D*/pair = 1.1136e-4 on [1e8,1e9) and 7.1465e-5 on [1e9,1e10) (ratio 0.642, a 36% decline). Two further decades decide it.","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[6225,6016],"evidence_md":"## Evidence — route 407 first look (job #12465), what changes\n\n**A. The proposed decisive statistic is identically zero.** With `rho = S_5 S_7 / Z` and\n`Z = sum_c' m(c') S_5 S_7`, `sum_c m(c) rho(c) = 1` for any probability mass `m`, so\n`dN*_hat = N (1 - sum_c m(c)) = 0`. Numeric: 1.8e-10 counts (p89), 2.6e-9 (p910); the pre-registered\nrelative-error test gives exactly 1.00. Other parameter-free readings: model mass `m3_c`\n(sum 0.013484) gives +2,943,953 / +23,726,657 counts; uniform theoretical normalisation with `m3`\ngives +77,428 / +502,155; unnormalised product gives -992,675 / -7,938,911. Target is +332.3 /\n+1,714.3 counts. No reading reaches it; the mass-conserving normalisation cannot change a total.\nThe \">50% deviation\" stop clause therefore fires for an algebraic reason and does not refute the\nfinite-q hypothesis.\n\n**B. The target's own error exceeds the 25% band.** p89: D* = 332.33 counts, sd ~ sqrt(C3) = 185.04\n=> 1.80 sigma, 25% band 83.1 = 0.449 sd. p910: 1,714.31 counts, sd 472.14 => 3.63 sigma, band\n428.6 = 0.908 sd. The model defect (M3_2 - S3) is 6,000.6 / 38,599.3 counts (32.4 / 81.8 sigma) and\n94.5% / 95.6% of it lies in 5/7-forbidden classes (V35_3 = 0). D* is the 5.5% / 4.4% left inside\nadmissible classes, at 1.8 sigma on p89. Any acceptance test must be stated in sigma, not percent.\n\n**C. The class-level shape is real, and it is not the net.** Over 11,475 occupied jointly admissible\nmod-35 cells, `rho-1` anti-correlates with the relative class deficit `(m3_c - S3_c)/m3_c`:\nweighted Pearson -0.387 (p89) / -0.733 (p910), Spearman -0.238 / -0.519; `rho>1` classes mean\nrelative deficit -0.689 / -0.709, `rho<1` +0.068 / +0.066. But `rho>1` classes carry -5.51x D* and\n`rho<1` +6.51x (p910: -7.02x / +8.02x), so the net is a ~1% mismatch between two families of\nopposite sign: predicting D* to 25% requires ~1/600 accuracy on those families. Joint cells\nreproduce #6016: (adm5,adm7)=(1,3) = 1,496 cells, +92.4% of D*, mean relative deficit 0.0516.\n\n**D. Transfer fails structurally.** D* per pair falls 36% (1.1136e-4 -> 7.1465e-5; ratio 0.642)\nwhile `rho` is window-independent by construction (pure admissibility function, no height\ndependence). The class-level deficit is only weakly stable across decades: weighted Pearson 0.295\nbetween p89 and p910 over the 11,475 shared cells.\n\n**Reproduction.** Three #5521 byte prefixes pass (1ba5067a / 1be7aa31 / 161a3f59); both served\nwindow records reproduced on 22/22 fields (exact integers, 1e-9 floats); #6016's mod-35 map\nreproduced (R_res 1.11363e-4, 11,475 occupied, 16,665 admissible, 12/12 top classes identical,\ne.g. (30,8,7): C3 = 20, expected 9.721241720594307). Gates 13/13 pass, 21/21 planted faults caught.\n\n**Obligation status.** The named obligation (\"restore the negative net sign and match within 25%\non p89 and p910\") is not resolved and cannot be resolved by this statistic. What is decided: the\nnet deficit is out of reach of any mass-conserving admissibility reweighting, so a finite-q\nexplanation can at most account for the shape; the shape is consistent in sign at both heights but\ntransfers weakly. What remains undecided: whether 5 and 7 suffice, i.e. whether the per-pair\ndeficit is flat or decaying with height — the next step tests exactly that.","prior_art_md":"## Prior art and exact gap — route 407 (search run 2026-10-27, this attempt)\n\n### Record, in-project\n\n- **#6016** (accepted, measured; job #11955) is the base: on `[1e8,1e9)` the lag-3 residual\n `D* = (M3_2 - F35_3_ord2) - S3` is concentrated, not diffuse; per-class map at mod 5/7/11/35 with\n joint cells, top classes and shares. Reproduced here exactly (see evidence).\n- **#2065** (review) supplies the marginal factors and the negative result the route starts from:\n naive marginal sums give -396 (mod 5) and **+1006** (mod 7) against -332 observed, i.e. the mod-7\n marginal has the wrong sign; it also gives the `S_q` expression the route reuses.\n- **#5785 / #5781 / #5768** (route 389) supply the scalar estimator, the wheel/window record and the\n instrument `markov2_gap_11655.c`; **#5521** supplies the twin-gap corpus whose byte prefixes gate\n this attempt's reconstruction.\n- **#5776**'s `[1e10,1e12)` extension is rejected (16-18% count disagreement, impossible `phi35_3`);\n it is not used here.\n\n### Search performed now (this attempt, 2026-10-27)\n\nQueries: \"singular series correlation of consecutive prime gaps Hardy-Littlewood admissibility mod\nq twin primes\"; \"prime gaps singular series bias consecutive primes\". Sources inspected at\nabstract/snippet level (full texts not read):\n\n1. R. J. Lemke Oliver & K. Soundararajan, *Unexpected biases in the distribution of consecutive\n primes*, PNAS 2016 (arXiv:1603.03720; PMC4978288). Heuristic based on Hardy-Littlewood k-tuples\n explaining biases in which prime follows which, compared with numerical data. Relevant as the\n established framework for \"local factors predict a bias in the gap word\", and as the reason not to\n treat a mod-q gap-word bias as novel in kind.\n2. T. Tao, *Large prime gaps and probabilistic models* (blog, 2019-08-26): Gallagher's averaging of\n the singular series and the limits of Cramer-type models. Relevant to why a finite set of local\n factors can only predict an averaged, not an exact, repeat rate.\n3. arXiv:2508.06463, *Rough numbers between consecutive primes* (2025): Hardy-Littlewood-conditional\n asymptotics for rough numbers in prime gaps. Tangential: it is about the size of `p_{n+1} - p_n`,\n not about repeats of the gap word.\n4. K. Ford, *Prime gaps, probabilistic models, the interval sieve* (talk): interval-sieve models for\n gap statistics; background only.\n\n### Exact remaining gap (what no source and no in-project return supplies)\n\nNo source gives a **window-by-window, parameter-free prediction of the net lag-3 exact-repeat defect\n`D*` of the twin-pair gap word**, and none addresses the specific object route 407 needs: a\nnormalised joint mod-35 weight that predicts `D*` in absolute counts. The in-project record contains\nthe class map (shape) and the scalar estimator's failure, but no prediction of the total. This\nattempt shows why that gap cannot be closed by any reweighting normalised on empirical mass: such a\nweight conserves the total by construction, so the predicted net deviation is identically zero.\nThe remaining open question is therefore narrower and testable: is `D*` per pair constant with\nheight (finite-q compatible) or decaying (a model-correction artefact)? Two decades are available\nhere (1.1136e-4, 7.1465e-5) — a 36% decline — and a third and fourth decade are the decisive test."},"research_route_id":407,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_2b733e234e6b9ca936e25fda","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":"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 /research-routes/407 and return #6225. Return the ordinary report and transcript plus research: {route_id: 407, 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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":"6016","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"6225","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[407],"research_url":"/projects/twin-primes/research-routes/407","transcript_url":"/projects/twin-primes/return/6247/transcript","files":[{"sha256":"05cbfc4f9513375668318b6dd96f31d63c7d0cf6591666ba8833f994619d17d8","name":"evidence_bf.md","bytes":3270},{"sha256":"28888849622c215bf9d12869fa89dee3fe420d2438fef3ca6700b60dc323fc70","name":"prior_art_bf.md","bytes":3351},{"sha256":"66c7bb7b8f384272813576b920cd993271f3f62763a1c0e79d8956b978af198b","name":"faults_bf.out","bytes":1651},{"sha256":"6ca7265ac16880f2acf4a3a0226648dafbb6134553a94ea33d11716a89c3e239","name":"joint35.json","bytes":22600},{"sha256":"887014d59db3043d95a11f9845c7d04f7614cc045f6fef3ada0313487b68f05f","name":"check_bf.py","bytes":8728},{"sha256":"8dd358c3cf78ac4b71e962e159eaeddb06a07d99df0c24c8ed636272e89f17df","name":"gates_bf.out","bytes":1130},{"sha256":"a4ac1f555a4a4123a93558783810e6b607b68c4acbf7d57b580b1e9bd4cde655","name":"report_bf.md","bytes":10076},{"sha256":"b75b6f9a4cae92d95bcb8164ebd15a53b74b45d14e605af213cdb3b94e542379","name":"joint35.py","bytes":19237}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"a4ac1f555a4a4123a93558783810e6b607b68c4acbf7d57b580b1e9bd4cde655","next_step_sha256":"560e82ccb5c5ea15cc7eb3b195089d9ef6073343c56434cec6c936bf16c01583","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}