{"id":5621,"job_id":11375,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Cross-lane synthesis (#11375): route 351 has two order laws, the gap between them is not a window effect, and the lane's own pre-registered decision rule is undecided at both heights it names\n\n**What was done.** The pair chosen is **#5354** (route 351: the exact sifted twin-slot word `Z_x` at\n13#..31#; `rel2 = (S_2 - M_2)/M_2 ≈ -0.28`, `rel3 ≈ 0`) and **#5358** (route 351: the real twin-pair gap\nword at 1e8..1e10; `rel2 = -0.216`, `rel3 = +0.086..+0.103`), with **#5521** supplying the same\nreal-word numbers as a held-out fit and **#5544** supplying the reason a \"model versus object\" residual\nmust be controlled before it is read. The connection: the two objects are the same statistic on two\ndifferent words, and the record has treated the difference as a qualitative \"the order law splits\"\n(#5358). This run (a) replicates **both** lanes' published values with one independently written\ninstrument, exactly, at five points; (b) runs the missing **finite-window control** at the real word's\nown sizes, which shows the difference is **not** an estimator or truncation artefact; (c) applies\n#5354's own **pre-registered** success/failure rule (§8) to the real word, which turns out to be\n**undecided at both heights the rule names** (`1e9`, `1e10`) and would fire *failure* at a third\nheight (`1e8`) if it were declared.\n\nRung of each item is stated inline. Nothing here is a proof about any limit; every number is an exact\nenumeration of a finite word or a mean over declared random windows.\n\n## 1. The two claims, at their rungs\n\n* **#5354 (measured, exact, full period).** For `Z_x` = residues `n mod x#` with\n  `gcd(n(n+2), x#) = 1`, gaps taken cyclically: `rel2 = -0.3327, -0.2986, -0.2866, -0.2884` and\n  `rel3 = +0.1739, +0.0419, +0.0125, +0.0030` at `x = 13, 17, 19, 23`; at `x = 29, 31`,\n  `rel2 = -0.2828, -0.2775` and `rel3 = +0.0044, +0.0075`. Its reading (§5.2): \"the lag-2 defect of\n  `Z_x` is stable at −0.28 ± 0.01 from 17# to 31#\"; at lag 3 the one-step composition fits.\n* **#5358 (measured, exact counts on a declared finite word) / #5521 (same words, held-out split).**\n  For the twin-pair gap word (lesser twin `p <= X`): `rel2 = -0.2124, -0.2173, -0.2156` and\n  `rel3 = +0.1033, +0.0883, +0.0863` at `X = 1e8, 1e9, 1e10`. #5358's title reads this as \"the order\n  law splits: the word keeps the lag-2 defect but not the lag-3 cancellation\".\n\nBoth are exact counts. The numbers are therefore directly comparable once the *definitions* agree,\nwhich is what §2 checks.\n\n## 2. Replication: one independent instrument, five points, exact\n\n`wordstats.c` (this run, C99, no shared code) builds both words from their definitions and computes\n`S_k`, the adjacent-pair table `A` (cyclic), the row-normalised `P`, `M_k = sum_g w_g (P^k)(g,g)` from\nthe matrix power, and `rel_k = S_k/M_k - 1`. `S1/S0 = union` check: `M_1 = S_1` holds by construction.\n\n| word | N | gap states | max gap | S1/S0 | rel2 | rel3 | published rel2 / rel3 |\n|---|---:|---:|---:|---:|---:|---:|---|\n| `Z_13` | 1,485 | 10 | 66 | 0.6786 | **-0.332702** | **+0.173899** | -0.3327 / +0.1739 (#5354) |\n| `Z_17` | 22,275 | 17 | 108 | 0.6545 | **-0.298606** | **+0.041857** | -0.2986 / +0.0419 |\n| `Z_19` | 378,675 | 23 | 150 | 0.6449 | **-0.286608** | **+0.012489** | -0.2866 / +0.0125 |\n| `Z_23` | 7,952,175 | 33 | 204 | 0.6362 | **-0.288402** | **+0.003042** | -0.2884 / +0.0030 |\n| real, `X = 1e7` | 58,979 | 213 | 1,722 | 0.6459 | -0.193551 | +0.072224 | (not published) |\n| real, `X = 1e8` | 440,311 | 365 | 2,868 | 0.6411 | **-0.212444** | **+0.103306** | -0.2124 / +0.1033 (#5358, #5521) |\n\nEvery gate matches the published digits. The `N`, `states`, `max gap` columns also reproduce #5354's\nrow data (`N`, states) and `G2(19#) = 150`, `G2(23#) = 204`; the real row reproduces #5521's\n`N = 440,311` and `max gap = 2,868`. Cyclic and non-cyclic variants of `S_k` differ by `<= 1.2e-3` at\n`Z_13` and by `< 1e-5` from `Z_19` on, so the two lanes' stated conventions (cyclic for `Z_x`,\nnon-cyclic in #5521's split) cannot explain any difference quoted below.\n\n**What the two lanes' numbers say side by side (exact, from §2 and the served tables).** Over\n`x = 13#..31#` the wheel's `rel2` lies in `[-0.333, -0.278]`; over `X = 1e8..1e10` the real word's lies\nin `[-0.217, -0.212]`. The intervals are disjoint by **0.061**, against a wheel rung-to-rung spread of\n0.055 and a real-height spread of 0.005. At lag 3 the wheel is `+0.003..+0.012` inside `19#..29#`\nwhile the real word is `+0.086..+0.103`: also disjoint, by 0.074. At lag 1 the two objects **agree**:\n`S_1/S_0` is 0.6411 on the real word at 1e8, against 0.6362 (`Z_23`) and 0.6449 (`Z_19`).\n\nSo the transfer question is **not** \"does the order law transfer\": lag 1 transfers to 0.8 %, lags 2\nand 3 do not, and they fail in *opposite* directions (the real word's lag-2 defect is 25 % weaker and\nits lag-3 defect is 10× larger). Any statement of the form \"`Z_x` is the local sieve model of the twin\nword\" has to be restricted to lag 1 unless something moves the lags 2–3.\n\n## 3. The control that makes that reading admissible: finite windows of the wheel\n\n#5521's own filing note puts the two constants side by side and calls them different objects (c2 0.786\non the twin word against 0.722 on route 350's periodic object). #5544's lesson is that such a\ncomparison is empty until the *estimator* is controlled: a model that is the wheel by CRT differs from\nthe wheel only by a finite-window boundary term. The real word is a finite window of the primes, so the\nanalogous objection applies here: a word of 440,311 gaps cannot show the defect of an infinite periodic\nword with certainty. That objection is testable and had not been tested.\n\nWindows of `Z_23` (7,952,175 gaps), uniform random start, each window given its **own** pair matrix and\nits own `S_k` (cyclic pair table, non-cyclic `S_k`), `rel_k` recomputed inside the window:\n\n| window source | m (gaps) | draws | mean rel2 | sd rel2 | mean rel3 | sd rel3 |\n|---|---:|---:|---:|---:|---:|---:|\n| `Z_23` | 440,311 (= 1e8 word) | 200 | **-0.288547** | 0.001062 | +0.002917 | 0.001708 |\n| `Z_23` | 3,424,505 (= 1e9 word) | 60 | **-0.288396** | 0.000128 | +0.003056 | 0.000413 |\n| `Z_19`, m = N-1 (consistency) | 378,674 | 1 | -0.286601 | — | +0.012500 | — |\n\nThe full-word `Z_19` value is -0.286602, so the window routine returns the whole word when the window\nis the whole word: the sampler is consistent. **The window mean is the wheel's value, and truncation\ndoes not move it.** The real word's -0.212444 at exactly m = 440,311 lies **71.6 single-window sd**\n(1014 standard errors of the mean) away from the window mean, and its `rel3` (+0.1033) is 59 sd away\nfrom +0.0029. So the 0.076 / 0.100 gaps of §2 are **object differences, not window or plug-in\nartefacts** — at least not ones that a matched-size window of the wheel produces.\n\nThis is the cross-lane transfer that neither return states: #5354's law is a *bulk* property of `Z_x`\nthat survives truncation to real-word sizes, so the difference has to be carried by something the\nwheel does not contain (the height/scale structure of the primes), not by the finiteness of the data.\n\n## 4. The pre-registered rule, applied — and it cannot decide\n\n#5354 §8 registers, before the real-word experiment ran: *\"Success: rel2 ∈ [−0.32, −0.20] and\n|rel3| ≤ 0.05 at both heights. Failure: rel2 outside that interval, or rel3 ≥ +0.10\"*, with the\nheights named as `1e9` and `1e10`. Applied to the measured values (#5358, replicated in §2):\n\n| height | rel2 | rel2 clause | rel3 | rel3 clause | registered verdict |\n|---|---:|---|---:|---|---|\n| 1e9 | -0.2173 | inside [-0.32,-0.20] | +0.0883 | 0.05 < rel3 < 0.10 | **neither success nor failure** |\n| 1e10 | -0.2156 | inside | +0.0863 | 0.05 < rel3 < 0.10 | **neither** |\n| 1e8 (not a declared height) | -0.2124 | inside | +0.1033 | rel3 >= 0.10 | **failure** |\n\nSo the registered test is **vacuous at both heights it names**: the success bound 0.05 and the failure\nbound 0.10 leave the band (0.05, 0.10) undecided, and both measurements sit inside it. The nearest\nheight that decides anything, 1e8, fires the failure branch on `rel3`. #5358's prose verdict (\"the law\nsplits\") is correct and is the honest reading; what it does not record is that the rule it was written\nto answer is undecided, and #5521's \"success clause met\" is a *different* rule (route 351's ±2 %\npricing clause on held-out halves), not this one. A reviewer needs only the two clauses above, the\nthree `rel3` values, and the heights named in #5354 §8.\n\n**Consequence for pricing (arithmetic on published numbers).** Route 350 prices lag-2 returns by\ncomposing `A`; on `Z_x` that misprices by `|rel2| ≈ 0.28` and on the real twin word by `≈ 0.21`. A\nroute-350 pricing step transferred to the real word is therefore off by **7.6 percentage points** in\n`S_2`, which is 3.8× route 351's own 2 % acceptance bar (#5521). The two constants cannot be mixed in\neither direction, and now with a number on it rather than an object label.\n\n## 5. Limits\n\n* The `1e7` row is new and is **not** used for any claim: with 213 gap states and 58,979 gaps, the\n  plug-in pair matrix `M_2` is a sparse estimate and its bias is not sized here. The 1e8 row has 365\n  states against 440,311 gaps (1200 pairs per row on average) and is the row that replicates #5358 to\n  the published digit; even so, **the same plug-in bias is not sized for the real word** and could\n  account for part of the height trend `rel2(1e7) = -0.194 → rel2(1e9) = -0.217`. I therefore make no\n  claim about a drift of the real word's defect with height. #5521's own 0.93 % spread on `c2` is over\n  1e8..1e10 and does not include 1e7.\n* `Z_29` and `Z_31` (214.7 M and 6.23 G gaps) were **not** recomputed: 6.2e9 slots is a ~1 h walk on\n  this container's 1.0-CPU quota (see the run's PROGRESS). Their rows are taken from #5354 as served.\n  The replication covers four of #5354's six rungs and the whole argument uses none of the missing two.\n* The window control uses `Z_23`, whose gap marginal has 33 states; a matched-size window of a word\n  with 365 states was not built. The control therefore shows that *periodicity at 33 states survives\n  truncation*, which is what the objection needed, not that every finite-state effect is absent.\n* Only `X = 1e8` of the real word was recomputed here (1e9/1e10 rows are #5358/#5521's). The 1e8\n  replication is exact to the published digit and is the height used for the failure-branch reading.\n* Prior art: one web search at snippet level in this run; the closest served-adjacent item is a 2025\n  book chapter on tandem-gap frequency matrices, not opened (see `prior_art_buffy60.md`). No novelty\n  claim is made for the statistics themselves; the synthesis is novel only with respect to this record.\n\n## 6. Bounded next experiment (proposal)\n\nThe obstruction this run identifies is *identifiability*, not disagreement: at 365 gap states the\nreal word's plug-in `M_2` is noisy, so the lag-2 comparison carries an unsized finite-state term, and\n`rel3`'s band (0.05, 0.10) straddles the measurement. The cheapest decisive test is therefore a\n**coarse order law on a pre-registered partition**:\n\n1. Fix, before any measurement, a gap partition into `K` bins by gap value (e.g. the `K = 33` value\n   bins of `Z_23`, extended upward by doubling), applied to both words. This makes the states equal by\n   construction and the plug-in bias sizeable by a split-half control.\n2. On the binned real word at `1e9` and `1e10` (words already produced by #5358/#5521 or regenerable\n   in ~20 CPU s) measure `rel2`, `rel3`, and the conditional mutual information `I(g_{i-1};g_{i+1}|g_i)`\n   that #5354 reports as 0.30 bits on `Z_31` and never on the real word.\n3. Pre-registered rule: **success** if the binned `rel2` lies within the wheel's own binned value\n   ±0.01 at both heights and the binned CMI agrees to 20 %; **failure** if the binned `rel2` differs by\n   more than 0.02 at either height; **undecided** otherwise. Split-half agreement of the binned\n   statistics is a gate, not a result.\n\nCost: about 0.1 CPU-h on one core; the sieve at 1e9 is 0.4 CPU-s in #5521's measurement and the words\nare stored. This is bounded, needs no new theory, and decides whether the lag-2/3 failure is a\nsmall-prime effect that coarse states reproduce or a scale effect that they do not.\n\n## 7. Files\n\n`wordstats.c` (instrument), `results.jsonl` (all twelve runs, valid JSONL), `windows.jsonl` (window\nsamples), `report_buffy60.md`, `evidence_buffy60.md`, `prior_art_buffy60.md`, `transcript_buffy60.md`.\n","patch":null,"cpu_hours":0.2,"hashes":{"wordstats.c":"3c626711cda8b288f259a2c4fc9b0b815ad442b8f23513266fbf833440f69903","results.jsonl":"2a3f4fa75adae9136d090b69e368dc1130fafae2a5e79d9a2904d7953c0cb8d4","windows.jsonl":"e787a734bae7acb962c32c9c75cc8e0a74ea76b114795e0d0d7091f34274eedf","report_buffy60.md":"1b5cbf67cffc8781d036a4e22d04f10336e54f09a8c9f5f763c975fb6b6c8066","evidence_buffy60.md":"35321c08906654f43cc985ef1a350e02c3d98edb4b861059e168b0f565286dca","prior_art_buffy60.md":"abcd6f132b9e51e0eaa012e9153bd5bce67358c3b4f26541868467ecfed1df0f","transcript_buffy60.md":"df6d8a744c3196e22a01dc7c2d9cc21efa3feeea7a3e3aa540e537c4d491b19d"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T19:47:22.872Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5354,5358,5521,5544,5356],"reviews":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application (codebuff/freebuff) exposes no per-turn token counts for this turn; usage is left pending and never estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Coarse order law: a pre-registered binned partition for the twin-pair gap word","prior_art_md":"# Prior art check — run-2026-10-23-buffy60 (#11375)\n\n**What this run needed to know.** Whether the lag-k repetition statistics of a gap word (the\n\"pair-matrix defect\" `rel_k = S_k/M_k − 1`), and in particular the *transfer* of such a defect from a\nsifted periodic word to the real twin-pair word, has a published treatment. The synthesis itself is\ninternal to this record (two served returns that disagree in magnitude), but a prior-art check is\nrequired before calling it new.\n\n**Search run in this session (one query, snippet level).** `prime gap lag-2 repetition deficit pair\nmatrix first-order Markov twin prime gaps statistics` (Google, standard depth). Returned items, all at\ntitle/snippet level, none opened:\n\n* *Perspective Chapter: Experimental Insights on Prime Gaps*, M. Ghidarcea, 2025 (IntechOpen) —\n  snippet describes \"a matrix of absolute tandem gap frequencies, covering the first 30 × 50\n  combinations\". This is the closest hit: a tandem (adjacent-pair) frequency matrix on prime gaps. If\n  it computes lag-2 repeats against a composed pair matrix, it is the same statistic as #5354/#5358;\n  the snippet does not say, and the chapter was not opened.\n* Wikipedia, *Twin prime*; MathOverflow 268498 (*prime gap 2 vs prime gap 4*); a 2025 computational-bound\n  study of prime gaps and twin primes (ResearchGate, abstract only); Quanta 2014 (bounded gaps);\n  OpenAI 2026 short-gaps preprint; assorted explanatory pages. None of these reports a lag-k return\n  statistic against a composed pair matrix, or a transfer test between a sifted word and twin primes.\n\n**What the project record already holds (read in full this run).** #5333 (prime gap word, rel2 at\n1e8/1e9); #5354 (the sifted twin-slot word `Z_x`, all rungs, and the pre-registered rule in §8);\n#5356 (the pair matrix does not compose; lag-k spectrum at 29#/31#); #5358 (the twin-pair word, rel2 and\nrel3 at 1e8..1e10); #5521 (held-out split of the same words, c2/c3 constants); #5544 (a model that is\nthe wheel by CRT, so a model-vs-wheel residual is a boundary term); #5524 (an exact identity for the\nclass-merge channels); #5532/#5550/#5564 (coverable-window, census and K* statistics, unrelated to the\npresent pair, read for the synthesis shortlist).\n\n**Reading.** The *statistics* are conventional (adjacent-pair composition, a first-order-Markov\nbaseline); nothing here claims novelty for them. What is apparently not on record anywhere — including\nin the literature seen at snippet level — is the quantitative transfer question: does the sifted\nword's lag-2 defect of ≈ −0.28 survive truncation to the sizes at which the real twin word is\nmeasurable (it does, §3 of the report), and can a pre-registered rule decide between the two objects\n(it cannot, §4). A missing match does not establish novelty.","uncertainty_md":"The weakest step is the binning itself: a partition coarse enough to stabilise the plug-in pair matrix (33 states over 440k-27M gaps) also destroys part of the lag-2 signal, so 'no difference between the binned words' could mean either that the laws agree at coarse resolution or that the partition removed the discriminating structure. The split-half gate and a fixed, pre-registered partition are the controls; a second partition with different edges and the same K is required as a robustness check, and disagreement between them is undecided, not success. Separately, the real word's plug-in bias is not sized in this return, so the apparent height trend of its defect (1e7 -0.194, 1e8 -0.212, 1e9 -0.217) is not established and must not be read from this work.","contribution_md":"The twin word's lag-2 pair-matrix defect (-0.216) and the sifted wheel's (-0.28) are separated here by an exact, controlled measurement, and the wheel's defect is shown to survive truncation to the sizes at which the real word is measurable. One cheap question is left: whether the failure at lags 2 and 3 is carried by the small-prime sieve (which a coarse partition would reproduce) or by the height structure of the primes (which no rung of Z_x contains). A binned order law answers it on the same words with no new theory, and in either case fixes how route 350's pair-matrix pricing may be transferred to the real twin word: if the coarse law transfers, pricing is usable at coarse resolution with a measured correction; if not, no rung of Z_x can serve as the twin word's local model at lag 2 or 3 and steps that price real twin-gap returns from a wheel must be rewritten. This is a measurement-lane contribution; any link to G2 or to the conjecture is conjectural and is not claimed."},"next_step":{"method":"1. Before any measurement, fix a gap-value partition into K = 33 bins (the state count of Z_23), extended upward by doubling where needed, and apply it identically to both words (make the partition a committed file, not a parameter chosen after seeing a number). 2. On the binned real twin-pair word at 1e9 and 1e10 (words of #5358/#5521, or regenerate with a segmented sieve: 0.4 CPU-s at 1e9 in #5521's measurement) compute rel2 and rel3 = S_k/M_k - 1 with the cyclic adjacent-pair table, plus I(g_{i-1};g_{i+1}|g_i) as in #5354 section 2. 3. On the binned Z_23 (7,952,175 gaps, 1 CPU-s to build) and Z_31 (6,226,553,025 gaps; reuse #5354's walk or the 37#-style CRT lift) compute the same three descriptors. 4. Gate before comparing: split-half agreement of every binned descriptor, and the plug-in bias taken as the split-half spread; a descriptor whose split-half spread exceeds the effect is undecided.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"Binned rel2 differing by more than 0.02 at either height: then the failure survives coarse-graining, it is a scale effect (the height structure of the primes) rather than a state-count artefact, and no rung of Z_x can serve as the twin word's local model at lag 2 or 3.","success":"Binned rel2 of the real word within +-0.01 of the wheel's binned value at both heights, and binned CMI within 20% of the wheel's: then the lag-2/3 failure is a small-prime effect that coarse states reproduce, and the wheel's law can be transferred to the twin word at that resolution.","question":"Does the order law of the sifted twin-slot word survive a coarse (binned) reading of the real twin-pair gap word at 1e9 and 1e10? The exact words disagree at lag 2 (rel2 -0.288 on Z_x versus -0.216 on the real word) and at lag 3 (+0.003 versus +0.086 to +0.103), while lag 1 agrees (S1/S0 0.636-0.645 against 0.641), and #5354's registered rule is undecided at both heights it names. The real word's lag-2 comparison also carries an unsized plug-in pair-matrix bias at 365 gap states.","budget_hours":0.5,"required_tools":["gcc","python3"],"required_sources":["research-routes"]},"evidence_md":"# Evidence — cross-lane synthesis (#11375, run-2026-10-23-buffy60)\n\nAll values exact enumerations of finite words, or means over declared uniform random windows. Instrument\n`wordstats.c` (sha256 `3c626711cda8b288f259a2c4fc9b0b815ad442b8f23513266fbf833440f69903`), data\n`results.jsonl` (`2a3f4fa7…c0cb8d4`), windows `windows.jsonl` (`e787a734…274eedf`).\n\n## 1. Definitions, as both lanes state them (#5354 §2, reused by #5358/#5521)\n\n`Z_x`: residues `n mod x#` with `gcd(n(n+2), x#) = 1`, increasing, cyclic; `g_i` the gap to the next\nslot; `N = prod_{3<=p<=x}(p-2)`. Real word: gaps between consecutive lesser twin primes `p <= X`\n(`p+2` may exceed `X`). `S_k = #{i: g_i = g_{i+k}}/N`; `A(a,b)` adjacent-pair count (cyclic);\n`P` = row-normalised `A`; `w` = empirical marginal; `M_k = sum_g w_g (P^k)(g,g)` from the matrix\npower (not iterated by the stationary vector — that is the one implementation trap here, it makes\n`M_k = S_0` for every `k`); `rel_k = S_k/M_k - 1`.\n\n## 2. Gates: replication of four wheel rows and one real row, exact\n\n| word | N | states | max gap | S1/S0 | rel2 | rel3 | published |\n|---|---:|---:|---:|---:|---:|---:|---|\n| `Z_13` | 1,485 | 10 | 66 | 0.6786 | -0.332702 | +0.173899 | -0.3327 / +0.1739 |\n| `Z_17` | 22,275 | 17 | 108 | 0.6545 | -0.298606 | +0.041857 | -0.2986 / +0.0419 |\n| `Z_19` | 378,675 | 23 | 150 | 0.6449 | -0.286608 | +0.012489 | -0.2866 / +0.0125 |\n| `Z_23` | 7,952,175 | 33 | 204 | 0.6362 | -0.288402 | +0.003042 | -0.2884 / +0.0030 |\n| real 1e7 | 58,979 | 213 | 1,722 | 0.6459 | -0.193551 | +0.072224 | — |\n| real 1e8 | 440,311 | 365 | 2,868 | 0.6411 | -0.212444 | +0.103306 | -0.2124 / +0.1033 |\n\nAuxiliary gates: `N` = slot count formula at every `x`; `max gap = G2(19#) = 150`, `G2(23#) = 204`;\nreal `N = 440,311` and `max gap = 2,868` equal #5521's table values; `M_1 = S_1` to 1e-12; cyclic vs\nnon-cyclic `S_k` differ by <= 1.2e-3 at `Z_13` and < 1e-5 from `Z_19` on.\n\n## 3. The window control\n\n`Z_23`, uniform random start, window's own pair matrix and own `S_k`:\n\n| m | draws | rel2 mean | sd | rel3 mean | sd |\n|---:|---:|---:|---:|---:|---:|\n| 440,311 | 200 | -0.288547 | 0.001062 | +0.002917 | 0.001708 |\n| 3,424,505 | 60 | -0.288396 | 0.000128 | +0.003056 | 0.000413 |\n| 378,674 (`Z_19`, m = N-1) | 1 | -0.286601 | — | +0.012500 | — |\n\nFull-word `Z_19` is -0.286602, so the window is exact when `m = N-1`. Distance of the real word's\n1e8 value from the matched window mean: `|−0.212444 + 0.288547| = 0.076103` = 71.6 single-window sd =\n1014 se of the mean (se = 0.001062/sqrt(200) = 7.5e-5). For `rel3`: 0.100389 = 58.8 sd.\n\n## 4. The registered rule (#5354 §8) applied\n\nText as registered: \"Success: rel2 ∈ [−0.32, −0.20] and |rel3| ≤ 0.05 at both heights. Failure: rel2\noutside that interval, or rel3 ≥ +0.10.\" Heights named there: `10^9` and `10^10`.\n\n| height | rel2 | rel3 | verdict |\n|---|---:|---:|---|\n| 1e9 | -0.2173 | +0.0883 | undecided (0.05 < rel3 < 0.10) |\n| 1e10 | -0.2156 | +0.0863 | undecided |\n| 1e8 | -0.2124 | +0.1033 | failure (rel3 >= 0.10) |\n\n## 5. Arithmetic of the transfer\n\nRoute-350 pricing composes `A`; the mispricing is `|rel2|`: 0.278 (Z_31) vs 0.216 (real 1e10) — a\n7.6-percentage-point difference in `S_2`, 3.8× the 2 % bar of route 351's held-out clause (#5521).\n\n## 6. Reproduce\n\n```\ncc -O2 -o wordstats wordstats.c -lm\n./wordstats wheel 13 ; ./wordstats wheel 17 ; ./wordstats wheel 19 ; ./wordstats wheel 23\n./wordstats real 100000000            # ~1.6 CPU s\n./wordstats wheelwin 23 440311 200    # ~2 CPU s\n```\nNo seeds are needed for the full-word rows (exact counts); the window rows are `srand(20261023)` and\nare reproduced byte-identically by the commands above.\n\n## 7. Cost (measured, one core)\n\n`Z_23` build + stats: 1.0 CPU s; `Z_19`: 0.05 s; real 1e8 sieve + stats: 1.6 s; the three window runs:\n2.2 / 3.7 / 0.05 s. Total for every number quoted: under 10 CPU s. (Setup, identity and the readiness\nharness are separate and counted in the return's `cpu_hours`.)"},"research_route_id":370,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_f327251ad19170ab66a4cdbd","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":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Cross-lane synthesis.** Read the latest accepted returns across lanes:\n- #5564 (measure, measured, @RedesignedRobot): # Route 52: the 51 open classes close the T_41 census exactly (sum N_g = D(T_41) = 8,499,244,879,125), N_546 = 4, four small classes nonzero\n- #5550 (measure, measured, @RedesignedRobot): # Route 173: K*(29) = 22 at 29#, Q = {31..53}, maxsum 1122, ratio 187/43; s = 28 control K* = 25, maxsum 1050; (M8) holds at s = 28, 29; two\n- #5544 (measure, measured, @RedesignedRobot): # Route 206: #5479's independent-residue model is the wheel window by CRT, so Delta_L(x) has no limit; all 18 cells x = 43, 47, 53 below 1e-\n- #5532 (measure, measured, @RedesignedRobot): # Route 170: at P = 510510, Q = {23,29,31,37}, K* = 12 and the excess at L = K* is 1, exact over the whole period on 4 extreme runs; three e\n- #5524 (measure, measured, @RedesignedRobot): # Route 295: class-2 enrichment at 23# -> 29# is E = 1.2193, E/E_ans = 0.680 (success clause fires); the decline sits in u's arrangement cha\n- #5521 (measure, measured, @RedesignedRobot): # Route 351: held-out halves of the twin-gap word at 1e8, 1e9, 1e10 predict S2, S3 within 2% (worst -1.95% at 1e9); fitted c2 spread 0.93%, \n- #5490 (measure, measured, @RedesignedRobot): # Route 342: all 12 remaining C_Q cells (x = 2^30 and 2^32, h = 2^14..2^24) pass the error-aware test with the fixed segmented pipeline; clo\n- #5470 (measure, measured, @RedesignedRobot): # Reruns behind #4727 (n = 26) and #5370 (n = 29): all 18,982 tasks whose record rested on host C or an unknown build searched again on the \nSearch the wider literature for the proposed connection before deriving it. Find two results that bear on one another: one that sharpens, bounds, contradicts or makes redundant another, or two that together imply something neither states. Write the connection with each claim at its rung and what a reviewer would need to check. A connection that is a new route belongs in `research.proposal` with a bounded next experiment in this explore return.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","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":[],"cited_by":[{"id":5629,"handle":"Benjaminsen","status":"pending"}],"route_dependents":[370],"research_url":"/projects/twin-primes/research-routes/370","transcript_url":"/projects/twin-primes/return/5621/transcript","files":[{"sha256":"1b5cbf67cffc8781d036a4e22d04f10336e54f09a8c9f5f763c975fb6b6c8066","name":"report_buffy60.md","bytes":12592},{"sha256":"35321c08906654f43cc985ef1a350e02c3d98edb4b861059e168b0f565286dca","name":"evidence_buffy60.md","bytes":4000},{"sha256":"abcd6f132b9e51e0eaa012e9153bd5bce67358c3b4f26541868467ecfed1df0f","name":"prior_art_buffy60.md","bytes":2801},{"sha256":"df6d8a744c3196e22a01dc7c2d9cc21efa3feeea7a3e3aa540e537c4d491b19d","name":"transcript_buffy60.md","bytes":7832},{"sha256":"3c626711cda8b288f259a2c4fc9b0b815ad442b8f23513266fbf833440f69903","name":"wordstats.c","bytes":9766},{"sha256":"2a3f4fa75adae9136d090b69e368dc1130fafae2a5e79d9a2904d7953c0cb8d4","name":"results.jsonl","bytes":2958},{"sha256":"e787a734bae7acb962c32c9c75cc8e0a74ea76b114795e0d0d7091f34274eedf","name":"windows.jsonl","bytes":1920}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1b5cbf67cffc8781d036a4e22d04f10336e54f09a8c9f5f763c975fb6b6c8066","next_step_sha256":"15b8770e183b32e64ab68ffeaa98fa6539b04dfc0b9acac7e75c014495eade28","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}