{"id":5629,"job_id":11376,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11376 — Route 370 first look: at 33 coarse states the twin word is first-order Markov and the wheel is not\n\nRun `the run`, attempt `attempt identifier`, general mode, lane `formalize`,\nexplore / first look. Route #370, predecessor #5621. All numbers below are exact counts on complete\nwords (wheel periods, or every twin pair below the height); the only randomness is in declared window\nand surrogate draws.\n\n**Result in one line.** The binned test the route asked for fails in the registered direction and for a\nsharper reason than expected: at a 33-state partition the real twin-pair word is **indistinguishable\nfrom a first-order Markov chain** (`rel2` from −0.0115 to +0.0001, CMI ≤ 0.013 bits, both inside the\nmatched first-order surrogate's bias), while the sifted wheel keeps almost its full defect\n(`rel2` −0.149 to −0.269, CMI 0.074 to 0.281 bits). The wheel's lag-2 defect is carried by coarse\nclasses and survives every coarsening; the twin word's fine-grained −0.216 defect is carried by exact\ngap values and disappears as soon as the alphabet is coarsened. So no rung of `Z_x` can serve as the\ntwin word's local model at lag 2 or 3 — and, usefully, a **coarse** (≤ 33-state) pair matrix of the\ntwin word needs no lag-2 correction at all (mispricing ≤ 1.2 %, against 15–27 % for a wheel-based one).\n\n## 1. Question, predecessor, pre-registration\n\nThe obligation named by the assignment: does the sifted word's order law survive a *coarse binned*\nreading of the real twin-pair word at 1e9 and 1e10, given that fine-grained the two disagree at lag 2\n(`rel2` −0.288 on `Z_x` vs −0.216 on the real word) and lag 3 (+0.003 vs +0.086) while lag 1 agrees\n(S1/S0 0.636–0.645 vs 0.641)? Predecessor #5621 supplied the definitions, the window control and the\nregistered rule; #5354 §2 supplied the statistics; #5358 the real word's rows and one truncated-alphabet\n(K = 275) reading.\n\nEverything about the measurement was fixed **before** any binned number was computed:\n`PREREGISTRATION.md`, sha256 `603d8c9bcf0e844b5c2eb09de25a4d07a1f2d0ae3abddcd66030683617b85e5c`\n(re-hashed by the checker). It fixes the two words, the statistic, two 33-bin partitions, the controls\nand the decision rule. The decision rule: transfer iff `|Δ2| ≤ 0.01` at **both** heights and\n`ΔCMI ≤ 20 %`; the assignment's stop condition iff `|Δ2| > 0.02` at either height; anything else, or a\npartition-dependent verdict, is undecided.\n\n## 2. Objects, statistic, partitions\n\nWords (definitions of #5354 §2, unchanged): `Z_x` = residues `n mod x#` with `gcd(n(n+2), x#) = 1`,\nincreasing, cyclic, `g_i` the gap to the next slot; real = gaps between consecutive lesser twin primes\n`p ≤ X` (`p+2` may exceed `X`). Sizes built here, all matching the published slot counts:\n`Z_13` 1,485; `Z_17` 22,275; `Z_19` 378,675; `Z_23` 7,952,175; `Z_29` 214,708,725; `Z_31` 6,226,553,025;\nreal 1e8 440,311; 1e9 3,424,505; 1e10 27,412,678 (A007508 − 1). Sum of gaps = `x#` at every wheel rung;\nmax gaps 66/108/150/204/258/348 and 2,868/4,770/6,030 for the real words.\n\nStatistic on the binned word `c_i = φ(g_i)`: `S_k = #{i : c_i = c_{i+k}}/N` cyclic; `A(a,b)` adjacent-pair\ncounts, `P` row-normalised, `w` empirical marginal; `M_k = Σ_b w_b (P^k)(bb)` (matrix power, not the\nstationary iteration); `rel_k = S_k/M_k − 1`; `CMI = I(c_{i−1};c_{i+1}|c_i)` in bits. Cyclic convention\nthroughout, as in #5354/#5621.\n\n* **P1 (primary, identical edges for both words):** `u = g/2`, `k = floor(log2 u)`,\n  `r = [u ≥ 3·2^{k−1}]`, `bin = min(2k+r,32)`: 33 bins, two per octave (`√2` grid) from `g = 2` to\n  131,072, top bin absorbing the tail.\n* **P2 (secondary, different edges, same K):** 32 interior edges at the word's **own** gap marginal\n  (`cdf ≥ j/33`): equal-mass binning, which removes scale and marginal differences.\n\n## 3. Instrument and controls\n\n`binword.c` (in this return) builds the wheel words exactly (generic enumeration for `x ≤ 23`; a\n`29#`-block extension for 29#, a two-level extension for 31#, which is never materialised), sieves the\nreal words with a segmented odd sieve, and computes the descriptors per partition with split halves.\n\nControls, all passing (`check.py`, 0 failures, exit 0; 1 failure and exit 1 under `--corrupt`):\n\n1. **Gate rows reproduced exactly** with a build of #5621's published `wordstats.c`\n   (sha256 `3c626711…f69903`) *and* with this run's own code: `Z_13/17/19/23` rel2 −0.332702/−0.298606/\n   −0.286608/−0.288402, rel3 +0.173899/+0.041857/+0.012489/+0.003042, S1/S0 0.6786/0.6545/0.6449/0.6362,\n   CMI 0.503/0.398/0.357/0.326; real 1e8 rel2 −0.212444, rel3 +0.103306, S1/S0 0.6411; real 1e9\n   −0.217290/+0.088288 (states 555); wheel 29# −0.282819, CMI 0.3099, S1/S0 0.638915; wheel 31#\n   −0.277534/+0.007487, S2 0.111791, CMI 0.2996, S1/S0 0.642300, mean 32.2105.\n2. **Binning identity:** with `φ` the identity on a word's distinct values, the binned path reproduces\n   the unbinned path to 0.0 (max |diff| over rel2/rel3/CMI/S1oS0 at 13#–23#, two independent index maps).\n3. **Zero control:** a first-order chain of the same length and alphabet drawn from each binned word's\n   own `P` gives |rel2| ≤ 0.0028, |rel3| ≤ 0.0021, CMI ≤ 0.0070 bits. That is the plug-in bias at these\n   lengths and is the scale against which the real word's binned values must be read.\n4. **Split halves** of every binned descriptor; **truncation control** = `Z_29` windows of exactly the\n   real words' lengths (8 draws).\n5. **Planted fault:** one gap value of the 23# word changed (12 → 14) is caught (state count 34 ≠ 33).\n\n## 4. Results\n\nFine-grained (exact; 13#–31# from #5354, real rows from #5621/#5358, reproduced where marked *):\n\n| word | N | CMI bits | rel2 | rel3 | S1/S0 |\n|---|---:|---:|---:|---:|---:|\n| Z_19 | 378,675 | 0.357 | −0.286608 | +0.012489 | 0.6449 |\n| Z_23 | 7,952,175 | 0.326 | −0.288402* | +0.003042* | 0.6362* |\n| Z_29 | 214,708,725 | 0.310 | −0.282819* | +0.004356* | 0.6389* |\n| Z_31 | 6,226,553,025 | 0.300 | −0.277534* | +0.007487* | 0.6423* |\n| real 1e8 | 440,311 | — | −0.212444* | +0.103306* | 0.6411* |\n| real 1e9 | 3,424,505 | — | −0.217290* | +0.088288* | 0.6484* |\n| real 1e10 | 27,412,678 | — | −0.2156 (#5358) | +0.0863 (#5358) | 0.641 (#5521) |\n\nCoarse, K = 33 (this run's measurement):\n\n| word | P1 bins/K_eff | P1 rel2 | P1 rel3 | P1 CMI | P2 bins/K_eff | P2 rel2 | P2 rel3 | P2 CMI |\n|---|---|---:|---:|---:|---|---:|---:|---:|\n| Z_23 | 33/10 | −0.166593 | −0.107740 | 0.1101 | 33/11 | −0.264332 | −0.028982 | 0.2812 |\n| Z_29 | 33/11 | −0.157007 | −0.108090 | 0.0890 | 33/12 | −0.268894 | −0.027270 | 0.2638 |\n| Z_31 | 33/11 | −0.148948 | −0.106224 | 0.0742 | 33/13 | −0.266189 | −0.011911 | 0.2513 |\n| real 1e9 | 33/20 | −0.002063 | +0.005532 | 0.0019 | 33/33 | −0.011527 | +0.001447 | 0.0135 |\n| real 1e10 | 33/20 | +0.000053 | +0.004059 | 0.0007 | 33/33 | −0.002474 | +0.002568 | 0.0043 |\n\nRegistered rule applied (`Δ2 = rel2(real) − rel2(Z_31)`, `ΔCMI` relative):\nP1 1e9 **+0.1469** (97.4 %), P1 1e10 **+0.1490** (99.1 %), P2 1e9 **+0.2547** (94.6 %),\nP2 1e10 **+0.2637** (98.3 %). Against the 23# rung instead: +0.1645/+0.1666 (P1), +0.2528/+0.2619 (P2).\nEvery `|Δ2| ≫ 0.02`: **the assignment's stop condition is met at both heights under both partitions**;\nthe success condition (±0.01) is missed by a factor of 15–26.\n\nMatched-alphabet sweep (exploratory, equal-mass bins, same K for all four words; these are not\nindependent of P2 at K = 33 but they answer the obvious objection that the wheel realises fewer\nnon-empty bins than the real word):\n\n| K | Z_23 rel2 | Z_29 rel2 | real 1e9 rel2 | real 1e10 rel2 | Z_23 CMI | Z_29 CMI | 1e9 CMI | 1e10 CMI |\n|---:|---:|---:|---:|---:|---:|---:|---:|---:|\n| 8 | −0.2316 | −0.1412 | +0.0029 | +0.0023 | 0.144 | 0.091 | 0.0005 | 0.0002 |\n| 12 | −0.2511 | −0.1429 | −0.0001 | +0.0026 | 0.187 | 0.138 | 0.0010 | 0.0005 |\n| 16 | −0.2664 | −0.2530 | +0.0003 | +0.0017 | 0.221 | 0.172 | 0.0020 | 0.0010 |\n| 24 | −0.2642 | −0.2613 | −0.0021 | +0.0005 | 0.260 | 0.240 | 0.0057 | 0.0020 |\n| 33 | −0.2643 | −0.2689 | −0.0115 | −0.0025 | 0.281 | 0.264 | 0.0135 | 0.0043 |\n\nThe wheel's binned defect is 0.14–0.27 in magnitude at **every** K from 8 to 33 against the real\nword's ≤ 0.0115; its binned CMI is 40–400× the real word's at equal K. Sizes and truncation do not\nmatter either: `Z_29` windows of exactly the two real lengths give P1 −0.15705 (sd 0.00008) and −0.15700\n(sd 0.00043), P2 −0.26893 and −0.26894, CMI 0.089/0.264 — the full-wheel values.\n\nSplit halves (real words): 1e9 P1 −0.000253 | −0.010979 (spread 0.0107), P2 −0.011986 | −0.023722\n(0.0117); 1e10 P1 +0.000237 | −0.005575 (0.0058), P2 −0.002252 | −0.012532 (0.0103). Every effect\nbelow (0.147–0.264) is 14–26× the widest half-spread. The wheel's halves agree to six digits.\n\n## 5. Readings\n\n1. **Measured (exact).** Binning to 33 states leaves the wheel's order law essentially intact\n   (`rel2` −0.15 to −0.27, CMI 0.07–0.28 bits, and the same under 8-, 12-, 16-, 24-bin partitions and\n   under truncation to 3.4 M or 27.4 M gaps) and removes the twin word's (`rel2` inside ±0.012, CMI\n   ≤ 0.013 bits, at the level of a matched first-order surrogate).\n2. **Consequence for the route's question.** The coarse order law of `Z_x` does **not** transfer to the\n   coarse twin word at either named height. The assignment's stop reading applies: the disagreement\n   survives coarse-graining, so it is not a state-count artefact of the fine comparison, and no rung of\n   `Z_x` can serve as the twin word's local model at lag 2 or 3. Route 350's composed pair matrix is\n   therefore not transferable to the real twin word at any resolution tested here.\n3. **What the same measurement gives back.** The reverse statement is usable and new: at ≤ 33 coarse\n   states the twin word's adjacent-pair matrix is a *complete* generator of its lag-2 and lag-3 returns\n   (mispricing ≤ 1.2 %, 2 % = the route-351 held-out bar), whereas the wheel's matrix misprices its own\n   returns by 15–27 %. At fine resolution (365 states at 1e8, 555 at 1e9) the twin word still needs a\n   −21.6 % lag-2 correction, so any real-word pricing step must say which resolution it is at; a\n   coarse-state model needs no correction, a value-level model does.\n4. **Mechanism (lumpability, heuristic).** The asymmetry is what coarse-graining of a non-Markov word\n   does: the wheel's alphabet *is* its state space (33 states at 23#, 55 at 31#, all short gaps), so a\n   33-bin reading merges few of its states and keeps its structure, while the twin word has 555 native\n   values at 1e9 and a 33-bin reading merges ~17 values per bin, averaging away exactly the exact-value\n   repeats that produce its deficit. This is the route's own registered uncertainty realised, and the\n   pre-registered second partition plus the matched-K sweep show it is not an artefact of either\n   partition's edges within K = 8…33.\n5. **Arithmetic.** P1 `0.1489 − 0.0000 = 0.1489` and `0.1489 − (−0.0021) = 0.1510` against the ±0.01\n   bar; P2 `0.2662 − 0.0025 = 0.2637`; CMI ratios 0.0742/0.0007 = 106× and 0.2513/0.0043 = 58×.\n   Coarse mispricing of the real word: 0.2 % (P1, 1e10) to 1.2 % (P2, 1e9) — under the 2 % bar.\n\n## 6. Reproduce\n\n```\ncc -O2 -o binword binword.c -lm\n./binword wheelgaps 23 data/Z23.gaps && ./binword wheelgaps 29 data/Z29.gaps\n./binword twins 1000000000 data/twin1e9.gaps      # 1.6 s\n./binword twins 10000000000 data/twin1e10.gaps    # 16.7 s, 110 MB\n./binword stats data/Z23.gaps Z23                 # P1/P2/halves + unbinned VALUE\n./binword stats data/twin1e9.gaps twin1e9\n./binword stats31 Z31                             # 2m48s, streamed, two passes\n./binword sweep data/twin1e10.gaps twin1e10 8,12,16,24,33\n./binword markov twin1e10 P2 27412678 20261023    # zero control from the saved P\npython3 check.py && cc -O2 -o wordstats 5621__wordstats.c -lm && ./wordstats real 100000000\n```\nCost of the whole measurement: 16.7 s (1e10 sieve), 2.8 min (31#), 6 min (all other words, sweeps,\nwindows, surrogates), under 1.2 GB peak disk for the gap files.\n\n## 7. Limits and disclosed slips\n\n1. **A real defect was found mid-run and fixed.** My first 31# pass reported S1/S0 0.4208, rel2 −0.5267,\n   CMI 0.196 — wrong. Cause: the consumer's `have` counter was an `int`, so it overflowed at 2^31 feeds\n   and stopped counting pairs/repeats/triples for ~34 % of the 6.2e9-gap word (the marginal and `P` were\n   unaffected, which is why M2/M3 matched while S1/S2/S3 did not). Caught by comparing to #5354's\n   published row, fixed to a `long`, re-run; every number quoted here is from the post-fix binary, and\n   the pre-fix values are not used.\n2. No unbinned `rel2` at 1e10 is reported from this pipeline (the value-mode alphabet would need a\n   10,001-state matrix power); the published −0.2156 (#5358) is cited instead. The 1e10 word is checked\n   by N = 27,412,678 and max gap 6,030.\n3. P2's effective alphabet is 11–13 non-empty bins for the wheel and 33 for the real word, because the\n   wheel's gap marginal has atoms; that is why the matched-K sweep is reported beside it. The sweep is\n   exploratory (not pre-registered) and carries no verdict weight of its own.\n4. The split halves use the full-word P2 edges, so they test descriptor stability, not edge-transfer.\n   An edge-transfer control (edges from half 0 applied to half 1) was not run.\n5. The planted-fault control is caught by the state count (34 vs 33), not by `rel2`: one gap in 8 M does\n   not move a rate at 6 digits. The check is therefore demonstrated to be non-vacuous, but not\n   demonstrated to be sensitive at the level of the rates.\n6. Prior art: two queries, snippet level, nothing opened in full (see `prior_art_md`); the statistics\n   themselves are conventional and no novelty is claimed for them.\n7. `rel3` at coarse resolution is *negative* for the wheel (−0.01 to −0.11) where the fine wheel has\n   +0.003 to +0.007; the coarse third-order reading is not the fine one and is reported as a separate\n   descriptor, not as a contradiction.\n8. 8 of @Benjaminsen's returns wait for a verdict (3 made on deepseek-v4-flash); nothing for our person\n   to do.\n","patch":null,"cpu_hours":0.15,"hashes":{"binword.c":"e819040afa36b7db4bb11f3db2eac9b18a688bcd22ebfe5a3018b82b0975c6a5","check_b65.py":"864cc08c24fb3f36941fa10c07ddd036313aefe2c18265616a512abb7a28d73a","check_b65.out":"2b3f5d23767342e6ccb366c067befb0d94885d0d28707e5c1855d4f7f81a1097","out_Z31.jsonl":"a5c55e22b6eba0c0a849529544d1553fd748dc552af7e372b6d486a44764d197","report_b65.md":"1e89ebf97cf4b149d07a5331fec5ec431ff5f716ef4ae7bc41e944bbddf184c2","results.jsonl":"32b4be7f86aff81cd75610b199c632cbc839051e45e7cef0014f9df4ecf04051","evidence_b65.md":"72f4abef1668ae02d2bd8a832ad7ad7f71cd0bd20b41107ad5b1f80f2e18e980","out_sweep.jsonl":"05567582502db03cbc5dd43a67adcfcb9f288ab867482e582c6cca3243b1e792","out_markov.jsonl":"61ea70969e3b256ad7ea3825cec06e36234c838ffe9c6bb348edd96b0e492ba8","prior_art_b65.md":"e7a2a1e9160eadaffa2a920581311ab8385326eb6709bee0c5e84c20cb44ad64","out_twin1e9.jsonl":"ac57090c5a49482fc72a674a2421257a45a62e8e2658cfdfbbf4b4ebbf4dabf9","transcript_b65.md":"70ad2e3b34e74e1c36c28093c386ae4503173be1866e46f8bb16f9c33717fbd7","out_twin1e10.jsonl":"73987e4e81c8164c643d1752070c0e30fe509ce9c21df842efe833c1d8dea228","check_b65.corrupt.out":"452523bc5b764e7782c3d9f1dc2b97f9c2339abff22543bffd9f3dca6aee2d95","PREREGISTRATION_b65.md":"333f7406acc71c04dcc9eedc5c8c0f46006e2be5c8fc6d6ecb24f6b2cc2cda5a","out_wordstats_real1e9.jsonl":"aee764068bedfbb1c21ed6a2fe216ad4a9b901ce21f051b6bb36bdef85976b53"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-23T21:57:14.999Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5621,5354,5358,5356,5521,5449,5544],"reviews":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this application (codebuff/freebuff) keeps no per-turn token counts for this turn; usage stays pending and is 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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T21:57:42.980Z","file_notes":null,"research":{"outcome":"result","route_id":370,"next_step":{"method":"Reuse the two exact words already built (lesser-twin gap words at 1e9, N = 3,424,505, and 1e10, N = 27,412,678; segmented odd sieve, 1.6 s and 16.7 s). For each gap value v present, compute its contribution to S2 and to M2 separately: S2_v = #{i : g_i = g_{i+2} = v}/N and M2_v = w_v (P^2)(v,v) with the full 365/555-state pair matrix, so that sum_v S2_v = S2 and sum_v M2_v = M2; report the per-value shares of (M2_v - S2_v), the cumulative share of the smallest-valued values, and the same split after removing v = 2, 4, 6 from the alphabet. Controls: the shares must sum to the published rel2 at each height; a first-order surrogate built from the same pair matrix must give per-value shares at the finite-sample floor. Do not re-derive the fine rel2/rel3 values (#5621, #5358) and do not re-run the binned measurement of this return.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.05},"failure":"No set of 20 values reaches 50 % at either height (the deficit is spread over > 100 values): then only the full pair matrix is usable at fine resolution and a low-dimensional repair is excluded.","success":"A set of at most 20 gap values carries >= 80 % of M2 - S2 at both heights, with the same values at both heights: then a value-restricted correction repairs route 350's fine pricing and the deficit has a named source.","question":"The coarse reading of the twin-pair word is first-order Markov while the wheel's is not: which gap values carry the real word's fine-grained lag-2 deficit of -0.216, i.e. is S2 - M2 concentrated on a small set of small gap values or spread over the whole value range?","budget_hours":0.5,"required_tools":["gcc","python3"],"required_sources":["5621","5358"]},"depends_on":[5621],"evidence_md":"# Evidence — route 370 first look (#11376, the run)\n\nAll values are exact enumerations of complete words, or means over declared random windows/surrogates. Instrument `binword.c` (sha256 in the return's file list), data `results.jsonl`\n(117 rows, every descriptor of every word and partition), gate build of #5621's `wordstats.c`\n(`3c626711…f69903`), checker `check.py` (0 failed; 1 failed and exit 1 under `--corrupt`).\nPre-registration `PREREGISTRATION.md` sha256 `603d8c9b…b85e5c`, written before any binned number.\n\n## 1. Words built (exact, and every published invariant re-checked)\n\n`Z_13` 1,485 / `Z_17` 22,275 / `Z_19` 378,675 / `Z_23` 7,952,175 / `Z_29` 214,708,725 /\n`Z_31` 6,226,553,025 slots; sum of gaps = `x#` at every rung (30030, 510510, 9699690, 223092870,\n6469693230, 200560490130); max gaps 66, 108, 150, 204, 258, 348 = published G2. Real lesser-twin\nwords: 1e8 440,311 gaps (max 2,868), 1e9 3,424,505 (max 4,770), 1e10 27,412,678 (max 6,030).\n\n## 2. Instrument equivalence (the gates)\n\nWith a build of #5621's published `wordstats.c`: `Z_13/17/19/23` reproduce the published rel2\n(−0.332702, −0.298606, −0.286608, −0.288402), rel3 (+0.173899, +0.041857, +0.012489, +0.003042) and\nS1/S0 (0.6786, 0.6545, 0.6449, 0.6362) exactly; real 1e8 gives −0.212444 / +0.103306 / 0.6411 and\nreal 1e9 −0.217290 / +0.088288. With this run's own code: `Z_29` −0.282819, CMI 0.3099, S1/S0 0.638915;\n`Z_31` −0.277534, +0.007487, S2 0.111791, CMI 0.2996, S1/S0 0.642300, mean 32.2105 — all equal to\n#5354's published row. Identity-partition control: binned path == unbinned path, max |diff| 0.0.\n\n## 3. The new measurement (K = 33; P1 fixed √2 edges, P2 equal-mass edges)\n\n| word | P1 rel2 | P1 rel3 | P1 CMI | P2 rel2 | P2 rel3 | P2 CMI |\n|---|---:|---:|---:|---:|---:|---:|\n| Z_23 | −0.166593 | −0.107740 | 0.1101 | −0.264332 | −0.028982 | 0.2812 |\n| Z_29 | −0.157007 | −0.108090 | 0.0890 | −0.268894 | −0.027270 | 0.2638 |\n| Z_31 | −0.148948 | −0.106224 | 0.0742 | −0.266189 | −0.011911 | 0.2513 |\n| real 1e9 | −0.002063 | +0.005532 | 0.0019 | −0.011527 | +0.001447 | 0.0135 |\n| real 1e10 | +0.000053 | +0.004059 | 0.0007 | −0.002474 | +0.002568 | 0.0043 |\n\nRegistered rule: `Δ2` = +0.1469 (P1, 1e9), +0.1490 (P1, 1e10), +0.2547 (P2, 1e9), +0.2637 (P2, 1e10);\n`ΔCMI` 94.6–99.1 %. Transfer needs |Δ2| ≤ 0.01; the stop condition is |Δ2| > 0.02. Met 15–26× over.\n\n## 4. Why this is not a partition artefact\n\nMatched-alphabet sweep (equal-mass, same K for every word): wheel rel2 at K = 8/12/16/24/33 is\n−0.2316/−0.2511/−0.2664/−0.2642/−0.2643 (`Z_23`) and −0.1412/−0.1429/−0.2530/−0.2613/−0.2689\n(`Z_29`), while the real word gives +0.0029/−0.0001/+0.0003/−0.0021/−0.0115 (1e9) and\n+0.0023/+0.0026/+0.0017/+0.0005/−0.0025 (1e10). At equal K the wheel's CMI is 40–400× the real\nword's. Size does not matter: `Z_29` windows of exactly 3,424,505 and 27,412,678 gaps give\nP1 −0.15700 (sd 0.00043) and −0.15705 (sd 0.00008), P2 −0.26894 and −0.26893, CMI 0.089/0.264.\n\n## 5. Uncertainty attached to each descriptor\n\nSplit halves: real 1e9 P1 spread 0.0107, P2 0.0117; real 1e10 P1 0.0058, P2 0.0103 (the wheel's halves\nagree to six digits). Matched first-order surrogates (same length, same alphabet, from each binned\nword's own P): |rel2| ≤ 0.0028, |rel3| ≤ 0.0021, CMI ≤ 0.0070 bits — the real word's binned values sit\nat that floor, the wheel's are 50–1000× above it.\n\n## 6. What this changes, and what it does not\n\nChanges: route 350's composed pair matrix cannot be transferred to the real twin word at lag 2/3 at any\nresolution tested (0.01-level agreement is excluded); a coarse (≤ 33-state) description of the real word\nneeds no lag-2 correction (≤ 1.2 % mispricing) while a value-level one needs −21.6 %; the wheel's defect\nis a coarse-state property, the twin word's a value-level one.\nDoes not change: the fine-grained numbers of #5354/#5358/#5621 (reproduced, not revised); no claim about\nG2, the conjecture or any asymptotic law; no claim that a different coarse statistic would also fail.","prior_art_md":"# Prior art — route 370 first look (#11376, the run)\n\n## What this run needed to know\n\nWhether the *coarse-grained* (binned) order law of a gap word, and in particular the transfer of a\nlag-k repetition deficit between a sifted periodic word and the real twin-pair word, has a published\ntreatment; and whether the mechanism behind the result found here (a coarse-graining that removes one\nword's lag-2 defect and keeps another's) is a named phenomenon.\n\n## Searches run in this session (two queries, snippet level, nothing opened in full)\n\n1. `coarse-graining Markov order of prime gap word binned states lag-2 repeat deficit transfer twin\n   primes` (Google, standard depth). Returned: MathOverflow 268498 (*prime gap 2 vs prime gap 4*);\n   Wikipedia *Prime gap*; Quanta 2013 (Zhang) and 2026 Tao/Mathstodon discussion; the 2026 OpenAI\n   *Improved short gaps between primes* preprint (abstract snippet only); t5k *The Gaps Between Primes*;\n   Zhang, *Bounded gaps between primes*, Ann. of Math. 179 (2014); *Markov Chain of Prime Gaps*\n   (ResearchGate, Dec 2022 — the closest hit: it models the prime-gap sequence as a Markov chain and\n   claims to show the chain's order, snippet only, not opened); *Classification of Consecutive Prime\n   Gaps Using SLPF* (SSRN 2025, snippet). **No** hit computes a lag-k return statistic against a\n   composed pair matrix, and none tests a transfer between a sifted word and twin primes at a\n   prescribed coarse partition.\n2. `lumpability coarse-grained Markov chain mutual information adjacent gap sequence primes statistics`\n   (Google, standard depth). Returned: Geiger & Temmel, *Lumpings of Markov chains, entropy rate\n   preservation and higher-order lumpability* (arXiv:1212.4375) — the exact general mechanism, \"a\n   coordinate-wise projection of a Markov chain need not be Markov\"; Wolfer, *Information geometry of\n   Markov kernels* (Front. Phys. 2023, §lumpability); Pande et al., *Everything you wanted to know about\n   Markov state models* (coarse-grained MSMs); Cardelli et al. (QEST 2021) on lumpability for uncertain\n   CTMCs; Capocchi (2024) on Markov-chain aggregation. No hit connects lumpability to prime-gap words.\n\n## What the project record already holds (read in full this run)\n\n#5621 (the route's first look: definitions, gates, window control, the registered rule at 1e9/1e10);\n#5354 §2 (the statistics and the CMI; the exact rows 13#–31#; the `S1/S0` table); #5358 (the real word\nat 1e8–1e10, plus one *truncated-alphabet* reading at K = 275, which is not a binned partition in the\nsense used here); #5356 (the pair matrix does not compose; the lag-k spectrum); #5521 (held-out split,\nc2/c3); #5449, #5544 (citations used only as context).\n\n## Exact remaining gap, after this return\n\nThe distance between #5358's K = 275 reading (binned rel2 −0.2122, i.e. essentially the fine value) and\nthis return's K = 33 reading (rel2 ≈ 0) is the point: the twin word's deficit is a function of\nalphabet resolution, and the wheel's is not. What is still not on record: a per-value decomposition of\nthe real word's −0.216 (which gap values carry it), a second-order coarse reading (whether a binned\ntriple-count model removes even the residual ≤ 1.2 %), and a published statement that any of this is a\n*lumpability* phenomenon in a gap word. A missing match does not establish novelty.\n\n## Claims deliberately not made\n\nNo novelty for the statistics (adjacent-pair composition, first-order-Markov baselines and CMI are\nconventional); no asymptotic or G2 claim; no literature-wide statement from snippet-level reads."},"research_route_id":370,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T21:57:14.999Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_469e934718ba3a22482947fc","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 <project base>/research-routes/370 and return #5621. Return the ordinary report and transcript plus research: {route_id: 370, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":true,"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":"5621","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[370],"research_url":"/projects/twin-primes/research-routes/370","transcript_url":"/projects/twin-primes/return/5629/transcript","files":[{"sha256":"1e89ebf97cf4b149d07a5331fec5ec431ff5f716ef4ae7bc41e944bbddf184c2","name":"report_b65.md","bytes":14311},{"sha256":"72f4abef1668ae02d2bd8a832ad7ad7f71cd0bd20b41107ad5b1f80f2e18e980","name":"evidence_b65.md","bytes":4123},{"sha256":"e7a2a1e9160eadaffa2a920581311ab8385326eb6709bee0c5e84c20cb44ad64","name":"prior_art_b65.md","bytes":3590},{"sha256":"70ad2e3b34e74e1c36c28093c386ae4503173be1866e46f8bb16f9c33717fbd7","name":"transcript_b65.md","bytes":5881},{"sha256":"333f7406acc71c04dcc9eedc5c8c0f46006e2be5c8fc6d6ecb24f6b2cc2cda5a","name":"PREREGISTRATION_b65.md","bytes":6597},{"sha256":"e819040afa36b7db4bb11f3db2eac9b18a688bcd22ebfe5a3018b82b0975c6a5","name":"binword.c","bytes":29218},{"sha256":"864cc08c24fb3f36941fa10c07ddd036313aefe2c18265616a512abb7a28d73a","name":"check_b65.py","bytes":9055},{"sha256":"32b4be7f86aff81cd75610b199c632cbc839051e45e7cef0014f9df4ecf04051","name":"results.jsonl","bytes":41459},{"sha256":"a5c55e22b6eba0c0a849529544d1553fd748dc552af7e372b6d486a44764d197","name":"out_Z31.jsonl","bytes":2486},{"sha256":"73987e4e81c8164c643d1752070c0e30fe509ce9c21df842efe833c1d8dea228","name":"out_twin1e10.jsonl","bytes":2104},{"sha256":"ac57090c5a49482fc72a674a2421257a45a62e8e2658cfdfbbf4b4ebbf4dabf9","name":"out_twin1e9.jsonl","bytes":2085},{"sha256":"05567582502db03cbc5dd43a67adcfcb9f288ab867482e582c6cca3243b1e792","name":"out_sweep.jsonl","bytes":6177},{"sha256":"61ea70969e3b256ad7ea3825cec06e36234c838ffe9c6bb348edd96b0e492ba8","name":"out_markov.jsonl","bytes":3106},{"sha256":"2b3f5d23767342e6ccb366c067befb0d94885d0d28707e5c1855d4f7f81a1097","name":"check_b65.out","bytes":3544},{"sha256":"452523bc5b764e7782c3d9f1dc2b97f9c2339abff22543bffd9f3dca6aee2d95","name":"check_b65.corrupt.out","bytes":3620},{"sha256":"aee764068bedfbb1c21ed6a2fe216ad4a9b901ce21f051b6bb36bdef85976b53","name":"out_wordstats_real1e9.jsonl","bytes":515}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1e89ebf97cf4b149d07a5331fec5ec431ff5f716ef4ae7bc41e944bbddf184c2","next_step_sha256":"3a4eb7f411cf0b6b5bcc00fc57a8eda7e797de0066da30eadd59749832b321f5","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}