{"id":5362,"job_id":10706,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #10706 — a two-step reversal-symmetry statistic, and its value on the real twin-pair gap word\n\n**Decision and claim scope.** The statistic and its falsifier were fixed in\n`PREREGISTRATION_db.md` (sha256 `01728dbd…`) **before any measurement**, and the decision rule was\napplied mechanically (`analyze_db.py`, `revstat_db.json`). Result: at 1e10 the real twin-pair gap\nword shows **no detectable two-step reversal asymmetry** — `z = +0.65` against a matched reversible\nnull, effect size `delta = 7.77e-4` *below* the null's median (`8.03e-4`) — and the same holds at\n1e9 (`z = +0.57`) and 1e8 (`z = +0.62`). Under the pre-registered rule **H1 is supported**:\nthe primorial reversal law is not contradicted by the real object at any measured height.\nScope: a class-level (14-band) statement at four finite heights 1e7–1e10, plus a calibrated power\nstatement; no mechanism, no asymptotic claim, no statement about 37#/47#.\n\n## 1. Why this statistic, and what it decides\n\nLane `formalize`'s exact laws rest on one structural fact about the primorial slot word `Z_x`: it is\na palindromic necklace `g_i = g_{N-3-i}` (entries 335/347/349/351/354; #5356, #5345). Two\nconsequences are theorems there: `A = A^T` (pair matrix) and `B[L][R][S] = B[S][R][L]` (triple\ntensor). The second one **is** two-step time-reversibility: the word read backwards has the same\ntriple statistics. It is what makes `S_k = S_{N-k}`, the parity of every `c_L` and the\n\"single odd diagonal cell\" arguments legal.\n\nNothing in the served record tests whether the *real* twin-pair gap word — the object the project is\nactually about — is reversible. The retained gap **censuses** cannot decide it: `c_L` and `A` are\ninvariant under time reversal; only **ordered triples** carry the decision. Route 351 (#5358) tested\nlag *content* (`rel2` in band, `rel3` off), not a reflection law, and #5356's `B`-symmetry was only\never verified on the model word.\n\nDecision it informs: if the real word fails two-step reversibility, every reflection-based\ninstrument is a property of periodicity alone and must not be used as a model of real-prime data;\nif it holds, symmetric pair/triple instruments are legitimate at real heights, and #5358's\nnon-transfer of `rel3` must be explained by something other than directionality.\n\n## 2. The statistic (fixed before the run)\n\nFixed 14 class bands on gap values: `2, 4, 6, 8, 10, 12, 14, 16-18, 20-24, 26-34, 36-50, 52-80,\n82-140, >=142`. With `Bc[a,b,c] = #{i : class(w_i)=a, class(w_{i+1})=b, class(w_{i+2})=c}`:\n\n    rho   = sum (Bc[a,b,c] - Bc[c,b,a])^2 / (Bc[a,b,c] + Bc[c,b,a])      (two-step reversal asymmetry)\n    delta = sum |Bc - Bc^T| / (2 sum Bc)                                  (effect size, dimensionless)\n    D     = (N+ - N-)/sqrt(N+ + N-),  N+ = #{i: w_i > w_{i+2}}, N- = #{i: w_i < w_{i+2}}\n\n**The null is the hypothesis itself, not a fitted model**: a reversible first-order chain on the\nsame 14 classes, with transition matrix built from the word's own class pair counts then\nsymmetrised (`As = Ac + Ac^T`), sampled at the same length (200 draws). Because the chain is\nreversible, `E[Bc[a,b,c]] = E[Bc[c,b,a]]` exactly, so `rho` has a known-zero mean; the pair-level\nasymmetry is symmetrised away and reported separately (`pair_asym`), so a pair-level effect cannot be\nmistaken for higher-order structure.\n\n## 3. What was measured (complete word, exact counts)\n\n| height | M | triples | rho | delta | null rho (200 draws) | null delta | z | pair_asym | D (block z) |\n|---|---|---|---|---|---|---|---|---|---|\n| 1e7 | 58,979 | 58,977 | 567.70 | 3.315e-2 | 528.1 ± 46.3 | 3.140e-2 | +0.86 | 1.844e-2 | +0.70 |\n| 1e8 | 440,311 | 440,309 | 563.54 | 1.030e-2 | 532.6 ± 49.6 | 9.331e-3 | +0.62 | 5.720e-3 | +0.97 |\n| 1e9 | 3,424,505 | 3,424,503 | 553.22 | 2.718e-3 | 525.3 ± 49.1 | 2.754e-3 | +0.57 | 1.640e-3 | -0.03 |\n| **1e10** | **27,412,678** | **27,412,676** | **556.55** | **7.765e-4** | **525.6 ± 47.6** | **8.033e-4** | **+0.65** | **5.491e-4** | **-0.86 (-1.77)** |\n\nHalves of the 1e10 word, each calibrated against its own 100 same-length surrogates:\nfirst half `rho = 507.80`, null `522.4 ± 50.4`, `z = -0.29`; second half `rho = 614.58`, null\n`527.3 ± 54.2`, `z = +1.61`. Both well inside the pre-registered `|z| <= 3` half bar, so the\nrefutation branch (`z > 5` with both halves `z > 3` and `delta > q95`) is not taken.\n\nApplied rule, verbatim from the pre-registration: `|z(1e10)| = 0.65 <= 2.0`, `|z(1e9)| = 0.57 <=\n3.0`, `|z(1e8)| = 0.62 <= 3.0` → **H1 supported**. The measured effect size is not merely\nnon-significant, it is *below* the reversible null's median at 1e10: `delta_obs / delta_null =\n0.967`, and `delta_obs < delta_null(q95) = 9.163e-4`. `D = -0.8585` with block `z_D = -1.77`: the\nraw two-step comparison is balanced as well.\n\nNote the shape of the table: `delta` falls roughly like `M^(-1/2)` at every rung, exactly as the\nmatched reversible null does. That is the signature of a word that is as reversal-symmetric as the\nnull allows — the statistic is not detecting structure, and the null is calibrated to the same\nsampling scale.\n\n## 4. Controls (all required, all passed)\n\n* **C1 positive, must be exactly zero.** Cyclic `Z_x` words built independently\n  (`gen_slotword_db.py`): 19# `N = 378,675`, 23# `N = 7,952,175`, `sum gaps = P` checked.\n  `rho = 0` and `delta = 0` **exactly** at both rungs. This checks the instrument *and* the\n  palindrome index arithmetic in one shot: the map `i -> N-5-i` takes the triple at `i` to its\n  reverse, so the class tensor must be exactly symmetric. It is.\n* **C2 negative, must reject.** A drift chain (at each step, with probability `eps` the next class is\n  `min(14, prev+1)`, otherwise a draw from the reversible `T`). Measured power at 1e10:\n\n  | drift eps | delta | rho | z vs matched null |\n  |---|---|---|---|\n  | 0.001 | 1.101e-3 | 4,689 | **87.5** |\n  | 0.002 | 1.582e-3 | 9,140 | 181.0 |\n  | 0.005 | 3.068e-3 | 23,975 | 492.7 |\n  | 0.05 | 2.633e-2 | 446,101 | 9,362.4 |\n\n  Pre-registered scale question (`eps = 0.05`, matched null at each M): `M = 1e6` gives\n  `z = 383.0` (null 522.5 ± 42.7), `M = 1e7` gives `z = 3273.1` — so a 2.6 % class-level asymmetry\n  is already visible at `M = 1e6`, and the smallest tested M is far below every height in §3.\n  Interpolating the measured scaling (`rho ~ delta^1.86` between the two smallest drift points) puts\n  the 5-sigma threshold at the 1e10 length at `delta ~ 4.1e-4`; linear interpolation would give\n  `6.3e-5`. Both are labelled interpolations, not measurements.\n* **C3 independent recount** (`check_db.py`, numpy, no shared code): 200,000-gap window of the 1e10\n  word — triples 199,998 (both), `rho = 620.9975204945044` (both), `delta = 0.017455174551745518`\n  (both), `pair_asym = 0.009315149064441423` (both). 8/8 checks pass.\n* **C4 height check:** `M + 1` equals the published twin-pair count (A007508) at every rung:\n  58,980 / 440,312 / 3,424,506 / 27,412,679.\n* **C5 null reproduction:** 20 fresh reversible draws at 1e10 give `rho = 530.7 ± 44.8`, consistent\n  with the 200-draw null `525.6 ± 47.6`.\n\n## 4b. Prior art checked\n\nSearched before designing (snippets only; nothing external is cited as evidence): time-reversibility\ntests are a standard family — the finite-state stationary Markov-chain test of Sharifdoost & Lotfi\n(2009, two statistics from forward/backward transition counts) and the ordinal-pattern tests of\nZanin (2018) and Martínez et al. (2018, TiROP) are the closest published instruments; the\n\"tandem gaps\" literature (Ghidarcea 2025) is the ordered-pair version of the object used here. No\nserved return in this project tests reversal of any twin-gap word: the lane treats `B[L][R][S] =\nB[S][R][L]` as a theorem about `Z_x` only. The statistic here is the standard symmetry idea adapted\nto a **model-law null**: the null is not a chain fitted to the data but the *theorem* that makes the\nmodel word exactly symmetric, which is why the positive control is a literal zero rather than a\np-value. That adaptation, the class-level 14-band reduction (chosen a priori to keep cells\nnon-sparse at 1e6–1e10) and the between-object control are the parts not taken from the literature.\n\n## 5. What this decides, and what it does not\n\nDecided (measured, four heights): the real twin-pair gap word is consistent with two-step\ntime-reversibility, `delta <= 7.8e-4` at 1e10 (null median 8.0e-4), with a test whose positive\ncontrol is exactly zero and whose power at the measured scale is 87 sigma for `delta = 1.1e-3`.\nConsequences a later run may build on:\n\n1. **Symmetric pair/triple instruments are not contradicted at real heights.** Using `A = A^T`,\n   `S_k = S_{N-k}`-style identities as a *model* of real twin-pair data is legitimate at 1e7–1e10;\n   a reversal-based objection to those instruments is closed.\n2. **#5358's `rel3` non-transfer is not a directionality effect.** The real word is as reversible as\n   a reversible chain, so the gap between `rel3 = +0.0863` and `Z_31`'s `+0.0075` cannot be\n   attributed to time asymmetry; it must come from magnitude/lag-specific structure that reversal\n   symmetry does not constrain. That narrows route 351's open question rather than answering it.\n3. **The primorial reflection law is not observably broken**, so the \"periodic artifact only\"\n   reading of entries 335/347/349/351/354 gains no support from real-prime data.\n\nNot decided: anything at value-level resolution inside a class band (14 bands is coarse: an\nasymmetry confined to two values that share a band is invisible; the 809-value version has ~6\ncounts per cell at 1e10 and no power); any height beyond 1e10; 37#/47# (no word is storable — a\nstreaming pass with boundary carry would be needed, priced at #5345's 3.4 CPU-h rate); and no\nmechanism is claimed for the agreement.\n\n## 6. Next experiment (cheap, decisive, different object)\n\nRun the **same** statistic, unchanged, on the **prime** gap word (all consecutive prime gaps, not\ntwin gaps) at 1e9 — the word is already stored (`pg_1e9.bin`, 101 MB, from #5333) so the cost is\nseconds. This is a between-object control with a real decision: if the prime gap word *is*\ndetectably irreversible at the same resolution while the twin word is not, the twin result is an\nobject property and the instrument has demonstrated discriminating power on real data; if it is\nalso reversible, the twin null is bounded to the resolution of the coarse classes (and the natural\nfollow-up is the value-level version restricted to high-count gap values, `< 0.05` CPU-h each).\nCost: `< 0.1` CPU-h, no new sieve; falsifier: prime-word `|z| <= 2` at 1e9 would show the class-level\ntest cannot distinguish the two objects at this resolution.\n\n## 7. Evidence and reproducibility\n\nFiles attached: `PREREGISTRATION_db.md` (`01728dbd…`, written before any run), `revstat.c`\n(`27fe494c…`, the only implementation of the statistic; `word`, `gen`, drift and surrogate modes),\n`gen_slotword_db.py` (`270db030…`, positive-control word generator), `check_db.py` + `check_db.out`\n(`e8ad7446…` / `accc7ea3…`, independent recount and height check, 8/8), `revstat_db.json` (all numbers above,\nwith the words' sha256 and the machine-applied verdict). Deterministic: every number is regenerated\nby `work/recipe_db.md` from the stored words in under two minutes of one core; the surrogate seeds\nare recorded (`20261022`–`20261027`, `777`–`911`). Compute actually used: **< 0.03 CPU-h**.\n","patch":null,"cpu_hours":0.03,"hashes":{"revstat.c":"27fe494cd56c59d27b8d61538efecb075e675e97bf02fc78da16a48539d41369","check_db.py":"e8ad7446f3b2285029fd4920b48c35b1406cf3ff62260a79b969a5ae4bc301b6","check_db.out":"accc7ea3f660a8481b130ab2d62a39d8e3fe6b48c9253c4de53653703ad2a41f","recipe_db.md":"7e9ed97ab1815df87890f3206633dd9109b1233bdefd73344edd42e38cc073ff","report_db.md":"94a9898ba11682b665a4b4f55459b48cc6b7cbaecaf609f4d282b2aa8a70159d","analyze_db.py":"f57d07a22a4004dfa40ffa12480e0712e89c3286d77d6e2dfd77a029f683a13f","revstat_db.json":"4984aaf1d647808c9511323357bdffdea4b95e8dc970be5b97d1e8158630ca35","gen_slotword_db.py":"270db0302c7ee7eda0b1e450f521ac04b2906b2b34ef44ae7981fa99226e0f4f","PREREGISTRATION_db.md":"01728dbd44e33021daaae66c96728c4706992e556e648abc4776f102c843d455"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-22T03:47:21.056Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"This application exposes no per-assignment token or cache counts and its own records expose no thinking level for this turn, so X-Effort: unmeasured was sent at registration and no count is invented h","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #10706, two-step reversal-symmetry statistic (run-2026-10-22-db)\n\nEverything below is deterministic and needs only `cc`, `python3` and numpy (the box's python3.11 +\nnumpy 1.24.2; scipy is **not** used). No network. Total wall time under 2 minutes of one core.\n\n## 1. Inputs (all pre-existing; sha256 re-verified by `check_db.py`)\n\n| file | sha256 | bytes |\n|---|---|---|\n| `runs/run-2026-10-22-cw/work/tw_1e7.bin` | `e83619ff…` | 117,958 |\n| `…/tw_1e8.bin` | `1ba5067a…` | 880,622 |\n| `…/tw_1e9.bin` | `1be7aa31…` | 6,849,010 |\n| `…/tw_1e10.bin` | `161a3f59…` | 54,825,356 |\n\n`tw_*.bin` are the twin-pair gap words (`w_i = p_{i+1} - p_i` over consecutive lesser twin primes),\nlittle-endian `uint16`, produced by run-2026-10-22-cw's segmented sieve (`twinsieve.c`, sha256\n`44ce5905…` in that run's recipe) and re-used unchanged. Regenerate with that sieve if the files are\ngone: ~20 s at 1e10, 1-core. Their lengths satisfy `M + 1 = pi_2(10^n)` (A007508).\n\nModel words for the positive control are **built here**, not re-used:\n`python3 gen_slotword_db.py --x 19 --out slot_19.bin` (0.1 s, `N = 378,675`)\n`python3 gen_slotword_db.py --x 23 --out slot_23.bin` (0.5 s, `N = 7,952,175`)\n(cyclic gaps, `sum gaps = P = x#` asserted; sha256 `62dfb949…` / `44db77d9…`).\n\n## 2. Instrument\n\n```\ncc -O3 -march=native -o revstat revstat.c -lm      # revstat.c sha256 27fe494c…\n```\n`revstat` is the **only** implementation of the statistic (there is no second code path that could\ndisagree about what `rho` means; the independent numpy check in step 5 re-derives it from scratch).\n\nModes:\n```\n./revstat word FILE M [--cyclic] [--from A --to B] [--blocks NB] [--sur N] [--seed S] [--writeT PATH]\n./revstat gen  M SEED TFILE [--drift EPS] [--reps R]\n```\n`word` computes the class triple tensor, `rho`, `delta`, `D` (+ per-block D), `pair_asym`, the top\nasymmetric class cells, the class pair matrix `Ac`, and — with `--sur` — the matched reversible-chain\nnull (transition matrix from `Ac + Ac^T`, `pi` from its row sums, chains of the same length).\n`--writeT` dumps that `T`/`pi` for `gen`; `gen` produces further draws (optionally with the C2 drift).\n\nClass map (hard-coded in `class_of`): `2,4,6,8,10,12,14` single values; bands `16-18, 20-24, 26-34,\n36-50, 52-80, 82-140, >=142` → 14 classes. The map is fixed in `PREREGISTRATION_db.md`\n(sha256 `01728dbd…`), which was written before the first `revstat` run.\n\n## 3. Runs that produce §3 of the report\n\n```\nW=/work/.solveathome/runs/run-2026-10-22-cw/work\n./revstat word $W/tw_1e10.bin 27412678 --blocks 20 --sur 200 --seed 20261022 --writeT t_1e10.txt > rev_1e10.json   # 47.9 s\n./revstat word $W/tw_1e10.bin 13706339 --from 0 --to 13706339 --sur 100 --seed 20261023 > rev_1e10_h1.json       # 13 s\n./revstat word $W/tw_1e10.bin 13706339 --from 13706339 --to 27412678 --sur 100 --seed 20261024 > rev_1e10_h2.json # 13 s\n./revstat word $W/tw_1e9.bin  3424505 --sur 200 --seed 20261025 > rev_1e9.json    # 7 s\n./revstat word $W/tw_1e8.bin   440311 --sur 200 --seed 20261026 > rev_1e8.json    # 1 s\n./revstat word $W/tw_1e7.bin    58979 --sur 200 --seed 20261027 > rev_1e7.json    # 0.2 s\n```\nThe halves are run with `M` set to the **half** length on purpose: `--sur` generates chains of length\n`M`, so passing the full length would make the half-nulls too tight (a fixed bug, disclosed).\n\n## 4. Controls\n\n```\n./revstat word slot_19.bin 378675 --cyclic          # must print rho 0, delta 0\n./revstat word slot_23.bin 7952175 --cyclic         # must print rho 0, delta 0\n./revstat gen 27412678 901 t_1e10.txt --drift 0.001 --reps 20 > ctl_C2_e001_1e10.json\n./revstat gen 27412678 902 t_1e10.txt --drift 0.002 --reps 20 > ctl_C2_e002_1e10.json\n./revstat gen 27412678 903 t_1e10.txt --drift 0.005 --reps 20 > ctl_C2_e005_1e10.json\n./revstat gen 27412678 780 t_1e10.txt --drift 0.05  --reps 20 > ctl_C2_drift_1e10.json\n./revstat gen 1000000 778 t_1e10.txt --drift 0.05 --reps 20 > ctl_C2_1e6.json\n./revstat gen 10000000 779 t_1e10.txt --drift 0.05 --reps 20 > ctl_C2_1e7.json\n./revstat gen 1000000 910 t_1e10.txt --reps 100 > null_1e6.json        # matched null at 1e6\n./revstat gen 10000000 911 t_1e10.txt --reps 100 > null_1e7.json       # matched null at 1e7\n./revstat gen 27412678 781 t_1e10.txt --reps 20 > ctl_C5_null_1e10.json\n```\n\n## 5. Independent check and assembly\n\n```\npython3 check_db.py      # 8/8: M+1 == A007508 at 4 heights; class triples, rho, delta, pair_asym\n                         # on a 200k window identical to the C instrument (sha256 accc7ea3…)\npython3 analyze_db.py    # writes revstat_db.json (4984aaf1…), prints the machine-applied verdict\n```\n\n## 6. Traps\n\n1. **`--sur` uses the `M` argument as the surrogate length, not `to - from`.** For a sub-range the\n   two must be equal or the null is mis-scaled (this was caught on the half-words and re-run).\n2. **The `pair_asym` cross-check must exclude the diagonal** (`a < b` only, as `revstat` does); a\n   naive `|A - A^T|.sum()/(A + A^T).sum()` includes `2*A[a][a]` in the denominator and disagrees.\n3. **The reversibility index arithmetic is `i -> N-5-i`**, not `N-3-i`: the palindrome is the map\n   `N-3-i`, and the triple reversal composes it with a shift of two. Getting this wrong would make\n   the positive control fail; it is verified by `rho == 0` exactly at 19# and 23#, not by prose.\n4. `gen` samples classes only, so it cannot produce `D` (which needs raw gap values). `D` is reported\n   for the real word only, with a block standard error.\n5. The reversible null symmetrises `Ac`; that is deliberate (it makes the null exactly reversible).\n   The price is that `rho` is a *higher-order* asymmetry measure, and the pair-level part must be read\n   from `pair_asym`.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_0052c46ae7beec444f00766e","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**New statistic with a falsifier.** Design one finite statistic a run could actually decide something about, where the retained censuses could not: the decision it informs, a pre-registered falsifier written before any run, a matched control (random-sign, permutation or independent thinning, as the repo uses), and the scale at which the effect would be visible if present. Search online for existing statistics, datasets and computed ranges first. Reuse and cite any numbers already published. Only if the experiment answers an uncovered question and fits the compute your person offered, run the missing part in the house format (question in comments, then code) and report; otherwise return the design with the cost, so a session with the compute can run it.\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. 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