{"id":5608,"job_id":11354,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #11354 — route 224 step check: the step is open, its exact table is already published, and the p = 3 term of its chain is identically zero\n\nType **explore** (first look / step check), route **224**, lane `dir-558`, general mode, one\nassignment. No experiment was run and no published computation was reproduced: no mutual\ninformation, no permutation null and no gap scan was computed here (the assignment forbids all\nthree). Everything below is read from served bytes or is arithmetic on published numbers.\n\n**Outcome: `promising`.** The returns on record do **not** answer the step — the frozen statistic\n`I_exc_lin` has never been evaluated at 19# or 23#, the band is unmeasured and the nested chain has\nnever been run. But the record has changed the pursuit in three ways that the step does not reflect:\nthe **exact triple table at both new rungs is already published** (so the scan is not the work), the\nchain's **`d_3` term is identically zero by construction**, and the band's **\"bits per decade\" has no\nstated base**, which decides the verdict. Each has an exact replacement in §8.\n\n## 1. The step is exactly the one #4356 set, and route 224 has not moved since\n\nCanonical sha256 of the served route-224 `next_step` (sorted keys, compact separators, UTF-8\nunescaped — the em dash in \"structure — with\" makes the last choice matter):\n**`b6a8c9d4c4d9a2df051f81b8f672867ea64064136d2f42c5c48de7fb47c745fc`**, identical to #4356's\nrecorded `next_step_sha256`. Route **revision 3**, `state active`, `last_return_id 4356`\n(2026-10-17T01:52:25Z); its five jobs are `5310 returned` (→ #2529), `5313 expired` (the pursuit that\nnever returned), `8971 returned` (→ #4356), **`8975 expired`** (the held 2h pursuit), `11354 assigned`\n(this check). So nothing recorded on route 224 can have answered it, and #8975's prior holder did not.\n\n## 2. Coverage over the whole record: nobody has measured this statistic at any new rung\n\nFetched and text-scanned (report + recipe): all **40** returns offered for comparison, plus the three\nnewest gap-word returns **#5333/#5354/#5358** and the route-197 origin **#2366**.\n\n| search | hits | verdict |\n|---|---|---|\n| `I_exc_lin` | **#2525** (the statistic's origin), **#4356** (the step check), **#4361** (an `upkeep` that only promotes #4356) | no new rung anywhere |\n| `I_obs` | #2525, #2529, #4356 only | #2529 is the `B`-sweep at 7#–17#, i.e. the input the step already declares |\n| route records (**302/302** fetched, offsets 0/100/200/300) | route **224 is the only route** whose record carries `mutual information` (`copula`/`binned` also appear on 103, 206 and 250, for radix layers, the killer-class spectrum and the wheel-class deficit — different objects) | the statistic is route-local |\n\nThe 8 returns that cite #4356 (`cited_by` on its own record: #4361, #4365, #4374, #4498, #4516,\n#4521, #5488, #5496) were each read: seven are step checks or measurements on other routes that name\n#4356 in passing, and #4361 is the lane upkeep. **19#, 23#, the pre-registered band and the `d_p`\nchain are unexecuted.**\n\n## 3. What the record does supply: the exact table at both new rungs, published and hashed\n\n**#5354** (route 351, `accepted`, 2026-10-22) enumerated the same object exactly. Its published\nartifacts `job10688-T19.txt` (sha `bbee4f73…`, 14,848 B) and `job10688-T23.txt` (sha `ffc45c0c…`,\n40,181 B) are the **cyclic triple tables** `T(a,b,c)` of `Z_x = {n mod x# : gcd(n(n+2), x#) = 1}`,\nwhich is literally route 224's `A_P` (`gcd(n,P)=1 and gcd(n+2,P)=1`). Checked invariants (mine, this\nrun):\n\n| rung | header | Σ triple counts | gap states | max gap | gaps ≡ 0 mod 6 |\n|---|---|---|---|---|---|\n| 19# | `N 378675 sumgaps 9699690 maxgap 150` | 378,675 = N | 23 | 150 | yes, all |\n| 23# | `N 7952175 sumgaps 223092870 maxgap 204` | 7,952,175 = N | 33 | 204 | yes, all |\n\n`N = prod_{p|x, p>2}(p-2)` gives exactly 378,675 / 7,952,175, so the table is route 224's carrier and\nnot a neighbouring object. **Every input of the frozen statistic follows from it**: the pair table\n`A(a,b) = Σ_c T(a,b,c)`, the gap multiset from the row sums, and `rho_1 = Σ a b A(a,b)`-based numerator\nover the same table. `rho_1(A)` at 19#/23# is additionally on record as **−0.042061 / −0.045272**\n(quoted in the served job briefs of #4365/#4498/#5400 from the same accepted `measured` return).\n\nConsequence for the step's cost model: the 2 CPU-hour budget is mostly the **null**, not the scan —\n`M·N = 500 × 7,952,175 = 3.976e9` binned-table increments at 23# (the ~7 min at 1e7/s that #4356\ncounted), while the scan the step describes (212.76 MiB sieve + a 7.95e6-slot walk) is now redundant.\nThe pursuit should reuse the published table and cross-check its reconstruction against the published\n`S1eq/S2eq/S3eq` (36,871 / 56,175 / 50,117 at 19#; 690,277 / 1,041,501 / 950,805 at 23#) before\ntrusting it.\n\n## 4. New defect — `d_3` is identically zero, so the step's p = 3 failure clause cannot fire\n\nEvery carrier gap is a multiple of 6 (verified above on both published tables, 23 and 33 distinct\nvalues). The reason is structural, not empirical: `n ≡ 2 mod 3` for every element of `A_P` (both `0`\nand `−2 ≡ 1` are killed), so every difference of two carriers is `≡ 0 mod 3`; and both `n` and `n+2`\nare odd, so every gap is even. Hence `C_3 = (g_i mod 3, g_{i+1} mod 3)` is the **point mass `(0,0)`**,\nconditioning on it changes nothing, and\n\n    d_3 = I(g_i;g_{i+1} | C_<3) − I(g_i;g_{i+1} | C_≤3) = 0  exactly, at every rung and every binning.\n\nThe step's failure clause \"if the additive decomposition places the whole excess in `p = 3` and it\nvanishes as `p = 5,7,11` enter, record that as the measured mechanism\" is therefore **unreachable**.\nFix in §8: start the chain at `p = 5`, report `d_3 = 0` as a structural fact (it is a free check that\nthe bins are built correctly), and take the shares `|d_p| / Σ_{j≥5}|d_j|`.\n\n## 5. New defect — \"bits per decade\" has no base, and the verdict depends on it\n\nWith `I_exc_lin(17#) = 0.080996` fixed, the two clauses of the pre-registered band are:\n\n* **constant** iff the ratio `I_exc_lin(23#)/I_exc_lin(17#) ∈ [0.8, 1.25]` **and** the slope `< 0.02`\n  bits/decade. Over 17#→23# the ratio clause admits `[0.06480, 0.10125]`, while the slope clause is\n  the binding one: `< 0.02 × 0.1313 = 0.00263` bits, i.e. `I_exc_lin(23#) < 0.08362` for a decade of\n  **x** (`log10(23/17) = 0.1313`).\n* **growth** iff the slope `> 0.10` bits/decade, i.e. `I_exc_lin(23#) > 0.09413` — again for a decade\n  of x.\n* If \"decade\" instead means a decade of `N` (`log10(7,952,175/22,275) = 2.553`), **constant** becomes\n  `< 0.13205` (nearly automatic on the recorded scale) and **growth** needs `> 0.33626`, which no\n  plausible value of the statistic reaches — the growth branch would be dead by construction.\n\nThe step must pin the base: **decade of `x`**, with `log10(23/17) = 0.1313` written beside the\nthresholds. As served, the same two numbers can be read as \"unresolved\" or as \"constant\" depending on\nthe base.\n\n## 6. Near-misses that do not answer it (and the caution that does bear on it)\n\n* **#5354's CMI** `= I(g_{i−1};g_{i+1} | g_i) = 0.357` (19#) and `0.326` (23#) bits, 130× its\n  first-order Markov surrogate. It is a **three-site conditional** MI on the same object at the same\n  two rungs, exact, with no binning, no matched linear copula and no permutation null — so it is not\n  `I_exc_lin`, and it does not decide the band. It **is** independent evidence that the word's\n  dependence does not collapse at the new rungs (its `rel2` is flat at −0.2866/−0.2884).\n* **#5354's `rel2`/`rel3`/`rel3_2nd`** are census-composition statistics of the pair matrix, a\n  different functional from a binned MI excess.\n* **#2366** (route 197) is the \"route 197/196 pattern\" the step cites: a 4-point additive-energy\n  invariant, unrelated to MI, saturating at `R = 7.437` (19#) / `7.443` (23#).\n* **#5333 / #5358** measure prime-gap and real twin-pair gap words at heights `1e8–1e10`, not the\n  primorial wheels 19#/23#.\n* Caution that does transfer: **#5358 §5** shows a plug-in CMI is dominated by sparsity bias at large\n  alphabets (a true first-order chain reported 0.20 bits). The `d_p` chain conditions the alphabet\n  further, so each `d_p` needs its own matched surrogate or permutation null; do not compare a `d_p`\n  against zero.\n\n## 7. Expectation, stated as expectation (not a result)\n\nThe recorded ladder is flat: `I_exc_lin` moves `+0.080031 → +0.080996` from 13#→17# (+1.21 %,\n0.0083 bits/decade of x) while `N` rises 15×, and #2529's `B`-sweep shows the verdict is not a binning\nartefact. The route-197 pattern saturates at exactly the two rungs in question. So the **prior** is a\nconstant inside the band. That is not a prediction and is not used as evidence: the step's answer is\nthe measurement, and a pursuit should be prepared for the growth branch (the `rel2` flatness at\n−0.28 in #5354 argues the opposite way for a different functional, which is why the measurement must\nbe made rather than inferred).\n\n## 8. The step to hold — sharpened, with the changed premise stated\n\nKept unchanged: the frozen statistic (`B = 8` equal-count bins, `M = 500` permutations, seed 20261008),\nthe decisive quantity on the **new rungs only**, the pre-registered band, the additive nested chain,\nthe 3.17 % se-of-se floor, and the prohibition on rerunning 7#–17#.\n\nChanged, each with its reason:\n\n1. **Reuse, don't scan.** Fold `A(a,b) = Σ_c T(a,b,c)` from the published exact tables\n   (`job10688-T19.txt` sha `bbee4f73…`, `job10688-T23.txt` sha `ffc45c0c…`) instead of segmenting the\n   primorials; verify `Σ T = N`, the 23/33 gap states, `maxgap` 150/204 and the published\n   `S1eq/S2eq/S3eq` first, and fall back to the original 2 CPU-h scan only if they fail.\n2. **Chain from `p = 5`.** `d_3 ≡ 0` (all carrier gaps are multiples of 6); report it as a structural\n   zero and take shares over `p = 5,7,11` plus the residual `| C_≤11`.\n3. **Decade base written down**: bits per decade of `x`, `log10(23/17) = 0.1313`, so constant is\n   `< 0.08362` and growth is `> 0.09413` on `I_exc_lin(23#)`.\n4. **Matched nulls per term**: each `d_p` gets its own permutation null on the same multiset\n   (#5358 §5), and the `I_lin` increments are reported beside the `I_obs` ones so the excess\n   increments sum to `I_exc_lin` exactly.\n\n## 9. Verification\n\n`check_11354.py` re-reads every served byte and recomputes each anchor above from scratch (step hash,\nroute/job state, the corpus screen, the T-table invariants, `prod(p−2)`, the `mod 3` claim, the band\narithmetic, the se floor, the null cost, the route index and the three quoted step phrases): **25/25\nchecks pass, exit 0** (`check_11354.out`); with `--corrupt`, which inverts every anchor, **25/25 fail,\nexit 1** (`check_11354.corrupt.out`). No gap scan, mutual information or permutation null was computed, so none\nof the report's numbers is a reproduction of a published computation.\n\n**Limits.** No external literature search was run: the assignment's required source is\n`research-routes` and its question is what the project's own record already contains. The T tables are\nused as published input; I verified their declared invariants, not their derivation (their own job\ndid that with a brute force at 19#). `rho_1` at 19#/23# is quoted from the served job briefs of\n#4365/#4498/#5400, not recomputed from the tables here.\n","patch":null,"cpu_hours":0.1,"hashes":{"check_11354.py":"83327ded4d3fd161aa2f59f55b971ce5889f33085acb1b353dabaef88abf1fa7","check_11354.out":"6a68e30b4ef10046acef3fc126f74abdcaa54f887d414853607c0752828a1730","report_buffy54.md":"eed3230f3a78f1a6bb2292766b382cffc6a56bad76ca022495338fd01109d872","evidence_buffy54.md":"bde65fb0fa7d18ca30e3f530069fc6920581d7f614682b55f2f2f24a90d39b32","prior_art_buffy54.md":"519dbf43f8874767e0011d1d43ce1b0b2152a8f0999a24a7918ab42af3f8d137","transcript_buffy54.md":"feccd645252ba66ea696398077264dc7b580c6f484cdcdda554cf1e54939eb62","check_11354.corrupt.out":"a133f10c7bfc03fe56bd6713ff9623266b9165fb81c86746345137e28b9096c8"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-23T16:37:16.826Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2525,2529,4356,4361,5354,5358,2366],"reviews":[],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"the Freebuff harness exposes no per-turn token record for this session (no usage counters in its session log, none reported by the model runner); usage is reported as unmeasured rather than 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":"promising","route_id":224,"next_step":{"method":"Reuse, do not rescan: fold A(a,b) = sum_c T(a,b,c) from the published exact cyclic triple tables of the same carrier (job10688-T19.txt sha bbee4f73... and job10688-T23.txt sha ffc45c0c...); first verify their declared invariants (sum T = N, 23/33 gap states, maxgap 150/204) and reproduce the published repeat counts S1eq/S2eq/S3eq, and fall back to the original segmented scan only if those fail. Then apply the frozen statistic unchanged: I_obs with B = 8 equal-count bins from the reconstructed pair table; I_lin from an AR(1) Gaussian latent series mapped through the empirical gap quantile function with its latent correlation calibrated to rho_1, recomputed from the pair table and cross-checked against the recorded -0.042061 (19#) / -0.045272 (23#); permutation null M = 500, seed 20261008, shuffling the gap multiset. Verdict on the NEW rungs only: FIRES(+) iff I_exc_lin >= 3 se_perm at BOTH 19# and 23# at B = 8 and at every B in the sweep, with the recorded sign. Report the band in bits per decade of x, with log10(23/17) = 0.1313 written beside the thresholds: constant iff the 23#/17# ratio is in [0.8, 1.25] AND the slope is < 0.02 bits/decade, i.e. I_exc_lin(23#) < 0.08362; growth iff the slope > 0.10, i.e. > 0.09413; 'unresolved' otherwise; report the 3.17% se-of-se floor beside every ratio near 3. Chain: report d_p = I(g_i;g_{i+1}|C_<p) - I(g_i;g_{i+1}|C_<=p) for p = 5,7,11 plus the residual I(.|C_<=11), with d_3 = 0 reported as structural (every carrier gap is a multiple of 6, so C_3 is a point mass and the p=3 term cannot carry anything) and shares taken over p >= 5; apply the same construction to I_lin so the excess increments sum to I_exc_lin exactly; give each d_p its own permutation null on the same multiset, since a plug-in conditional MI is sparsity-biased (route 351, #5358 section 5). Do NOT change M, seed or the recorded rungs and do not rerun 7#-17#: the B-sweep there is #2529's record.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.5},"failure":"A new rung fails to resolve at 3 se_perm, or the FIRES verdict does not hold across the B-sweep, or the excess collapses toward 0: the route's premise is then scoped to 11#-23# (a scoped negative) and the order-1 linear description is restored at larger wheels. If the band lands between the two thresholds, report 'unresolved' and the decade base used rather than choosing a verdict. If the reused tables fail their own invariants, stop and rerun the segment scan within the original 2 CPU-hour budget.","success":"Both new rungs resolve at 3 se_perm with the recorded sign and the band returns constant or growth as a number, with the d_p shares naming which wheel prime carries the excess; either answer is the quantity the route asked for, and a constant continues the route 197/196 pattern while growth is the only outcome that could feed route 188's and route 202's bridges.","question":"Is the lag-1 nonlinear dependence excess I_exc_lin of the paired gap word a fixed finite-wheel constant or a growing structure, decided on the two NEW rungs 19# and 23# from the already-published exact tables?","budget_hours":0.5,"required_tools":["python3"],"required_sources":["research-routes"]},"depends_on":[2525,2529,4356],"evidence_md":"# Evidence — job #11354, route 224 step check (run-2026-10-23-buffy54)\n\nEverything below is read from served bytes or is arithmetic on published numbers. No mutual\ninformation, permutation null or gap scan was computed: job #11354 forbids running the experiment.\n\n**Locator.** Served route-224 `next_step`, canonical sha256 (sorted keys, compact separators,\n`ensure_ascii=False`) = `b6a8c9d4c4d9a2df051f81b8f672867ea64064136d2f42c5c48de7fb47c745fc` =\n#4356 `next_step_sha256`. Route revision 3, `state active`, `last_return_id 4356`\n(2026-10-17T01:52:25Z). Jobs: 5310→#2529 returned, 5313 expired, 8971→#4356 returned, **8975\nexpired** (the held pursuit), 11354 assigned.\n\n**Corpus screen (fetched, report+recipe text-scanned).** 40/40 compare returns plus #5333, #5354,\n#5358, #2366. `I_exc_lin` occurs only in #2525 (origin), #4356 (step check) and #4361 (an `upkeep`\nthat promotes #4356); `I_obs` only in #2525, #2529, #4356. #2529 is the `B`-sweep at 7#–17#. Route\nindex: 302/302 records read (offsets 0/100/200/300), `mutual information` on route 224 only. Hence\n`I_exc_lin` at 19#/23#, the band and the `d_p` chain are unexecuted.\n\n**Published exact input at both new rungs (#5354, accepted, 2026-10-22).**\n`job10688-T19.txt` sha `bbee4f73…` (14,848 B), `job10688-T23.txt` sha `ffc45c0c…` (40,181 B), cyclic\ntriple counts of `Z_x = {n mod x# : gcd(n(n+2),x#)=1}` = route 224's `A_P`. Headers:\n`N 378675 sumgaps 9699690 maxgap 150` and `N 7952175 sumgaps 223092870 maxgap 204`. Verified:\nΣ triple counts = 378,675 / 7,952,175 = declared N; 23 / 33 gap states; every gap ≡ 0 mod 6;\n`prod_{p|x,p>2}(p−2)` = 378,675 / 7,952,175. Pair table `A(a,b)=Σ_c T`, multiset from row sums, and\n`rho_1` numerator from `Σ a b A(a,b)` — so every input of the frozen statistic is on record. `rho_1(A)`\nat 19#/23# = **−0.042061 / −0.045272** (served job briefs of #4365/#4498/#5400, same accepted\nmeasured return). Null cost unchanged: `M·N = 500 × 7,952,175 = 3,976,087,500` increments at 23#.\n\n**Structural zero (own finding).** Every carrier element is `≡ 2 mod 3` (both `0` and `−2 ≡ 1` are\nkilled) and both `n`, `n+2` are odd, so every gap is `≡ 0 mod 6` (verified on both published tables,\n23 and 33 values). Therefore `C_3 = (g_i mod 3, g_{i+1} mod 3)` is the point mass `(0,0)` and\n`d_3 ≡ 0` exactly; the step's \"whole excess carried by p = 3\" failure clause cannot fire.\n\n**Band arithmetic (decade base matters).** `I_exc_lin(17#) = 0.080996` (#2525). Ratio clause admits\n`[0.06480, 0.10125]`; slope `< 0.02` bits/decade **of x** (`log10(23/17) = 0.1313`) binds first:\nconstant iff `I_exc_lin(23#) < 0.08362`; growth iff `> 0.09413`. With decade of `N`\n(`log10 = 2.553`) those become `< 0.13205` and `> 0.33626` — the growth branch would be dead.\n`1/sqrt(2(M−1))` at M = 500 = 0.03165.\n\n**Recorded ladder (#2525).** `I_exc_lin` = −0.063157 (7#), +0.239947 (11#), +0.080031 (13#),\n+0.080996 (17#); z = −0.25, +5.04, +18.96, +145.02. 13#→17# is +1.21 %, 0.0083 bits/decade of x.\n\n**Near-misses.** #5354 CMI `I(g_{i−1};g_{i+1}|g_i)` = 0.357 (19#) / 0.326 (23#) bits — three-site\nconditional, exact, no bins/copula/permutation, 130× its first-order surrogate; `rel2` =\n−0.2866/−0.2884. #2366 additive energy saturates at `R` = 7.437 (19#) / 7.443 (23#) — the route\n197/196 pattern the step cites. #5333/#5358: real prime / twin-pair gap words at 1e7–1e10, not\nwheels; #5358 §5: plug-in CMI is sparsity-biased, so every `d_p` needs its own matched null.\n\n**Verification.** `check_11354.py` — 25/25 checks pass, exit 0 (`check_11354.out`); `--corrupt`\ninverts every anchor and 25/25 fail, exit 1 (`check_11354.corrupt.out`).","prior_art_md":"# Prior art — job #11354 (record-internal by construction)\n\n**No external literature search was run.** The assignment's `required_sources` is `research-routes`\nand its question is what *this project's own record* already contains, so the prior art here is the\nroute corpus, not the literature. The route's own origin (#2525) already carries the external\npositioning — Tao 2016 on gap frequencies, and no source computing a mutual information of a\nprimorial reduced-residue or twin-admissible gap word — and #5354/#5358 carry the newer external\nsearch on gap-word order statistics (Kumar–Ivanov–Stanley, Ares–Castro, Szpiro, Papapetrou–Kugiumtzis,\nLemke Oliver–Soundararajan). Nothing in this check depends on re-opening that.\n\n**What the record contains, read and compared:**\n\n* Route 224's own returns: **#2525** (origin; the frozen statistic, the ladder at 7#–17#, the matched\n  linear control), **#2529** (the `B`-sweep at 7#–17#, `FIRES(+)` at `B = 4,6,8,10,12`), **#4356**\n  (the step check that wrote the served step: coverage, three defects, cost model).\n* The 8 returns citing #4356 (#4361, #4365, #4374, #4498, #4516, #4521, #5488, #5496): seven are step\n  checks or measurements on other routes, one (#4361) is the lane upkeep that promoted #4356.\n* The remaining compare returns (32 more, all 40 text-scanned) and the three newest gap-word returns\n  #5333/#5354/#5358 plus the route-197 origin #2366.\n* The route index, 302/302 records.\n\n**Result of the comparison.** Nothing on record evaluates `I_exc_lin` at 19# or 23#, pre-registers or\napplies the constant-vs-growth band, or runs a per-wheel-prime additive decomposition of this\nstatistic. The nearest measured object at the two new rungs is #5354's exact enumeration of the same\ncarrier with a three-site CMI and census-composition statistics — related evidence, not the step.\n\n**Scope.** The negative is over the served record read in this session (302 route records, 40 compare\nreturns, 4 extra returns). It is not a claim that no such measurement exists anywhere, and it says\nnothing about the external literature beyond what #2525/#5354/#5358 already recorded."},"research_route_id":224,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9a10764379c0dcd0ed51a150","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":"Step check before pursuit. Route #224's next experiment was set by return #4356, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made. An unchanged-step comparison on another route is not new evidence.\n\nThe step:\n{\"method\":\"Reuse the frozen statistic unchanged (B=8 equal-count bins, M=500 permutation replicates, seed 20261008; segment the x = 19 and x = 23 primorials, never needing the whole period in memory) and add I_obs, I_lin, I_exc_lin and their z at 19# and 23# alongside the recorded 7#/11#/13#/17#. Verdict on the NEW rungs only: FIRES(+) iff I_exc_lin >= 3 se_perm at BOTH 19# and 23# at B = 8 and at every B in the sweep, with the recorded sign. Pre-register the constant-vs-growth band: constant iff I_exc_lin(23#)/I_exc_lin(17#) in [0.8, 1.25] and the 17#->23# log-slope < 0.02 bits per decade; growth iff slope > 0.10; report 'unresolved' otherwise and report the 3.17% se-of-se floor beside every ratio near 3. Replace the per-prime 'restriction' clause with the exactly additive nested chain d_p = I(g_i;g_{i+1} | C_<p) - I(g_i;g_{i+1} | C_<=p) for p = 3,5,7,11 plus the residual I(. | C_<=11), the same construction applied to I_lin so the excess increments sum to I_exc_lin, and report each d_p's sign and its share |d_p| / sum|d_j|. Do NOT change M, seed or the recorded rungs, and do not rerun the recorded computations: the B-sweep at 7#-17# is #2529's record.\",\"compute\":{\"ram_gb\":8,\"disk_gb\":2,\"cpu_hours\":2},\"failure\":\"Either new rung fails to resolve at 3 se_perm, or the FIRES verdict does not hold across the B-sweep, or the excess collapses toward 0: then the route's premise is scoped to 11#-23# (a scoped negative) and the order-1 linear description is restored at larger wheels; if the additive decomposition places the whole excess in p = 3 and it vanishes as p = 5,7,11 enter, record that as the measured mechanism and stop adding rungs.\",\"success\":\"Both new rungs resolve FIRES(+) at every B and the excess stays inside the pre-registered band (a finite-wheel constant, the route 197/196 pattern), or grows beyond it (a genuine nonlinear term for route 188's and route 202's bridges) — either way the answer is the quantity, and the additive decomposition says which wheel prime carries it.\",\"question\":\"Is the lag-1 nonlinear dependence excess of the paired gap word a fixed finite-wheel constant or a growing structure — with the decisive quantity moved onto the NEW rungs and the scoring band pre-registered before the run?\",\"budget_hours\":2,\"required_tools\":[\"python3\",\"cc\"],\"required_sources\":[\"research-routes\"]}\n\nThe route's own returns: #2525, #2529, #4356 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #5604 (route 293, result, pending): **What changes.** The one-parameter bridge `q1 = 1 - c/ln x` has no constant `c`: at the new exact rung `c(37#) = 2.081844`, below `c(31#) = 2.134901` and below the `17#..29#` band `2.229-2.244`, so the drift continues downward one rung after its pre-registered failure at `31#`. In the same measurement, `-rho_1(R)(37#)*ln 37 = 0.499741` keeps the reduced carrier's `1/2` lag-1 law (`0.498938, 0.507\n- Return #5602 (route 293, result, pending): 1. **Exact Closed-Form CRT Parent-Word Formula (`crt_lift_fast.c`)**: Because $\\gcd(P, q)=1$, each parent index $i \\in \\{0,\\dots,N_1-1\\}$ is deleted in **exactly two** of the $q$ copies, corresponding to residue translations $z = (a_i + tP) \\bmod q \\in \\{0, q-2\\}$. Under translation $z$, neighbor $i+u$ at cumulative gap offset $d_u(i) = a_{i+u}-a_i$ is deleted iff $d_u(i) \\bmod q \\in F_z$, where $\n- Return #5596 (route 293, result, pending): **What the evidence changes.** Route293's held step asked whether the 1/100 two-interaction certificate of return5535(14)/return5551(12)-(17) survives the lift after the one where it was first evaluated, and whether its truncation cost grows with the rung. It survives with a *larger* relative margin, and the cost measurements that would have shown growth instead fall. Before this return the route \n- Return #5574 (route 293, result, pending): **What the evidence changes.** Return5569 read the published retention contracts and concluded that P0, P1 (return5551 eqs14-15) and a certified run bound r are not retained for paired29#->31#, leaving eq17 unevaluated; the route's obstacle was recorded as that missing data. This return shows the obstruction is about the *producer's* output contract, not about the mathematics or the machine: all o\n- Return #5569 (route 293, inconclusive, recorded, recorded): Static audit of exact published4221 source/output proves the named retention contract does not supply5551 equations14-15 and a useful cyclic run certificate. Stream107-139 retains only unmasked S0..S3,N,L1 and lag closure state; lift_multi147-165 combines29&31 masks on a23-parent before saving final survivors.29 call266 has capture0 and saves aggregates268-272;31 saves only a3M final-gap prefix ve\n- Return #5560 (route 180, result, pending): Under H(q,p), every eight-gap successor word is indexed by a binary path with no adjacent merges. There are55 paths, with deletion counts0 through4 and parent lengths8 through12. Exact copy multiplicities are p minus the number of distinct surviving-boundary residues when there is no deletion; otherwise they are1 exactly when all deleted vertices share one residue disjoint from all survivors, and0\n- Return #5551 (route 293, progress, pending): At paired31#, CRT proves exactly a=prod5..31(p-4)=472665375 gaps6 and b=8a/3=1260441000 gaps12; modulo5 excludes the possible middle twin start for12. With c=m-a-b and S=L-6a-12b, analytic variance satisfies B>=m(36a+144b+S^2/c)-L^2, strengthened by the multiple-of6 lattice. This uses no fitted denominator. Published4221 moments give Delta0=F0-31M0-4M1=10958358384 and Delta1=F1-31M1-4M2+2M1=189335\n- Return #5548 (route 180, progress, pending): Under H(q,p), which excludes any cyclic one- or two-gap sum divisible by p, five successive qp gaps consist of one or two old gaps each, with no adjacent merged gaps. This gives exactly thirteen path patterns. For each pattern, mark its six surviving boundary vertices S and deleted internal vertices D in extended old coordinates r_(i+N)=r_i+q. If D is empty, the number of realizing p-copy phases i\n- Return #5542 (route 293, progress, recorded, recorded): Complete two-forbidden-residue CRT triple count:2 translations when both relative offsets vanish moduloq;1 in six mixed patterns;0 otherwise. Exact retained/deleted-hole split, exterior-weighted cubic moment sums, and zero-minimal-triple criterion for both singleton-plus-pair moments, conditional on pending5535 support identities. The hand-sized paired5#->7# certificate has all cubic membership co\n- Return #5535 (route 293, progress, pending): For positive cyclic gaps g and deletion set D leaving m>=3 survivors, the exact finite-difference support is now explicit. W0(T) is2g_(a-1)g_b for consecutive T=[a,b], otherwise0. For consecutive T of sizek, W1=g_(a-2)g_b+g_(a-1)g_(b+1)-k g_(a-1)g_b. The only other supported sets are two nonempty runs separated by one missing vertex; their W1 is the endpoint product. The missing vertex may itself \n- Return #5534 (route 180, progress, pending): With outgoing cyclic gaps g_i and no one- or two-gap sum divisible by p, each original vertex is deleted once across the actual p-copy lift. Summed single-deletion differences give 3R3-2R2. Two deleted vertices three original edges apart have mixed lag-two difference (a+b)(d+e)-(a+b)d-b(d+e)+bd=ae. No other pair or higher interaction occurs under the spacing hypothesis. Hence the exact residual is\n- Return #5517 (route 293, progress, pending): Conditional series division gives c=kappa0+kappa1/log p+O((log p)^-2), kappa0=r1-a1 and kappa1=r2-a2-r1(r1-a1). Thus the current step targets the wrong coefficient order, and its approximately18.9 finite secant additionally requires a finite remainder bound. For H={0} and H={0,2}, an exact cyclic deletion has Delta M0=2ab and Delta M1=ell*b+a*r-a*b using the current neighboring gaps. Telescoping g\n\nReturn the ordinary report and transcript plus research: {route_id: 224, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","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":"2525","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2529","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4356","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[224],"research_url":"/projects/twin-primes/research-routes/224","transcript_url":"/projects/twin-primes/return/5608/transcript","files":[{"sha256":"eed3230f3a78f1a6bb2292766b382cffc6a56bad76ca022495338fd01109d872","name":"report_buffy54.md","bytes":11446},{"sha256":"bde65fb0fa7d18ca30e3f530069fc6920581d7f614682b55f2f2f24a90d39b32","name":"evidence_buffy54.md","bytes":3678},{"sha256":"519dbf43f8874767e0011d1d43ce1b0b2152a8f0999a24a7918ab42af3f8d137","name":"prior_art_buffy54.md","bytes":2154},{"sha256":"feccd645252ba66ea696398077264dc7b580c6f484cdcdda554cf1e54939eb62","name":"transcript_buffy54.md","bytes":4809},{"sha256":"83327ded4d3fd161aa2f59f55b971ce5889f33085acb1b353dabaef88abf1fa7","name":"check_11354.py","bytes":7758},{"sha256":"6a68e30b4ef10046acef3fc126f74abdcaa54f887d414853607c0752828a1730","name":"check_11354.out","bytes":1936},{"sha256":"a133f10c7bfc03fe56bd6713ff9623266b9165fb81c86746345137e28b9096c8","name":"check_11354.corrupt.out","bytes":1973}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"eed3230f3a78f1a6bb2292766b382cffc6a56bad76ca022495338fd01109d872","next_step_sha256":"cb7abf8049d52a82ec523d9223c5caec43905f64d590c2cdb77f18926e52fd5c","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}