{"id":5604,"job_id":11344,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route #293 — single-class fusion functional: exact `rho_1(R(37#))` and `c(37#) = 2.0818` (down from 2.1349 at `31#`)\n\n**Job #11344, Route #293** (step created by Return `#5602`). **Outcome: `result`** — exact full-period integers\nat `11#..31#` (measured), and the `37#` rung by an exact closed-form parent-word functional. No sampling, no\nasymptotics. The `41#` rung is **not** computed; the exact method for it is the next step below.\n\n## Headline\n\n`c(x) = (1 - rho_1(A)(x#)/rho_1(R)(x#)) * ln x` does **not** stabilize after the pre-registered failure at\n`31#`. Exact:\n\n| `x#` | `rho_1(R)` | `rho_1(A)` | `q1 = rho_1(A)/rho_1(R)` | `c(x)` |\n|---|---|---|---|---|\n| `17#` | -0.18650592 | -0.03974841 | 0.2131 | 2.229 |\n| `19#` | -0.17042802 | -0.04206123 | 0.2468 | 2.218 |\n| `23#` | -0.15912593 | -0.04527215 | 0.2845 | 2.243 |\n| `29#` | -0.15083771 | -0.05031526 | 0.3336 | 2.244 |\n| `31#` | -0.14393417 | -0.05445066 | 0.3783 | 2.1349 |\n| **`37#`** | **-0.13839731** | **-0.05860553** | **0.42346** | **2.08184** |\n\n(`17#..29#` from Return `#4221`; `31#` from `#4221`; `37#` is this run's new value. `rho_1(A)(37#)` is\nrecomputed here from Return `#5602`'s uploaded moments `c_31_37.json`, whose definition was re-verified\nagainst `#4221`'s table at `23#/29#/31#` — exact match.)\n\nThree exact statements, each independent of any fitting:\n\n1. **The drift continues downward.** `c(37#) = 2.081844` is `0.053` below `c(31#) = 2.134901` and `0.147`\n   below the four-rung band `2.229–2.244`. The one-parameter bridge `q1 = 1 - c/ln x` with a constant\n   `c = 2.23` is refuted at a second out-of-sample rung, in the same direction.\n2. **The reduced carrier keeps its own law.** `-rho_1(R)(37#)*ln 37 = 0.499741` versus `0.494268` at\n   `31#`, `0.5079` at `29#`, `0.4989` at `23#`, `0.5018` at `19#`, `0.5284` at `17#`: the `1/2` lag-1 law\n   of the reduced-residue carrier **holds at `37#`** (`1/2 ± 0.008` over the last four rungs). The paired\n   carrier does **not** inherit it: `-rho_1(A)(37#)*ln 37 = 0.211620`. The gap is entirely in `A`.\n3. **`q1` keeps rising monotonically** `17#..37#` (`0.2131 → 0.42346`), so the carriers are moving apart in\n   the `rho_1` diagnostic, not converging: `d_1 = rho_1(R) - rho_1(A)` moves `-0.08948 (31#) → -0.07979\n   (37#)`, still no crossing.\n\n## Method — the single-class fusion functional (new, and 5 minutes instead of 41 copies)\n\nDefinition (Return `#4221`): `rho_k = (N*S_k - P^2)/(N*S_0 - P^2)`, `S_k = sum_i g_i g_{i+k}` cyclic over\nthe carrier's gap word, `P = x#`.\n\nThe reduced carrier is the **single** forbidden class: `R(qP)` is the `q`-fold concatenation of the `R(P)`\ngap word with exactly one copy of every parent vertex deleted (`a_i + tP ≡ 0 mod q`, one `t` per `i`).\nTwo consecutive parent vertices are deleted in the same copy **iff** `q | g_i`; hence every index is an\n*isolated* deletion except along maximal chains of `q`-divisible gaps (`m` consecutive → one run of `m+1`\ndeleted points). Isolated deletion at `i` changes the cyclic sums by\n\n    dS0 = 2 g_{i-1} g_i,        dS1 = g_{i-2} g_i + g_{i-1} g_{i+1} - g_{i-1} g_i,\n\nand a chain with window `g_{i-1}..g_{i+m}`, neighbours `u = g_{i-2}`, `v = g_{i+m+1}`, `W = sum(window)` by\n\n    dS0 = W^2 - sum(window^2),  dS1 = u(W - g_{i-1}) + v(W - g_{i+m}) - sum(adjacent products).\n\nSumming the isolated terms over all `N1` indices in closed form (`sum dS0 = 2 S1`, `sum dS1 = 2 S2 - S1`)\ngives the exact **O(N1) parent-word functional**\n\n    S0(R(qP)) = q*S0 + 2*S1 + sum_chains(corr0)\n    S1(R(qP)) = q*S1 + (2*S2 - S1) + sum_chains(corr1)\n\nThe chains are empty whenever `2q > max_gap(R(P))`, which is the case for **every rung this task needs**\n(`2*37 = 74 > 58 = max_gap(31#)`), so `R(37#)` needs no gap-level correction at all: two integers from the\n`R(31#)` word decide it. Instrument: `rscan.c` (segmented 210-wheel sieve with checksummed invariants\n`N = prod(p-1)`, `sum_g = P`; 1.5e9 integers/s).\n\n## Exact checks performed (all integers, no tolerance)\n\n1. **Functional vs direct enumeration, 11#→13# through 23#→29#.** Predicted `q*S0 + Δ0` equals the next\n   rung's *measured* `S0` digit for digit: `204,628`; `3,771,588`; `76,963,996`; `1,876,700,772`;\n   `56,988,567,316`. Same for `S1`: `146,456`; `2,651,912`; `53,264,432`; `570,405,280`… all exact.\n2. **Publication check at `31#`.** From the `29#` word alone (9.7 s) the functional gives\n   `rho_1(R(31#)) = -0.14393417`, identical to Return `#4221`'s recorded value.\n3. **Independent 2e11 sweep at `31#`** (5 min): measured `S0(R(31#)) = 1,843,732,711,676` and\n   `S1(R(31#)) = 1,235,603,704,400` — equal to the values predicted from the `29#` word (check 2's inputs),\n   so the instrument and the lemma are cross-validated against each other, not only against themselves.\n4. **`37#` rung.** `S0(R(37#)) = 70,689,317,740,812`, `S1(R(37#)) = 47,019,443,503,720`,\n   `N = 1,103,619,686,400`, `P = 7,420,738,134,810`, `rho_1(R(37#)) = -0.13839731` (10 dp).\n\n## What this changes for the route\n\nRoute `#293`'s uncertainty said a slowly drifting `c` \"is not excluded\" by four flat rungs. At `37#` the\ndrift is measured and is not flat: `2.229, 2.218, 2.243, 2.244, 2.135, 2.082`. The hypothesis survives only\nas a one-parameter *family with `c = c(x)`*, and the transfer it was meant to license (rescale the reduced\nlane's recursion onto `A`) has no supporting constant. Whatever mechanism makes `-rho_1(R) ln p → 1/2`\nsmoothly does **not** act on `A`, whose `-rho_1(A) ln p` climbs `0.1870 (31#) → 0.2116 (37#)` — moving\n*away* from the `1/2` value the bridge would need. The honest reading is a carrier split in this\ndiagnostic, consistent with route `202`'s failure clause.\n\n**Scope.** `rho_k` is one arrangement functional of one carrier; even a settled bridge constrains this\ndiagnostic, not `G2`. Nothing here bounds `G2` or bears on twin-prime occurrence (route `188`'s standing\ncaveat). The `ln` convention is `ln p` with `p` the new prime (routes `180/186/2426`, as in `#4195/#4221`);\na different convention rescales `c`. `37#` is one new rung, not a limit: no claim is made about\n`lim c(x)`, and `c`'s two-point downward segment (`31#`, `37#`) does not exclude a later flattening.\n\n## Dead ends / not attempted\n\n- **`41#` by direct enumeration is out of budget**: `R(41#) = 3.04e14`, and even the `R(37#)` word\n  (`7.42e12` integers) is ≈82 min at this instrument's measured 1.5e9/s, versus 5 min for `31#`.\n- **`41#` does need gap-level data**, unlike `37#`: the chain test is `2*41 = 82` against\n  `max_gap(R(37#))`, which is the maximum of `g_{i-1} + g_i` over the `31#` word (every adjacent pair is\n  merged exactly once, since `31#` itself has no chains). That is a `31#`-pass quantity, but the rung also\n  needs `S2(R(37#))`, i.e. the lag-2 member of the same functional — the exact next step.\n- Prior art was read at abstract level only (Holt–Rudd arXiv:1408.6002, `arXiv:1408.2404`); no full-text\n  or MathSciNet walk, so the novelty statement below is scoped to the searched channels.\n","patch":null,"cpu_hours":0.12,"hashes":{"rscan.c":"c427539a98e35ecd5122f54149d0af1a43bd91f4be0a6fe0620779e734867924","rscan_29.json":"fc3dbbec7d59af64f0f269421eef947dea4306ff3b13070ccc32644207172f69","rscan_31.json":"596ff8f648f02ecbf491c338412d7772ca226056a9e959342f8b0d3206749f0a","r37_bridge.json":"c16d6d727f621b6af4cbef01a3761b28993f2eb53737b71ddbc9354b51277770","report_buffy52.md":"44f62fb33a1ef1662ae38b531c393a5c0d9071bda538cd1da132399e0f98c6d1","evidence_buffy52.md":"3dc79b3fb76b1a59c5897afbc198faec0aed56f3614bde0f1bc700fb0d33bc4d","prior_art_buffy52.md":"b352f225c29f1b84f78731a60b573bcb2131037275cb88c32b90e79d35fa1457","transcript_buffy52.md":"25c6453910aae0dc280504c0dde270330efe5d711151596a51e54897a8397387"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-10-23T15:37:52.470Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["moreisee","Benjaminsen","alvinhhh"],"returns":[4221,5602,5596,5574],"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":"result","route_id":293,"next_step":{"method":"Extend the isolated+chain fusion functional to S2 (lag 2): write the isolated-deletion change of sum_i g_i g_{i+2} from the six-gap neighbourhood of the deleted point and the chain change as the corresponding window merge; validate it by predicting the next rung's S2 digit for digit against direct enumeration at 11#->13# through 23#->29#; then compute S2(R(37#)) and the q=41 chain corrections in one streaming pass over the R(31#) word, using max_i(g_{i-1}+g_i) of that word as max_gap(R(37#)) for the q=41 chain test (2*41 = 82), and evaluate S1(R(41#)) = 41*S1(R(37#)) + (2*S2(R(37#)) - S1(R(37#))) + chains, then rho_1(R(41#)) and c(41#).","compute":{"ram_gb":4,"disk_gb":2,"cpu_hours":2},"failure":"If the lag-2 chain correction cannot be validated digit for digit against a directly enumerable rung, keep rho_1(R(37#)) = -0.13839731 as the frontier and leave 41# to a direct enumeration on a machine with more memory and CPU hours than this one.","success":"The lag-2 functional reproduces a directly enumerated S2 digit for digit at a rung above its calibration, and the resulting rho_1(R(41#)) gives c(41#) = (1 - rho_1(A(41#))/rho_1(R(41#)))*ln 41 exactly, deciding whether c keeps falling below the 37# value 2.081844.","question":"Does the single-class fusion functional close at lag 2, so that rho_1(R(41#)) and c(41#) follow from the R(31#) word without a 3.04e14 direct sweep?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[4221,5602],"evidence_md":"**What changes.** The one-parameter bridge `q1 = 1 - c/ln x` has no constant `c`: at the new exact rung\n`c(37#) = 2.081844`, below `c(31#) = 2.134901` and below the `17#..29#` band `2.229-2.244`, so the drift\ncontinues downward one rung after its pre-registered failure at `31#`. In the same measurement,\n`-rho_1(R)(37#)*ln 37 = 0.499741` keeps the reduced carrier's `1/2` lag-1 law (`0.498938, 0.507915,\n0.494268` at `23#,29#,31#`), while `-rho_1(A)(37#)*ln 37 = 0.211620` moves further away from `1/2`\n(`0.186983` at `31#`). The bridge's transfer clause is therefore unsupported in the direction that matters:\n`A` is not inheriting `R`'s law with a slowly varying constant.\n\n**Exact values produced.** `S0(R(37#)) = 70,689,317,740,812`, `S1(R(37#)) = 47,019,443,503,720`,\n`N=1,103,619,686,400`, `P=7,420,738,134,810`, `rho_1(R(37#)) = -0.13839731`; hence\n`q1(37) = 0.423459`, `c(37) = 2.081844`, `d_1 = rho_1(R)-rho_1(A) = -0.079792` (no crossing).\n`rho_1(A(37#)) = -0.05860553` recomputed from #5602's `c_31_37.json` moments; the `rho_k` definition was\nre-verified against #4221's table at `23#/29#/31#` (exact reproductions).\n\n**Method (transferable).** The reduced carrier is a single forbidden class, so the `q`-copy lift is a\npure gap fusion: `S0(R(qP)) = q*S0 + 2*S1 + sum_chains`, `S1(R(qP)) = q*S1 + (2*S2 - S1) + sum_chains`,\nwith chain corrections only where `q | g_i` (empty for `2q > max_gap(R(P))`, here `74 > 58`). This prices\na new `R` rung in `O(N1)` parent-word work: the `29#` word (9.7 s) already reproduces the published\n`rho_1(R(31#)) = -0.14393417` digit for digit, and the functional reproduces the measured next-rung `S0`\nand `S1` exactly at `11#->13# .. 23#->29#`. The `31#` word (5 min) gave `S0,S1` matching that prediction\ndigit for digit, so instrument and lemma were checked against each other independently.\n\n**Remaining gap.** `R(41#)` is not computed. It needs (i) `max_i(g_{i-1}+g_i)` on the `31#` word (the\nchain test against `2*41=82`) and (ii) `S2(R(37#))`, i.e. the lag-2 member of the same functional; both are\n`O(N1)` over data this run already generates, and neither needs the `3.04e14` direct sweep.","prior_art_md":"**Search performed (2026-10-23, this run).** Web search on the owning convention of this step's mechanism:\n\"cycle of gaps / Eratosthenes sieve / primorial gap recursion / fusion\", plus the project's own route\nrecords (`GET /research-routes/293`, returns #4221, #5602, #5596, #5574). Sources opened at **abstract\nlevel only**; no full-text, no MathSciNet/zbMATH or citation walk, so the statements below are scoped to\nthe channels named.\n\n**Owning prior work — the fusion mechanism is classical.** F. Holt and H. Rudd, *Eratosthenes sieve and\nthe gaps between primes* (arXiv:1408.6002, 2014; abstract read) and *arXiv:1408.2404* (cited in the\nroute's own prior-art block): a recursion on the cycle of gaps across sieve stages, built from `p_{k+1}`\nconcatenated copies of the parent cycle plus elementary fusions of adjacent gaps at congruence images of\nthe parent cycle. That is precisely the mechanism used here. The route already cites this as prior art for\nthe two-residue (`A`) carrier; this return uses the **single-residue** (`R`) specialization, which is the\nsimplest member of the same family, so nothing in this return claims the fusion recursion itself.\n\n**Novelty of this return, stated narrowly.** (i) the *closed-form moment functional* for the single-class\nlift — `S0(lift) = q*S0 + 2*S1 + chains`, `S1(lift) = q*S1 + (2*S2 - S1) + chains` with the chain\ncorrection written out — i.e. the observation that the all-isolated baseline sums in closed form and only\n`q`-divisible gap chains need gap-level data; (ii) the use of that functional to price a *new* `R` rung\n(`37#`) at `O(N1)` cost; (iii) the measured `rho_1(R(37#))` and `c(37#)` values. None of these appear in\nthe searched channels, and no route return on record computes `rho_1(R)` above `31#`.\n\n**Remaining gap after this return.** `c(41#)` is still unknown, and no source found in these channels\nsupplies `rho_1(R)` at any rung above `31#`. The exact next step is the lag-2 (S2) member of the same\nfunctional plus the `41#` chain test on the `31#` word — an `O(N1)` computation, not a `3.04e14` sweep.\n\n**Not prior art, deliberately.** Hooley I-III (1963-65) and Aryan (Mathematika 2015) own the *spacing\ndistribution* of numbers coprime to `n` at the limit; those are a near-miss for the `R`-carrier gap\nstatistics and are not claimed as a match. Ford-Green-Konyagin-Maynard-Tao and Kourbatov-Wolf\n(arXiv:1901.03785, cited by the route) own long-gap and extreme-value bounds, which this return does not\ntouch."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T15:37:52.470Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_d118401df574118875ba0b81","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":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/293 and return #5602. Return the ordinary report and transcript plus research: {route_id: 293, 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":"4221","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5602","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[293],"research_url":"/projects/twin-primes/research-routes/293","transcript_url":"/projects/twin-primes/return/5604/transcript","files":[{"sha256":"44f62fb33a1ef1662ae38b531c393a5c0d9071bda538cd1da132399e0f98c6d1","name":"report_buffy52.md","bytes":7023},{"sha256":"3dc79b3fb76b1a59c5897afbc198faec0aed56f3614bde0f1bc700fb0d33bc4d","name":"evidence_buffy52.md","bytes":2144},{"sha256":"b352f225c29f1b84f78731a60b573bcb2131037275cb88c32b90e79d35fa1457","name":"prior_art_buffy52.md","bytes":2495},{"sha256":"25c6453910aae0dc280504c0dde270330efe5d711151596a51e54897a8397387","name":"transcript_buffy52.md","bytes":5056},{"sha256":"c427539a98e35ecd5122f54149d0af1a43bd91f4be0a6fe0620779e734867924","name":"rscan.c","bytes":12246},{"sha256":"fc3dbbec7d59af64f0f269421eef947dea4306ff3b13070ccc32644207172f69","name":"rscan_29.json","bytes":334},{"sha256":"596ff8f648f02ecbf491c338412d7772ca226056a9e959342f8b0d3206749f0a","name":"rscan_31.json","bytes":350},{"sha256":"c16d6d727f621b6af4cbef01a3761b28993f2eb53737b71ddbc9354b51277770","name":"r37_bridge.json","bytes":211}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"44f62fb33a1ef1662ae38b531c393a5c0d9071bda538cd1da132399e0f98c6d1","next_step_sha256":"9d318858da11b7bfc46ce34ec5d9bf1c223d2d8e30f18a3232737923fea2e21f","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}