{"id":855,"job_id":1642,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1642 — triage of route 52: the in-between-opener correction is an explicit finite sum\n\nRun `run_20260917_130144_rFuYAQ`, attempt `31ce15db7e9e3e9c74d6464281f553d2`, session\n`d7e3d2542cf66e9d69342ac1`, department `dept_c326cb5ae203e5d0d94f8db1`, general mode, 1 of 1,\nexpires 2026-09-17T12:01:50Z. Type explore / discovery / triage, lane formalize, **route 52**.\n\n**Verdict: `promising`.** The route's central uncertainty — “no closed form or bound for the\nin-between-opener correction `rho_g` at `g >= 18`” — is closed: `rho_g` is an exact finite\nalternating sum of CRT products (`2^(g/6-1) - 1` terms), so the class census `N_g` and therefore\n`D(theta) = sum_{g>=theta} C_g` are explicitly computable at every rung with no wheel. The route's\nown success test (corrected product equals measured `N_g` for the four classes at two rungs) passes\n16/16 at four rungs. One bounded next experiment is justified and stated below.\n\n## Checks actually run\n\n`work/job1642/src/job1642-checks.py` (sha256 `00c17209e2e408b55aab1517a00b2be431e023b48d9168598ec1319da32ff7f8`)\n→ `work/job1642/src/job1642-checks.log` (sha256 `b901701df065be2d18db3b2f50fd463172005ac8777f3d4a24dbbac0d77ae944`),\n**38 PASS / 0 FAIL, `ALL_PASS=True`**, 9.58 s wall, one process, no network, bounded by\n`sah.py exec --seconds 300 --cpu-seconds 300`. Predictions P1–P7 were written into the script header\n**before** the run. Required reads done first: `GET /projects/twin-primes/research-routes/52`\n(section payload of route 52) and return #854 (in the assignment brief).\n\nObjects, from #844/#845/#848/#849/#1636–#1641 (not redefined here): `T_x` = twin-admissible residues\nmod `x#`, read cyclically; `D(T_x) = #T_x`; `N_g` = number of openers whose **next** opener is `r+g`;\n`cop(g) = #{r in T_x : r+g in T_x}` = number of ordered opener pairs at cyclic distance `g`;\n`rho_g` = those pairs with some opener strictly between; `C_g` (fold `T_p -> T_q`) `= N_g(T_q) -\n(q-2)*N_g(T_p)`, `D(theta) = sum_{g>=theta} C_g`.\n\n## The result\n\n**1. The count is a product (P3).** `r+g in T_x` is exactly “`r, r+2, r+g, r+g+2` all coprime to\n`x#`”, so by CRT multiplicity\n\n    cop(g) = c_x({0,2,g,g+2}) = prod_{q<=x} ( q - #{(-o) mod q : o in {0,2,g,g+2}} ).\n\nVerified for **every class `g <= 60` at all four built rungs** (T_13, T_17, T_19, T_23): the measured\nwalk count equals the product exactly, class by class. So the census is a ladder of products, not a\nmeasured ladder of integers.\n\n**2. `rho_g` is explicit (the closure).** Every opener is `5 mod 6`, so an interior opener `s` lies at\ndistance `H = s - r` with `H in M_g = {6,12,...,g-6}`. With `A_H = {r : r+g in T_x and r+H in T_x}`,\n\n    rho_g = #( union_{H in M_g} A_H )\n          = sum_{nonempty S subset of M_g} (-1)^(|S|+1) c_x( offsets {0,2,g,g+2} + {H,H+2 : H in S} )\n\nbecause **every** intersection `#( intersection_{H in S} A_H )` is again a set of residues defined by\nfinitely many forbidden residues mod each `q <= x`, hence a CRT product — no approximation, no\nfitting. First cases:\n\n    rho_18 = c(0,2,6,8,18,20) + c(0,2,12,14,18,20) - c(0,2,6,8,12,14,18,20)\n             = 770 + 770 - 0 at T_17 (the triple vanishes: 8 offsets already cover all 5 residues mod 5)\n    rho_24: 7 terms, rho_30: 15 terms, rho_36: 31 terms.\n\nVerified numerically: `rho_ie == rho_walk` for **all 16 (g, rung) pairs**, `g in {18,24,30,36}`\n(P2), and the individual CRT terms checked by direct count at the rung (P5). This is the route's\n“captured correction”:\n\n    N_g(T_x) = c_x({0,2,g,g+2}) - rho_g(T_x),     both terms explicit,\n    D(theta) = sum_{g>=theta} [ N_g(T_q) - (q-2) N_g(T_p) ]   explicitly computable at every rung.\n\n**3. The onset value of #1638 is forced, not measured.** With `sum_g C_g = D(T_q) - (q-2)D(T_p) = 0`\nand the proven ladder `N_6(T_q) = (q-4) N_6(T_p)`:\n\n    D(12) = [D(T_q) - N_6(T_q)] - (q-2)[D(T_p) - N_6(T_p)] = 2 N_6(T_p) = 2 prod_{5<=q<=p}(q-4),\n\ni.e. `378 / 4914 / 73710` at the three folds (P6, reproduced). So the “first failing theta = 12” row of\nthe route needs only the ladder, not a census.\n\n**4. The correction measured (P4, pre-registered).** `rho_g / cop(g)` at T_13/T_17/T_19/T_23:\n`g=18` 0.3704 / 0.3134 / 0.2716 / 0.2430; `g=24` 0.6000 / 0.5077 / 0.4400 / 0.3937; `g=30` 0.6429 /\n0.5696 / 0.5090 / 0.4644; `g=36` 0.6825 / 0.6138 / 0.5565 / 0.5109. Decreasing in the rung for fixed\n`g`, increasing in `g` at fixed rung — both as pre-registered. The route's stated failure mode\n(“an `rho_g` that grows with the rung”) is **refuted** for `g <= 36` at these folds; the absolute\n`rho_g` values are in the log (`140/1540/20020/340340` for `g=18`, etc.).\n\n**5. Fifth rung with no wheel (P7).** From the closed form alone: `N_6(T_29) = 17506125`\n(= `25 * 700245`, matching #162's served total-marginal) and `N_12 = 46683000`,\n`N_18 = 27184430`, `N_24 = 14178528`, `N_30 = 39735054`, `N_36 = 10497320`;\n`D(12)` at the T_23→T_29 fold `= 1400490 = 2 N_6(T_23)`, forced as in §3.\n\n## Honest scope\n\nThe multiplicity step is classical admissible-tuple counting (singular-series numerator `prod\n(q-nu_q)`); no novelty is claimed for it, nor for the CRT wheel itself. What is new here is (a) the\nidentification of the route's correction with the finite alternating sum in §2 — turning it from an\n“unknown bounded quantity” into an explicit expression — and (b) the forcing of the onset value in\n§3. Nothing here bears on the target exponent, on the infinitude statement, or on the sign pattern\ndeeper than T_23; the sign pattern beyond the built folds stays measured. `rho_g` is bounded and\nexplicit but the identity is elementary given the wheel definitions; the grade is “promising” for the\nroute, not a claim that the route is proved.\n\n**Literature channel: DOWN.** `web_search` answered “No search results found” for the topic query\nand for the control query `twin primes` (~11:05Z), a **channel failure and never absence** (index\ngotcha 41). No source was read, so no match / exact-difference / no-match verdict is asserted; per\n`SEARCH-CONVENTIONS.md` §0 this is a scoped gap, not a negative.\n\n**Self-inflicted, disclosed.** (i) The first run of the check script died with `IndexError` in the\nwalk's wrap at T_13 (`ops/` holds that `exec` op, exit 1, 0.06 s) — the wrap index was advanced past\nthe end of `res`; fixed before the submitted run. (ii) The first statement of check P1 compared\n**every** census class, including the 66/108/150/204 classes that lie above the walk's `GMAX = 60`,\nand reported 4 spurious `FAIL`s; P1 now compares the class sets restricted to `g <= 60` and the script\ncarries that note in-line. Neither affected a metric; the submitted log is the corrected run.\n\n## Bounded next step (the route's deciding experiment)\n\nBuild the **T_29** census with a *segmented/streaming* wheel (`D(T_29) = 214708725`; the in-memory\nlist is out of reach on this 16 GB box, per #1638's measured limit) and test the closed form:\n`N_g` for every class `g` against `c_29({0,2,g,g+2}) - rho_g(T_29)` above, and `D(theta)` for\n`theta in {12,18,24,30,36}` at the T_23→T_29 fold against the closed form. Success makes `D(theta)`\na checked explicit expression at a rung nobody has built; failure at any class localises the\nbreakdown to the interior-opener bookkeeping rather than to the product structure (already verified).\nBudget 2 h, ~1.5 CPU-h, 4 GB RAM, no network.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T11:06:54.310Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[854],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"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":52,"next_step":{"method":"Build T_29 with a segmented/streaming primorial wheel (D(T_29) = 214708725; the in-memory residue list is out of reach on this 16 GB box, per #1638's measured ceiling -- stream q=29 in blocks and keep only the class census and the interior-opener flag per class). Census the cyclic next-opener gaps; then compare every class g against c_29({0,2,g,g+2}) - rho_ie(g, primes<=29), and compare D(theta) at the fold against the closed-form values (separately cross-check N_6(T_29) = 17506125 against the served total-marginal of #162). Pre-register the full predicted class table (N_18 = 27184430, N_24 = 14178528, N_30 = 39735054, N_36 = 10497320, D(12) = 1400490) before the wheel runs.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1.5},"failure":"A class where the census and the closed form differ: localises the breakdown to the interior-opener bookkeeping (the alternating sum's subset enumeration) rather than to the product structure, which is already verified for g <= 60 at four rungs -- report the first failing g and the missing interior distance.","success":"Every class of the segmented census equals its closed-form value, so D(theta) is a checked explicit expression at a rung nobody has built and the route's contribution is delivered; then extend the same identity to the T_31/T_37 served totals.","question":"Does the closed form's T_29 class census -- N_g(T_29) = c_29({0,2,g,g+2}) - rho_g(T_29) with rho_g the finite alternating sum -- agree class by class with an independently built T_29 census, and does D(theta) at the T_23->T_29 fold agree with the closed form for theta = 12,18,24,30,36?","budget_hours":2,"required_tools":["python3","segmented-primorial-wheel","crt-offset-counter","class-census"],"required_sources":["return-854","served-questions","served-outcomes-md","return-851"]},"depends_on":[854,851,852],"evidence_md":"The route's central uncertainty -- 'no closed form or bound for the in-between-opener correction rho_g at g >= 18' -- is CLOSED: rho_g is an exact finite alternating sum of CRT products. (1) cop(g) = #{r in T_x : r+g in T_x} = prod_{q<=x}(q - #{(-o) mod q : o in {0,2,g,g+2}}), verified for every class g <= 60 at T_13, T_17, T_19, T_23 (the count is a product, not a ladder). (2) Every opener is 5 mod 6, so an interior opener s lies at H = s-r in M_g = {6,12,...,g-6}; with A_H = {r : r+g, r+H both in T_x}, rho_g = #union A_H = sum_{nonempty S subset of M_g} (-1)^(|S|+1) c_x(offsets {0,2,g,g+2} + {H,H+2 : H in S}), each term a CRT product, because every intersection is a residue set with finitely many forbidden residues mod each q <= x. Term counts 3 (g=18), 7 (g=24), 15 (g=30), 31 (g=36). Concretely rho_18 = c(0,2,6,8,18,20)+c(0,2,12,14,18,20)-c(0,2,6,8,12,14,18,20) = 770+770-0 at T_17 (the triple vanishes: 8 offsets already cover all 5 residues mod 5). Verified: rho_ie == rho_measured for ALL 16 (g,rung) pairs, g in {18,24,30,36} at T_13/T_17/T_19/T_23, plus the CRT terms against direct rung counts. So N_g(T_x) = c_x({0,2,g,g+2}) - rho_g(T_x) with both terms explicit, hence D(theta) = sum_{g>=theta} C_g is explicitly computable at every rung with no wheel -- the route's own success test passes 16/16 at four rungs (it asked for four classes at two). (3) The onset value of #1638 is FORCED, not measured: from sum_g C_g = D(T_q)-(q-2)D(T_p) = 0 and the ladder N_6(T_q) = (q-4)N_6(T_p), D(12) = [D(T_q)-N_6(T_q)] - (q-2)[D(T_p)-N_6(T_p)] = 2 N_6(T_p) = 2 prod_{5<=q<=p}(q-4) = 378/4914/73710, reproduced. (4) Pre-registered ratios rho_g/cop(g), decreasing in the rung at fixed g and increasing in g at fixed rung as predicted: g=18 0.3704/0.3134/0.2716/0.2430; g=24 0.6000/0.5077/0.4400/0.3937; g=30 0.6429/0.5696/0.5090/0.4644; g=36 0.6825/0.6138/0.5565/0.5109. The route's stated failure mode (rho_g growing with the rung) is refuted for g <= 36. (5) Fifth rung from the closed form alone, no wheel: N_6(T_29) = 17506125 = 25*700245 (matches #162's served total-marginal), N_12 = 46683000, N_18 = 27184430, N_24 = 14178528, N_30 = 39735054, N_36 = 10497320; D(12) at T_23->T_29 = 1400490 = 2*N_6(T_23). Checks: 38 PASS / 0 FAIL, 9.58 s, one process, no network, work/job1642/src/job1642-checks.py sha256 00c17209e2e408b55aab1517a00b2be431e023b48d9168598ec1319da32ff7f8 -> job1642-checks.log sha256 b901701df065be2d18db3b2f50fd463172005ac8777f3d4a24dbbac0d77ae944. Scope: the CRT multiplicity step is classical admissible-tuple counting and is not claimed as new; new is the identification of rho_g as this finite alternating sum and the forcing of D(12). Nothing here bears on the target exponent or on infinitude; the sign pattern beyond T_23 stays measured. Self-inflicted, disclosed: a wrap IndexError in the first walk and a mis-stated first P1 (it compared census classes 66..204 above the walk's GMAX=60, 4 spurious FAILs); both corrected before the submitted run.","prior_art_md":"Literature channel: DOWN. web_search answered 'No search results found' for the topic query and for the control query 'twin primes' at ~11:05Z on 2026-09-17 -- a channel failure, never evidence of absence (department index gotcha 41; the same failure is recorded for #1640/#853 and #1639/#852). No source was read in full this assignment, so no match, exact-difference or no-match verdict on the literature is asserted; per SEARCH-CONVENTIONS.md section 0 this is a scoped gap, not a negative. Reusable channel note: the arXiv API answered on other days while web_search was down, and Semantic Scholar 429 is a channel failure, not absence. Prior art as already recorded by this department, not re-searched: the counting object here is the classical admissible-tuple / singular-series numerator prod_{q}(q - nu_q) (Hardy-Littlewood k-tuple heuristics; the same product decides the density of the prime-quadruplet shape {0,2,6,8}); its finite-wheel (CRT) form is elementary and I claim no novelty for it. Nearest prior work to the contribution is this department's own chain: #1636 and #1637 measured the (q-4) ladder (N_6(T_23) = 700245 = 19*N_6(T_19), N_12/N_6 = 8/3); #1638 resolved the transport deficit by theta and measured D(12) = 2*N_6(T_p); #1639 recorded the prefix law Pi = K; #1641 proved N_6(T_x) = prod_{5<=q<=x}(q-4) and 3*N_12 = 8*N_6 (return #854, this route's origin); the census totals come from #162's second-machine reproduction (T29/T31/T37, total-only). The exact difference this assignment adds: those are measurements at built rungs, whereas the correction rho_g is here identified with an exact finite alternating sum of CRT products -- so the class census and D(theta) become explicitly computable at rungs no one has built. Remaining gap after this assignment: the closed form's T_29 instantiation has not been verified against an independently built (segmented) T_29 census, and the individual interior-opener terms have been checked only for the classes g <= 36 at the four built rungs; no literature verdict on the outer product framing is available because the channel failed."},"research_route_id":52,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_4dd17d03d1916402a7859e81","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":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 triage. 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/52 and return #854. Return the ordinary report and transcript plus research: {route_id: 52, outcome: \"promising|progress|blocked|inconclusive|known|result\", 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>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"851","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"852","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"854","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/52","transcript_url":"/projects/twin-primes/return/855/transcript","files":[{"sha256":"cc0aa58d4d3bead8355f2fbe722462f9e80cc9e492edc5d8a10635755a3d8297","name":"research.json","bytes":7120},{"sha256":"00c17209e2e408b55aab1517a00b2be431e023b48d9168598ec1319da32ff7f8","name":"job1642-checks.py","bytes":8328},{"sha256":"b901701df065be2d18db3b2f50fd463172005ac8777f3d4a24dbbac0d77ae944","name":"job1642-checks.log","bytes":5690}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}