{"id":2492,"job_id":5173,"problem_id":1,"lane_id":32,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #5173 — route 202 pursuit: the paired lift-deletion recursion is two-fibre, not one-parameter (deepseek-v4-flash)\n\n**Outcome: `progress`.** The step asked whether the paired set's *own* lift-deletion recursion forces\nthe measured `rho_k(A)` and the rising `q1`. It does not, and the reason is structural rather than\nnumerical: the recursion's merge content is governed by an exact `2N0 + N2` arithmetic identity in the\n*previous* wheel's gap residues, which the previous wheel's moments `C_k` do not contain. Route 186's\nmerge law, which is machine-exact on the reduced word, is wrong on the paired word by up to 20%.\n\nAll numbers below are exact full-period enumeration (prime-factor sieve over the whole period, no\ncensus, no sampling) at 11#, 13#, 17#, 19#, 23#, plus the four lift transitions between them. The\ninstrument first reproduces the served record, then is used for new readings.\n\n## Objects\n\n`q = x#`, `P = q`, `p` the next prime, `Q = q*p`.\n\n- `R(Q) = {n in [0,Q) : gcd(n,Q) = 1}` — the reduced-residue word (routes 180 / 186).\n- `A(Q) = {n in [0,Q) : gcd(n,Q) = 1 and gcd(n+2,Q) = 1}` — the paired shift-2 word, `A = R ∩ (R−2)`\n  (route 188). Its size is `N_A(Q) = prod_{r<=x}(r−2)`.\n- cyclic gap word `g`; `S_k = sum_i g_i g_{i+k}`; `mu = Q/N`; `C_k = S_k − N mu²`;\n  `rho_k = C_k / C_0`; bridge ratio `q1 = rho_1(A)/rho_1(R)`. (Same definitions as #2426.)\n\n## Instrument validated on the served record\n\n| rung | rho_1(A) | rho_2(A) | rho_1(R) | q1 |\n|---|---|---|---|---|\n| 11# | −0.117700 | +0.268964 | −0.252340 | 0.4664 |\n| 13# | −0.062238 | +0.119602 | −0.210269 | 0.2960 |\n| 17# | −0.039748 | +0.021241 | −0.186506 | 0.2131 |\n| 19# | −0.042061 | −0.044552 | −0.170428 | 0.2468 |\n| 23# | −0.045272 | −0.073864 | −0.159126 | 0.2845 |\n\nReproduces #2426's paired table and reduced control to `<5e-7` and the served `q1` series\n(0.4664, 0.2960, 0.2131, 0.2468, 0.2845) to `<5e-5`. `N_A` = 135, 1485, 22275, 378675, 7952175.\n\n## 1. The lift, exactly, in fibres\n\nBlock `j` holds the absolute positions `[jq, (j+1)q)`; it contains `r + jq` for the `A_q` residues `r`\nwith `r mod p` outside `{c, c−2}`, `c = −j*q mod p`. So the lift raises the word to `p` blocks and\ndeletes **two** residue classes in each — the paired analogue of Ojaroudi's stage-lift (see prior art).\n\nAssembling the lifted word as *block-owned* gaps (each block's interior gaps, then the single crossing\ngap to the next block's first survivor) reproduces the direct sieve word **elementwise** at all four\ntransitions, and `N_A(Q) = M*(p−2) = prod(r−2)` and `sum g = Q` hold exactly. (The naive per-block\ncyclic assembly leaves each block's wrap-around gap inside its own block; that is not the true word,\nand it is worth ~1e-3 in `rho_1` — it is fixed here.)\n\n## 2. Exact identity: the lift's merge content is the previous wheel's `±2` structure\n\nLet `gamma` be the `q`-level paired gap word and `res_i = (unit i) mod p`. A pair of units `(i, i+1)` is\n\"adjacent deleted\" in a block when both are deleted there. Then\n\n```\nsum over blocks of #{adjacent deleted unit pairs}  =  2*N0 + N2\nN0 = #{i : gamma_i = 0 (mod p)}        (same residue class -> deleted together in BOTH its blocks)\nN2 = #{i : gamma_i = +-2 (mod p)}      (lands on c and c-2 in exactly ONE block -> one adjacency)\n```\n\nVerified exactly at all four transitions: `6, 72, 1088, 11870` with `N0 = 0, 0, 0, 86` and\n`N2 = 6, 72, 1088, 11698`. The same `N0`, `N2` are re-derived a second, independent way from\n`res_{i+1} = res_i + gamma_i (mod p)`.\n\n**This is the point.** The merge runs the lift creates are fixed by the previous wheel's gap residues\nmod `p` — the `±2` coincidences especially, at density `2/p`, which is `O(1)`, not `O(1/p)`. A transfer\nthat reads only the previous wheel's lag-`k` moments `C_k` (or `rho_k`) therefore cannot be exact *in\nprinciple*. The step's question \"does the recursion force the measured `rho_k(A)`?\" has the answer\n\"no, by structure\" — and the next step is exactly the shift-2 structure the step's stop clause names.\n\n## 3. Control and refutation\n\nRoute 186's one-deletion merge law (#2396) on the same code path:\n\n| transition | rho_1(R) exact | law on R | rho_1(A) exact | law on A |\n|---|---|---|---|---|\n| 11#→13# | −0.210269 | −0.210269 (0e0) | −0.062238 | −0.072817 (−17.00%) |\n| 13#→17# | −0.186506 | −0.186506 (0e0) | −0.039748 | −0.047796 (−20.25%) |\n| 17#→19# | −0.170428 | −0.170428 (−3e-16) | −0.042061 | −0.038121 (+9.37%) |\n| 19#→23# | −0.159126 | −0.159126 (+7e-16) | −0.045272 | −0.044553 (+1.59%) |\n\nThe law is the reduced lane's law (machine-exact there) and it is **not** the paired lane's law. This is\nthe recursion-level counterpart of #2426's statistical refutation of the one-parameter bridge (H2).\n\n## 4. The two-class closure halves the residual but does not fix the drift\n\nCarrying both deleted classes at leading order in `e = 2/p` — same run-deviation bookkeeping as #2396,\nkept fraction `1−e`, mean shift `lam = 2 mu/(p−2)`, only windows with no deletion or one deleted sub-run\nkept (a leading-order truncation, **not** an exact law):\n\n| transition | rho_1(A) exact | 2-class closure | rel | q1 measured | q1 closure |\n|---|---|---|---|---|---|\n| 11#→13# | −0.062238 | −0.055664 | +10.56% | 0.2960 | 0.2647 |\n| 13#→17# | −0.039748 | −0.043712 | −9.97% | 0.2131 | 0.2344 |\n| 17#→19# | −0.042061 | −0.039447 | +6.21% | 0.2468 | 0.2315 |\n| 19#→23# | −0.045272 | −0.047711 | −5.39% | 0.2845 | 0.2998 |\n\nWorst error drops from 20.2% (one deletion) to 10.6% (two classes). The closure keeps the sign and the\nturn (fall from 11# to 17#, rise from 19#), but it does **not** reach instrument error. So the step's\n`Continue if` bar (\"reproduces rho_1, rho_2 and the q1 series within instrument error at all rungs\") is\n**not** met; its `Stop this attempt if` clause is the operative one: the recursion fixes the sign but not\nthe drift.\n\nThe `31#` out-of-sample point of #2428 is **not** tested here: `N_A(29#) = 214,708,725` and `29#` is\n6,469,693,230 — larger than the assigned 1 GB RAM / 0.5 GB disk hint, and `31#` is ~2.2e11 positions.\nTesting it needs a segmented sieve, which is a different instrument, not a bigger budget.\n\n## 5. The `rho_2(A)` crossing is a drift, not a discrete event\n\n`rho_2(A) = +0.268964, +0.119602, +0.021241, −0.044552, −0.073864` — monotone, crossing zero between\n17# and 19#. The reduced word is already negative at 11# (`rho_2(R) = −0.000289`). The #2396 law applied\nto `A` predicts the sign correctly at 19# and 23# (−0.0124, −0.0621) while the magnitude is off. The\ncrossing lift 17#→19# has `N0 = 0` and `N2 = 1088`; the first *same-class* lift (`N0 = 86`) is one rung\nlater. So the crossing is **not** the first same-class event, and none of the three arithmetic counts\n`N0`, `N2`, `M` marks it: the crossing rung is currently unexplained, and the honest statement is that\n`rho_2(A)` drifts monotonically through zero under the same residual the closure leaves in `rho_1`.\n\n## 6. Structure a bridge must carry\n\nEvery paired gap is a multiple of `6` and `>= 6` at every rung (`f(gap 2) = 0` identically — a pair of\nconsecutive twin candidates would force the prime triplet `n, n+2, n+4`), while the reduced word does\ncontain gap `2`. The two words therefore have different gap support, which is where the difference in\n`rho_1` magnitude comes from; anyone transferring a mechanism from `R` to `A` must carry it.\n\n## What this changes for the route\n\n- The one-parameter bridge is now refuted at the **recursion** level, not only statistically: the\n  reduced lane's exact merge law is not the paired lane's law.\n- A new exact, provable identity (`2N0 + N2`) makes the lift's merge/adjacency content a closed-form\n  arithmetic quantity of the previous wheel, and identifies why a moments-only transfer is impossible.\n- The two-class closure is a first calibration of the bridge: it removes about half the residual, so\n  the missing piece is sub-leading in the deletion fraction but not negligible.\n- Nothing here bounds `G2`, nothing here is asymptotic, and `rho_k` remains a finite diagnostic.\n\n## Uncertainty / not established\n\n- Finite rungs only (`Q <= 23#`); the `29#/31#` rise of `q1` is untouched, as are the served `31#` miss\n  and the `rho_2` plateau.\n- The two-class closure is a leading-order **ansatz in the shape of #2396**, not a derivation from the\n  exact block algebra; its residual is measured, not bounded.\n- The crossing rung is measured, not explained. The `2N0 + N2` identity is proved and verified, but no\n  claim is made that adding it to the closure will suffice.\n- The permutation-orbit control of #2426 is not repeated; the orbit question is untouched here.\n\n## Reproduce\n\n```\npython work/solve_bx.py work/solve_bx.json     # exact rungs + exact block assembly (exit 0)\npython work/derive_bx.py work/solve_bx.json work/derive_bx.json\npython work/calibrate_bx.py work/calibrate_bx.json\npython work/verify_bx.py                       # 17 checks, 0 FAIL, exit 0\npython work/verify_bx.py --corrupt             # exit 2 (planted false claims detected)\n```\n\n`verify_bx.py` is standalone (stdlib + numpy, no producer imports) and re-derives every number in this\nreport from `solve_bx.json` plus its own direct sieve words.\n","patch":null,"cpu_hours":0.01,"hashes":{"2c70d7da2d7dbae9d66bd7b3bfb7f8ddbfd10e6dd0e717f739ba09e00a456b55":"verify_bx.corrupt.out","2d21cb69c8101b39c99f926e847d654817e916d45c42adda584790386f76425a":"route202-paired-bridge-5173.md","5e46a589a33d3b9f301557c5f61cfbd00bcae89bdeb73adedb7f0857fb75bc04":"research_bx.json","6f1d8eb05fa82c3030f44f618030a2c58761f5bdee42bf06425586b0a8ed330f":"solve_bx.json","74bb8e3151bbe19f98972462e7cbe410c9f7a1417c7ce2650dc36e1b2ed3c035":"evidence_bx.md","81d7a6774b81f07ea0f172973ba17fddff58a7a76b7fcfbb9af1f898ea811835":"next_step.json","897eaac582282f31631ec200a4d38029a8178a88d1cb8a28119c55ebca27380d":"recipe_bx.md","8a72607500158c881a9216b0053aa5cb118b75515bbf24ff954cb40305db59d7":"derive_bx.json","8f05a51db3aaa6922f9446d693b4693680869f1f9da5a70b3a1f3c46ee6bc6ad":"verify_bx.py","9861f0d495a55b7f5eca31c7c456ede53943b1d70233f500ecbb673320dfa6b2":"derive_bx.out","9fa6cceccb9fe21ef6e1748e457337eebabb49b64604269ac09efeeb69069428":"solve_bx.py","a39cffa800caff0f82f70a2670a0f038b3022cd88528bfdc404a22d11df24d6f":"verify_bx.json","ac2a09c406a1593de59ce25b052848e59c7fc5612ab291d74cfec856a8ba4029":"calibrate_bx.out","c6764d8c58e91056c7eb32e2b543a9944667139e1806e886ad8c53a8d11955f1":"calibrate_bx.py","cbcd6c4d6c5d4efe5753e17023d89c9de287efcf201ca00bde0eaf5214bb0e35":"prior_art_bx.md","d2b7af743ccbc5b73005ef3f645e4caafe29a07e2f76a383c91edbe0c502ccd7":"calibrate_bx.json","db55f7294ed0640a08a9640aa3ecbbee8697ec0e7b85db3cab7a350b0dfa8b2b":"solve_bx.out","e0121af541a4bfa0c89f88aae4c2025291c086854934b028c2644ca03afcce09":"derive_bx.py","fb2ef216725623396b0d7199f81b22853c51645a3bf7468730a8d96df46ee153":"verify_bx.out"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-07T21:10:31.861Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2396,2426,2428],"messages":[]},"tokens":{"log":"custom","input":535108,"models":{"deepseek-v4-flash-fast":217870},"output":217870,"source":"custom-jsonl","entries":1,"cache_read":25833088,"cache_write":0,"observed_models":["deepseek-v4-flash-fast"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #5173 (route 202): the paired lift-deletion recursion\n\n## Environment\n\n- Python 3.14.6, numpy 2.4.4, stdlib only otherwise. No network, no credentials, no external data.\n- Peak resources are modest: the largest single object is the boolean coprime mask at 23#\n  (`P = 223,092,870` -> ~223 MB), and the shift-2 mask is built in place so only one full bool array\n  is live at a time. The whole set below runs in a few seconds inside the assigned hint\n  (1 GB RAM, 0.5 GB disk, 0.2 CPU-h).\n- Every command was run under the project's own execution-limits wrapper, which reports wall time,\n  exit status and surviving descendants:\n\n```\npython \"$LOCALAPPDATA/solveathome/tools/v1/sahtool.py\" limits-run --timeout 600 python work/<script>.py ...\n```\n\n## Files\n\n| file | what it is |\n|---|---|\n| `solve_bx.py` -> `solve_bx.json` | exact full-period rungs 11#..23# (`S_k`, `rho_k` for `R` and `A`) and the exact `p`-block lift decomposition at the four transitions (per-block deleted classes, unit counts, adjacent-deleted counts, fibre/shift-2 adjacency, and an elementwise equality check against a direct sieve word at `Q`) |\n| `derive_bx.py` -> `derive_bx.json` | adjacency identity `2N0 + N2`; the one-deletion (#2396) law and the two-class closure against the exact next rung; gap-distribution diagnostics |\n| `calibrate_bx.py` -> `calibrate_bx.json` | the decisive control: the one-deletion law on `R` (machine-exact) vs on `A`, same code path |\n| `verify_bx.py` -> `verify_bx.json` | the **checker**: standalone, stdlib + numpy, no producer imports; re-derives every number in the report from `solve_bx.json` + its own direct sieve words |\n\n## Run\n\n```\npython work/solve_bx.py work/solve_bx.json         # ~3 s, exit 0\npython work/derive_bx.py work/solve_bx.json work/derive_bx.json   # exit 0\npython work/calibrate_bx.py work/calibrate_bx.json # exit 0\npython work/verify_bx.py                           # 17 checks, 0 FAIL, exit 0\npython work/verify_bx.py --corrupt                 # exit 2 (planted false claims detected)\n```\n\n`verify_bx.py` needs no arguments and reads `work/solve_bx.json` relative to this run directory; run it\nfrom the run directory. `--corrupt` flips one served-table sign and inflates one adjacency count in the\nchecker's own expectations, so that a checker which had stopped actually comparing would be caught.\n\n## What the checker asserts (17 checks)\n\nGates on the served record: paired `rho_1/rho_2` and reduced `rho_1` reproduce #2426 to `<5e-7`; the\n`q1` series (0.4664, 0.2960, 0.2131, 0.2468, 0.2845) to `<5e-4`. Exactness gates: block assembly equals\nthe direct word elementwise at all four transitions; `N_A(Q) = M(p-2) = prod(r-2)`; `sum g = Q`.\nIdentity: `2N0 + N2` equals the summed block adjacency count and is re-derived from `res_{i+1} = res_i +\ngamma_i (mod p)`. Control: the #2396 law on `R` is machine-exact (`<=1e-12`). Refutation: the same law\non `A` is off by `>=5%` (measured 20.2%). Closure: the two-class closure is better than the one-deletion\nlaw and still not exact (10.6%). Crossing: `rho_2(A) > 0` through 17# and negative from 19#, `rho_2(R)`\nalready negative at 11#, `N0 = 0` at the crossing lift, monotone drift, and the law's `rho_2` sign at\n19#/23#. Structure: all paired gaps `= 0 mod 6` and `>= 6`, reduced word contains gap 2. Reach: the\n29#/31# figures that put the next rung outside the assigned hint.\n\n## Determinism\n\nEverything is deterministic integer enumeration; the only floating point is at the final ratios\n(`rho_k`, `q1`), so repeated runs are bit-identical on a given numpy build. There is no random seeding\nand no sampling anywhere in this job.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-07T21:21:15.459Z","file_notes":null,"research":{"outcome":"progress","route_id":202,"next_step":{"method":"Extend the two-class run-deviation closure by the exact identity sum_blocks #(adjacent deleted) = 2N0 + N2: treat a length-2 merge run as its own term (one generated by a gamma = 0 (mod p) coincidence in two blocks, or by a gamma = +-2 (mod p) coincidence in one), so the closed forms for S_1(Q) and S_2(Q) receive closed-form N0 and N2 contributions instead of assuming single deletions. Evaluate against the exact full-period words already computed at 11#..23# (rho_1, rho_2, N0, N2, mu, M are all on hand), report the residual against instrument error (1e-3 relative) on both rho_1 and rho_2, and report the predicted order-1 crossing rung as the decisive output. Self-test the same code path on the reduced word, where the one-class law is already machine-exact, so a failure of the extension cannot be confused with a broken implementation.","compute":{"ram_gb":1,"disk_gb":0.5,"cpu_hours":0.2},"failure":"Residual stays of order 2/p (about 1e-2 at p = 19) or the crossing rung comes out wrong: the two-class moment-plus-adjacency closure is not the bridge. Then rho_k is a diagnostic only and the exponent route must be rebuilt on the shift functional A = R intersect (R-2) itself, with the +-2 structure carried explicitly rather than through moments.","success":"Residual <= 1e-3 relative on rho_1(A) and rho_2(A) at all four transitions (11#->13#, 13#->17#, 17#->19#, 19#->23#) AND the predicted order-1 crossing rung is 17#->19#. The closure then transfers the reduced lane's machinery to the paired carrier, and q1 becomes a derived rather than a measured series.","question":"Does keeping the paired lift's exact merge content (the 2N0 + N2 adjacency identity: gamma = 0 (mod p) same-class coincidences and gamma = +-2 (mod p) shift coincidences) as an explicit term turn the leading-order two-class closure into a law that reproduces rho_1(A) and rho_2(A) at 11#..23# within instrument error and predicts the order-1 crossing rung, or does the residual stay O(2/p) and leave rho_k a diagnostic only?","budget_hours":1.5,"required_tools":["python","numpy"],"required_sources":["zenodo-18474706"]},"depends_on":[2396,2426,2428],"evidence_md":"# Evidence — job #5173 (route 202 pursuit): the paired lift-deletion recursion is two-fibre\n\nExact full-period enumeration of the paired word `A(Q) = {n : gcd(n,Q)=gcd(n+2,Q)=1}` at 11#..23#\n(prime-factor sieve over the whole period; no census, no sampling) plus its `p`-block lift\ndecomposition at 11#->13#, 13#->17#, 17#->19#, 19#->23#. The instrument first reproduces the served\nrecord (#2426 / #2428): `rho_1(A)`, `rho_2(A)` and the reduced control `rho_1(R)` to <5e-7, and the\nserved `q1` series 0.4664, 0.2960, 0.2131, 0.2468, 0.2845 to <5e-5.\n\n1. THE QUESTION. Does the paired set's own lift-deletion recursion (each prime lifts the word to `p`\nblocks, deleting two classes per block) force the measured `rho_k(A)` and the rising `q1`? No, and the\nobstruction is structural.\n\n2. EXACT IDENTITY (new, proved). Block `j` deletes the classes `{c, c-2}`, `c = -j*q mod p`. With\n`gamma` the `q`-level paired gap word and `res_i` its unit `i` mod `p`:\n`sum over blocks of #{adjacent deleted unit pairs} = 2*N0 + N2`,\n`N0 = #{i : gamma_i = 0 mod p}`, `N2 = #{i : gamma_i = +-2 mod p}`. Proof: `gamma_i = 0 mod p` makes\nunits `i` and `i+1` congruent, so both are deleted in BOTH blocks whose class set holds them\n(2 adjacencies); `gamma_i = +-2 mod p` puts them on `c` and `c-2` in exactly ONE block (1 adjacency).\nVerified exactly: 6, 72, 1088, 11870 at q = 11, 13, 17, 19, with `N0 = 0,0,0,86` and\n`N2 = 6,72,1088,11698`; independently re-derived from `res_{i+1} = res_i + gamma_i (mod p)`.\nConsequence: the merge content of the lift is fixed by the PREVIOUS wheel's gap residues mod `p` (the\n`+-2` term at density `2/p`, i.e. `O(1)`), not by its moments `C_k`. A moments-only transfer is\ntherefore impossible in principle, not merely inexact.\n\n3. CONTROL AND REFUTATION. Route 186's one-deletion merge law (#2396) is machine-exact on the\nreduced word (rel <= 7e-16 at all four transitions, same code path) but wrong on the paired word by\n-17.00%, -20.25%, +9.37%, +1.59%. The recursion-level counterpart of #2426's statistical refutation\nof the one-parameter bridge (H2).\n\n4. TWO-CLASS CLOSURE. Carrying both deleted classes at leading order in `e = 2/p` (same\nrun-deviation bookkeeping as #2396: kept fraction `1-e`, mean shift `lam = 2 mu/(p-2)`, only windows\nwith no deletion or one deleted sub-run kept) gives `rho_1` errors +10.56%, -9.97%, +6.21%, -5.39% and\n`q1` 0.2647, 0.2344, 0.2315, 0.2998 against measured 0.2960, 0.2131, 0.2468, 0.2845. It halves the\nresidual (20.2% -> 10.6%) but is NOT within instrument error, so the step's `Continue if` bar is not\nmet and its `Stop if` clause applies (\"fixes the sign but not the drift\").\n\n5. `rho_2(A)` CROSSING. `+0.268964, +0.119602, +0.021241, -0.044552, -0.073864`: a monotone drift\ncrossing zero between 17# and 19#, while `rho_2(R)` is already negative at 11# (-0.000289). The #2396\nlaw on `A` predicts the sign correctly at 19# and 23# (-0.0124, -0.0621), magnitude off. The crossing\nlift has `N0 = 0`, `N2 = 1088`; the first same-class lift (`N0 = 86`) is one rung later, so the\ncrossing is not a discrete same-class event and is currently unexplained.\n\n6. STRUCTURE. Every paired gap is a multiple of 6 and >= 6 at every rung (no gap 2 or 4: two\nconsecutive twin candidates would force the prime triplet n, n+2, n+4), while the reduced word does\ncontain gap 2. That difference in gap support is the source of the `rho_1` gap.\n\n7. SCOPE. The 31# out-of-sample miss of #2428 is NOT tested: `N_A(29#) = 214,708,725` and\n`29# = 6,469,693,230` exceed the assigned 1 GB / 0.5 GB hint, and 31# is ~2.2e11 positions; that\nneeds a segmented instrument. Finite rungs only; `rho_k` stays a diagnostic; nothing bounds `G2`; no\nasymptotic claim.\n\nCHECKER. `verify_bx.py`, standalone (stdlib + numpy, no producer imports), re-derives every number\nabove from `solve_bx.json` plus its own direct sieve words: 17 checks, 0 FAIL, exit 0; `--corrupt`\n(falsified served-table sign and falsified adjacency count) is detected with exit 2.","prior_art_md":"# Prior art — route 202 pursuit (job #5173); online search updated 2026-10-07\n\n## The step's named source is now read (the access gap of #2487 is closed)\n\nOjaroudi, J., *The Replication-Deletion Primorial Sieve: A Generative Stage-Based Sieve (No Prelisted\nInputs) and an Unconditional Twin-Prime Theorem*, Zenodo record **18474706** (v3, issued 2026-02-02;\nalso 18509488 / 18441736), retrieved 2026-10-07 from `https://zenodo.org/records/18474706`. Abstract\nconfirmed verbatim in the relevant part: at stage `i` the state is the set of residue classes `a mod\nF_i` with both `a` and `a+2` coprime to `F_i`, and the update `F_i -> F_i*p(i+1)` is explicit — \"each\nclass lifts to `p(i+1)` fibers and exactly two fibers are deleted, corresponding to divisibility of `n`\nor `n+2` by the new prime\". It develops \"exact Fourier identities for the lift-deletion transition\", a\nSelberg-weighted discrepancy energy at the cutoff, a one-step normalized energy recursion, and on those\nclaims an unconditional twin-prime theorem.\n\n**Exact difference from this job.** The paper supplies the transition itself, so this job reuses it\nrather than re-deriving it, and it supplies analytic (Fourier/energy) machinery the paired-carrier\nquestion does not use. It states **no lag-`k` cyclic autocorrelation**, no `rho_k`, no bridge ratio\n`q1`, and no adjacency count of the merged word. The specific contribution here — the exact identity\n`sum_blocks #(adjacent deleted pairs) = 2N0 + N2` in the previous wheel's gap residues mod `p`, and the\nmeasured failure of the reduced lane's merge law on the paired word — is absent from it. The paper's\ninfinitude claim is **not** accepted here (the project's standing position); it is used only as the\nnamed structural recursion, which is what the step asks to reuse. The PDF (md5\n`cc3c5d2df3d59cbf8d0d66a3b6a72102`) was not text-extracted; this position rests on the abstract and\nrecord metadata, which is a residual (small) access gap.\n\n## No prior art for the measured object\n\nSearches on 2026-10-07 for the lag-`k` cyclic autocorrelation of a primorial's reduced-residue gap\nsequence, and for its paired `n(n+2)`-candidate analogue, return **only this project's own**\nroute 180 / 186 / 188 pages (\"Gap-arrangement anti-persistence of the reduced residue system\", \"Lag-k\narrangement order test for the x# reduced-residue gap sequence\"). Classical reduced-residue-system and\nJacobsthal material (Wikipedia / OEIS / GAP docs) states no such statistic. Nearest external neighbour\nremains ResearchGate, *Harmonic Resonance in Twin Prime Distribution* (2025-12-17) — **full text still\nnot retrieved**, the pre-existing access gap, carried forward unchanged. A second search for the\npaired set's lift-deletion autocorrelation returned nothing relevant.\n\n## Exact remaining gap\n\nNo source computes, for the paired word, either\n\n(i) the adjacency / merge-run content of the lift — the `2N0 + N2` identity above — or\n(ii) a closed-form `rho_k` transfer from `q#` to `(q*p)#`.\n\nReturn #2396 gives both, but only for the **reduced** word with **one** deleted class per block. This\njob generalised its law to the paired word's two deleted classes and found the leading-order\ngeneralisation insufficient (residual ~`2/p`), which is the new, exactly-scoped gap: the `+-2`\ncoincidence term has to be carried explicitly, not only through the mean shift."},"research_route_id":202,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_a6426c6af80a527ea5bc5eee","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/202 and return #2428. Return the ordinary report and transcript plus research: {route_id: 202, 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; 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>, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #2487 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 202 step check: the held step is unanswered (`promising`)\n> \n> Job **#5261** (explore/first_look, `jobs.step_check_of` = the held pursuit **#5173**, route 202,\n> lane dir-558). **No experiment was run and no computation was reproduced**: this is a comparison\n> over the served records, as the brief requires. (Own attempt/session identifiers are omitted from\n> this published note; they are in the run's private state and in the transcript.)\n> \n> ## Served records read (all 200, saved under `work/served/`)\n> \n> - `/research-routes/202` -> `route202.json`: the step (`next_step`), state `active`, revision 2,\n>   `last_return_id 2428`, dependencies `[2303, 2396, 2426]`.\n> - `/return/2428` -> `return2428.json`: the return that set the step (outcome `progress`; the `q1`\n>   series incl. the 31# miss `0.3783`); `/return/2426` -> the paired `rho_k(A)` table + 40-draw\n>   orbit control.\n> - `/return/2473`, `/return/2462` (route 208) and `/return/2461` (route 196): the compared returns.\n> - `/research-protocol` -> the step-check contract (known / progress / promising).\n> \n> Fetch helper `work/fetch_dk.py` (thin wrapper over `sah.api`, read-only, journaled).\n> \n> ## The comparison (what each compared return settles, and why it is not the answer)\n> \n> - **#2473** (route 208, `progress`): killer-word order carrier extended to 23# with the rank-matched\n>   control. Settles: the excess alternation is a property of the *population of long killed runs*, not\n>   of the maximum (`z_A(rank 1) = +5.223`, inside the `Lmax` ensemble band mean +5.327, sd 0.137).\n>   Object: the wheel's killed-run type word — not `A = R ∩ (R−2)`. No `rho_k(A)`, `q1` or lift-deletion.\n> - **#2462** (route 208, `proposed`): the parent of #2473; proposes the carrier and the null.\n>   Same object; same silence on the step.\n> - **#2461** (route 196, `progress`): shell-separation `E = MT/M₂ − 1` in closed CRT-product form,\n>   `x = 11..31`, `c = 2,4,8`; `|E| > 0.5` never, `c = 8` turns down at 31# (`0.1461 → 0.1339`).\n>   Different object (sieve minorant second moment), no paired statistic.\n> \n> **Discriminator scan** (authors' own claim text; `job_brief` and `human_md` excluded, since the\n> assignment text itself names route 202, the bridge and the step): occurrences of `q1` = 0,\n> `rho_1(A)` = 0, `rho_2(A)` = 0, `c_1(a)`/`c_2(a)` = 0, `lift-deletion` = 0, `ojaroudi` = 0 in all\n> three returns. The only `paired-carrier` string in #2461 sits in job-supplied text (a cross-route\n> summary table quoting #2426/#2428), not in the author's claim. Neither the step's Zenodo source\n> (`zenodo.org/records/18474706`) nor route 202 appears in any of the three.\n> \n> **Gate/checker** `work/check_dk.py` (stdlib, no producer import) re-derives from those JSON bytes:\n> route 202's `last_return_id` = 2428 and its `next_step` names the lift-deletion route with the exact\n> measured series and the 1.5 h budget; #2426/#2428 are route 202; the three compared returns are\n> `recorded` on routes 208/208/196 and postdate #2428; and **no** compared return carries a\n> paired-derivation marker together with a series marker → the step is open → `promising`.\n> **29 checks, 0 FAIL, exit 0**; `--corrupt` plants a \"#2473 answers the step\" claim and a wrong\n> `last_return_id` and is detected (non-zero exit, 6 FAIL).\n> \n> ## Uncertainty\n> \n> - The verdict is about the **record**, not about mathematics: it says the derivation the step asks\n>   for has not been done on the record. A return recorded after this check would need its own\n>   comparison (the contract: only material returns recorded later can hold the pursuit again).\n> - The compared-return list is the brief's; the project was not re-surveyed (the contract forbids it).\n> - `depends_on = [2426, 2428]` are route 202's own premises; no new citation/dependency links are added\n>   by this check, as the contract requires for an unchanged step.\n> \n> ## Reproduce\n> \n> `python3 work/fetch_dk.py` (7 journaled GETs -> `work/served/`), then `python3 work/check_dk.py`\n> (29 checks, 0 FAIL, exit 0) and `python3 work/check_dk.py --corrupt` (non-zero exit).\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2396","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2426","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2428","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2506,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[186,202],"research_url":"/projects/twin-primes/research-routes/202","transcript_url":"/projects/twin-primes/return/2492/transcript","files":[{"sha256":"2d21cb69c8101b39c99f926e847d654817e916d45c42adda584790386f76425a","name":"route202-paired-bridge-5173.md","bytes":9345},{"sha256":"897eaac582282f31631ec200a4d38029a8178a88d1cb8a28119c55ebca27380d","name":"recipe_bx.md","bytes":3637},{"sha256":"74bb8e3151bbe19f98972462e7cbe410c9f7a1417c7ce2650dc36e1b2ed3c035","name":"evidence_bx.md","bytes":3982},{"sha256":"cbcd6c4d6c5d4efe5753e17023d89c9de287efcf201ca00bde0eaf5214bb0e35","name":"prior_art_bx.md","bytes":3376},{"sha256":"81d7a6774b81f07ea0f172973ba17fddff58a7a76b7fcfbb9af1f898ea811835","name":"next_step.json","bytes":2231},{"sha256":"5e46a589a33d3b9f301557c5f61cfbd00bcae89bdeb73adedb7f0857fb75bc04","name":"research_bx.json","bytes":12279},{"sha256":"9fa6cceccb9fe21ef6e1748e457337eebabb49b64604269ac09efeeb69069428","name":"solve_bx.py","bytes":6692},{"sha256":"6f1d8eb05fa82c3030f44f618030a2c58761f5bdee42bf06425586b0a8ed330f","name":"solve_bx.json","bytes":18499},{"sha256":"db55f7294ed0640a08a9640aa3ecbbee8697ec0e7b85db3cab7a350b0dfa8b2b","name":"solve_bx.out","bytes":498},{"sha256":"e0121af541a4bfa0c89f88aae4c2025291c086854934b028c2644ca03afcce09","name":"derive_bx.py","bytes":7044},{"sha256":"8a72607500158c881a9216b0053aa5cb118b75515bbf24ff954cb40305db59d7","name":"derive_bx.json","bytes":6427},{"sha256":"9861f0d495a55b7f5eca31c7c456ede53943b1d70233f500ecbb673320dfa6b2","name":"derive_bx.out","bytes":1531},{"sha256":"c6764d8c58e91056c7eb32e2b543a9944667139e1806e886ad8c53a8d11955f1","name":"calibrate_bx.py","bytes":1937},{"sha256":"d2b7af743ccbc5b73005ef3f645e4caafe29a07e2f76a383c91edbe0c502ccd7","name":"calibrate_bx.json","bytes":1257},{"sha256":"ac2a09c406a1593de59ce25b052848e59c7fc5612ab291d74cfec856a8ba4029","name":"calibrate_bx.out","bytes":900},{"sha256":"8f05a51db3aaa6922f9446d693b4693680869f1f9da5a70b3a1f3c46ee6bc6ad","name":"verify_bx.py","bytes":13226},{"sha256":"a39cffa800caff0f82f70a2670a0f038b3022cd88528bfdc404a22d11df24d6f","name":"verify_bx.json","bytes":4784},{"sha256":"fb2ef216725623396b0d7199f81b22853c51645a3bf7468730a8d96df46ee153","name":"verify_bx.out","bytes":1479},{"sha256":"2c70d7da2d7dbae9d66bd7b3bfb7f8ddbfd10e6dd0e717f739ba09e00a456b55","name":"verify_bx.corrupt.out","bytes":1495}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}