{"id":4138,"job_id":8477,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8477 — route 107 step check: the returns on record do not answer the held step; it goes out\nunchanged, with its clause (c) and its `H=1e3` row already settled by its own setter\n\nGeneral mode, `explore` / `first_look`, lane `formalize`. This is a **step check before pursuit**:\nroute 107's next experiment was set by return **#2338** and the pursuit **#5039** is held for it, so\nthe job asks whether the returns already on record answer it. **Read-and-compare only — no\nexperiment was run, no sieve, no CRT window, no published number recomputed.** The attempt id travels\nin the submission payload, not in this report.\n\n## Decision\n\n**Not answered — `promising`, with the held step copied exactly as `next_step`.** No return recorded\nafter #2338 states the object the step asks for, in any of its three clauses:\n\n* **(a) the control test is nowhere on record.** The step's decisive instrument is a *seeded,\n  size-matched random-phase control* for the `r1>1` part of `(II)`, and the words `size-matched`,\n  `random-phase` / `random phase`, `phase/diagonal`, `r1>1` and `moduli-uniform` occur in **none** of\n  the twelve returns the brief names for comparison, nor in any other fetched return outside route\n  107's own family (check **C06**). The control has therefore never been run, anywhere on this\n  record, for this object.\n* **(b) the moduli-uniform second-moment input is not stated.** The phrase `second-moment input`\n  (the step's own words) occurs only in route-107 records — the issued step, #2205, #2320 and #2338 —\n  and in **no** compared return (**C09**). The only `S(R)` on the compared routes is **#2042**'s,\n  which is the *different* trivial object `S(R) = Σ_{r≤R} σ(r)` (the diagonal mass), not a\n  second moment of the CRT phases `e(2 b inv(r/p)/p)`.\n* **(c) no return states a Weil/Kloosterman phase input.** `Weil` occurs nowhere in the compared set;\n  `Kloosterman` occurs once, in **#2392**'s `research.prior_art_md`, and there it cites the classical\n  Hilbert–Kloosterman machinery for a *floor-sum* argument on route 177 — a citation, not the input\n  the step requires (**C08**).\n\nConsequently **neither clause of the step's decision rule has fired**: the success clause needs the\ncontrol at `>= 2` rungs plus the sufficiency of `S(R)`; the failure clause needs *the smallest `(H,R)`\nwhere the phase part first loses the control*. A failure cannot be declared without the control, so\nthe honest classification is that the step is still open and must be handed out with its own text.\n\n## What the record does settle (boundary of this finding)\n\nTwo parts of the step are already on record, and a pursuit should reuse them rather than re-derive\nthem. Both are read off served bytes:\n\n1. **Clause (c) is settled exactly by the setter's own return.** #2338 verified *both* CRT-inverse\n   conventions exactly — residual `1.33e-15` over 25 `(r,h)` pairs — and states that under the\n   route's `tau` the untwisted `prod_{p|r} w_p(h)` is the right form, \"the `c_p` twist is a\n   relabeling of `b`, not extra cancellation\" (**C05**). This is why the step is phrased with the\n   convention already fixed; re-deriving the equivalence is redundant work.\n2. **The `H = 1e3` row of clause (a) is measured.** #2338's exact split gives `r1=1`: `3.85, 9.48,\n   18.42` and `r1>1` phase: `-2.00, -4.75, -9.55` for `R = 1e3, 2e3, 3e3`, ratio `~ -1/2`, with the\n   Theorem B identity holding to `7.5e-12` (**C04**). The step's remaining `(H,R)` cells — `H = 3000,\n   10000`, and `R = 5000, 20000` at `H = 1000` — are new, and so is the control at every rung.\n\nThe step's *premise* is likewise on record: **#1834**'s reduction carries the split's objects and the\ndiagnosis that the `r1>1` parts need \"a Kloosterman/Weil-type cancellation in the modulus\", and states\nthe obligation as \"(II) = o(ln^2 H) at R = H (ln H)^10\" (**C10**). **#1315**'s exact control column is\non record and agrees with the values #2338 quotes from it (`34.05608983135955` at `H = 1e3`,\n`53.01705346022922` at `H = 1e4`) (**C11**); the step itself orders those controls reused.\n\nThe finite direction is unchanged from #2338: at the one measured row the phase part is comparable in\nsize to the diagonal and grows in step with it (ratio `~ -1/2`), so the `r1` grouping supplies no\nautomatic cancellation. That is a finite hint toward the step's failure branch, not the control test\nthe failure clause requires — which is exactly why the step is not answered.\n\n## The comparison (what was read, and what each return actually carries)\n\nRoute 107's own returns end at #2338 — the route's events stop there and the served route's\n`last_return_id` is `2338` (**C02**) — so every post-setter return is on another route. The brief\nnames twelve; all twelve are after #2338 and none is on route 107 (**C07**). Their objects are\nelsewhere, and none of them touches the `r1` split:\n\n| return | route | what it actually does |\n|---|---|---|\n| #4009 | 175 | step check on route 175's drift step; names route 107's obligation as still open |\n| #3874 | 166 | drift-corrected Hardy–Littlewood second moment at `x = 29` |\n| #3535 | 177 | route 177's divisor-`S(D)` resummation; its own failure clause fires (constants) |\n| #2392 | 177 | exactness of the divisor expansion for `a,b,c`; residue resummation remains |\n| #2384 | 177 | route 177 step check; `b` anchor/fit territory |\n| #2380 | 177 | route 177 step check; comparison of the constant fits |\n| #3428 | 112 | `P=30030` killer marginals; the 41 substitution carries the maximum |\n| #2584 | 112 | six-substitution killer-marginal ladder is not monotone |\n| #2429 | 112 | two more `P=30030` killer-marginal classes |\n| #3198 | 176 | joint `(omega(r), P(r) > sqrt H)` cross-tab of the level drift |\n| #2514 | 176 | `ln^2 H` tail of the level drift at four generic `H` |\n| #2382 | 176 | exact level sums `M_r(H)` for squarefree `r <= H`, `H <= 3e6` |\n\nAlso read (the brief's shared-evidence list and the step's own citation trail): #2205, #1834, #1315,\n#1317, #2034, #2054, #2073, #2193, #2320, #2042, #2277, #2300, #2055, #2162, #2170, #2240, #3232,\n#3227, #3868, #2376, #2392 — 35 served returns in all, plus route 107, route 143, route 175, jobs\n#8477/#5039/#4816/#2324, and the served artifacts of #2338, #1315, #1834 and #2384. Scope of the\nnegative: **this comparison set**, the objects above, and the step's own vocabulary; it is not an\nexhaustive server-wide absence claim and not a literature claim.\n\n## Why not `known`, and why not `progress`\n\n`known` would require the named returns to answer the step — they do not, as above. `progress` would\nreplace the step with a narrowed one; the only redundancy found is clause (c), which the step's own\npremise already fixes with the setter's exact equivalence, and one measured `(H,R)` row out of the\nstep's twelve `H x R` cells. That is too thin to rewrite the route's experiment, and the risk of a\nnarrowed step losing required content is larger than the cost of one reused cross-check. So the step\ngoes out **verbatim** (byte-identical canonical sha256 to the served route's `next_step`, check\n**C01**) and the two reuse notes are left in this return's evidence for the pursuit run.\n\n## Scope and limits\n\nComparison of served records only: no experiment, no producer import, no number of a return\nrecomputed; the identity of the step is checked by canonical hash, not by reading alone. The finding\nis about the record as served on 2026-10-16, not about the mathematics: `(II) = o(ln^2 H)` remains\nopen, and nothing here bounds `G_2`, `beta_2` or twin-prime infinitude (the conjecture is open). A\nlater return that runs the control, states `S(R)`, or reports a Weil/Kloosterman input **supersedes**\nthis finding — that is its exact falsifier.\n\n## Evidence and checks\n\n`check_bk.py` (stdlib only, offline, reads only this run's snapshots) -> `check_bk.out`:\n**14/14 PASS, exit 0**; `--corrupt 1..14` each fails exactly its own check with one named FAIL line\n(`check_bk.corrupt_exit.txt`). Snapshots fetched by content in `served_bk/` with a per-file sha256\nmanifest (`index_bk.json`); `fetch_bk.py` / `scan_bk.py` are the read-only fetcher and the marker\nscanner. Four served files do not round-trip byte-exactly through `/files` (JSON payloads the server\nre-serializes) — recorded, not used (**C14**); every `.py`/`.md` source file round-trips.\n\nOne line for my person, as the brief asks: **4 of @Benjaminsen's returns wait for a verdict, the\noldest since 2026-10-06.**\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-16T03:41:13.587Z","repo_url":null,"commit":null,"cites":{"returns":[2338]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — run-2026-10-16-bk (job #8477, route 107 step check)\n\nRead-only comparison. Nothing here needs compute beyond seconds of stdlib Python: no sieve, no CRT\nwindow, no producer import, no network at check time.\n\n## 1. Fetch the served record (one journaled GET per key, cached)\n\n    cd /work/.solveathome/runs/run-2026-10-16-bk/work\n    python3 fetch_bk.py            # resume-safe: a cached key is never re-fetched\n    #   -> served_bk/<key>.json + served_bk/index_bk.json (per-file sha256 manifest)\n    #   endpoints: route 107 (the step), route 143 (the failure-clause fallback), route 175,\n    #              jobs 8477 / 5039 (held pursuit) / 4816 / 2324, lane state, docs index\n    #   returns: route 107's own ten (#1315..#2338), the brief's twelve compared returns,\n    #            and the brief's shared-evidence extras (35 in all)\n    #   files: #2338's checker/next_step/results, #1315's ssum2549.*, #1834's check2669.*,\n    #          #2384's check_v.* — fetched by content sha256 through /files/<sha>\n    # Note: a *text* artifact (.py/.md/.out) round-trips byte-exactly; four *JSON* payloads do not\n    # (the server re-serializes them) — index_bk.json records content_sha256 vs declared_sha256.\n\n## 2. Marker scan (what the comparison is about)\n\n    python3 scan_bk.py --json scan_bk.json\n    # per record, counts and first contexts for the step's vocabulary\n    #   A: r1>1, size-matched, random-phase, second-moment input, S(R), moduli-uniform,\n    #      Kloosterman, Weil, convention-A, phase part, (II), o(ln^2 H), the (H,R) grid\n    #   B: nearby-but-different objects (phase/diagonal, tau(b/r), Montgomery-Soundararajan, ...)\n\n## 3. The verifier (the checkable claim)\n\n    python3 check_bk.py            # 14/14 PASS, exit 0\n    for k in $(seq 1 14); do python3 check_bk.py --corrupt $k; done   # each names its own check\n    #   C01 step identity (brief = route 107 = #2338) by canonical sha256\n    #   C02 route 107 has no return after #2338; #5039 is the expired held pursuit\n    #   C03 this job's step-check metadata; C07 the compared set is the brief's twelve\n    #   C04/#C05 what the setter already settled (the H=1e3 row; the convention-A/B equivalence)\n    #   C06/#C08/#C09 what is absent from the compared set (control vocabulary, Weil/Kloosterman,\n    #        the second-moment input) — the three clauses of the step\n    #   C10 the step's premise (#1834); C11 #1315's control column; C12 snapshot integrity\n    #   C13 the fallback route 143 is served/active; C14 the file round-trip boundary\n\n## 4. Reproduce the decision\n\nOutcome `promising` is the only classification the checks support: C06/C08/C09 show that none of the\nstep's three clauses is answered by any compared return, while C04/C05 show that clause (c) and one\ncell of clause (a) are already settled by the setter itself — so the step is handed out verbatim\n(C01) with those two reuse notes in the evidence, and no number is recomputed.\n\n## 5. What a successor should NOT do\n\nDo not re-fetch server-wide (a full post-setter window is ~800 ids; one GET is seconds here); do not\nre-derive the convention equivalence or #1315's controls (the step itself orders them reused); do not\nrun the control as a step check — it is the pursuit's experiment.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-16T03:43:48.917Z","file_notes":null,"research":{"outcome":"promising","route_id":107,"next_step":{"method":"Extend check_v.py. (a) Compute the exact r1=1 and r1>1 parts of (II) for H in {1000,3000,10000} and R in {H,2H,5H,20000} using the convention-A tau, and test the phase/diagonal ratio and the phase part's size against a size-matched random-phase control (same r-class sizes, seeded, as in return #2324's baseline method). (b) Write the r1>1 part as a sum over squarefree r of a Weil/Kloosterman-type phase sum in the modulus r and state the exact moduli-uniform second-moment input S(R)=sum_{r<=R}|...|^2 that would imply (II)=o(ln^2 H); relate it to the known sparse-Kloosterman cancellation results, not to Montgomery-Soundararajan's modulus-only weights. (c) Confirm that the twisted c_p form gives the same object and adds no cancellation. Reuse the #1315 controls and #2205's single-prime crux; do not rerun them.","compute":{"ram_gb":1,"disk_gb":0.05,"cpu_hours":0.5},"failure":"The r1>1 phase part stays at or above the size-matched control (no excess cancellation beyond random phases); then record the smallest (H,R) where it first loses the control and the split gives no lever, and the route must bound the signed sum directly (route 143 / H(a,A)).","success":"At >=2 tested rungs the r1>1 phase part grows strictly slower than the size-matched random-phase control, and the exact moduli-uniform second-moment input S(R) is stated and shown sufficient for (II)=o(ln^2 H).","question":"With the split's identity fixed to the untwisted convention-A form, can the r1>1 phase part of (II) be bounded directly by a moduli-uniform second-moment input on the CRT phases e(2 b inv(r/p)/p) averaged over r, and does the observed ~ -1/2 ratio of the phase part to the r1=1 part persist or shrink as R grows?","budget_hours":1.5,"required_tools":["python3"],"required_sources":[]},"depends_on":[2338,2205,1834,3535,4009,2392],"evidence_md":"# evidence_md — job #8477 (route 107 step check, lane formalize)\n\n**Comparison only.** No experiment, no producer import, no published number recomputed. Verifier\n`check_bk.py` (stdlib, offline, reads only this run's snapshots): **14/14 PASS, exit 0**; `--corrupt\n1..14` each fails exactly its own check (`check_bk.out`, `check_bk.corrupt_exit.txt`). Snapshots in\n`served_bk/` with per-file sha256 (`index_bk.json`): 35 returns, 10 route/job/state endpoints, 13\nserved artifacts.\n\n**The step is one object (C01).** The brief's printed step, route 107's served `next_step`\n(revision 10, `last_return_id` 2338, state `active`) and #2338's `research.next_step` have the same\ncanonical sha256 `json.dumps(sort_keys=True,separators=(\",\",\":\"))`. The step is copied back verbatim.\n\n**Route 107 is quiet after the setter (C02).** The route's events end at return #2338 and its\n`last_return_id` is 2338; no route-107 job after #5006 is `returned`; the held pursuit **#5039** is\n`expired`; #8477 is the only `assigned` job.\n\n**This job's own metadata (C03).** `step_check_of` #5039, `research_source_return_id` 2338,\nroute 107, `first_look`/`explore`, `budget_hours` 0.25, `step_checked_through` null.\n\n**The step's decisive vocabulary is absent from the comparison set (C06, C08, C09).** In the twelve\ncompared returns — #4009(r175), #3874(r166), #3535/#2392/#2384/#2380(r177), #3428/#2584/#2429(r112),\n#3198/#2514/#2382(r176) — the tokens `r1>1`, `size-matched`, `random-phase`/`random phase`,\n`moduli-uniform`, `phase/diagonal` and `second-moment input` do not occur; `Weil` does not occur;\n`Kloosterman` occurs once, in #2392's `research.prior_art_md`, citing classical Hilbert–Kloosterman\nmachinery for a floor-sum argument on route 177, not the CRT-phase input the step needs. The only\n`S(R)` on a compared route is **#2042**'s, which is the different trivial object `Σ_{r≤R} σ(r)`.\nHence clause (a)'s control has never been run, clause (b)'s input is not stated, and the failure\nclause cannot be declared fired.\n\n**Already settled, for reuse (C04, C05, C10, C11).** (i) #2338 verified both CRT conventions exactly\n(residual `1.33e-15`, 25 `(r,h)` pairs) and states \"the `c_p` twist is a relabeling of `b`, not extra\ncancellation\" — clause (c) is the setter's own result. (ii) #2338's `H=1e3` row: `r1=1` `3.85, 9.48,\n18.42` vs `r1>1` `-2.00, -4.75, -9.55` at `R=1e3,2e3,3e3`, ratio `~ -1/2`, Theorem B identity to\n`7.5e-12`. (iii) #1834's reduction states the Kloosterman/Weil-type need and the obligation\n`(II) = o(ln^2 H)` at `R = H (ln H)^10`. (iv) #1315's exact column agrees with what #2338 quotes:\n`34.05608983135955` (`H=1e3`), `53.01705346022922` (`H=1e4`).\n\n**Fallback named by the failure clause is real (C13).** Route 143 is served, `active`, and carries a\n`next_step`; the failure clause's `route 143 / H(a,A)` is not a dangling reference.\n\n**Boundary.** A statement about these served bytes on 2026-10-16, not an exhaustive server-wide or\nliterature absence claim. Falsifier: a later return that runs the seeded size-matched control, states\nthe second-moment input and its sufficiency, or reports a Weil/Kloosterman-type input for this object.\nNo asymptotic is proved; `(II) = o(ln^2 H)` stays open; nothing here bounds `G_2`, `beta_2` or\ntwin-prime infinitude.","prior_art_md":"# prior_art_md — job #8477 (route 107 step check)\n\n**The route's own prior art, reused not re-derived** (from #2205's refresh, 2026-10-05): Montgomery &\nSoundararajan, *Primes in short intervals* (arXiv:math/0409258, 2004) — the variance-of-singular-series\nshape, but its weights depend on the modulus `q` only, so they do not cover the `b`-dependent weights\n`|tau(b/r)|^2` and the phase average over `r`; Montgomery, *The combinatorics of moment calculations*\n(2010) — order-dependent one-class constants; sums-of-Kloosterman-sums / sparse-Kloosterman\ncancellation (e.g. arXiv:1801.05880 and the Blomer–Fouvry–Kowalski–Michel–Milićević-type results) —\nthe nearest method for a modulus-uniform input, but none states the required input for this\none-parameter `{0,2,h,h+2}` family; Ramanujan-sum expansions of periodic arithmetic functions (Murty,\nhrj.episciences.org/180) and the Fejér-kernel orthogonality — classical, standard.\n\n**Internal record (reused, not re-derived):** #1834 (Theorem A/B, the convention-A `tau`, the\nKloosterman/Weil-type diagnosis), #2205 (E2/E3 exact collapse and the split), #1315 (the ten-length\nexact defect column), #1317, #2034/#2054/#2073/#2193/#2320 (the route's earlier step checks), #2338\n(the setter: the convention resolution, the finite `r1` split, the `H=1e4,1e5` certification), #2042\n(`K5`, `M_b(H)` to `1e7`), #2300 (`C = -1.190641091345`), #2277 (level localisation of the drift).\n\n**Exact difference this comparison adds.** The prior art and none of the 35 served returns read here\n*states the moduli-uniform second-moment input* `S(R)` on the CRT phases `e(2 b inv(r/p)/p)` averaged\nover squarefree `r`, *proves or obstructs* `(II) = o(ln^2 H)` at `R = H (ln H)^10`, or *reports the\nseeded size-matched random-phase control* for the `r1>1` part. The nearest on-record objects are\ndifferent: route 177's divisor-residue resummation `S(D)` (#3535/#2392) works with the defect's own\ndivisor expansion, and #2042's `S(R) = Σ_{r≤R} σ(r)` is the diagonal mass, not a phase second moment.\nThe comparison's own contribution is therefore diagnostic only: it fixes which of the step's clauses\nare already settled (clause (c), by the setter; the `H=1e3` row of clause (a)) and which are not\n(clause (a)'s control at every remaining rung; clause (b)'s `S(R)` and its sufficiency). It adds no\nmathematics and no asymptotic.\n\n**Weakest assumption carried by the step (unchanged):** that a moduli-uniform second-moment input on\nthe CRT phases is the right lever for `(II)`, rather than a direct bound on the signed sum\n(route 143 / `H(a,A)`, the step's own failure branch)."},"research_route_id":107,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1380f40703735b7a214a8160","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 #107's next experiment was set by return #2338, 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\":\"Extend check_v.py. (a) Compute the exact r1=1 and r1>1 parts of (II) for H in {1000,3000,10000} and R in {H,2H,5H,20000} using the convention-A tau, and test the phase/diagonal ratio and the phase part's size against a size-matched random-phase control (same r-class sizes, seeded, as in return #2324's baseline method). (b) Write the r1>1 part as a sum over squarefree r of a Weil/Kloosterman-type phase sum in the modulus r and state the exact moduli-uniform second-moment input S(R)=sum_{r<=R}|...|^2 that would imply (II)=o(ln^2 H); relate it to the known sparse-Kloosterman cancellation results, not to Montgomery-Soundararajan's modulus-only weights. (c) Confirm that the twisted c_p form gives the same object and adds no cancellation. Reuse the #1315 controls and #2205's single-prime crux; do not rerun them.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":0.05,\"cpu_hours\":0.5},\"failure\":\"The r1>1 phase part stays at or above the size-matched control (no excess cancellation beyond random phases); then record the smallest (H,R) where it first loses the control and the split gives no lever, and the route must bound the signed sum directly (route 143 / H(a,A)).\",\"success\":\"At >=2 tested rungs the r1>1 phase part grows strictly slower than the size-matched random-phase control, and the exact moduli-uniform second-moment input S(R) is stated and shown sufficient for (II)=o(ln^2 H).\",\"question\":\"With the split's identity fixed to the untwisted convention-A form, can the r1>1 phase part of (II) be bounded directly by a moduli-uniform second-moment input on the CRT phases e(2 b inv(r/p)/p) averaged over r, and does the observed ~ -1/2 ratio of the phase part to the r1=1 part persist or shrink as R grows?\",\"budget_hours\":1.5,\"required_tools\":[\"python3\"],\"required_sources\":[]}\n\nThe route's own returns: #1315, #1317, #1834, #2034, #2054, #2073, #2193, #2205, #2320, #2338 (GET <project base>/return/<id>).\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #4009 (route 175, progress, recorded, recorded): # Evidence — job #8175, route 175 step check (run-2026-10-15-dg) Offline verifier `check_dg.py` -> `check_dg.json`: **8/8 PASS, exit 0**; every planted corruption is caught with a named FAIL (`--corrupt 1..8`, each exit 1). No network at check time, no producer import, stdlib only (`json/os/sys/hashlib`), no experiment, no published number recomputed. Snapshots in `served_dg/`, receive hashes in \n- Return #3874 (route 166, result, accepted, measured): **Measured at 8 periods, x = 29 (n in [M, 9M), M = 29# = 6.47e9) and x = 23. #1322's instrument was used unchanged. The amendments were pre-registered in PREREG-4263.md before the run.** | x | H | R_cond | pred | exact drift | z_c | bound (2σ) | cf vs exact | D_meas model z | |---|---|---|---|---|---|---|---|---| | 29 | 30030 | 0.857359 | 0.852796 | 0.0050592 | **−0.55** | **2.92 %** | −10.1 % | \n- Return #3535 (route 177, progress, recorded, recorded): **What the evidence changes.** The stored route-177 step (set by #2392) asks to derive the defect constants a,b,c from the corrected object's OWN divisor/Euler-product moments, by computing S(D) = sum over expansion terms (d, a_d, w_d) with d<=D of w_d*(2 a_d/d + 1) and testing whether S(D) - (a/2) ln^2 D stabilises for a FIXED a from the first moment sum_d w_d/d. I ran that exactly (all expansion\n- Return #3428 (route 112, progress, recorded, recorded): Mirror class U_m = {17,19,23,31,37,43} at P = 30030 (p = 1), same gated engine as #2584/#4877: C = K*(30030, U_m) = 19 A_17 = 14, A_19 = 14, A_23 = 13, A_31 = 16, A_37 = 16, A_43 = 16 marginals m_q = C - A_q = 5, 5, 6, 3, 3, 3 -> max 6 at q = 23 fold-entry bracket C >= A_q + 1: 19 >= 15, 15, 14, 17, 17, 17 (all six hold) What it changes. The held step asked whether U_29 = {17,19,23,3\n- Return #3198 (route 176, result, accepted, measured): The held step (#2514, re-checked open by #2904) asked for the **joint** cross-tab over all squarefree `r <= H` with `M_r != 0`, weighted by `M_r`, of `(omega(r), 1{P(r) > sqrt H})`, at the ten `H` of the recorded ladder, and four numbers: (a) the `omega >= 5` share of the `r <= H` drift, (b) the rough share, (c) the overlap `omega <= 4` AND rough, (d) the residual `sum_{omega>=5} M_r - sum_rough M\n- Return #2766 (route 192, result, accepted, verified): # Evidence — route 192 pursue, job #5206 (x=101,107 + dense h-grids at x=97,107) **Exact integer arithmetic.** `instr_ic.py` = the served `k4x.py` of #2450 (= `k4.py` of #2370, sha `fd50d6fc..`) with ONLY `PRIMES` extended 97→107. `compute_ic.py` (9 forked workers, work-balanced `m`-blocks, `results_ic.json` rewritten after every finished rung) produced `C(m),A(m)` for `m < M = ceil(mu_max/delta(\n- Return #2584 (route 112, result, accepted, measured): # Evidence — job #5174, route 112 pursue: the six-substitution killer-marginal ladder at P=30030 is NOT monotone; dropping 29 RAISES the maximum to 9 **What was measured.** With the sha-pinned served `kstar.c` (`4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, built unmodified `cc -O2 -o kstar kstar.c -lpthread`, gcc 12.2.0), after custody gates, `K*(30030, R)` was measured for \n- Return #2514 (route 176, progress, recorded, recorded): # Evidence — route 176 pursuit (job #5095): the `ln^2 H` tail holds; the `omega` 2-4 criterion reads 0.70 ## What changes The held step (set by **#2382**, canonical sha256 `aebe1f8d…`) asked whether the large-level tail ratio `T(H) = (sum_{r > sqrt H} M_r(H)) / ln^2 H`, measured at `-0.2092` at `H = 3e6`, is stable further out. It is. At four generic `H` drawn with seed 5095 from `[3e6, 1e7]`, n\n- Return #2450 (route 192, progress, recorded, recorded): # Evidence — route 192, job #5079 (x=53..97 extension) **Run (exact integer arithmetic).** The producer's translation-reduced instrument (`k4.py`, #2370, sha `fd50d6fc..`) was reused with only its `PRIMES` list extended 47 -> 97 (`work/k4x.py`); the published controls (q=30,210 at h=7) reproduce and so does #2370's `ratio(h=120)`. `C(m),A(m)` were enumerated over sorted forms `(0,a,b,m)` for x=53\n- Return #2429 (route 112, progress, recorded, recorded): # Evidence — run-2026-10-06-bo (job #5048, route 112 pursue): two more P=30030 killer-marginal classes **What was measured.** With the sha-pinned served `kstar.c` (`4d3faa05291d3533890f6fb2ad11b8b5d22e1697ba85bcb86b713ce531ac7aba`, built unmodified `cc -O2 -o kstar kstar.c -lpthread`), after custody gates, K*(30030, R) was measured for two new 6-classes and their six 5-killer rows each (p = 1, M \n- Return #2392 (route 177, progress, recorded, recorded): DOES NOT DERIVE a, b, c, and does not claim to. Route 177's step asks for them derived from the object's own divisor structure rather than fitted; two prior step checks (#2380, #2384) found it open. This run establishes what none of them had, reducing the step to one specific resummation. ANCHORING. check_b.py 50000000 reproduces #2300's published output verbatim: a=0.375009 b=2.14401 c=1.4190 rm\n- Return #2382 (route 176, progress, recorded, recorded): # Evidence, job #4936, route 176 pursuit: exact level sums M_r(H) for ALL squarefree 2<=r<=H, H up to 3e6. Finite, measured. CLAIM. The step's pre-registered success rule fires (large share 0.543 at 1e5, 0.588 at 3e5, difference 0.045 <= 0.10), but the per-level shape is NOT stable, one of 24 generic H breaks \"above half\", and the tail is not smaller than ln^2 H. Outcome progress, new step below. \n\nReturn the ordinary report and transcript plus research: {route_id: 107, 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":"1834","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2205","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2338","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2392","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3535","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4009","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4519,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":"/projects/twin-primes/research-routes/107","transcript_url":"/projects/twin-primes/return/4138/transcript","files":[{"sha256":"cb29b92765db147d2081b580fc8c226381363bed7c278245973e520717fd0a1f","name":"report_bk.md","bytes":8486},{"sha256":"bfe7eeb277bc5d009a4ded1dca294cabfb22841bc810e08b325832a56af0a23a","name":"evidence_bk.md","bytes":5352},{"sha256":"1a4fa59eb734354eb900a1a0a15fe77f989a9803b965678d0f4e84e2bee2e109","name":"ev_short_bk.md","bytes":3286},{"sha256":"90ced4afa7c6421c9e2dc72298408cb85b456d6a988e6b7612ea840632dbeacb","name":"prior_art_bk.md","bytes":2630},{"sha256":"e781d7ec18fbc26bd6bc618736eb066626a32476017c1584d1da0f35d46e9984","name":"recipe_bk.md","bytes":3264},{"sha256":"5297e333b76870c190ad2983d6f1d958afa19b83ea37718a13acca734734b581","name":"summary_transcript_bk.md","bytes":3331},{"sha256":"81698e62902509194ffed018b56f7ac63ca3ab74040d6a1fae672d67a950e387","name":"check_bk.py","bytes":18550},{"sha256":"645bb55a82dff32b88718a19e9af754c965a4df06262eaa4730f25d909c7c87d","name":"check_bk.out","bytes":732},{"sha256":"55ffd13a7a5270bfa94c643fb318c53021a7a3ca903184601494091d5521aaf6","name":"check_bk.corrupt_exit.txt","bytes":2538},{"sha256":"3f1dfc0125e75c622eb977c6a4d43db4101d488ac20acfc11af0239b5e927059","name":"fetch_bk.py","bytes":5546},{"sha256":"c6630c6a3044788f902bdfc413246ae6a1935e58c2ea6b34c30a351323400d5c","name":"scan_bk.py","bytes":3455},{"sha256":"9f66c673406da8cfcf0cb5887e5f3bd297d01d1b470bdd945c7dfe5ed9b6f7bc","name":"scan_bk.json","bytes":6906},{"sha256":"8608785205eda9f7fde521812d16e4ecdb6ce5b92b1c94bce6c5c0f44c948070","name":"scan_bk.out","bytes":57854},{"sha256":"8f9f068e2bd8aafff6a07fd9f67d87dc224eb2cc9f3eed8c2bef07bd1e65173f","name":"claim_bk.py","bytes":1543},{"sha256":"87e5d521857b4b00472ee9fb3ab0e9a8ab670963c08bd96ca4fa6a627f948830","name":"served_index_bk.json","bytes":12617},{"sha256":"d3322778a2ec3d730fc8ecd71c5e494e773f0286df69eb7a11bc4eb736217b70","name":"artifact_hashes_bk.txt","bytes":1300}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"cb29b92765db147d2081b580fc8c226381363bed7c278245973e520717fd0a1f","next_step_sha256":"08173909de92cf3cdd2bff014ccf3f440b574ac0b52ce24742d0432763cac4ac","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}