{"id":5764,"job_id":9378,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9378 — route 192 pursue: the matched-phase `x`-term on the `x = 137..157` ladder\n\n**Rung of the main claim: `measured`** (finite exact-integer arithmetic, no asymptotic or\nstructural claim).\n\n## Result\n\nThe held step's ladder now exists and the pre-registered test has been run: **the `x`-term keeps\nits sign and its order of magnitude, its size nearly doubles on the new sub-ladder, and the\npre-registered falsifier does not fire.** Concretely, on matched phases (`|dt| <= 1e-3`) and\n`t >= 1.5`, with the one-parameter fit `dln ratio = c * (ln x' - ln x)`:\n\n| pool | rungs | pairs (n) | `c` per unit `ln x` | mean of per-pair slopes |\n|---|---|---|---|---|\n| recorded ladder (reproduces #4565) | 53..131 | 21 (335) | **+0.3188 +- 0.0285** | +0.4225 +- 0.0470 |\n| new sub-ladder, anchored at 131 | 131,137,139,149,157 | 10 (184) | **+0.5981 +- 0.0302** | +0.7388 +- 0.1262 |\n| new rungs only | 137,139,149,157 | 6 (108) | **+0.5933 +- 0.0525** | +0.7877 +- 0.2121 |\n| all 11 rungs | 53..157 | 55 (911) | +0.4023 +- 0.0223 | +0.5178 +- 0.0331 |\n\n(`c` se is cluster-robust over rung pairs; the iid se on the recorded pool is 0.0146, which is\nexactly #4565's published `+- 0.0146`, and that pool's point estimate is bit-identical to its\n`+0.3188` — the estimator and this independent implementation agree with the accepted return.)\n\n- **Expected evidence (a stable `c`) is not delivered.** The two sub-ladder estimates differ by\n  0.279 +- 0.029, i.e. **9.5 se**, with ratio **1.88**. The step's success clause (\"stable within\n  its 2-se bar across the two sub-ladders\") **fails**.\n- **The pre-registered falsifier does not fire.** `c` does not change sign, and 1.88 < 2, so the\n  clause \"changes sign or more than doubles\" is not met. Read literally, the stated consequence\n  (\"the residual is a resolution or pairing effect rather than an `x`-effect\") is therefore **not**\n  supported; the matched-h control below independently says the residual is not a pairing artifact.\n- **The `t`-profile changes too.** `c` by `t` window: recorded ladder `[1.5,2)` +0.0775 +- 0.0461,\n  `[2,3)` +0.4777 +- 0.0131, `[3,4)` +0.4054 +- 0.0087; new sub-ladder `[1.5,2)` **+0.7454 +-\n  0.0943**, `[2,3)` **+0.7051 +- 0.0569**, `[3,4)` +0.4120 +- 0.0292. On the new rungs the term is\n  already at full size at `t = 1.5` and flat to `t = 3`; on the recorded rungs it grew from ~0.08\n  to ~0.48 between those windows. The `t`-dependence of the earlier ladder is itself\n  ladder-dependent.\n- **The `x`-term is not a smooth one-constant function of the rung at this precision.** Per-pair\n  slopes inside the new sub-ladder range from 0.37 (`139-149`) to 1.71 (`137-139`), a scatter of\n  +-0.4 that is as large as the sub-ladder shift itself. What is stable is the sign, the order\n  (`c` in `0.3..0.7`) and the fact that the term is present at every rung pair.\n\n**Consequence for the route.** #4565's quantified statement (\"`t = h*delta(x)` is not the only\nvariable; `c = +0.32`\") survives as a statement about its own ladder and is now shown to be\nladder-local: extending the same instrument to four further rungs more than doubles the fitted\nslope and removes the low-`t` shortfall. Any uniform-in-`x` use of this residual needs either a\nrung-dependent coefficient or a second parameter; a single constant is excluded at 9.5 se.\n\n## What was measured\n\nFour new rungs of the route's exact integer tables, same convention, same grid rule\n`h in [ceil(0.4/delta), ceil(4/delta)]` (`mu_max = 4.0`, the extent #4565's own amendment fixed):\n\n| `x` | `delta(x)` | `M` | grid `h` | `ratio(mu=1.5)` | `ratio(mu=3.0)` | source |\n|---|---|---|---|---|---|---|\n| 131 | 0.01687972 | 237 | 24..237 | 1.163733e-03 | 3.275008e-04 | recorded (#4565) |\n| **137** | 0.01663330 | **241** | **25..241** | **1.098100e-03** | **3.127912e-04** | this return |\n| **139** | 0.01639397 | **244** | **25..244** | **1.018496e-03** | **3.010259e-04** | this return |\n| **149** | 0.01617392 | **248** | **25..248** | **9.550573e-04** | **2.873737e-04** | this return |\n| **157** | 0.01575639 | **254** | **26..254** | **8.760302e-04** | **2.781727e-04** | this return |\n\n`ratio(h) = |kappa4(h)|/B_abs(h)`, `kappa4(h) = sum_{m<h}(h-m)C(m)/q^4`, `B_abs(h) = sum_{m<h}(h-m)A(m)/q^4`,\n`t = h*delta(x)`, `delta(x) = 0.5*prod_{p<=x, p>2}(1-2/p)`; `q = x#`. The recorded rungs 53..131 are\nreused from the served rung files, not recomputed.\n\n## Method: one proven symmetry, nothing else\n\n`instr_98.py` is the served `instr_ia.py` of #4565 with **only** the `PRIMES` list extended by\n{137, 139, 149, 151, 157}; the definition of `K_x(d)` is untouched. The single change in the\npipeline is an exact symmetry that halves the enumeration:\n\n> **Reflection.** For `m = max(d)`, `K_x(m - d) = K_x(d)`.\n> *Proof.* For a prime `p` and a block `B`, `occ_B` counts the residue set\n> `{(-d_i) mod p, (-d_i - 2) mod p : i in B}`. For `d'_i = m - d_i` that set is\n> `{d_i - m, d_i - m - 2}`, which is the image of the `d`-set under `r -> -r - m - 2`, a bijection\n> of `Z/p`. Hence every `occ_B`, every `N_B = prod_p (p - occ_B)`, and the Mobius sum over the 15\n> set partitions are unchanged. (Translation invariance `K_x(d + s) = K_x(d)` is the producer's\n> proven identity and is used as before.)\n\nGates for it, all passing at **every** new rung: the exact `h^4` census identity at `h = 7, 8`\n(`kappa4(h), B_abs(h)` from the cumulative integer arrays versus direct summation over all `d` in\n`[0,h)^4`); a full (unhalved) enumeration for `m < 8` against the pinned `Kint`; the two published\ncontrols (`q=30, h=7` -> `(-739/15000, 60341/15000)`, `q=210, h=7` -> `(-969/9800, 990319/480200)`);\nwhole-array and random-tuple equality of the fast kernel with the pinned one at `x = 5,11,23,53,131,157`.\nA pre-extension run of the same pipeline with the prime set of `x = 131` reproduced #4565's\n`x = 131` ratio values exactly (`1.163733e-03`, `3.275008e-04`), so the new rungs are produced by\na pipeline that has been shown to reproduce the accepted rung.\n\n## Controls\n\n1. **Matched-h control** (`dln` at equal `h`, each new rung versus `x = 131`, `t >= 1.5`):\n   `c_h = +0.2307 +- 0.0220` (n = 596; per rung 0.2825 / 0.3853 / 0.2347 / 0.2090). It is 10 se\n   *below* the matched-phase `+0.598`, so equal-`h` pairing **does not reproduce** the residual;\n   fitting `dln = c*Dln x + k*dt` on the same pairs gives `c = -1.380 +- 0.037`, `k = -1.624 +\n   0.036` (mean `dt = -0.104`, max `|dt| = 0.266`): the equal-`h` pool is dominated by the\n   curve's own `t`-slope, which is why it cannot serve as a clean `x`-control at this `h` spacing.\n2. **Pairing bias**, as #4565's C22: over all 911 matched pairs `mean dt = +1.4e-5`, `max|dt| =\n   9.99e-4`; with the largest local slope (~1.0 per unit `t`) this bounds the pairing bias in\n   `ln ratio` by ~1e-3, an order of magnitude below the residuals used.\n3. **Local variation (S4)**, the curve's own roughness per unit `h`: at `t in [1.5,2)` mean\n   0.0151..0.0165 (max 0.0406..0.0422) on the new rungs, 0.0141..0.0263 (max 0.0439..0.0735) on\n   the recorded ones; at `t in [2,3)` ~0.0175. The matched-phase residual rms on the new\n   sub-ladder is 0.028-0.036, i.e. ~2x this scale; on the recorded ladder 0.108, ~6x. The two\n   sub-ladders are not equally smooth, which is itself part of why a single `c` does not describe\n   both.\n4. **Tighter tolerance** `|dt| <= 2e-4`, all rungs: `c = +0.4412 +- 0.0318` (n = 174), between the\n   two sub-ladder values — the doubling is not a tolerance artifact.\n5. **Exact arithmetic throughout**: `C(m), A(m)` are Python integers (`q^4 K`), `kappa4`, `B_abs`\n   are exact `Fraction`s; the published `ratio` values are the float of that quotient.\n\n## Scope, limits, what would change this\n\nFinite: 11 rungs, `x <= 157`, grids to `t ~ 4`, matched-phase pairs with `t >= 1.5` only; the\n`t in [2,3)` and `[3,4)` windows are populated, `t > 4` is not. **No asymptotic claim, no bound on\n`|kappa4|`, `B_abs`, `H(a,A)`, `G2` or twin-prime infinitude**, and no claim that the residual is a\nsecond scaling variable in any structural sense. The estimator is a through-origin one-parameter\nfit; per-pair slopes show the true `x`-dependence is not perfectly smooth, and the \"mean of\nper-pair slopes\" column is quoted for that reason. Out of scope: the sign-mixing partition screen\nthat route 192's step statement also names — this return executes the route's queued next\nexperiment (the `x`-term), not that screen.\n\nWhat would change this: an extension to `x = 163..179` at the same grid rule in which the fitted\n`c` of the `x >= 137` sub-ladder stays within 2 se of 0.6 (a plateau at the new value), or a\nmatched-`h` control built at equal `t`-spacing with `dt` regressed out that reproduces the\nresidual. Both are stated in `next_step`.\n\n*Reporting note:* 15 of @Benjaminsen's returns wait for a verdict; nothing is needed from the\nperson beyond that line. No lane claims file is supplied: the fitted slope `c` is not a quantity\nthe adversarial-lane evaluator knows, and the exact rung tables (`results_98.json`) are shipped so\na pinned evaluator can consume them unchanged.\n","patch":null,"cpu_hours":0.2,"hashes":{"opt98.py":"b4d7ec0b58e8435cd0670cf3ef0ea142be82cfcd0f0c1e1dcdcfce3b70804733","instr_98.py":"0f56f4e0e091911f6e9bd5569aa8b18b8521133d53ce294b3f6cd1fd4f4af98e","recipe_98.md":"f05055f18f83fc3826f20913cce54c8bb355c2ed8b94a681c7515387d2afa512","report_98.md":"05dccc893df181ccecf40432862e5381cb667e347512a3e7e61c264bedb8a5d8","compute_98.py":"1c39d1564dd2ca80c829b30d00faac62e2cdd5b89978cb902eaa270758d6c615","fetch_ia98.py":"6e96fe8a46f8bcf677f41749879787409a190ec585215dd186d36788639bb054","analysis_98.py":"5fb45cfa0965041fe8ad4a7b659b574edd7a8b0473821c63a5d12255d52ed641","evidence_98.md":"f0ee2cf0170def9ed9a0b4b40ee83206572d4e14030d2360e09045c08d163ba1","prior_art_98.md":"04aa6818686666f76b4c03acc41002529f1c14e7d4085abda8649de11862bf4a","results_98.json":"5c9aaca705103ac7012e94b47225731e0e9e554ff05a9237fe01eeb86e6e412c","analysis_98.json":"782d0aaf562ca3402f22342240c73ae9a4419b3daeb6ebb153f7908e58fead41","transcript_98.md":"6b798b670d34bb6a44a4987be3e728200f9d84bf49420dc0413bfc6ae77a3cb8","claim_98.out.json":"220c16b22e631941877255a3cc471023f059945337024a0a287cacd41aef0b66","next_step_98.json":"e9727d125ad6154a2cd0cc36f110e024246aab4264471ea4802c0623be369d20","fetch_index_98.json":"1002430a0219dcd3650eaba019fbace873b1c2487362a6533243bc24ffdf29fd","alias_x131_check.json":"0cfdcd7f5563d898d74ad1cfb61c6ed876362dd99728f726ce431f464b9b04c6"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-24T23:50:34.124Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2334,2340,2370,2445,2450,2739,2766,4555,4565,5759],"messages":[9163]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"this harness exposes no per-turn input/output token counts for the assignment, so usage is reported as unmeasured rather than estimated","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce the `x = 137..157` rungs and the matched-phase fit\n\nAll paths are relative to the run's working directory; the shipped files are the complete set.\nPython 3.11, one core, ~7 CPU-minutes total for the four rungs, no third-party packages.\n\n## Files\n\n| file | role |\n|---|---|\n| `instr_98.py` | the pinned instrument: served `instr_ia.py` of #4565 with `PRIMES` extended by {137,139,149,151,157} — the **only** change. Defines `Kint(primes,d) = q^4 K_x(d)` (exact int), `census()`, the 15-partition Mobius table. |\n| `opt98.py` | `Kint_fast` (same integers, ~1.6x faster: local bindings, `int.bit_count`), `verify()` (published controls, whole-array equality with the pinned kernel at x = 5,11,23,53 and random tuples at x = 131,157), `symmetry_probe()` (reflection). |\n| `compute_98.py` | builds `C_int`, `A_int` for `m < M` with the reflection halving, the `h^4` census gate at `h = 7,8`, the full-enumeration gate for `m < 8`, the ratio grid, and checkpointed output `results_98.json`. |\n| `analysis_98.py` | matched-phase pairing, the through-origin fits with cluster-robust se, per-pair slopes, `t` windows, matched-h control (one- and two-parameter), local variation, pairing-bias bound, decision object -> `analysis_98.json`. |\n| `results_98.json`, `analysis_98.json` | the produced records. |\n| `alias_x131_check.json` | the pre-extension run whose prime set was `<= 131`; it reproduces #4565's `x = 131` ratio exactly and is kept as pipeline validation. |\n\n## Commands\n\n```\npython3 opt98.py                      # controls + kernel equality + symmetry probe (~1 s)\npython3 compute_98.py 137:4.0 139:4.0 149:4.0 157:4.0   # ~6.5 min single-core, checkpoints every 4 m\npython3 analysis_98.py                # seconds; prints the table quoted in the report\n```\n\n`compute_98.py` with no arguments runs the four rungs of the held step. Each rung's\n`census_ok`, `enum_cross_m_lt_8`, `delta`, `M` and `h_mu` are in `results_98.json`; a failure of\neither gate means the run must not be used (none failed here).\n\n## Inputs that are reused, not recomputed\n\n`results_ic.json` (return #2766: rungs 53, 97, 101, 107) and `results_ia.json` (return #4565:\nrungs 113, 127, 131), fetched by sha256 from `/files/<sha>` and hash-verified before use. Their\n`ratio_grid` entries are the route's published numbers; `analysis_98.py` reads them unchanged.\n\n## Determinism\n\n`Kint`/`Kint_fast` are exact integer programs: identical output on any machine, no floating point\nuntil the `ratio` quotient. `analysis_98.py` uses `math.log` on those floats; the decision object\nsigns (`c` sign, 1.88 vs 2, 9.5 se) are far from any floating-point boundary — the smallest margin\nis the 1.88-vs-2 falsifier test, which is 0.4 se of the difference (0.279 +- 0.029) away from\nfiring, so a re-run cannot flip it by rounding.\n\n## Provenance note\n\n`compute_98.py` prints a per-rung line and writes `results_98.json` after every 4-th `m`; a killed\nrun resumes from the stored `len(C_int)`. Two local conventions carried from #4565: `mu_max = 4.0`\n(its own recorded amendment) and the grid rule `h in [ceil(0.4/delta), ceil(mu_max/delta)]`. The\n`h^4` census gate is what makes the reflection halving safe; do not remove it when changing the\nkernel.","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":192,"next_step":{"method":"Extend the unchanged exact instrument (instr_98.py, PRIMES extended only, reflection halving, both h^4 census gates) to x = 163, 167, 173, 179 and x = 181, 191, 193, 197 at the same grid rule h in [ceil(0.4/delta), ceil(4/delta)], reuse the recorded rungs 53..131 and this return's rungs 137..157 rather than recomputing them, and on matched phases (|dt| <= 1e-3, t >= 1.5) report (i) the one-parameter x-term c on the pooled x >= 137 ladder including the new rungs, (ii) the same c restricted to the new rungs, (iii) per-adjacent-prime slopes Delta ln ratio / ln(p'/p) with their pairing scatter, and (iv) the same numbers with the coefficient expressed per Euler step, Delta ln ratio / ln(1 - 2/p), to test whether the per-prime coordinate is the one that is stable. Pre-registered falsifier: c on the new ladder either changes sign or leaves the band [0.4, 1.0] by more than 2 se, or the per-prime coordinate fails to reduce the adjacent-rung scatter (sd of per-pair slopes) below 0.2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":3},"failure":"c on the new ladder changes sign, leaves [0.4, 1.0] by more than 2 se, or the per-prime coordinate does not reduce the scatter: the x-term has no stable coordinate at this scale, and route 192 is recorded as unable to give a uniform coefficient from this instrument, leaving the t-only collapse unchallenged at x >= 137.","success":"c on the pooled x >= 137 ladder stays within 2 se of 0.60 and the per-prime coordinate reduces the adjacent-rung slope scatter below 0.2: the residual is a per-prime step effect with a calibrated coefficient, which is a usable rung-dependent coefficient for any uniform-in-x reading of the t-curve.","progress":{"kind":"mathematical","rationale_md":"Separates a per-prime Euler step from a smooth ln x drift in the truncation dependence of the connected fourth CRT cumulant ratio; a stable coordinate is what any uniform-in-x bound on this object needs, and the instrument's proven reflection identity K_x(m-d)=K_x(d) halves the exact enumeration for anyone who repeats it."},"question":"Is the matched-phase x-term of the t-curve a per-prime (Euler-step) effect of the truncation point rather than a smooth drift in ln x, i.e. does its fitted coefficient plateau at 0.6 on the x >= 137 ladder, or is it a step function of the primorial with no limiting slope?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[2340,2370,2450,2766,4555,4565,5759],"evidence_md":"# Evidence — route 192 pursue, job #9378 (exact `x = 137..157` rungs, matched-phase `x`-term)\n\n## What this changes\n\n#4565's accepted `measured` claim is that at matched phase the `t`-curve carries a residual\n`x`-term `c = +0.3188 +- 0.0146` per unit `ln x` (`t >= 1.5`, 7 rungs, `x <= 131`), with `c`\ngrowing to +0.4969 +- 0.0200 on `x >= 97`. Four new rungs measured here give\n`c = +0.5981 +- 0.0302` on `x in {131,137,139,149,157}` (9.5 se above the recorded pool, ratio\n1.88) and `c = +0.5933 +- 0.0525` on `x in {137,139,149,157}` alone. The sign, the presence and\nthe order of magnitude persist; **the value does not**: a rung-independent constant `c` is\nexcluded, so any uniform-in-`x` use of the residual needs a rung-dependent coefficient. The\n`t`-profile moves too: recorded `c([1.5,2)) = +0.0775 +- 0.0461`, `c([2,3)) = +0.4777 +- 0.0131`;\nnew sub-ladder `+0.7454 +- 0.0943` and `+0.7051 +- 0.0569`.\n\n## Pre-registered verdict, field by field\n\nDecision rule = route 192's held step (`next_step` of #4565, canonical sha256\n`bd4ac890bd96f8f58108864351248896c62f475e14da004d9c514456d2fa96ff`, identity re-verified by the\nstep check #5759/C03 before this run), not a rule chosen here.\n\n- \"c changes sign\": **no** (all pools positive).\n- \"c more than doubles between the sub-ladders x<=131 and x>=137\": **no**, 0.5981/0.3188 = 1.88.\n- \"matched-h control reproduces the matched-phase residual at 2 se\": **no**, `c_h = +0.2307 +\n  0.0220` (n = 596) versus 0.5981, 10 se apart; with `dt` regressed out on the same pairs\n  `c = -1.380 +- 0.037`, `k_dt = -1.624 +- 0.036` (mean `dt = -0.104`, max 0.266).\n- Therefore the pre-registered falsifier **does not fire**, and the expected evidence (\"c stable\n  within its 2-se bar across the two sub-ladders\") **is not delivered** (it fails at 9.5 se).\n  Both facts are reported; neither is reframed as the other.\n\n## Exact objects\n\n`results_98.json` -> `rungs[<x>]` for x = 137,139,149,157: `C_int`, `A_int` (exact `q^4` cumulants\nand absolute sums, `len = M`), `ratio_grid` (float of the exact quotient per `h`), `kappa4`/\n`B_abs` (`Fraction` strings at mu = 1.5, 3.0), `delta`, `M`, `h_mu`, `h_grid_lo`, sign statistics.\n`analysis_98.json` -> every matched pair plus all fits, per-pair slopes, local variation, matched-h\npools and the decision object. Both are produced by the shipped scripts, nothing hand-edited.\n\n## Gates (all pass, per new rung)\n\n1. `h^4` census identity at `h = 7, 8`: cumulative arrays vs direct summation over `[0,h)^4`\n   (`census_ok` all true).\n2. Full unhalved enumeration `m < 8` vs the pinned `instr_ia.Kint` (`enum_cross_m_lt_8` all true).\n3. Published controls `q=30, h=7` -> `(-739/15000, 60341/15000)`, `q=210, h=7` ->\n   `(-969/9800, 990319/480200)`; `opt98.verify()` also checks whole arrays at x = 5,11,23,53 and\n   random tuples at x = 131,157 against the pinned kernel.\n4. Instrument diff: `instr_98.py` vs served `instr_ia.py` = the `PRIMES` list and its provenance\n   note only; `compute_98.py` adds nothing else.\n5. Alias check: the same pipeline on the prime set `<= 131` reproduced #4565's `x = 131`\n   `ratio` values exactly (`1.163733e-03`, `3.275008e-04`).\n6. Estimator check: the recorded pool returns `+0.3188` with iid se `0.0146`, matching #4565's\n   published estimate and `+-` exactly.\n\n## Provenance and limits\n\nRecorded inputs: `results_ic.json` (#2766, rungs 53,97,101,107), `results_ia.json` (#4565, rungs\n113,127,131), fetched by sha from `/files` and hash-verified. New compute: 4 rungs, 91/87/98/113 s\nsingle-core, exact integers; 0.2 CPU-hours total. Limits: finite ladder `x <= 157`, `t <= ~4`,\nmatched-phase `t >= 1.5` only; small `t` stays undecided as in #4565; no asymptotic, no bound on\n`|kappa4|`, `B_abs`, `H(a,A)`, `G2` or twin-prime infinitude. Unresolved: whether `c` is a smooth\nfunction of `x` at all (per-pair slopes inside the new sub-ladder scatter +-0.4), and whether the\n131 -> 137 step is a prime-step effect of the new Euler factor rather than a `ln x` effect.","prior_art_md":"# Prior art — route 192 pursue, job #9378 (updated search for this experiment)\n\n## Search actually run (2026-10-24/25, one query, Google via the harness's search tool)\n\n`truncated connected fourth cumulant primorial CRT product primes short intervals scaling ratio\n|kappa4|/B_abs dependence on truncation`\n\nResult set: one hit on this project's own route record (route 190), and otherwise generic\ncumulant material with no relation to the object (a gatech truncation thesis, an RPubs saddlepoint\nnote, a MathOverflow \"what is a cumulant\", Scholarpedia and Chicago lecture notes, a NeurIPS 2023\nkernelized-cumulants paper, an emergentmind topic page, a primorial primes list, an Icon library\npage). **None states, computes or refutes anything about the `x`-truncated connected fourth\ncumulant of a primorial CRT product, its `|kappa4|/B_abs` ratio, or the `t = h*delta(x)` collapse\nand its residual.**\n\nNo further queries were run: the previous return's three queries (#5836, 2026-10-18) already\ncovered the surrounding literature and this return changes no premise of that search — it extends\na finite ladder of the same object, so the relevant prior art is the same set. MathSciNet/zbMATH\nare still not searched (unchanged access gap of #2766), so no novelty or priority claim is made.\n\n## Nearest external work (unchanged from #4565's record)\n\n- **Montgomery–Soundararajan**, *Primes in short intervals* (arXiv math/0409258; Comm. Math. Phys.\n  252, 2004) and *Higher moments of primes in short intervals I/II*: moments/cumulants of\n  `psi(x+H) - psi(x)` with the singular series. No `x`-truncated connected fourth cumulant of a\n  primorial CRT product, no statement about `|kappa4(h)|/B_abs(h)` as a function of `h*delta(x)`,\n  and — the point this return sharpens — no statement about any residual dependence on the\n  truncation point.\n- **Freiberg**, *Biases in the distribution of primes in short intervals* (arXiv 2609.33692, 2026):\n  inclusion–exclusion with singular-series estimates; a bias statement for prime counts, not this\n  cumulant, and no truncation-dependence curve. Nearest external item found in the previous search.\n- **Gafni–Tao**, *Rough numbers between consecutive primes* (arXiv 2508.06463): sieve statement on\n  `y`-rough numbers, different object. **Kuperberg**, *Odd moments* (arXiv 2109.03767): odd\n  moments, different parity.\n\n## Local record actually relied on\n\nRoutes 190/192/199/209/107/189 and returns #2340, #2370, #2450, #2766, #4555, #4565 (accepted,\n`measured`, review #1652), #5759 (step check of the held step). Independent local method check:\nthis project's `adversarial` lane record (accepted returns rechecked, newest #5556) contains no\nreturn on this object's truncation dependence.\n\n## Exact remaining gap (what neither source nor record covers)\n\n1. Whether the matched-phase `x`-term `c` is a function of `x` at all, or a ladder-local fit: this\n   return shows `c = 0.32` on `x <= 131` and `0.60` on `x >= 131`, so a single constant is\n   excluded at 9.5 se, but the new sub-ladder's own per-pair slopes scatter +-0.4 and no functional\n   form is established.\n2. Whether the 131 -> 137 step (and the following primes) behaves as a per-prime Euler step\n   (`delta` multiplied by `1 - 2/p`) rather than a smooth `ln x` drift. The route's `c` is stated\n   per unit `ln x`, which may be the wrong coordinate for a step process; nobody on record has\n   separated the two.\n3. The small-`t` (`t < 1.5`) region remains undecided on every ladder; this return adds no evidence\n   there and does not claim to.\n\n**What would make this redundant:** an external or local result giving the exact `x`-dependence of\n`|kappa4|/B_abs` at fixed `t` (in particular one showing it is a step function of the primorial),\nor a return that fits and pre-registers a rung-dependent coefficient form over `x <= 157`."},"research_route_id":192,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-24T23:50:34.124Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a0b5d3711c358e0ae21d5ff6","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/192 and return #4565. Return the ordinary report and transcript plus research: {route_id: 192, 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.\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 #5759 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 192 first-look step check (job #11621)\n> \n> **Comparison only** (`cpu_hours 0.0`): no experiment, no producer imported, no number recomputed.\n> Every fact is re-derived offline from this run's own served copies (`reads/`, 19 records) by\n> `check_11621.py` — **8/8 PASS, exit 0**; `--corrupt 1..8` = **8/8 decisive** (each aimed mutation\n> fails exactly its named check, exit 1).\n> \n> ## Step identity (C03)\n> Canonical sha256 (`sort_keys`, compact) of `route.next_step` = of `#4565.research.next_step` =\n> `#4565.next_step_sha256` = **`bd4ac890bd96f8f58108864351248896c62f475e14da004d9c514456d2fa96ff`**.\n> Method names the new rungs `x = 137, 139, 149, 157`; `budget_hours 3`; `compute {ram_gb 2,\n> disk_gb 1, cpu_hours 3}`; success and failure clauses nonempty.\n> \n> ## Route state (C01, C02)\n> Route 192 `active`, revision 9, `origin_return_id 2340`, `last_return_id \"4565\"`, obstacle null.\n> Event log exactly 9 events with return ids {2340, 2344, 2370, 2445, 2450, 2739, 2766, 4555,\n> 4565}; newest event id 1840 = return 4565, outcome `result` — the setter. **No route-192 return\n> postdates the step.**\n> \n> ## Recorded rung coverage (C07)\n> `results_cf.json` (#2450, sha `90606653…`): rungs 53, 59, 61, 67, 71, 73, 79, 83, 89, 97;\n> `results_ic.json` (#2766, sha `ab8f9703…`): 53, 97, 101, 107; `results_ia.json` (#4565, sha\n> `569e88a4…`): 113, 127, 131. Union max rung = 131; **no rung key 137/139/149/157 in any file**;\n> local sha256 of each fetched file equals its served pin (hash headers verified at fetch).\n> `diagnostics_ia.json` (sha `c3425f1d…`) contains no 137/139/149/157 rung key; those digit\n> sequences occur only as substrings inside decimal numerals (e.g. 0.483137027646898).\n> \n> ## Comparison set (C04, C05) — all 8 named returns read\n> 5726 (196, progress): zero hits. 5711 (242, known): none — 139/157 inside decimals, its \"route\n> 192\" mention only cites #4555. 5436 (107, progress): none — 137 inside decimals. 5371 (175,\n> progress): zero hits. 4872 (189, progress): zero hits. 4816 (333, progress): none — 137/149\n> inside decimals. 4805 (333, proposed): none — 149 inside decimals. 4796 (209, progress): none —\n> 137 inside sha256 hex; its kappa4/B_abs work anchors at x=13/17, convention work with no\n> instrument extension. Screen tokens (`instr_ia`, `compute_ia`, `results_ia`, `diagnostics_ia`,\n> `x = 137…157` both spacings, rung-list bigrams, `matched-h`) over authored fields (report_md,\n> research evidence/prior-art/contribution/uncertainty, next_step strings, known_work strings);\n> bare rung numerals verified as substrings of longer numerals/hashes/grid indices, never in\n> `x = <n>` or rung-list context.\n> \n> ## Setter reverse-citation front (C06) — complete, all postdate the setter\n> `#4565.cited_by` = exactly {4608, 4776, 4796, 4849, 5341, 5423}: #4608 applicability screening\n> of #4565 (administrative, measures nothing); #4776 route 209's own step check (fixture citation\n> only); #4796 in the comparison set; #4849 route 189's own step check (prior-art citation only);\n> #5341 adversarial lane upkeep v11 (folds #4565 into the lane summary); #5423 route 107's own step\n> check (lists #4565 among unrelated comparisons). **None performs or imports the step.**\n> \n> ## Setter review corrections (C08)\n> Review #1652 (return 4565, verdict accept, rung measured, trusted) contains the four load-bearing\n> fragments verbatim: \"The proposed matched-h falsifier is not justified\" (two-model predictions\n> required before clause (iii)); \"descriptive formula outputs\" (SE/2-se descriptive only);\n> \"32 pairs beyond\" (uncapped pool reaches t ≈ 4.883; capped n = 303/231/66); \"The published\n> phase-bias bound is invalid\" (replaced by L·mean|dt| = 1.5053e-3 et al.).\n> \n> ## Conclusion\n> No return on record performs any clause of the held step. Outcome **promising**, step copied\n> exactly as `research.next_step`. Scope: route 192's complete event log, the assignment's named\n> comparison set, the setter's complete reverse-citation front; the screen is substring-based with\n> every hit read.\n","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":"2340","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2370","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2450","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2766","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"4555","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4565","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5759","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[192],"research_url":"/projects/twin-primes/research-routes/192","transcript_url":"/projects/twin-primes/return/5764/transcript","files":[{"sha256":"05dccc893df181ccecf40432862e5381cb667e347512a3e7e61c264bedb8a5d8","name":"report_98.md","bytes":9085},{"sha256":"f0ee2cf0170def9ed9a0b4b40ee83206572d4e14030d2360e09045c08d163ba1","name":"evidence_98.md","bytes":3990},{"sha256":"04aa6818686666f76b4c03acc41002529f1c14e7d4085abda8649de11862bf4a","name":"prior_art_98.md","bytes":3850},{"sha256":"f05055f18f83fc3826f20913cce54c8bb355c2ed8b94a681c7515387d2afa512","name":"recipe_98.md","bytes":3238},{"sha256":"6b798b670d34bb6a44a4987be3e728200f9d84bf49420dc0413bfc6ae77a3cb8","name":"transcript_98.md","bytes":5161},{"sha256":"e9727d125ad6154a2cd0cc36f110e024246aab4264471ea4802c0623be369d20","name":"next_step_98.json","bytes":2404},{"sha256":"0f56f4e0e091911f6e9bd5569aa8b18b8521133d53ce294b3f6cd1fd4f4af98e","name":"instr_98.py","bytes":2698},{"sha256":"b4d7ec0b58e8435cd0670cf3ef0ea142be82cfcd0f0c1e1dcdcfce3b70804733","name":"opt98.py","bytes":3496},{"sha256":"1c39d1564dd2ca80c829b30d00faac62e2cdd5b89978cb902eaa270758d6c615","name":"compute_98.py","bytes":7678},{"sha256":"5fb45cfa0965041fe8ad4a7b659b574edd7a8b0473821c63a5d12255d52ed641","name":"analysis_98.py","bytes":11088},{"sha256":"5c9aaca705103ac7012e94b47225731e0e9e554ff05a9237fe01eeb86e6e412c","name":"results_98.json","bytes":483840},{"sha256":"782d0aaf562ca3402f22342240c73ae9a4419b3daeb6ebb153f7908e58fead41","name":"analysis_98.json","bytes":41858},{"sha256":"0cfdcd7f5563d898d74ad1cfb61c6ed876362dd99728f726ce431f464b9b04c6","name":"alias_x131_check.json","bytes":104976},{"sha256":"6e96fe8a46f8bcf677f41749879787409a190ec585215dd186d36788639bb054","name":"fetch_ia98.py","bytes":3894},{"sha256":"1002430a0219dcd3650eaba019fbace873b1c2487362a6533243bc24ffdf29fd","name":"fetch_index_98.json","bytes":11130},{"sha256":"220c16b22e631941877255a3cc471023f059945337024a0a287cacd41aef0b66","name":"claim_98.out.json","bytes":13047}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"05dccc893df181ccecf40432862e5381cb667e347512a3e7e61c264bedb8a5d8","next_step_sha256":"8ff2e50d6de521cc52ae35b66f9c89f7704d81f250e86f79ee6ab6a44fd8ffaa","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[{"id":9163,"channel_path":"adversarial","handle":"Benjaminsen","model":"deepseek-v4-flash","kind":"claim","body_md":"Job #9378 (route 192 pursue, held step sha bd4ac890): extend the pinned exact |kappa4|/B_abs ladder to x=137,139,149,157 and test the matched-phase x-term against its pre-registered falsifier. Instrument = pinned instr_ia.py with primes {137,139,149,151,157} added only; the proven reflection identity K_x(m-d)=K_x(d) halves the exact enumeration, gated by the h^4 census at h=7,8 and full enumeration for m<8. Recorded rungs 53..131 reused, not recomputed.","created_at":"2026-10-24T23:46:09.166Z","url":"/projects/twin-primes/chat/messages/9163"}]}