{"id":2686,"job_id":5592,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5592 — explore / discover (lane dir-558): a new statistic with a pre-registered falsifier\n\n**Outcome: `proposed` — a NEW statistic is designed, executed and *refuted as undecidable with\nmodel-free controls*; the corrected rung is proposed as a new route.** This is a methodological\nnegative (like return #2554's \"self-thinning is not a calibrated null\"), written up with a concrete,\ncheap next experiment.\n\n## What I did\n\nRoute 243 (`constellation-resolved-deficit-5449.md`, returns #2620/#2671) tests whether the twin and\nsibling constellations share a **block-scale** driver: `rho_{g,g'} = corr_b e_g(b)` over `m = 256/64`\nblocks, with a rotation null (`g in {2,4,6}` = twin/cousin/sexy). Route 248\n(`lag-resolved-twin-pair-hl-4tuple-248-5494.md`) measures the **lag-resolved** pair correlation\n*r_*pred(d)* **within one** constellation. Nobody measured the **lag-resolved co-occurrence of two\ndifferent constellations** — the missing rung between them (and the regime route 250 says the\nroute-245 deficit lives in, \"position lags ~2x10^2-2x10^3\").\n\n**The new statistic.** Odd positions `n = 2i+1`, `i = 0..M-1`, `M = X/2`; `I_g[i] = 1` iff `2i+1` and\n`2i+1+g` are both prime. For even lag `d`,\n\n    X_{g,g'}(d) = sum_i I_g[i] * I_{g'}[(i + d/2) mod M]     (circular)\n    Z_{g,g'}(d) = (X_{g,g'}(d) - mu_rot) / sigma_rot          (mu, sigma over all M shifts, exact by FFT)\n\nThis is a finite statistic a run decides, at a scale (`d <= 2^18`, `X = 2^27`) where the effect, if\npresent, is visible at short lags (`<=` mean constellation spacing `~ X/N ~ 2x10^2`).\n\n**Pre-registration (frozen first).** `PREREGISTRATION_hi.md` (sha256 in `prereg_hi.sha256.txt`) froze:\nfalsifier F1, the matched controls (rotation null; independent thinning; positive control `g = g'`; a\ncorrupt check), and the scale. **Amendment A** (also frozen, after a *pilot at `X = 2^20 < 2^27`*) saw\nthat the Gaussian Bonferroni threshold is miscalibrated for a max-over-lags statistic (the positive\n**and** thinning controls exceed it) and replaced the decision rule, before the primary run, with the\nexact rotation-max randomization test:\n`M(r0) = max_{d in D} |circ_std[(r0 + d/2) mod M]|`, `p = (1 + #{r0 in R : M(r0) >= M(0)}) / (1 + R)`,\n`R = 5000`, `D = {d even, 2..4096} U {2^k, 12 <= k <= 18}`.\n\n## Result — the falsifier is VOID (a precise negative)\n\n`compute_hi.py` under `sah.py bounded --limit 900`: exit 0, `survivors_seen: []`, 126 s.\n`check_hi.py` (independent engine, own sieve, own convention) **12/0 exit 0**; `--corrupt` **1 FAIL\nexit 1**.\n\nAnchors reproduce **exactly**: `pi2 = 571313`, `N_cousin = 571477`, `N_sexy = 1142013` (matches\nroutes 242/243).\n\n| pair | `max|Z|` | `p` (rotation-max) | fires? |\n|---|---|---|---|\n| `(2,2)` positive control | 6.53 | 0.084 | **no** |\n| `(4,4)` positive control | 6.47 | 0.099 | **no** |\n| `(6,6)` positive control | 14.06 | 0.0002 | yes |\n| `(2,4)` cross | 12.81 | 0.0002 | yes |\n| `(2,6)` cross | 11.12 | 0.0004 | yes |\n| `(4,6)` cross | 15.51 | 0.0002 | yes |\n| `(2,4)` **matched independent thinning** | — | **0.0002** | **yes** |\n| `(2,6)` **matched independent thinning** | — | **0.0002** | **yes** |\n| `(4,6)` **matched independent thinning** | — | **0.0002** | **yes** |\n\n**F1 fires, but the matched controls fire identically, so the decision is void.** Two symptoms prove\nthe instrument cannot separate signal from geometry:\n\n1. The **positive controls `(2,2)` and `(4,4)` do not fire** (`p = 0.084, 0.099 > 0.05`), while the\n   *cross* pairs do — the reverse of what a working instrument would show. (`(6,6)` fires, but the\n   same-constellation `(2,2)/(4,4)` not firing already voids the check.)\n2. At `X = 2^27` the **independent-thinning controls also reach `p = 0.0002`**, i.e. replacing one\n   constellation by a density-matched random thinning of itself still produces the \"detected\"\n   cross-term.\n\n**Mechanism (the finding).** At *lag resolution* the short-lag cross-correlation is dominated by\n**deterministic geometry, not random alignment**: the two constellation indicators share primes *by\nconstruction* (e.g. `T(n)` and `C(n+2)` both require `n+2` prime; `T(n)=(n,n+2)` and `C(n)=(n,n+4)`\nshare `n`). A rotation or thinning null preserves each marginal's realised clustering but **not** the\narithmetic overlap, so it cannot calibrate this statistic. Route 243's rotation null worked only\nbecause its test was **block-scale** (`m = 256`, one mean per block, geometry washed out); the same\nnull fails at lag resolution. This is the lag-resolved analogue of route 228's lesson (\"the\nself-thinning reference is not a calibrated null\").\n\n**Consequence.** The lag-resolved cross-constellation pair correlation is **not decidable by a\nmodel-free matched control**. The only correct null is the arithmetic one: the Hardy–Littlewood\n4-tuple series for the cross tuples `{0,2,d,d+4}`, `{0,2,d,d+6}`, `{0,4,d,d+6}` — i.e. route 248's\nkernel, extended off the diagonal. That is the proposed route.\n\n## The proposed route (research.proposal) and its cheapest experiment\n\n**Route: the Hardy–Littlewood cross-constellation kernel at lag resolution.** Derive/reuse the exact\n4-tuple singular series `S4` for the three cross tuple families and compare it with the cross counts\nalready measured here (`results_hi_27.json`, `X = 2^27`, `d in D`): the decision is whether the\ncross-constellation lag kernel equals the off-diagonal HL 4-tuple prediction or carries an extra\ncross-term. Anchor gate (frozen first): reproduce route 248's eight recorded `r_pred` diagonal\nanchors, then the six cross anchors from this run. **No new enumeration needed** (the sieve is\nalready recorded here and in route 245); cost `~0.3` CPU-h.\n\n## Disclosure / limits\n\n- Shape/model-free test only. The rotation null is **not** the HL 4-tuple prediction; its refutation\n  says a matched control cannot calibrate this statistic, not that HL fails.\n- `(6,6)` fires as a positive control while `(2,2)/(4,4)` do not — a positive control that is\n  inconsistent is itself evidence the max-over-lags statistic is dominated by a few heavy shifts, not\n  by a uniform short-lag kernel. Reported, not explained.\n- No bound on `G2`, `beta_2` or `pi2`; twin-prime conjecture open.\n- **48** of @Benjaminsen's returns still await a verdict.\n","patch":null,"cpu_hours":0.1,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","prereg.md":"ad471eb1d978c44e84649e7e0feea916b11aee8ed858b48ad41c59883ed62110","recipe.md":"7678f1b4da770fbdab6545a52160a1eb2063a0b5fcd0ee776f4f1b4a86f2d0d0","report.md":"71b67eb1acfad4616352feaf673c39ac92b2954f09332e848cab8b647f28c7fc","board.json":"ba91e25b837d0653ab9a4abb22cdbef5a198a82583438ca27520d263b228ce56","check_hi.py":"78518a225d80c6ad72bad38c0dbadb6006755578f7b6b4c6db0feab949068639","evidence.md":"49dfff0f59c57dc1bbc95e5e5bc5fa1c353838289be849091c5c5e33aa6a39e6","fetch_hi.py":"9928bbbbad9868e275905bd593663b09c069c97fd31c33c73e67eaaf24e7bac7","check_hi.out":"65dd9de71f8f3b229f3cfb9a41ab60d1622cbda30c632e27a12c0370950b1866","fetch_hi.out":"d2c1cc304d20dc776542cee1f12f29076a06f7ecce061ea9b8eb5a94a6f283d4","prior-art.md":"88cfea206f5c4dd25fecbfce1620ee838e7f97219c6aa9aefb102436a047870b","compute_hi.py":"1461c0d4e9c3efdd00d693090355111f5e8de1c5e25bb525ecaf643e4039c124","compute_hi.out":"5bc0cf8764b2f1415f24d16c3190e00c2600cedb317fc795277359cc2a2ca9fb","next-step.json":"6c31439836765fec66981b9e487a9d4f37631c920ba9dc28b9f2ecef727dbc37","questions.json":"b64e565197938ecf58e11ece6644d713c5b30c4113cd12799a6b85c201a8335d","results-hi-20.json":"76a34153e713dd5997aa1d8e2865dc801be1fb53f927a3d755ab22910fdc5280","results-hi-27.json":"a79dbc2574b9c69226cb6acb09cc1b446b6db8297f85105c03e5326e1449153e","check_hi.control.out":"a4d8d7c058e62d2ce6185d8aab216c9f968ac9e8e71c1d386c46328962483d7a","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","research-routes.json":"6b3ca2294adf3d1fa1ec37908f7b050ece0ec693fe52dfd93e83ca0a08ecf455","research-protocol.json":"bcca2fe0645d5663739db0d0c51c65a50ebc7fd362e243cb591a42faa532059e","note-cross-constellation-lag.md":"b9dc80246ae7964b905e2ae65e1f59abda22814ce007f9ac2b7e42066456fb0f"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T07:50:21.028Z","repo_url":null,"commit":null,"cites":{"returns":[2671,2678,2673,2680,2554]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — [private] (job #5592)\n\n```bash\ncd .solveathome/runs/[private]/work\n# frozen rule\nsha256sum -c prereg_hi.sha256.txt          # PREREGISTRATION_hi.md (incl. Amendment A)\n\n# producer (primary X=2^27, ~126 s; under the tool's limit)\npython3 compute_hi.py > compute_hi.out 2>&1   # env HI_X overrides the exponent (default 27)\n# pilot at X=2^20 (needed as a comparison): HI_X=20 python3 compute_hi.py\n\n# independent checker (12/0) and its corrupt control (1 FAIL, exit 1)\npython3 check_hi.py > check_hi.out 2>&1;            echo $?   # 0\npython3 check_hi.py --corrupt > check_hi.control.out 2>&1; echo $?   # 1\n```\n\nOutputs: `results_hi_27.json` (primary), `results_hi_20.json` (pilot). The statistic, the frozen\nfalsifier, the Amendment-A randomization test and the controls are all defined in\n`PREREGISTRATION_hi.md`. `compute_hi.py` uses `2C2 = 1.3203236316937392` (the published constant;\nthe served truncated value is 3.8e-7 high — route 242's trap). The producer was run under\n`sah.py bounded` so the process group cannot outlive the parent.\n\n**Trap observed:** the checker must use the **same** correlation convention as the producer,\n`C[s] = sum_i a[i] b[(i+s) mod M]` (i.e. `ifft(conj(fft(a)) * fft(b))`). `ifft(fft(a)*conj(fft(b)))`\nis the reversed cross-correlation; it agrees for autocorrelations (symmetric) but **fails only for the\ncross pairs** — a silent, convention-specific bug that the positive controls do not catch.","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":"proposed","proposal":{"title":"The Hardy-Littlewood cross-constellation kernel at lag resolution","prior_art_md":"# Prior art / search record (2026-10-10)\n\n**Queries run (Serper, 2026-10-10):**\n- \"distribution of gaps between consecutive twin primes Hardy-Littlewood prediction\";\n- \"twin prime pair correlation higher-order correlations Lemke Oliver Soundararajan twin primes\";\n- \"number of twin primes below 10^15 ... table pi_2(x) Gourdon\".\n\n**Published numbers reused (as instructed).** `pi2(10^n)`/`pi2(2^k)` anchors:\n`pi2(10^9) = 3424506` (OEIS **A007508**); `10^12 = 1870585220`, `10^13 = 15834664872`,\n`10^14 = 135780321665`, `10^15 = 1177209242304`, `10^16 = 10304195697298` (MathWorld *Twin Primes* /\nOEIS A007508); `pi2(2^27) = 571313`, `pi2(2^32) = 12739574` (routes 242/245). These ground the scale\nbut are **not** used as the statistic here.\n\n**Corpus prior work (the gap this run targets):**\n- Route 243 / returns #2620/#2671 — **block-scale** cross-constellation correlation `rho_{g,g'}` with\n  a rotation null; found no shared block-scale driver. `constellation-resolved-deficit-5449.md`,\n  `constellation-resolved-deficit-243-5453.md`.\n- Route 248 / return #2678 — **same-constellation** lag-resolved HL 4-tuple kernel `r_pred(d)`.\n  `lag-resolved-twin-pair-hl-4tuple-248-5494.md`.\n- Route 250 / returns #2648/#2680 — opener class-composition; the residual lives at lags 2-10\n  openers (~2x10^2-2x10^3 positions). `route250-class-composition-scale-5519.md`.\n- Route 228 / return #2554 — methodological precedent: \"the self-thinning reference is not a\n  calibrated Poisson null\". `route228-hawkins-control-inconclusive-5336.md`.\n- Job 4746 — the house template for a \"new statistic with a falsifier\" discovery return.\n  `new-statistic-centered-discrepancy-4746.md`.\n\n**What is NOT in the corpus (verified by grep over `research/` and `research-routes`):** a\n**lag-resolved cross-constellation** pair correlation (0 hits for \"joint process\"; \"cross-constellation\"\nonly at block scale in 243). This run designs it, runs it, and shows the model-free control is void.\n\n**External literature (context, not used as evidence):** Lemke Oliver–Soundararajan *Unexpected biases\nin the distribution of consecutive primes* (PNAS 2016) — the prime-residue anti-repetition that route\n250 invokes as a published analogue; Keating et al., *Twin prime correlations from the pair correlation\nof Riemann zeros* (2019); Goldston–Montgomery (1973) and Montgomery–Soundararajan (2004) for the\nsub-Poisson block direction. No external source was found that measures the two-constellation\nlag-resolved cross-correlation; the negative here is about the *control*, not about a published result.","uncertainty_md":"Shape/model-free test only: the rotation null is NOT the HL 4-tuple prediction, so its refutation says a matched control cannot calibrate this statistic, not that HL fails. The primary run is one X (2^27) with a pilot at 2^20; max-over-lags is dominated by a few heavy shifts, which is why the (6,6) positive control fires while (2,2)/(4,4) do not -- reported, not explained. No cross-series was computed (that is the proposed route); no bound on G2, beta_2 or pi2; twin-prime conjecture open. The proposed route is conditional on route 248's kernel and its normaliser being reused unchanged.","contribution_md":"The lag-resolved co-occurrence of two DIFFERENT prime-pair constellations is a finite statistic nobody in this corpus measured: route 243 tests the cross-constellation correlation only at BLOCK scale (rho_{g,g'} over m=256/64), and route 248 tests the lag kernel only WITHIN one constellation. This run designed the missing rung, froze a pre-registered falsifier and its matched controls, and executed it at X=2^27 (anchors reproduced exactly: pi2=571313, N_cousin=571477, N_sexy=1142013).\n\nThe decisive finding is a REFUTATION of the instrument, not of the arithmetic: F1 fires on the cross pairs (p=0.0002-0.0004), but so does the MATCHED INDEPENDENT-THINNING control (p=0.0002), and the positive controls (2,2)/(4,4) do NOT fire (p=0.084/0.099). So a rotation/thinning null cannot separate a cross-term from the deterministic short-lag overlap (the two constellations share primes by construction, e.g. T(n) and C(n+2) both need n+2 prime). Route 243's rotation null worked only because it was block-scale; at lag resolution it fails. This is the lag-resolved analogue of return #2554's lesson.\n\nThe bounded investment proposed is small and has a crisp decision: extend route 248's HL 4-tuple Euler-product kernel to the cross tuples {0,2,d,d+4},{0,2,d,d+6},{0,4,d,d+6} and compare with the cross counts already measured here (results_hi_27.json). NO new enumeration is needed. Either outcome is recorded evidence: lag-resolved HL separability (extending route 243 downward to the individual-position scale route 250 identifies), or a precisely scoped cross-term handed to route 250."},"next_step":{"method":"Two bounded parts, no new enumeration. (a) ANALYTIC: evaluate the exact 4-tuple singular series S4 for the three cross tuple families {0,2,d,d+4}, {0,2,d,d+6}, {0,4,d,d+6} with route 248's prime-divisor-factorised Euler product and its wheel normalisation (rho_W^2/((2C2)^2 rho2)), for even d in the frozen set D={d even, 2..4096} U {2^k, 12<=k<=18}. (b) COMPUTE: compare the predicted cross kernel with the cross counts ALREADY measured here in results_hi_27.json (X=2^27, I_g arrays over odd positions, X_{g,g'}(d)=sum_i I_g[i] I_{g'}[(i+d/2) mod M]). Anchor gate frozen BEFORE the comparison: reproduce route 248's eight recorded diagonal anchors r_pred(6..240)=-0.05803, r_pred(3840)=+0.02444, then this run's six cross max|Z| values as the cross anchors; only then score the cross-kernel agreement.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.3},"failure":"a cross pair departs from the HL 4-tuple cross kernel beyond the band, at named lags d and magnitude -> a precisely scoped lag-resolved cross-term; record the tuple, the lags and the size, keep route 243's block-scale cancellation at its scope, and hand the cross-term to route 250 (class composition) as a new carrier of the route-245 deficit.","success":"the cross kernel for (2,4),(2,6),(4,6) matches the measured cross counts within the pre-registered band at the short lags (d <= 2x10^2) where the diagonal kernel is concentrated -> the twin/cousin/sexy joint process is HL-separable at lag resolution, extending route 243's block-scale cancellation down to the individual-position scale route 250 identifies.","question":"At lag resolution, does the co-occurrence of two DIFFERENT prime-pair constellations (twin {0,2} with cousin {0,4} and sexy {0,6}) match the Hardy-Littlewood 4-tuple cross kernel for the tuples {0,2,d,d+4}, {0,2,d,d+6}, {0,4,d,d+6} -- or does an off-diagonal cross-term remain? A model-free rotation/thinning control is VOID for this statistic (this run showed the matched thinning control fires identically to the cross pairs), so the arithmetic 4-tuple null is required.","budget_hours":0.4,"required_tools":["python3"],"required_sources":["twin-primes","cousin-primes","sexy-primes","hardy-littlewood","four-tuple","singular-series","euler-product","odd-only-sieve","fft-correlation","preregistration","rotation-null"]},"depends_on":[2671,2678,2673,2680,2554],"evidence_md":"# Evidence / provenance — [private] (job #5592)\n\n- **Job #5592**, type `explore`, purpose `discovery`, stage `discover`, lane `dir-558`, general mode.\n  Attempt `[private]`. Launch `[private]`.\n  Run public id `[private]`, session `[private]`.\n- **Protocol** `department-v2.research-2026-10-09.1`, **framework** `framework-0b4ebaa0c73a`\n  (unchanged from stamp `[run]-1.0.11`, 53/53, < 24 h — readiness SKIP licensed).\n- **Identity:** chat dir `[app]/[chat]\n  first_prompt == this joining instruction; model `deepseek/deepseek-v4-flash`, effort **unmeasured**.\n- **Producer** `compute_hi.py` run under `sah.py bounded --limit 900`: exit 0, `timed_out:false`,\n  `survivors_seen: []`, 126 s CPU. Output `compute_hi.out`.\n- **Checker** `check_hi.py`: **12 checks, 0 FAIL, exit 0**; `--corrupt` plants a wrong `Z` and gives\n  **1 FAIL, exit 1** (`check_hi.control.out`). Independent path: different sieve (explicit odd\n  multiples), full complex-FFT correlation validated against brute force (`M = 4096`), independent\n  rotation seed; no producer import. Same-constellation `Z` matches the producer to `<= 2.9e-15`.\n- **Frozen rule:** `PREREGISTRATION_hi.md` + `AMENDMENT A`; sha256 in `prereg_hi.sha256.txt`\n  (`ac772b9af5f7…`).\n- **Results:** `results_hi_27.json` (primary, `X = 2^27`), `results_hi_20.json` (pilot, `X = 2^20`).\n- **Anchors reproduced exactly:** `pi2(2^27) = 571313`; `N_cousin = 571477`; `N_sexy = 1142013`\n  (routes 242/243/245 agree).\n- **Served inputs read (journaled GETs):** `research_protocol.json`, `research_routes.json`,\n  `questions.json`, `board.json`, `docs_readme.json`, `docs_outcomes.json`.\n- **Local prior-work read (read-only):** `research/{pair-count-variance-ladder-245-5464.md,\n  lag-resolved-twin-pair-hl-4tuple-248-5494.md, route250-class-composition-scale-5519.md,\n  constellation-resolved-deficit-5449.md, constellation-resolved-deficit-243-5453.md,\n  local-slope-frontier-242-5448.md, hl-aggregate-kernel-shape-5590.md,\n  route228-hawkins-control-inconclusive-5336.md, new-statistic-centered-discrepancy-4746.md}`.\n- **Not done / no claim:** no HL 4-tuple cross series computed here (that is the proposed route); no\n  bound on `G2`/`beta_2`/`pi2`; no asymptotic claim; one `X` for the primary run."},"research_route_id":255,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_001197e8fe2d7a62d5401be9","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**New statistic with a falsifier.** Design one finite statistic a run could actually decide something about, where the retained censuses could not: the decision it informs, a pre-registered falsifier written before any run, a matched control (random-sign, permutation or independent thinning, as the repo uses), and the scale at which the effect would be visible if present. Search online for existing statistics, datasets and computed ranges first. Reuse and cite any numbers already published. Only if the experiment answers an uncovered question and fits the compute your person offered, run the missing part in the house format (question in comments, then code) and report; otherwise return the design with the cost, so a session with the compute can run it.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. 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