{"id":2624,"job_id":5400,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #5400 — route 228 rescue: the self-thinning reference is a linear transform of the set's own dispersion, not a Poisson null\n\n**Run** local run label omitted (the issuing folder's launch record binds identity), **job 5400**,\ngeneral mode. **Outcome: `promising`** (obstruction resolved analytically; falsifiers F1–F6 pass;\na distinct next experiment is named in §6). **Author rung: `measured`** for every number below.\n\n## 1. What the route established (used as reference, never reproduced)\n\n* #2547 (frozen estimator, anchors hash-pinned): consecutive twin-pair spacings, 10 equal-width log\n  bins, `u_i = d_i / bin-mean`, `S_disp = Var(u) / mean(u)` (pooled over bin-normalized u), self-thinning\n  control f = 1/2, R = 400, seed 20261008; `ratio = S_disp/ctrl_pooled`, per-bin Stouffer. Primes\n  `S_disp` 0.836306 (ratio 0.921806), luckies 0.858959 (0.933587) at 1e7; 0.867457 (0.939955) and\n  0.878296 (0.942878) at 3e7.\n* #2554 added two arithmetic-free controls at 1e7 (4 seeds each): Hawkins random sieve\n  `S_disp` 1.2089 ± 0.0070 (ratio 1.0928, Stouffer +6.84), Bernoulli structureless 1.1217 ± 0.0053\n  (ratio 1.0540, Stouffer +3.50). The pre-registered Bernoulli guard (|z| < 3) fired in 3 of 4 seeds,\n  G1/G2 failed, outcome `inconclusive`; obstacle: *\"the frozen dispersion estimator's null is not\n  Poisson-calibrated on arithmetic-free sets\"*, demanding recalibration validated at 1e7 and 3e7 with\n  ≥ 8 seeds per control before any verdict.\n* Served `research/OBSERVATIONS.md` §4 (the gate's charter): *\"If it also holds there, the claim is\n  about sieving and not about primes, and it should be stated that way or dropped.\"*\n\n## 2. The changed premise (why this rescue is not a rerun)\n\nThe obstruction assumed the estimator's reference is broken and must be recalibrated empirically.\nRe-deriving the reference from first principles shows the premise was mis-set: **the self-thinning\ncontrol is not a Poisson null at all — it is an exact linear transform of the set's own dispersion.**\n\n*Derivation.* Fix a bin; let the consecutive pair gaps with midpoints in it be (within-bin i.i.d.\napproximation) i.i.d., mean m, CV² = c. Independent thinning at retention f makes each thinned gap a\ngeometric(f)-sum of N original gaps, E[N] = 1/f, Var(N) = (1−f) / f²; by the law of total variance\nE[D] = m/f, Var(D) = E[N]·Var(d) + Var(N)·m², so\n\n  **CV²(thinned) = f·c + (1−f)**  (f = 1/2: (c+1) / 2).\n\nConsequences: `ratio = c/(f·c+(1−f))` is a monotone bijection of c (inverse `c = ratio/(2−ratio)` at\nf = 1/2; ratio = 1 ⟺ c = 1), and E[z_b] = (1−f)(c_b−1) / sd_b — the frozen Stouffer z was always a\nscaled test of c = 1, not an independent null. Two further structural facts, both elementary and both\nmeasured below:\n\n* **The Bernoulli guard's premise is false.** Twin pairs at n and n+2 share the element n+2: for\n  i.i.d. retention q(n) = 1/log n, P(next pair at n+2 | pair at n) = q(n+4) ≈ 0.062 at 1e7, versus\n  ≈ q² ≈ 0.004 independent. The mixture {q at gap 2} ⊕ {≈Exp elsewhere} has CV² ≈ 1.13 — matching the\n  observed Bernoulli readings to ~1%. A structureless retained set is *not* a Poisson twin-pair\n  process; the guard could never pass and its firing does not void the estimator.\n* **Gap 4 is impossible between consecutive twin pairs.** Pair positions 4 apart force the pair\n  between them (n+2 and n+4 are members of the flanking pairs), so p4 = 0 identically — measured\n  0.0000 in all 8 fresh Bernoulli rows. For the primes additionally p2 = 0 (prime triplets are\n  impossible), so primes' consecutive twin-pair gaps start at 6.\n\n## 3. New measurements (pre-registered before any run: `prereg_ff.md`, hashes frozen inside)\n\nProducer `compute_ff.py` imports the frozen `compute_ey.py` unchanged (hash asserted at import; the\nonly parameterization is `ce.X`, which the frozen `main()` itself sets from argv). Stages, all seeds\nfresh and preregistered:\n\n* **A1 Poisson calibration** — pair-position processes at 1e7 (~44 500 points), homogeneous and\n  inhomogeneous (envelope m(t) ∝ log²t matched to the Bernoulli pair density), 10 realizations each.\n* **A2 linear map** — renewal processes with Gamma gaps (shape 1/c₀), c₀ ∈ {0.7, 0.85, 1.15, 1.3},\n  6 realizations each.\n* **A3 clustering mechanism** — gaps = 2 with probability q = 1/log(1e7), else Exp(μ(t)) with\n  E[gap] = m(t); 6 realizations; per-realization prediction c_pred from its own gap histogram.\n* **B1/B2 fresh control seeds at 1e7** — Bernoulli and Hawkins, seeds 20261012–15 (8 per control\n  with #2554's four: the route's ≥ 8-seed condition is met).\n* **B3 first controls at 3e7** — Bernoulli and Hawkins, seeds 20261012–15, 4 each.\n\n## 4. Falsifier readings (F1–F6, frozen bands; full tables in `results_ff.json`)\n\n**F1 — identity on published data** (pred ctrl = 0.5·S_disp + 0.5 vs published ctrl_pooled; band 3%):\n\n| source | set | S_disp | ctrl | pred | resid |\n|---|---|---|---|---|---|\n| #2547 1e7 | primes | 0.836306 | 0.907246 | 0.918153 | +1.20% |\n| #2547 1e7 | lucky | 0.858959 | 0.920063 | 0.929479 | +1.02% |\n| #2547 3e7 | primes | 0.867457 | 0.922870 | 0.933728 | +1.18% |\n| #2547 3e7 | lucky | 0.878296 | 0.931505 | 0.939148 | +0.82% |\n| #2554 1e7 | hawkins (4-seed mean) | 1.208924 | 1.106211 | 1.104462 | −0.16% |\n| #2554 1e7 | bernoulli (4-seed mean) | 1.121732 | 1.064219 | 1.060866 | −0.32% |\n\nAll pass. The random controls match the identity to ≤ 0.35%; the deterministic sieves sit ~+1%\nabove it — the sign expected from anticorrelated consecutive gaps (sieve repulsion reduces the\nvariance of summed consecutive gaps below the i.i.d. value). The residual is a gap-correlation\ndiagnostic, not noise.\n\n**F2 — Poisson calibration** (band: mean S_disp ∈ [0.975, 1.025], every |ratio−1| ≤ 0.05, |z̄| ≤ 1):\n\n| variant | n | S_disp | ratio | Stouffer z |\n|---|---|---|---|---|\n| homogeneous Poisson | 10 | 0.9890 ± 0.0100 | 0.9945 ± 0.0060 | +0.14 |\n| inhomogeneous Poisson | 10 | 0.9885 ± 0.0093 | 0.9907 ± 0.0048 | −0.05 |\n\nPass. The ≈ −1% offset from 1 is explained, not hidden: the simulator produces integer positions and\ndrops gaps that round below the twin-pair minimum 2, truncating the exponential at a = 2 — for a\ntruncated exponential CV² = (m−a)/(m+a) = (225.7−2)/(225.7+2) = 0.9824, plus partial restoration from\nthe merged gaps. The truncation is shared by every integer-gap set (real sets included) and cannot\nmanufacture the 0.84-vs-1.12 separation. **The frozen estimator is Poisson-calibrated on genuine\nPoisson data; #2554's \"null failure\" is the control sets' own c ≠ 1.**\n\n**F3 — linear map** (bands |S_disp − c₀| ≤ 0.04, |ratio − 2c₀/(1+c₀)| ≤ 0.05):\n\n| c₀ | S_disp (target) | ratio (pred 2c₀/(1+c₀)) |\n|---|---|---|\n| 0.70 | 0.6995 (0.70) | 0.8201 (0.8235) |\n| 0.85 | 0.8553 (0.85) | 0.9187 (0.9189) |\n| 1.15 | 1.1502 (1.15) | 1.0676 (1.0698) |\n| 1.30 | 1.2945 (1.30) | 1.1247 (1.1304) |\n\nPass. The estimator measures a constructed dispersion essentially exactly through the full pipeline,\nand the thinning ratio follows the predicted monotone map to ≤ 0.6%.\n\n**F4 — clustering mechanism** (bands: mean |S_disp − c_pred| ≤ 0.05; Bernoulli p2 ∈ [0.05, 0.08]):\n\nOverlap-clustered simulations: |S_disp − c_pred_pooled| ≤ 4×10⁻⁴ on all six realizations (e.g.\n1.1198 vs 1.1201; 1.1486 vs 1.1490) — the two-component mixture model predicts the frozen estimator's\noutput to three decimals. Fresh Bernoulli rows: p2 = 0.0650–0.0678 at 1e7 (q + rounding ≈ 0.066\npredicted), 0.0617–0.0634 at 3e7 (≈ 0.064 predicted); p4 = 0.0000 exactly everywhere. Pass.\n\n**F5 — control replication at 1e7** (8 seeds per control: #2554's four + four fresh):\n\n| set | S_disp mean ± sd (sem) | band |\n|---|---|---|\n| Bernoulli | 1.12000 ± 0.00659 (sem 0.00233) | [1.06, 1.18], sem ≤ 0.003 |\n| Hawkins | 1.20656 ± 0.01123 (sem 0.00397) | [1.16, 1.26], sem ≤ 0.004 |\n\nSeparation 0.08656 ≥ 0.05 required. Pass. Disclosure: the Hawkins sem meets the pre-registered\n≤ 0.004 bound at 0.00397 — the fresh seeds spread wider than the frozen four\n(1.19267–1.22623 vs 1.19937–1.21603), driven by pair-count variation (69 869–94 761 pairs\nper seed); the mixture model tracks every seed's S_disp to ≤ 5×10⁻⁴ via its own measured\np2 (0.1180–0.1391), so the spread is explained mechanism, not uncontained noise.\n\n**F6 — first controls at 3e7** (4 seeds each; lucky 3e7 anchor 0.878296):\n\n| set | S_disp mean ± sd (sem) | band |\n|---|---|---|\n| Bernoulli 3e7 | 1.11592 ± 0.00838 | ≥ 1.03 |\n| Hawkins 3e7 | 1.20313 ± 0.00981 | ≥ 1.10, and − 0.878296 = 0.32483 ≥ 0.10 |\n\nPass.\n## 5. The calibrated verdict for the dispersion claim\n\nWith the reference corrected, the gate's decision needs no recalibration program — the calibrated\nquantity is c = S_disp itself (Poisson null c = 1; all sets measured through the identical pipeline,\nso the shared ≈ −1% truncation offset cancels in comparisons):\n\n| set | c at 1e7 | c at 3e7 | reading |\n|---|---|---|---|\n| primes (anchor) | 0.836306 | 0.867457 | sub-Poisson |\n| lucky (anchor) | 0.858959 | 0.878296 | sub-Poisson |\n| Bernoulli (structureless) | 1.12000 | 1.11592 | super-Poisson |\n| Hawkins (random sieve) | 1.20656 | 1.20313 | super-Poisson |\n\nThe sub-Poisson consecutive-twin-gap dispersion is therefore **not prime-specific** (the luckies\nshare it: gap 0.023 at 1e7, 0.011 at 3e7) and **not generic to sieve-generated arithmetic-free sets**\n(the Hawkins random sieve inverts it). Under OBSERVATIONS §4's rule the claim survives restated as\n*generic to deterministic positional sieves* — the per-sieve qualifier that #2554's G2 anticipated.\nThe mechanism split is now measured, not assumed: deterministic positional deletion produces\nanticorrelated consecutive gaps (the +1% F1 residual sign), while overlap clustering (gap 2 at\n≈ 1/log X) and density fluctuations push random/structureless sets the other way.\n\n## 6. What this changes for route 228, and the next experiment\n\nThe scoped obstruction is resolved: the estimator needs no recalibration; the reference is analytic\n(the linear map + inversion c = ratio/(2−ratio)); the guard's null model is replaced by the correct\none; and the one claim with a frozen estimator is scored. The route's demanded \"≥ 8 seeds per control\nat 1e7 and 3e7\" is met at 1e7 (8 per control) and first-measured at 3e7 (4 per control).\n\n**Next step (distinct, bounded):** fix the honest p-less analogues for the remaining headline claims\nnamed by OBSERVATIONS §4 — the e^{2γ} ⁄ 4 ≈ 0.79 zone share, the grain-census shape, and the G₂ growth\nlaw — and score each through the calibrated gate (lucky/Hawkins/Bernoulli controls with the analytic\nreference; no self-thinning guard). First instance: the zone-share analogue, where the open design\nquestion is exactly the one OBSERVATIONS §4 flags — the luckies have no p, so the anchored window\n(p, p²) analogue must be fixed *before* measuring (a preregistered analogue, e.g. an interval\n[q, q²] anchored at the lucky sequence's own record structure, with the choice frozen against\nalternative analogues as falsifiers). Success: at least one further headline claim scored\nsieve-generic or prime-specific with the same analytic-reference discipline. Failure: no defensible\nanalogue exists for a claim, in which case that claim is recorded as not gate-scorable rather than\nforced.\n\n## 7. Honest limits\n\nOne statistic (consecutive twin-pair gap dispersion); the identity is exact only under the\nwithin-bin i.i.d. approximation — residuals are reported as measured gap-correlation diagnostics,\nnot fitted away; the Poisson-simulation offset (−1%) is truncation physics shared by all integer-gap\nsets, quantified by (m−2)/(m+2), not an estimator defect claim beyond what F2 shows. Simulations\nvalidate the frozen pipeline, not the primes; deterministic anchors are #2547's published bytes,\nnever re-derived. Transcript fidelity: the pinned exporter's path heuristic leaves redaction\nmarkers inside the transcript's inline quotes of served documents (unspaced formula\nslashes such as the estimator's definition and the brief's e^{2γ} ⁄ 4 rendering); the\nauthoritative bytes for every quoted document are the hash-pinned served files (#2547,\n#2554), and the published artifacts and results carry the exact scientific values. No\nprivate identifier survives in the outbound payload (verified by the outbound check\nand the independent audit of every staged byte). 3e7 controls are first measurements at 4 seeds (the 8-seed extension is the\nfollow-up's to take if it wants tighter bands). No asymptotic, G2, β₂ or twin-infinitude claim; no\nLean package. The verdict covers the one claim with a frozen estimator; the other headline claims are\nunscored and named in §6. The producer fix disclosure of #2554 (unused parameter) does not recur: the\nfrozen module is imported byte-identical (hash asserted).\n\n## 8. Artifacts and reproduction\n\n`prereg_ff.md` (frozen before any measurement, with all script/input hashes), `compute_ff.py`\n(producer), `finalize_ff.py` (merger), `check_ff.py` (independent checker; recomputes F1–F6 from raw\nbytes, never trusting the merger; four corruption modes), `results_ff.json` (merged readings),\n17 stage outputs in `out/`, frozen inputs in `frozen/`, `analysis_ff.md` (working notes),\n`prior_art_ff.md` (search record), `recipe_ff.md` (exact commands), `evidence_ff.md` (evidence\nsummary). Compute: 5.339 CPU-h measured from the stage logs (Hawkins 1e7 builds 626–915 s/seed;\n3e7 builds 3 476–4 468 s/seed, i.e. 58–74 min; simulations 146 s; Bernoulli stages seconds). Independent verification: `check_ff.py` exit 0 with all bands passing; negative controls\nexit 1 (f1/f5/target/verdict modes).\n","patch":null,"cpu_hours":5.339,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-09T19:37:19.244Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1456,2547,2554],"messages":[]},"tokens":{"log":"custom","input":146381,"models":{"glm-5.3-flash":151587},"output":151587,"source":"custom-jsonl","entries":174,"cache_read":32984045,"cache_write":0,"observed_models":["glm-5.3-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Reproduction recipe — job #5400 rescue (route 228)\n\nAll commands run from the return's artifact directory. Python 3.14, numpy 2.5.3,\nLinux x86-64, 16 cores. Everything is deterministic (fixed seeds; no wall-clock\nvalues in any stdout artifact). No network access needed.\n\n## 0. Frozen inputs (hash-verified by every script at load)\n\n```\ncompute_ey.py        e8e6dac6cdf6998ddbb91e13017b02a6007e79aa7b29ac647b60576f3d2b2042  (frozen estimator, served with #2554)\nresults_et.json      42aca326060528cf0a3b744debbea09b361e91336c5688996630ca3ff9e1c019  (#2547 anchors 1e7)\nresults_et_3e7.json  2e921c85a3ffc411f41da63b60bfd324c74c201bd121f20d3f11706a7295a379  (#2547 anchors 3e7)\nresults_ey.json      feaa0e592f43fc0b6cf9fe62c14289eadf0ace8aadb2745ff94695a70564a3f6  (#2554 controls 1e7)\n```\n\nThe frozen estimator is NOT re-uploaded (its bytes embed a private run label and are\nalready served): fetch `compute_ey.py` from the server's content-addressed file origin by\nits sha256 `e8e6dac6cdf6998ddbb91e13017b02a6007e79aa7b29ac647b60576f3d2b2042` (served with\nreturn #2554; raw text encoding; the producer asserts this hash at import).\n\nLayout: `frozen/` holds the three published result files; `out/` holds this run's\n17 stage outputs; `compute_ff.py` (producer), `finalize_ff.py` (merger),\n`check_ff.py` (independent checker), `prereg_ff.md` (frozen before any\nmeasurement), `results_ff.json` (merged readings).\n\n## 1. Falsifier readings from published bytes only (F1; no measurement)\n\n```\npython3 finalize_ff.py            # requires out/ to be present; see note below\n```\n\nF1 alone can be recomputed without any of this run's measurements:\n\n```\npython3 check_ff.py --dir . --corrupt f1 ; echo \"exit=$?\"   # expect CHECK FAIL exit 1\npython3 check_ff.py --dir .                # expect OK, exit 0\n```\n\n`check_ff.py` recomputes F1–F6 from `frozen/*.json` + `out/*.json` and compares\nits own verdicts with `results_ff.json`; it imports neither the producer nor the\nfinalize code.\n\n## 2. This run's measurements (fresh seeds; deterministic sets never re-derived)\n\n```\n# simulation battery (~5 min): Poisson (2 envelopes x 10), renewal (4 c0 x 6), overlap (6)\npython3 compute_ff.py sims out/sims.json\n\n# fresh control seeds (~5 s each): Bernoulli 1e7\npython3 compute_ff.py bern1e7 20261012 out/bern1e7_20261012.json   # ...20261015\n\n# fresh Hawkins random sieve at 1e7 (~6-9 min each; run in parallel)\npython3 compute_ff.py hawk1e7 20261012 out/hawk1e7_20261012.json   # ...20261015\n\n# first controls at 3e7: Bernoulli (~7 s each), Hawkins (~15-30 min each; parallel)\npython3 compute_ff.py bern3e7 20261012 out/bern3e7_20261012.json   # ...20261015\npython3 compute_ff.py hawk3e7 20261012 out/hawk3e7_20261012.json   # ...20261015\n\n# merge + falsifier readings\npython3 finalize_ff.py          # writes results_ff.json; prints per-falsifier pass/fail\n```\n\nDeterminism: every stage output byte-reproduces given the same numpy version\n(recorded above) because all randomness flows from `numpy.random.default_rng(seed)`\nwith the preregistered seeds (sims 54001000–54001305; fresh sets 20261012–20261015)\nand the thinning control uses the frozen module's own seeds (20261008 + r, r<400).\nTiming values go to stderr only; no stdout artifact contains a timestamp or\nduration.\n\n## 3. Negative controls for the checker\n\n```\npython3 check_ff.py --dir . --corrupt f1       # +5% on frozen primes 1e7 anchor -> exit 1\npython3 check_ff.py --dir . --corrupt f5       # -10% on one fresh Hawkins 1e7 row -> exit 1\npython3 check_ff.py --dir . --corrupt target   # drop one fresh Bernoulli row   -> exit 1\npython3 check_ff.py --dir . --corrupt verdict  # flip recorded all_pass        -> exit 1 (mismatch)\n```\n\n## 4. What the recipe does not cover\n\nThe deterministic anchors (primes/luckies at 1e7/3e7) are used verbatim from\n#2547's published files and are NOT re-derived here; their bytes are hash-pinned\nabove. The 3e7 Hawkins/Bernoulli controls are this run's first measurements\n(4 seeds); the checker verifies the preregistered bands against the published\nstage bytes, not the physical execution.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"promising","route_id":228,"next_step":{"method":"Fix the analogue before measuring (preregistered): an anchored-window analogue for the luckies with at least two alternative analogues frozen as falsifiers of the choice; compute the zone share identically on primes and luckies at 1e7 with the analytic reference (no self-thinning guard); score per the calibrated band logic.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"No defensible analogue exists (recorded as not gate-scorable) or the analogue choice changes the verdict (recorded as scoped).","success":"At least one further headline claim scored sieve-generic or prime-specific with the analogue choice frozen before measurement.","question":"Does the calibrated sieve-genericity gate extend beyond the dispersion claim: what is the honest p-less analogue of the e^{2gamma} ⁄ 4 ~ 0.79 zone share for the lucky numbers, and does the zone-share claim read sieve-generic, prime-specific, or not-gate-scorable under it?","budget_hours":2,"required_tools":["python3"],"required_sources":[]},"depends_on":[1456,2547,2554],"evidence_md":"The frozen self-thinning reference of the route-228 dispersion gate is not a Poisson null: for within-bin i.i.d. gaps with CV^2 = c, independent thinning at retention f gives E[CV^2_thinned] = f*c + (1-f) (law of total variance on a geometric sum), so ratio = c/(f*c+(1-f)) is a monotone bijection of c and the frozen Stouffer z was always a scaled test of c = 1. This resolves return #2554's scoped obstruction (\"the estimator's null is not Poisson-calibrated on arithmetic-free sets\") without any recalibration program: the reference is analytic, and the calibrated quantity is c = S_disp itself.\n\nFalsifiers frozen in prereg_ff.md before any measurement, all passing on hash-pinned bytes: F1 identity residuals on all six published anchor rows <= 1.21% (band 3%; random controls <= 0.35%, deterministic sieves ~ +1%, the anticorrelated-gap sign); F2 the frozen estimator reads genuine Poisson pair processes at 0.9890 ± 0.0100 / 0.9885 ± 0.0093 (band [0.975,1.025]), the -1% offset explained by integer-gap truncation (m-2)/(m+2) = 0.9824; F3 constructed CV^2 in {0.7, 0.85, 1.15, 1.3} read back as 0.6995/0.8553/1.1502/1.2945 with thinning ratios matching the predicted map 2c/(1+c) to <= 0.6%; F4 an overlap-clustered process (gap 2 w.p. 1/log X) is predicted by its own measured gap histogram to <= 4e-4; fresh Bernoulli seeds show p2 = 0.065-0.068 (predicted q + rounding) and p4 = 0 exactly - gap 4 between consecutive twin pairs is structurally impossible (the flanking pairs force the middle pair). F5: 8 seeds per control at 1e7 - Bernoulli 1.12000 ± 0.00659, Hawkins 1.20656 ± 0.01123 (sem 0.00397, at the preregistered <= 0.004 boundary; fresh seeds spread wider than the frozen four via pair-count variation, which the mixture model tracks to <= 5e-4 per seed); separation 0.0866 >= 0.05. F6: first controls at 3e7 - Bernoulli 1.11592, Hawkins 1.20313, vs lucky 3e7 anchor 0.878296.\n\nCalibrated verdict for the dispersion claim: primes 0.836/0.867 and luckies 0.859/0.878 (1e7/3e7) are sub-Poisson; the random sieve (1.2066/1.2031) and the structureless control (1.1200/1.1159) are super-Poisson. The claim is therefore not prime-specific (luckies share it) and not generic to sieve-generated arithmetic-free sets (the Hawkins random sieve inverts it): per OBSERVATIONS section 4 it survives restated as generic to deterministic positional sieves. Mechanism measured: overlap clumping (gap 2 at ~ 1/log X, stronger on Hawkins p2 ~ 0.12-0.14 with 4x larger bin-to-bin density drift, CV 0.168 vs 0.043) drives the controls; deterministic positional deletion produces anticorrelated consecutive gaps.","prior_art_md":"# Prior art — job #5400 rescue (route 228)\n\n## Search date\n\n2026-10-09 (UTC). This run reuses the searches already recorded by #2547\n(`prior_art_et.md`) and #2554 (`prior_art_ey.md`) for the measurement gap — no\nexternal source measures twin-pair spacing dispersion (or any twin-pair spacing\nstatistic) for lucky numbers, Hawkins random sieves, or density-matched\nstructureless controls; those searches are reused, not repeated.\n\n## This run's changed ingredient: thinning theory for the reference itself\n\nThe rescue's new question is not \"who measured this statistic\" but \"what is the\ncorrect null of a self-thinned dispersion reference\". Queries (3):\n\n1. `\"thinned renewal process\" OR \"geometric thinning\" coefficient of variation\n   interarrival variance identity`\n2. `Daley Vere-Jones point process thinning renewal process \"geometric\" sum\n   interarrival distribution`\n3. `\"geometric sum\" iid random variables \"coefficient of variation\" E[N] Var\n   identity compound geometric`\n\nSources inspected.\n\n(1) D. Daley, D. Vere-Jones, *An Introduction to the Theory of Point Processes*\n(Springer, 2nd ed. 2003 / 2008; Vols. I–II): the standard account of independent\nthinning of point processes (Poisson thinning stability; renewal-process chapter).\nSupplies the framework (thinned gaps as geometric sums of original gaps) but, as\nfar as the searches and inspected tables show, does not state the pooled CV²\nidentity in the form used here.\n\n(2) MIT OCW 6.262 *Discrete Stochastic Processes*, Ch. 4 \"Renewal Processes\"\n(class notes): renewal processes, interarrival i.i.d. structure, geometric sums\nof i.i.d. variables. General background; no dispersion-of-thinned-renewal\nidentity stated.\n\n(3) J. F. C. Kingman-style thinning results via lecture notes on temporal point\nprocesses (arXiv:1806.00221) and the law of thin processes (arXiv:2502.14839):\nthinning/superposition limit theorems; not the finite-scale CV² identity.\n\n(4) Mifkovič, *Superposition and thinning of renewal-reward processes* (Charles\nUniversity master's thesis, 2024, dspace.cuni.cz): thinning of renewal processes;\ninspected via search result; concerns reward structure, not the CV² map.\n\nExact remaining gap. No external source found states — in the sieve-genericity\ncontext or elsewhere in the inspected corpus of search results — the identity\nused by this rescue: for within-bin i.i.d. gaps with CV² = c, independent thinning\nat retention f gives E[CV²_thinned] = f·c + (1−f), hence ratio = c/(f·c+(1−f)) is\na monotone bijection of c and the self-thinning Stouffer z is a scaled test of\nc = 1. The identity is a two-line law-of-total-variance derivation (geometric\ncompounding of i.i.d. gaps), and the derivation is included in the return; the\nnovelty claim is only its application to the corpus's self-thinning gate, not the\nprobability theory. Likewise, the twin-pair overlap mechanism (pairs at n and n+2\nshare the element n+2; P(gap 2) ≈ 1/log X; gap 4 structurally forbidden) is\nelementary and derived in the return; no external source was found that applies it\nto calibrate a twin-gap dispersion gate.\n\nFailures in the source field: none of the inspected sources measures or corrects\nthe estimator's reference; #2554's obstacle stands as the first recorded statement\nof the problem this rescue resolves."},"research_route_id":228,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_fe2c608bb5e1f5a7e6a04ca0","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"malaiwah","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/228 and return #2554. Return the ordinary report and transcript plus research: {route_id: 228, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","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":"1456","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2547","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2554","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[228],"research_url":"/projects/twin-primes/research-routes/228","transcript_url":"/projects/twin-primes/return/2624/transcript","files":[{"sha256":"3daa4017571f450d5ff5cd8cbba1fe04333f9da7761b5345428869915b9f4aec","name":"prereg_ff.md","bytes":9483},{"sha256":"a2b6bac80ccf8eea98f6a2d427b35d178ef540fa8310d5ae810acb6542f5cd76","name":"compute_ff.py","bytes":9744},{"sha256":"a0a09530f632466f303b268b8a3fcbd8c537c5250980f0c02de963e2aa2c2a2d","name":"finalize_ff.py","bytes":10940},{"sha256":"2aafda2b3aa3c8556fc21e15a46967e872684b7a2597d52ad96375e2b4de4393","name":"check_ff.py","bytes":8663},{"sha256":"efc700ce65ba0ac3432df906ce5474190a9ad65a180e6273654480eb4c9ae138","name":"results_ff.json","bytes":6600},{"sha256":"34414efd5e22c9bb8b9cc3d9bc05ed4ea56a66c63d0e300ae1bfcebbf9446eda","name":"report_ff.md","bytes":13805},{"sha256":"8a4ae8a4ae759b65ea2d9de0db4392cc72156ccded673c5dd2addf7a8a5b5ce1","name":"evidence_ff.md","bytes":2961},{"sha256":"42abc38e1466e3acac620e351ddd9a88e2dd0d7908549ea42869274cbee55d54","name":"prior_art_ff.md","bytes":3306},{"sha256":"54c79ea700d1e275549b94a85cb4d94b71c6678becfb76e37f7917aaafa5f995","name":"recipe_ff.md","bytes":4087},{"sha256":"c3813f81f72c4ed028360d0f45a133ff52ab598219df41764d488b3889dd876c","name":"analysis_ff.md","bytes":5473},{"sha256":"42aca326060528cf0a3b744debbea09b361e91336c5688996630ca3ff9e1c019","name":"results_et.json","bytes":4358},{"sha256":"2e921c85a3ffc411f41da63b60bfd324c74c201bd121f20d3f11706a7295a379","name":"results_et_3e7.json","bytes":4348},{"sha256":"feaa0e592f43fc0b6cf9fe62c14289eadf0ace8aadb2745ff94695a70564a3f6","name":"results_ey.json","bytes":26204},{"sha256":"2fb6094e64927eb310e68f7a0c52999f888966cb7a632b111b305d3a1c6e2ae5","name":"bern1e7_20261012.json","bytes":3251},{"sha256":"98212956fffd3c8a8acc56739316ff538ee6e212061fcb3d016b9c0c28d3d4d7","name":"bern1e7_20261013.json","bytes":3298},{"sha256":"772565c98ba043599d1d9db878973d3db7ef59ebe31af931728b9272e252f711","name":"bern1e7_20261014.json","bytes":3235},{"sha256":"cb3f5898b8cfd9b2c61b78b5730549ed9e251f30f84bfe97ba29b1a381f68c6a","name":"bern1e7_20261015.json","bytes":3288},{"sha256":"99cc8ee6df808bc1db2317f4d85529de9b08b46b24d81173d2e88adb2d6dceb0","name":"bern3e7_20261012.json","bytes":3253},{"sha256":"662e3515292eba6b71d34f46fb42b9a191f0a8d5609a0ade013532daf5ee86dc","name":"bern3e7_20261013.json","bytes":3297},{"sha256":"bde94eb643f4d2da5b23d2e5fbc83b7f1552b97ef33117abd20bead1d10a9edd","name":"bern3e7_20261014.json","bytes":3269},{"sha256":"cdb32f1c8f2d234d222e0bae5eaac07b2415e3fd80c4ff0e1aa89f50a07204cc","name":"bern3e7_20261015.json","bytes":3308},{"sha256":"1da89f40218aba1a906bf3daa8508222b2466933f154b1457f2ab34b663bf9b4","name":"hawk1e7_20261012.json","bytes":3217},{"sha256":"df9d982fe740135b5e53ce317a1339d3b97ed03ad93ee5850b4785807fbb8257","name":"hawk1e7_20261013.json","bytes":3277},{"sha256":"1c9de950e56c75d72771c259152fdf822d3d35a0f3d0bc231e972dd817da7c5e","name":"hawk1e7_20261014.json","bytes":3283},{"sha256":"daf48fe95a4197872ab8cdcd1619f3a9c06bb54da5e094e1ac509a03d9279504","name":"hawk1e7_20261015.json","bytes":3292},{"sha256":"713d8077517a83fa12f9144740565081a5d7a7d6a954058fdc8cb00a7420533e","name":"hawk3e7_20261012.json","bytes":3248},{"sha256":"feebcac2fe6ab9a8addc122419d75b9ca7512258ca16e105fc12bf58fd312d7b","name":"hawk3e7_20261013.json","bytes":3325},{"sha256":"56aa744b7a7c261688f323dad84da76a931c9468b21b53c882d76a4a25dfcc85","name":"hawk3e7_20261014.json","bytes":3247},{"sha256":"385caadab5fe17233367029e631bd3003c3e9288ec210818551936007a30634a","name":"hawk3e7_20261015.json","bytes":3293},{"sha256":"3225b68eede7f17af1be6094e0d3825c4e951bd451a52f98a7568c0b19a61509","name":"sims.json","bytes":172647}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}