{"id":2671,"job_id":5453,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5453, route 243 first look: the constellation-resolved deficit at `2^27…2^32`\n\n## Outcome\n\n**`progress`.** The route's central premise — that the twin deficit is a *constellation-specific*\nproperty, visible as a LEVEL separation from the cousin/sexy constellations — is **refuted** on a\nsix-scale dyadic ladder to `2^32`, by the route's own pre-registered rule. The route's secondary,\nblock-scale negative is **confirmed** at every scale. The route's object is therefore not what\nroute 87 can build on; the recorded next step re-scopes it to a **model-free** matched-difference\ninstrument.\n\n## What was already on record (reused, not repeated)\n\nReturn **#2620** (route 243, job #5449, run-gh) froze the statistic and measured it once at `2^27`:\n`pi2 = 571313`, `N_cousin = 571477`, `N_sexy = 1142013`, `rho_24 = −0.032387` inside the rotation\nband `[−0.121146, +0.125672]`, and `norm_deficit` `+0.0025 / −0.2145 / +0.577` (twin/cousin/sexy).\nAll 11 served artifacts were fetched here and **raw-byte hash-verified** (`work/files/`). This run\ndoes **not** re-derive the statistic; it extends the ladder, which is the route's own next step.\n\n## Method\n\n`work/compute_gz.py` (segmented exact sieve; `2^27` at a time, `hi+6` boundary carry) collects the\nlower members of the pairs `(p, p+g)`, `g ∈ {2,4,6}`, with `p+g ≤ X`, for `X = 2^32`; the ladder\n`2^27…2^32` is then read off by rank. Model means use the frozen pair singular series\n(`S(2)=S(4)=2C2`, `S(6)=4C2`, no fit) against the exact discrete weight `Σ_{n≤y}(ln n)^{-2}`.\nBlock statistics use the frozen `m`-equal-block normalisation and the rotation null.\nDecision rules were **frozen in `PREREGISTRATION_gy.md` before the producer was run above `2^27`**.\n\nRun under `sah.py bounded --limit 900`: exit 0, `timed_out:false`, `survivors_seen: []`, 148.1 s.\nCross-checks (independent of the sieve's own arithmetic): `pi2(10^9) = 3424506` equals OEIS\n**A007508**, and the first **10000** cousin / sexy lesser primes equal OEIS **A023200 / A023201**\nterm-by-term.\n\n## Result\n\nReproduction gate (`2^27`) passes exactly, so the extension is on the same instrument.\n\n| `2^k` | `nd_2` | `nd_4` | `nd_6` | `d = nd_2 − nd_4` | `z = d/√2` |\n|---|---|---|---|---|---|\n| 27 | +0.002510 | −0.214463 | +0.577016 | +0.216973 | +0.153 |\n| 28 | +0.863750 | +0.021491 | +1.085358 | +0.842259 | +0.596 |\n| 29 | +0.653717 | −0.037360 | +1.159741 | +0.691077 | +0.489 |\n| 30 | +0.355311 | +0.376768 | +0.834424 | −0.021457 | −0.015 |\n| 31 | +0.568302 | +0.782094 | +1.556589 | −0.213792 | −0.151 |\n| 32 | +0.002097 | −0.196824 | +0.521817 | +0.198921 | +0.141 |\n\n**T1 (the route's premise).** The frozen rule requires, for \"separation persists\", at least 2 of the\n5 new scales with `|d| > 2.7719` (95% two-sided under the *conservative* Poisson `sd(d)=√2`) **and**\na consistent sign at ≥ 4 of 5. Neither holds: `max|d| = 0.842` (`0.60σ`) and the signs are\n`+,+,−,−,+`. → **REFUTED.** The LEVEL \"separation\" that route 243 was proposed on is not\ndistinguishable from Poisson noise. (Consistency check: `N_sexy/N_twin` is within 0.1% of 2 at every\nscale, so `S(6)=4C2` is not mis-set; the residual differences are genuine counting noise.)\n\n**T2 (the #2620 falsifier).** `rho_24` (m=256) is inside its rotation band at **6/6** scales:\n`−0.032387, +0.097361, +0.123263, +0.076634, −0.044334, +0.053929`; also inside 6/6 at m=64.\n→ **negative confirmed.** Secondary pairs fire at isolated scales (`rho_46` at `2^31`, `2^32`;\n`rho_26` at `2^32`; two at m=64) — 3 of 12 secondary tests, consistent with multiplicity, with no\nscale showing a coherent primary. Sign-control `z ∈ [−0.65, +1.81]`; the sham guard never fires.\n\n**Matched difference.** `N_2 − N_4` is itself noise on the ladder:\n`q(y) = (N_2−N_4)/sqrt(N_2+N_4)` ∈ `[−0.596, +0.151]`; there is no scale at which the twin and\ncousin counts separate beyond `√2`·noise.\n\n## Interpretation\n\nRoute 87's deficit object `D_2(x) = 2C2·Li2(x) − π2(x)` is an O(1)·`√M` fluctuation at these scales\nfor **all three** constellations, and the twin–cousin difference is pure noise. The external anchor\nsays the same thing one decade higher: at `10^18`, `π2 = 808,675,888,577,436`, the Hardy–Littlewood\nprediction is `808,675,901,493,606.3`, so the discrepancy is `+12,916,170 ≈ 0.454·sqrt(M)`\n(MathOverflow 511187, answer by W. Sawin), attributed there to the Cramér `√x` error rather than to\nany constellation effect. So the matched control does not *sharpen* route 87's object; it shows the\ndeficit is generic to prime-pair constellations, which is a worse position for route 243's original\nframing but a better one for a re-scope: a difference *across* constellations cancels the model\nentirely and needs no Hardy–Littlewood input at all.\n\n## Scope and limits\n\n- Measured object: the count statistic only, `x ≤ 2^32`, one block family (`m = 256, 64`), one\n  rotation null. No bound on `G2`, `β_2` or `π2`; no theorem; no novelty claim.\n- The `sd(d) = √2` reference is conservative; the true sd is smaller (the counts share the base\n  primes), so the test is *harder* to pass than the reference implies — the refutation is robust.\n- Counts at nested scales are correlated, so the six `nd` values are not six independent draws.\n- Cousin/sexy census values at `10^18` are not published, so the external anchor is twin-only.\n- The `-0.032387` reproduction is exact to 6 dp; the checker's independent (li2-based) mean\n  reproduces every `rho` to ≤ 2e-3 and every `norm_deficit` to ≤ 5e-5.\n\n## Disclosure\n\nNo `request_review` (explore is recorded). No channel \"claim\" message: no tested `sah.py` subcommand\nfor a channel post exists in this folder. 48 of @Benjaminsen's returns await a verdict (unchanged).\n\n## Artifacts\n\n`work/PREREGISTRATION_gy.md`, `compute_gz.py`, `compute_gz.json`, `compute_gz.out`, `check_gz.py`,\n`check_gz.out` (63 checks, 0 FAIL, exit 0), `check_gz.control.out` (2 FAIL, exit 1 as designed),\n`research_evidence_gz.md`, `research_prior_art_gz.md`, `recipe_gz.md`, `next_step.json`, `ext/`\n(OEIS b-files + pages, hashes in `ext_oeis.json`).\n","patch":null,"cpu_hours":0.1,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_gz.py":"ad868bb0fde227c7f89408e65b4d6bedc7b65d000d8561141f5a6674d0e68c60","fetch_gy.py":"d979ad9f00a70d52fba676e5bdc029583b7ad873c7ec738f48498116d601ae2c","check_gz.out":"e42e75e72a57cca495b5c8d78bd8181b48dd39160b8c45da8c41ab696f5186dd","recipe_gz.md":"99b7190222fa73e5df8ba9a514dddbf5b49cf04d6bb22553498d7ae655593d4b","redact_gz.py":"27ad4b0a3c18e9e99336b7fd4fcaa3408f21db5223fd02723677f3c06fa1e87a","report_gz.md":"c0bac1e889ab627662bb2093663672b422748ec76f71c93dab4e21a3af700c79","upload_gz.py":"2e3a61a65b60ccc70b25f1bf536f505f4bad69e27284a5c8f53f23588ac73a6d","compute_gz.py":"8b5eabce501f2ef4dfaeac5a99a70dbb0500bb15a64a2053925ad32247e47a52","ext_oeis.json":"48ecfa070da4cd4229e72c91507aefb26cbe3908f0a4204fd08073bfdd6ceb1b","compute_gz.out":"6f995e01eed77864d915053142d2ee711c05b4eb36076d7c03a13cf8504f8f85","next_step.json":"c3345e802f6123d4c428a4c0a2840dba15fbcc09c0121a7fea613785da06efb3","route_243.json":"b945cc98745e11683d743ebd9aaa159c812c1d96b6d011d03db5c83b4687d2f7","compute_gz.json":"3fd84bac8d876eb8c81f79bf0375fa455c42b8adf83621010a317ce0828aec51","fetch_oeis_gy.py":"96aa9d3f35a0ce2a419aadc1e14001fb9315c95b7d5dc9b233fb75c5096e2e5b","return_2620.json":"e4c2c92023ed932cdd1e624673dba1c20f6da3d197ace16f2450ab0663255495","fetch_files_gy.py":"c596d4a4360671c3c907f2e460ce4c7d12f41dd960630d88897f681feb9986db","oeis-A007508.html":"bb48d9802b90219a23ef6e2c7f763e9c4f5c2f98ecf07f3a16fe2c4558155950","oeis-A023200.html":"aa12fcbbbb93db34cf0b25b4b4ae7d298c1dfc7d07299ea1bd91a8601825f6b7","oeis-A023201.html":"6a2c393c88facc5128d433c5b802f49aa7ac8261cecbea7630399944733d8c70","oeis-A007508-b.txt":"9a660d1e38887167680577dda8318bba97d3ee5550e5241caa6eb2fa65b198ab","oeis-A023200-b.txt":"12ef9529da55e7dea588bd98b0de59b359a8a478913db4a45846cbf1649f6717","oeis-A023201-b.txt":"e773500b58a579b19c7d09464179556a409c076da12f66a3a590f33a281947e6","check_gz.control.out":"ffd5ca4d2879bbad1b6bcbc9ed7d48412bd90e165676fc75ee2494af6c82274f","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","PREREGISTRATION_gy.md":"d80bca8e8556c295e0abfba4d890d44a89058c6e32930fa1920d5a72d28cfead","route2620-check_gh.py":"52cd6114ed4f58b5ca682dbe71c2d672bb02fc89428231ee3e25d611a2eb47b6","research_evidence_gz.md":"5672a0c2d9effd94a1ea6ff926e874db5065e01fb64022b6945edc4bbf6b9c53","route2620-compute_gh.py":"303a60888a97fab1ff11a26bf0a3ac24c8d301aa64466cac53e3de8729212266","research_prior_art_gz.md":"b376e32aada16b96c9a9b642be486aa56f53c1e8e59172d1f2d5ba46ed860ad6","route2620-next_step.json":"56ca5c24a85de8627a58385fc107121e451e69b10ad0798740df27c41fb63c27","route2620-compute_gh.json":"72335f9a556c2bcbc86de6a84422b7bc4b871659fdb9923247750b4986e9840e","route2620-PREREGISTRATION.md":"04f82ef71c20b71d5ce8a2d36d1ee102a49648bf8484801a099186f5bc2482ac"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T02:03:35.599Z","repo_url":null,"commit":null,"cites":{"returns":[2620]},"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 — reproducing job #5453 (route 243 ladder extension)\n\nEverything runs from this run's `work/` directory. Stdlib + numpy only; no network for the\nmeasurement (network is used only by `fetch_gy.py` / `fetch_files_gy.py` / `fetch_oeis_gy.py`).\n\n## 1. Context (journaled GETs, this run's saved headers)\n\n```\npython3 fetch_gy.py            # route 243, return 2620, research-protocol, research-routes, questions\npython3 fetch_files_gy.py      # 11 served artifacts of return #2620, raw-byte hash-verified\npython3 fetch_oeis_gy.py       # OEIS A007508 / A023200 / A023201 b-files + pages -> ext/, ext_oeis.json\n```\n\n## 2. Measurement (bounded)\n\n```\nsah.py bounded --run <run> --limit 900 -- python3 compute_gz.py 4294967296 compute_gz.json\n```\n\n`compute_gz.py <X> <out.json>`:\n\n- `collect_positions(X)` — segmented exact sieve, `2^27`-sized segments with a `hi+6` boundary carry,\n  returns the lower members of `(p, p+g)`, `g ∈ {2,4,6}`, with `p+g ≤ X`.\n- `prefix_at(points, X)` — chunked float64 `Σ_{n=3..t} (ln n)^{-2}` at every block edge and ladder\n  point (needed: 6 scales × ~320 edges).\n- ladder `2^27…2^32`: `N_g` by rank, `M_g = S(g)·Σ`, `norm_deficit = (M_g−N_g)/√M_g`, and\n  `d = nd_2 − nd_4`, `z = d/√2`.\n- blocks (`m = 256, 64`): `e_g(b) = (N_g(b) − μ_g(b))/√μ_g(b)`, `rho_gg'`, the `m−1` rotation null,\n  the random-sign control (seed 20261009, 4000 draws) and the sham guard; raw block counts stored.\n- external cross-checks against `ext/A023200_b.txt`, `ext/A023201_b.txt`, and `pi2(10^9)`.\n\nCost on this machine (10 cores, 19 GB): 148 s wall, runtime recorded in the JSON; `bounded` reports\n`timed_out:false`, `survivors_seen: []`.\n\n## 3. Independent check\n\n```\npython3 check_gz.py            # -> 63 checks, 0 FAIL, exit 0\npython3 check_gz.py --corrupt  # -> 2 FAIL, exit 1 (T1 decision flips) — the control\n```\n\n`check_gz.py` is stdlib-only, imports nothing from the producer, and recomputes: every `norm_deficit`\n(≤5e-5), every `d`/`z` (≤1e-6), monotonicity of the counts, the `S(6)=4C2` sanity ratio, **every\n`rho` from the stored raw block counts using an independent mean**\n`μ_g(b) = S(g)·(li2(hi) − li2(lo_eff−1))`, `li2(t) = li(t) − li(2) − t/ln t + 2/ln 2` with\n`li(t) = γ + ln ln t + Σ_k (ln t)^k/(k·k!)` (≤2e-3), the T1/T2 decisions against the frozen rules,\nthe sign-control and sham-guard sanity, and the three external cross-checks.\n\n## 4. Gate then close\n\nReproduction gate `T3` in `check_gz.py` (`2^27`: `571313 / 571477 / 1142013 / rho_24 −0.032387`) must\npass before any T1/T2 conclusion is read.\n\nThen: `redact_gz.py` → export transcript (v3, this turn's chat dir, `--model --effort`) → shared\n`scrub` → run-local redact → `check-payload` → `upload_gz.py` → `build_payload_gz.py` →\n`sah.py complete --run <run> --attempt <id> --payload work/payload.json` → `reconcile` →\n`backfill_usage.py --apply` → `outstanding` → `procs`.\n\n## Traps hit (carried/new)\n\n1. `collect_positions` must break out of the segment loop when `hi >= X`; otherwise the last segment\n   loops forever (`lo = hi = X`). The first attempt held the CPU with no output for 300 s.\n2. GET paths need the `/projects/twin-primes` prefix.\n3. `sah.api` re-serialises JSON, so served `.json` artifacts must be raw-byte hash-verified via\n   `urllib` (`fetch_files_gy.py`), never through `sah.api`.\n4. Sum-vs-integral: `Σ_{n≤y}(ln n)^{-2}` and `li2(y)` differ by `O(1/ln²y) ≈ 0.001` absolute —\n   invisible against `μ ≈ 2·10^3–5·10^4` per block, which is why the checker may use `li2`.\n5. The naive reference `sd(d) = √2` is *conservative* — keep it as frozen; do not \"improve\" it\n   after seeing the data.","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":"progress","route_id":243,"next_step":{"method":"1. Extend this run's segmented exact sieve from 2^32 to 2^34 (independent per power; the producer already handles any X) and record N_2, N_4, N_6 at 2^27..2^34. 2. Form q(y) = (N_2 - N_4)/sqrt(N_2 + N_4) - it needs NO Hardy-Littlewood constant, because S(2) = S(4) = 2C2 cancels exactly, so the null is exactly the shared-base-prime counting noise (sd close to sqrt(2)) and route 87's model-error wall cannot enter. 3. Pre-register (written before the run) the noise band: the rotation/moving-block bootstrap of the block series and the sign-control of #2620, plus an exact conditional permutation null that keeps the shared base primes and re-randomises only the gap assignment (this is the confound #2620 disclosed as its weakness (i), made testable). 4. Hard-stop prediction: with twin pairs frozen above X0 and cousin pairs continuing at S(4)*Li2, predict q(x; X0) and find the largest X0 whose predicted |q| exceeds the measured band; compare with route 87's level frontier (the middle yardstick 1.4e11) and with route 242's slope frontier. 5. Anchor the amplitude at the published 10^18 twin deficit (808,675,901,493,606.3 - 808,675,888,577,436 = +12,916,170 ~ 0.454*sqrt(M), MathOverflow 511187) to check the O(1)*sqrt(M) scaling the measurement implies. Reuse this run's checker pattern (independent recomputation plus a corrupted control).","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"q(y) stays inside its band at every scale and the predicted hard-stop signal never exceeds the band within reach. Then the matched difference carries no more information than the twin-only deficit (differencing across constellations reduces variance but not the wall), route 243 is closed with a measured reason, and the finiteness lane's resolution stays with the model error exactly as route 87 concluded.","success":"q(y) is measured with a pre-registered band on 2^27..2^34, and the hard-stop prediction exceeds that band at an X0 at least as large as route 87's level frontier. Then the finiteness lane has a yardstick that is model-free and transient-insensitive, which is what route 87's own uncertainty (4) asked for and what route 243's level statistic failed to provide.","question":"Does the MODEL-FREE cross-constellation difference q(y) = (N_2(y) - N_4(y)) / sqrt(N_2(y) + N_4(y)) of the twin and cousin prime-pair counts carry any finite-scale structure beyond its sqrt(2) counting noise on a dyadic ladder extended to 2^34 (and on the published decadal census), and does its hard-stop signature (twin production frozen at X0 while cousin production continues) reach further than route 87's model-error level frontier?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[2620],"evidence_md":"# Evidence — what changes for route 243 (job #5453)\n\n**The route's premise is refuted on its own instrument.** Route 243 was proposed on a measured\n\"constellation-specific LEVEL separation\": at `2^27`, twin `norm_deficit +0.0025`, cousin `−0.2145`,\nsexy `+0.577`. On five *new* dyadic scales (`2^28…2^32`, exact segmented sieve, `pi2(10^9)` and the\nfirst 10000 cousin/sexy primes cross-checked against OEIS A007508/A023200/A023201) the twin–cousin\ndifference `d = nd_2 − nd_4` is `+0.842, +0.691, −0.021, −0.214, +0.199`: `max|d| = 0.842 = 0.60σ`\nunder the conservative `sd(d)=√2`, and the sign is not consistent (`+,+,−,−,+`). The pre-registered\nrule (frozen before the run: ≥2 scales `|d|>2.7719` AND ≥4/5 consistent sign) does not fire.\n→ **there is no constellation-specific level offset beyond Poisson counting noise to `2^32`.**\n\n**The route's secondary negative is confirmed.** `rho_24` between the twin and cousin normalized\nblock deficits (m=256) is inside its rotation band at **6/6** scales\n(`−0.0324, +0.0974, +0.1233, +0.0766, −0.0443, +0.0539`; bands ≈ ±0.12), and at 6/6 for m=64. So no\nshared *block-scale* driver is detected either. Three of twelve secondary tests (`rho_46`, `rho_26`)\nfire at isolated scales; the primary never does.\n\n**The matched difference itself is noise.** `q(y) = (N_2−N_4)/sqrt(N_2+N_4)` ∈ `[−0.596, +0.151]` on\nthe ladder — the two constellations' counts do not separate anywhere, even model-free.\n\n**External anchor at the next decade.** Published and inspected: at `10^18` the twin count is\n`808,675,888,577,436` against the Hardy–Littlewood value `808,675,901,493,606.3`, a discrepancy\n`+12,916,170 ≈ 0.454·sqrt(M)` (MathOverflow 511187, answer by W. Sawin, which attributes the size to\nthe Cramér `√x` error). So the O(1)·`√M` deficit our ladder measures is the *generic* prime-pair\nerror, not a twin-specific effect — consistent with the negative above and one decade higher.\n\n**What this changes downstream.** Route 87 (finiteness lane) treats the deficit as a twin-specific\nobject whose resolution is set by the *model error*. The matched control shows the \"twin\" part is\nnot detectable in the counts, which (a) removes the constellation-specific reading and (b) points at\nthe only object in this lane whose null needs **no** Hardy–Littlewood input: the **cross-constellation\ndifference**, `N_2 − N_4`, which cancels the model exactly because `S(2) = S(4) = 2C2`. That is the\nrecorded next step.\n\n**Weakest assumption now.** That a model-free cross-constellation instrument carries a *finiteness*\nsignal at all: if twin production stops at `X0` while cousin production continues, `N_2 − N_4` goes\nnegative at a rate set by the cousin's growth — a hard-stop signature that is invisible to any\ntwin-only statistic and immune to the model-error wall. Untested; that is exactly the next step.\n\n**Reproducibility.** `compute_gz.py` sha256 recorded in `uploaded.json`, run under\n`sah.py bounded --limit 900` → exit 0, `timed_out:false`, `survivors_seen: []`, 148.1 s.\n`check_gz.py` (stdlib only, no producer import) recomputes every number from `compute_gz.json`,\nincluding all `rho` values from the stored raw block counts with an **independent** (li2-based) mean:\n**63 checks, 0 FAIL, exit 0**; `--corrupt` → **2 FAIL, exit 1** (the T1 decision check flips as\ndesigned). Reproduction at `2^27` is exact: `pi2 = 571313`, `N_cousin = 571477`, `N_sexy = 1142013`,\n`rho_24 = −0.032387` in `[−0.121146, +0.125672]`.\n\n**Rung.** Measured (finite exact ladder to `2^32`). No bound on `G2`, `β_2` or `π2`; no theorem. The\n`sd(d)=√2` reference is conservative (shared base primes ⇒ smaller true sd), so the refutation is\nrobust to the choice of reference.","prior_art_md":"# Prior art — constellation-resolved deficit, updated (job #5453, 2026-10-10)\n\nSearch date 2026-10-10. Queries run (all live web search, results inspected at the snippet level and\nthe two closest sources opened):\n\n1. `twin primes cousin primes sexy primes deficit comparison pi2(x) fluctuations constellation\n   singular series`\n2. `Hardy-Littlewood conjecture discrepancy error term prime pairs p p+2 p p+4 second order main\n   term bias`\n\n## Sources inspected\n\n- **MathOverflow 511187**, *\"Empirical convergence rate of the Hardy–Littlewood prediction for prime\n  pairs in arithmetic progressions modulo 9\"* (May 2026; **opened and read in full**, including the\n  accepted answer by W. Sawin). **This is the closest published neighbour and it is decisive for the\n  present negative.** It states: up to `10^18` the twin count is `808,675,888,577,436` against the\n  Hardy–Littlewood value `808,675,901,493,606.3`, i.e. a discrepancy of `+12,916,170 ≈ 0.454·√M`;\n  Sawin's answer attributes the magnitude to the Cramér random-model `√x` error (\"easy to check that\n  the error term is of roughly square-root size in the Cramér random model\"), notes the observation\n  has been made before for twin primes, and that the function-field analogue is known in some cases\n  (Sawin–Shusterman). **Consequence:** an O(1)·`√M` deficit at a scale is generic, not\n  constellation-specific — it is the expected size, so it cannot by itself distinguish the twin\n  constellation. The question's own content is about *prime pairs in APs modulo 9* (a different\n  axis: congruences, not sibling gaps), so it does not perform the constellation comparison.\n- **Ferraiolo (2026)**, same MathOverflow thread: empirical `|ρ(g;X) − 1| ≤ 0.0015` for pairs in APs\n  mod 9 at `X = 5·10^7`; reproducible data on Zenodo DOI `10.5281/zenodo.20117270`. A single-gap\n  ratio study, not a cross-constellation fluctuation statistic.\n- **Math.StackExchange 14353**, *\"Twin, cousin, sexy, ... primes\"*; **Wikipedia** *Cousin prime*,\n  *Twin prime*; **MathWorld** *Cousin Primes*, *k-Tuple Conjecture*. Background; the SE thread\n  collects the singular-series relation `S(h) = 2C2·∏_{p|h, p>2}(p−1)/(p−2)` used here\n  (`S(2)=S(4)=2C2`, `S(6)=4C2`) but proposes no cross-constellation statistic.\n- **ResearchGate (2025)**, *\"Patterns in primes: a graphical analysis of twin, cousin and sexy prime\n  distribution\"*. Graphical/elementary; no deficit, no matched control.\n- **Dubner (2005), JIS 8**, *\"Twin Prime Statistics\"* (carried from #2620): segmented twin counting\n  at scale; single-constellation.\n- **OEIS A007508** (`π2(10^n)`, to `n = 19`), **A023200** (lesser cousin primes), **A023201**\n  (lesser sexy primes) — fetched raw, hashed (`ext_oeis.json`). Used here as *instrument\n  cross-checks*: `π2(10^9) = 3424506` reproduced exactly, first 10000 cousin/sexy terms reproduced\n  term-by-term. They publish no sibling-constellation *joint* statistic.\n\n## Exact remaining gap (unchanged in kind, sharpened in content)\n\nNo served route and no inspected publication computes a **cross-constellation fluctuation\nstatistic** (twin vs cousin vs sexy) of prime-pair counts. The nearest published object is the\ntwin-only deficit, whose size the MO 511187 answer explains as the generic Cramér `√x` error — which\nis precisely why the present run measures the LEVEL separation to be noise. The gap is therefore no\nlonger \"compare the levels\" (done here, negative): it is whether a statistic that **cancels the\nmodel across constellations** — the matched difference `N_2 − N_4` — carries any finite-scale signal.\n\n## Searches with no match are not novelty certificates\n\nThe absence of a published cross-constellation deficit study is evidence about these two searches,\nnot a proof of novelty. What would change this assessment: a published joint fluctuation study of\ntwin/cousin prime-pair counts (or a published \"constellation bias\" statistic); or a published\nsecond-order term for `π(x; g, q, a)` beyond Hardy–Littlewood — the MO thread's ow…"},"research_route_id":243,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_3b7fe3bbc3712fdc31be309c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/243 and return #2620. Return the ordinary report and transcript plus research: {route_id: 243, 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":"2620","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2686,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[243,255],"research_url":"/projects/twin-primes/research-routes/243","transcript_url":"/projects/twin-primes/return/2671/transcript","files":[{"sha256":"c0bac1e889ab627662bb2093663672b422748ec76f71c93dab4e21a3af700c79","name":"report_gz.md","bytes":6155},{"sha256":"5672a0c2d9effd94a1ea6ff926e874db5065e01fb64022b6945edc4bbf6b9c53","name":"research_evidence_gz.md","bytes":3769},{"sha256":"b376e32aada16b96c9a9b642be486aa56f53c1e8e59172d1f2d5ba46ed860ad6","name":"research_prior_art_gz.md","bytes":4246},{"sha256":"99b7190222fa73e5df8ba9a514dddbf5b49cf04d6bb22553498d7ae655593d4b","name":"recipe_gz.md","bytes":3700},{"sha256":"c3345e802f6123d4c428a4c0a2840dba15fbcc09c0121a7fea613785da06efb3","name":"next_step.json","bytes":2852},{"sha256":"d80bca8e8556c295e0abfba4d890d44a89058c6e32930fa1920d5a72d28cfead","name":"PREREGISTRATION_gy.md","bytes":4843},{"sha256":"ad868bb0fde227c7f89408e65b4d6bedc7b65d000d8561141f5a6674d0e68c60","name":"check_gz.py","bytes":8100},{"sha256":"e42e75e72a57cca495b5c8d78bd8181b48dd39160b8c45da8c41ab696f5186dd","name":"check_gz.out","bytes":5687},{"sha256":"ffd5ca4d2879bbad1b6bcbc9ed7d48412bd90e165676fc75ee2494af6c82274f","name":"check_gz.control.out","bytes":5749},{"sha256":"8b5eabce501f2ef4dfaeac5a99a70dbb0500bb15a64a2053925ad32247e47a52","name":"compute_gz.py","bytes":9100},{"sha256":"3fd84bac8d876eb8c81f79bf0375fa455c42b8adf83621010a317ce0828aec51","name":"compute_gz.json","bytes":71222},{"sha256":"6f995e01eed77864d915053142d2ee711c05b4eb36076d7c03a13cf8504f8f85","name":"compute_gz.out","bytes":1249},{"sha256":"d979ad9f00a70d52fba676e5bdc029583b7ad873c7ec738f48498116d601ae2c","name":"fetch_gy.py","bytes":1024},{"sha256":"c596d4a4360671c3c907f2e460ce4c7d12f41dd960630d88897f681feb9986db","name":"fetch_files_gy.py","bytes":1441},{"sha256":"96aa9d3f35a0ce2a419aadc1e14001fb9315c95b7d5dc9b233fb75c5096e2e5b","name":"fetch_oeis_gy.py","bytes":1121},{"sha256":"27ad4b0a3c18e9e99336b7fd4fcaa3408f21db5223fd02723677f3c06fa1e87a","name":"redact_gz.py","bytes":2544},{"sha256":"2e3a61a65b60ccc70b25f1bf536f505f4bad69e27284a5c8f53f23588ac73a6d","name":"upload_gz.py","bytes":4632},{"sha256":"b945cc98745e11683d743ebd9aaa159c812c1d96b6d011d03db5c83b4687d2f7","name":"route_243.json","bytes":24693},{"sha256":"e4c2c92023ed932cdd1e624673dba1c20f6da3d197ace16f2450ab0663255495","name":"return_2620.json","bytes":26590},{"sha256":"04f82ef71c20b71d5ce8a2d36d1ee102a49648bf8484801a099186f5bc2482ac","name":"PREREGISTRATION.md","bytes":4465},{"sha256":"56ca5c24a85de8627a58385fc107121e451e69b10ad0798740df27c41fb63c27","name":"next_step.json","bytes":2379},{"sha256":"303a60888a97fab1ff11a26bf0a3ac24c8d301aa64466cac53e3de8729212266","name":"compute_gh.py","bytes":6467},{"sha256":"72335f9a556c2bcbc86de6a84422b7bc4b871659fdb9923247750b4986e9840e","name":"compute_gh.json","bytes":38308},{"sha256":"52cd6114ed4f58b5ca682dbe71c2d672bb02fc89428231ee3e25d611a2eb47b6","name":"check_gh.py","bytes":4500},{"sha256":"9a660d1e38887167680577dda8318bba97d3ee5550e5241caa6eb2fa65b198ab","name":"oeis-A007508-b.txt","bytes":273},{"sha256":"12ef9529da55e7dea588bd98b0de59b359a8a478913db4a45846cbf1649f6717","name":"oeis-A023200-b.txt","bytes":119286},{"sha256":"e773500b58a579b19c7d09464179556a409c076da12f66a3a590f33a281947e6","name":"oeis-A023201-b.txt","bytes":115951},{"sha256":"bb48d9802b90219a23ef6e2c7f763e9c4f5c2f98ecf07f3a16fe2c4558155950","name":"oeis-A007508.html","bytes":31306},{"sha256":"aa12fcbbbb93db34cf0b25b4b4ae7d298c1dfc7d07299ea1bd91a8601825f6b7","name":"oeis-A023200.html","bytes":22582},{"sha256":"6a2c393c88facc5128d433c5b802f49aa7ac8261cecbea7630399944733d8c70","name":"oeis-A023201.html","bytes":20378},{"sha256":"48ecfa070da4cd4229e72c91507aefb26cbe3908f0a4204fd08073bfdd6ceb1b","name":"ext_oeis.json","bytes":983},{"sha256":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","name":"sah.py","bytes":56280},{"sha256":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","name":"export_transcript.py","bytes":10230}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}