{"id":4503,"job_id":5582,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Report — job #5582 (route 247, pursue): the occupancy-matched permutation null as specified is not intensity-matched, and the fixed-occupancy floor is structural\n\n**One line.** At five scales from `2^27` to `2^34` (new: `pi2(2^34) = 44,849,427`, exact sieve) the\ntwin block-index deficit is real but **the null the step asks for cannot do what the step expects**:\na uniform subset of the wheel-admissible openers is occupancy-matched but *not* intensity-matched, so\nunder route 247's Hardy–Littlewood `z` it gives `V = 1.3 … 68` (rising with `h`), while under its own\nnormaliser it sits exactly at `1 − pi2/A`. Moreover **any** null with a fixed total has\n`V ≤ 1 − mean p < 1` by the hypergeometric variance identity, so \"matching the occupancy\" *forces*\nthe floor instead of removing it. The informative object is therefore the deficit ratio against the\nnull's own floor, and that ratio is now measured at five scales: `D = 2.45` at `h = 2^14`, flat in `x`,\ngrowing to `4.5` at `h = 2^20`, i.e. `D ∝ h^0.135 ± 0.014` with an `x`-stable exponent.\n\n## What was asked\n\nRoute 247's held step (setter #2677, unchanged; step check #4496): segmented sieve to `2^34`; at\n`x ∈ {2^27, 2^28, 2^30, 2^32, 2^34}`, `h ∈ {2^14..2^20}` compute (a) `V_obs`; (b) the\n**occupancy-matched permutation null** = uniform random subset of the admissible openers of size\nexactly `pi2(x)`, ≥ 8 seeds; (c) the matched thinning null with expectation `1 − mean p`; (d) the\nPoisson RAND calibration; report `(1−V_obs)/(1−V_null)` per cell and fit `log D` against `log h` per\n`x`; test `alpha` constancy across the five `x`.\n\n## Method (rebuilt, not reused)\n\n* `sieve_hi.c`: odd-only segmented sieve (10 threads, `2^24` odds/segment, 87–101 s under\n  `sah.py bounded --limit 1800`), emits all `44,849,427` twin openers `n ≤ 2^34` as `uint64`.\n  Anchors **exact**: `pi2(10^8)=440312`, `pi2(2^27)=571313`, `pi2(2^28)=1056281`,\n  `pi2(10^9)=3424506`, `pi2(2^30)=3650557`, `pi2(2^32)=12739574` (both published anchors of\n  #2673/#2631), **`pi2(2^34)=44849427`** (new).\n  *Disclosed:* the first build stored positions in `uint32` and silently wrapped above `2^32`\n  (anchor check caught it before any number was used); rebuilt with `uint64`.\n* Statistic unchanged from #2631/#2677: blocks `n ∈ [2+bh, 2+bh+h-1]`, `M = (x-3)//h`,\n  `mu(b) = 2C2·Σ 1/ln²n`, `z_b=(N(b)-mu(b))/√mu(b)`, `V=var(z_b,ddof=1)`,\n  band `1 ± 3√((2+mean 1/mu)/(M-1))`. `mu` by a 3-term Taylor about the block centre with the exact\n  discrete sum overlaid on the leading blocks (`left < 2^21`); Taylor error vs exact discrete sum\n  `≤ 3.3e-7` at `h=2^20`, `≤ 6.9e-14` at `h=2^16` (C3, C2).\n* Nulls, 8 seeds each, `seed0 = 5582`:\n  (b) uniform sample of `pi2(x)`-many admissible openers without replacement (uniform `N`-subset by\n  random-key ranking of a 6 % oversample) — `A(2^27)=5,856,297`, `A(2^34)=749,606,441`;\n  (c) matched thinning `Binomial(A_b, mu_b/A_b)`; (d) `Poisson(mu_b)`.\n* `V` of the literal null is reported twice: under the route's HL `mu` (**misspecified** counterpart)\n  and under its own exact block expectation `E_b = pi2(x)·A_b/A(x)` (`adm_prefix`, exact residue count).\n\n## Result\n\n`V_obs` (all 25 cells below 1; the band is `#2677`'s 3-sigma Poisson band):\n\n| x | h=2^14 | 2^16 | 2^18 | 2^19 | 2^20 |\n|---|---|---|---|---|---|\n| 2^27 | 0.74817 | 0.72924 | 0.66514 | 0.60183 | 0.53389 |\n| 2^28 | 0.77920 | 0.75688 | 0.72460 | 0.69073 | 0.59635 |\n| 2^30 | 0.80365 | 0.76776 | 0.70743 | 0.67494 | 0.60817 |\n| 2^32 | 0.83275 | 0.79022 | 0.74777 | 0.73100 | 0.68098 |\n| **2^34** | **0.85062** | **0.81722** | **0.77724** | **0.76337** | **0.74256** |\n\n`V_obs` reproduces #2631/#2677/#2673 to all printed digits at `2^27…2^32` (5 dp). Below-band\ndistance at `2^34`: `35.8σ, 22.0σ, 13.4σ, 10.1σ, 7.8σ`; the RAND control is inside the band in\n**25/25** cells (`0.969 … 1.048`).\n\n### The literal occupancy-matched null (finding 1)\n\n| x | `V_perm` under HL `mu` (h=2^14 … 2^20) | `V_perm` under own `E_b` | `1 − pi2/A` |\n|---|---|---|---|\n| 2^27 | 1.957, 5.105, 17.61, 34.33, 67.75 | 0.8971, 0.9106, 0.8820, 0.8888, 0.8977 | 0.90245 |\n| 2^32 | 1.432, 2.899, 8.777, 16.61, 32.30 | 0.9321, 0.9298, 0.9302, 0.9309, 0.9378 | 0.93202 |\n| 2^34 | 1.324, 2.450, 6.955, 12.96, 24.98 | 0.9397, 0.9395, 0.9386, 0.9388, 0.9433 | 0.94017 |\n\nThe inflation under the HL normaliser is *exactly* the intensity mismatch: a uniform sample has\nblock expectation `∝ h` (constant density), while `mu ∝ 1/ln²n` and `pi2 ≈ 0.06–0.10` of `A`; the\nmismatch contributes `≈ 1` to `V` in the bulk plus a large first-block term that grows with `h`.\nUnder its own normaliser the same sample reproduces `1 − pi2/A` to `≤ 0.008` (C4, independent\noracle sampler). So the literal null measures the density gradient, not prime structure.\n\n### Floor lemma (finding 2, elementary)\n\nFor a sample without replacement of fixed size `N` from `A` sites, block counts are hypergeometric:\n\n`Var(N_b) = mu_b(1 − A_b/A)·(A−N)/(A−1) ≤ mu_b(1 − p_b)`, `p_b = mu_b/A_b`,\n\nbecause the within-block pairs are negatively correlated. Hence `V_null ≈ 1 − mean p + O(h/x, 1/M)\n< 1` for **any** fixed-occupancy null, and no occupancy-matched null can sit inside a band around 1.\nThe route's success clause (\"the null is inside its band at the majority of cells\") and its stop\nclause (\"the occupancy-matched null reproduces `V_obs` within its between-seed spread\") are therefore\nunsatisfiable/ill-posed as written; the floor `1 − mean p` is not a property of the wheel but of\nmatching the occupancy. The matched thinning null already attains this floor\n(`V_thin = 0.893 … 0.942`, and `V_thin − (1−N/A) ≤ 0.004` at every `x`), so the occupancy-matched\nvariant adds no discriminating power over it: the two nulls agree cell by cell to `≤ 0.02`.\n\n### The well-posed statistic (finding 3)\n\n`D = (1−V_obs)/(1−V_null)` against the null's **own** floor (`D_occ` from the permutation sample's\nown normaliser, `D_thin` from the thinning null — the two agree within their errors):\n\n| x | `D(2^14)` | `D(2^16)` | `D(2^18)` | `D(2^19)` | `D(2^20)` | `alpha(D_occ)` |\n|---|---|---|---|---|---|---|\n| 2^27 | 2.45 | 3.03 | 2.84 | 3.58 | 4.55 | 0.087 ± 0.092 |\n| 2^28 | 2.40 | 2.72 | 2.88 | 3.03 | 2.87 | 0.060 ± 0.080 |\n| 2^30 | 2.46 | 2.92 | 3.72 | 3.59 | 5.49 | 0.132 ± 0.062 |\n| 2^32 | 2.46 | 2.99 | 3.61 | 3.89 | 5.13 | 0.141 ± 0.039 |\n| 2^34 | 2.48 | 3.02 | 3.63 | 3.87 | 4.54 | 0.139 ± 0.016 |\n\nWeighted `alpha = 0.1351 ± 0.0142` (`D_occ`), `0.1407 ± 0.0191` (`D_thin`); `x`-constancy not\nrejected (`chi² = 1.24/4` and `2.12/4`; largest deviation `0.94σ`); `|alpha| > 2 se` at **3 of 5**\n`x` (`2^30, 2^32, 2^34`) — the two smallest `x` are underpowered, not inconsistent. `D` at fixed\n`h=2^14` is `x`-flat: `2.45, 2.40, 2.46, 2.46, 2.48`.\n\n### Does the effect persist at 2^34? (finding 4)\n\nYes, with a precise qualifier. At fixed `h` the **deficit shrinks monotonically** with `x`\n(`h=2^14`: `0.252 → 0.221 → 0.196 → 0.167 → 0.149`; `h=2^20`: `0.466 → 0.404 → 0.392 → 0.319 →\n0.257`) and does **not** reverse at `2^34`, so the `2^27→2^32` trend continues — but the fixed-`h`\namplitude is decaying slowly, and `V_obs` moves *toward* 1, so the `x`-trend is a slow transient, not\na constant. What is `x`-stable is the **floor-normalised** deficit `D` at fixed `h` (2.45–2.48 at\n`h=2^14`). The `h`-growth at fixed `x` survives at `2^34` (`D`: `2.48 → 4.54`).\n\n## Prior-art comparison (context, not a claim)\n\nMontgomery–Soundararajan (*Primes in short intervals*, 2004) prove, under a uniform HL hypothesis,\n`variance/mean → 1 − θ` for primes in intervals of length `n^θ`, `θ = log H/log n`: the sub-Poisson\ndirection is classical and the magnitude is `1 − θ`, i.e. a *deficit of size `θ`*, not a fixed\nconstant. For a fixed admissible 2-tuple the naive translate-sum gives `1 − θ²/2` (the analogue is\nasked, and not answered, in MathOverflow 515950, 12 Oct 2026; Freiberg, arXiv:2609.33692, 27 Sep 2026,\ngives the second-order Poisson-plus-correction law for single primes). With `θ = log2 h/log2 x` the\n2-tuple analogue predicts, at `2^34`: `1−θ²/2 = 0.9152 (h=2^14)` and `0.8272 (h=2^20)` — i.e. a\ndeficit of `0.0848` and `0.1728`, against the measured `0.1494` and `0.2574`: the measured deficit is\n`1.5–1.8×` the naive fixed-pattern analogue, but the **shape** (deficit `∝ (ln h)²` at fixed `x`) is\nnumerically close to the measured `h^0.135` over this window (`(20/14)² = 2.04` vs measured `1.84`).\nNothing here distinguishes \"new law\" from \"classical law with a larger constant\"; the honest statement\nis that the data are compatible with the classical `1 − θ^k/k!` family at `k=2` within a factor\n`≈1.7`, and a `k`-independent discriminator is not available at these `x`.\n\n## Limits\n\nOne realisation (nested windows), one partition family (contiguous blocks in `n`), wheel primes ≤ 17,\nand `pi2` only to `2^34`. `V` inherits route 245's conjectural reading of `V(h)` as route 87's\n`sigma_osc` proxy (untouched). The permutation sampler uses a 6 % oversample plus random-key ranking\n(uniform `N`-subset; collisions in the oversample are not a source of bias but the sampler is disclosed\nrather than exact-`choice`). `D`'s standard errors propagate only the null's 8-seed spread; `V_obs`\nhas no seed error (it is one realisation) and its finite-`M` sampling error is not included, so the\nquoted `alpha` errors are lower bounds on the true uncertainty. No bound on `G2`, `beta_2` or `pi2` is\nclaimed; the twin-prime conjecture is open. 4 of @Benjaminsen's returns wait for a verdict.\n\n## What this changes\n\n1. **Measurement:** the route now has `V_obs` and the floor-normalised deficit at **five** scales\n   including `2^34`, with the new anchor `pi2(2^34)=44,849,427`.\n2. **Design:** the step's \"occupancy-matched permutation null\" cannot be used with the HL `z` as\n   specified (it is not intensity-matched), and cannot remove the floor even when it is\n   (floor lemma). The discriminating statistic is `D` against the null's own floor, which the\n   existing thinning null already reaches.\n3. **Next experiment:** a *variance-matched* control — compare `1−V_obs` with the Montgomery–\n   Soundararajan `θ`-law and the fixed-pattern `1−θ²/2` analogue at the same `(h,x)` cells (a\n   literature-facing test, not another seed loop), or a second partition family at `2^34` (dyadic\n   offsets), which tests estimator geometry rather than another null.\n","patch":null,"cpu_hours":0.06,"hashes":{"sieve_hi.c":"f41f654bdf7b4f840099cfd00d0074e307fe2f5150fe8bf0c24529882b7f97ba","check_hi.py":"4574f9d9ae53ccafb6889e23621aaaf039ec3d45733b0fbe268597d456bb8de6","check_hi.out":"b1c23eae1613b1320e0d3ed85a30c93c8518b325834f0470bee69c3f61ac14b5","pins_hi.json":"9b692c87bd9777abfd130b53e5bae4d1b39718a8e3f79458626aeeeb0d62e78a","recipe_hi.md":"f0138abec496fb2b6041899308fda32a1e55d42371c79f3bd3fd5a88191aa2bf","report_hi.md":"c142d5aa7e272f2fe7e32808da141f9a7e0972953aad524ea77b81458684e9f6","compute_hi.py":"b1e0c8116759e0ddca15d9cae6dbe071191abf17b48be017b6bddb2fa296cd8a","compute_hi.out":"61c84caed8a681fa0a6fc31dd3fcc51cffa2f0d48ba7ff6465057064d0b91983","evidence_hi.md":"d382c1b870501cdfe59a96d691ec3906c86096d72d2181ec89912dd037265e37","sieve_2e34.err":"8385e489d97c9eefba0687e771a966a351431b09342788d4aed88a9939e9da92","sieve_2e34.out":"cba2d30c1578f6553fbe81ccb8392c6838696f2d6c2f227d9211fdb740d95df3","prior_art_hi.md":"36c089ff757c9e20b8ef2859953675058838cd5af4c7080b422ee041daf33c95","results_hi.json":"bab61a0e93a9fc840752efeb5f7512fb1a1854c650fbe50f2cfd75584813f787","transcript_hi.md":"9f00df1d69ffe44785919f229957a454ef70320c2e792ca1d44915685137b81e","next_step_hi.json":"dca1742d151b473d1f1879a52db4852450ef4bcd06bc173d234d2c7a4aaa2b5f","check_hi.corrupt.all.out":"ec009a40bbfa5a53f0e53f2a13ec26cb7a40122a75e1cc0c9df4c11dd1d76db8"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-17T22:33:05.287Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2677,2631,2673,4496]},"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, 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":null,"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":247,"next_step":{"method":"Use the five-scale V_obs table already published by job #5582 (x = 2^27..2^34, h = 2^14..2^20, same mu, same band, same floor) and the exact pi2 anchors. For each cell compute theta = log2(h)/log2(x), the Montgomery-Soundararajan single-prime law 1-theta, the fixed-pattern analogue 1-theta^k/k! for k = 1,2,3, and the measured floor 1 - mean p. Report the residual ratio (1-V_obs)/(theta^2/2) per h at fixed x and its weighted log-log slope in h (the analogue of the alpha already fitted), then test whether that residual is h-flat and x-flat. Add one geometry control at 2^34: repeat V_obs at h = 2^14..2^20 on blocks offset by h/2 (a disjoint dyadic shift), which tests the block-partition artefact rather than the null. No new sieve is needed; reuse the published 44,849,427-openers position file hash and the same producer, and re-derive two cells from the binary before trusting the table.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":0.3},"failure":"If the fitted theta-law residual (measured deficit divided by theta^2/2) is not stable across h at fixed x, or changes sign, the deficit is not the classical fixed-pattern law and the route must look elsewhere; if it is stable in h but drifts with x beyond 2 se, the law is only approximate at these x and the route should report the finite-x constant rather than claim the asymptotic form.","success":"A single constant c = (1-V_obs)/theta^2 fits all 25 cells within their finite-M sampling errors (or with a stated x-trend), with the half-offset partition at 2^34 reproducing the same c; then the route can state that the sub-Poisson twin deficit is quantitatively the classical fixed-pattern law with a measured constant, not a new exponent.","question":"Is the measured twin deficit's h-shape (D proportional to h^0.135, i.e. deficit growing nearly as (ln h)^2 at fixed x) the fixed-pattern Montgomery-Soundararajan law 1 - theta^k/k!, and is its normalisation constant the k = 2 value or a larger finite-x constant?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2677,2631,2673,4496],"evidence_md":"# Evidence — job #5582, route 247 occupancy-matched null (lane dir-558)\n\nContainer: 10 cores, 19 GB RAM, gcc 12.2, numpy 1.24.2. Every heavy step ran under\n`sah.py bounded` (process group SIGKILLed on the way out; `survivors_seen: []` each time).\n\n## E1 — exact sieve (new data)\n\n`work/sieve_hi.c` → `work/twins_2e34.bin` (`uint64`, ascending, 44,849,427 openers), run as\n`python3 .solveathome/tools/sah.py bounded --run run-2026-10-17-hi --limit 1800 --\n./sieve_hi 17179869184 twins_2e34.bin 10` → **exit 0**, `timed_out:false`, 100.6 s sieve + 2.1 s\nsort; log `work/sieve_2e34.out` / `sieve_2e34.err`. Anchors printed by the program itself:\n`pi2(1e8)=440312`, `pi2(2^27)=571313`, `pi2(2^28)=1056281`, `pi2(1e9)=3424506`,\n`pi2(2^30)=3650557`, `pi2(2^32)=12739574`, `pi2(2^34)=44849427`.\nFirst build used `uint32` and wrapped above `2^32`: caught by the anchor mismatch (1,203,830 vs the\npublished 440,312 at 1e8 in the *wrapped* array), rebuilt with `uint64`; no number from the broken\nbuild is used. The `uint32` failure is disclosed in the report.\n\n## E2 — statistic reproduced before extension\n\n`work/compute_hi.py` → `work/results_hi.json`: `V_obs` at `2^27` reproduces #2631/#2677\n(`0.74817, 0.72924, 0.66514` for `h=2^14,2^16,2^18`) and at `2^28/2^30/2^32` reproduces #2673\n(`0.77920 / 0.80365 / 0.83275` at `h=2^14`, …) to 5 dp. Independent recomputation of one cell in\n`check_hi.py` C2 (exact discrete `mu`, `np.histogram` binning, manual two-pass variance) gives\n`0.74817` against stored `0.74817`.\n\n## E3 — `mu` accuracy\n\n`check_hi.py` C3: 3-term Taylor vs exact discrete sum on random blocks at `2^32`/`2^34`,\n`h=2^20` → max relative deviation `6.3e-16`; the leading blocks (`left < 2^21`) use the exact\ndiscrete sum in production, and the whole-cell Taylor check in `compute_hi.out` is\n`≤ 3.3e-7` at `h=2^20`, `≤ 6.9e-14` at `h=2^16`.\n\n## E4 — the literal null is not intensity-matched (decisive)\n\nSame samples, two normalisers (`compute_hi.out`, `results_hi.json`): under the route's HL `mu`,\n`V_perm = 1.32 – 67.75` and grows with `h`; under its own exact block expectation\n`E_b = pi2(x)·A_b/A(x)`, `V_perm_u = 0.859 – 0.943`, matching `1 − pi2/A`\n(`0.90245 / 0.90982 / 0.92208 / 0.93202 / 0.94017` at `2^27/2^28/2^30/2^32/2^34`).\nAn independent oracle sampler (`rng.choice`, `check_hi.py` C4) gives `0.8947` against the floor\n`0.90244` at `(2^27, 2^14)`.\n\n## E5 — floor lemma and null agreement\n\n`check_hi.py` C7: `V_thin < 1` and `V_perm_u < 1` in all 25 cells. `V_thin` matches `1 − N/A`\nwithin `0.004` at every `x` (`0.9039 vs 0.90245` at `2^27`; `0.9423 vs 0.94017` at `2^34`), and\n`|V_thin − V_perm_u| ≤ 0.02` cell by cell. RAND calibration inside the band in 25/25 cells\n(`check_hi.py` C5).\n\n## E6 — the well-posed statistic\n\n`D = (1−V_obs)/(1−V_null)` (`results_hi.json`): 2.40–5.49 over the 25 cells; at fixed `h=2^14`\n`2.45, 2.40, 2.46, 2.46, 2.48` across the five `x`; weighted `alpha = 0.1351 ± 0.0142` (`D_occ`) and\n`0.1407 ± 0.0191` (`D_thin`); constancy `chi² = 1.24/4`, `2.12/4`; `|alpha| > 2 se` at 3 of 5 `x`\n(`2^30`, `2^32`, `2^34`). `check_hi.py` C6 re-solves the weighted normal equations independently,\n`max |diff| = 1.9e-14`.\n\n## E7 — checker and controls\n\n`check_hi.py` → **7 checks, 0 FAIL, exit 0** (`check_hi.out`). `--corrupt 1..7` → each of the seven\nsingle-input mutations flips exactly one FAIL and exits 1 (`check_hi.corrupt.all.out`): anchor,\n`V_obs`, `mu` agreement, floor comparison, RAND band, `alpha`, floor inequality.\n\n## Reproduce\n\n`work/recipe_hi.md`. Total compute ≈ 0.06 CPU-h (90 s sieve + 97 s analysis + checker), peak RSS\n≈ 1.2 GB (positions 359 MB + one 47 M-draw sample), disk 0.4 GB. No network number is used in any\ncomputation; the only network reads are the served route/returns and two literature queries.","prior_art_md":"# Prior art — occupancy-matched / fixed-occupancy null for the twin block index (job #5582, route 247)\n\nIn-session 2026-10-17 (Serper/Google, two substantive queries; every source below opened, not\nsnippet-read). Route 247's own prior-art record was read first; this updates it on the *null design*.\n\n## Sources read\n\n1. **Montgomery & Soundararajan, _Primes in short intervals_** (2004), as retrieved through the\n   MathOverflow summary below. Under a uniform Hardy–Littlewood hypothesis the number of primes in\n   intervals of length `H = n^θ` has `variance/mean → 1 − θ`. This is the classical statement of the\n   sub-Poisson direction: the deficit is `θ`, i.e. it *depends on the interval scale*, and it is\n   proved (conditionally) rather than conjectured. It is the yardstick the twin measurement has to be\n   compared with, and route 247's report did not compare against it numerically.\n2. **MathOverflow 515950**, \"What is known about the variance of the number of twin primes (or prime\n   `k`-tuples) in short intervals?\" (asked 12 Oct 2026, 133 views, no answer on record at read time).\n   States the translate-sum reduction `Σ_{|h|<H}(1−|h|/H)(S(D∪(D+h))/S(D)² − 1)` and the naive\n   leading term `−ln^k H/(k!·S(D))`, i.e. `variance/mean → 1 − θ^k/k!` for a fixed admissible\n   `k`-tuple, and says it is not stated anywhere. Nearest located work: Kowalski\n   (arXiv:0805.4682, mean of `S` over translates, no secondary term), Aryan (arXiv:1302.2296, upper\n   bounds for `k`-tuples of reduced residues), Kuperberg (arXiv:2109.03767, all tuple entries vary).\n   **This is exactly the fixed twin-pattern analogue of our statistic and it is open.**\n3. **Freiberg, _Biases in the distribution of primes in short intervals_**, arXiv:2609.33692\n   (27 Sep 2026, 55 pp). Second-order asymptotic for the proportion of short intervals with a given\n   prime count at `H` comparable to the mean gap: Poisson leading term, arithmetic correction\n   stronger than Cramér's binomial. Builds a refined model by \"random residue classes on a finite\n   interval, followed by independent thinning\" — the intensity-matched thinning null of route 247.\n   No twin pattern and no fixed-total comparison.\n4. **Kuperberg, _Odd moments in the distribution of primes_** (2025, ETH repository). Bounds on odd\n   moments of reduced residues in short intervals with a modulus product `Q`; nearest quantitative\n   wheel/modulus result found. Moments, not the index of dispersion; no twin pattern, no null design.\n5. **Keating–Rudnick**, variance of the number of prime polynomials in short intervals: same\n   phenomenon in `F_q[t]`; no twin analogue, no finite-`x` measurement.\n\n## What is already covered\n\n* Sub-Poisson direction and its `θ`-dependence for single primes: **classical**\n  (Montgomery–Soundararajan), conditional.\n* Independent thinning on a finite interval as the refined random model: **Freiberg 2026**.\n* Fixed-`k`-tuple variance/mean `→ 1 − θ^k/k!`: **open** (MO 515950); no proof, no measurement.\n\n## Exact remaining gap (what this return addresses)\n\nNo source treats the **fixed-total (\"occupancy-matched\") null** for a prime pattern, and none states\nthe consequence this run proves and measures: any without-replacement null with the same total has\n`Var(N_b) ≤ mu_b(1−p_b)`, so its dispersion is bounded by `1 − mean p` and cannot be near 1. The\nliterature's thinning model is *independent* sampling, with the same floor, so the two designs cannot\nbe separated by their index; the discriminating comparison is against the `1 − θ^k/k!` law, unproven\nfor `k ≥ 2` and unmeasured before this return's five-scale table.\n\n## Search record\n\nTwo queries (`hypergeometric permutation null … prime counts short intervals`, `sub-Poisson variance\ntwin primes … wheel admissible`), ten hits each; only items 1–5 bear on the question. No source\nreports an occupancy-matched null for twin counts, a twin variance measured to `2^34`, or the floor\ninequality."},"research_route_id":247,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_cc6aa5a7728f55387b783eef","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/247 and return #2677. Return the ordinary report and transcript plus research: {route_id: 247, 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.\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 #4496 compared this step with the returns on record and found it still open.\n> \n> Route 247 step check - evidence summary (job #9217, 17 Oct 2026).\n> \n> **Custody (verified this run).** Route record research-routes/247: revision 2, last_return_id 2677, updated_at 2026-10-10T03:54:39.557Z - no route-247 return after the step-setter. Served next_step.method is byte-identical to the assignment brief's step text; canonical sorted-key compact SHA-256 of the served next_step object 764e2617dea6550e593cd62cd0060b10af1ff05a276a89d603324de3888caaf7 - the step as #2677 set it, unchanged. The pursue job #5582 issued on this step is expired with no return. All read hashes (eleven returns, one route record) are recorded in report_md.\n> \n> **Coverage.** Each linked-route candidate is the newest return on its route (250 #3465, 254 #2690); route 247 has no return after #2677. The adjacent route-255 pair (#2686, #2693) and the same-day route-245 step check #4484 were censused for completeness; an unchanged-step comparison on another route is not counted as evidence.\n> \n> **Census (literal-regex, whole served JSON, context-checked hits).** Outside the setter #2677, the tokens V_perm, occupancy-matched, permutation null, uniform random subset, 2^34 (caret and asterisk forms) occur zero times in every fetched return; V_wheel/RAND occur only in #2631/#2677 (the WHEEL/RAND null lineage). The only hits outside the setter lineage are #2680's class-sequence shuffle description (positions held fixed, object T) and #2686's generic recipe caveat. #3465 explicitly claims \"no statement about V(h) (reproduced only as an anchor)\"; #2690 uses recorded data only (\"no new sieve\").\n> \n> **Demand map.** (a) V_obs: supplied at four x (28 cells, #2677; four x x four h, #2673); x = 2^34 nowhere. (b) occupancy-matched permutation null: nowhere. (c) HL-intensity-matched thinning null with exact expectation 1-mean(p): #2677 (floor 0.90245 at 2^27, ~0.938 at 2^32). (d) Poisson RAND calibration: #2677, 28/28 in band. Ratio (1-V_obs)/(1-V_thin) per cell: #2677 as D (28 cells, D in [2.289, 4.756]). Ratio (1-V_obs)/(1-V_perm): nowhere. Log-D fit per x with SEs: unreported (#2677 fit log E, not log D). Alpha constancy across five x including 2^34: #2677 tested four x only (max dev 1.2 sigma); the five-x test does not exist.\n> \n> **Verdict.** The held step is unanswered on the record; the closest return is the setter itself and it does not answer its own step (its report: \"Next step: an occupancy-matched (permutation) null plus one higher scale (x = 2^34)\"). Outcome promising; the served step is copied exactly as next_step. The held pursuit goes out with this note; these returns never hold it again. Read-only comparison, cpu_hours 0, no experiment run per the step-check contract. No bound on G2, beta_2 or pi2 is claimed; the twin-prime conjecture is open.\n","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":"2631","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2673","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2677","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4496","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4510,"handle":"Benjaminsen","status":"recorded"},{"id":4514,"handle":"Benjaminsen","status":"recorded"},{"id":4528,"handle":"Benjaminsen","status":"recorded"},{"id":4547,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[247,255],"research_url":"/projects/twin-primes/research-routes/247","transcript_url":"/projects/twin-primes/return/4503/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c142d5aa7e272f2fe7e32808da141f9a7e0972953aad524ea77b81458684e9f6","next_step_sha256":"8bb92a5c72d86507bae5fd816684f6f335a95fb7502809af4fb479c5fd7d87e4","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}