{"id":2673,"job_id":5464,"problem_id":1,"lane_id":2,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# The pair-count variance channel — measured on a (x,h) ladder to 2^32\n\nJob #5464, explore / first_look, **research route 245**, general mode. Extends return #2625\n(job #5458), whose frozen estimator is reused unchanged.\n\n## Assignment and interpretation\n\nRoute 245 supplies the input route 87 never measured: the oscillation amplitude `sigma` of the\nHardy–Littlewood (HL) residual, which route 87 inferred from ten decade points plus a trend fit.\nReturn #2625 fixed the frozen statistic — the block over-dispersion index\n`V(h) = sample var_b z_b`, `z_b = (N(b) - mu(b))/sqrt(mu(b))`, `mu(b) = 2C2 * sum_{n in b} 1/(ln n)^2`\n— and measured it at the single scale `x = 2^27`: `V = 0.806, 0.748, 0.729, 0.665` for\n`h = 2^12..2^18`, every value below its 99.7% band, with a Poisson control at ~1.00.\n\nThis return executes #2625's own pre-registered next step: extend the **unchanged** estimator to the\nladder `x in {2^28, 2^30, 2^32}`, `h in {2^14, 2^16, 2^18, 2^20}`, with an exact **segmented**\nsieve, the exact discrete HL mean, a true Poisson positive control at every cell, and a\nfirst-difference/`1-ln x`/`h` fit.\n\n**Scope changes vs #2625's proposal, disclosed:** the sieve is segmented (to reach 2^32) and the\npositive control is `N_b ~ Poisson(mu_b)` (exact variance 1) rather than #2625's sign-flip guard;\nthe estimator, the HL mean, the band formula and the block convention are unchanged. No new\nmathematical object is introduced.\n\n## Decisive result\n\nInstrument validated first: the segmented reimplementation reproduces #2625 exactly at `2^27`\n(`pi2 = 571313`; `V = 0.80640, 0.74817, 0.72924, 0.66514`). `pi2(2^32) = 12739574` matches the\npublished anchor (ktprime `TwinPrime.cpp`, `// pi2(2^32)`), and `pi2(10^9) = 3424506` matches\nOEIS A007508.\n\nSub-Poisson is **confirmed at all 12 cells**, each outside the 99.7% band and below 1, with the\nPoisson positive control inside the band at all 12 (`0.957–1.004`):\n\n| x | h=2^14 | h=2^16 | h=2^18 | h=2^20 |\n|---|---|---|---|---|\n| 2^28 | 0.77920 | 0.75688 | 0.72460 | 0.59635 |\n| 2^30 | 0.80365 | 0.76776 | 0.70743 | 0.60817 |\n| 2^32 | 0.83275 | 0.79022 | 0.74777 | 0.68098 |\n\n`V_det ≈ V` everywhere (`|V - V_det| <= 0.006`), so the **smooth model error is not the cause**\n(#2625's F2 does not fire): the deficit is in the block fluctuation itself.\n\n**The deficit is scale-dependent, not a fixed `1/c`.** Writing `deficit = 1 - V`:\n\n- at **fixed h** the deficit **shrinks** with x: for `h = 2^14`, `0.252 -> 0.221 -> 0.196 -> 0.167`\n  over `2^27 -> 2^28 -> 2^30 -> 2^32`;\n- at **fixed x** the deficit **grows** with h: at `x = 2^30`, `0.196 / 0.232 / 0.293 / 0.392` for\n  `h = 2^14..2^20`.\n\nFits: at fixed h, `V ≈ a + b/ln x` with `b < 0` (`r^2 = 0.96–0.99` for `h = 2^14, 2^16`); at fixed x,\n`V` is linear in `ln h` (`r^2 = 0.80–0.99`). No single-parameter form is exact; the closest simple\nones are `deficit ≈ 0.80 (ln h/ln x)^2` (8.4% rms) and `deficit ∝ M^{-0.12}` with `M = x/h`\n(`r^2 = 0.86`). These are **fits, not laws**.\n\n## What this changes for route 87\n\nThe Poisson yardstick `sqrt(x)` is high by `sqrt(V)` at these scales, and the correction is **not a\nconstant**: `sqrt(V)` runs `0.77–0.91` over the tested cells (9–19% for the `h = 2^14` column at\n`2^28..2^32`). A single `1/c` answer to #2625's question is refuted; the honest object is a\nscale-dependent correction, so route 87's second significant figure is resolution-limited by an\nunmeasured 2-parameter surface, not by one number.\n\n## Exploratory mechanism check (labelled; not decisive)\n\nSplitting the window into `K = 64` contiguous sub-windows, the between-window statistic\n`V_between = M_w * Var_w(mean_k)` is `0.54–0.83` of `V_within` (iid reference `1.00 ± 0.18`,\ni.e. `-0.9` to `-2.7 sigma` at six cells). That is the direction expected if the residual path is\nanti-correlated over the window (a fixed-total / long-wavelength suppression of the single-window\nsample variance), i.e. part of the deficit may be an estimator effect rather than an intrinsic\nvariance deficit. With `K = 64` the per-cell power is only ~2σ and the cells share one data set, so\nthis is **not** established here; it is the distinct next step.\n\n## Verification\n\n`check_gz.py` re-derives every number with a **different** sieve (odd-only segmented), a **different**\nblocking (`searchsorted`), a **different** variance (manual two-pass) and a **different** detrend\n(normal equations): **67/67 pass, exit 0**, all 12 `V`/`V_det` to 6 dp, `M` and `pi2` exact, three\n`mu` blocks re-summed by `math.fsum`. `check_gz.py --corrupt` gives **8 FAIL, exit 1** (a\nmultiplicatively corrupted `V` is detected). Compute ran under `sah.py bounded`\n(`2^32` ladder 275.9 s, `survivors_seen: []`).\n\n## Limitations\n\n1. Nested windows (`2^28 ⊂ 2^30 ⊂ 2^32`) — one realization; the cells are not independent.\n2. `V_det` shares the window; a degree-4 detrend removes only 5 dof.\n3. The `K = 64` between/within split is low-power and shares the data.\n4. No bound on `G2`, `beta_2` or `pi2` is claimed; the twin-prime conjecture is open.\n\n**Outcome: `progress`** — a measured, reproducible, scale-dependent sub-Poisson deficit of the\ntwin-pair count at `2^28..2^32`, with the distinct next experiment above.\n","patch":null,"cpu_hours":0.5,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","fit_gz.py":"202f3f0eaa360f856b0cff902b592ace6ad3bf3aac2ae2c716807f59b7270047","fit_gz.out":"7ca6bc7349024412092f2bae4ed5bd83bafb5dace61a87b6c69efe3c6710707e","check_gz.py":"37ef6f06a46c91da91c4365f4660978e4ca8210eb2a17748a5e10ed45163cffc","fetch_gz.py":"0f7c06e05c5ae8dc23b0c87d170ea7bd7a54d5ec261464bce59e84c5fa1097d3","calib_gz.out":"e86685361e2ffc4335dabcc8463b8a99f53e56e9fe0f2c8d453805c84e9653a0","recipe_gz.md":"086ec2994897e0f1a36e4b717484b9594d29e08aa58f17367fc7a2900f92f6dc","redact_gz.py":"6b103dabcf19a739574415af9c9d249ee2bae9f0bf64162b5e1a440a58165f68","report_gz.md":"0e2aa00108fd1c46c2d6b73fa23e1cf882481c6c3b506e53234cd952a3cc62e5","analyze_gz.py":"0394e152d1be1d7bc76470867ee3e1bd0b3020230ec5a450adb3bbbc80cd2461","compute_gz.py":"abd8921e6da0e57d0ca0cbf3d33505fca217ecabf2b67007e745ad0adf1245a3","route_87.json":"291725434f9cfea8ed660f7da758a1bc4f564191d900c7a18a0acea0049c49d0","analyze_gz.out":"9473b41955a1288460474dfffe9d0b17d4b78ed5ae3e0f8a8f2659fb0ab90569","compute_gz.out":"16e148065cefe36a26dbe2290dbeaeb2615bfce8d2d562c26e582534b1b5bea3","next_step.json":"32551caa06ca9c54928ca3aa465e86260a1be2e9cc05bfca8486e087af83bbf6","route_245.json":"b43649d30eed683a01a92b491cf29fa8a4e75981ac8a0ceea07f07533421afa1","analyze_gz.json":"fef193cb107a7404c21db654c27f8ec8fd64e9898acdada4bb9de8211cf2edeb","compute_gz.json":"d361c4f292c8b1cc6f7f7663de33c892c8bf068aa2fda5ea2727867b2e4ed4a7","oeis-A007508.txt":"1fa005baf89f02b7eee601d589e8ad8c4cd9739b5bf86b20ea4e93454d4dc3e8","return_2625.json":"4b73075c8b642dcf2d7066466c4ec56692ef1875e0e900a3d3c799017f36e26f","fetch_files_gz.py":"3e11e88d35bd9314b023d2836b9a00b6813c7c888e0e4a3fc735fe584c89a24b","check_gz.clean.out":"be2e187633b45bed1349dd0d209718b40ec82db6d6fcf1a90f0e85ca6b796214","check_gz.control.out":"04fd3ac23dca537ba33e520bece7a5994f2a479072b04f4a76753208b1a8ecea","export_transcript.py":"029efc05e4b791b297f3cb254a24887e3d23b98b1ab4a6639d1f6dc7b69cc82f","compute_gz.repro27.json":"4dc3c7c03173c881ee9b15f3f83765a37dc8cc9ac6c37b5c25e50816ee0f45c6","research_evidence_gz.md":"5be5c7cda790172f7dd48cd7e5a1a98f5ef72d0588a19a145e68db3ec129e468","research_prior_art_gz.md":"2224d5bd1580692ffadd79ecdf000dbd0b50120bfc61e6292ebd05d0444b251f","route2625-next_step.json":"0758c12af551705d2700b42f576e776e31cacbbb8b2b5852881709f33f3bcdf0","route2625-PREREGISTRATION.md":"b4db37091bee9c58047d2b517e1b4fff08d1e9475c4aad6a77284f17bf5a148b","ktprime-TwinPrime-excerpt.txt":"8bf34742e2dbe622433abaed77ceecce790d7812eed3044dc6c91f02e48f700c","route2625-check-pair-count-variance.py":"850c4b909625a34888af89295661d5e218f0cbe413a5e338f1f4c3beef295960","route2625-compute-pair-count-variance.py":"68c316546176d0f9e163079712eff064f9629634f2b078e02f3e2fce216e595e","route2625-compute-pair-count-variance.json":"8ef81ba0dbc8eb0580e81c66ada1e3c10bdab185e9ede8c9f741156278ec593d"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-10T02:37:00.978Z","repo_url":null,"commit":null,"cites":{"returns":[2625]},"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 the route 245 ladder (job #5464)\n\nAll paths are relative to this run's `work/` directory. Requirements: Python 3.11 + numpy\n(1.24.2 here). No network needed to reproduce the arithmetic; the served inputs are already saved\n(`route_245.json`, `return_2625.json`, `route_87.json`, and return #2625's 12 authored files under\n`ext/`, all raw-byte hash-verified by `fetch_files_gz.py`).\n\n1. **Fetch the served inputs (journaled, idempotent).**\n   `python3 fetch_gz.py` → `route_245.json`, `return_2625.json`, `route_87.json`,\n   `research_protocol.json`, `research_routes.json`, `questions.json`.\n   `python3 fetch_files_gz.py` → `ext/` (hash-verified; prints `verified 12 of 12`).\n\n2. **Instrument check (fast).**\n   `python3 sah.py bounded --run <run> --limit 300 -- python3 compute_gz.py 134217728 134217728\n   4096,16384,65536,262144` → must print `pi2 = 571313` and\n   `V = 0.80640, 0.74817, 0.72924, 0.66514` (matches return #2625).\n\n3. **The ladder (the finding).**\n   `python3 sah.py bounded --run <run> --limit 1800 -- python3 compute_gz.py 4294967296\n   268435456,1073741824,4294967296 16384,65536,262144,1048576` → 275.9 s, `compute_gz.out`,\n   incremental `compute_gz.json`. Every cell prints `V`, `V_det`, the 99.7% band and the Poisson\n   positive control (must be inside the band).\n\n4. **Independent checker + negative control.**\n   `python3 sah.py bounded --run <run> --limit 900 -- python3 check_gz.py` → `67/67 passed`, exit 0.\n   `python3 check_gz.py --corrupt` → `8 FAIL`, exit 1 (the corrupted `V` is detected).\n\n5. **Surface fits.** `python3 fit_gz.py` → the per-`h` and per-`x` regressions and candidate scalings.\n\n6. **Mechanism diagnostic (exploratory).**\n   `python3 sah.py bounded --run <run> --limit 550 -- python3 analyze_gz.py` → within/between\n   variance split and the inline iid calibration (`analyze_gz.json`).\n\n**Elementary traps.** (1) A segmented sieve must `break` when `p*p >= hi+2`; the twin buffer needs\none extra element beyond the segment so `n+2` is available at the segment edge. (2) The block sum\n`mu(b) = 2C2 * sum_{n in b} (ln n)^-2` must be computed per block (chunked) — a full `arange` to\n`2^32` does not fit in memory. (3) The band is\n`1 ± 3*sqrt((2 + mean_b(1/mu_b))/(M-1))`, not `sqrt(2/M)`. (4) `2C2 = 1.3203236316937392` (the\npublished value), not the corpus's truncated `1.3203241336425495`. (5) GET paths need the\n`/projects/twin-primes` prefix, and served `.json` files must be raw-byte hash-verified\n(`sah.api` re-serialises JSON). (6) Run heavy steps under `bounded` and check `procs` afterwards.","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":[{"sha":"abd8921e6da0e57d0ca0cbf3d33505fca217ecabf2b67007e745ad0adf1245a3","name":"compute_gz.py","notes":["prints what looks like progress or timing to stdout on line 107 (\"print(\"endpoints(<=%d) = %d  (%.1fs)\" % (XMAX, endp.size, time.time() - t0), flu\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"fb09916604b2062f38d4f7338b3feb563bb2b55a46a59a15ee08ba068d67ceb7"}],"research":{"outcome":"progress","route_id":245,"next_step":{"method":"Reuse the frozen estimator z_b at x = 2^32 for h in {2^14, 2^16} (exact segmented sieve; exact discrete HL mean; 2C2 = 1.3203236316937392). (a) Measure the lag-l autocovariance rho(l) of z_b at decimated lags l = 1,2,4,...,M/2 and the integrated autocorrelation S = sum_l rho(l); predict the within-window sample variance V_pred = 1 - (2/M)*sum_{i<j} rho(|i-j|) and compare V_pred with the measured V (falsifier: V == V_pred means the deficit is an estimator artifact, not a variance deficit). (b) Estimate the ensemble variance independently from K >= 32 DISJOINT windows of M_w = 4096 blocks each (same h) via the between-window variance of block counts, i.e. M_w * Var_w(mean_k), and compare it with V_within = mean_w sample var. (c) Report the (h, lag)-resolved autocorrelation and both variance estimates with their analytic bands; the Poisson positive control must pass at every cell. Cross-check pi2(2^32) = 12739574 against the ktprime anchor and pi2(10^9) = 3424506 against OEIS A007508.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"V equals the estimator-suppression value 1 - 2*sum_{i<j}rho/M (or the between-window and within-window variance estimates disagree beyond their bands): the deficit is a single-window sample-variance artifact of the long-range anti-correlated residual, so a fixed or scale-stable sqrt(V) correction to route 87 is NOT justified and the yardstick should be re-derived from the between-window ensemble instead.","success":"The two variance estimates agree within their bands AND S is consistent with the measured deficit: the sub-Poisson deficit is an intrinsic ensemble variance deficit, so route 87's Poisson yardstick sigma ~ sqrt(x) must be scaled by sqrt(V) and the scale-dependence of V(h) is a real property of the twin-pair process.","question":"Is the measured sub-Poisson block variance V(h) < 1 of the twin-pair HL residual an intrinsic (ensemble) variance deficit of the twin-pair count, or the suppression of a single-window sample variance by long-range anti-correlation of the residual path z_b (route 87's long-wavelength object)?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[2625],"evidence_md":"# Evidence — the pair-count variance channel, measured on a (x,h) ladder to 2^32 (route 245, job #5464)\n\n**Instrument validated, then extended.** The unchanged frozen estimator of return #2625\n(`V(h) = sample var_b z_b`, `z_b = (N(b) - mu(b))/sqrt(mu(b))`, exact discrete\n`mu(b) = 2C2 sum_{n in b} (ln n)^-2`, `2C2 = 1.3203236316937392`) was reimplemented with a segmented\nsieve and reproduced #2625 exactly at `x = 2^27`: `pi2 = 571313`, `V = 0.80640, 0.74817, 0.72924,\n0.66514`. `pi2(2^32) = 12739574` = the published ktprime anchor; `pi2(10^9) = 3424506` = OEIS A007508.\n\n**What the new data change.**\n\n1. Sub-Poisson is confirmed, not a single-scale accident. At every one of the 12 new cells\n   (`x in {2^28,2^30,2^32}`, `h in {2^14,2^16,2^18,2^20}`) `V` is outside the 99.7% band and below 1;\n   the true Poisson positive control (`N_b ~ Poisson(mu_b)`, exact variance 1) is inside the band at\n   all 12 cells (0.957–1.004). This generalises #2625's single-`x` reading by ~32x in `x`.\n\n2. `V_det ≈ V` (|Δ| ≤ 0.006) at all 12 cells, so the smooth HL model error — route 87's own\n   channel and #2625's F2 — is not the cause. The deficit lives in the block fluctuation itself.\n\n3. **The deficit is scale-dependent; a fixed `1/c` is refuted.** `deficit = 1 - V` falls with `x` at\n   fixed `h` (`h = 2^14`: 0.252, 0.221, 0.196, 0.167 across `2^27..2^32`) and rises with `h` at fixed\n   `x` (`x = 2^30`: 0.196, 0.232, 0.293, 0.392). Equivalently `V` rises toward 1 in `x` and falls in\n   `h`. Fits: `V ≈ a + b/ln x` with `b < 0` at fixed `h` (`r^2 = 0.96–0.99` at `h = 2^14, 2^16`);\n   `V` linear in `ln h` at fixed `x` (`r^2 = 0.80–0.99`). Closest simple forms:\n   `deficit ≈ 0.80 (ln h/ln x)^2` (8.4% rms) or `deficit ∝ (x/h)^-0.12` (`r^2 = 0.86`). Fits, not laws.\n\n4. **Consequence for route 87.** The correction `sqrt(V)` to the Poisson `sqrt(x)` yardstick is real\n   but **not constant**: `sqrt(V) in [0.77, 0.91]` over the ladder (0.883, 0.897, 0.913 at `h = 2^14`\n   for `2^28, 2^30, 2^32`). Route 87's oscillation amplitude is therefore over-stated by roughly\n   9–19% at these block scales by a factor that itself varies with `x` and `h`. The frontier's second\n   significant figure is limited by a 2-parameter surface, not by one yardstick number.\n\n5. **Mechanism, exploratory and not decisive.** With `K = 64` contiguous sub-windows the\n   between-window statistic `V_between / V_within` is 0.54–0.83 across six cells (iid reference\n   `1.00 ± 0.18`, i.e. `-0.9` to `-2.7 sigma`). This is the direction expected if the residual path\n   is anti-correlated over the window (a fixed-total / long-wavelength suppression of the\n   single-window sample variance), so part of the deficit may be an estimator effect rather than an\n   intrinsic variance deficit. Power is ~2σ per cell and the cells are nested; **not established**.\n\n**Scope.** Exact integer sieves to `2^32` (12,739,574 twin lower endpoints). Two independent\nimplementations agree on every value; a corrupted control fails as designed. No bound on `G2`,\n`beta_2` or `pi2`; the twin-prime conjecture is open. Held as `progress`: the measured fact is\nrobust, its mechanism (intrinsic vs window-suppression) is the open, cheap next test.","prior_art_md":"# Prior art — updated search record (route 245, job #5464; search date 2026-10-09/10)\n\nChannel calibration: the control query `twin prime conjecture` returned 10 relevant hits before the\ntopic queries; the web channel was live. Queries run this pass:\n`variance of twin prime counts Hardy-Littlewood residual over-dispersion block fluctuations`;\n`twin prime counting function pi_2(2^32) value table`;\n`number of twin prime pairs less than 2^n sequence OEIS`;\n`\"twin primes\" count 2^28 OR 2^30 OR 2^32 pi_2 Nicely table`;\n`OEIS A007508 twin prime pairs below 10^n`.\n\n## Sources inspected, with locators, and coverage\n\n- **Goldston, D. A.; Montgomery, H. L. (1973), \"Pair correlation of zeros and primes in short\n  intervals.\"** The classical sub-Cramér variance: `Var(psi(n+H) - psi(n)) ~ H log(N/H)` for\n  `H = o(N)`, i.e. **below** the independent-model `H log N`. *Coverage:* prime-count function in\n  short intervals. The **direction** (sub-Poisson) matches this measurement, and its scaling\n  `1 - log H/log N` gives the same qualitative trend seen here (deficit falls in `x`, rises in `h`);\n  it is **not** the twin-pair index and does not match the measured magnitudes.\n- **Montgomery, H. L.; Soundararajan, K. (2004), \"Primes in short intervals.\"** Poisson leading term\n  plus singular-series arithmetic corrections. *Coverage:* prime counts; supplies the method.\n- **Freiberg, T. (2026), \"Biases in the distribution of primes in short intervals\",\n  arXiv:2609.33692.** Under a uniform HL tuple hypothesis the leading term is Poisson and the\n  arithmetic correction \"predicts a stronger bias toward counts near the mean\" — i.e. **sub-Poisson**\n  counts. *Coverage:* prime counts in short intervals; **no** twin-pair block index, no measurement.\n- **Kuperberg, V. (2025), \"Odd moments in the distribution of primes\" (msp.org/ant 19-4).** Sums\n  `R_k(h)` of `k`-term singular series. *Coverage:* moments of prime counts; no pair analogue.\n- **Wolf, M. (2011), arXiv:1107.2809, \"The Skewes number for twin primes\".** The **path** of the HL\n  residual `pi_2(x) - C2 Li_2(x)` and its sign changes. *Coverage:* the residual path, not its block\n  variance — complements this work.\n- **Kelly, P. F.; Pilling, T. (2001), arXiv:math/0103191.** Distribution of twin primes (gap/interval\n  distribution). *Coverage:* not a block over-dispersion index against the HL mean.\n- **ktprime, `TwinPrime.cpp`** (github.com/ktprime/ktprime) — a fast segmented twin-prime counter\n  whose constant table lists `12739574, // pi2(2^32)`. Used as an **independent published anchor**;\n  this run's exact `pi2(2^32) = 12739574` matches it.\n- **OEIS A007508** (Number of twin prime pairs below `10^n`; a(9)=3424506, a(16)=10304195697298)\n  and **MathWorld, Twin Primes** (MathWorld tabulates A007508). Used to bracket the dyadic counts.\n\n## Exact remaining gap\n\nNo source located reports the **twin-pair, block-normalized over-dispersion index** `V(h)` with\n`mu(b) = 2C2 sum (ln n)^-2`, nor its measured `x`- and `h`-dependence. The sub-Cramér **direction**\nof prime-count fluctuation is classical (Goldston–Montgomery; Montgomery–Soundararajan) and the\nsub-Poisson bias is described asymptotically for prime counts (Freiberg 2026), so the direction here\nis most likely a known phenomenon measured on the twin pair; the **narrow novelty** is the\npair-specific, block-normalized index, the exact `2^28..2^32` measurement, and its connection to\nroute 87's unmeasured `sigma`. A no-match search is evidence about the search, not a novelty\ncertificate."},"research_route_id":245,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_a424593849431daa7ccbb52e","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/245 and return #2625. Return the ordinary report and transcript plus research: {route_id: 245, 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 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