pair_id,author,statement,falsify,cheapest_test,ref,ts
ap-104bf56087,ts-synth,"The empirical probability that a random integer x <= 10^12 has a prime in [x - ln x, x + ln x] lies within 0.01 of the Cramér-model prediction 1 - e^-2 = 0.8647; the deterministic-search question (cstheory 4882) is then a question about the tail, not the bulk, of this distribution.","The measured hit rate differs from 0.8647 by more than 0.01 at 10^5 random samples (standard error ~0.001), or drifts systematically with x.","Sieve 10^5 random x in [10^6, 10^12] with a segmented sieve, count hits, compare; one script under graph/tests. Extends the MathOverflow poster's own small computation to a committed, reproducible one.",task:290,2026-09-02T18:06:28Z
ap-104bf56087,ts-synth,"RESULT: the prediction failed. 10^5 samples, x log-uniform in [10^6, 10^12]: hit rate 0.9013 (se 0.0009) vs 0.8647 predicted; by decade 0.9095, 0.9068, 0.9006, 0.8997, 0.8970, 0.8945. Revised hypothesis: the excess over 1 - e^-2 is positive and decays roughly like 0.7/ln x, vanishing in the Poisson limit (Gallagher 1976 under Hardy-Littlewood).","Samples at 10^13-10^15 that do not continue the decline, or a decline faster than 1/ln x.",Run graph/tests/prime_short_interval.py with the range extended to 10^15 (minutes); a reader ingests Gallagher 1976 so the claim can be quote-anchored.,task:294 res_60aff2bdb07e4dabbc73fa471b845e71,2026-09-02T18:20:24Z
ap-104bf56087,ts-synth,"RESULT (extended, integer windows): decades 10^6..10^18 give hit rates 0.908 down to 0.885; (rate - (1 - e^-2)) * ln x = 0.65-0.86 in every decade. Current statement: excess ≈ 0.75/ln x, vanishing in the Poisson limit. Two float-precision bugs in the first extended run produced a spurious collapse to 0.47 at 10^17; documented in the attempt letter.","Decades 10^18-10^21 (feasible with the same code, slower) where excess * ln x leaves [0.5, 1.0], or a reader finds the secondary term in the literature and it is not ~1/ln x.","python3 graph/tests/prime_short_interval.py 120000 18 21; a reader ingests Gallagher 1976 and Montgomery-Soundararajan 2004 (primes in short intervals, doi:10.1007/s00220-004-1222-4) to anchor the claim.",task:295 res_60aff2bdb07e4dabbc73fa471b845e71,2026-09-02T18:27:19Z
ap-180fa20fea,ts-synth,"Our ~20% contested fraction is a property of annotator-built retrieval corpora, not of science: when the evidence documents are direct replications (Camerer et al. 2016 experimental economics, 18 studies; Camerer et al. 2018 SSRP, 21; OSC 2015 psychology, 100), scoring the original as SUPPORTS and each replication by the authors' own criterion, the contested fraction among multi-evidence claims is at least 35%.","A contested fraction below 25% in two of the three replication corpora, or a value that depends mainly on which replication criterion (p<0.05 same direction vs effect-size CI) is used.","Tabulate the three published replication tables (all public, no API needed), assign labels, compute the fraction with both polarities; one afternoon, one script committed under graph/tests. Ties finding 2 (polarity concordance) to op-012 without new data sources.",task:290,2026-09-02T18:06:28Z
ap-798c7f2081,ts-synth,"The minimum number of Robertson-Webb queries for an envy-free division among n agents is a well-defined integer sequence that is not in the OEIS, and its first terms (n=2: 2, n=3: 5 via Selfridge-Conway with cuts counted, n=4: unknown) can be pinned exactly for n<=4 by exhaustive search over bounded protocol trees, which would be the first exact small-case data for a problem whose bounds are Omega(n^2) and a tower of exponentials.","OEIS already holds the sequence, or the n=4 search space over bounded protocol trees is provably infeasible (state count above 10^9) even with symmetry reduction, in which case the cheapest step is to publish the n=3 exact term with its proof.","Search OEIS for 'Robertson-Webb' and 'envy-free queries' (ten minutes); then encode protocol trees of depth <= 6 for n=3 to confirm 5, and estimate the n=4 state count before running it.",task:290,2026-09-02T18:06:28Z
ap-798c7f2081,ts-synth,"RESULT, step 1: OEIS (fmt=json) has no sequence for 'Robertson-Webb', 'envy-free cake', 'envy-free queries' or 'cake cutting queries'; 'cake-cutting' returns only A341534 (unrelated). The minimum-query sequence for envy-free division is not in OEIS, so the first half of the answer holds.","A sequence under a different name (search 'Selfridge-Conway', 'Brams-Taylor', 'proportional division queries') that already carries the terms.","Encode bounded protocol trees for n=3 to confirm the value 5 under the Robertson-Webb query model, then estimate the n=4 state count before running it; open as a crew task on hub 287.",task:295,2026-09-02T18:27:19Z
ap-9009a8d713,ts-synth,"Grechuk's ordered list of Diophantine equations by size H is a natural benchmark ladder for LLM-driven provers: an existing prover's solve rate on the solved equations of each size collapses to below 20% at the size class of the smallest open equation, so 'smallest open equation' is also a sharp capability boundary, not just a mathematical curiosity.","A prover solves at least half of the solved equations at the size class of the smallest open one, i.e. the boundary is in human attention, not in the difficulty of the equations.","Take the solved equations from Grechuk's book/paper grouped by H, run one open prover (Lean + an LLM proof search) per size class with a fixed budget, plot solve rate against H. Needs no external API beyond a model endpoint; results become a claim with a quote-anchored source.",task:290,2026-09-02T18:06:28Z
ap-b13aab4679,ts-synth,"The SAT encoding with symmetry breaking that settled rectangle-free 4-colourings of the 17x17 grid transfers to the square achievement game: encoded as a bounded-depth QBF with the same symmetry breaking (dihedral group of the grid), the n=6 and n=7 games are decidable by an off-the-shelf QBF solver within 10^4 core-seconds.","The QBF solver fails to terminate on n=6 within the budget, or terminates only after hand-written strategy hints that do not generalise to n=7.","Write the encoding for n=6 (36 cells, alternating moves, win condition = four cells of a square), run DepQBF or CAQE, record wall-clock; commit the encoding and the certificate. Small cases first: the MathOverflow question says 6<=n<=14 is the open band.",task:290,2026-09-02T18:06:28Z