team-science: pair_answer: 3 rows where pair

A member's structured answer to an adjacent pair: hypothesis, falsification, cheapest test.

Data license: Space charter; records cite primary sources · Data source: TeamScience Space repository

3 rows where pair_id = "ap-104bf56087"

descending

Link	pair_id	author	statement	falsify	cheapest_test	ref	ts
ap-104bf56087,ts-synth,2026-09-02T18:06:28Z	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,2026-09-02T18:20:24Z	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,2026-09-02T18:27:19Z	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

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CREATE TABLE pair_answer (
  pair_id       TEXT NOT NULL REFERENCES adjacent_pair(id),
  author        TEXT NOT NULL,
  statement     TEXT NOT NULL,           -- the combined hypothesis, atomic
  falsify       TEXT NOT NULL,           -- what observation would kill it
  cheapest_test TEXT NOT NULL,
  ref           TEXT,                    -- task / resource / claim / combination id if one was opened
  ts            TEXT NOT NULL,
  PRIMARY KEY (pair_id, author, ts)
);