Ford receipt — Erdős #357 / OEIS A364132: a(28) = 75 lands in the Erdős Frontier Atlas
Recorded 2026-10-29 (America/Denver), Ford rotation (sprintBL search, sprintBN/sprintBO atlas and board).

Claim: A364132(28) = 75, i.e. 75 is the least N such that {1..N} holds an increasing 28-term sequence with all 406 segment
sums (sums of consecutive terms) distinct. An exhaustive computer search by two independent engines, not a proof.
New: OEIS A364132 lists a(1..22), ending 50, 52; re-checked 2026-10-29 03:51 MDT (22 terms). The atlas certificate had
already decided a(23..27) = 56, 60, 63, 67, 69 (contract erdos-357-a364132-a364153-new).

Method: Ford's separately written engine ss_cleanroom.c (sha256 cc5d0992e8856f440c1fb851b3498eb09bcf23e84fdd9e20df6a6c28ff77d514),
mode m (increasing), the search at N split into 38 parts that together cover it. Answers: NONE on 38 of 38 parts at each
N = 70, 71, 72, 73, 74 (about 3.9e10, 6.0e10, 9.5e10, 1.5e11, 2.3e11 nodes); at N = 75, FOUND on parts 29 and 36, with
2, 3, 4, 6, 11, 25, 38, 40, 41, 47, 52, 54, 56, 57, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75
(28 strictly increasing terms, max 75, 406 of 406 segment sums distinct by brute force; a second sequence with max 75 also
passes). Every run reproduced the control m27@68 part 6/14 = NONE with 1,260,135,573 nodes exactly; every run exited 0.
Second engine: the atlas's unchanged segsum.c (sha256 b0228c31c7ac9ea83a9ba7793f614a1c7450b75592cb66bc7d831672f068d145),
unpartitioned, answers none at 70, 71, 72, 73 and 74 and found at 75 (the sequence above); its m27@68 node count
8,443,190,723 equals the one pinned in RESULT.json.

Atlas PR https://github.com/techno-optimist/erdos-frontier-atlas/pull/187, merged as a12a8488d038da17b7119a22bdcaf14aebb8684e.
  New files in certificates/erdos-357 (RESULT.json, verify.py, segsum.c, ss_cleanroom.c unchanged):
    ford_searches_a364132_28.txt  sha256 6670a13cd18933778e404522c11453ce4ac55b27b1f5a65e7a0560211ffff143
    g4_crosscheck_a364132_28.txt  sha256 9be7af59dfdc2a396564114a4209312a45df1a6ba083b396eca044d93740145c
    verify_ford_a28.py            sha256 c3b21fe0b6aeae74a6b6a7da493fe551a1989f274a06c618c5ac14c2bd8ddadf
  New promoted contract erdos-357-a364132-28. Gap map #357: next cell A364132(29) >= 76, no upper bound recorded; the
  confidence class moves C2 -> C1 as computed by the atlas validator from the evidence list.
  Atlas gates PASS locally on the PR commit (validators, views, contracts, fast replay, pytest 261 passed / 2 skipped,
  verify-certs, receipt drift); GitHub fast-claim-gate passed before merge.

Disclosed: the two engines are independent code using the same segment-sum bitset method, not independent mathematics.
The certifying --full replay (all 229 + 7 searches, about 1.1e12 nodes) was not re-run before merge, per Kevin's
direction to publish verified results promptly. Says nothing about the asymptotic question of Erdős #357. Not submitted
to OEIS; Kevin submits OEIS extensions himself.