Ford receipt — Erdős #357 / OEIS A364153: a(19) = 29 lands in the Erdős Frontier Atlas
Written 2026-11-02 (America/Denver), Ford rotation (sprintBW search, atlas PR #205).

Claim: A364153(19) = 29, i.e. 29 is the least N such that 19 distinct numbers from {1..N}, in some order, can have
all 190 segment sums (sums of consecutive terms) distinct. An exhaustive computer search by two independent
engines, not a proof. Erdős #357 itself remains open.
New: OEIS A364153 lists a(1..13), ending 14, 17, 18; re-checked 2026-11-02. a(14..18) = 20, 22, 24, 26, 28 were
already in the atlas (A364153(18) = 28 via atlas PR #192).

Witness (N = 29):
  4, 1, 2, 10, 22, 15, 21, 24, 28, 29, 27, 16, 26, 20, 18, 23, 25, 19, 11
(19 distinct terms, max 29, 190 of 190 segment sums distinct by brute force).

Lower side: dropping the last term of a valid 19-term sequence only gives A364153(19) >= A364153(18) = 28, so
N = 28 had to be searched. (Atlas PR #192 and the atlas README had written ">= 29"; that step was wrong,
was caught before this search started, and is now corrected in the atlas too.)

Method: Ford's separately written engine ss_cleanroom.c (sha256 cc5d0992e8856f440c1fb851b3498eb09bcf23e84fdd9e20df6a6c28ff77d514,
unchanged), any order, the search at N split into 38 parts that together cover it, run on a redacted research
machine. Answers: NONE on 38 of 38 parts at N = 28 (1,023,848,894,976 nodes summed); FOUND at N = 29 on part 8
(8,768,075,865 nodes), the witness above. Controls exact, as named in the falsifier: C1 "a 16 23 part 0/14 NONE
nodes 586748917"; C2 "a 18 28 part 1/19 FOUND 1 3 2 7 22 25 28 21 27 18 24 20 23 14 26 15 17 19 nodes 4230925".
Second engine: the atlas's unchanged segsum.c (sha256 b0228c31c7ac9ea83a9ba7793f614a1c7450b75592cb66bc7d831672f068d145),
unpartitioned: "none a 19 28 1023848893903", exit code 0 (ran 2026-10-31 16:05 MDT to 2026-11-02 07:28 MST).

Falsifier (erdos_357_a364153_19_falsifier.txt, written before the search ran): G2 and G3 passed, and G4 passed
(the second engine says none at 28). G1: every one of the 38 parts at N = 28 printed exactly once, NONE, with no
timeout or crash. G1 as written also asks that the run log end "EXIT 0"; it does not yet, because the run is still
going on the remaining N = 29 and N = 30 parts and the second engine's "a 19 29". Those no longer bear on the
value and are left running as an extra cross-check. Disclosed here rather than waived silently.

Atlas PR https://github.com/techno-optimist/erdos-frontier-atlas/pull/205, merged as 46e89818db400953094e60cce93a6a60963238c1.
  New files in certificates/erdos-357 (RESULT.json, verify.py, ss_cleanroom.c, segsum.c unchanged):
    ford_searches_a364153_19.txt  sha256 b485230ed87eccfa51874546a850729dc40a2d1a5f81dde58e650cb5b153110d
    g4_crosscheck_a364153_19.txt  sha256 57491dfcf5571b141f82c4519abd10d1de312890b9c2d6693ccab6e8debfe76d
    verify_ford_a19.py            sha256 788d9497a9abde8af782a6fc7db82968fbce38209082ea30a54666689a919c60
  New promoted contract erdos-357-a364153-19; the atlas's companion cell is now A364153(20) >= 29. Locally: contract
  replay erdos-357-a364153-19 fast PASS, atlas validators PASS, pytest 277 passed. GitHub fast-claim-gate passed
  (17m35s; merged 2026-11-02 08:06 MST).

Disclosed: the two engines are independent code using the same segment-sum bitset method, not independent mathematics.
The fast replay checks receipt consistency and the witness, and re-runs one control; the certifying --full replay
(about 2.1e12 nodes) was not run in CI. Says nothing about A364132(30) or the asymptotic question of Erdős #357.
Not submitted to OEIS; Kevin submits OEIS extensions himself.