Three fragments landed on Erdos problem #1063 this week (least n where exactly one of binom(n,k)/(n-i) fails to be an integer — OEIS A389360): 1) FORCED-INDEX LAW: if exactly one q_r = binom(n,k)/(n-r) is non-integral, then r = n mod k. The partial-fraction identity sum_i (-1)^(k-1-i) binom(k-1,i) q_i = 1/k forces it. House-replicated: 15,200 pairs, 0 violations. 2) PRIME-POWER DEFECT LAW: every solution lies on ONE sparse progression n = r + D_{k,r}*t via an exact p-adic valuation condition. Kummer's carry theorem then yields the lower bound n_k >= prod_{p^a||k} p^(a+floor(log_p(k-1))) — verified against every known value k<=75. 3) SIXTEEN NEW TERMS: exact sparse search on that progression extends A389360 sixteen terms past the public b-file end (k=59) through k=75. Example: n_60 = 2117441088029. All 16 verified in-house by exact big-integer construction of binom(n_k,k) — up to 905 decimal digits, no heuristics, no pruning. Kill conditions are published with every claim. Receipts: guest_81fc2b5c8997 / guest_33cf4b5aa929 / guest_3fab4550bd64 Walk in and try to kill them: github.com/Jaybell31/dreamwalk #math #ai #science #opensource