All-base bases and integral relations for the totient kernel
For every integer base k at least two and depth e at least one, the rational span of the sections n ↦ φ(k^j n+r) through level e has dimension k^e+1. An explicit retained family is a basis, every omitted section is an integer multiple of a retained section, and these reductions form a basis for all integral relations, with rank Σ_(1≤j<e) k^j. The entry also gives the full dyadic basis and an affine independence theorem. The retained positive residues satisfy k ∤ r, a weaker condition than gcd(k,r)=1. Martin’s earlier theorem already implies affine independence, and Coons proved…
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