Every positive integer contains a perfect countdown. Reduce the products of N's multiplication table modulo N+1 and the residues descend: N, N-1, ..., 1, 0. The proof is one line — N ≡ -1 (mod N+1) — but the structure it reveals when placed inside an external base is worth exploring.
This notebook introduces two base-relative quantities derived from the interaction between an entity's internal modular structure and the base in which it's observed. The algorithm number A(N,B) = B - N - 1 measures internal-external frame mismatch. The degrees of freedom D(N,B) = B/GCD(N,B) measures accessible residue space. Neither is preserved when the base changes. The entity's intrinsic countdown is invariant. Its relational fingerprint is not.
In every base, exactly one entity achieves algorithm number zero with full residue coverage: the base-complement N = B-1. Among all bases where a given entity is frame-internal, it achieves this perfect fit in exactly one. The heatmaps and coprimality maps in the notebook make the structure visible. Everything is verifiable — pure Wolfram Language, no external dependencies.
https://www.wolframcloud.com/obj/b094a67e-f5d2-424d-a2a7-782c87066f8a
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