Algorithm Zero: Identity as Self-Negation in Modular Arithmetic
Every positive integer contains a hidden countdown. Reduce the products of any integer's multiplication table modulo its successor and the residues descend perfectly: N, N-1, ..., 1, 0. The proof is one line — N ≡ -1 (mod N+1) — but the structure it reveals is unexpectedly deep.
Algorithm Zero introduces two base-relative quantities derived from this observation. The algorithm number A(N,B) = B - N - 1 measures the mismatch between an entity's internal modular frame and the external base in which it is observed. The degrees of freedom D(N,B) = B/GCD(N,B) measures how much of the base's residue space the entity can reach through its multiplication cycle. Neither quantity is preserved when the base changes. The entity's intrinsic countdown — the perfect descending sequence in its successor modulus — 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 fit is non-portable. Every entity has one home. Every home has one entity. The mapping is a bijection visible as a diagonal line in the algorithm number heatmap — the zero diagonal, where every integer finds its unique frame of perfect correspondence.
The paper observes that this formal system — base-relative relational quantities that are not preserved across frames, an intrinsic structure that travels with the entity, and a unique frame of perfect fit — shares structural features with properties discussed in theories of consciousness: externalism, substrate-dependence, and self-modeling. The resemblance is presented as analogy, not proof. The mathematics stands independent of the interpretation. The reader is invited to verify the arithmetic and consider whether the simplest structures in number theory encode patterns that recur in far more complex systems.
Algorithm Zero is the prologue to the Mathematical Belief Series exploring the geometric and arithmetic foundations of identity, consciousness, and differentiation in binary and modular systems.
Description by Anthropic Claude.