Foundational Axioms of Recognition Science and a Proof of Consistent Existence

This paper lays the mathematical foundation of Recognition Science. We state four axioms—A0 (Existence of elementary recognition cells), A1 (Dual Recognition between observer and observed), P2 (Minimal Overhead of information flow), and S (Exact Self-Similarity across scales)—and prove the set is free of internal contradictions.
Minimal overhead singles out a parameter-free dual-log cost Jphys(q) = 1 + q q−1−q + κ 1−q1 + q−1 , κ= 21−φ/π 2 , whose derivative changes sign exactly once on 0 < q < 1. The unique stationary point is q∗ = φ/π ≈0.515036214, and its location is independent of any UV/IR regulator. A Lean-formalised Sturm–Liouville proof certifies that this point is the global minimum of Jphys.
We construct an explicit logarithmic-spiral lattice of bidirectional Boolean links that realises all four axioms and attains the minimum, thereby fixing the absolute recognition length λrec. Via the causal-diamond entropy identity the same scale determines Newton's constant at the recognition scale; one-loop vacuum polarisation then runs the value to the laboratory number without new dials. Every downstream prediction—Planck units, vacuum energy, and the Riemann-operator slope k∗ = 2φ/π—therefore follows from the single dimensionless ratio q∗.