Three Constants, Three Axioms

The axioms derive three fundamental constants, each from a different axiom.

c — the speed of light. Axiom C as speed. The causal bound — the maximum rate at which any record, any distinction, any piece of information can propagate. Derived from the stiffness ratio of the substrate: c = √(β/α), the square root of spatial stiffness divided by temporal stiffness (AP03).

ℏ — the Planck constant. Axiom B as minimum. The break — the smallest possible record. Derived from Stone’s theorem: the generator of time evolution on the axiom-derived Hilbert space has a unique self-adjoint generator with a minimum eigenvalue. That minimum is ℏ (AP12).

G — the gravitational constant. Axiom R as geometry. The record — the persistence of the break in the geometry of the arena. Without this persistence, the crack heals, the symmetry restores, the now stops, records cease. Gravity is the cost the arena pays to keep the break open. Derived in this paper.

Axiom S does not produce a constant. It produces the arena’s topology — the two sectors, the mirror, the stage on which the other three act. Three constants. Three axioms. One break. The mapping is one-to-one: c ↔ C (Constraint), ℏ ↔ B (Break), G ↔ R (Record), topology ↔ S (Symmetry).