On the Extension of Peano’s Axioms to Total Functions: Triadic Information Dynamics and Geometric Invariants Within Specialized Physical Systems

John Archibald Wheeler’s celebrated phrase, "It from Bit," proposed that every physical quantity derives its meaning from information, the binary yes/no act of measurement. The hypothesis remains evocative but mathematically under-specified: the prevailing formal systems of arithmetic and logic, grounded in Peano’s axioms, describe static entities rather than dynamical information exchange. This paper introduces a completion of Peano’s axioms through two additional principles, the Total Function Axiom and the Relativistic Closure Axiom, that admit three-valued flux and enforce path-independent conservation. The result, termed It from Trit, demonstrates how triadic information systems naturally generate geometric structures (Pascal’s triangle, the Fano plane, Yang–Baxter relations) and numerical invariants on curved manifolds. We show that a minimal physical model, three interacting balls under inverse-square equilibrium within a spherical boundary, yields exactly 42 stable positions whose curvature invariants (glyph numbers) include the fundamental constants of mathematical physics: π, e, ϕ, algebraic roots, special functions, and notably 137 (the inverse fine-structure constant). These constants emerge not by assignment but as necessary projections from an eight-dimensional flux space to four-dimensional observables under triadic closure. The framework resolves the Monty Hall paradox as a natural consequence of triadic conservation and suggests that the 26 free parameters of the Standard Model may be geometric necessities rather than adjustable inputs. We provide algorithmic implementations and discuss the importance of further development with an emphasis on falsifiable predictions.