The Information Pixel Universe: Axiomatic Foundations of Generative Space

                                                                                            Abstract

     The edifice of traditional mathematics is built upon the real number continuum and ZFC set theory, which have supported two centuries of brilliant development. Yet deep within this edifice lies a question that has never been formally asked: why are only closed paths, analytic functions, and stable configurations the protagonists of research? Why is "being" tacitly assumed to precede "becoming"?

Starting from five axioms free of real-number presuppositions, this paper reduces "being" to "becoming" and demotes "closure" from a tacit mathematical premise to an inevitable consequence of physical dynamics. Core conclusions include: the Even Modulus Survival Theorem—odd-modulus universes self-destruct due to dynamical skew, and the observer resides in an even-modulus universe as a survivor bias; the emergence of π and sin—not a priori constants, but the macroscopic signatures of discrete coupling under emergent continuity; the intrinsic boundary of the Planck scale—ρ > 0 is a necessary requirement for the self-consistency of the axiom system; the failure of renormalization is not a divergence, but the non-existence of the basic mathematical objects of continuous field theory below the Planck scale; the energy-scale dependence of parity violation—the parity non-conservation of the weak interaction may originate from the dynamical elimination of odd-modulus transient states, containing testable predictions for the LHC and FCC-hh.

All core theorems are derived purely algebraically on ℤ_q, without reliance on the continuum presupposition, actual infinity, or any unresolved mathematical conjecture. Rigor stems from regression rather than abstraction: discrete information pixels and coupling dynamics constitute a naturally physically compatible mathematical cornerstone—mathematics emerges directly from the minimal units of physical processes, rather than as an approximation to the continuum.

Keywords: information pixel; generative space; axiomatization; continuum hypothesis; even modulus survival; topological charge emergence; generalized Rayleigh quotient; parity non-conservation; endogenous observer; construction of natural numbers; potential infinity; ZFC as effective theory; heat death theorem