Universal Axiomatic Algebra for Dissipative Information Systems on Compact S³ Manifolds

Existing information geometry on compact manifolds [1, 2] and recent IJGMMP S³ spectral geometry papers [49, 51, 53] construct static statistical metrics but lack a complete, minimal axiom system for open dissipative information systems with critical self-reference bifurcations. The STC-S3 framework [S3] provides domain-geometric analytic solutions but lacks unified operator algebra and complete axiom foundations. This paper establishes three minimal complete axioms based on the energy-sea ontology [T1–T4], constructs a full operator algebra — boundary projection ℬ, gate switch G, encoding mapping ℱ, residual entropy accumulation ΔS, and critical saturation threshold K_c — and derives the unique inner-mapping solution that serves as the mathematical foundation for self-reference (the "first driving force"). The axiom system is embedded into the S³ Hopf coordinate geometry [S1], yielding manifold-corrected critical thresholds κ_dom(Γ) and κ_crit(Γ), together with an improved closed-boundary iteration scheme for Gadget-4 N-body simulations that eliminates periodic-box topological artifacts. The framework is unified with Shannon channel theory [7] and Landauer information-erasure thermodynamics [8], proving that the present construction is the compact-manifold generalization of classical information geometry [1, 2]. Both positive (self-reference emergence) and negative (zero self-reference) complete criteria are derived, distinguishing reversible temporary self-reference from irreversible linguistically-fixed self-reference via two distinct entropy thresholds. This paper serves as the shared mathematical foundation for the entire STC series — V2 (quotient manifolds), V3 (4D FLRW dynamics), and the PLR carbon-silicon self-reference papers — providing a non-repetitive, standardized toolset across all scales.