PFUSRC‑166 Topological Unfolding from 0 to 1: Mathematical Validation and Closed‑Loop Confirmation of the PFUSRC System's Foundational Axioms

Abstract

This paper takes the infinite product constant c_{Pell } ≈0.58058 from the Stevenhagen conjecture as its point of entry, conducting a reverse structural isomorphism comparison from a mathematical phenomenon back to the foundational axioms of the PFUSRC system. The constant is defined as: c_{Pell }=1-\prod_{j=1}^{\infty}\left(1-\frac{1}{2^{2 j-1}}\right) \approx 0.580577 ... This paper demonstrates: 1. The convergence behavior of this infinite product precisely corresponds to the PFUSRC ontological definition of "0 = coupling interface.” Each term lies between 0 and 1 and tends toward 1, causing the product to converge to a non‑zero value — not to nothingness, but to a stable fusion state at the interface achieved through infinite coupling. This is not an approach to vacuity, but an approach to complete fusion — structural fragments, through infinite coupling, ultimately disperse into the whole structure. 2. The fragmentation process starting from "1/2" validates the "division invariance” theorem of the “Unary System” in PFUSRC‑002. Any complete structural unit, when divided, yields subunits that remain 1 under topological equivalence. These subunits are topologically complete, but when infinitely many subunits are multiplied after fragmentation, their cumulative numerical effect produces a non‑trivial limit. This limit is neither 0 nor 1, but an “intermediate value” determined by the fragmentation rule — precisely the essence of c_{Pell}. 3. The constant c_{Pell}, understood as “half a biconical structure plus a residual perturbation,” validates the projection mapping of the biconical steady‑state structure in PFUSRC‑00. The 0.5 corresponds to half a breathing cycle; the 0.08058 corresponds to the residual phase difference direction generated by helical non‑integrability. Whether this residual value exactly equals 0.08058 requires establishing a precise mapping in subsequent work. This paper’s contribution is to identify the topological structural origin of 0.58058 (0.5 + residual), not to claim that this precise derivation has been completed. 4. The complete chain of reasoning traces back from the number‑theoretic constant to the “Theory of Logical Origin (013),” “Original Topological Steady‑State Proof (00),” “Physics Volume (001),” “Ontological Reconstruction of Zero (01),” and “Mathematics Volume (002)” — completing a structural isomorphism comparison from a number‑theoretic constant back to the system’s foundational axioms. This paper argues that the PFUSRC system is not merely a set of internally derived theoretical postulates; it can undergo reverse structural isomorphism comparison using external mathematical phenomena, establishing a self‑consistent closed chain from logical foundations to numerical projection phenomena. Some numerical mapping relations fall within the system’s topological hypotheses; rigorous analytical derivation is reserved for future work.