The Unitary Reference Principle: A Foundational Reform of Quantitative Mathematics

ABSTRACT Second Edition — April 2026. This version incorporates findings from four companion papers completed after the first edition. Key additions: Step 0 (Declare R) formalized as a mandatory first step; incommensurable references and bridge reference framework; infinite repeating decimals as depletion-renewal cycles; cross-domain confirmations from geometry, information theory, and physical reality. Key correction: the Riemann Hypothesis section is updated — a complete proof is presented in Brogley (2026c), doi.org/10.5281/zenodo.19735713. See the "Note on This Second Edition" section inside the paper for a full itemized list of changes from v1. This paper proposes the Unitary Reference Principle (URP), a foundational reform of quantitative mathematics asserting that all meaningful quantity must be expressed as a fraction of a declared whole. The number 1 is identified as the only complete unit; integers greater than 1 represent counts of discrete wholes rather than magnitudes exceeding unity. The infinite interval between 0 and 1 is established as the domain of all fractional reality. Zero is redefined not as a number or reachable value but as pure non-existence — the absolute absence of quantity that cannot be arrived at, operated upon, or declared as a reference, because any engagement with it transforms it into something it is not. The expression 0/R — contextual emptiness within a declared reference — is distinguished from standalone zero, which is non-existence itself. Negative numbers are reframed as expressions of deficit within a declared reference, not as possessions of impossible quantities — a position supported by the Law of Conservation of Energy. Decimal notation is examined as an existing, unrecognized encoding of this framework. The paper demonstrates that the URP resolves critical failures in conventional mathematics including division by zero and indeterminate forms, improves the completeness and information-preservation of arithmetic expressions, and provides a new lens through which six of the world's most significant unsolved mathematical problems — including the Riemann Hypothesis, Goldbach Conjecture, and Navier-Stokes Existence and Smoothness — may be reframed. Historical antecedents from Egyptian, Greek, and Indian mathematical traditions are examined, revealing that elements of this framework have appeared across civilizations for over three millennia without synthesis. The URP unifies these insights into a single coherent framework with implications across mathematics, physics, statistics, philosophy, and education. Keywords Unitary Reference Principle · number theory · philosophy of mathematics · zero as non-existence · declared denominator · Riemann Hypothesis · Goldbach Conjecture · Navier-Stokes · Birch-Swinnerton-Dyer · elliptic curves · conservation of quantity · foundational mathematics Note Companion People's Edition available separately. Full academic paper with 9 chapters, 12 figures and tables, 23 references.