What Mathematics Is: Constraint, Object, and Derivation in the Foundations of Mathematics

Abstract Mathematics is the most secure of the sciences, yet its foundations remain contested after more than a century of sustained inquiry. This paper argues that the persistent failure of foundational programmes, from logicism and formalism to intuitionism and set theory, arises from a common structural error: each programme privileges a single aspect of mathematics while suppressing the others. Drawing on the Triaxial Existential Field (TEF), this paper analyses mathematics under three irreducible registers: mathematical objects as C-loci (determinate identities under constraint), mathematical structure as R-constraint (admissibility conditions defining what exists and how it relates), and mathematical proof as P-derivation (sequential actualisation accessing truth under cost). This triaxial framework is applied to six foundational problems: infinity, Gödel's incompleteness theorems, the continuum hypothesis, the Liar paradox, the unreasonable effectiveness of mathematics in physics, and the Banach-Tarski theorem. In each case, the paradox or puzzle is shown to arise from a register error and to yield to correction once the register assignment is identified. The paper distinguishes three grades of resolution: dissolution (where the paradox rests on a structural error and no residual puzzle remains), reframing (where a forced dichotomy is removed but a genuine open question may persist), and structural interpretation (where an existing result is given a register reading that unifies it with cross-domain phenomena). The central claim is that mathematics is the study of R-constraint in its pure form, and that the relationship between mathematics and physics is one of register identity (the same register studied in abstract and concrete modes) rather than mysterious correspondence. Explicit falsification conditions are provided for each case.