The Categorical Mixing Problem: Axiomatic Collision as a Formal Phenomenon (The Categorical Mixing Problem: A Research Programme, Paper III of III)

This paper provides the formal completion of the Categorical Mixing research programme initiated in Papers I and II. Where Paper I established the philosophical framework distinguishing Truth as the meta-axiomatic ground from Reality as the physical-axiomatic domain (DOI: 10.5281/zenodo.18945326), and Paper II applied the framework diagnostically to the P vs NP problem (DOI: 10.5281/zenodo.18946375), this paper formalises the central concepts of the programme within the language of formal logic and the foundations of mathematics. Precise definitions are introduced for a formal system, an axiomatic level, an axiomatic boundary, and a categorically mixed formula. The central theorem of the programme is stated and proven: that the presence of categorical mixing in a formula is necessary and sufficient for its irresolvability from within any single consistent formal system. This theorem is applied to three canonical cases — the Liar's Paradox, Gödel's Incompleteness Theorems, and the P vs NP problem — demonstrating in each case that categorical mixing is formally present and constitutes the structural explanation for the observed irresolvability. The paper concludes by examining the implications of the framework for set theory, type theory, and the foundations of mathematics more broadly, and identifies a class of open problems — including the Continuum Hypothesis and large cardinal axioms — that the framework predicts are instances of categorical mixing awaiting correct reformulation. This is the third and final paper of the research programme.