Tier-0: A Fixed-Point Admissibility Grand Unified Theory of Mathematics

This paper proposes a candidate grand unified theory of mathematics in a strict structural sense: a single, auditable principle that generates a universal admissible universe of mathematical objects, explains when objects arising in different branches are genuinely the same, and forces agreement of invariants across admissible realizations. The unifying mechanism is Tier-0 admissibility, formulated entirely in order-theoretic and fixed-point terms. Starting from a universal carrier, a complete lattice obtained canonically via MacNeille completion of a preorder of mathematical presentations. The framework specifies three monotone idempotent operators: • normalization (presentation/gauge reduction),• robust-core extraction (removal of non-stable content), and• closure/completion (stability under required limits or universal properties). Their composite recursion defines admissible mathematics as the fixed-point universe of the system. By the Knaster–Tarski theorem, this admissible universe is guaranteed to exist and itself forms a complete lattice, yielding canonical least and greatest admissible envelopes and a precise three-stage failure taxonomy. Unification is achieved through a correspondence calculus: the paper defines Tier-0 correspondences between domains as monotone maps that intertwine admissibility up to admissible equivalence, proves that such correspondences compose, and shows that admissible objects in any participating domain determine canonical universal admissible normal forms. Under a mild reflectivity (Galois connection) hypothesis, the paper proves a canonical realization theorem: every universal admissible object admits a canonical admissible realization in each reflective domain, unique up to domain admissible equivalence. This yields a precise, non-rhetorical meaning of “the same mathematical object appears across different branches.” The paper further defines Tier-0 invariants as quantities stable under the admissibility recursion and proves an invariant consensus lemma, guaranteeing that such invariants agree across all admissible realizations in all domains. To demonstrate non-vacuity and breadth, the framework is instantiated across four major pillars, foundations (logic / HoTT), geometry (correspondences and motives), arithmetic (anabelian reconstruction), and analysis (spectral rigidity), organized by a single operator-role ledger rather than domain-specific axioms. The result is an operational Math-GUT schema: to integrate a new branch, one specifies a carrier, a Tier-0 triple, and a correspondence into the universal carrier; all unification consequences then follow formally. No physical axioms, empirical parameters, or domain-privileging assumptions are required.