A Relational Ontology Grounded in the Moonshine Module: ZFC-Equivalent Foundations via a Single Primitive Binary Relation

This paper addresses a foundational question in the philosophy of
mathematics: what is the minimal ontological primitive from which the
full machinery of set theory can be derived? We construct a
ZFC-equivalent mathematical ontology grounded in a single primitive
binary relation (x ← y) — a directed edge in a graph — and demonstrate
that all nine axioms of Zermelo–Fraenkel set theory with Choice (ZFC)
translate directly into the topology of this relational graph. Set
membership, well-foundedness, and the axiom of choice all receive
explicit graph-theoretic interpretations, collected in a unified
translation table.

The mathematical structure hosting this ontology is the Moonshine
module V♮, the unique infinite-dimensional algebraic object of central
charge c = 24 whose symmetry group is the Monster group M — the
largest sporadic finite simple group. V♮ is constructed over the
24-dimensional Leech lattice Λ₂₄, the densest known sphere packing in
24 dimensions, embedded in a 26-dimensional conformal field theory.

The framework — called Monolit (M) — establishes six axioms (A0–A6)
and three structural postulates (P0–P2). Key results include: (1) the
positive-definiteness of the structural tension form τ̃ is derived as
a theorem via the Cartan–Killing form of semisimple Lie algebras,
rather than assumed as an axiom; (2) well-foundedness of the relational
system is not postulated but follows geometrically from the
sphere-packing property ‖λ‖² ≥ 4 of Λ₂₄, distinguishing this
framework from Aczel's non-well-founded set theory; (3) the Monster
group's action provides a finite upper bound on state transitions from
any given node, partitioning the infinite state space into finite local
orbits.

The framework explicitly rejects any empirical interpretation of
bosonic string theory; the 26-dimensional domain is used exclusively
as a mathematically rigorous target space. The projective dynamics,
observer mechanics, entropic time, and connection to General Relativity
are developed in a companion paper.

This work was conducted independently, without institutional
affiliation or funding.