This is Version 4 of Paper I of the VIBE Meta Theory series. It constructs a minimal preontological operator algebra that serves as the foundational substrate for a five-part programme on structural foundations of pre-geometric physics. The paper builds a free monoid of operator words over a four-element generator set and establishes: (i) a structural minimality theorem showing that the cardinality four is forced under a two-axis classification together with a type-separating evaluation into the endomorphism monoid of a state set; (ii) four explicit sufficient conditions under which the generated endomorphism class is structurally non-commutative (Absorber, Projector, Involution–Contractor, and Fixed-Point-Shift), each accompanied by a concrete witness model; (iii) a monoidal Banach fixed-point theorem for contractive operator words under a minimal technical metric assumption; (iv) operator-induced kernel partitions, a refinement lattice, a derived information functional monotone under refinement, and a pre-chronological ordering on operator words arising from compositional refinement without presupposing temporal primitives; (v) a universal invariance theorem for the partition family generated by operator kernels, established internally to the class of evaluation-preserving homomorphisms; (vi) an additive generator cost functional, a κ-induced pseudometric on the state set, and an observational quotient structure; and (vii) a Structural Closure Theorem with an explicit no-smuggling audit, showing that the modules of free composition, dynamics, partition and ordering, and cost and quotient are jointly sufficient to define differentiation, ordering, cost, and observational quotients without introducing geometric, probabilistic, informational, or temporal primitives. The paper is, by design, the foundational substrate of the series. It does not derive specific physical field equations, does not reconstruct spacetime, does not claim emergence of quantum mechanics, and makes no empirical predictions. Its contribution is a minimal, non-circular, and structurally auditable algebraic platform together with four explicit conditions under which the framework would be structurally falsified. Papers II–V of the series develop, on this substrate and under additional explicitly declared assumptions, the emergence of geometric structure, linear representation regimes, their joint compatibility, and the empirical projection and calibration layer. This paper constitutes Part I of the VIBE Meta Theory research programme. The broader programme is documented under the Master DOI 10.5281/zenodo.19139920. Keywords: preontological operator algebra, free monoid, structural non-commutativity, operator-induced partitions, pre-chronological ordering, structural closure, pre-geometric physics, foundations of physics, mathematical physics.
Tomas Leinich (Fri,) studied this question.