Geometry in Algebra. Eversion: An Ontological Framework for Quantum Mechanics and Special Relativity

We propose an ontological framework for the formalism of quantum mechanics and special relativity, based on three principles: Conciliation (the projective work of bringing multiple perspectives into a single coherent state), Fluidity (the irreversible, anti-quantized continuity of process, identified with Time), and Eversion (an operator, denoted IW, which turns a structure inside out, generating a boundary between an inner and an outer domain).

The framework yields several formal correspondences. The successive application of the Eversion operator generates the standard hierarchy of normed division algebras: ℝ → ℂ ≅ Mat(2 × 2, ℝ) → ℍ ≅ Mat(2 × 2, ℂ) (along the XY axis) and ℝ → ℂ → 𝕆 ≅ Mat(2 × 2 × 2, ℝ) (along the Z axis). The first branch is identified with the gauge groups U(1) and SU(2) of the electromagnetic and weak interactions; the second branch with SU(3) of the strong interaction, in which the multiplication law of the Fano plane appears naturally as a consequence of the rank-3 tensor structure (and explains the loss of associativity for octonions). On this basis SU(3) is more accurately represented as Mat(2 × 2 × 2, ℝ) than as Mat(8 × 8).

In the macroscopic limit the framework reproduces the Minkowski metric and the Lorentz transformations through quaternionic representation of spacetime, with the coefficient c interpreted as the rate of curvature of the Eversion of Time into Space. Mass acquires the form m² = T² − X². Dark matter and dark energy are interpreted as artifacts of incomplete Eversion of Matter and Space respectively. Spin ½, quantum entanglement, and the spherical wavefront of a photon receive natural geometric descriptions as projections of inner Eversion structures into observable space.

The framework is offered as a working ontology — a language in which the existing formalism of quantum mechanics becomes meaningful, and in which it extends naturally to a broader class of phenomena involving Conciliation between multiple perspectives.