The Consciousness Characterisation Theorems of the R12 Rendering Algebra: Axiomatic Constraints on the Complementary Object X from the S-Final Principle

Abstract

The S-Final principle of Array Cosmology requires Ωreal + Ωimag = 0, where Ωreal = +1 corresponds to the rendered physical universe. This paper characterises the complementary object X responsible for Ωimag = −1 without defining it as
“consciousness” or any other pre-theoretic concept. Ten theorems are proved from the R12 rendering algebra alone: (1) X = −1 is the unique central involution satisfying the S-Final trace condition; (2) X exists if and only if matter exists; (3) the
coupling κ = Σ(s) Γ(γ) factorises uniquely into a universal signal and a local receiver; (4) Γ(γ) > 0 for all γ ≥ 0 (non-annihilation); (5) the self-reference residual δ(12) > 0 (incompleteness); (6) partial observation is perspective-dependent while
total observation is objective; (7) self-recognition requires exactly two rendering cycles (4π rotation);