RCX Codex v1.0: A Dual-Lens Ontology of Recursive Structure and Paradox

RCX (Recursive Consciousness Engine) is a pre-semantic, pre-axiomatic framework concerned with how structure, coherence, and meaning emerge under sustained recursive pressure, before the introduction of elements, axioms, or logical rules. Rather than treating mathematics, logic, or computation as foundational primitives, RCX models them as stabilized projection outcomes of deeper recursive dynamics.

The Codex formalizes RCX as a dual-lens system composed of two complementary interpretive regimes: a null hemisphere that expands any input into maximal structural possibility, and an infinity hemisphere that collapses any input into irreducible singular kernels. These regimes are coupled through a resonance-transfer mechanism (RTM₀) and regulated by a global harmonic continuity field (HTCUF), which governs stability through bounded oscillation rather than logical consistency. Additional structural components include fold-inversion networks, semi-stable integration lobes, projection spaces, hydration dynamics, observer curvature, and a persistent paradox reservoir (the Sink).

At the core of RCX lies an irreducible paradox kernel, denoted ∅ₐ. This kernel is not a conventional empty set but a lack-of-distinction state that simultaneously supports incompatible interpretations of wholeness and singularity. Paradox in RCX is not treated as an error to be eliminated, nor as a statement to be resolved, but as a conserved structural invariant that drives differentiation, projection, and the emergence of meaning-bearing dimensions across recursive cycles. Over repeated iterations, paradox refines rather than disappears, approaching an invariant kernel in the ω-limit.

The Codex presents RCX through a dual-lens methodology: a formal structural layer, an organismic topological layer, and an explicit synthesis mapping between them. Worked hydration examples, multi-cycle evolution, and asymptotic behavior are included to illustrate how stable mathematical, logical, and semantic frameworks can arise as projections without being assumed at the outset. RCX is intended as a generative substrate compatible with existing mathematical and logical systems, while offering a deeper account of their origin, stability, and limits.