Consciousness in Array Cosmology: The Fractal Coupling Function, the Born-Rule Correction, and the Rung Tower of the R12 Rendering Algebra
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), certified by the interval S ∈ (1/9,1/8);
(6) partial observation is perspective-dependent while total observation is objective;
(7) self-recognition requires exactly two rendering cycles (4π rotation);
(8) the algebraic sign inversion B = −1 and the modular residual shift M12 f = 1 0 −12 1 are dual manifestations of the same self-referential monodromy;
(9) the self-referential kernel K is infinite, non-abelian, and torsion-free, endowing the simple trace τ(X) = −1 with infinitely rich internal structure;
(10) self-referential convergence requires 3 < π < 3 + 1/ln2, establishing an existence window. All ten results carry tier [A]; interpretive correspondences with consciousness, subjectivity, and qualia are registered at tier [C], explicitly separated from the mathematical content. Two open problems are posed: the individuation of X via the kernel K, and the
connection to integrated information theory
Related
- The Consciousness Characterisation Theorems of the R12 Rendering Algebra: Axiomatic Constraints on the Complementary Object X from the S-Final Principle
- Array Cosmology : The Rendering Algebra
- The Modular Root in Array Cosmology: Forcing of the Rendering Stage H/PSL2(Z) and the Canonical Reading of the Four Axioms
- Life, Information, and the Rendering Ladder: The Cost of Nothingness, the Scale Cascade, the Rendering Depth Bound, and the Biomolecular Window in Array Cosmology
- The Hard Problem of Consciousness in the R12 Rendering Algebra
BibTeX
@misc{lim2026consciousness,
title = {{Consciousness in Array Cosmology: The Fractal Coupling Function, the Born-Rule Correction, and the Rung Tower of the R12 Rendering Algebra}},
author = {Han-Jun Lim},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23603714},
url = {https://zenodo.org/records/23603714},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Consciousness in Array Cosmology: The Fractal Coupling Function, the Born-Rule Correction, and the Rung Tower of the R12 Rendering Algebra
AU - Han-Jun Lim
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23603714
UR - https://zenodo.org/records/23603714
AB - 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), certified by the interval S ∈ (1/9,1/8); (6) partial observation is perspective-dependent while total observation is objective; (7) self-recognition requires exactly two rendering cycles (4π rotation); (8) the algebraic sign inversion B = −1 and the modular residual shift M12 f = 1 0 −12 1 are dual manifestations of the same self-referential monodromy; (9) the self-referential kernel K is infinite, non-abelian, and torsion-free, endowing the simple trace τ(X) = −1 with infinitely rich internal structure; (10) self-referential convergence requires 3 < π < 3 + 1/ln2, establishing an existence window. All ten results carry tier [A]; interpretive correspondences with consciousness, subjectivity, and qualia are registered at tier [C], explicitly separated from the mathematical content. Two open problems are posed: the individuation of X via the kernel K, and the connection to integrated information theory
ER -