We rigorously analyze the spectral properties of signed non-backtracking operators in random 3-SAT instances and distinguish between linear detection (belief propagation thresholds) and nonlinear detection (SAT thresholds) via cluster counting.

We formalize functionals, functors, and λ-calculus operators that encode solution clusters and demonstrate the limitations of local linear operators.

We provide proofs using a Dialectical Frame (¬, ↔ | Boundary Loop) and explore extensions to topological and categorical spaces. A new thought experiment introduces shape-synesthesia functors and twistor spaces linking η, τ, and cohomology classes.

Practical applications, counterexamples, and algorithmic predictions are presented alongside minimal Lisp implementations.

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Abstract

We rigorously analyze the spectral properties of signed non-backtracking operators in random 3-SAT instances and distinguish between linear detection (belief propagation thresholds) and nonlinear detection (SAT thresholds) via cluster counting. 

We formalize functionals, functors, and λ-calculus operators that encode solution clusters and demonstrate the limitations of local linear operators. 

We provide proofs using a Dialectical Frame (¬, ↔ | Boundary Loop) and explore extensions to topological and categorical spaces. A new thought experiment introduces shape-synesthesia functors and twistor spaces linking η, τ, and cohomology classes. 

Practical applications, counterexamples, and algorithmic predictions are presented alongside minimal Lisp implementations.