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module module high

IndisputableMonolith.Core.Complexity

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The Core.Complexity module aggregates the Recognition Science treatment of computational complexity, reducing the P versus NP question to whether Boolean circuits can simulate J-cost gradients on the hypercube. Researchers deriving complexity bounds from the J-functional equation would cite it as the central assembly point. The module contains no local proofs and consists entirely of imports from submodules that define the J-cost Laplacian, frustration measure, and circuit lower bounds.

claimHigh J-frustration on a CNF formula $\phi$ implies that any feed-forward Boolean circuit simulating the global J-cost gradient requires super-polynomial size; JFrust$(\phi)=0$ precisely when $\phi$ is satisfiable.

background

The module sits inside the Recognition Science derivation of physics and computation from the single functional equation J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). It imports the J-cost Laplacian construction on the Boolean hypercube {0,1}^n, where vertices are assignments and edge weights equal the absolute difference in satJCost. J-frustration is defined as the binary indicator that is 0 exactly on satisfiable formulas and 1 on unsatisfiable ones, measuring the topological depth of the J-cost barrier around the satisfying region.

CircuitLedger supplies the motivating reduction: the P versus NP gap is equivalent to asking whether a Turing-equivalent feed-forward circuit can reproduce the global J-cost gradient that resolves SAT in linear recognition steps. NonNaturalness records that J-frustration satisfies the three Razborov-Rudich conditions for a natural property, thereby placing the lower-bound argument outside the natural-proofs barrier.

proof idea

This is a definition module, no proofs. Its structure is an import aggregator that wires together the three-step argument of CircuitLowerBound: (1) definition of the weighted hypercube graph via JCostLaplacian, (2) extraction of the binary JFrust measure from JFrustration, and (3) derivation of super-polynomial size lower bounds once frustration is shown to be non-natural.

why it matters in Recognition Science

The module supplies the formal core for the P versus NP program inside Recognition Science, feeding the assembly in PvsNPAssembly and the circuit-size claim in CircuitLowerBound. It extends the T0-T8 forcing chain (J-uniqueness, phi fixed point, eight-tick octave, D=3) into computational complexity by showing that the same J-cost functional forces non-natural barriers that separate polynomial from super-polynomial circuit resources.

scope and limits

depends on (11)

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