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module module high

IndisputableMonolith.Core.Complexity

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Core.Complexity assembles the Recognition Science treatment of P versus NP by importing submodules on J-cost Laplacians, J-frustration, circuit lower bounds, and non-naturalness. Researchers working on the RS resolution of P versus NP cite this module as the central collection point. The module itself contains no proofs and instead coordinates the imported results on frustration measures and their circuit-size consequences.

claimJ-frustration on CNF formulas, defined via the weighted hypercube graph with edge weights $|satJCost(φ,a)-satJCost(φ,a')|$, implies super-polynomial circuit size when the frustration value is 1.

background

The module Core.Complexity imports twelve submodules that together formalize the P versus NP gap inside Recognition Science. CircuitLedger motivates the question whether feed-forward Boolean circuits can simulate the global J-cost gradient that resolves SAT in linear recognition steps. JCostLaplacian defines the weighted graph on the Boolean hypercube whose vertices are assignments and whose edges carry the absolute difference in satJCost values. JFrustration introduces the binary measure JFrust that equals 0 precisely on satisfiable formulas and 1 on unsatisfiable ones, capturing the topological depth of the J-cost barrier. NonNaturalness verifies that this property meets the three Razborov-Rudich criteria for a natural proof.

proof idea

This is a module that imports and organizes submodules; no proofs are contained here. The overall structure is obtained by importing BalancedParityHidden, RSatEncoding, CircuitLedger, JCostLaplacian, SpectralGap, JFrustration, NonNaturalness, CircuitLowerBound and PvsNPAssembly, thereby assembling the chain from J-cost landscape to circuit-size lower bounds.

why it matters in Recognition Science

The module supplies the core machinery that feeds PvsNPAssembly and the broader RS P versus NP program. It supplies the three-step argument of CircuitLowerBound that high J-frustration forces super-polynomial circuit size, together with the non-naturalness result that blocks standard natural-proof obstructions. The setting directly addresses the T0-T8 forcing chain by embedding the J-uniqueness and phi-ladder structure into complexity measures.

scope and limits

depends on (11)

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