IndisputableMonolith.Complexity.CircuitLowerBound
The CircuitLowerBound module states the Algebraic Restriction hypothesis deriving circuit depth lower bounds from J-frustration measured on the J-cost landscape. Complexity researchers examining P vs NP through Recognition Science would cite it to connect RS operators to Boolean circuit models. The module presents the claim as a direct hypothesis modeled on Karchmer-Wigderson bounds but specialized to multiplicative J-cost differences.
claimFor any Boolean circuit $C$ of size $S$ computing a function $f$ whose J-frustration on the J-cost landscape is at least $τ$, the depth of $C$ satisfies depth$(C) ≥ log₂(τ/S)$.
background
The module operates in the setting where the P vs NP gap reduces to whether feed-forward Boolean circuits can simulate the global J-cost gradient that R̂ uses to resolve SAT in O(n) recognition steps, as stated in the CircuitLedger module. It relies on the J-Cost Laplacian, which equips the Boolean hypercube with edge weights |satJCost(φ, a) - satJCost(φ, a')| for Hamming-distance-1 assignments. J-Frustration quantifies the topological depth of the J-cost landscape barrier around a formula's satisfying region, while RSatEncoding supplies the non-natural polytime certifier property of R̂ for satisfiable k-CNF instances.
proof idea
This is a hypothesis module presenting the Algebraic Restriction without formal proof. The argument rests on the observation that the multiplicative structure of J-cost differences forces any evaluation to access Ω(n) input bits simultaneously, yielding the stated logarithmic depth bound as an analog of the Karchmer-Wigderson relation specialized to the J-cost structure.
why it matters in Recognition Science
The module supplies the Algebraic Restriction hypothesis that feeds directly into the PvsNPAssembly module, where it supports the conditional Path A for P ≠ NP by implying exponential circuit size from high J-frustration. It closes one link in the chain from J-frustration non-naturalness to circuit lower bounds while leaving the uniform topological obstruction path to sibling modules.
scope and limits
- Does not prove the Algebraic Restriction hypothesis.
- Does not supply explicit numerical constants or tighter bounds.
- Does not treat topological obstructions or uniform variants.
- Does not extend to circuit models outside the Boolean feed-forward setting.
used by (2)
depends on (5)
declarations in this module (11)
-
structure
AlgebraicRestrictionHyp -
structure
TopologicalObstructionHyp -
theorem
circuit_lower_bound_algebraic -
theorem
circuit_lower_bound_topological -
structure
UniformTopologicalObstructionHyp -
theorem
exp_dominates_poly -
theorem
p_neq_np_conditional -
structure
AlgebraicRestrictionProofSketch -
def
the_proof_sketch -
structure
CircuitLowerBoundCert -
def
circuitLowerBoundCert