nonNaturalnessCert
The nonNaturalnessCert assembles the three properties that certify high-depth J-frustration properties cannot serve as natural proofs in the Razborov-Rudich sense. Circuit-complexity researchers cite it when constructing the master P-vs-NP certificate. The definition is a direct record construction that wires the zero-fraction theorem for constant-true functions, the unit upper bound on false-point fractions, and the barrier-bypass lemma.
claimA certificate consisting of three assertions: (i) for every $n$, the constant-true function on $n$ bits has false-point fraction zero; (ii) every Boolean function on $n$ bits has false-point fraction at most one; (iii) for every threshold $τ>1$, the high-depth property (false-point fraction $≥τ$) cannot be simultaneously constructive, large, and useful.
background
FalsePointFraction(f) measures the fraction of inputs on which Boolean function f evaluates to false. HighDepthProp(τ) is the predicate asserting this fraction is at least τ. A property is natural precisely when it is constructive (computable in poly(2^n) time), large (holds for a 1/poly(n) fraction of all truth tables), and useful (implies no poly-size circuits). The upstream theorem jfrust_not_natural states that for τ>1 the high-depth property fails naturalness because it is not large.
proof idea
The definition is a one-line record construction. It supplies the const_true_zero field from const_true_zero_fraction, the fraction_le_one field from falsePointFraction_le_one, and the barrier_bypass field from jfrust_not_natural.
why it matters in Recognition Science
This definition supplies the non_natural field of pvsNPMasterCert in the PvsNPAssembly module, closing the non-naturalness leg of the master certificate that combines J-cost Laplacian, spectral gap, J-frustration, and circuit lower bounds. It formalizes the bypass of the Razborov-Rudich natural-proof barrier for high-depth J-frustration properties.
scope and limits
- Does not prove P ≠ NP.
- Does not exhibit an explicit large and useful natural property.
- Does not address existence of one-way functions.
- Does not compute false-point fractions for non-constant functions.
Lean usage
non_natural := nonNaturalnessCertformal statement (Lean)
158def nonNaturalnessCert : NonNaturalnessCert where
159 const_true_zero := const_true_zero_fraction
proof body
Definition body.
160 fraction_le_one := fun n f => falsePointFraction_le_one f
161 barrier_bypass := jfrust_not_natural
162
163end -- noncomputable section
164
165end NonNaturalness
166end Complexity
167end IndisputableMonolith