polynomial_time_3sat_algorithm_hypothesis
The definition encodes the hypothesis that a polynomial-time algorithm exists for deciding 3-SAT on n variables whenever the ProgramTarget predicate holds. Complexity researchers examining geometric isolation and backpropagation approaches to SAT would cite it when tracing potential P=NP arguments. The definition directly states the required algorithm properties without any derivation or construction steps.
claimFor a natural number $n$, the hypothesis states that if the program target of size $n$ holds, then there exists an algorithm mapping CNF formulas over $n$ variables to optional assignments such that every satisfiable formula yields a satisfying assignment $x$ with evalCNF$(x,φ)=$ true and every unsatisfiable formula yields none.
background
The module builds fully-determined backpropagation states from total assignments for SAT instances. An Assignment for $n$ variables is a total Boolean function on the variables (Fin $n$ → Bool or Var $n$ → Bool). CNF is a structure consisting of a list of clauses. evalCNF evaluates the conjunction of clauses under an assignment, returning true for the empty formula. Satisfiable asserts existence of an assignment making evalCNF true.
proof idea
This is a direct definition of the hypothesis proposition. No lemmas or tactics are applied; the body simply assembles the quantified statement from the upstream CNF and Assignment primitives.
why it matters in Recognition Science
The definition supplies the hypothesis premise for the downstream theorem polynomial_time_3sat_algorithm. Documentation records its falsification via failed deterministic isolation, provably incomplete propagation on expander-Tseitin formulas, and the Baker-Gill-Solovay relativization barrier showing no relativizing proof of P=NP is possible.
scope and limits
- Does not construct or prove existence of any polynomial-time 3-SAT algorithm.
- Does not overcome relativization barriers to P=NP.
- Does not supply geometric structure intrinsic to arbitrary SAT instances.
- Does not discharge the propagation-completeness premise.
formal statement (Lean)
179def polynomial_time_3sat_algorithm_hypothesis (n : Nat) : Prop :=
proof body
Definition body.
180 ProgramTarget n →
181 ∃ (alg : CNF n → Option (Assignment n)),
182 (∀ φ, Satisfiable φ → ∃ x, alg φ = some x ∧ evalCNF x φ = true) ∧
183 (∀ φ, ¬Satisfiable φ → alg φ = none)
184