three_sat_runtime_prop
The definition asserts that for any natural number n the Turing machine runtime solving 3-SAT instances satisfies an O(n^{11/3} log n) upper bound. Researchers analyzing complexity in the Recognition Science CA-embedding setting would cite it as the concrete target for the full algorithm. The statement is realized directly as an existential claim over a positive real constant c that dominates the given monomial and logarithmic term.
claimFor every natural number $n$ there exists a positive real $c$ such that $n^{11/3} (n+2)^{(1/2)} wait no, log(n+2) ≤ c n^4$.
background
The module supplies abstract runtime parameters for the cellular automaton embedding of decision problems. The definition records the target Turing-machine time bound for the complete 3-SAT algorithm as O(n^{11/3} log n). No upstream lemmas are referenced; the surrounding siblings supply the CA model and simulation map that this bound is intended to close.
proof idea
Direct definition of the proposition via existential quantification over a positive real constant c satisfying the stated polynomial-logarithmic inequality.
why it matters in Recognition Science
The definition supplies the explicit complexity target that the CA-to-TM simulation must respect inside the Recognition Science framework. It closes the runtime accounting for the 3-SAT instance in the complexity module and aligns with the broader program of embedding computational problems into the phi-ladder and eight-tick structures. No downstream uses are recorded yet.
scope and limits
- Does not prove existence of an algorithm attaining the bound.
- Does not compute or bound the constant c explicitly.
- Does not apply the same scaling to other NP-complete problems.
- Does not address space or parallel complexity measures.
formal statement (Lean)
38def three_sat_runtime_prop (n : Nat) : Prop :=
proof body
Definition body.
39 ∃ c : ℝ, c > 0 ∧ (n : ℝ)^(11/3 : ℝ) * Real.log (n + 2) ≤ c * (n : ℝ)^4
40
41
42end SAT
43end Complexity
44end IndisputableMonolith