rs_adversarial_lower_bound
Any decoder g that inspects only a proper subset M of variables can be fooled by suitable choice of secret bit b and full mask R. Complexity theorists studying natural proofs would cite this as the Recognition Science formulation of the barrier against local certificates for unsatisfiability. The proof reduces immediately to the adversarial_failure result in BalancedParityHidden.
claimFor any natural number $n$, any finite subset $M$ of the indices $[n]$, and any function $g$ from assignments on $M$ to a bit, there exist a bit $b$ and a full assignment $R$ on $[n]$ such that $g$ applied to the restriction of the encoded mask (flipped according to $b$) differs from $b$.
background
The BalancedParityHidden module supplies the encoder enc(b, R) that returns R when b is false and the bitwise negation of R when b is true, together with the restrict operation that projects a full assignment onto the coordinates in M. These definitions formalize a hidden-mask encoding whose local views can be adversarially controlled. The present theorem lives inside the R̂ SAT Encoding module, whose MODULE_DOC states that R̂ supplies a non-natural polytime certifier separating recognition-time complexity from Turing computation time, with the natural-proof-barrier connection listed as a hypothesis.
proof idea
The proof is a one-line wrapper that directly invokes BalancedParityHidden.adversarial_failure on the supplied M and g.
why it matters in Recognition Science
This supplies the combinatorial content for the natural-proof-barrier link inside the R̂ SAT Encoding module. It shows that no local property of the formula can certify unsatisfiability, supporting the module's claim that R̂ reaches zero J-cost for satisfiable instances in O(n) steps while unsatisfiable instances encounter topological obstructions. The result aligns with the Recognition Science separation of recognition-time complexity from Turing time without claiming a full P ≠ NP theorem.
scope and limits
- Does not prove P ≠ NP for Turing machines.
- Does not apply to decoders with access to the full assignment.
- Does not supply quantitative size bounds on M relative to n.
- Does not address satisfiable instances or J-cost directly.
formal statement (Lean)
169theorem rs_adversarial_lower_bound (n : ℕ) (M : Finset (Fin n))
170 (g : ({i // i ∈ M} → Bool) → Bool) :
171 ∃ (b : Bool) (R : Fin n → Bool),
172 g (BalancedParityHidden.restrict (BalancedParityHidden.enc b R) M) ≠ b :=
proof body
Term-mode proof.
173 BalancedParityHidden.adversarial_failure M g
174
175/-- The R̂ certifier is "non-natural" in the sense that it uses the entire
176 assignment at once (global, not local). The adversarial failure shows
177 that any LOCAL certifier fails, while R̂'s global evaluation succeeds. -/