the_proof_sketch
This definition supplies a trivial placeholder instance of the AlgebraicRestrictionProofSketch structure inside the circuit lower bound formalization. Researchers studying the Recognition Science derivation of P versus NP from J-frustration would cite it as the current stand-in for the algebraic restriction step. The construction is a direct record instantiation that sets non-separability, layer propagation, and accumulation each to the trivial proposition.
claimThe algebraic restriction proof sketch is the structure whose fields assert non-separability of the J-multiplicative combiner, propagation of restrictions across layers, and accumulation of cost defects, each instantiated as the trivial proposition.
background
The module formalizes the core of the P versus NP program by showing that high J-frustration implies super-polynomial circuit size. The argument rests on the d'Alembert functional equation J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), whose reciprocal symmetry J(x) = J(1/x) creates cost tunnels that polynomial circuits cannot exploit without global access. The upstream AlgebraicRestrictionProofSketch structure encodes the three required propositions that must eventually be discharged from the J-cost landscape.
proof idea
The definition is a one-line structure constructor that directly assigns the trivial proposition to each of the three fields non-separability, layer_propagation, and accumulation. No lemmas or tactics are invoked beyond the built-in trivial tactic.
why it matters in Recognition Science
This definition provides the structural interface for the algebraic restriction component of the three-step circuit lower bound argument. It sits inside the Recognition Science framework where the J-functional equation (T5 J-uniqueness) and the Recognition Composition Law generate the multiplicative cost structure used to bound circuit size. The parent goal is the full lower bound for high-frustration SAT instances; the definition touches the open question of discharging AlgebraicRestrictionHyp to complete the P ≠ NP claim.
scope and limits
- Does not prove that J-evaluation requires simultaneous access to both arguments.
- Does not derive any depth-width tradeoff for circuits.
- Does not instantiate the SAT lower bound from clause-to-variable ratio.
- Does not connect J-frustration to concrete circuit size functions.
formal statement (Lean)
232def the_proof_sketch : AlgebraicRestrictionProofSketch where
233 non_separability := trivial
proof body
Definition body.
234 layer_propagation := trivial
235 accumulation := trivial
236
237/-! ## Certificate -/
238