AlgebraicRestrictionProofSketch
AlgebraicRestrictionProofSketch outlines the non-separability of J-cost evaluation, per-layer propagation limited by depth, and accumulation to linear size for high-frustration functions. Complexity researchers working on circuit lower bounds would cite the sketch when discharging the algebraic restriction hypothesis in the P versus NP argument. The definition is a record type whose three fields are instantiated downstream with trivial.
claimA structure asserting non-separability of J-cost under the d'Alembert relation $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$, layer propagation of multiplicative structure to at most $2^d$ variables at depth $d$, and accumulation yielding circuit size at least $n$ for functions with J-frustration at least 1.
background
The module develops circuit lower bounds from J-frustration by showing that the Recognition Composition Law forces super-polynomial size. The d'Alembert equation endows the J-cost landscape with multiplicative structure and reciprocal symmetry that creates cost tunnels requiring global access. AlgebraicRestrictionHyp encodes the needed tradeoff: for any UNSAT formula f on n variables, every deciding circuit c satisfies gate_count(c) >= LandscapeDepth(f) * n.
proof idea
The declaration defines a structure whose three fields are set to True as placeholders. No lemmas or tactics are applied inside the definition. The downstream the_proof_sketch supplies the concrete instance by assigning trivial to non_separability, layer_propagation, and accumulation.
why it matters in Recognition Science
This sketch supplies the algebraic restriction step that the_proof_sketch depends on to certify the overall lower bound. It advances the module program of deriving P ≠ NP from J-frustration on 3-SAT instances and connects directly to the Recognition Composition Law. The open question it touches is full discharge of AlgebraicRestrictionHyp to obtain the super-polynomial bound.
scope and limits
- Does not discharge AlgebraicRestrictionHyp.
- Does not derive an explicit Omega(n) size lower bound.
- Does not instantiate the argument for concrete SAT formulas.
Lean usage
def the_proof_sketch : AlgebraicRestrictionProofSketch where non_separability := trivial layer_propagation := trivial accumulation := trivialformal statement (Lean)
227structure AlgebraicRestrictionProofSketch where
228 non_separability : True
229 layer_propagation : True
230 accumulation : True
231