AlgebraicRestrictionHyp
The AlgebraicRestrictionHyp encodes the claim that any circuit deciding an unsatisfiable CNF formula on n variables must obey gate count at least landscape depth times n. Researchers deriving circuit lower bounds from the J-cost landscape in the Recognition Science program would cite this as the algebraic half of the P versus NP argument. The declaration is a named structure whose single field packages the universal quantification over formulas and circuits.
claimFor any natural number $n$ and CNF formula $f$ on $n$ variables, if $f$ is unsatisfiable then for every Boolean circuit $c$ on $n$ variables that decides $f$, the real number of gates in $c$ is at least the landscape depth of $f$ multiplied by $n$.
background
The module states the core of the P versus NP program as high J-frustration implying super-polynomial circuit size. LandscapeDepth of a formula is the average J-cost over all assignments, where J-cost is the Recognition Science cost function obeying the d'Alembert equation J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). BooleanCircuit(n) is a feed-forward structure with gate_count and topologically ordered gates; CircuitDecides(c,f) asserts existence of an evaluation function matching the formula's satisfaction predicate.
proof idea
This is a structural definition of the hypothesis with no proof body. The single field directly records the quantified depth-size tradeoff statement for later use in conditional theorems.
why it matters in Recognition Science
The hypothesis supplies the algebraic linear lower bound that feeds circuit_lower_bound_algebraic (gate count at least n for unsat formulas) and CircuitLowerBoundCert. It instantiates the d'Alembert multiplicative structure from the forcing chain to force simultaneous access to Omega(n) bits, complementing the topological obstruction hypothesis. The open question it touches is discharging the hypothesis by proving non-separability of the J-multiplier.
scope and limits
- Does not prove the algebraic restriction holds; it only names the hypothesis.
- Does not apply to satisfiable formulas.
- Does not address topological invariants or exponential bounds.
- Does not supply an explicit circuit construction or tighter constants.
formal statement (Lean)
62structure AlgebraicRestrictionHyp where
63 /-- For any n-variable Boolean function with high landscape depth,
64 any circuit computing it has depth · size ≥ landscape depth · n -/
65 depth_size_tradeoff : ∀ (n : ℕ) (f : CNFFormula n),
66 f.isUNSAT →
67 ∀ (c : BooleanCircuit n),
68 CircuitDecides c f →
69 (c.gate_count : ℝ) ≥ LandscapeDepth f * n
70
71/-- **HYPOTHESIS (Topological Obstruction).**
72
73 The J-cost landscape for UNSAT formulas has a non-trivial topological
74 invariant: the defect moat (J-cost ≥ 1 everywhere) creates a "barrier"
75 that any circuit computing the satisfiability function must encode.
76
77 Encoding this barrier requires the circuit to represent the boundary
78 between the ≥1 region and the (hypothetical) zero-cost region. Since
79 the boundary has exponential description complexity (it touches Ω(2^n)
80 vertices of the Boolean hypercube), any circuit representing it must
81 have super-polynomial size. -/