landscape_linear
For any SAT instance the sum of variables and clauses is at least the number of variables. Researchers assembling certificates for the Recognition Science to Turing complexity bridge cite this linear size relation. The proof is a direct one-line application of the standard natural-number addition inequality.
claimFor every SAT instance with $n$ variables and $m$ clauses, $n + m$ is at least $n$.
background
In the TuringBridge module a SATInstance is a structure carrying a positive natural number of variables and a positive natural number of clauses; the module encodes such an instance as a J-cost landscape whose minimum cost is zero precisely when the clauses are satisfied. The surrounding context is the open bridge from R-hat minimization on the full Z^3 ledger to classical Turing-machine complexity classes, with encoding faithfulness and non-naturality of the R-hat certificate already in place. Upstream results supply the J-cost definition and the ledger factorization that make the landscape encoding possible.
proof idea
The proof is a one-line term that applies the Nat.le_add_right lemma to the two natural numbers n_vars and n_clauses.
why it matters in Recognition Science
The declaration supplies the landscape_linear field inside the TuringBridgeCert structure that collects the minimal facts needed for the P versus NP bridge. It fills the basic size relation required by the module's strategy while the superpolynomial simulation overhead from R-hat to Turing tape remains the open gap identified in the module documentation and the referenced paper section on Q3.
scope and limits
- Does not claim any bound on the time required to minimize J-cost.
- Does not address satisfiability or the spectral gap.
- Does not invoke the phi-ladder or Recognition Composition Law.
- Does not depend on any upstream forcing-chain step T0-T8.
Lean usage
def turingBridgeCert : TuringBridgeCert where encoding_faithful := encoding_faithful non_natural := rhat_is_non_natural landscape_linear := landscape_linear gap_identified := the_open_gapformal statement (Lean)
123theorem landscape_linear (sat : SATInstance) :
124 sat.n_vars + sat.n_clauses ≥ sat.n_vars := Nat.le_add_right _ _
proof body
Term-mode proof.
125
126/-! ## What Remains Open -/
127
128/-- The genuinely open piece: proving that the simulation overhead
129 from R-hat (global lattice minimization) to Turing machine
130 (local tape operations) is superpolynomial.
131
132 This would require showing that no polynomial-time TM can simulate
133 the convergence of degree-normalized SpMV on an n-variable
134 J-cost landscape. The spectral gap argument (from
135 SpectralGapContraction) gives convergence in O(1/λ₂) octaves,
136 but translating "octaves on Z³" to "steps on a Turing tape"
137 is the missing bridge. -/