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Lemma · interaction-depth bound

CRD-SAT-10a

Strict width bounds the first possible unit propagation

Exact statement

If every clause in a strict rr-CNF has rr distinct literals, then every consistent seed assigning fewer than r−1r-1 variables has trivial unit-propagation closure: no clause can yet be unit or false.

StatusProved for strict finite r-CNF databases
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The lemma identifies the earliest seed size at which unit propagation can distinguish databases consisting only of full-width clauses.

Definitions

  • A strict rr-clause contains exactly rr distinct literals and is neither tautological nor shortened by repetition.
  • The signature σt(D)\sigma_t(D) records unit-propagation closure and conflict status for every consistent seed assigning at most tt variables.

Hypotheses and scope

  • Every stored clause has exactly rr distinct literals.
  • The seed is consistent and assigns at most r−2r-2 variables.

Proof or evidence

After fewer than r−1r-1 seed assignments, every unsatisfied strict rr-clause retains at least two unset literals. Therefore no unit clause or falsified clause exists from which propagation or conflict could begin.

Strict-width lemma →

Verification notes

The counting argument was reconstructed independently of the finite checker. Its strict-width hypothesis is necessary and is stated explicitly.

Limitations

  • Shorter clauses, repeated literals, or auxiliary clauses can make propagation begin earlier.
  • Trivial low-depth closure is a coarse observation, not a solver bisimulation.

Open work

Compare this seed-depth formulation with standard notions of propagation hardness and absorption.