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Corollary · exact extremal costs

CRD-SAT-14b

The best legal first-UIP gap is exactly m

Exact statement

The priority-independent minimum first-UIP costs are cmin⁡(Fm)=m+1c_{\min}(F_m)=m+1 and cmin⁡(Gm)=1c_{\min}(G_m)=1, giving exact gap mm; the maximum costs are equal: cmax⁡(Fm)=cmax⁡(Gm)=m+1c_{\max}(F_m)=c_{\max}(G_m)=m+1.

StatusProved from the all-schedule classification
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Minimum cost measures proof work available in the retained database; maximum cost measures what a legal but unhelpful reason schedule can still realize.

Hypotheses and scope

  • The trace space and cost function are exactly those of CRD-SAT-14a.

Proof or evidence

Taking minima and maxima of the two exact spectra in CRD-SAT-14a gives the formulas immediately.

Minimum and maximum cost corollary →

Verification notes

The source's distinction between available proof work and guaranteed online work was retained. The minimum is existential and does not include the cost of discovering the best trace.

Limitations

  • A smaller minimum does not guarantee that an online solver realizes it.
  • The common maximum shows that the shortcut does not improve every legal schedule.

Open work

Test a different proof resource, such as minimum resolution width or clause space, rather than extending the classified path family.