No fixed seed depth determines this branching cost

Exact statement

Across unbounded clause width, no fixed finite unit-propagation seed depth determines the fixed-policy DPLL decision count: for any $t\ge1$ , choosing $r=t+2$ gives semantically equivalent databases with equal signatures through depth $t$ but different decision counts.

StatusProved from the arbitrary-width family

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This is the negative half of the later exact completeness-radius result.

Hypotheses and scope

Clause width is allowed to grow with the requested observation depth.
The branching resource is the fixed-policy decision count of CRD-SAT-11b.

Proof or evidence

Given $t$ , take $r=t+2$ . CRD-SAT-11a gives semantic equivalence and signature equality through $r-2=t$ , while CRD-SAT-11b gives decision counts $r$ and $r-1$ .

Fixed-depth obstruction →

Verification notes

The quantifier over seed depth and the dependence of width on that depth were made explicit; the result is not stated for fixed-width formulas.

Limitations

Unbounded width is essential to this construction.
At fixed maximum added-clause width, CRD-SAT-12a gives a finite complete radius.

Open work

Use CRD-SAT-12a for the complementary positive theorem at fixed added-clause width.