# Positive focal pencil verification report

The general proof establishes equivalence of positive focal compatibility
and a strictly positive kernel vector, a division-free positive-circuit
predicate, Hermite sign counting and a complete projective-pencil split.
Identically singular pencils require both tail tests and derivative-root
samples of bounded intervals. The complementary count excludes rank-one
matrices instead of calling them invalid focal pairs.

The exact regression suite contains 836 checks: 489 kernel checks,
159 root-free checks, 177 boundary/design checks and 11 complementary
counts. Positive-circuit feasibility is compared with a separate exact
Fourier-Motzkin implementation. Camera-generated examples, repeated roots,
infinity, irreducible sign patterns, GL2 reparameterizations and the
bounded feasible interval |z|<1 are included. These computations do not
replace the written general argument.

Normal and optimized executions agree after excluding only runtime and
optimization metadata. The release build additionally requires all eight
normal/optimized runs from the extracted ZIP to reproduce the reference
mathematical output with Python isolated mode. Hashes and outputs are
retained in the source archive.

The seven-page PDF compiled with the built-in compiler and cached Tectonic.
Every final page was visually inspected. Formulae, references, the
seven-correspondence table and author footnote are legible without clipping
or overlap. Long reference links were shortened for layout; no mathematical
content changed. The final compiler log has no errors, undefined references,
missing glyphs or box warnings.

This is originating-researcher self-audit, not independent review or formal
verification. Classical inputs and the inherited generic formula are
credited. The specified pencil theorem is complete; practical speed,
the informal full source question and absolute novelty are not certified.