# Verification of the collapsed-boundary counterexample

The originating researcher accepts the written proof after a detailed
self-audit. The source class includes spatial and value dependence. The
quasiconvexity calculation checks arbitrary affine multiplicity blocks via
their barycenters, without presupposing Sobolev labels. The direct method
uses the precise metric Sobolev compactness and trace theorems; existence
of actual minimizers is established separately from the root competitor.

For arbitrary scalar competitors, almost-everywhere circular slicing gives
periodic H^1 loops. The half-angle gauge is antiperiodic. Its derivative
cross term is bounded, not discarded. The transition lemma covers complex
detours and nonmonotone crossings via an absolutely continuous primitive.
The radial measure dr/r and both sheets' factors are explicit. At lambda
equal to 2^20, elementary rational bounds prove a strict energy gap.

The upper scalar construction has compatible angular endpoints, finite
energy at the origin and zero outer trace. Global Sobolev labeling would
split the energy; trace and derivative behavior on coincidence sets are
accounted for. The argument excludes every such labeling and hence collapse.
It does not describe the singular set or identify the comparison map with
an actual minimizer.

The exact standard-library checker performs 28 symbolic/constant checks
and 2,050 checks on 1,025 rational annular radii. It uses fractions, no
floating-point arithmetic, and retains its checks under Python -O. Normal
and optimized reports agree. These finite diagnostics verify algebraic
constants, not the analytic theorem or novelty.

The desktop editor compiler and local Tectonic compilation succeeded. All
five final PDF pages were opened and visually inspected: equations,
references, author footnote and page numbers are legible, without clipping
or overlaps. An initial bibliography underfull-box warning was corrected by
ragged-right references; its source, PDF and log remain in the private
checkpoint. The final log contains no undefined references or box warnings.
Portable source compilation and both checker modes are recorded separately
in build_record.json before the package is registered as ready.

The final manuscript retains the full accepted analytic argument. The
general value-dependent scope and established prior work are explicit.
No independent review, proof-assistant verification or absolute priority
is asserted. No external publication action occurred for this package.