# Verification report

Manuscript: Exponential Rarity of Multiple Likelihood Modes in Bivariate
Seemingly Unrelated Regression. Author: Alper Ferudun, Mercury Software GmbH.
Date: 29 September 2026.

## Accepted mathematical scope

A complete analytic proof of a deterministic unimodality certificate, an
explicit fixed-design Gaussian SUR multiple-stationary-point probability
bound, the common-regressor extension, a random-Gaussian-design bound and
an exponential-order lower bound for the random-design model. The source
MANOVA theorem is classical and credited. No optimal rate constant or exact
finite-sample probability is claimed.

The full derivation, source recovery, proof audit and retained failed
approaches are in reproducibility/. The proof was self-audited by the
originating researcher, not independently reviewed or formally verified.

## Exact diagnostics

Python 3.9.6 and 3.13.5 each passed 16,440 assertions, with identical reports
apart from runtime version. This is the same test set, not twice as many
different mathematical cases.

- 3,969 rational normal forms: symbolic determinant and quintic identities,
  plus exact Sturm real-root counts.
- 869 certificate-positive cases: all have exactly one real root.
- 105 simple-condition cases: all satisfy the stronger certificate.
- 500 exact nonunit normalization cases.
- Constant, covariance-domination, degenerate-boundary and nonconvexity checks.
- Root histogram: 3,225 one-root and 744 three-root cases. The grid does not
  include five-root examples and does not rule them out.

The original checker incorrectly expected its finite grid to contain a
five-root example. That diagnostic expectation was removed, with the original
script and failure retained internally. No theorem was changed to pass a test.
The Sturm implementation also has an independent generic five-root polynomial
control. Finite checks are not a replacement for the all-parameter proof.

## Manuscript and publication status

The native LaTeX compiler and final exported build succeed. The final build
has no TeX warnings; all seven page images have been inspected for legibility,
equations, glyphs, links and clipping. The package uses only the author's
manuscript and original supporting artifacts; cited third-party PDFs and
their text are not redistributed.

This is an unrefereed preprint. Targeted primary-literature searches did not
find the displayed finite-sample bound or certificate, but do not certify
absolute priority. The accompanying release receipt and manifest bind the
exact checked files. Local readiness is not a public DOI or publication.