# Verification report

Manuscript: Strict Likelihood Multimodality and Septic Equations for
Two-Component Birkhoff Models. Alper Ferudun, Mercury Software GmbH.
28 September 2026. Corpus identifier: AIM-COMPUTATION-0073.

## Strict distribution-space modes

The 2013-observation data have counts 219/95 on even/odd permutations fixing
4 and 49/70 on even/odd permutations not fixing 4. The displayed gauge-fixed
component matrices have upper 3-by-3 blocks with diagonal/off-diagonal entries
3/2 and 0/5, respectively, last row/column 1 and relative raw weight 200/31.

Exact rational differentiation proves all 16 active scores vanish, all 16
Sylvester minors of the negative face Hessian are positive and each of the
three inward derivatives is -31/966240 for normalized log-likelihood.
The row-cycle orbit contains three distinct full-support distributions.

Parameter strictness alone would not suffice. The specific nonnegative fiber
is proved unique up to component swap: 18 polar relations leave six variables;
20 independent lifted relations leave a one-parameter pencil. A rank-one
minor gives s = 0 or 3565000/53. The latter forces a negative weighted
component because z_id^2 - (q_id/2)^2 = 22458880000/54537 > 0. The proof
then uses continuous monomial-coordinate recovery and compactness to exclude
improving image branches. No generic-identifiability assumption is used.

## Further exact results

- A determinant-sign obstruction excludes nonuniform parity-constant vectors
  from the closed two-component model for every n >= 4. Symmetry, compactness
  and a positive directional derivative at uniform imply at least three
  global maximizers for the parity datasets. An n=3 example shows why the
  restriction n >= 4 matters.
- All 475020 degree-six monomials, partitioned into 132724 multidegree blocks,
  have full-rank substitution matrices. Two independent implementations use
  different codings, elimination orientations and prime fields. Thus no
  nonzero homogeneous equation exists in degrees 1 through 6.
- Exact rational reduction supplies a primitive 96-term septic. A separate
  20-variable formal expansion cancels identically. Its 24 symmetry images
  have different multidegrees and gradient rank four at the mode; the mixture
  Jacobian has rank 19. This proves local, not global, defining equations.

## Reproduction and limitations

The core checks are 1516 in verify_parity_obstruction.py, 298 in
verify_boundary_mode.py and 57 in audit_fiber_certificate.py: 1871 named checks
per runtime. Supplied receipts pass on Python 3.13.5, Python 3.12.14 and
optimized Python 3.13.5. Degree-six injectivity and septic expansion are
separate complete finite certificates. See the source archive's README for
commands and the distinction between discovery outputs and final proof.

The written compactness, inverse-chart and all-n arguments remain essential;
test counts do not replace them. No independent peer review, formal proof
assistant certification or absolute novelty certificate is claimed. The
source's whole defining-ideal question remains open here. No positive-component
interior strict-mode theorem or isolation of the parity global maxima is claimed.

The eight-page PDF compiled successfully with the built-in compiler and
Tectonic. Every page was visually reviewed; no clipping, overlaps, missing
glyphs or unresolved references were found. This report describes the package,
not an external-publication receipt.