# Verification report

Manuscript: A Degree-24 Obstruction to Hopf Structures on Symmetric-Group
Supercharacter Spaces. Alper Ferudun, Mercury Software GmbH, 28 September 2026.

Theorem scope: the explicitly defined fine, coarse and maximal supercharacter
families of S_n, degree n with V_0=V_1=k, ordinary connected graded Hopf
structures, characteristic zero. Whole-source-family classification not claimed.

## Finite certificate

- Fine-family dimension at n=24: 794 = (1575+11)/2+1.
- First negative Euler exponent: a_24=-1; all earlier exponents nonnegative.
- Forced coefficient in degree 24 from the lower-degree Euler factors: 795.
- Coarse and maximal families: first negative exponent a_4=-1.
- Ordinary Sym control: all exponents equal 1 through the tested degree.
- The polynomial Hopf algebra on the first 23 exponent multiplicities actually
  realizes the entire lower-degree prefix, so positivity of that prefix alone
  cannot exclude a Hopf structure.

The standard-library program compares generating-function DPs against recursive
cycle-partition enumeration and compares logarithmic-derivative inversion
against exact factor multiplication. Self-conjugate partitions provide another
check of the distinct-odd count. All operations are exact integers; there is no
numerical optimization or floating-point threshold. The program uses explicit
failure checks, not assertions that disappear in optimized mode.

All 333 checks passed on Python 3.13.5 and 3.12.14, and on Python 3.13.5 with
`-O`. Repeating a check across runtimes does not multiply the number of distinct
claims and is not independent mathematical review.

## Non-computational dependency and scope

The implication from a negative exponent to nonexistence is the published
Hilbert-series theorem of Zhou-Shen-Lu, Proposition 3.5, with their attribution
to Brown-Gilmartin-Zhang. Its hypotheses were checked in the primary source:
no finite-GK or algebraically-closed-field assumption is imposed. The superclass
families follow from Hendrickson's star-product. The manuscript gives all
dimension and generating-function arguments.

The evidence is a written proof plus an exact arithmetic certificate. It is not
formal verification or a peer-review certificate. Bounded literature searching
does not certify absolute novelty. Neither positive characteristic nor signed,
ungraded or regraded Hopf structures are covered.

The four-page PDF compiled successfully with the desktop compiler and Tectonic;
all pages were visually inspected. Publication status remains unpublished.