# Verification report

**Quantum Symmetric Algebras on Multiple Standard Copies: Structure and Categorical Non-Equivalence**

Alper Ferudun, Mercury Software GmbH - 7 September 2026.

Status: author-approved unrefereed preprint, containing claimed complete
proofs. This report is primary's self-audit. It is not independent peer review,
formal verification, or a priority certificate.

## Claim and hypotheses

The field is C(q) for indeterminate q; U is an ordinary multiplicity space of
finite dimension m. The ambient categories are the split semisimple generic
polynomial Hecke category and its classical symmetric-group analogue, with
Ind-completions for all internal polynomial right modules.

The homogeneous-component and fiber-product results hold for m >= 1 and
degree n >= 2, with degree zero and one separately stated. The categorical
non-equivalence holds for m >= 2 and requires no linearity, grading,
augmentation, tensor-action or evaluation compatibility from an equivalence.
The paper does not claim results at roots of unity, on torsion quotients,
or for arbitrary q-tca generators.

## Load-bearing obligations checked by primary

| Obligation | Self-audit conclusion |
|---|---|
| Quadratic relation space | It is exactly the negative eigenspace of the BZ normalized commutor on the direct sum, not a quantum matrix algebra with two deformed factors. |
| Multiplicity quotient | Each degree-n relation sum is the sum of the adjacent negative eigenspaces; no global Hecke invariance of that ideal is assumed. |
| H_3 braid defect | Exact 2-by-2 rational-function matrices satisfy the Hecke relations; the cube defect determinant is 4(q^4+q^2+1)(q^2-1)^4/(q^2+1)^6. |
| Dual transfer | Fixed-vector equations reverse words across the tensor factors, and ordinary adjacent permutations have product of order three. |
| All-degree propagation | Local H_3 semisimplicity removes every (2,1) restriction component. The span(v,w) lemma proves stability before using simplicity. |
| Row and column multiplicities | Remaining quotients are the ordinary symmetric and exterior powers of U. Degrees zero and one are not double-counted. |
| Algebra fiber product | Both generator maps kill the original quadratic space. Degreewise subdirect surjectivity uses disjoint row/column simple supports, followed by equality of multiplicities. |
| m=2 square-zero kernel | The exterior multiplicity vanishes above degree two. This is an annihilated ideal, not a claimed splitting of algebras. |
| All simple modules | Least polynomial degree forces every simple to have augmentation action. No additional simple modules are omitted from the graph. |
| Projective covers | Free adjunction is exact; any submodule surjecting onto the augmentation simple contains the degree-minimal generator and hence the entire cover. |
| Intrinsic identification | The unique source of the directed Ext^1 graph is empty; (2,1) is the unique depth-three vertex with two distinct predecessors. |
| Final invariant | Hom from the (2,1) cover to the empty cover vanishes quantumly and has classical dimension m(m^2-1)/3. Only zero/nonzero is used. |

These checks are the same primary researcher's analysis. They do not become an
independent review merely because the argument has two formulations.

## Executable evidence

Four original programs are supplied under reproducibility/:

1. check_hecke_braid_defect.py: exact rational-function arithmetic over Q(q),
   including Hecke relations, the nonzero generic determinant, q=2 value
   6804/15625, and vanishing at q=1. No floating-point interpolation.
2. check_cubic_matrix.py: the original six doubled-standard relations,
   an exact maximal-minor specialization and classical rank control.
3. check_cubic_blocks.py: polynomial determinant identities over Z[q] for
   the small blocks of the maximal minor, whose product is
   plus or minus q^12(q^2-1)^8. It imports the adjacent cubic matrix checker.
4. check_all_degree_dimensions.py: sparse rank elimination from the original
   degree-n quadratic relation matrices, with sixteen explicit cases.

Each program passed in Python 3.12.14 and Python 3.13.5, with identical
standard output between runtimes. The raw JSON outputs and SHA-256 records
are included. These are two executions of the same code, not two independent
implementations.

The sixteen higher-degree cases comprise:

- (d,m)=(2,2), q=2, degrees 2,3,4,5 over F_101;
- classical q=1 controls at degrees 3,4,5 for (d,m)=(2,2);
- (d,m,n)=(3,2,3),(2,3,3),(3,3,3),(2,3,4),(3,2,4),(1,3,4),(3,1,4)
  at q=2 over F_101;
- (d,m,n)=(2,2,4), q=3 over F_103;
- (d,m,n)=(2,2,4), q=2 over Q using exact fractions.

In particular, (d,m)=(2,2) gives quantum dimensions 10,16,25,36 in degrees
2 through 5, compared with classical dimensions 20,35,56 in degrees 3 through
5. Finite-field and rational checks are finite regression evidence only.
The all-degree and all-multiplicity claims rest on the manuscript's proofs.

To reproduce, copy the source package to a scratch directory, keep the four
programs together, and run each with python3 -B. No third-party Python
packages, credentials, network requests, or proof assistant are needed.

## Source fidelity, novelty, and release

The paper targets the doubled-standard instance named in the AIM source
remark and its multiple-copy extension. Known cubic nonflatness is attributed
to Tucker-Simmons's 2013 thesis, not claimed as new.
See source_and_novelty_review.md for the limited literature search and the
unread Michigan thesis lead. Novelty has not been independently certified.

The author explicitly authorized release without waiting for Claude.
Independent-review and formal-verification flags remain false; canonical
research classification remains CANDIDATE_PROOF. Repository publication is
not an independent confirmation that the source problem is solved.