# Verification report

Paper: Polynomial Rigidity and a Sharp Degree Bound for Free Quadratic Variation

Author: Alper Ferudun (Mercury Software GmbH). Date: 28 September 2026.

## Written proof

1. In the free difference quotient of a degree-d polynomial, only last-letter
   removals have left degree d-1. Hence the degree-(2d-2) part of its covariance
   is the sum of the last-letter derivatives times their adjoints.
2. The coefficient of each word followed by its reverse is a sum of squared
   moduli. Unique equal-length word splitting prevents cancellation. This
   proves the exact degree identity for every unital moment functional.
3. Injective polynomial evaluation lifts operator unit-covariance identities
   to formal polynomial identities. The degree identity gives affine maps;
   the full covariance matrix gives coisometry, not merely unit rows.
4. In the stated pathwise setting, norm continuity extends the almost-everywhere
   covariance equality to the algebraically free time. Noise rotation, literal
   same-noise coupling and endpoint-law steering are kept distinct.
5. At a standard semicircular tuple, the highest Wick coefficients form a
   positive Gram matrix of rank at most the number of variables. The
   Hilbert-Schmidt/trace inequality gives the bound. A sum of squares attains
   equality and proves sharpness. Quadratic stability is its direct corollary.

The detailed argument and adverse boundary tests are in the reproducibility
folder. No known gap remains in these explicitly stated theorems. This is an
originating-researcher self-audit, not independent review or formal verification.

## Exact checks

The checker uses exact Gaussian rationals and explicit runtime checks (not
Python `assert` statements). All 4003 checks passed with Python 3.13, Python
3.13 with optimization, and bundled Python 3.12. The same script across
runtimes is not three independent proofs.

- 1227 Fock creation-annihilation moments compared to noncrossing-pair recursion.
- 240 polynomial cases in one to four variables and degree at most five,
  including 120 self-adjoint samples.
- Raw Itô contraction compared with the independent highest-degree Gram formula,
  including an arbitrary nonpositive unital functional.
- Sharp examples in one to eight variables.
- Noninjective evaluation, traced-only normalization, duplicate rows and
  imaginary self-adjoint commutator boundary tests.

These finite tests supplement, but cannot replace, the all-degree proof.

## Sources and novelty

The exact AIM page, Speicher's free Itô statements and the hypotheses of the
adjacent Diez rigidity theorem were inspected. Standard free stochastic
calculus, Fock-space tools and Gram inequalities are credited, not claimed
as new. The inherited dataset's one-variable obstruction and affine converse
are also credited. A bounded search did not locate the exact degree and
quantitative statements; that is not a priority certificate.

## Artifact review

The built-in LaTeX compiler and Tectonic export succeeded. All six final PDF
pages were rendered and visually checked; no clipping, broken formula,
undefined reference or compiler warning remains. The packaged checker is
rerun in isolation, and file hashes and ZIP integrity are recorded in the
release build receipt and manifest.

## Claims not made

The whole AIM-PROBABILITY-0108 source record is not closed. Neither arbitrary
terminal-law steering, the microstates limsup/liminf question nor pressure
equilibrium classification is settled. General-degree stability of all
nonlinear chaos components is not proved. Publication and DOI assignment
do not establish correctness, peer review, search indexing or priority.