Afterword: FPRD Lab’s contribution

A tensor-opposite proof of the anyon-proliferation identity

Methodological and contribution addendum. The paper’s mathematical scope and review status remain as stated in the paper.

The Lab’s approach

FPRD means Finitely Presented Recognition Dynamics. Its starting point is to distinguish a finite description, the possibly infinite process it generates, and what an observer can learn from that process. Equal answers need not come from equal internal structures. Conversely, an infinite state space may have a short, exact description and useful coordinates; failure to compress it into finitely many states need not end the investigation.

A small example: balanced parentheses

The grammar S → (S)S | ε presents the language of balanced parentheses. To recognize a prefix, keep the number of unmatched opening parentheses, or a failure state if a closing parenthesis arrived too early. An opening parenthesis increments the count. A closing parenthesis decrements a positive count; at zero it enters failure. Failure persists. At the end, accept exactly when the count is zero.

The prefixes ( and ()( leave the same future task: any suffix that completes one completes the other. The prefix (( leaves a different task: ) completes the first two prefixes but not this one. These future tasks are residuals. There are infinitely many counts, yet a finite set of update rules generates their dynamics. This is a classical automata example, not a discovery of FPRD Lab. It illustrates the research question: which coordinates preserve all relevant futures, and what does updating or reading those coordinates cost?

From viewpoint to proof

In practice, the Lab fixes the observable and the allowed operations, compares presentations of the same problem, and seeks an exact translation or invariant. A failed shortcut is useful when a counterexample identifies the missing coordinate or hypothesis. Small computations test particular uncertainties; a proof must explain why the result holds beyond those tests. Costs, relation sides, boundary conditions, and the direction of each translation remain explicit.

AI-assisted exploration, source comparison, executable checks, and adversarial reconstruction support this work. Their agreement is not automatically independent evidence: shared assumptions must be exposed. A formal development certifies its stated formal theorem, not every interpretation attached to it. Each paper retains its own evidence and review status.

The approach is most useful when a poor representation hides a reusable construction, a compatibility condition, or a sharp obstruction. Outside recognition theory, it can guide the search for a structural reformulation without producing a literal residual dynamical system. The account below distinguishes that methodological influence from the ingredients of the proof.

FPRD’s contribution to this paper

The search began with a presentation question: can the required cancellation be made exact while preserving the braiding information? Cheng supplied the anyon-proliferation identity to be investigated. The Lab initially considered transparent-subalgebra and relative-center machinery, including work of Yadav and Shimizu, as a possible route to expose the surviving structure.

That exploratory route did not become the proof printed here, and it did not produce a residual-dynamics model of the problem. Its influence belongs in the discovery account, not in the list of formal dependencies. The successful candidate argument instead uses the tensor-opposite category in a Morita-center and alpha-induction construction.

The decisive calculation keeps two operations distinct: reversing the tensor product and reversing the braiding. Transporting the evaluation half-braiding through the chosen monoidal equivalence yields exactly the negative central embedding required by the diagonal algebra. The proof then identifies the induced algebra, establishes dominance, and computes the centralizer. Matching dimensions alone would not identify the specified algebra or its braiding.

Inherited ingredients and the boundary of the contribution

Cheng owns the proposed identity. The symmetric-case argument of Lan–Kong–Wen, the condensation and central-functor results of Davydov–Müger–Nikshych–Ostrik, Schauenburg’s Morita-center equivalence, and the other cited fusion-category theorems supply the structural machinery. Yadav–Shimizu influenced the initial investigation but is not an essential premise of the final proof.

The Lab’s contribution is the search for the appropriate presentation and the specific tensor-opposite convention calculation and assembly offered in this manuscript. Calling this an FPRD investigation does not establish that residual dynamics was necessary for the discovery or that the proof is independent of its substantial categorical sources. This is a useful boundary case for the method: a presentation-oriented question can lead to an ordinary structural proof without realizing the hoped-for dynamical translation.

The manuscript remains a candidate proof without external specialist review. It concerns a categorical identity, not microscopic Hamiltonians or a dynamical explanation of a physical phase transition.

Where to inspect the contribution: the paper’s chirality calculation, induced-algebra lemma, and centralizer argument; the initial research packet records the transparent-subalgebra direction.