EXPERT_REVIEW.md

# Independent-review brief

## The claim worth challenging first

For a finite or compact group acting on an M-dimensional logical space, with arbitrary physical unitary representations constrained to commute with a finite-sector Hamiltonian, the exact optimal phase-registered group-averaged squared covariance defect is `2(1-C/M)`. Here C is the maximum dimension of a logical subrepresentation that can be packed as whole irreducible copies into the physical energy degeneracies. Total physical dimension is assumed to be at least M.

This is not a claim about arbitrary approximate physical group actions, phase-insensitive gates, general quantum channels, or error-correcting codes. Compact-group representations are continuous and averages use normalized Haar measure.

## Proof path

The semantic quotient merges states only when every future protected observation agrees. Its minimality is the standard fiber-refinement argument. In a linear atlas, the pullback-closed row space computes that quotient. Review the finite-horizon warning: a horizon-L code need not be a stationary quotient for all subsequent edges.

For exact physical covariance, decompose each energy sector into irreducibles. An equivariant isometry requires sufficient multiplicities, which are exactly an integer allocation of irreducible copies to sectors. The physical representation is freely chosen; this freedom is a major assumption.

For the deficit, twirl the isometry in the Hilbert space of maps. The squared defect is twice its squared distance from the intertwiner space, divided by M. The twirled map is an operator-norm contraction whose rank is at most the matched representation dimension. Attainment comes from an isometric match on every shared irreducible and an isometric extension between the residual complements, which share no irreducible type.

The key steps to audit are the rank bound, attainability of the matched-dimension bound, and optimization over all sector representations. The extension must use the hypothesis that physical dimension is at least logical dimension. The proof allows cross-sector superpositions; it does not assume each codeword has definite photon number.

## Counterexamples already built in

A profile `(1,1,4,4)` cannot fit into sectors `(1,2,3,4)`, despite passing both total-dimension and largest-irrep checks. An exact finite-horizon quotient can fail closure under one more instruction. The H6 code's photon-number-twirled states have pairwise trace overlap 1/18. Its projected first-mode occupation operator is not scalar, so it fails a necessary exact one-photon-loss correction condition. Rationally independent mode frequencies give a nondegenerate energy spectrum and exclude exactly energy-preserving nonabelian logical actions on the register alone.

## Priority review

Compare first with Denys–Leverrier, arXiv:2306.11621, and St-Amand–Burelle–Royer, arXiv:2609.26660. The logical/physical representation framework and twirl construction are prior art. Search specifically for optimization over free physical representations with fixed sector degeneracies, the exact Hilbert–Schmidt defect optimum, and its relation to partial irrep packing. Also compare task-relative quantum compression, representation-theoretic universal coding, conserved-quantity resource theories, and equivariant approximation in Hilbert spaces.

The regular simplex, Helmert matrix, natural permutation decomposition, and singlet–triplet fusion example are not proposed as new mathematical objects. A distinctive thematic combination does not establish theorem-level novelty.

## Experimental review

The 44-test report verifies finite examples and implementation consistency. It is not statistical evidence that arbitrary unseen tasks or physical devices work. The Cayley spectral gap is a floating-point eigensolver result. No noisy-device simulation, laboratory test, passive-optical circuit, or general compression comparison is included.

A valuable next validation would fix a realizable optical control representation and a specified loss channel, charge the phase reference and ancillas, then compare memory energy and recovery fidelity with existing covariant codes. That is a new experiment, not an accomplished result of this release.