A Covariant, Completely Positive Channel from an Exact Sp(3) Relational Construction, Trace-Preserving and Action-Determined Under an Explicit Convention Premise
We ask whether a construction’s invariances can be derived from the defined structure of a relational object rather than imposed on it, and whether the physical action such an object induces is determined within the stated convention rather than chosen among admissible placements. Working over ℚ(i) with an exact USp(6) = Sp(3)- covariant construction, we build an effective completely-positive map N = D ∘ E that compresses a thirty-five-dimensional carrier through a six-dimensional memory, and we show its content is gauge-independent — invariant under the memory’s multiplicity basis — and completely positive outright; it is trace-preserving, and so a channel in the strict sense, only under the convention premise stated below. We then separate three objects routinely conflated in constructions of this kind: the stored tensor, the bilinear scoring functional it validates, and the induced channel action. That separation is unconditional, and it dissolves a standing L/L^{T} ambiguity as a category inversion rather than an arithmetic error. Identifying the induced action requires one explicit premise, P, fixing the dual-leg reading of the documented Choi convention; under P, and only under P, the induced action is uniquely the realignment L^{T} — a map that is, in any case, trace-preserving, fully Sp(3)-covariant on the whole opera- 1 tor algebra, and real, with A_physical = [[1/7, 3/14], [4/7, 6/7]]. We keep the plain facts and the premise-relative result distinct throughout, mark the boundary at which a purely structural formalism cannot decide a question formulated as concerning what structure does not represent, and state the remaining frontier as concrete computations.
We ask whether a construction’s invariances can be derived from the defined structure of a relational object rather than imposed on it, and whether the physical action such an object induces is determined within the stated convention rather than chosen among admissible placements. Working over ℚ(i) with an exact USp(6) = Sp(3)- covariant construction, we build an effective completely-positive map N = D ∘ E that compresses a thirty-five-dimensional carrier through a six-dimensional memory, and we show its content is gauge-independent — invariant under the memory’s multiplicity basis — and completely positive outright; it is trace-preserving, and so a channel in the strict sense, only under the convention premise stated below. We then separate three objects routinely conflated in constructions of this kind: the stored tensor, the bilinear scoring functional it validates, and the induced channel action. That separation is unconditional, and it dissolves a standing L/L^{T} ambiguity as a category inversion rather than an arithmetic error. Identifying the induced action requires one explicit premise, P, fixing the dual-leg reading of the documented Choi convention; under P, and only under P, the induced action is uniquely the realignment L^{T} — a map that is, in any case, trace-preserving, fully Sp(3)-covariant on the whole opera- 1 tor algebra, and real, with A_physical = [[1/7, 3/14], [4/7, 6/7]]. We keep the plain facts and the premise-relative result distinct throughout, mark the boundary at which a purely structural formalism cannot decide a question formulated as concerning what structure does not represent, and state the remaining frontier as concrete computations.
An exact Sp(3)-covariant relational construction stores a single four-index object C_R on a 35-dimensional carrier and reads two different scalars out of it: an induced channel action and a score. The induced action is the trace-preserving map realign(C_R) = L^T, with sector-transfer matrix A_phys = [[1/7, 3/14],[4/7, 6/7]]; the score is the diagonal table A_ext = diag(1/7, 6/7). We prove, exactly over Q(i), three things. First, A_ext is exactly the diagonal of A_phys: the score keeps the sector-preserving coefficients of the induced channel and discards its sector-mixing terms. Second, the operator that produces the score is not self-adjoint — in the standard inner product and in the physical invariant metric G_W its Hermiticity deviations are nonzero (1/14, 3/7 and 3/14, 9/7), and its symmetrized spectrum runs outside [0,1] — so it is not a POVM effect on the carrier. Third, and positively, the same number is a positive figure of merit on the channel: the physical transfer matrix is itself a positive readout A_phys[s,t] = Tr[P_s N(P_t)]/d_s with P_s the self-adjoint (G_W-orthogonal) sector projector, so the score Tr[P_s N(P_s)]/d_s is a nonnegative sector-return fidelity — the trace of two positive operators. The certificate's non-self-adjoint operator merely represents this positive number through the oblique readout M_s = P_s^T = G_W P_s G_W^{-1}, which fails self-adjointness only because [P_s, G_W] != 0. The score is therefore not a measurement effect on states but a positive sector-return figure of merit of the induced CP channel; its non-self-adjoint appearance is a representation artifact of the oblique readout, not intrinsic signedness. All claims reproduce from a standard-library-only Python verifier and a frozen exact-operator file (integrity-checked by SHA-256), with no external dependencies. A naming caveat is stated explicitly: what is established is a G_W-positive sector-return functional, not a process effect or tester in the standard Ziman–CDP sense, which would additionally require standard-metric positivity and a tester normalization not claimed here.
This is the successor to the USp(4) result (doi:10.5281/zenodo.21694560) in the Satyalogos operational-integration program, extending it to the first nontrivial rank step of the symplectic tower, USp(6) = Sp(3). In the USp(6) = Sp(3) covariant compression model the carrier is W = V14 ⊕ V21 = sl(6) (dimension 35), encoded through a six-dimensional covariant memory M = V6 and decoded. Working entirely over the Gaussian rationals Q(i), we reduce the effective-channel physical- and PPT-positivity cones to seven Hermitian isotypic blocks by exact Gram-dual congruence, identify the correct 36-dimensional real metric-Hermitian outer span, certify a coherent lower witness by exact construction, and certify an outer-PPT upper bound by an exact rational sum-of-Hermitian-squares dual whose defining identity holds with zero residual. The result is the exact chain F*_effEB,V6 ≤ F*_outerPPT,36 ≤ 1/100 < 1/40 < 39587/1382976 ≤ F*_coh(Sp3), so a specific coherent covariant channel strictly outperforms every entanglement-breaking effective channel through the six-dimensional memory V6. Two standalone verifiers — one dependency-free (Python standard library only), one a full reconstruction — recheck the certificates with no optimizer and no floating point in the proof path; both return ALL PASS. Three registers are kept visibly separate: [T] exact theorem, [N] numerical diagnostic, [O] open/conditional. The exact optima, the tightness of the 1/100 bound, and any general Sp(n) or capacity interpretation remain open. The Sp(2) and Sp(3) objectives are analogous covariant functionals — shown distinct by an exact normalization-bridge computation — so the tower correspondence is structural rather than one identical functional carried up the tower. The attached bundle contains the manuscript, both verifiers, the exact-rational datasets, run scripts, and SHA-256 manifests for full reproducibility.
We study the canonical USp(4)-covariant quantum compression problem in which the fifteen-dimensional carrier W = V5 ⊕ V10 is encoded through a four-dimensional covariant memory M = V4 and decoded, and we ask for the optimal entanglement fidelity F*(p) of the composite channel on a trace-one, sector-weighted reference state ρ_p. This problem is the operational face of a structural fact: the same pair of sectors 5 ⊕ 10 that makes USp(4) on C^4 state-homogeneous but not observable-uniform is exactly the modal content a minimal covariant memory is asked to carry. Our contribution is not a resolution of the underlying capacity question but a demonstration that this problem admits an exact reduction and a machine-checkable global bound: the reduction and the final lower- and upper-bound certificates are exact, while only the intermediate relaxation-solving step is numerical. We give an exact representation-theoretic reduction — Wedderburn splitting of the covariant commutant over Q(i), an induced-involution positivity bridge, and an exact trace-preservation reduction to fifteen free rational coordinates — turning F*(p) into a bilinear polynomial optimization over a product of two explicit semialgebraic bodies, each the affine preimage of the product cone (R+ × SOC4 × SOC4). Exploiting correlative sparsity we build an order-two sum-of-squares relaxation, solve it numerically for discovery, reconstruct the resulting SOS upper bounds in exact rational arithmetic, and separately certify exact rational feasible lower bounds from explicit witnesses. At p = 1/2 and p = 7/8: 0.0749 ≤ F*(1/2) ≤ 0.21, and 0.1546 ≤ F*(7/8) ≤ 0.643 (exact rationals in the paper). The upper bounds are Positivstellensatz certificates whose polynomial identity and positive semidefiniteness are verified with no optimizer in the loop by a standalone verifier we provide. Throughout we keep three visibly separate registers — exact theorems, numerical diagnostics, and open or conditional interpretations — and we are explicit that the exact optimum, the exactness of the relaxation, and the width of the certified interval remain open. Package: this record includes the manuscript, a single 30-page combined archival PDF (cover + manuscript + auditable reduction-supplement listing) inside the package, and a verification package with two independent machine-checkable obligations (the exact reduction chain and the certified two-sided enclosure). Manuscript licensed CC BY 4.0; code licensed MIT.
This record gives an exact, machine-checked resource analysis of a specific USp(6) = Sp(3)-covariant quantum compression construction, and the reproducible certificates behind it. The carrier is W = V14 ⊕ V21 (dimension 35), the memory is the sixdimensional irreducible V6, the source is the balanced sector-mixed state with weights (1/28, 1/42), and the figure of merit is the Schumacher entanglement fidelity 𝜔. The construction and its coherent witness are inherited from the companion compression paper (DOI 10.5281/zenodo.21744306), which certifies F*_effEB,V6 ≤ 1/100 < 1/40 < 39587/1382976 = 𝜔(witness): a coherent covariant channel through the six-dimensional memory beats every entanglement-breaking channel through that memory. Four results, each verified exactly over the rationals, answer one coherent question — what preserves the compression advantage in this symmetry-constrained construction: (1) Sector-dephasing benchmark. Removing all coherence between the two output sectors drops the fidelity from 39587/1382976 to the within-sector value 55/3528, which still exceeds the 1/100 entanglement-breaking upper bound; the certified advantage survives, and the coarse output-sector label is blind to the input. (2) Branch entanglement. The two normalized output-sector channels N14 and N21 (output sector restricted, input/reference kept full) are both completely positive, trace-preserving, and have a negative partial transpose; each retains reference entanglement, so neither is entanglement-breaking. (3) Branch replacement. Replacing one branch at a time by an entanglement-breaking channel, at fixed sector probabilities and through the same memory, keeps the V14 branch (M14, 𝜔 = 835/74088 > 1/100) above the bound while the V21-retaining channel (M21, 𝜔 = 263/49392 < 1/100) does not clear it; both complete channels remain NPT. (4) Entanglement-breaking comparator. An explicit admissible covariant entanglementbreaking channel through the same memory strictly outperforms M21, with 𝜔 = 1544075/288004752 > 263/49392 (exact margin 5261/144002376 > 0). This resolves the previously open comparison and brackets the entanglement-breaking optimum as 1544075/288004752 ≤ F*_effEB,V6 ≤ 1/100. The overall lesson is objective-specific: retaining reference entanglement and obtaining a task-specific quantum advantage are distinct here. Every claim is checked by an exact rational verifier (python-flint): Choi positivity and partial-transpose tests use the certificate’s own Gram metric, and the comparator’s score is confirmed by reconstructing the actual channel and re-evaluating the original objective. Scope and caveats: the results have been audited for correctness by an independent reviewer; this record does not claim comprehensive novelty relative to the existing literature on quantum-memory resource theories and operational benchmarks, nor establish journal suitability. Any connection to the author’s broader “Presence Hypothesis” program is
We observe that the ingredients a relational construction already requires contain a gauge structure, and that its group is the one the frame cannot see. LetM= Gbig/H be a homogeneous space. Three independently established facts close into a statement when placed together. An operational quantum reference frame is a covariant POVM on a homogeneous space Σ ∼= G/Hx, and the relativization map expressing a system observable relative to it is defined only on the Hx-invariant part of the system algebra. The canonical connection of a reductive homogeneous space is h-valued: the projection of the Maurer–Cartan form onto the isotropy algebra. And the isotropy directions carry zero Fisher–Bures weight, which, that metric being the maximal monotone one, is the first statement in the vocabulary of the geometry. Four consequences. Gauge invariance is not imposed on the relational description; it is the condition under which an observable can be expressed relative to a frame at all. The Cartan decomposition splits what the frame cannot resolve from what it measures. The structure is non-abelian whenever the isotropy is, and for Sp(2n,R)/U(n) the isotropy is U(n). And the size of that isotropy and the multiplicities of its content are fixed by the field content, NB = 2 dimG+nH, a gauge sector carrying multiplicity two because the adjoint is real and a charged scalar one because the charge is complex. We then measure what the gauge algebra’s commutant inside the isotropy retains of it: the double centraliser is the full type-preserving algebra L λ u(dλ), of dimension 82 against the gauge algebra’s 12. The commutant does not remember the gauge group; it remembers the isotypic decomposition.No gauge group is derived here, and nothing is identified with anything physical.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.