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.
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.
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.
A completely positive channel N that is covariant for a group G acting on a carrier W = ⊕_s V_s (pairwise inequivalent irreducibles) admits two natural bilinear read-outs of its sector structure: a "realign reading" A_phys[s,t] = Tr[P_s N(P_t)]/d_s, the channel's coefficient matrix on the sector algebra, and a "no-realign reading" A_ext[s,t] = (1/d_s) sum_{a,A} <a|P_s N(|a><A|) P_t|A>, a score. We prove three things. First, covariance forces N(P_t) to be block-scalar, so A_phys is the coefficient matrix c_{st} in N(P_t) = sum_s c_{st} P_s (Schur's lemma). Second, the score is the Gram matrix of the channel's sector-overlap vectors, A_ext[s,t] = (1/d_s) <w_s, w_t> with w_s = (Tr[P_s K_k])_k; hence its diagonal is a manifest sum of squares (nonnegative for any channel) and D·beta is Hermitian positive semidefinite in the sector-trace metric D = diag(d_s), where beta = A_ext. Third, the score equals the sector-return diagonal, A_ext = diag(A_phys), if and only if two independent conditions hold: (I) the overlap vectors of distinct sectors are orthogonal, and (II) within each sector a trace/Frobenius balance sum_k |Tr K_k^{(ss)}|^2 = sum_k ||K_k^{(ss)}||_F^2. Condition (II) has a Schumacher entanglement-fidelity reading: the maximally-mixed entanglement fidelity of the sector-s diagonal sub-channel equals A_phys[s,s]/d_s. We call channels meeting (I)+(II) sector-return. A specific Sp(3) = USp(6)-covariant channel on the 35-dimensional sl(6) adjoint (split 14 + 21) is sector-return exactly, while generic covariant channels — the identity among them — are not; in the correct metric its diagonal sub-channels are scalar-unital and sit exactly at the (II)-balance, and a numerical spectral pass from exact-rational operators resolves them as Hermitian positive semidefinite of Choi rank D^2 = 36 (D = 6 the memory dimension), with dimension-indexed spectra whose leading eigenvalue is the sector-return coefficient. We close with the open problem this sharpens: whether covariant compression through a finite memory (a D-dimensional auxiliary memory, not a minimal Stinespring environment) forces sector-return.
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
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.
On a homogeneous space M = Sp(2n,R)/U(n) carrying a compact group action, a kinematic no-go establishes that no quantity derivable from the manifold, the metric, the base point and the group can label the members of a certain triple. This paper measures in the layer that premise does not reach: the layer in which a clock and a system are separated and one is conditioned on the other. The central result is structural — the triple and the clock are one consequence, not two. A single input, an irreducible of the compact group occurring with multiplicity m, forces at once that the sector is Sym2(Cm), that the commutant is u(m), and that the available clock is the Cartan of su(m) inside it, of dimension m−1. At m = 2: three lines, a four-dimensional commutant, a torus of rank two, a clock unique up to scale, weights {+1, 0,−1}. At m = 1: one line and no clock at all, measured in the same block, as the arena’s own falsifier. The clock is not assumed periodic; its compactness and minimal period are measured, and its overlap function, its three orthogonal readings and its sensitivity to the clock-state phase freedom follow in closed form from covariant-POVM formulae. With a distinguished tangent direction admitted, one exact measurement lifts the degeneracy: the projection of the adjoint square, restricted to the sector, is exactly diagonalwith integer equispaced spectrum. Two limitation theorems then show that admitting the direction buys classes but not one invariant direction, and that no spectral invariant can supply the one remaining bit. A family index is therefore a reading: the value a line takes on a clock derived from the same multiplicity that produced the line, with a residual bit gauge-fixed relative to an orientation the construction does not supply.
A common self-consistent channel diagnostic uses a reference contrast matrix R and a processed contrast matrix P. Under a fixed d-dimensional carrier, a common preparation/readout interface, and an invertible reference, one obtains W = Tr(P R^{-1}) = Tr T, where T is the transfer block on the d^2 − 1 traceless operator coordinates. Every entanglement-breaking d-dimensional channel obeys W ≤ d − 1. The inequality is useful only if the dimension and interface assumptions are part of the experimental contract. We construct an exact countermodel when they are not. For every d ≥ 2, set m = d^2 − 1 and let n_1, …, n_{m+1} be the vertices of a regular simplex in R^m. An affine encoding of a local m-dimensional signal ball into the m + 1 = d^2 classical simplex, paired with binary linear readouts, gives an invertible isotropic contrast matrix R = β I_m. The stochastic memory transition K_q = q I_{m+1} + (1 − q) 11^T/(m+1), 0 ≤ q ≤ 1, acts as r ↦ q r on the encoded local chart. Hence for every depth n the observed contrasts satisfy P_n = q^n R, all semigroup identities P_{n+k} = P_n R^{-1} P_k, and the apparent transfer spectrum is exactly that of the d-dimensional depolarizing channel. Yet the physical realization is a measure-and-prepare channel on a d^2-state classical memory and is therefore entanglement breaking for every q. A naive d-dimensional trace witness returns W = (d^2 − 1)q, which exceeds the legitimate entanglement-breaking threshold d − 1 whenever q > 1/(d + 1). At q = 1 the apparent violation factor is d + 1 although the physical memory is classical. Under affine mixture-preserving encodings, d^2 classical states are also minimal for faithfully representing an open (d^2 − 1)-dimensional interface chart. The result does not contradict valid dimension witnesses or trusted-input quantum-memory tests. It identifies a precise failure mode of transferring a dimension-conditioned trace certificate outside its interface contract: repetition consistency, spectral consistency, and arbitrarily deep Markov consistency do not by themselves recover the missing dimension assumption.