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.
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 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.
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
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state A, published as a 1,472-byte witness, with certified Ic>0 at the exact rational p=16239/250000=0.064956 (per-Pauli convention; total error rate 3p=0.194868) -- beyond both the best value printed in the Agarwal et al. paper (0.064657) and the strongest state in the same authors' public repository (≈0.064911), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in p on [0,1/4], so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than 1/25000 -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.
This record contains the manuscript and complete verification artifact for the Bell inequality J62_4422 of Pál and Vértesi.It certifies the exact rational bracket750075557472719639 / 10^18 ≤ β_Q ≤ 750091086083765047288227625 / 10^27,and exact attainment of 3/4 by the stated two-qubit strategy. The lower endpoint is an exact rational strategy. The upper endpoint is certified by a sum-of-squares chain for the retained 365-word relaxation level.The Krawczyk certificate localizes the underlying KKT zero and establishes a strict local maximum, including negativity of the border form in all 256 boundary strata; it does not establish global optimality of that localized point. Accordingly, this work does not claim the exact quantum value, qubit-restricted optimality, minimum optimizer dimension, or global optimality of the localized KKT point.The archive includes SHA-256 manifests, standard-library replay programs, independently written certificate checks, mutation controls, and pinned full Krawczyk replays. The acceptance programs are fail-closed under optimized Python. The manuscript and certificates are licensed CC BY 4.0; verification code is licensed MIT.Version 1.1 is a hygiene-only update: it replaces one machine-local default path in a helper with the portable repository root ".", updates the corresponding checker and certificate source bindings, and leaves the mathematical calculation, inputs, claims, and replay outputs unchanged.
Cubitt, Harrow, Leung, Montanaro, and Winter (CHLMW) exhibited an explicit pair of quantum channels whose minimum output Rényi entropy is nonadditive at order zero, and reported numerical violations at positive orders close to zero. Their paper states that a semidefinite-programming argument yields a rigorous positive-order interval for the same example, without printing the endpoint or a verifiable certificate; a recent paper by Leung, Lovitz, and Wu records that for this pair "no rigorous endpoint was obtained". We close that gap for the trace-preserving normalization of the printed pair fixed in Section 2. Small rational witness matrices prove that every output of either channel has all eigenvalues between 301/100000 and 2/3; a single explicit entangled input has an exact rational joint output spectrum of rank eight; and two independent elementary interval arguments turn these three facts into a proof of strict additivity violation, Spmin(NR⊗NSˉ)<Spmin(NR)+Spmin(NSˉ), for every real order 0<p≤1/22. Every step of the verification reduces to comparisons of integers, and the complete certificate is a few small rational matrices that a reader can check with a short program -- or, for any single order, by hand. To our knowledge, consistent with the assessment of Leung, Lovitz, and Wu, this is the first printed, computer-verifiable certified positive-order endpoint for this explicit pair. We claim no novelty for the phenomenon or for the eigenvalue-floor mechanism, both due to CHLMW, and no optimality of the endpoint.
We study a minimal coexistence task: a single qubit of memory must be used, through one unassisted encode–decode factorization, both to report a classical binary label (which of two orthogonal "sectors" a carrier occupied) and to preserve quantum information within one sector, in the presence of a Z₂ symmetry. We quantify the first by a label accuracy L — which we show is exactly a two-state discrimination success probability for a decoder-induced measurement — and the second by the unconditional same-sector entanglement fidelity F_same. Our main result is the exact optimal trade-off: for every required accuracy τ in [1/2, 1], the maximum attainable F_same is a single closed-form curve Φ(τ), and this maximum is the same whether or not the factorization is required to be covariant under the symmetry (at the fixed memory representation V = Z). An explicit two-parameter covariant CPTP family attains Φ at every threshold, so covariance imposes no optimal-fidelity penalty anywhere on the trade-off. The frontier has a structural transition at τ = 5/8, above which every optimal code-restricted encoder is necessarily nonunitary, and the attainable score region is nonconvex. A directly applicable prior hybrid-memory inequality (Kuperberg) specializes here to F_same ≤ (5 − 2τ)/4, which is valid but strictly loose for τ > 1/2 and tight at τ = 1/2. Whether the exact frontier, the transition, or covariant optimality follow from earlier results remains open. The results are proved analytically, with selected exact symbolic identities, explicit constructions, and regression tests. This record includes the manuscript and a verification supplement of seven standalone SymPy scripts, a README, and a runner (run_all.sh) that reproduce the exact checks (each exits 0 on success; tested with Python 3.11.15, SymPy 1.14.0, NumPy 2.4.4). The results are mathematical; any interpretive applications are developed separately and are not part of this work.