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 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 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.
The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps ΦA,B,C and ΦD,E,F on Md, the entrywise bound (AD)ij≥2⌊d⌋AiiAjjDiiDjj for all i,j guarantees that ΦA,B,C∘ΦD,E,F is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with a=1 is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.
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.
Paper 3 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops an exact finite local-to-global analysis of the six-dimensional Hamming-cube foundation Atlas underlying the TUS OS construction. Every admissible completed foundation carries the same exact local qubit prepare–measure representation, while the complete repeated-common-effect Atlas has exact minimum complex Hilbert dimension four. The paper classifies the full ambient family of completed four-state foundations in the six-dimensional Hamming cube. Every foundation is uniquely an affine coset associated with a perfect matching of the six coordinates. There are exactly 240 unordered foundations and 5,760 ordered codebooks. Every foundation induces the same exact 27-class tetrahedral-SIC qubit operational quotient, with carrier-fibre profile 1⁸, 2¹², 4⁶, 8¹ and the corresponding Bloch-radius structure of eight pure states, twelve states of radius √(2/3), six states of radius 1/√3, and one maximally mixed state. Each ambient chart reduces losslessly to a signed perfect matching. The chart-coloured union of two matchings obeys an exact component-parity collision law, yielding the complete seven-value image spectrum over all 28,680 ambient foundation pairs. Exactly 10,800 pairs jointly reconstruct the full 64-state carrier, giving an ambient mathematical chart-cover number of two. The frozen TUS selector defines a canonical subatlas of 36 distinct foundations and 64 anchor-indexed chart instances across nine matching classes. Its selected-foundation reconstruction graph is bipartite and eight-regular with 144 edges, while selector multiplicity yields exactly 400 reconstructing pairs among the 2,016 canonical chart-instance pairs. Only this canonical subatlas inherits the native LANDING and GUIDANCE structure. At the sequential level, all 384 active map/chart descent tests fail: no active LANDING map descends to a state-only transformation on a fixed 27-class local chart quotient. Under the complete principal future language, every canonical chart refines to all 64 deterministic predictive states, with maximum separating depth two. The subordinate GUIDANCE witness is also classified: its endpoint map has image size 34, eight two-cycle attractors and maximum transient depth five, while the additional GUIDANCE-selected chart raises principal path reconstruction from 138/192 contexts to 169/192. The paper then establishes the exact global static common-effect Hilbert dimension. Although every individual foundation admits an exact qubit representation, no unrestricted qutrit state/common-effect family can reproduce all 4,096 Atlas probabilities while simultaneously satisfying all 240 overlapping foundation POVM identities. Together with an explicit chart-independent dimension-four construction, this proves d_q^atlas = 4 for the complete static Atlas. The attaining construction uses two declared qubit sectors; unrestricted uniqueness or rigidity of minimum representations is not claimed. The result therefore exhibits a precise local-to-global separation: exact qubit prepare–measure form exists on every individual foundation, but an exact chart-independent repeated-common-effect representation of the complete Atlas requires complex Hilbert dimension four, while the complete deterministic future language preserves all 64 predictive identities. Local operational state, global relational state and deterministic predictive state are consequently distinct mathematical structures. The paper explicitly keeps these distinct from the still-unresolved whole-position process state. The scope boundary is explicit. The paper does not establish a physical quantum carrier, operational access to all 240 ambient foundations, LANDING as a quantum channel, GUIDANCE as a quantum observable, endpoint channel or instrument, contextuality, nonlocality, quantum advantage, unrestricted rigidity of minimum representations, or whole-position process synthesis. Its conclusions are exact at the static Atlas, canonical-chart and deterministic predictive scopes defined in the manuscript. Supporting General Theory Programme This paper is Paper 3 in the TUS OS quantum-operational and Answerability research programme. It extends the results of Papers 1–2 from the canonical local geometry and exact qubit prepare–measure behaviour to the complete finite relational Atlas. It classifies the full family of admissible foundations, proves that every local chart carries the same exact 27-class qubit operational quotient, identifies the canonical TUS subatlas, and shows that the complete native future language refines those local quotients back to all 64 deterministic predictive states. Its principal global result is that the complete common-effect Atlas has exact minimum quantum dimension d=4, while remaining distinct from the larger 64-state predictive structure and from later process-level dynamics. The paper also classifies the subordinate GUIDANCE witness and its finite reconstruction contribution without promoting it into a channel or instrument. The wider programme then develops complete-future and process closure in Paper 4, the Anchored Analog Quantum System framework in Paper 5, contract-relative resource and operational-observability theory in Papers 6–7, generic Answerability theory in Papers 8–9, complete TUS OS formal and executable realisation in Paper 10, and culminates in Paper 11 with A General Theory of Answerability. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations under one foundation-defined tetrahedral-SIC Harness, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSExtends the local prepare–measure construction to the complete ambient family of 240 completed foundations and the canonical TUS OS subatlas. Every completed foundation has the same 27-class qubit operational quotient; pair relations between charts provide exact carrier-reconstruction structure; the complete repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4; and the frozen complete principal future language has deterministic predictive-state minimum 64. The dimension-4 result is a static representation theorem and does not by itself supply common quantum process dynamics. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleFixes the formal native domain, complete principal terminal future-experiment language and one-common-word experiment class; establishes complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results and complement reachability; and proves that the rational invariant probability module has dimension 33. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, yielding the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, with cross-parity forcing and the exact state-only common-CPTP minimum remaining unresolved. It further establishes recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational results through a typed access-contract resource law, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification