This work establishes the first theorem-level connection between a structure derived internally by the Sigma calculus and a published experimental quantum system. The Sigma calculus predicts a canonical finite transition structure generated by four equivalent source directions. Until now, this structure had been established only within the internal mathematical framework. Here it is shown that the complete structure is realized, at the level of unweighted nearest-neighbour transition incidence, in the published site-resolved (⁸⁸Sr) quantum-walk experiment of Young et al. The result is stronger than a visual or geometric similarity. An exact finite equivalence is constructed between the Sigma source complex and the experimentally documented transition section. This equivalence preserves the full outgoing transition star, admissible paths, face relations, discrete derivatives, and the associated Noether data. It can be verified by explicit finite matrix identities and therefore provides both a reproducible certificate and a clear criterion by which the proposed registration could be falsified. After the experimental transition structure is registered, the canonical Sigma projection produces a three-dimensional carrier whose normalized Gram matrix is exactly that of a regular tetrahedral simplex: its diagonal entries are (1), all off-diagonal entries are (-1/3), and the resulting simplex residual vanishes exactly. No spatial angles, geometric coordinates, external point cloud, or fitted reconstruction are introduced to obtain this result. The significance of this package is therefore that Sigma has crossed an important boundary: a nontrivial structure derived from its own finite transition and Noether principles is no longer confined to the abstract calculus but has been identified exactly within a source-locked section of real experimental data. The scope of the result is deliberately precise. It establishes equivalence for the published unweighted nearest-neighbour effective-transition section. It does not claim microscopic identity with the experimental apparatus, exact equality of tunnelling rates, reconstruction of laboratory space, or exclusion of additional longer-range interactions.
This work establishes the first theorem-level connection between a structure derived internally by the Sigma calculus and a published experimental quantum system. The Sigma calculus predicts a canonical finite transition structure generated by four equivalent source directions. Until now, this structure had been established only within the internal mathematical framework. Here it is shown that the complete structure is realized, at the level of unweighted nearest-neighbour transition incidence, in the published site-resolved (⁸⁸Sr) quantum-walk experiment of Young et al. The result is stronger than a visual or geometric similarity. An exact finite equivalence is constructed between the Sigma source complex and the experimentally documented transition section. This equivalence preserves the full outgoing transition star, admissible paths, face relations, discrete derivatives, and the associated Noether data. It can be verified by explicit finite matrix identities and therefore provides both a reproducible certificate and a clear criterion by which the proposed registration could be falsified. After the experimental transition structure is registered, the canonical Sigma projection produces a three-dimensional carrier whose normalized Gram matrix is exactly that of a regular tetrahedral simplex: its diagonal entries are (1), all off-diagonal entries are (-1/3), and the resulting simplex residual vanishes exactly. No spatial angles, geometric coordinates, external point cloud, or fitted reconstruction are introduced to obtain this result. The significance of this package is therefore that Sigma has crossed an important boundary: a nontrivial structure derived from its own finite transition and Noether principles is no longer confined to the abstract calculus but has been identified exactly within a source-locked section of real experimental data. The scope of the result is deliberately precise. It establishes equivalence for the published unweighted nearest-neighbour effective-transition section. It does not claim microscopic identity with the experimental apparatus, exact equality of tunnelling rates, reconstruction of laboratory space, or exclusion of additional longer-range interactions.
This work extends the theorem-level connection established in the Sigma Transition/Noether Same-Section Reviewer Package (doi:10.5281/zenodo.21639274) from the unweighted transition-incidence structure to the additional anisotropic rate metric measured on the same source-locked ⁸⁸Sr quantum-walk section. For a positive opposite-channel rate assignment (Jₓ, Jᵧ, Jₓ, Jᵧ), compression to the canonical rank-three Sigma augmentation carrier produces the exact spectrum Jₓ, Jᵧ, and (Jₓ + Jᵧ)/2. The prescribed source quarter-turn is an isometry if and only if Jₓ = Jᵧ, whereas the corresponding half-turn remains an isometry for every positive pair. Consequently, unequal measured axis rates do not invalidate the previously established unweighted C₄ same-section registration. Instead, they reduce the symmetry of the weighted carrier metric canonically from C₄ to C₂. An explicit rank-three metric realization, its edge-length classes, and a scale-independent anisotropy certificate are derived without fitting spatial directions, angles, auxiliary points, or laboratory coordinates. For the published Young et al. central tunnelling rates, Jₓ/ℏ = 2π × 178.1 Hz, Jᵧ/ℏ = 2π × 148.3 Hz, the resulting residual is Ωᵣₐₜₑ = 22201/1331712 ≈ 0.01667102196270665. Using only the reported marginal standard errors, the exact weighted-C₄ condition Jₓ = Jᵧ is incompatible with the central estimates by at least 149/11 standard-error units for every admissible correlation. The already registered finite transition-complex equivalence transports the supplied rate-induced Dirichlet form to the Sigma-labelled transition star. This closes the weighted application-form route once the experimental rate assignment has been registered. It also provides an exact falsification interface: the unweighted source symmetry, the weighted metric symmetry, and their respective residuals are mathematically distinct and can be tested independently. The accompanying reproducibility layer reconstructs the published central rates from the public analysis material and evaluates a disjoint even/odd repetition split of the official experimental record. The split supports conditional same-experiment reproducibility of the anisotropic rate shape. It is not an independent experimental confirmation, because it uses the same experiment and retains the published nearest-neighbour opposite-axis model. The principal significance is that the Sigma-to-experiment correspondence now survives a nontrivial weighted deformation: the experimentally registered transition section retains its exact source-generated combinatorial structure while its measured rate metric acquires a rigorously determined anisotropic normal form. The tunnelling rates Jₓ and Jᵧ enter this application as externally supplied effective-Hamiltonian characterization data; predicting their numerical values independently of the chosen experimental preparation is neither required nor intended here. The result does not identify the carrier with laboratory space, establish microscopic identity with the apparatus, exclude longer-range interactions, or constitute an independent experimental validation of the Sigma framework.
We establish an exact realization of the structural drift construction of the Negative-Entropy framework in a standard two-level quantum system. The abstract constraint operator J, its kernel L=\ker J, and the associated orthogonal projector \Pi_L are realized precisely by the spectral eigenspace structure of the two-level Hamiltonian, with the identification J_-=H-E_-I=2d\,\Pi_+. The projector-drift identity of the earlier framework reduces exactly to \dot\Pi_L=\dot\Pi_-, and the resulting operator chain J_-\to\Pi_-\to\dot\Pi_-\to K\to(g_{ij},F_{ij})\to\Gamma_J connects the abstract construction to quantum geometric data through the Kato parallel-transport generator, the quantum metric, and the Berry curvature. We further prove a non-equivalence result within the original K_3 network-flow model: at a fixed active-set region where the structural projection is locally constant, the structural drift strength vanishes identically, while the multiplicative dynamical feedback strength remains strictly positive. This shows that structural drift is not a necessary mechanism for dynamical feedback in the original K_3 construction, and no linear operator factorizes the feedback response through the directional structural-change operator. The paper makes no empirical claims about consciousness, and does not assert a unified theory. The exact realization is an operator-level identification within the two-level spectral geometry. It does not introduce a new quantum dynamical law, nor does it establish a physical equivalence between the quantum geometric response and the network-flow feedback strength. The paper establishes a mathematically verifiable bridge between the abstract structural framework and a standard physical system, together with a rigorous boundary on where the bridge does not extend. Note on notation. The constraint matrix of the K_3 network-flow model is denoted \mathsf{B}, while the scalar entry B of the dynamical matrix M_q retains the unadorned symbol B. Note on operator conventions. The framework is developed in finite-dimensional real or complex Hilbert spaces. The adjoint of a linear operator A is denoted A^\dagger; in the real case this coincides with the transpose A^\top. Projectors are orthogonal projectors with respect to the Hilbert-space inner product.
Paper 3 of a four-paper program on generalized quantum measurement, realization geometry, and informational invariance. Generalized quantum measurements admit multiple purification and dilation realizations of the same operational state–measurement data. This paper studies the geometric structure induced by that representation freedom and develops a transport theory on the resulting realization space. On each finite-dimensional regular fixed-rank state–POVM stratum, framed purification amplitudes and framed Naimark realizations are shown to assemble into a smooth principal bundle whose fibers are precisely compact auxiliary gauge orbits. The paper establishes complete-POVM robustness results controlling states, labeled measurement architectures, fixed coarse events, canonical normalization, and smooth operational paths. A framed Hilbert–Schmidt response geometry is then introduced on the realization bundle. The associated minimum-response construction yields a smooth gauge-equivariant principal connection and an induced positive response metric on the operational base manifold. This provides a geometrically distinguished transport law and establishes relative canonicity within the framed response geometry. The manuscript further introduces Quantum Informational Identity (QII) and Dynamical Quantum Informational Identity (DQII) as explicit foundational identity assumptions. Exact gauge transformations are shown to preserve the generated informational profile of a realization, while smooth gauge paths preserve the corresponding dynamical profile history. Under QII and DQII, all quantum-informationally admissible principal connections determine the same represented quantum-physical transport content. Relative to the framed Hilbert–Schmidt response geometry, the minimum-response connection is the unique minimum-response representative of that common content. The paper refers to this combination of: uniqueness of represented quantum-physical transport content, and uniqueness of the framed minimum-response representative, as absolute canonicity within the generated realization framework. The results are restricted to finite-dimensional generated realizations on regular fixed-rank strata with fixed outcome labels. The paper does not claim that the framed response geometry is uniquely derived from informational principles, nor does it extend the transport theory across singular or rank-changing loci.
Paper 2 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper proves that a deterministic classical six-bit engine can realise exact finite qubit prepare–measure behaviour while retaining the classical path, role and receipt structure of the underlying engine construction. Building on the tetrahedral qubit geometry established in Paper 1, the paper supplies the operational layer connecting the receipt-custodied engine architecture to exact qubit prepare–measure form. For every lawful anchor, TUS OS Engine 1 constructs a canonical three-state face and unfolds it through three principal LANDING paths, each comprising a realised first step, intermediate-state re-evaluation and same-role continuation. Independently computed GUIDANCE witnesses provide subordinate contextual evidence. The operative object is therefore a receipt-custodied, LANDING-led expansion, not a collection of six co-equal transitions. The paper defines a Coordinate-Incidence Operational Harness, formally distinct from TUS Domain Binding. For each lawful anchor, its foundation-defined four-outcome construction maps all 64 classical six-bit carrier states onto exactly 27 distinct anchor-relative qubit preparations under one tetrahedral-SIC POVM. A separately declared three-outcome face Harness reproduces the exact qubit-trine table, while the canonical four-state foundation sector reproduces the exact tetrahedral-SIC table. The 27-state qubit preparation image consists of eight pure states, twelve states of Bloch radius 2/3, six states of radius 1/3, and one maximally mixed state. Across the complete canonical census of 192 LANDING paths, every first transition changes the qubit preparation image, every second transition changes it relative to the intermediate image, and exactly 58 complete two-step paths return to their initial preparation image. The paper further establishes exact classical and qubit factorisations of the Harness response law, positive finite entrywise isolation margins for the target trine and tetrahedral-SIC kernels, and bounded rigidity of the corresponding qubit realisations under the stated dimension-two, pure-preparation and rank-one-effect assumptions. The result separates three structures that remain distinct throughout the construction: the deterministic classical carrier and transition architecture, the exact qubit prepare–measure image induced by the Harness, and the receipt-custodied LANDING/GUIDANCE path structure. Every state visited by the LANDING-led expansion is resolved within the same exact qubit prepare–measure scenario, while path order, rank role, transition history and receipt identity remain classically carried. The qubit representation neither replaces the underlying classical carrier nor promotes the LANDING dynamics themselves to quantum channels. Current TUS receipts custody the realised source architecture and its path identities; they do not presently certify Coordinate-Incidence Harness trials or outcomes. The Harness remains a separately declared, computationally non-driving operational interface. The claim boundary is explicit. The paper establishes an exact qubit prepare–measure representation of a deterministic classical finite-state architecture under the declared Operational Harness. It does not claim physical quantum execution, contextuality, classical impossibility, quantum computational advantage, device-independent self-testing, or that LANDING transformations themselves constitute qubit channels. The tested natural lift explicitly does not establish the latter. Supporting General Theory Programme This paper is Paper 2 in the TUS OS quantum-operational and Answerability research programme. Building on Paper 1’s exact tetrahedral geometry, it develops the operational calculus that turns the canonical TUS structure into exact qubit prepare–measure behaviour across the full 64-state carrier, including the 27-state qubit preparation image, tetrahedral-SIC and trine realisations, and LANDING/GUIDANCE state resolution at the paper’s declared scope. Paper 3 extends this to the complete relational Atlas and exact global static quantum dimension. Paper 4 develops the complete-future, orthogonality, process, LANDING/GUIDANCE, and native Triadic synthesis theory. Paper 5 introduces the Anchored Analog Quantum System framework, while Papers 6–7 generalise the contract-relative resource and operational-observability structure. Papers 8–9 then develop generic Answerability theory, Paper 10 proves the complete protected and executable realisation of TUS OS in the Answerability Calculus, and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 2 therefore provides the operational bridge between the geometry established in Paper 1 and the later relational, process, resource, executable, and General-Theory results. 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 three-outcome face Harness yields the exact trine table. The Harness is computationally non-driving: it represents the already-fixed classical carrier structure rather than supplying inputs to the native engine. 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 construction to the completed-foundation family and canonical TUS OS subatlas, establishes exact carrier-reconstruction structure, proves that the repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4, and establishes deterministic predictive-state minimum 64 for the frozen complete principal future language. 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 theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold d
Physics may have mistaken a quotient for an origin. Quantum theory represents a preparation by a trace-one positive operator ρ. This is exact for relative outcome weights, but it is produced by the many-to-one map N(σ)=σ/Tr(σ), for which N(λσ)=N(σ) for every λ>0. If total response scale is inert under every admissible continuation, the identification loses nothing relevant. If one declared continuation can distinguish σ from λσ, however, normalisation has erased a future-relevant distinction. The trace-one state remains an exact terminal quantum reading, but it cannot by itself be a faithful future-sufficient carrier. The reason this order is easy to miss is structural: once the quotient is taken as the complete input, every downstream construction that factors through it is necessarily blind to the positive-ray scale already identified away. The mathematical ingredients used below are largely established and are not presented as isolated novelties. Jordan–von Neumann, Frobenius, and finite-dimensional Gleason–Busch-type results serve instead as consecutive links in a conditional dependency-order theorem. Equality of complete future-response profiles first fixes the minimal predictive quotient. Under explicit locks, positive refinement fixes square-root access; quadratic recomposition fixes a real inner-product carrier; and the stated return-algebra, positive-involution, and central exhaustive-phase locks then select its complex Hilbert representation. Positive additive effect weights acquire the unique pre-normalised trace form Wω(E)=Tr(σωE); and only after the non-zero total sω=Tr(σω) is closed does one obtain ρω=σω/sω and the Born reading. The familiar finite-dimensional Hilbert–trace–Born structure therefore reappears, but in a different dependency position. Two no-go results make this dependency claim exact within their stated domains: canonical ray normalisation cannot by itself faithfully carry active scale, and, in dimension at least two, a normalised invertible congruence is affine on the full state space only when its transport is scale-uniform. A common-profile theorem states when independently reconstructed readings are uniquely profile-equivalent over one fixed common future-profile image, rather than merely compatible descriptions. Under locks L0–L9 on one typed finite-dimensional assembly datum, these results form a single conditional genealogy; the paper neither derives quantum theory from NIR alone nor asserts that nature realises the locks. Its central result is not a new quantum rule. If two realised or admissibly generated representatives σ and λσ on the same positive ray, with σ≠0 and λ≠1, have different complete declared future-response profiles, no faithful representation of those profiles can factor solely through canonical ray normalisation N. The trace-one state is not wrong: it is the exact answer to a later, conditional question.
Paper 1 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper establishes an exact tetrahedral qubit-state geometry within the deterministic six-bit construction underlying the TUS OS Triad. For every lawful six-bit anchor A, the full-state-space maximin construction selects companions B,C such that the resulting triple has exact pairwise Hamming separation (4,4,4). Every such triple has a unique equidistant completion in the same six-bit state space, given by D = A ⊕ B ⊕ C. Under the normalised sign embedding, the four completed states have unit norm, pairwise inner product −1/3, zero centroid and rank three. They therefore form the vertices of a regular tetrahedron. The paper constructs an exact isometric representation of this geometry as four qubit Bloch vectors forming a tetrahedral symmetric informationally complete structure, while affine centring and common renormalisation of the original three-state face yield exact qubit-trine geometry. The analytical construction is verified exhaustively over the complete finite state space. The census covers all 64 six-bit anchors, 249,984 ordered anchor-conditioned candidate pairs, 41,664 unordered triples, 960 mutual-distance-four triples, and 240 completed four-state sets. The same geometric recovery is demonstrated in two frozen public TUS receipt packages. The result separates the mathematical completion from the protocol object. Only the three states (A,B,C) are emitted as active Triad starts; the fourth state (D) is a uniquely determined mathematical completion and does not become an additional receipt-bearing protocol member. The tetrahedral and trine constructions therefore provide exact qubit-geometric representations of an already-fixed finite classical structure rather than additional states or operations in the native TUS OS protocol. The claim boundary is explicit. The paper does not claim that TUS OS physically prepares qubits, performs SIC or trine measurements, exhibits contextuality or other nonclassical behaviour, or obtains quantum computational advantage. Its result is an exact finite geometric correspondence and protocol linkage between the completed six-bit Triad structure and canonical qubit tetrahedral and centred-face trine geometry. Supporting General Theory Programme This paper is Paper 1 and the opening theorem paper in the TUS OS quantum-operational and Answerability research programme. It proves that the canonical six-bit TUS OS Triad has exact (4,4,4) Hamming geometry, a unique XOR completion, regular-tetrahedral geometry, an exact qubit-SIC representation, and centred-face trine geometry. Paper 2 develops the exact operational calculus and full-carrier qubit prepare–measure representation. Paper 3 extends the result to the complete relational Atlas and exact global static quantum dimension. Paper 4 develops the complete-future, orthogonality, process, LANDING/GUIDANCE, and native Triadic synthesis theory. Paper 5 introduces the Anchored Analog Quantum System framework, while Papers 6–7 generalise the contract-relative resource and operational-observability structure. Papers 8–9 then develop generic Answerability theory, Paper 10 proves the complete protected and executable realisation of TUS OS in the Answerability Calculus, and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 therefore provides the mathematical starting point for the complete programme. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS Triad XOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion and exact qubit-SIC representation. The canonical maximin construction yields a unique same-space completion whose normalised sign embedding forms a regular tetrahedron, while the centred and commonly renormalised emitted face has exact qubit-trine geometry. The emitted three-state face itself is not asserted to be a trine before that affine centring and normalisation. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureExtends the Paper 1 geometry to 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, 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 construction to the completed-foundation family and canonical TUS OS subatlas, establishes exact carrier-reconstruction structure, proves that the repeated-common-effect static Atlas has exact minimum complex Hilbert dimension 4, and establishes deterministic predictive-state minimum 64 for the frozen complete principal future language. 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 BreakdownGeneralises the access-dependent resource phenomenon 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 theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold dual-Poisson transparency. Paper 8 — The Resource Geometry of Answerable InferenceIrreducible Answerability Structure, Minimal Sufficient Cores, and Certified Inference-Resource BoundariesDevelops the resource geometry of answerable inference. It formalises lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions and irreducible sufficient cores, while keeping semantic structure, certified lower bounds and operational feasibility independently constituted. Gate E establishes a local answerability-to-resource threshold result; globally, exact closure identifies the Pareto-minimal inference-resource boundary. On the generated exact branch, L = ↑F, and under well-partial-order conditions the resulting lower-resource geometry has a unique finite canonical antichain normal form. Paper 9 — The Answerability CalculusLawful Reduction, Contract–Resource Representation, Composition and ActivationDevelops a general mathematical calculus of Answerability as a constituted, typed relation. Its Base Theorem states that no terminal proposition counts as an answer unless it lawfully discharges the independently constituted contract. The paper develops lawful reduction, contract–resource geometry, extremal representation, repai