Subjectivity-Intersection Mathematics asks whether structures treated as objective can be composed as stable consequences of lawful intersections among differentiated standpoint roles rather than supplied as standpoint-free primitives. We study one restricted homogeneous operator representation of that principle. Objective mathematics is used here as a representational instrument downstream of registered standpoint-relative intersections; no formal datum or objective reconstruction is identified with subjectivity or Subjectivity Intersection itself, and no such representation is promoted into an exhaustive objectification of its generative principle. Let H be a complex Hilbert space, e in H a unit vector, and R in U(H tensor H) a checked-braid solution satisfying R(e tensor e) = lambda(e tensor e), with |lambda| = 1. At every finite order, the canonical braid-orbit isometries have constant ordered overlaps equal to the phase-normalized right endpoint slice; leg reversal gives the left family. Within the declared Subjectivity-Intersection representation, we call this neutral canonical overlap property all-orders intersection consistency; the interpretive name adds no mathematical conclusion. The standard constant-overlap Gram criterion forces both phase-normalized slices to be positive contractions, while the pointing condition makes both fix e. No finite-dimensionality, separability, involutivity, self-adjointness, or trace assumption is used. Together with separately proved pointed-unitary covariance and compatibility with the declared shuffled tensor law, this verifies the stipulated technical compositional-objectivity criterion in this representation class. The sole general-mathematics novelty candidate is the complete implication from a pointed unitary braid to canonical all-orders overlaps and bilateral endpoint positivity. Its Gram, localization, compression, covariance, and tensor ingredients are cited rather than claimed; historical priority remains open pending claim-level search and specialist review. Separately, complete population response laws on finite registered sets determine source-relative quotients, and a supplied pair kernel descends exactly under fibre constancy. These facts specify an ideal-population prospective certificate contract, not recovery of its inputs or a finite-sample natural-identification theorem. The atomic, NMR, and FPR1 records are component-level compatibility checks. The cryo-EM check yields a byte-identical canonicalized candidate-matrix artifact over heterogeneous class averages, while its exact-path baseline remains incomplete. No natural certificate is accepted and no result identifies the formal variables with phenomenal subjectivity.
Subjectivity-Intersection Mathematics asks whether structures treated as objective can be composed as stable consequences of lawful intersections among differentiated standpoint roles rather than supplied as standpoint-free primitives. We study one restricted homogeneous operator representation of that principle. Objective mathematics is used here as a representational instrument downstream of registered standpoint-relative intersections; no formal datum or objective reconstruction is identified with subjectivity or Subjectivity Intersection itself, and no such representation is promoted into an exhaustive objectification of its generative principle. Let H be a complex Hilbert space, e in H a unit vector, and R in U(H tensor H) a checked-braid solution satisfying R(e tensor e) = lambda(e tensor e), with |lambda| = 1. At every finite order, the canonical braid-orbit isometries have constant ordered overlaps equal to the phase-normalized right endpoint slice; leg reversal gives the left family. Within the declared Subjectivity-Intersection representation, we call this neutral canonical overlap property all-orders intersection consistency; the interpretive name adds no mathematical conclusion. The standard constant-overlap Gram criterion forces both phase-normalized slices to be positive contractions, while the pointing condition makes both fix e. No finite-dimensionality, separability, involutivity, self-adjointness, or trace assumption is used. Together with separately proved pointed-unitary covariance and compatibility with the declared shuffled tensor law, this verifies the stipulated technical compositional-objectivity criterion in this representation class. The sole general-mathematics novelty candidate is the complete implication from a pointed unitary braid to canonical all-orders overlaps and bilateral endpoint positivity. Its Gram, localization, compression, covariance, and tensor ingredients are cited rather than claimed; historical priority remains open pending claim-level search and specialist review. Separately, complete population response laws on finite registered sets determine source-relative quotients, and a supplied pair kernel descends exactly under fibre constancy. These facts specify an ideal-population prospective certificate contract, not recovery of its inputs or a finite-sample natural-identification theorem. The atomic, NMR, and FPR1 records are component-level compatibility checks. The cryo-EM check yields a byte-identical canonicalized candidate-matrix artifact over heterogeneous class averages, while its exact-path baseline remains incomplete. No natural certificate is accepted and no result identifies the formal variables with phenomenal subjectivity.
A Kernel-Role Determinacy Theorem for Bell-Witness Fixation, No-Signaling, and Tensor-Level Compatibility This manuscript develops a fixed-domain AASC analysis of Bell-nonlocal quantum entanglement as a theorem of reusable witness determinacy. It does not introduce hidden variables, modify quantum mechanics, derive Hilbert-space formalism from below, or propose a new signaling mechanism. Instead, it asks a prior structural question: once a Bell-type operational family is used as the same non-degenerate witness across settings, lawful transformations, retained records, comparison, and later classification, what admissible same-domain standing form is forced? The governing claim is that AASC is not an optional interpretive overlay on physics. Once an operational Bell family is asserted as a determinate reusable witness, admissibility, standing, reference, and irreversibility are already active as minimal necessity conditions of same-witness objecthood. Target instantiation therefore carries governance occupancy: an alleged determinate Bell witness with no admissible governance realization is not an empty alternative, but a refusal or retyping of the target. The manuscript defines finite operational Bell objects with explicit setting carriers, outcome families, channel/interface data, standing-token valuations, continuation and report/reuse structure, comparison profiles, and act-time status. It then defines object-indexed query sorts, dependent response families, exactly recoverable operational simulations, lawful Bell groupoids, six governance-effect functors, and target-active extensional governance equivalence. The result is a typed interface in which arbitrary native formalisms can be compared by their target-bearing governance effects rather than by their internal syntax. The central Bell-specific theorem states that every same-target candidate-admissible Bell-governance construction is forced component by component into the certified boundary-compatible standing form on the complete target-active practice. Candidate admissibility does not assume the canonical form at the outset. Instead, target occupancy, lawful transport, standing-flow exclusion, compatibility certification, coordinate-deviation exhaustion, and componentwise closure force exact target-bearing realization and governance equivalence to the canonical representative CP∗C^{*}_{P}CP∗. The target-induced governance quotient therefore has exactly one element. The Bell carrier is treated as a compatible empirical model on a finite measurement cover. Bell-local completion is identified with the existence of a global section, and Bell nonlocality is the obstruction to such a section. The manuscript separates operational marginal invariance, absence of controllable communication, and exclusion of same-domain standing transport. No-signaling is therefore preserved without retyping Bell nonlocality as superluminal communication, metric transport, distant repair, or hidden standing transfer. The finite-dimensional quantum specialization uses the standard bipartite tensor-product and local-POVM carrier. Every separable realization in that carrier is Bell-local; therefore, if the fixed Bell-nonlocal empirical model has such a finite-dimensional quantum realization, every realizing state in that carrier is entangled. Post-quantum Bell-nonlocal no-signaling models remain within the general witness theorem but outside the quantum-entanglement corollary. The accompanying Lean 4 development, release v0.8.0, directly verifies the finite Bell-side witness carrier: denominator-32 vector/Born weights, structural no-signaling, CHSH numerator 808080, all sixteen deterministic local extrema, finite denominator-32 Bell-local-envelope exclusion, and conditional endpoint routing. Its direct witness certificate is vectorBorn_flagship_spine, and its deepest released endpoint is reasonableDeepestFiniteEntanglement_spine. The manuscript-level typed governance interface, empirical lawful groupoid, compatibility equivalence, coordinate-deviation exhaustion, completed role collapse, candidate-to-canonical derivation, and finite-dimensional quantum specialization are not attributed to the released Lean layer. This paper is the Bell-witness member of the paired AASC quantum determinacy arc. The record-side manuscript, Admissible Record Construction in Physically Non-Selective Quantum Regimes, fixes when local physical record occurrences stand as the same evidential record objects. The present manuscript fixes when the Bell operational family sampled by such records stands as the same reusable witness. The paired endpoint is therefore: record-role fixation+Bell-witness compatibility.\text{record-role fixation} + \text{Bell-witness compatibility}.record-role fixation+Bell-witness compatibility. The manuscript is classificatory and eliminative rather than mechanistic. It does not claim that AASC generates the Bell probabilities or supplies a hidden physical influence. Its claim is that reusable Bell-witness determinacy strongly constrains how Bell-nonlocal correlations may be coherently classified on a fixed domain: not as independently complete local states later connected by a mechanism, not as Bell-local hidden-variable value assignments, not as superluminal signaling, and not as same-act repair, but as boundary-level compatibility carried by one standing-preserving Bell witness.
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
When the conditions under which a record is formed remain fixed, a physical theory may represent that record completely by a unary law of the state observed. This paper asks what structure is forced when the same independently certified physical identity gives different complete records under different formation conditions. The initial datum distinguishes two typed empirical roles, while the state identity of the formation side is derived from its complete behavior. Failure of factorization through the first projection proves that the formation-side role cannot be deleted. The complete row and column behaviors of the primitive law then determine two kernel pairs and their coequalizers. Unique descent retains every observable distinction while removing every redundant label, and the resulting realization is terminal among all exact role-preserving realizations. The product pairs the reduced roles; a pullback extracts the combinations on which records can actually be formed. Only after these steps does exponential transposition identify each surviving formation class with one complete response map. The resulting bijection supplies the state criterion and establishes the observation-state object. Compatibility with the antecedent theory preserves the independently given physical state space and yields the physical–observation law in which a fact is generated jointly by the state observed and the state of observation. The same construction gives exact criteria for conditional, label-free, and joint identifiability and selects the same-source pullback on which transport may subsequently be defined. A nonzero coherent response on that pullback yields the phase-transport fundamental equation, the first fundamental equation written directly on observation space. Quantum theory subsequently enriches this canonical interface. Coherent response and full circular phase select the complex field; reversible binary mixing yields CP1 ∼= S2 ; independent composition creates the four-dimensional event domain; response-event identity glues probabilities across complete contexts; and Gleason representation with marginal consistency gives the Born trace pairing. The two-qubit correlation tensor then yields the singlet cosine law, the CHSH inequality, the Tsirelson bound, and the Horodecki spectral criterion. Bell violation is thereby identified with the absence of a positive global answer section across incompatible observation states. The quantum formalism is thereby closed into one architecture: complex state geometry describes coherent physical possibility, complete observation states specify the ways in which facts may form, the Born interface joins them, and quantum instruments carry a realized fact into subsequent dynamics. Fixed observation sections recover classical unary laws, while smooth equivariant enrichment lifts relativistic frame geometry and internal gauge geometry to the same joint domain. The central result is ontological. Physobser Theory admits observation into objective physical reality not as a mind, an external readout, or a perturbation, but as an irreducible state-bearing axis; a fact is generated at its lawful interface with physical state. Category theory is the proof instrument. The discovered physical content is the dual-axis constitution of facts, which supplies a common foundation for quantum, classical, relativistic, and gauge physics.
Paper 5 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper identifies and formalises a system-level phenomenon in exact quantum representation: an independently constituted deterministic native system can admit multiple exact quantum-operational views whose minimum representation resources can depend sharply on a precommitted access contract, while the native system itself remains unchanged. The quantum representation is therefore neither constitutive of the native system nor a post hoc encoding selected to reproduce a prescribed quantum target. It is an exact operational representation of behaviour exposed by a declared contract on an independently fixed structural target. We introduce two mathematical definitions. An Analog Quantum View (AQV) is the distinguished operational-equivalence class containing every exact permitted finite-dimensional quantum realisation of the behaviour exposed by a nonvacuous, precommitted access contract, under complete persistent-resource accounting. An Anchored Analog Quantum System (AAQS) is an independently constituted, provenance-bearing native system admitting a nonempty contract-indexed family of such views while preserving native identity, native quotient independence and protected provenance outside the quantum representation. The definition excludes circular construction, candidate-dependent post-selection, hidden side information and untyped resource comparisons; the additional properness criterion excludes recognition based solely on the universal orthogonal-sector embedding. TUS OS is proved to be a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus. Its independently defined native structural quotient is canonically isomorphic, as a complete deterministic transition-output system, to deterministic predictive equivalence and to the canonical six-bit carrier, with exact deterministic predictive minimum 64. On this same carrier-faithful structural target, within a fixed no-persistent-side-memory quantum representation category, the minimum Hilbert dimension depends sharply on the permitted operational access. The complete static Atlas has exact minimum d_Atlas^min = 4, whereas recovery-complete LANDING selected-line instrument access and recovery-complete GUIDANCE selected-line readout each have exact minimum d_L,instr^min = d_G,read^min = 64. These yield two exact factor-16 fixed-target, fixed-category resource separations. Recovery-complete state-only common-CPTP access remains bounded by 32 ≤ d_CPTP^min ≤ 64. A separate exact classical–quantum construction retains the certified four-dimensional quantum component together with a 32-state complement-orbit memory. Under the declared componentwise resource order, this hybrid and the 64-sector no-persistent-memory construction are Pareto-incomparable rather than competing estimates of a single resource minimum. The resulting TUS OS Anchored Analog Quantum System Theorem composes native recognition, predictive identity, contract-indexed quantum representation and conservative downstream interpretation without collapsing their domains. Predictive Synthesis and Analog Quantum Views are sibling constructions from independently fixed native identity: neither constructs the other, and neither reconstructs the Native Witness. Provenance and receipt custody remain outside the quantum target, while lawful Domain Binding and downstream human, rules-engine or AI interpretation, operating under independently fixed interfaces and gates, preserve protected native identity and source authority and cannot strengthen the governing evidential ceiling. No claim is made of physical quantum execution, quantum ontology, encoding of question text as a quantum state, quantum advantage, unrestricted domain coverage or model-independent correctness. Realised-instance inheritance remains conditional and target-specific rather than an unrestricted theorem over all TUS OS executions. The paper reports a retrospective prior-art audit through 11 August 2026 that identified no earlier or contemporary public disclosure whose documented properties were found to satisfy the complete AAQS conjunction. Subject to that audited public-record scope, TUS OS is the first formally demonstrated Anchored Analog Quantum System identified in the record examined. This is a system-class priority claim rather than a claim that the constituent ideas—quantum representations of classical processes, predictive equivalence, provenance, resource orderings or noninterference—are themselves novel. Supporting General Theory Programme This paper is Paper 5 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–4, it organises the established geometry, operational structure, complete-future theory, process closure, and native Triadic synthesis through the theory of Anchored Analog Quantum Systems, establishing TUS OS as a proper instance of that framework. Papers 6–7 extend the programme through contract-relative quantum realisation, resource separation, operational observability, and contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and calculus; Paper 10 proves the protected and executable realisation of TUS OS; 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 — 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 family of 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 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. 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 and the exact parity-resolved pairwise total-variation optimum; determines complement reachability; and proves that the rational invariant probability module generated by the complete terminal probability language 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, giving the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, while leaving cross-parity forcing and the exact state-only common-CPTP minimum unresolved. It further establishes recovery-complete LANDING and GUIDANCE process/readout results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact four-dimensional-quantum-plus-32-state-classical hybrid construction, joint LANDING–GUIDANCE compatibility, 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 and proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus. It elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational representations through typed access contracts and resource accounting, and composes native recognition, predictive identity, contract-indexed quantum representation and conditional downstream claim-custody preservation into the TUS OS Anchored Analog Quantum System Theorem. The 64 inputs are an abstract canonical corpus, not 64 asserted receipt executions. 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 O
Paper 7 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a structural theory for classifying how representation-resource requirements change when the operational obligation imposed on an independently fixed native system is lawfully strengthened. Building on the contract-relative resource law of Paper 6, it asks which source-level features of a strengthening determine or constrain the resulting resource transition. The theory fixes the native system, weak and strong contracts, realisation category, tolerance and charged resource before optimisation. A finite-baseline gate requires the weak contract to possess a finite-resource realisation, preventing pre-existing category nonrealisability from being misclassified as a strengthening-induced transition. The source-level object is a Contract Distinction System (CDS), which records the native configurations challenged by the stronger obligation, required distinctions or recovery targets, success rules, inspection boundaries and interface/commonality structure. Semantic challenge refinement and CRC cofinal/refinement equivalence control dependence on alternative finite challenge presentations. A Certified Observability Transport Theory (COTT) then maps licensed source-level facts into necessary resource lower bounds inside a fixed representation category and tolerance regime. The resulting directed finitary demand profile supports two distinct mechanisms. Internal divergence occurs when a single strengthened contract contains finite restrictions of arbitrarily large certified demand and can therefore exclude every finite-resource strong realisation in the declared category. Family divergence occurs when every member of a parametrised family remains finitely realisable while the additional resource induced by strengthening grows without bound across the family. Lower recognition is kept logically separate from upper-side existence: bounded certified demand is not treated as an existence theorem, and finite regimes require independent construction, compactness/gluing or another valid upper certificate. Conditional on finite baseline feasibility, the resulting pointwise classifier distinguishes resource-neutral, strict finite, finite-resource breakdown and UNRESOLVED transitions, with incrementally unbounded finite separation carried separately as a family-scaling modifier. Several exact witnesses separate these regimes. With three native labels, the same coarse distinction graph K3, representation category, Hilbert-dimension resource and discrimination threshold fixed, changing only interface commonality produces an exact minimum-dimension transition from 2 to 3 throughout 2/3 < τ ≤ 1/2 + √3/4. Thus interface structure can itself be representation-resource-bearing. For the canonical TUS OS target, the complete static Atlas has exact minimum Hilbert dimension 4, whereas recovery-complete selected-line instrument access has exact minimum 64, giving an exact strict-finite transition 4→64. The weaker state-only common-CPTP problem remains unresolved within 32≤d≤64, demonstrating that the stronger interface theorem cannot be transferred backward through a forgetful map. For periodically modulated decay processes, every finite member remains finitely realisable while the contract-induced capacity gap diverges with family size. For the fixed dual-Poisson process, every positive worst-case transparency strengthening eliminates all finite-resource strong realisations in the declared autonomous one-copy classical–quantum category. A synthetic Countable Pairwise Identity Ladder supplies a cross-interface internal-divergence test, while the Fanizza–Lumbreras–Winter result marks the separate outer boundary of baseline category nonrealisability. Together, the results establish that operational observability is a resource-bearing input to quantum realisability without being invariably resource-increasing. Resource transition depends on the typed structure of the operational obligation—not merely native-state count, predictive-state cardinality, distinction count or coarse distinction graph. Supporting General Theory Programme This paper is Paper 7 in the TUS OS quantum-operational and Answerability research programme. Building on Paper 6’s contract-relative resource theory, it develops operational observability and a structural classification of contract-induced resource transitions, distinguishing resource-neutral, strict finite, finite-resource breakdown and unresolved pointwise outcomes, with family divergence carried separately as a family-scaling modifier while separating lower-bound certification from constructive realisability. Papers 8–9 then move to Answerability as a generic mathematical object, developing the resource geometry of answerable inference and the Answerability Calculus; Paper 10 proves the protected and executable realisation of TUS OS within that 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 — 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, 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 OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and 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 ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, 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. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. 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, 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 the structural classification theory for contract-induced representation-resource transitions. It introduces the Contract Distinction System (CDS) as a representation-independent source description of the additional distinctions, recovery targets, challenge structure and interface/commonality obligations introduced by lawful contract strengthening; semantic challenge refinement and CRC cofinal/refinement equivalence to control presentation dependence; and a Certified Observability Transport Theory (COTT) that maps licensed source-level structure into necessary category- and tolerance-specific resource lower bounds. The resulting directed finitary-demand theory separates internal divergence within one fixed strengthened contract from family divergence across finitely realisable members, while keeping lower recognition logically independent from upper-side construction or compactness/gluing. Conditional on finite baseline feasibility, the pointwise classifier distinguishes resource-neutral, strict finite, finite-resource breakdown and UNRESOLVED transitions, with incrementally unbounde
added; unclosed consequences remain explicitly OPEN. This paper develops the interaction, phase-transport and identification components of the All-Domain Framework (ADF), which seeks a common account of microscopic and macroscopic formation and maintenance. The framework takes multi-frequency immaterial angular-momentum dynamic structures as the sole fundamental constituents of existence and states: dynamics is existence; without dynamics, there is no existence. Within a declared lowest-order local two-axis second-order tensor class, seven independent operator forms are obtained for non-collinear axes. Their unital real associative algebra is the full real three-by-three matrix algebra; this is not a completeness theorem for quantum states. Integrable real linear evolution has positive fundamental-matrix determinant at finite times, a constraint distinct from algebraic generation and from time irreversibility. The ideal phase–vector interface gives an inverse geometry, a chord-factorization obstruction and a complete conditional circle-map family. On a fixed-axis radial branch, an explicit formation-phase correction transports supplied initial phase data while retaining its winding. Oriented finite cross geometry identifies sampled relative phases; finite samples do not identify an unrestricted phase function or its winding. Tensor-side descriptors are complete in the declared class, but the frequency-side audit does not supply the matching layer actions. The physical layer–form mapping therefore remains OPEN / STOP. Coefficient selection, actual initial preparation, global probability weights, experimental correlations and single-record generation are not derived. The front comparison states precisely which answers are mathematical, which are conditional and which remain research objectives. Keywords: All-Domain Framework; quantum-generation interface; multi-frequency immaterial angular-momentum dynamic structures; helical self-organization; operator-form classification; phase transport; identifiability Preprint release: v1.0, 11 September 2026. This release is derived from research candidate v14. It aligns the series name with All-Domain Framework (ADF), removes the framework name from the title, and states the series-wide no-phenomenon-specific-patching rule. It does not upgrade any mathematical, physical or experimental result; all declared OPEN interfaces remain OPEN