The companion paper on effective composition characterises, for a finite substrate without a supplied subsystem factorisation, the possibilistic support layer completely and the preparation layer for two reference classes, and explicitly leaves open the states-and-effects layer of the resulting composition contract. We resolve that layer's finite, mono-system, and bipartite-composition questions, keeping the general preparation layer and the tensor product exactly as open as the companion paper left them. The primitive family arising from cylindrical conditioning of a strictly positive, rank-one reference measure is not itself convex: it is finite while its own convex hull is a continuum, unconditionally, for any local outcome set of size at least two. An explicit classical-randomisation postulate closes it onto the full classical probability simplex, independently of the reference measure's rank; rank governs composition of the primitive family, not the shape of its closure once randomisation is added. Simplicial geometry fixes the mathematical representation of affine effects as vectors in a cube, but not their physical availability as measurements; a weak coarse-graining axiom already yields the entire Boolean skeleton of sharp effects for free, while the full continuum of effects requires a further, dual randomisation postulate on the measurement side. Under two further, distinct, and logically independent axioms — one for a common classical control resource, one for the physical existence of a composed bipartite effect — the pointwise product of two local effects is proved unique, not postulated, and local tomography holds exactly when the local effect families linearly span their ambient functional spaces, equivalently when they separate their local states and contain the unit effect in their linear span. All results are finite, exact, and reproduced by committed scripts using rational arithmetic. Interpretive reading (structural, not a result): the finite hypotheses examined here force, successively and for free, a classical, simplicial operational skeleton — convex states once mixing is supplied, a Boolean effect algebra once coarse-graining is supplied, and exact local tomography once spanning local families are supplied — under precisely identified imports, and produce neither a quantum state geometry nor the Born rule. Locating exactly how much operational structure a finite admissibility contract yields for free, and where every further richness must instead be imported, is the intended service of this result to the non-injective foundations programme: it marks the boundary this classical skeleton cannot cross on its own, and hands that boundary, precisely typed, to whatever module or phase law is proposed next.
The companion paper on effective composition characterises, for a finite substrate without a supplied subsystem factorisation, the possibilistic support layer completely and the preparation layer for two reference classes, and explicitly leaves open the states-and-effects layer of the resulting composition contract. We resolve that layer's finite, mono-system, and bipartite-composition questions, keeping the general preparation layer and the tensor product exactly as open as the companion paper left them. The primitive family arising from cylindrical conditioning of a strictly positive, rank-one reference measure is not itself convex: it is finite while its own convex hull is a continuum, unconditionally, for any local outcome set of size at least two. An explicit classical-randomisation postulate closes it onto the full classical probability simplex, independently of the reference measure's rank; rank governs composition of the primitive family, not the shape of its closure once randomisation is added. Simplicial geometry fixes the mathematical representation of affine effects as vectors in a cube, but not their physical availability as measurements; a weak coarse-graining axiom already yields the entire Boolean skeleton of sharp effects for free, while the full continuum of effects requires a further, dual randomisation postulate on the measurement side. Under two further, distinct, and logically independent axioms — one for a common classical control resource, one for the physical existence of a composed bipartite effect — the pointwise product of two local effects is proved unique, not postulated, and local tomography holds exactly when the local effect families linearly span their ambient functional spaces, equivalently when they separate their local states and contain the unit effect in their linear span. All results are finite, exact, and reproduced by committed scripts using rational arithmetic. Interpretive reading (structural, not a result): the finite hypotheses examined here force, successively and for free, a classical, simplicial operational skeleton — convex states once mixing is supplied, a Boolean effect algebra once coarse-graining is supplied, and exact local tomography once spanning local families are supplied — under precisely identified imports, and produce neither a quantum state geometry nor the Born rule. Locating exactly how much operational structure a finite admissibility contract yields for free, and where every further richness must instead be imported, is the intended service of this result to the non-injective foundations programme: it marks the boundary this classical skeleton cannot cross on its own, and hands that boundary, precisely typed, to whatever module or phase law is proposed next.
A recurring assumption in emergent-physics programmes is that once a fundamental description is given, the decomposition of the world into independently preparable subsystems comes for free. We show that on a substrate not supplied with a subsystem factorisation this assumption splits into two separate, checkable layers. In the finite case we characterise the possibilistic layer completely, and we resolve the preparation layer for two reference classes: the full simplex of preparations, and the strictly positive cylindrical-conditioning families; the general preparation layer remains open. The arena is a finite configuration set Ω carrying a reference admissible subset, two effective projections ΠA:Ω→OA and ΠB:Ω→OB that are not assumed injective, and cylindrical admissibility conditions combined by intersection; no factorisation Ω≃ΩA×ΩB is assumed at any point. At the possibilistic layer, we prove that the pairs of local outcome sets on which joint interrogation is margin-stable and jointly rectangular — the independence blocks — are exactly the nonempty rectangles (bicliques) contained in the projected support relation. At the preparation layer, we prove a benchmark and a criterion. The benchmark: on any block, if all probability measures on the underlying support are admissible preparations, every joint outcome law, in particular every product law, lifts to the substrate; the absence of a supplied substrate factorisation alone therefore obstructs nothing. The criterion: for the family of normalised cylindrical conditionings of a strictly positive reference weight matrix W, closure under products of independently locally conditioned states holds if and only if W has rank one. A minimal countermodel exhibits a complete rectangular support whose induced preparations are nevertheless correlated: support rectangularity does not imply preparation independence. A type discipline runs through both layers: states of one side obtained by conditioning on the other are steered conditional states, not local preparations, and admitting them manufactures spurious witnesses. The individual mathematical ingredients are elementary or standard; the contribution is their typed assembly into a proposed finite prerequisite contract of support and preparation criteria for emergent effective subsystems over an admissibility structure without a supplied subsystem factorisation. The hypotheses abstract the composition problem motivated by the non-injective foundations programme, but their realisation from the programme's axioms is not derived here, and the physical necessity of the contract itself is not derived from an operational theory. Interpretive reading (structural, not a result): the absence of a supplied subsystem factorisation does not forbid independent effective subsystems; it relocates their independence from an inherited property of the substrate to an emergent property of the projected support and of the preparation weights.
The Admissibility of Natural Physics, Part I begins where an admissibility-first reconstruction of single-system quantum mechanics ends. Its central question is not whether the familiar tensor-product formalism can be assumed, but whether any physical composition law can be derived without importing composite systems, global state spaces, entanglement, locality, or signalling as primitives. The book develops a disciplined reconstruction of binary quantum composition from operational and structural requirements. It first records the exact content and limitations of the inherited single-system closure, establishes dependency and non-circularity statutes, and introduces the possible object of natural physics only as a provisional target ledger rather than as an assumed ontology. It then examines joint preparation, product states and effects, marginal consistency, operational no-signalling, minimal and maximal probabilistic composites, local tomography, and the classification of admissible tensor-product representations. Particular attention is given to the distinction between a mathematical representation theorem and the uniqueness of a physical composition rule. Real, complex, quaternionic, Jordan-algebraic, minimal, maximal, restricted, and non-tomographic alternatives are compared under explicitly declared admissibility conditions. The complex Hilbert-space tensor product is recovered only on branches satisfying the required factorization, coherence, scalar, and operational closure conditions. Where several inequivalent composites survive, the result is a classified family rather than a falsely asserted uniqueness theorem. Entanglement is introduced only after composition has been commissioned. It is defined as the failure of an admitted joint state to belong to the convex closure of admitted product preparations. The analysis proves, within the resulting predictive grammar, that entanglement is a property of composite-state geometry, not an additional force, substance, interaction mechanism, or signalling channel. Throughout, the argument enforces quotient primacy, refinement stability, finite diagnostic closure, regeneration stability, explicit dependency receipts, local termination, and the possibility of Undefined as a legitimate scientific outcome. Part I therefore supplies neither a speculative ontology nor a presupposed many-world formalism. It provides a conditional classification theorem for binary physical composition and identifies precisely what remains to be established before interaction, repeated composition, particle identity, locality, fields, symmetries, and particle content may be addressed.
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
In a finite-dimensional causal convex operational theory with closed physical effect sets and essentially unique purification, we derive pure products, general purity preservation and effect non-restriction. Exact binary steering follows from an informative noisy pointer; support order intervals, pure dilations and probabilistic returns then force unrestricted effects without local tomography, ideal compression, product domination or a bound on purifying partners. Finite conditioned control is also derived. Spectrality, pure support projections and symmetric transitions yield a Jordan representation. For an essentially small catalogue, we then classify compositions without local tomography. Exact pure-ancilla faces restrict simple factors to a common real or complex matrix field. Sector translations embed in a locally finite abelian group. Complete event representations and simultaneous phase normalizations identify the full sequential and symmetric monoidal structure. At fixed field, equivalence classes are determined by the effective charge group, the actual product-closed object catalogue and a parity character. The proof retains all rectangular event spaces and does not add charge objects or compressions to an incomplete catalogue. Local tomography selects the ordinary complex quantum case on the available multiplicative catalogue. Exact one-way teleportation supplies an operational comparison. In the specified protocol class, one state is shared on two copies of the input system, without an additional correlated record. Exact preservation of all reference correlations forces a pure balanced resource and reversible corrections on nonzero outcomes. A system with three rank-one sectors and Z3 fusion has exact one-copy outcome minima twelve over the reals and nine over the complexes, with corresponding message entropies, although both alphabets fit in four binary digits. Joint protocols have the same regularized rate. Alternative geometric proofs, restricted-model obstructions, optimal physical returns, a quantum-pointer application and an open-effect counter-theory delimit the assumptions and conclusions. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.
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
This paper develops a reconstruction of probability that does not begin by assuming a probability space, a probability simplex, or the Born rule. Instead, it asks a prior foundational question: under what physical and mathematical conditions are a set of alternatives entitled to carry positive, additive probabilities at all? The construction begins with lawful histories, future interfaces, and retained records. It distinguishes coherent alternatives from exclusive empirical events, preventing multiple interfering routes from being mistaken for separate probabilistic outcomes. A future-sufficient quotient identifies distinctions that no admitted future test can detect, while a record-qualification step determines when alternatives have become genuine empirical events. Dagger-compatible comparison and positive record factorization then produce a positive comparison kernel, from which normalized event probabilities arise only after the relevant record sectors are certified as exclusive. The framework is extended beyond individual measurements. Compatible finite record refinements generate a global probability measure under explicit realizability conditions; stochastic kernels provide a compositional calculus for carrying probabilistic information through unresolved future structure; and locally classical measurement contexts are assembled into a single positive state on a generally non-Boolean effect structure. This allows globally coherent probability to exist even when no context-independent deterministic assignment of outcomes exists. On a reflection-positive Hilbert-space branch, admissible future contexts are represented by positive effects and positive normalized states by density operators. The Born state-effect law then appears as the finite-dimensional representation of a probability structure that has already been earned through record qualification, positivity, exclusivity, refinement, and contextual consistency rather than introduced as a primitive postulate. The paper also separates two mechanisms that are often conflated: quadratic Born-type weighting arises from dagger self-comparison, whereas equal probabilities for symmetric alternatives arise from symmetry of the outcome structure. A further state-effect bridge analysis classifies residual coordinate freedom and identifies an exceptional rank-two adjugate transformation with Lorentzian signature, which is removed when positive self-dual identification is required. The central interpretation is that probability can function as a lawful form of global completion: it can carry future-relevant law through unresolved structure even when deterministic uniqueness is unavailable. The paper does not derive a mechanism selecting one actual outcome and does not claim that every physical sector satisfies the stated hypotheses. Instead, it provides an explicit dependency-controlled architecture connecting lawful completion, reflection positivity, event qualification, contextuality, stochastic continuation, Hilbert reconstruction, and the Born rule.