The Resource Geometry of Answerable Inference
Year: 2026
Type: preprint
Abstract: Paper 8 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a mathematical theory of how the independently constituted structure of an answerability problem determines the inference resources required for its lawful discharge. Building on the contract-relative resource framework of Papers 6 and 7, it moves from representation-resource transitions to inferential resource while keeping representation resource and inference resource formally distinct. The theory begins from a frozen typed problem comprising the question, lawful Binding, admitted evidence, inference and evaluation semantics, answerability contract and target, realisation class, declared resource order and tolerance. Lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions and irreducible sufficient cores are constituted independently of candidate implementations and resource optima. Resource accounting separately charges representation, preprocessing and online inference so that apparent savings cannot be manufactured by offloading work into hidden support. The lower-resource and operational branches are then developed independently. Certified finite transport determines which resource budgets can be proved insufficient, while the feasible set F records resource vectors actually realised by lawful target-achieving systems. Under finite transport-completeness, independent upper-side attainment and explicit order conditions, the certified and operational Pareto-minimal boundaries coincide. On the stronger generated exact branch, L = ↑F, so the independently constituted answerability-side resource object is exactly the dominance closure of lawful operational feasibility. Under well-partial-order conditions, this lower-resource geometry has a unique finite canonical antichain normal form representing the Pareto-minimal resource requirements of lawful target achievement. A logically separate Gate E theorem provides the local semantic-to-resource bridge. On its contrastively sharp branch, the point at which an answerability threshold becomes irreducible coincides with a sharp local resource transition, without identifying semantic cores with the global Pareto frontier. The paper also develops resource transport, robustness and theorem-directed computational falsification, including explicit countermodels separating boundary equality from region equality, Pareto efficiency from positive-linear supportedness, and semantic core structure from resource geometry. Together, the results establish a precise sense in which, at the scope proved here, a constituted question determines the resources required for answerability: independently constituted answerability-side mathematics recovers the canonical lower-resource geometry of lawful operational feasibility without defining that geometry from an implementation optimum. Supporting General Theory Programme This paper is Paper 8 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 6–7, it moves from system-specific quantum realisability to Answerability as a generic mathematical object, developing the resource geometry of answerable inference by fixing the question, evidence, entitlement, evaluation language, admissible inference structure, and resource obligations before optimisation. Paper 9 then constitutes the Answerability Calculus; Paper 10 returns to TUS OS to prove its protected and executable realisation 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 unbounded finite separation retained as a separate family-scaling modifier. 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, repair and typed composition, while separating structural from resource activation. It then closes the theory with a
Author: Whitlock, Mark
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204261200
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