The Resource Geometry of Answerable Inference

Abstract The Resource Geometry of Answerable Inference 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 target, realisation class and declared resource order. 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. This work is Paper 8 in the TUS OS quantum-operational research programme. Supporting mathematical papers Papers 1–7 develop the programme from canonical geometry through prepare–measure structure, global representation, process closure, system-level recognition, contract-relative resource theory and structural resource-transition classification. Paper 8 extends that trajectory to the mathematical structure and resource geometry of answerable inference. 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 contract-relative resource phenomena: exact strict finite separation, incrementally unbounded finite separation across a family, and finite-resource breakdown. These are instantiated respectively by the TUS OS 4→64 separation, periodically modulated decay processes with finite memberwise realisations and growing prediction-to-transparency cost, 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 8 moves from representation-resource transitions to the resource geometry of answerable inference itself. It formalises how the concerns carried by a question become mathematically tractable through a frozen answerability problem comprising the lawful Binding, admitted evidence, inference and evaluation semantics, answerability target, realisation class and declared resource order. It develops lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions, minimal sufficient cores and certified lower-resource geometry, while keeping operational feasibility and upper-side construction independently constituted. Under explicit completeness and exact-attainment conditions, it proves exact closure of the Pareto-minimal inference-resource boundary; on its generated exact branch, the answerability-side resource object equals the dominance closure of lawful operation