The Answerability Calculus

Abstract

Abstract The Answerability Calculus develops a general mathematical theory of Answerability as a constituted, typed relation and establishes laws governing its lawful transformation, resource representation, repair, composition, activation, obstruction and conservation. Its governing result is the Base Theorem of Answerability: Answerability is a constituted, typed relation. No terminal proposition counts as an answer unless it lawfully discharges the independently constituted contract. A typed Answerability problem therefore fixes its semantic frame and contract, admitted realisation category, resource structure and tolerance regime before candidate success or optimisation is inspected. This preserves distinct levels of semantic satisfiability, admitted-category realisability and budgeted feasibility, preventing terminal truth, implementation existence and resource affordability from being silently identified. The paper develops a lawful reduction calculus with separately typed semantic transformations, operational lifts and resource lifts; standalone and contextual equivalence; and a contract–resource representation theory based on exact attained resources and principal-budget feasibility. Under explicit generation hypotheses, the complete lower-budget geometry admits extremal reconstruction from irreducible frontiers. Lawful repair is set-valued in general, with a canonical greatest repair only on a stronger order-theoretic branch. Typed composition is then introduced through independent semantic, operational and resource data. Structural activation is defined by genuinely new mandatory composite obligations and is kept distinct from resource activation. Independent obstruction certificates provide the bridge from new obligations to resource consequences, yielding exact feasible-resource activation only under additional stated comparison conditions. The resulting theory allows structural activation without resource penalty, resource advantage without structural activation, and incomparable multi-resource composition regimes. The impossibility theory closes the same object from the opposite direction. Capability and obstruction form a complementary principal-budget geometry: lawful reductions transport feasibility forward and obstruction contravariantly backward. Composite impossibility separates inherited constituent obstructions from activation-induced obstructions, while the cycle theory distinguishes frontier preservation, pointwise fixation, profile conservation and full mixed-geometry invariance. Forty explicit countermodels and conservativity results protect these theorem boundaries and prevent weaker equivalences, construction failures or resource bounds from being promoted into stronger claims. The accompanying historical comparison is logically separate from the mathematics. Under a reproducible whole-object comparison protocol, it identified no earlier public source or coherent corpus among sources publicly available before 4 March 2026 that independently constitutes a structure-preserving equivalent of the complete Answerability organism, nor an earlier identified complete formal calculus of that organism at comparable material scope. This is a bounded and falsifiable historical finding, not a priority claim over the constituent mathematics. At the scope proved here, Paper 9 therefore constitutes and closes Answerability as a general mathematical object whose lawful discharge, realisation, resource geometry, transformation, failure and conservation structure are explicit and independently testable. This work is Paper 9 in the TUS OS quantum-operational research programme. This paper is the ninth paper in the TUS OS quantum-operational research programme. Papers 1–8 develop the programme from canonical geometry and exact prepare–measure behaviour through global relational and predictive structure, complete-future and process closure, system-level Anchored Analog Quantum System recognition, contract-relative resource theory, structural classification of contract-induced resource transitions, and the resource geometry of answerable inference. The present paper generalises and closes that trajectory as the Answerability Calculus: a typed mathematical theory of lawful reduction, contract–resource representation, repair, composition, activation, impossibility, obstruction and conservation, governed by the principle that no terminal proposition counts as an answer unless it lawfully discharges the independently constituted contract. 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 cons