We formulate a contract-relative operational test for whether future-predictive quantum information lies in the post-processing capability of any admissible present description. For a declared present-access family and typed recovery policy, the present-completion deficiency is the diamond-norm distance between the predictive target and a contract-defined present-capability set. The underlying quantum channel deficiency and sufficiency mathematics is established prior art; the contribution is its use inside a path-sensitive black-hole reconstruction framework that separates coarse endpoint from rich present state, claim descent from witness descent, information insufficiency from failure of joint realizability, and local recovery from path monodromy. We prove access- and target-monotonicity under explicit typed closure conditions, give a pairwise distinguishability-gap lower bound, state a finite-dimensional attainment criterion for finitely many compact channel-type components, and derive the single/joint/virtual access hierarchy. The exact [[5,1,3]] perfect-code benchmark covers all ten three-retained and all ten two-retained access channels: every three-retained channel is exactly reversible, every two-retained channel is completely depolarizing with optimal diamond-norm deficiency 3/2 to the logical identity, and a Bell-projector discrimination certificate attains that lower bound. Two distinct three-retained erasure histories have the same coarse endpoint but different rich-present channels while both remain exactly recoverable, providing a direct null calibration of the endpoint split. The Bousso-Kaya-Lin-Shahbazi-Moghaddam finite-region family is retained as a coarse-data insufficiency benchmark, not yet as a no-present-completion theorem. The next black-hole-specific target is a certified positive fiberwise completion lower bound surviving operational quotienting, joint-realizability controls, and declared contract variations.
We formulate a contract-relative operational test for whether future-predictive quantum information lies in the post-processing capability of any admissible present description. For a declared present-access family and typed recovery policy, the present-completion deficiency is the diamond-norm distance between the predictive target and a contract-defined present-capability set. The underlying quantum channel deficiency and sufficiency mathematics is established prior art; the contribution is its use inside a path-sensitive black-hole reconstruction framework that separates coarse endpoint from rich present state, claim descent from witness descent, information insufficiency from failure of joint realizability, and local recovery from path monodromy. We prove access- and target-monotonicity under explicit typed closure conditions, give a pairwise distinguishability-gap lower bound, state a finite-dimensional attainment criterion for finitely many compact channel-type components, and derive the single/joint/virtual access hierarchy. The exact [[5,1,3]] perfect-code benchmark covers all ten three-retained and all ten two-retained access channels: every three-retained channel is exactly reversible, every two-retained channel is completely depolarizing with optimal diamond-norm deficiency 3/2 to the logical identity, and a Bell-projector discrimination certificate attains that lower bound. Two distinct three-retained erasure histories have the same coarse endpoint but different rich-present channels while both remain exactly recoverable, providing a direct null calibration of the endpoint split. The Bousso-Kaya-Lin-Shahbazi-Moghaddam finite-region family is retained as a coarse-data insufficiency benchmark, not yet as a no-present-completion theorem. The next black-hole-specific target is a certified positive fiberwise completion lower bound surviving operational quotienting, joint-realizability controls, and declared contract variations.
Paper 6 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a general theory of how the exact resource cost of representing an independently fixed native system depends on the operational access contract imposed on that system. Rather than assigning a single intrinsic “dimension” to a process or operational object, the paper treats realisation as a fully typed problem determined jointly by the native system, the access contract, the admitted realisation category and the tolerated error. The framework defines contract-indexed achievable resource regions, declared-memory closure, lawful contract restriction and resource monotonicity. Strengthening a contract can require a realisation to preserve or recover distinctions that a weaker behavioural contract is permitted to compress, while the independently fixed native system itself remains unchanged. Lawful contract restriction induces reverse inclusion of achievable regions, but strict separation requires additional lower and upper certificates. For stochastic processes, the paper introduces causal-state transparency as a recovery obligation imposed on an exact autonomous predictor at a declared inspection boundary. All persistent quantum, classical, controller, history and source-side information available at that boundary is included in the resource account. A direct-sum classical–quantum identity-capacity theorem proves that uniform worst-case recovery probability τ across N intrinsic causal classes requires identity capacity at least ⌈τN⌉. The theorem is then applied to two external stochastic-process families. Periodically modulated decay processes exhibit an unbounded finite separation: exact autonomous prediction has minimum identity capacity 2, exact causal-state transparency has minimum capacity N, and every fixed positive transparency threshold forces a lower bound growing as ⌈τN⌉. For the dual-Poisson process, an exact qubit predictor exists under the weak prediction contract, while every positive uniform worst-case transparency threshold makes the finite-capacity achievable region empty. This is a contract-induced loss of finite feasibility on the same native process, not a claim that the process itself ceases to exist or that no larger infinite-capacity realisation could exist. The paper then returns to the TUS OS discovery instance from which the general problem arose. The same independently fixed 64-state canonical target admits a complete static quantum-operational Atlas of exact minimum Hilbert dimension 4, while recovery-complete LANDING selected-line instrument access, GUIDANCE selected-line readout and GUIDANCE selected-line-and-endpoint joint-instrument access each have exact minimum Hilbert dimension 64 within the same declared representation category. The native target does not change; the operational access obligation does. Together, the TUS OS, periodically modulated decay and dual-Poisson results establish three distinct exact contract-relative regimes: strict finite separation: 4 → 64; unbounded finite separation: 2 → N; finite-capacity breakdown: 2 → ∅finite for every τ > 0. These are three instances of one general structural law, not three identical recovery mechanisms. TUS OS uses admitted experiment families that collectively separate its canonical carriers, whereas the stochastic-process results use direct causal-state decoding at a declared inspection boundary. The results establish that finite-capacity behavioural sufficiency does not imply positive uniform worst-case recoverability of native predictive identity, and that exact representation cost is not a context-free property of the native system alone. It depends on the complete typed obligation imposed on an admitted realisation. The paper does not claim physical quantum execution, quantum advantage, a universal scalar resource, equivalence of TUS OS recovery with causal-state transparency, or finite-capacity breakdown for prior-weighted dual-Poisson transparency. Supporting General Theory Programme This paper is Paper 6 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–5, it develops the general theory of contract-relative quantum realisations of classical processes, showing that quantum resource requirements are not intrinsic to a process alone but depend on the governing observational, recovery, and transparency contract. Paper 7 extends this into operational observability and the classification of contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and 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 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 Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold dual-Poisson transparency. 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
We construct a supported local spacetime adaptation of the observation-boundary contract of VII-E and the finite hybrid-channel contract of VII-G. The two contracts are proved independently within one laboratory on I-B’s prescribed conical Lorentzian product. For the observation boundary, we prove the effect-pullback criterion, its reference-stable form, the exact sharp-measurement coherence criterion, and a qubit separation of trace distance 1/2 between ideal residual targets with identical records. For hybrid evolution, we prove a complete branch representation, history-sensitive observation closure, spectator consistency, and the exact recovery–constant-complement criterion. An informative classical record consequently excludes recovery of every unknown quantum input, even when the declared residual is also available. The additional contribution is an end-to-end finite observation package that keeps these distinctions visible in supplied spacetime. We provide a compactly supported 3+1-dimensional lift of a four-face transverse adjacency exchange, an exact guarded reversible encoding, and a separately constructed qubit instrument selecting strictly ordered slices without changing the conical metric. A two-step specialization has two histories with the same terminal slice but different conditional residuals. After coarse aggregation, the record remains unchanged while two complete residual targets have the exact half-diamond gap 4(1 − 1/√2)/25. An explicit residual audit enlarges the required input observation space from the diagonal algebra to the full qubit algebra. A destructive finite clock, complete-channel approximation bounds, and a causal route for both payload and classical control complete the local Grade S construction. The package may serve as a precise testbed for distinguishing record-level claims from instrument-level claims in geometric feedback models. The channel and recovery principles themselves are established finite-dimensional facts; the claimed contribution is their supported assembly and the explicit certificates and separations. No metric emergence, gravitational dynamics, field instrument, radiative coefficient, or quantum Ward identity is constructed.
We distinguish complete unitary preservation, subsystem quantum recovery, destructive observation, and operational discard in a reference-preserving channel formulation. A finite acyclic typed fragment represents allocation, specified unitary transformations, destructive computational readout, and classical forgetting. Quantum recovery returns a quantum register; terminal readout returns a classical record and is a different task. For a complete isometry V from C to R tensor B, we give an explicit global recovery channel and prove that exact recovery from R is equivalent to constancy of the complete complementary channel to B. A process retaining only classical outcomes cannot reconstruct an arbitrary d-dimensional quantum code with half-diamond error below 1 − 1/d. Unobserved readout of B, followed by discarding its outcomes, is exactly partial trace over B and leaves the radiation channel unchanged. Its coherent dilation retains the information omitted from the classical interface. Independently, a finite-band quartic source–scalar-reservoir Hamiltonian is constructed and approximated by finite-dimensional proxies with explicit complete-channel allowances. Separate qubit dictionaries preserve the accessible–inaccessible partition. For a commuting four-Majorana control, the exact radiation recovery error is (3 − √(1 − c_R(t)²))/4, the residual recovery error is (1 − c_R(t))/2, and the complete residual leakage is the maximum of 1/2 and (1 + 2c_R(t))/4. Thus, the complete output is reversible although neither output alone need recover the diary. Access-preserving endpoint continuity and fixed-proxy recurrence are proved separately from syntactic termination. The gravitational implication remains conditional on a same-model unitary dilation, outgoing access construction, and constant endpoint residual. No positive noncommuting recovery window, physical field-to-memory receiver, quantum-gravity unitarity theorem, or gravitational evaporation endpoint is established.
Here “operational objectivity” is used only in a certificate-relative sense: satisfaction of a selected public-audit standard, not quantum Darwinism, spectrum-broadcast objectivity, or complete intersubjectivity. A declared family of complete queries determines an access-relative public class. We first state an elementary fiber-stability proposition: when a certificate’s target dependence factors through the public evaluation map and its auxiliary records do not encode which member of a public fiber was realized, a pass cannot identify the underlying physical state if that map is noninjective. We then propose a typed audit certificate C = (W,I,Cal,G,D,R,U) with six requirements: declared access, calibrated interventions, relative gauge alignment, a dependence-valid evidence pedigree, reproducibility with independently generated error oppor- tunities, and minimal epochal public revision. Construction J1 tests both the joint consistency and the inferential limit of this architecture inside standard real-amplitude quantum theory with ordinary real tensor composition. Ideal real-trine effects are nominal targets, while symmetrically relabeled noisy trines are the implemented effects and span each full local rebit effect space. Nevertheless I4 ±(3/5)J⊗J ρ±= 4 give probability 1/9 for every outcome of every implemented epoch-0 product context. A separately declared and calibrated global projective measurement outside that interface distinguishes them with total-variation distance 3/5 and equal-prior success 4/5. The witness depends on an independently trusted, hidden-direction-sensitive calibration source χH = (I4 + J⊗J)/4; calibration-source access is not target-query access, and no preparation–measurement duality or joint-SPAM self-calibration is assumed. Supporting results record exact transcript invariance for finite adaptive protocols on fresh preparations, the bound 1−(1−ε)n for an ε single-round discrepancy, and a distribution-free finite-sample upper certificate. Thus the selected O1–O6 pass can coexist with noninjectivity of the full separated-product map in J1. The claimed contribution is the fiber-stability diagnostic, typed synthesis, and joint witness, not a new operational quotient, local-tomography theorem, rebit defect, general objectivity criterion, or selection of complex quantum theory.
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 preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.