Physical theories treat a state as the antecedent of responses. This paper asks a scope question: which distinctions must that representation retain? In a restricted continuation–response model, two histories are future-equivalent when every admitted continuation gives them the same response profile. Their quotient is the minimal response-sufficient carrier. A projection is insufficient whenever its kernel contains a pair separated by an admitted continuation. To connect this established minimisation schema to physical evidence, the paper defines non-identical restitution (NIR): proper transport of an effective contrast with a predictively distinct endpoint, independently controlled provenance, and carried form recoverable through a predeclared cross-stage decoder. An informative-acquisition lemma shows that when a downstream channel is initially neutral and becomes contrastive through such a calibrated transport, the source histories were already distinct in the complete continuation carrier. The terminal record reveals rather than generates that source-level predictive distinction. Behavioural minimisation, sufficient statistics, predictive representations, operational theories, and traceable transmission all have established precedents. No new minimisation or transmission theorem is claimed. The contribution is their joint use in a continuation-relative scope audit that separates predictive identity from independently certified genealogy. A reduced quantum state supplies an active-kernel example; a controlled transducer instantiates NIR; trace normalisation is secondary. The kernel condition is necessary and sufficient for response-sufficiency; the audit does not decide metaphysical fundamentality. Histories, continuation action, response map, and directed composability are inputs; no temporal order, quantum theory, or spacetime theory is derived.
Physical theories treat a state as the antecedent of responses. This paper asks a scope question: which distinctions must that representation retain? In a restricted continuation–response model, two histories are future-equivalent when every admitted continuation gives them the same response profile. Their quotient is the minimal response-sufficient carrier. A projection is insufficient whenever its kernel contains a pair separated by an admitted continuation. To connect this established minimisation schema to physical evidence, the paper defines non-identical restitution (NIR): proper transport of an effective contrast with a predictively distinct endpoint, independently controlled provenance, and carried form recoverable through a predeclared cross-stage decoder. An informative-acquisition lemma shows that when a downstream channel is initially neutral and becomes contrastive through such a calibrated transport, the source histories were already distinct in the complete continuation carrier. The terminal record reveals rather than generates that source-level predictive distinction. Behavioural minimisation, sufficient statistics, predictive representations, operational theories, and traceable transmission all have established precedents. No new minimisation or transmission theorem is claimed. The contribution is their joint use in a continuation-relative scope audit that separates predictive identity from independently certified genealogy. A reduced quantum state supplies an active-kernel example; a controlled transducer instantiates NIR; trace normalisation is secondary. The kernel condition is necessary and sufficient for response-sufficiency; the audit does not decide metaphysical fundamentality. Histories, continuation action, response map, and directed composability are inputs; no temporal order, quantum theory, or spacetime theory is derived.
Physics may have mistaken a quotient for an origin. Quantum theory represents a preparation by a trace-one positive operator ρ. This is exact for relative outcome weights, but it is produced by the many-to-one map N(σ)=σ/Tr(σ), for which N(λσ)=N(σ) for every λ>0. If total response scale is inert under every admissible continuation, the identification loses nothing relevant. If one declared continuation can distinguish σ from λσ, however, normalisation has erased a future-relevant distinction. The trace-one state remains an exact terminal quantum reading, but it cannot by itself be a faithful future-sufficient carrier. The reason this order is easy to miss is structural: once the quotient is taken as the complete input, every downstream construction that factors through it is necessarily blind to the positive-ray scale already identified away. The mathematical ingredients used below are largely established and are not presented as isolated novelties. Jordan–von Neumann, Frobenius, and finite-dimensional Gleason–Busch-type results serve instead as consecutive links in a conditional dependency-order theorem. Equality of complete future-response profiles first fixes the minimal predictive quotient. Under explicit locks, positive refinement fixes square-root access; quadratic recomposition fixes a real inner-product carrier; and the stated return-algebra, positive-involution, and central exhaustive-phase locks then select its complex Hilbert representation. Positive additive effect weights acquire the unique pre-normalised trace form Wω(E)=Tr(σωE); and only after the non-zero total sω=Tr(σω) is closed does one obtain ρω=σω/sω and the Born reading. The familiar finite-dimensional Hilbert–trace–Born structure therefore reappears, but in a different dependency position. Two no-go results make this dependency claim exact within their stated domains: canonical ray normalisation cannot by itself faithfully carry active scale, and, in dimension at least two, a normalised invertible congruence is affine on the full state space only when its transport is scale-uniform. A common-profile theorem states when independently reconstructed readings are uniquely profile-equivalent over one fixed common future-profile image, rather than merely compatible descriptions. Under locks L0–L9 on one typed finite-dimensional assembly datum, these results form a single conditional genealogy; the paper neither derives quantum theory from NIR alone nor asserts that nature realises the locks. Its central result is not a new quantum rule. If two realised or admissibly generated representatives σ and λσ on the same positive ray, with σ≠0 and λ≠1, have different complete declared future-response profiles, no faithful representation of those profiles can factor solely through canonical ray normalisation N. The trace-one state is not wrong: it is the exact answer to a later, conditional question.
A Kernel-Role Determinacy Theorem for Record Fixation, Fixed-Domain Quotients, and Boundary-Trace Normal Form This paper specializes the AASC kernel to the construction and use of quantum measurement records. Its central question is not initially how outcomes are produced, whether a wavefunction collapses, or which interpretation of quantum theory is correct. It asks what must be fixed for a measurement record to function as the same determinate physical and evidential object across retention, prospective use, deliberation, comparison, and report. For any non-degenerate determinate physical object, state, carrier, transformation, record, comparison, or endpoint, the paper takes the governing kernel to be: NDObj(x) ⇒ K(x) = (Adm, Stand, Ref, Irr). Admissibility, standing, reference, and irreversibility are therefore treated as necessary role conditions of determinate record use, not as an optional interpretive overlay added to an otherwise complete physical description. Measurement-record determinacy The manuscript constructs a fixed-domain measurement-record target in which a record must preserve: reference to the same construction token; standing-bearing evidential eligibility; admissible reuse; original act-time qualification; lawful redescription; retained and prospective record use; deliberative and comparative use; reportability and downstream inheritance. The resulting assignment space is quotiented by equivalence of standing-bearing measurement-record content. A rival quotient claiming exactly the same content collapses to the canonical measurement-record quotient. Representational differences that preserve the complete role content are treated as skin; differences affecting record identity, evidential ordering, admissibility, continuation, or downstream use remain physically relevant. Physically non-selective quantum regimes The paper then studies a physically non-selective quantum regime containing unitary branching, entanglement, decoherence, branch-local records, and stable local reports. The central result is conditional and target-specific: The bare unitary-entangling-decohering package can support multiple stable branch-local records without supplying a unique standing evidential record for the fixed construction-relative target. This is not a claim that branch-local records are unreal, unstable, or physically useless. It is a distinction between: physical occurrence of branch-local record structures; local branch-relative reportability; evidential role fixation for a record reused as the same standing object across the measurement practice. The paper shows that non-selective structures may agree on branch weights, density content, decoherence, correlations, instrument sums, channel closure, effect weights, projector orthogonality, and repeatability without selecting one fine record as the standing evidential record. Fixed-domain quotient and fixation commutation The theorem ladder establishes: fixed-domain assignment-space forcing; exact quotient construction for standing-bearing record content; same-scope equivalence collapse; fixation commutation under admissible transformation; inheritance of the original qualification requirement; persistence-bearing reuse; asymmetry exhaustion; anti-weakening; selector exhaustion. If evidential ordering varies across admissible assignments while the target requires one fixed-domain ordering, the ordering is not defined on that scope merely because numerical values or branch-local reports happen to coincide. The paper therefore treats measurement-record fixation as prior to the theory-level evidential reuse that depends on it. Boundary-trace normal form After internal and provenance/indexical routes have been exhausted, the residual same-domain route is a singleton stable boundary trace. Such a trace is not collapse dynamics, hidden-variable selection, or a new physical mechanism. It is an admissibility-preserving role anchor that can fix a record-bearing assignment only when its channel, incidence, transport law, original authority, and singleton-positive condition are independently warranted. The result distinguishes: sound partial selection from complete role fixation; unique trace value from unique original occupant; class-valued record fixation from raw-token fixation; later discovery of an original qualification from retroactive repair; physical record occurrence from standing evidential identity. Finite quantum specialization The paper occupies the abstract structure with an explicit finite two-branch quantum measurement domain containing: two outcomes; two fine records and one merged record; denominator-two branch weights; a balanced integer-amplitude state; density-table realization; selective and non-selective instrument maps; environment-record maps; projective branch maps; complementary effects; zero-product projectors; repeatability operations. On the declared balanced witness, amplitude weights, density weights, branch-map traces, effect weights, pointer records, instrument records, environment records, and channel records agree on the same branch architecture. Selective surfaces preserve the fine-record distinction. Non-selective surfaces agree only on the merged record. The fine records therefore remain separated, while the declared non-selective surfaces supply no singleton fine-record selector. The finite selector-boundary theorem eliminates the disjunction of candidate hidden selectors across the state, instrument, environment, channel, effect, and repeatability surfaces. The resulting diagnosis is precise: the missing structure is admissible fine-record role fixation, not further branch production, weighting, decoherence, stabilization, or repeatability. Scope and non-claims The manuscript establishes a finite, conditional, manuscript-facing theorem. It does not claim: to derive Hilbert-space quantum mechanics; to derive general POVM theory; to derive full projection-valued-measure theory; to reconstruct the full operator algebra; to provide a general Kraus representation theorem; to reconstruct complete positivity; to derive the continuum Born rule; to derive universal physical decoherence theory; to introduce a new collapse dynamics; to prove that no physical collapse process occurs in nature; to prove physical branch selection; to establish interpretive truth; to classify every possible physical measurement mechanism. The absence of a hidden selector on the declared finite non-selective surfaces is therefore not a theorem that no physical collapse or selection process can occur elsewhere. It is the finite theorem that those declared surfaces do not themselves supply the required fine-record selector. Relation to quantum interpretations The result does not deny the practical success of quantum theory, decoherence, branch-relative reporting, instrumental updates, or decision-theoretic methods. A practice aimed only at prediction, branch-local reporting, or instrumental updating may be a different target from the construction-relative record use studied here. If a theory claims to underwrite retained, prospective, deliberative, and evidential reuse of the same record-bearing construction token, it must preserve the relevant kernel roles and admit a standing-bearing fixation regime. Bare Everettian dynamics, without additional dynamics, ontological privileging, or admissible restriction to one constructional class, supplies no fine-record selector on the declared target. A strategy that preserves non-selective physics while restricting admissible record-fixation assignments belongs to a restriction-to-one-constructional-class route. Instrumentalist and QBist strategies instead reduce or redirect the target explanatory burden. Role in the AASC programme This paper provides the record-fixation layer of the AASC Quantum Determinacy Pair. Its companion Bell-scope work addresses the compatibility and no-signaling side of quantum witness determinacy. Together, the pair treats quantum records and witnesses as kernel-governed standing structures rather than as automatically determinate physical occurrences. The paper’s principal contribution is to identify the first failure point in a physically non-selective measurement regime: The initial problem is not necessarily the production of branch-local outcomes. It is the admissible fixation of the record role through which outcomes are retained, reported, remembered, compared, and used as evidence. The public Lean 4 formalization is synchronized with the finite measurement-boundary release v0.9.1 and provides the finite artifact for the declared effect, projector, compatibility, closed-stack, boundary-certificate, and audit layers. Keywords AASC; measurement problem; quantum measurement; measurement-record fixation; physically non-selective regimes; Everettian non-selection; record-role determinacy; fixed-domain quotient; standing equivalence; boundary-trace fixation; selector exhaustion; decoherence; branch-local records; admissibility and irreversibility; Lean 4 formalization
When the conditions under which a record is formed remain fixed, a physical theory may represent that record completely by a unary law of the state observed. This paper asks what structure is forced when the same independently certified physical identity gives different complete records under different formation conditions. The initial datum distinguishes two typed empirical roles, while the state identity of the formation side is derived from its complete behavior. Failure of factorization through the first projection proves that the formation-side role cannot be deleted. The complete row and column behaviors of the primitive law then determine two kernel pairs and their coequalizers. Unique descent retains every observable distinction while removing every redundant label, and the resulting realization is terminal among all exact role-preserving realizations. The product pairs the reduced roles; a pullback extracts the combinations on which records can actually be formed. Only after these steps does exponential transposition identify each surviving formation class with one complete response map. The resulting bijection supplies the state criterion and establishes the observation-state object. Compatibility with the antecedent theory preserves the independently given physical state space and yields the physical–observation law in which a fact is generated jointly by the state observed and the state of observation. The same construction gives exact criteria for conditional, label-free, and joint identifiability and selects the same-source pullback on which transport may subsequently be defined. A nonzero coherent response on that pullback yields the phase-transport fundamental equation, the first fundamental equation written directly on observation space. Quantum theory subsequently enriches this canonical interface. Coherent response and full circular phase select the complex field; reversible binary mixing yields CP1 ∼= S2 ; independent composition creates the four-dimensional event domain; response-event identity glues probabilities across complete contexts; and Gleason representation with marginal consistency gives the Born trace pairing. The two-qubit correlation tensor then yields the singlet cosine law, the CHSH inequality, the Tsirelson bound, and the Horodecki spectral criterion. Bell violation is thereby identified with the absence of a positive global answer section across incompatible observation states. The quantum formalism is thereby closed into one architecture: complex state geometry describes coherent physical possibility, complete observation states specify the ways in which facts may form, the Born interface joins them, and quantum instruments carry a realized fact into subsequent dynamics. Fixed observation sections recover classical unary laws, while smooth equivariant enrichment lifts relativistic frame geometry and internal gauge geometry to the same joint domain. The central result is ontological. Physobser Theory admits observation into objective physical reality not as a mind, an external readout, or a perturbation, but as an irreducible state-bearing axis; a fact is generated at its lawful interface with physical state. Category theory is the proof instrument. The discovered physical content is the dual-axis constitution of facts, which supplies a common foundation for quantum, classical, relativistic, and gauge physics.
Paper 5 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper identifies and formalises a system-level phenomenon in exact quantum representation: an independently constituted deterministic native system can admit multiple exact quantum-operational views whose minimum representation resources can depend sharply on a precommitted access contract, while the native system itself remains unchanged. The quantum representation is therefore neither constitutive of the native system nor a post hoc encoding selected to reproduce a prescribed quantum target. It is an exact operational representation of behaviour exposed by a declared contract on an independently fixed structural target. We introduce two mathematical definitions. An Analog Quantum View (AQV) is the distinguished operational-equivalence class containing every exact permitted finite-dimensional quantum realisation of the behaviour exposed by a nonvacuous, precommitted access contract, under complete persistent-resource accounting. An Anchored Analog Quantum System (AAQS) is an independently constituted, provenance-bearing native system admitting a nonempty contract-indexed family of such views while preserving native identity, native quotient independence and protected provenance outside the quantum representation. The definition excludes circular construction, candidate-dependent post-selection, hidden side information and untyped resource comparisons; the additional properness criterion excludes recognition based solely on the universal orthogonal-sector embedding. TUS OS is proved to be a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus. Its independently defined native structural quotient is canonically isomorphic, as a complete deterministic transition-output system, to deterministic predictive equivalence and to the canonical six-bit carrier, with exact deterministic predictive minimum 64. On this same carrier-faithful structural target, within a fixed no-persistent-side-memory quantum representation category, the minimum Hilbert dimension depends sharply on the permitted operational access. The complete static Atlas has exact minimum d_Atlas^min = 4, whereas recovery-complete LANDING selected-line instrument access and recovery-complete GUIDANCE selected-line readout each have exact minimum d_L,instr^min = d_G,read^min = 64. These yield two exact factor-16 fixed-target, fixed-category resource separations. Recovery-complete state-only common-CPTP access remains bounded by 32 ≤ d_CPTP^min ≤ 64. A separate exact classical–quantum construction retains the certified four-dimensional quantum component together with a 32-state complement-orbit memory. Under the declared componentwise resource order, this hybrid and the 64-sector no-persistent-memory construction are Pareto-incomparable rather than competing estimates of a single resource minimum. The resulting TUS OS Anchored Analog Quantum System Theorem composes native recognition, predictive identity, contract-indexed quantum representation and conservative downstream interpretation without collapsing their domains. Predictive Synthesis and Analog Quantum Views are sibling constructions from independently fixed native identity: neither constructs the other, and neither reconstructs the Native Witness. Provenance and receipt custody remain outside the quantum target, while lawful Domain Binding and downstream human, rules-engine or AI interpretation, operating under independently fixed interfaces and gates, preserve protected native identity and source authority and cannot strengthen the governing evidential ceiling. No claim is made of physical quantum execution, quantum ontology, encoding of question text as a quantum state, quantum advantage, unrestricted domain coverage or model-independent correctness. Realised-instance inheritance remains conditional and target-specific rather than an unrestricted theorem over all TUS OS executions. The paper reports a retrospective prior-art audit through 11 August 2026 that identified no earlier or contemporary public disclosure whose documented properties were found to satisfy the complete AAQS conjunction. Subject to that audited public-record scope, TUS OS is the first formally demonstrated Anchored Analog Quantum System identified in the record examined. This is a system-class priority claim rather than a claim that the constituent ideas—quantum representations of classical processes, predictive equivalence, provenance, resource orderings or noninterference—are themselves novel. Supporting General Theory Programme This paper is Paper 5 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–4, it organises the established geometry, operational structure, complete-future theory, process closure, and native Triadic synthesis through the theory of Anchored Analog Quantum Systems, establishing TUS OS as a proper instance of that framework. Papers 6–7 extend the programme through contract-relative quantum realisation, resource separation, operational observability, and contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and calculus; Paper 10 proves the protected and executable realisation of TUS OS; 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 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 family of 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 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. 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 and the exact parity-resolved pairwise total-variation optimum; determines complement reachability; and proves that the rational invariant probability module generated by the complete terminal probability language 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, giving the certified same-parity support graph and the bound 32 ≤ d_CPTP^min ≤ 64, while leaving cross-parity forcing and the exact state-only common-CPTP minimum unresolved. It further establishes recovery-complete LANDING and GUIDANCE process/readout results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact four-dimensional-quantum-plus-32-state-classical hybrid construction, joint LANDING–GUIDANCE compatibility, 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 and proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus. It elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, organises the inherited exact quantum-operational representations through typed access contracts and resource accounting, and composes native recognition, predictive identity, contract-indexed quantum representation and conditional downstream claim-custody preservation into the TUS OS Anchored Analog Quantum System Theorem. The 64 inputs are an abstract canonical corpus, not 64 asserted receipt executions. 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 O
We construct a source-matched reference calibration for distinguishing the coherent law h = 0 from the specified added generator h = J/25 in a finite quantum-record model. A complete-carrier transfer bound connects an accessible reference contrast to a delayed target contrast while retaining the original drift and actual law. On reference coverage and measured admission, restricting both hypotheses by the same observed reference band yields a directed target-set separation of at least 0.0071230619 whenever both compatible classes are nonempty. Rival worlds may have different admitted nuisance parameters. The constants are uniform over independently known finite lengths m >= 65, with tag 65, access prefix 1:32 and target deadline 25.02/J. Finite reference preparation and work/source reset satisfy the declared response budgets. Reset acts on the actual joint input and consumed supply, with the retained reference R exactly inert. It does not assign a universal reduced channel to correlated supplies. Under almost-sure conditional resource and supply bounds, the comparison extends to history-dependent acquisition. A prespecified rule, including abstention below the calibration threshold, has each-law unconditional wrong-declaration probability at most 2 exp(-6) + exp(-85/16) < 0.01. This is neither a posterior probability nor an error bound conditional on admission. The statistical step uses standard bounded-martingale concentration. Global target states and specified auxiliary supplies remain resources. Reset requires additional globally conditioned controls and has very large finite costs. The result is a conditional theorem for the two declared laws; it does not establish practical implementation or a unique diagnosis of fundamental collapse. A supporting Lean 4 companion contains 85 theorem declarations and 30 supporting definitions across Calibration, Operations and Risk. It provides checked scalar, sufficient-bound, resource-arithmetic and finite-history risk consequences under explicit premises. The companion has an executed Linux preflight and accepted returned macOS compilation, named-type and transitive-axiom evidence. The quantum constructions and conditional moment, filtration, tower and tail derivations remain written proofs or external premises; this is not an end-to-end formalization of the physical theorem chain. The original checked companion targeted author v0.4 and is carried forward unchanged, with its scope and execution chronology preserved.
A formal framework is developed for causal-relational closure (CRC), the principle that a physical relation remains open while an admissible future can still complete, distinguish, or reroute it. At a causal cut, histories are compared only through the probabilities generated by physically admissible future protocols. Two histories are future-equivalent when every admissible continuation gives the same outcome statistics. The resulting future quotient is the coarsest prediction-preserving partition of past histories, and therefore the minimal operational object relative to the declared contract. Several structural results follow. Enlarging the admissible future can only refine the quotient; combining future contracts intersects their equivalence relations; distinctions that have become future-indistinguishable cannot reappear under a fixed complete contract; and exact quotient dynamics exists whenever forward evolution respects future equivalence. For quantum realizations, future equivalence becomes equality modulo the annihilator of the admissible effect space, yielding a contract-dependent distinguishability seminorm. A past-future response matrix provides a finite-dimensional linear representation whose rank defines an operational relational dimension without identifying quotient cardinality with Hilbert-space size. This structure supports quantitative definitions of closure capacity, active causal width, and epsilon-causal depth. Quantum erasure, delayed-choice entanglement swapping, reversible observer records, and complexity-limited black-hole decoding illustrate the same calculus. The mathematics overlaps with causal states, predictive state representations, quantum combs, process tensors, quantum Markov order, and statistical sufficiency; the distinctive claim is the physical inversion that the admissible future helps define the operative relational object rather than merely compress a pre-assumed instantaneous ontology.