This preprint introduces the concept of an observer-relative undifferentiated probability envelope(观察者相对的未分化概率包络) as a minimal operational description for situations in which a family of candidate underlying states cannot yet be distinguished by a specified observer, interface, or intervention set. Rather than treating “undifferentiated” as an absolute absence of structure, the framework defines it through an observer-induced equivalence relation on a candidate state space. When all candidate states belong to a single operational equivalence class, the observer has access only to a coarse probability-level description. The paper distinguishes two interpretations that must remain separate. Model A(模型A), adopted as the current operational description, treats the probability distribution itself as the minimal empirically justified state description. Model B(模型B) is retained as an open ontological hypothesis in which each individual realization may already possess a unique underlying actual state, while the observer has only incomplete knowledge of that state. Model B is neither accepted nor rejected; it is reserved for future tests that could produce predictions distinguishable from Model A. The analysis formalizes the first-distinction boundary using equivalence classes and partition refinement, shows that a minimal nontrivial refinement naturally produces two distinguishable classes without implying physical binary ontology, and emphasizes the distinction between operational indistinguishability and ontological identity. Quantum coherent superposition is deliberately not assumed as the deepest interpretation of the undifferentiated layer. Instead, quantum-state descriptions are treated provisionally as higher-level effective operational representations whose deeper interpretation remains open. This work freezes a temporary lower boundary in a broader research chain connecting observer-relative distinguishability, admissible state spaces, relational structure, dynamics, projection, coarse-graining, and finite observational readout. The next stage is to determine whether a deeper ontological completion can generate experimentally distinguishable consequences, or whether the probability-envelope description is operationally complete.
This preprint introduces the concept of an observer-relative undifferentiated probability envelope(观察者相对的未分化概率包络) as a minimal operational description for situations in which a family of candidate underlying states cannot yet be distinguished by a specified observer, interface, or intervention set. Rather than treating “undifferentiated” as an absolute absence of structure, the framework defines it through an observer-induced equivalence relation on a candidate state space. When all candidate states belong to a single operational equivalence class, the observer has access only to a coarse probability-level description. The paper distinguishes two interpretations that must remain separate. Model A(模型A), adopted as the current operational description, treats the probability distribution itself as the minimal empirically justified state description. Model B(模型B) is retained as an open ontological hypothesis in which each individual realization may already possess a unique underlying actual state, while the observer has only incomplete knowledge of that state. Model B is neither accepted nor rejected; it is reserved for future tests that could produce predictions distinguishable from Model A. The analysis formalizes the first-distinction boundary using equivalence classes and partition refinement, shows that a minimal nontrivial refinement naturally produces two distinguishable classes without implying physical binary ontology, and emphasizes the distinction between operational indistinguishability and ontological identity. Quantum coherent superposition is deliberately not assumed as the deepest interpretation of the undifferentiated layer. Instead, quantum-state descriptions are treated provisionally as higher-level effective operational representations whose deeper interpretation remains open. This work freezes a temporary lower boundary in a broader research chain connecting observer-relative distinguishability, admissible state spaces, relational structure, dynamics, projection, coarse-graining, and finite observational readout. The next stage is to determine whether a deeper ontological completion can generate experimentally distinguishable consequences, or whether the probability-envelope description is operationally complete.
This structural preprint freezes the current lower boundary reached in a continuing coarse-to-fine research program on observation, distinguishability, probability, admissible state spaces, dynamics, and finite readout. The paper introduces the provisional Chinese term Yuan (元) solely as a low-commitment placeholder for the presently unresolved side of the first expressible distinction. Yuan is not proposed as a state, object, field, observer, intrinsic capacity, or physical entity, and its existence is not asserted. The first layer that can currently be formalized is a relational distinction: a test, comparison, interaction, or reference yields at least two operationally different responses. From this layer the paper freezes a chain through primitive cut, event algebra, observer/test family, probability envelope, observer-resolution refinement, latent-state disclosure, relational/admissible state structure, dynamics/flow, projection/coarse-graining, and finite readout. The operational descriptive horizon is placed between Yuan and the first mathematically expressible relational distinction. Below that horizon, assigning a property to Yuan already reintroduces a distinction; above it, distinctions can be represented and tested. The horizon is therefore a limit of present descriptive resolution, not a claim that reality ends there. Quantum coherent superposition remains PARKED / UNMAPPED, and an observer-independent unique-actual-state hypothesis remains OPEN / UNPROVEN. The next phase is upward validation: refine observation or intervention and test whether a coarse equivalence class can be split into substructures with independently measurable responses. 中文描述 本文固化当前“由粗到细、再向更深层约化”研究所达到的暂时描述边界。论文暂时采用中文技术占位词 “元”,只用于标记“第一次可表达区分之前,目前仍无法被现有语言和数学无循环刻画的未知位置”。“元”不被预设为状态、对象、场、观察者、能力、物质、信息或任何已知物理实体,本文也不宣称“元”一定存在。当前第一个可以被数学化的层级是 关系性区分:某种测试、比较、相互作用或参照产生至少两个不同的操作响应。由此向上依次形成原始切分、事件代数、观察/测试族、概率包络、观察分辨率细化、隐态显化、关系/允许态结构、动力学/流、投影/粗粒化及有限读出。本文把 操作描述视界 放在“元”与“第一次可表达关系性区分”之间:视界以下,一旦试图给“元”赋予性质,就已经重新引入区分;视界以上,区分、事件、概率、状态和实验差异才开始能够被表示和检验。因此,该视界是当前描述分辨率的边界,不是存在的边界。量子相干叠加继续保持“暂缓 / 未映射”,观察者无关的唯一实际态假说继续保持“开放 / 未证实”。下一阶段转向向上验证:增加新的观察维度或干预,检验粗粒等价类能否裂分出具有独立可测响应的细结构。
Quantum experiments constrain physical descriptions with extraordinary precision, yet several disagreements about quantum reality remain unresolved. Their persistence has different causes that require different responses. This article distinguishes exact agreement of predictions, inadequate statistical resolution, uncertain background dynamics, restricted access to records, conditional exclusion by no-goresults, and incomplete predictive specification. It develops a comparative framework in which distinguishability belongs to specified formulations and experimental domains rather than to interpretation names. A critical review connects philosophical work on underdetermination, Pessoa’s account of relations between experiments and interpretations, and recent statistical approaches to quantum tests. Likelihood equality clarifies the limits of differential confirmation, while identification of individual claims explains how experiments can establish physical structure without determining a complete ontology. Two analytic examples make the distinction practically consequential: a collapse contrast with bounded background uncertainty, and an observer-record model in which inaccessible information limits discrimination. The examples show when additional data help, when calibration or coherent access is required, and when no improvement within the stipulated domain can decide the comparison. The resulting assessment rejects both unrestricted empirical equivalence and claims that particular experiments uniquely establish a favored interpretation. It specifies the additional predictions, controls, and outcomes needed to make those claims evidentially meaningful.
Abstract We present Kunato Core Ver.5.0.0, a general framework for ontological admission, typed representation transport, empirical identifiability, effective-structure validation, and provenance-preserving epistemic state transition. The framework addresses a recurrent inferential problem across mathematical physics, statistical inference, inverse problems, effective theories, information theory, complex systems, quantum theory, and scientific realism: mathematical definability, empirical adequacy, model identification, and physical realization are distinct achievements and must not be silently promoted into one another. Kunato formalizes this separation through the constraint Formal(M) AND Emp(M; O) DOES NOT IMPLY Ontic(M). No representation receives automatic ontological privilege. Finite observation provides a first source of non-identification. Distinct formal models can agree exactly on a finite empirical record; in particular, uncountably many models can be constructed that coincide on any prescribed finite set of sample points. This establishes non-uniqueness within the relevant formal model class without promoting mathematical alternatives into claims about distinct physical causal histories. Finite measurement precision is treated through fibers of explicit resolution maps rather than by identifying metric neighborhoods with equivalence classes. The framework further distinguishes finite-dimensional state spaces from finite-state ontologies: finite-dimensional complex Hilbert spaces can contain a continuum of pure-state rays, whereas finite operational distinguishability requires additional assumptions such as a uniform positive separation scale on a compact state space. For stochastic observations, Kunato separates distributional identifiability, asymptotic inference, finite-sample discrimination, and zero-error identification. Positive overlap between competing observational laws prevents zero-error discrimination from the corresponding finite record. Increasing statistical evidence can therefore strengthen discrimination without automatically supplying an ontological realization warrant. Identification within a declared candidate class likewise remains conditional on that class and does not establish that physical reality belongs to it. The mathematical core is a typed property-transport calculus. Given a representation map Phi: X -> Y and a property P: X -> Z, P factors through Phi precisely when P is constant on every fiber of Phi. Equivalently, the kernel equivalence relation induced by Phi is contained in the kernel equivalence relation induced by P. This criterion separates state identification from property identification: a non-injective representation may fail to recover individual source states while still determining every property invariant on its fibers. Representation types, identified by their induced fiber equivalence relations, are organized by refinement within the complete lattice of equivalence relations on X. This provides a common mathematical structure for coarse-graining, quotient representations, projections, information loss, recoverability, and comparison of effective descriptions. Refinement is a partial order, so representations may also be incomparable. Effective structures consequently require property-specific transport warrants. Compression of a unitary parent operator, for example, need not preserve unitarity or normality. Conversely, non-normality alone does not entail a non-real spectrum, irreversibility, non-Markovianity, or absence of characteristic timescales. Observable symmetry, effective geometry, statistical regularity, entropy, information structure, temporal scales, and other representation-dependent properties cannot be transported automatically to a parent ontology. Kunato operationalizes these restrictions through methodological admission protocols. The Metric Admission Protocol regulates inference from effective metric structure to fundamental geometry. The Statistical-Invariant Admission Protocol separates existence of a mean from typicality, concentration, and self-averaging. The Timescale Admission Protocol distinguishes observed decay and hazard structure from claims about fundamental temporal dynamics. The Informational-Semantic Admission Protocol constrains inference from probability, entropy, information geometry, and semantic structure to latent ontology. The Criticality Admission Protocol separates phenomenological scaling from stronger claims of critical structure by requiring appropriately scoped diagnostics such as finite-size behavior, exponent stability, data collapse, robustness analysis, and negative controls. These protocols are methodological admission criteria rather than asserted laws of nature. Ontological admission is governed by differentiated semantic, fit, identification, scope, realization, revocation, and provenance warrants. These warrant types prevent empirical fit, interpretation, candidate-class identification, physical realization, and evidential history from being collapsed into an undifferentiated notion of support. Epistemic judgments are represented as contextual and versioned states, allowing changes in evidence, candidate classes, semantics, scope, and warrants to be reconstructed as explicit state transitions rather than hidden revisions. A central distinction is that UNRESOLVED is not REJECTED. Failure to establish an adequate promotion warrant blocks the requested inference but does not establish falsity or nonexistence. Rejection requires an independent refutation condition. Revocation acts on claims and accepted epistemic states while retaining their provenance; it does not imply any operation on authors or other persons. Self-application closes the framework. Continuum models, discrete models, cellular automata, XOR-based constructions, effective theories, quantum representations, and Kunato itself receive no exemption from the admission rules. Failure to identify a continuum ontology does not identify a discrete ontology, empirical adequacy of a constructive model does not establish its fundamental realization, and formal validity of Kunato does not confer physical ontological status on Kunato. The framework therefore entails neither a privileged fundamental model nor ontological nihilism. Not identified does not imply false, and not admitted does not imply nonexistent. Kunato Core Ver.5.0.0 provides a unified formal and computational vocabulary for ontological admission, finite-data non-identification, finite-record discrimination, inverse problems, coarse-graining, property factorization, representation refinement, effective dynamics, statistical invariants, temporal inference, information-theoretic interpretation, criticality assessment, model-class identification, provenance, epistemic-state revision, and scientific realism. Its objective is not to determine a preferred ontology by methodological fiat, but to make the inferential boundaries between representation, observation, identification, and physical realization explicit and auditable. The framework is summarized by four constraints: no hidden epistemic transition; no hidden ontological promotion; no untyped representational transport; and no self-exemption. Its terminal methodological principle is: no representation receives automatic ontological privilege.
Abstract We investigate the minimal structural conditions under which representational accessibility and irreversibility-like asymmetry arise in physical description. Starting solely from the existence of distinguishable states and a class of admissible transformations, we analyze how accessibility of distinctions changes under non-injective representational updates without introducing prior notions of space, time, probability, measurement, or thermodynamic structure.Admissible non-injective transformations naturally induce equivalence relations on the state space, leading to a distinction between manifest difference, corresponding to representationally accessible distinctions, and latent difference, corresponding to distinctions that remain structurally present but inaccessible within a given representation. We show that accessibility and recoverability of distinctions are determined intrinsically by the admissible transformation structure itself rather than by externally imposed observational limitations or representational scales.A central result of the framework is that irreversibility-like asymmetry arises when latent distinctions become non-recoverable within the admissible transformation class. In this perspective, irreversibility-like behavior is interpreted not as a consequence of probabilistic uncertainty or thermodynamic entropy increase, but as a structural restriction on recoverability of representational accessibility.The present work does not introduce new mathematical constructions. Rather, it provides a conceptual reinterpretation of equivalence structures, representational accessibility, and recoverability within a minimal framework of physical description. This framework identifies structural preconditions for accessibility restriction and recoverability failure prior to the introduction of statistical, thermodynamic, or observer-dependent assumptions.
This paper is Part II of The Pre-Emergence Origin of Probability trilogy. Part I develops the structural requirements for finite recurrent stability before spacetime. Part II asks when persistent boundary-readable structure can support numerical probability. Part III examines what probability-status remains after boundary loss, before numerical weighting, and under what narrower conditions Born-form weighting may follow. The paper analyzes how probability-measure status may arise within a closed deterministic description rather than being assumed as a primitive numerical rule. A closed description cannot appeal to an external observer, hidden weighting convention, retained label, post-outcome selection, or independently supplied probability measure to explain its own probability assignments. Numerical probability must instead be supported by structures and relations admitted within the described system. The analysis begins with the requirements for persistent structured identity. Persistent identity is treated as recoverable cross-state relational distinction rather than exact sameness, repeated naming, scalar aggregation, or externally preserved memory. Within the tested structural registry, persistent refinable identity requires ordered dependence, a sufficient stabilization architecture, distributed relational structure, recurrence, bounded recoverability, and noncollapse of the distinctions being preserved. Bounded recurrent relational correction is classified as weak oscillatory form in a deliberately structural sense. The classification does not establish a physical wave, sinusoidal motion, exact periodicity, frequency, propagation, dynamics, mechanism operation, or physical carrier. It identifies a recurrent recoverability architecture that a later realization would have to preserve. The paper then develops the transition from persistent structure to boundary observability. Representation-sensitive bulk descriptions may depend on phase origin, refinement history, decomposition, internal labels, subpart identity, or other distinctions that do not survive admissible boundary readout. Under admissible-observable indistinguishability, boundary-readable observables factor through an emergence-boundary quotient that preserves only the distinctions carried by the admitted observable family and surviving the declared transport relation. The boundary quotient does not itself provide probability. It supplies a local architecture of boundary-readable alternatives and permits the admission of candidate nonnegative contribution scalars. Neither the alternative carrier nor a candidate scalar establishes scalar uniqueness, scalar closure, additivity, measure realization, normalization, or probability interpretation. Numerical probability-measure status becomes available only under a narrower cumulative condition package. The package requires an exact finite carrier, a finite positive event measure, a normalized scalar component on the same carrier, exact scalar-measure compatibility, one-scalar factorization of every admitted weight-bearing map, exclusion of independent weight-bearing invariants, pre-outcome assignment, absence of an external weighting source, and a declared scalar-only predictive regime. Additivity, commensurability, regularity, positive scale, finite nonzero total, and normalization remain separately controlled conditions within that route. Under those conditions, normalized scalar weights may receive conditional probability-measure status inside the declared local regime. This is not an unconditional derivation of probability from determinism. The result does not establish unrestricted probability, universal scalar representation, a global probability law, objective chance, stochastic dynamics, collapse, measurement theory, physical realization, or a final physical theory. Within the trilogy, Part II therefore supplies the middle structural layer: persistent recurrent structuretoboundary-readable alternativestocandidate scalar contenttoconditional numerical probability-measure status. Part III begins from a different question. It examines what remains when a boundary-readable description loses the ability to guarantee which admissible alternative obtains, before any numerical scalarization is supplied. Part II provides the scalar, measure, normalization, compatibility, and realization-burden architecture inherited by that later analysis, but it does not pre-establish the boundary-loss results of Part III. ### Revision note This revision is a substantial reconstruction and trilogy integration rather than a minor editorial update. The derivational architecture, definitions, condition packages, failure boundaries, dependency order, and claim-scope controls have been revised throughout. The manuscript has also been rewritten in new prose to present the argument as Part II of The Pre-Emergence Origin of Probability trilogy. Major changes include a clearer separation of persistent identity, ordered dependence, stabilization, distributed relational recurrence, weak oscillatory form, boundary observability, candidate scalar status, scalar closure, measure realization, normalization, and conditional probability-measure status. The emergence-boundary quotient and transport-survival architecture have been clarified, and the numerical-probability result has been narrowed to an expressly declared scalar-only predictive regime satisfying the complete carrier, measure, compatibility, factorization, pre-outcome, and independent-invariant exclusion conditions. The revision also strengthens the distinction between candidate weights, normalized weights, conditional measures, and probability interpretation. It adds more precise failure localization, preserves the distinction between conditioned failure and universal impossibility, and expands the realization-burden architecture separating mechanism satisfaction, physical scalar selection, boundary enforcement, branch generation, refinement implementation, physical realization, theory-specific realization, empirical validation, and final physical-theory status. The paper is now explicitly positioned between Part I, which establishes the structural requirements for finite recurrent stability before spacetime, and Part III, which develops boundary-unresolved and pre-numerical probability-status after recoverability loss. The central research program remains mechanism-neutral. No physical mechanism, collapse process, specific realization theory, unrestricted probability law, or empirical validation is claimed.
This paper argues that the quantum measurement problem is not a single explanatory gap but a collection of distinct tasks. We separate detector representability, probability structure, single-event actuality, and outcome-conditioned refinement, and distinguish all four from causal registration, the physical criterion by which an already-instantiated event becomes a completed measurement record. These are explanatory roles within one measurement phenomenon rather than successive temporal stages. The analysis begins from the contrast between a global quantum description containing several detector-correlated components and the single entry that appears in a realized measurement history. Rather than compressing this contrast into an unexplained transition often labeled “collapse,” we ask which specific explanatory constraints must be satisfied before the relation between the two descriptions can be diagnosed precisely. A finite-resolution alternating position–momentum history is used throughout to separate questions of record formation, probability, actuality, causal registration, and outcome-conditioned refinement. A continuum analysis shows that making the registered history arbitrarily dense does not by itself recover a classical trajectory. Several parts of the architecture are exact once their inputs are specified. A detector representation induces an exact quotient; a specified acquisition channel and record family induce system-side effects; and a specified quantum instrument determines a family of subnormalized conditional branches before any one outcome is instantiated. Given a realized outcome label, the same instrument determines the corresponding normalized outcome-conditioned state. A finite-dimensional acquisition model explicitly realizes the representability chain, while a record-algebra indistinguishability result shows that removing coherence between detector-record sectors can leave all expectation values in the lifted record algebra unchanged even when that coherence remains detectable in a richer context. These results sharpen, but do not dissolve, the measurement problem. Record structure does not derive Born weights; a probability law does not itself constitute the event realized in one run; a mathematically available conditional branch family does not select one branch condition as actual; and causal onset, stabilization, registration, and conditional refinement all presuppose or follow that event rather than produce it. The resulting constraint audit tests—without assuming—the hypothesis that global quantum multiplicity and detector-record definiteness may belong to different representational levels. The main conclusion is not a new collapse dynamics but a localization of the remaining explanatory burden. Once representability, probability, causal registration, and conditional refinement are assigned their proper roles, the unresolved actuality transition can no longer be hidden inside the single label “collapse.” In the typed measurement architecture considered here, no pre-event operation has the mathematical role of producing an individual event token without additional event-producing structure. This is version 1.0 of the manuscript.