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。
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。
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 an intermediate layer in a downward research decomposition. It separates an admissible pre-projection state structure, dynamical flow, observation/coarse-graining maps, and finite readouts. A planar relational-rotation benchmark shows that finite binary, eight-class, or twelve-sector readouts need not be identified with the dimension or ontology of the admissible state space. Common absolute orientation cancels from purely relational angle observables. The note explicitly does not claim a hidden-variable model, a new quantum state space, or a final ontology. It defines transition-relation reconstruction and discrete-versus-continuous symmetry as the next death gates.
This preprint proposes a strong-form, falsifiable four-layer physical hypothesis in which ordinary binary outcomes are coarse-grained projections of deeper physical organization. The Yuan pre-generative layer consists of three jointly necessary conditions: a carrier/possibility domain, boundary and constraint conditions, and internal coupling/organizational relations. Their joint realization is hypothesized to generate eight physically distinguishable quantum process sectors. In the strong form, these sectors are organized as four relational directions and two orientations with a D4-type composition structure. The generative layer projects into a hidden-binary structural layer containing ordered complementary pairs 01/10, A/B/C/D source or directional labels, an independent orientation attribute, and a 3-to-6-to-96 fine-grained relational record space, before conventional measurement coarse-grains the structure to 0/1. The hypothesis is deliberately frozen before experimental validation. It predicts recoverable hidden information beneath identical binary outcomes, a minimal eight-process description, D4 noncommutative composition fingerprints, resolvable fine-grained relational records, necessity of the three Yuan conditions, and cross-platform consistency. The next stage is a preregistered experimental program designed to support, modify, or falsify each layer independently.
Horizon Monism proposes a unified theoretical framework in which quantum structure, operational time, and spacetime geometry emerge from finite reference horizons embedded within a single, scale-free relational–interaction totality. Rather than treating spacetime, fundamental time, and quantum probabilities as primitive, the framework investigates their possible reconstruction through finite distinguishability, relational compression, and effective projection structures. The paper establishes a strict mathematical hierarchy of finite-horizon resolutions, deriving monotonic distinction deficits and a directed fine-to-coarse order as a candidate foundation for operational time. This construction is extended to multiple reference horizons through order-preserving transport and conditional shared-blindness theorems. Within a conditionally constructed complex groupoid algebra, the minimal noncommutative effective structure is identified as M₂(ℂ). Projection compression is shown to generate noncommuting effective operators, while smoothly varying projections induce connection curvature. Under specified geometric conditions, this curvature reconstructs the Levi-Civita curvature of an emergent Lorentzian spacetime. Explicit de Sitter and nonlinear Green-spectral constructions provide conditional examples of Einstein geometry and stable nonvanishing projected curvature. A further reconstruction of quantum probability establishes that normalized, context-independent, coarse-graining-consistent effect assignments admit a unique density-matrix representation, recovering the Born rule for pure states. The Principle of Minimal Sufficient Realization is additionally applied to a restricted class of Standard-Model-type representations, yielding conditional minimality results for color, fermion generations, and Higgs structure. The framework distinguishes exact mathematical theorems from conditional physical realizations. Its principal unresolved questions concern the microscopic origin of quantum probability consistency, physical state selection, continuum dynamics, and experimentally distinguishable predictions. Horizon Monism is presented as a mathematical and theoretical research program toward the possible co-emergence of quantum physics and gravitation.
From Distinction to Quantum Structure Compiling Physical Geometry from the Change Axiom Chain Driven by Dean Kulik July 2026 Abstract We present a foundational axiom chain that compiles physical geometry from a single primitive: distinction exists. Beginning with C0 (distinction) and C1 (every distinction admits lawful continuation), we derive — without importing spatial geometry, coordinates, time, or quantum mechanics as primitives — the necessity of a linear state space, a complex inner product, Hilbert completion, self-adjoint observables, and the structure of a probability measure on observable subspaces. We further derive the unique free-field Lagrangian consistent with the axiom constraints, the gauge principle as a structural consequence of independent classification, and the quantization bridge connecting the classical field to operator algebras. The central result is the Division Algebra Theorem (candidate): among the three finite-dimensional division algebras admitted by Frobenius’s theorem, only the complex numbers simultaneously satisfy the continuation independence requirement (C1a) and the dual-encoding requirement of C5. The paper maintains a strict five-category epistemic discipline — primitive, definition, compiled theorem, candidate theorem, empirical calibration — throughout, ensuring every claim carries its exact epistemic status. 1. Introduction: The Foundational Inversion Standard physical theory begins with space. Objects occupy locations in a pre-existing spatial manifold; motion is defined relative to that manifold; forces act within it. Newton’s mechanics, Einstein’s general relativity, and the Standard Model of particle physics all share this architecture at the foundational level: a stage is assumed, and the physics plays out on it. The program presented here inverts this order. We propose that space, time, geometry, and quantum structure are not primitives but derived objects — outputs of a more fundamental algebraic structure built from the minimal concept of distinguishable change. The chain begins: rather than: This is not merely a philosophical reordering. It is a concrete research program with specific mathematical obligations at each step: each layer must be derived from the previous layers without importing external structure, and every claim must be categorized by its epistemic status. The framework is falsifiable — if any compiled theorem fails its formal derivation, either the derivation or one of the primitives must be revised. 1.1 Motivation: SHA-256 as Physical Evidence The SHA-256 hash function provides an unexpected but precise illustration of the C1-first claim. SHA-256 is field-balanced in both axes: • Horizontal (input → output): Bit content redistributes across the compression function. The total information content of the input is preserved in the digest — different shape, identical content. The mapping is a redistribution, not a creation. • Vertical (round → round): The carry palindrome property holds: KL divergence between forward and backward carry distributions equals zero. The AHRC (Algebraic Hash Round Consistency) result shows to machine epsilon across all 64 rounds. The geometry is self-balancing at every step. Neither property was engineered into SHA-256. The designers specified diffusion targets; the field obeyed the redistribution constraint underneath and the two-axis conservation emerged. This is the claim of the axiom chain in concrete form: the transition algebra runs without an observer, and we arrive afterward to read what it did. SHA-256 does not choose to balance. It cannot not balance. Law 3 — redistribution conserves total — runs underneath every hash computation invisibly, and we measure it as a mathematical law about SHA-256. The chain argues this is the universal situation: what we call physical law is the transition algebra’s self-accounting, read from inside by instruments that are themselves running the same algebra. 1.2 Program Overview This paper establishes: 1. The primitive constraint set (Section 2) 2. Definitions required for precision (Section 3) 3. Compiled theorems through the free-field Lagrangian (Sections 4–6) 4. The internal symmetry and gauge principle (Section 7) 5. The quantization bridge — six elements of the Quantum Kinematics Theorem (Section 8) 6. The Division Algebra Theorem — the central new argument of this paper (Section 9) 7. Candidate theorems and open problems with proof strategies (Section 10) 8. Full dependency graph (Section 11) 1.3 Epistemic Categories Throughout this paper, every claim is labeled with one of five categories: Label Meaning [P] Primitive — assumed, irreducible [D] Definition — terminology, requires no proof [C] Compiled theorem — formally derived from prior P/D/C entries [T] Candidate theorem — strong evidence, formal proof incomplete [E] Empirical calibration — requires measurement, not deduction This discipline has a structural consequence: if a [C] theorem fails, either the derivation contains an error or one of the primitives must be revised. If a [T] theorem fails, the program survives intact — that conjecture was incorrect, and the framework needs refinement at that node only. If an [E] calibration disagrees with experiment, the structural theory may remain correct while the mapping to physical units requires revision. 2. The Primitive Constraints We assume the following as irreducible primitives. They are not derived from anything else in the framework. Their justification is that each is genuinely minimal — no proper subset generates the others. [P] C0 — Distinction Exists At least two states exist that are not identical. This is the logical precondition for any other constraint. C1 cannot operate on a single undifferentiated object. C0 is the one irreducible prior: something must be distinguishable from something else. Without C0, the entire chain has no content. [P] C1 — Every Distinction Admits Lawful Continuation Every state in the system must admit at least one transition to a successor state. This is the change axiom. No state is a dead end. No distinction terminates. The system cannot freeze, because freezing would require a state with no admissible successor — and that state would constitute a privileged terminal configuration, which is ruled out by C1 itself. Every local rule must be well-defined on every admissible state: this immediately excludes absorbing dead states from the dynamics. [P] C1a — Independence Preservation Independent distinctions remain independently continuable. This is stronger than C1. It says not only that every state continues, but that the continuation of one distinction cannot be blocked or modified by the simultaneous continuation of an independent distinction. Two systems that are independent at time must continue to be independently continuable. This axiom is what forces linearity (see Section 4) and what later eliminates quaternionic algebras (see Section 9). [P] C2 — Continuation Preserves Reachable History Every continuation preserves the set of states reachable from any prior state in the history. Nothing in the past is erased. The execution history — the accumulated record of all prior transitions — remains reachable. This is not a claim about memory in any computational sense; it is a structural requirement that the transition graph not delete nodes from the accessible state space. C2 forces completeness of the state space (Section 4) and the one-directionality of the ordering relation (Section 5). [P] L3 — Redistribution Conserves Total Every lawful continuation redistributes the total content of the field without changing that total. The field never gains content; it never loses content. Every transformation is a redistribution — a shift in the configuration of a fixed total. This is the deepest form of the conservation principle. It does not assume conservation of energy, momentum, or charge; it says something more primitive: the field’s total is an invariant of continuation. Those physical conservation laws are derived from L3 operating in specific geometric contexts. [P] L4 — Redistribution Is Local Redistribution propagates through local adjacency. Content cannot shift from location A to location B without occupying every intermediate location. This is the locality axiom. It follows from C2 applied to intermediate locations: if content jumps from A to B without passing through the intervening field, every intermediate location is denied its C2-guaranteed reachable history. The inheritance constraint — every new location must inherit all primitives — makes this inadmissible. L4 is therefore arguably compilable from C2, but we list it as a primitive for clarity because it plays an independent role throughout the chain. [P] C5 — The Observable Event Primitive An observable event is the minimum continuation that simultaneously propagates (extends through at least one L4 step) and leaves a distinguishable residue. C5 defines the minimum unit of observation. Three constraints are built into it: • Propagation (C1 minimum): the event must extend. A continuation that fails to propagate is not a continuation. • Residue (C0 requirement): the event must leave a mark distinguishable from prior state. Without residue, C0 is not satisfied at the event level — nothing registered. • Simultaneity: propagation and residue are not sequential. They coexist in the same elementary event. This simultaneity requirement is the key structural feature of C5. It will force complex amplitudes (Section 9) and the uncertainty principle (Section 8). 3. Definitions Definitions introduce terminology. They do not require proof; they establish the language
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.