Principles of the Natural Formation Mechanism of Quantum Structures: Key Data on Closed Dynamical Motion, Orbital Statistics, Correlation Geometry, and Related Aspects
Macroscopic and microscopic organization are studied here as parts of one proposed relational formation– maintenance framework, the All-Domain Framework (ADF). Its foundational claim is that multi-frequency immaterial angular-momentum dynamic structures are the sole fundamental constituents of existence: dynamics is existence; without dynamics, there is no existence. Across the series, no phenomenon-specific physical postulate, outcome-specific operator or post hoc parameter is added; unclosed consequences remain explicitly OPEN. This paper establishes mathematical consequences of specified premises, not empirical confirmation of that ontology. Theorem C gives a bijection between finite-common-return frequency classes modulo common positive scaling and primitive positive integer representatives; the quotient is at most countable while within-class scales remain continuous. On specified closed orbits, invariant measures and Fourier projection statistics are derived. Constant-rate phase–vector compatibility is separated from mere phase recurrence. Conditional formation transport in the companion paper relates the remaining circle-map function to supplied initial data. Synchronized maintained-circle families obey an exact positive-semidefinite, unit-diagonal, rank-at-most-two Gram characterization. These results connect recurrence, maintained motion and geometric statistics without identifying a tensor count with quantum states or an orbit measure with Born probabilities. A front-loaded comparison distinguishes established quantum calculations from the generative questions addressed here. Global preparation weights, two-body experimental correlations, single-record generation and the physical frequency-layer-to-form mapping remain open.
Macroscopic and microscopic organization are studied here as parts of one proposed relational formation– maintenance framework, the All-Domain Framework (ADF). Its foundational claim is that multi-frequency immaterial angular-momentum dynamic structures are the sole fundamental constituents of existence: dynamics is existence; without dynamics, there is no existence. Across the series, no phenomenon-specific physical postulate, outcome-specific operator or post hoc parameter is added; unclosed consequences remain explicitly OPEN. This paper establishes mathematical consequences of specified premises, not empirical confirmation of that ontology. Theorem C gives a bijection between finite-common-return frequency classes modulo common positive scaling and primitive positive integer representatives; the quotient is at most countable while within-class scales remain continuous. On specified closed orbits, invariant measures and Fourier projection statistics are derived. Constant-rate phase–vector compatibility is separated from mere phase recurrence. Conditional formation transport in the companion paper relates the remaining circle-map function to supplied initial data. Synchronized maintained-circle families obey an exact positive-semidefinite, unit-diagonal, rank-at-most-two Gram characterization. These results connect recurrence, maintained motion and geometric statistics without identifying a tensor count with quantum states or an orbit measure with Born probabilities. A front-loaded comparison distinguishes established quantum calculations from the generative questions addressed here. Global preparation weights, two-body experimental correlations, single-record generation and the physical frequency-layer-to-form mapping remain open.
added; unclosed consequences remain explicitly OPEN. This paper develops the interaction, phase-transport and identification components of the All-Domain Framework (ADF), which seeks a common account of microscopic and macroscopic formation and maintenance. The framework takes multi-frequency immaterial angular-momentum dynamic structures as the sole fundamental constituents of existence and states: dynamics is existence; without dynamics, there is no existence. Within a declared lowest-order local two-axis second-order tensor class, seven independent operator forms are obtained for non-collinear axes. Their unital real associative algebra is the full real three-by-three matrix algebra; this is not a completeness theorem for quantum states. Integrable real linear evolution has positive fundamental-matrix determinant at finite times, a constraint distinct from algebraic generation and from time irreversibility. The ideal phase–vector interface gives an inverse geometry, a chord-factorization obstruction and a complete conditional circle-map family. On a fixed-axis radial branch, an explicit formation-phase correction transports supplied initial phase data while retaining its winding. Oriented finite cross geometry identifies sampled relative phases; finite samples do not identify an unrestricted phase function or its winding. Tensor-side descriptors are complete in the declared class, but the frequency-side audit does not supply the matching layer actions. The physical layer–form mapping therefore remains OPEN / STOP. Coefficient selection, actual initial preparation, global probability weights, experimental correlations and single-record generation are not derived. The front comparison states precisely which answers are mathematical, which are conditional and which remain research objectives. Keywords: All-Domain Framework; quantum-generation interface; multi-frequency immaterial angular-momentum dynamic structures; helical self-organization; operator-form classification; phase transport; identifiability Preprint release: v1.0, 11 September 2026. This release is derived from research candidate v14. It aligns the series name with All-Domain Framework (ADF), removes the framework name from the title, and states the series-wide no-phenomenon-specific-patching rule. It does not upgrade any mathematical, physical or experimental result; all declared OPEN interfaces remain OPEN
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.
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
Dynamic Quasi-Closure and Relational Organization of Quantum Systems Residual Certificates, Geometric Holonomies, Environmental Correlations, and Structural Transfer Conditions Robert Kupski · BKT–74–QREC · Edition 2.6 Abstract A mathematical and physical account of organizational retention in finite-dimensional quantum systems is developed within the Law of One Mechanism (LOM) research programme. Organization is specified by a distinguished family of states, while its retention depends on the dynamical generator, preparation, interaction with the environment, and observation interval. State recurrence, endpoint retention, quasi-closure of the entire trajectory, family invariance, and attraction are distinguished. Error certificates based on the residual of the equation of motion, their geometric refinements, invariance conditions, and bounds for quantum channels are derived. An exactly solvable two-coupling model relates protection of a selected sector to destructive interference between transition amplitudes. At resonance, for coupling frequencies of 0.30 and 0.40 MHz, leakage after 0.25 µs is 0 for the dark state, 0.2304 after complete dephasing of the preparation before evolution, and 0.4608 after a relative-phase shift of π. All three preparations have the same initial populations. In the ideal resonant model, keeping leakage at or below the one-percent threshold throughout evolution allows a maximum absolute phase deviation from the dark preparation of approximately 0.2087119452 rad. In an open three-qubit model, with the same Hamiltonian and preparation and local relaxation and coherence times T₁ = 80 µs and T₂ = 40 µs, replacing independent dephasing with common dephasing increases the time to one-percent total organizational loss from 0.483439 to 0.804027 µs, or by 66.31%. For these two realizations, the lower bound on the worst-case absolute error of a deterministic threshold-time prediction based solely on their shared local data is 0.1602938517 µs. An independent measurement of normalized relative-coherence decay identifies the correlation parameter within the model family and determines the coefficient of cubic leakage onset into the third qubit without fitting its subsequent trajectory. This transfer assumes an unchanged environmental generator across protocols and specified conditions of Markovian dynamics. The account is extended to the quantum geometric tensor, dark-state transport, and interferometric readout. A full coupling-phase cycle produces an adiabatic geometric phase of −0.72π and an ideal interference probability of 0.1812880051. Exact evolution determines finite-duration corrections. In the open model, return fidelity is bounded by excitation survival: for one initial excitation, no pumping, equal local relaxation times, and an excitation-preserving Hamiltonian, the upper bound is exp(−T/T₁), where T denotes the cycle duration. Geometric records and environmental memory are distinguished, and a conditional two-cycle order test is formulated. LOM postulates that fields and particles emerge from a more fundamental relational and informational organization and constitute its effective realizations. The article gives selected elements of this hypothesis an operational form through states, couplings, generators, and measurement relations. The results are conditional theorems and calculations for specified models; they are neither new experimental data nor evidence establishing the existence of a metafield. Additional constitutive content of the programme may take the form of independent constraints linking admissible generators and protocols. The study establishes a common account of retention, perturbation tolerances, environmental correlations, geometric transport, and dynamical identifiability. Introduction Persistence of organization in a quantum system is a property of the relations between the state, a distinguished class of configurations, and the dynamics. Two different, even orthogonal states may belong to the same organizational class, whereas recurrence to the initial state may follow an earlier departure from that class. Retention at one instant, control of the distance of an entire trajectory from a state family, and attraction to that family therefore describe distinct physical properties. Their quantitative separation identifies the contributions of autonomous evolution, preparation, perturbations, and measurement interventions to observed organizational retention. The study develops the LOM–GTSFC–USC–GTCW programme. The Law of One Mechanism (LOM; Polish: Prawo Jednego Mechanizmu, PJM) provides the overarching framework; GTSFC denotes the Global Theory of Shared Fundamental Couplings, USC the Universal Structural Code, and GTCW the Gravitational–Thermodynamic Cyclicity of Multiverses. Within this framework, the metafield is a hypothetical structure of admissible relations, couplings, and tolerance bands. USC describes their admissibility rules, while GTSFC organizes couplings and their sectoral realizations. The hypothesis of a common organizational principle permits different effective Hamiltonians and environmental channels. Structural information concerns the organization of relations that sustain a selected class of states; preservation of a bit record and state entropy do not exhaust this concept. A possible LOM principle more fundamental than the informational layer itself remains a separate source hypothesis. The article continues the BKT–74 line of research on memory, retention, and stabilization. BKT–74A distinguishes organization from instantaneous geometry; BKT–74B separates preparation and return memory from the stabilizing response; and BKT–74C relates retention and first exit to the system–environment coupling operator. BKT–74–QREC develops these distinctions through explicit state families, leakage operators, trajectory certificates, and relations between protocols. The protection analysis concerns a specified preparation and generator. The existence of an invariant subsector and the formation of such organization from general initial conditions are separate questions. Experimental context is provided by Dong and co-workers’ studies of regular trajectories in a variational description of an interacting 24-qubit system and their search and stabilization through hybrid feedback control (Dong et al., 2026; DOI: 10.1038/s41567-026-03431-z). Exact quantum evolution, dynamics on a variational manifold, and a loop comprising measurement, optimization, and re-preparation represent different levels of description. The two-coupling model considered here is a self-contained analytical construction. It provides an explicit example of interference-based protection, and its parameters and results refer to that model, without identifying it with the qubit-ladder Hamiltonian or the data of the cited experiment. Dynamic quasi-closure bounds the distance of a trajectory from a distinguished family of states at every instant of the interval considered. The residual of the equation of motion measures the mismatch between an approximate trajectory and a specified generator, yielding certificates for state and observable errors. Removing the component associated with the global phase gives an angular certificate in ray space. For organization described by a subspace, the operator K = QHP/ℏ, where P projects onto that subspace, Q = I − P, I is the identity operator, H is the Hamiltonian, and ℏ is the reduced Planck constant, identifies direct leakage directions. Full organizational retention also depends on the action of the generator within the distinguished sector. This approach connects protection properties to the dynamics and separates instantaneous cancellation of leakage, invariance, and attraction. In the resonant two-coupling model, the dark-state condition is g₁a₁ + g₂a₂ = 0. The quantities g₁ and g₂ are angular coupling frequencies, related to ordinary frequencies by gⱼ = 2πfⱼ, while a₁ and a₂ are preparation amplitudes. For f₁ = 0.30 MHz and f₂ = 0.40 MHz, the dark state is |D⟩ = 0.8|1⟩ − 0.6|2⟩, where |1⟩ and |2⟩ are orthonormal states spanning the distinguished sector. Comparing the dark preparation, its completely dephased mixture, and a preparation with reversed relative phase isolates the role of coherence at unchanged populations. Phase-error and coupling-direction tolerances quantify the robustness of this protection. Certificates based on the integrated residual norm provide sufficient conditions; loss of information about phase cancellations can limit their sharpness. The extension to open systems uses Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) generators with specified jump operators. In the three-qubit model considered, identical local relaxation and coherence times T₁ and T₂ do not specify dephasing correlations and therefore do not uniquely determine retention time. The parameter c interpolates between independent noise, c = 0, and common noise, c = 1. At T₁ = 80 µs and T₂ = 40 µs, the difference between the times to one-percent total loss yields a lower error bound for predictions based on local data that do not distinguish these realizations. Total loss includes both leakage into the third qubit and relaxation to the state with no excitation. This non-identifiability concerns the scope of the calibration data, while the predictions of a fully specified model remain unambiguous. An additional measurement of relative coherence provides information about environmental correlations. With the couplings switched off, the decay rate of population-normalized coherence determines c within the family considered, provided that the pure-dephasing rate is positive. The calibration result then determines the coefficient A₃ in the expansion P₀₀₁(t) = A₃t³ + O(t⁴), where P₀₀₁ denotes the population of the state with the third qubit excited, and t is the time since dark-state preparation and activation of the couplings. The relation holds for th
This preprint develops a discrete, fully relational framework in which spacetime structure, classical dynamics, and quantum-like behavior emerge from three minimal principles: relational existence without background manifold, local deterministic update, and finite per-element update capacity. The central organizing primitive is finite update capacity. At each update step, elements can realise only a bounded number of relational changes. Competition among admissible updates induces “blocked realisations,” which are defined micro-ontologically as energy. Proper time is identified with realised record growth. Spatial distance and effective dimension arise from intrinsic graph structure and shell scaling. Under broad locality conditions, marginal frontier stability constrains viable scaling regimes. Requiring compatibility between flux balance and the observed inverse–square law selects an effective three-dimensional scaling regime among integers. Classical force structure emerges from anisotropies in blocked realisation. A statistical layer appears through coarse-graining of deterministic microhistories. Composition and norm stability require unit-modulus complex weights; refinement invariance and classical coherence fix the phase rule θ=ℏ−1∫Edτ, with the same blocking-defined energy that generates intrinsic force. In decohered regimes, quadratic (Born-type) probability scaling follows from additive consistency under random-phase aggregation. The framework is programmatic: it does not yet derive general relativity or quantum field theory from explicit microscopic rule families. However, it establishes structural coherence linking geometry, inertia, inverse–square interaction, phase accumulation, and probabilistic outcomes under a single finite-capacity relational principle. The work also identifies structural empirical constraints, including compatibility between shell scaling and classical force behavior, stability of inertial response, and the possible appearance of capacity-induced high-energy dispersion within the intrinsic causal-cone bound. Explicit construction and simulation of proposal/selection rule families constitute the next stage of investigation.
This paper derives the angular correlation law and constructs joint probabilities for quantum entanglement within the All-Domain Framework (ADF). It presents explicit mathematical results, readout exclusion results, and experimental analysis procedures. Main results Angular correlation law. Starting from the invariant graph d = Kx and the Riccati compatibility condition C + EK − KA − KBK = 0, the paper derives the cross-moment identity Qxd = QxKᵀ. On the anti-relational isotropic branch K = −I and Qx = sI, the normalized continuous correlation is exactly ρ(a,b) = −a·b. Joint-probability construction. A deterministic measurable partition of the maintenance circle is constructed with four outcome-region measures μ(RAB) = (1 − ABc)/4. The construction gives unbiased local marginals and ⟨AB⟩ = −c. Under the identification c = a·b, it reproduces the angular correlation and the standard CHSH value 2√2. Readout exclusion results. Seven candidate mechanisms are audited under their stated assumptions. The results identify why passive isotropic averaging, local hard thresholds, and the examined dynamical mechanisms do not independently generate the required four-sector outcome law. Experimental analysis methods. The paper provides graph reconstruction from measured cross moments, anti-relational residual tests, independent isotropy tests, and continuous-correlation reconstruction. Calibration is separated from testing against the target correlations. Synthetic numerical verification. The reported tests demonstrate branch discrimination, graph-inversion convergence, and sensitivity to noise. A hard-sign negative control identifies a systematic discrepancy between local threshold readout and the target joint probabilities. The continuous-correlation derivation and joint-partition existence theorem are established mathematical results under the specified conditions. Physical analyzer-setting transduction and dynamical generation of individual binary outcomes remain explicitly identified interfaces for further work. The numerical results use synthetic data. This illustrated edition contains 20 pages, including six vector figures.