Causal-model arguments establish that a classical causal model reproducing Bell correlations while exhibiting no-signalling must be unfaithful: its no-signalling independences are not entailed by d-separation in the causal graph. Applied to time-symmetric, retrocausal, and all-at-once models, this is the fine-tuning objection. The resulting debate has largely concerned whether faithfulness is the appropriate norm, without a criterion for deciding particular cases or an associated measurement.This paper supplies a proposed criterion and establishes a precise symmetry result for the balanced binary singlet sector. If a nonnegative \(2\times2\) compatibility table has the simultaneous-outcome-flip symmetry \[ m_{++}=m_{--}, \qquad m_{+-}=m_{-+}, \] then every common scalar entrywise map \[ m_{xy}\longmapsto f(m_{xy}) \] with nonnegative output and nonzero normalization preserves uniform local marginals. No monotonicity, continuity, differentiability, or power-law assumption is required. Within two-qubit quantum kinematics with ideal analyzer projectors, joint rotational invariance and perfect same-axis anticorrelation force the singlet state and hence this pairing symmetry. The theorem establishes robustness relative to the symmetry-restricted deformation class; it does not confer causal faithfulness or robustness against arbitrary symmetry-breaking cellwise perturbations.The same singlet-plus-common-map structure has a deformation-independent surplus consequence: \[ E^f(\pi-\Delta)=-E^f(\Delta), \qquad E^f(\pi/2)=0. \] These angular identities are distinct from no-signalling and can fail experimentally. They provide the surplus consequence required by the proposed criterion in the balanced sector.A separate constant-flux completion law predicts an exactly flat ideal joint completion rate. We compare it with a Bell-local sign--cosine model followed by a joint two-bin selection rule. The pointwise maximal acceptance envelope for reproducing the singlet correlator has symmetric minima \[ Z^\ast\simeq0.878567 \] near \(46.44^\circ\) and \(133.56^\circ\). A flat calibrated joint acceptance rate above this value excludes that two-bin class. More generally, total variation gives \[ S_{\mathrm{obs}} \le \min\left\{ 4,\, 2+2\sum_{q\in Q}(1-Z_q) \right\}, \] so a flat rate above \[ \frac{5-\sqrt2}{4}\simeq0.896447 \] excludes every Bell-local measurement-independent selection model covered by this bound when \(S_{\mathrm{obs}}=2\sqrt2\).Finally, an exact eight-state reconstruction of a published post-processing model has acceptance rate \(3/4\) in every setting context while attaining the algebraic CHSH value \(4\). Its accepted laws satisfy \[ \Delta_Q=1, \qquad D_Q=\frac13, \] saturating both the selection-inflation lower bounds and the upper bounds permitted by the absolute rate. Thus rate flatness is not a certificate of fair sampling, although the absolute rate remains quantitatively useful. The quantity requiring structural certification is the conditional acceptance function, or a sufficient experimentally accessible proxy for it.
Causal-model arguments establish that a classical causal model reproducing Bell correlations while exhibiting no-signalling must be unfaithful: its no-signalling independences are not entailed by d-separation in the causal graph. Applied to time-symmetric, retrocausal, and all-at-once models, this is the fine-tuning objection. The resulting debate has largely concerned whether faithfulness is the appropriate norm, without a criterion for deciding particular cases or an associated measurement.This paper supplies a proposed criterion and establishes a precise symmetry result for the balanced binary singlet sector. If a nonnegative (2\times2) compatibility table has the simultaneous-outcome-flip symmetry [ m_{++}=m_{--}, \qquad m_{+-}=m_{-+}, ] then every common scalar entrywise map [ m_{xy}\longmapsto f(m_{xy}) ] with nonnegative output and nonzero normalization preserves uniform local marginals. No monotonicity, continuity, differentiability, or power-law assumption is required. Within two-qubit quantum kinematics with ideal analyzer projectors, joint rotational invariance and perfect same-axis anticorrelation force the singlet state and hence this pairing symmetry. The theorem establishes robustness relative to the symmetry-restricted deformation class; it does not confer causal faithfulness or robustness against arbitrary symmetry-breaking cellwise perturbations.The same singlet-plus-common-map structure has a deformation-independent surplus consequence: [ E^f(\pi-\Delta)=-E^f(\Delta), \qquad E^f(\pi/2)=0. ] These angular identities are distinct from no-signalling and can fail experimentally. They provide the surplus consequence required by the proposed criterion in the balanced sector.A separate constant-flux completion law predicts an exactly flat ideal joint completion rate. We compare it with a Bell-local sign--cosine model followed by a joint two-bin selection rule. The pointwise maximal acceptance envelope for reproducing the singlet correlator has symmetric minima [ Z^\ast\simeq0.878567 ] near (46.44^\circ) and (133.56^\circ). A flat calibrated joint acceptance rate above this value excludes that two-bin class. More generally, total variation gives [ S_{\mathrm{obs}} \le \min\left{ 4,, 2+2\sum_{q\in Q}(1-Z_q) \right}, ] so a flat rate above [ \frac{5-\sqrt2}{4}\simeq0.896447 ] excludes every Bell-local measurement-independent selection model covered by this bound when (S_{\mathrm{obs}}=2\sqrt2).Finally, an exact eight-state reconstruction of a published post-processing model has acceptance rate (3/4) in every setting context while attaining the algebraic CHSH value (4). Its accepted laws satisfy [ \Delta_Q=1, \qquad D_Q=\frac13, ] saturating both the selection-inflation lower bounds and the upper bounds permitted by the absolute rate. Thus rate flatness is not a certificate of fair sampling, although the absolute rate remains quantitatively useful. The quantity requiring structural certification is the conditional acceptance function, or a sufficient experimentally accessible proxy for it.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
We present a machine-checked Bell-consistency analysis for the finite-dimensional singlet sector of Constraint-Surface Dynamics (CSD), in which realised outcomes are deterministic almost everywhere and reproduce Bell-violating singlet correlations, while the finite measurement dynamics remain tensor-factorised and the observable marginals satisfy no-signalling. Given a singlet preparation sector and measurement independence, deterministic context-indexed outcome maps on a shared probability space reproduce Pst(a,b)=41−sta⋅b, and attain the maximal quantum CHSH value 22. We formally prove that no compatible setting-local four-response assignment A1,A2,B1,B2 can reproduce the singlet correlations at the four specified CHSH contexts, namely that the realised maps cannot all satisfy Fij(x)=(Ai(x),Bj(x)). The Bell obstruction therefore excludes a compatible setting-local four-response representation of the singlet correlations at the four specified CHSH contexts, not determinism of the realised outcome. The formal development also studies the structure of the finite measurement interaction in the same singlet sector, without assuming that the contextual outcome family is derived from that finite measurement dynamics. Local setting transformations UA(a)⊗UB(b) and local detector couplings VA⊗VB compose as (VA⊗VB)(UA(a)⊗UB(b))†=(VAUA(a)†)⊗(VBUB(b)†), so the complete finite measurement chain remains tensor-factorised. Pointer-volume probabilities reproduce the singlet law for all detector settings, including the perfect-correlation and perfect-anticorrelation limits. Operational no-signalling is verified at both the singlet-kernel and pointer-volume levels, with each local marginal equal to 1/2 and invariant under the remote setting. The singlet preparation sector and measurement independence are explicit assumptions, and no general no-signalling theorem for arbitrary non-factorising ontic dynamics is asserted. The contribution is not a new Bell inequality or a new general result that deterministic models can reproduce Bell-violating correlations. Rather, it is a CSD-specific consistency construction and a machine-checked separation of contextual determinism, Bell-local response factorisation, finite measurement factorisation, and operational no-signalling. Within this stated scope, the Lean 4 development provides a formally verified example in which deterministic realised outcomes almost everywhere, Bell-violating singlet correlations, tensor-factorised finite measurement dynamics, Bell-nonfactorisable joint outcomes, and operational no-signalling coexist, while any compatible Bell-local four-response assignment reproducing the singlet correlations at the four specified CHSH contexts is mathematically excluded.
This is Paper 142 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops a finite mechanical account of quantum entanglement within pq-spacetime. Entanglement is identified with the persistence of a non-factorable family of admissible future histories. Two measurement regions can therefore appear as separate local systems in emergent spacetime while still remaining projections of one joint finite closure problem. Bell inequality violation follows because the joint outcomes are not generated by two independent local completion families. No-signalling is nevertheless preserved because complete changes to a distant measurement boundary do not alter the local marginal outcome weights. The paper establishes the Entanglement Closure Theorem and distinguishes microscopic non-factorability from observable entanglement after coarse-graining into stable measurement records. It also gives an explicit minimal singlet construction using two opposed history fibers carrying the relevant four-phase weights. The resulting finite ledger is antisymmetric, nonseparable, and exactly balanced at each local wing. Complete changes of basis on the distant side preserve those local marginals. Using the previously derived Hilbert-space and beta-Dirac bridges of pq-spacetime, the paper recovers the familiar spinor overlap law, the quantum singlet correlation, the maximal Bell-CHSH value, monogamy of maximal entanglement, and decoherence through environmental boundary enlargement. The analysis also explains why no preferred measurement ordering is required and why entanglement remains compatible with relativistic causality. The resulting picture is direct. Entangled systems do not exchange superluminal messages, carry classical answer sheets, or require a force acting across space. Their local outcomes arise from one shared finite admissibility structure whose joint correlations are nonlocal while its controllable local statistics remain causal. Entanglement is therefore physically real, nonclassical, and nonlocal in correlation, but not a signalling mechanism.
The Born rule assigns a probability law to a physical state and a complete observation state, but a probability law is not the event realized in one trial. Building on the physical–observation dual-axis state space, we construct an event layer that separates these two mathematical objects. Every measurable finite Born kernel admits a deterministic circle lift whose event cells have the required normalized Haar volumes. On a principal U(1) selector bundle, cell membership is gauge invariant and Haar reduction returns the Born trace law exactly. A gauge-invariant action yields both selector transport and an observation-axis equation in which mixed curvature is the cross-axis force. The Fisher metric of the Born response supplies its minimal kinetic geometry, while a least-interface principle fixes connected binary event cells. Curvature determines relative transport but cannot select an absolute event without an initial selector; pure principal transport preserves Haar equilibrium but cannot create it. Bell violation further requires the deterministic event map to be contextual on the implemented joint setting. The event law leads to a concrete test of single-event continuity. The physical control is an azimuthal cycle and the observation coordinate is calibrated independently from the binary Born response. The benchmark states three additional closures explicitly: a reduced spherical completion with rotationally invariant unit-Chern curvature, minimal free response dynamics, and a selector-transparent returning instrument. The fixed dimensionless protocol X = 0.100 and ωS/Ω = 4 has a unique balanced periodic response orbit and predicts the protocol constant κM = 0.03080966337. No event record enters this calculation. A returning measure-and-prepare instrument then predicts a triangular mismatch recurrence: 0.980702% after one cycle, 99.9684% after 102 cycles, and 0.0632% after 204 cycles, although every single-cycle marginal remains balanced. The memoryless realization of the same instrument gives 50% mismatch at every nonzero lag. Separate response, phase-calibration, and event ensembles prevent the predicted orbit from being programmed into the test, while causal-break and selector-scrambling controls bound ordinary apparatus memory. This numerical “M-event fingerprint” converts the century-old question of objective single-event continuity into a preregisterable, falsifiable serial-correlation experiment. It is a protocol-specific prediction, not a new natural constant; a null result falsifies the stated minimal realization rather than every possible deterministic completion.
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.
For local hidden-variable accounts of the CHSH experiment that are faithful -- reproducing the observed singles and coincidence rates and unbiased marginals of a polarization singlet with symmetric detector efficiency ηeff -- I determine the minimal measurement dependence M (Hall's measure) needed to reach a CHSH value S. Linear programming over all local strategies yields M(S,ηeff)=max{0,ηeff((S+2)ηeff−4)/6}, whose edges reproduce Hall's bound, the Garg-Mermin threshold, and the postselection ceiling. The quantum point is certified exactly: M(22,9/10)=(272−33)/100, by primal and dual certificates in Q(2). I solve the detection profile under which a deterministic sign model reproduces the singlet exactly, and prove exact quantum correlations and angle-independent coincidence rates jointly impossible in this class. Mixtures of this profile with measurement dependence trade M against a cos4(a−b) modulation of the fourfold coincidence sum, up to 12.4% in relative amplitude, identical at the two CHSH angles; existing data exclude its full amplitude and bound its mixtures with flat detection to a detection share of order 15%; its sub-percent content is unbounded. A two-setting cross-ratio and a settings-torus log-Fourier protocol, calibrated against channel-efficiency imbalance, reach 5σ at 0.1% within hours; a flat result excludes the solved family to a detection share of order 1%. A local mechanism, moiré phase locking, with deterministic phase evolution and quenched randomness, attains the certified floor exactly at ηeff=1 and, with the solved profile, reaches M=0 at ηeff≈0.80. Its floor-attaining census shares a low-order Fourier skeleton across independent solutions and is not realizable by station-local offsets on the terrains tested.