Phenomenological--Velocity Relational Event Completion(\(\PV\text{-}\REC\))is formulated as a non-Hilbert, time-symmetric,source--absorber boundary-event mechanics.Its native ontology contains no wavefunction,no Hilbert-space state postulate,no unitary state evolution,no Born-rule actualization,and no collapse law.The theory instead uses real spin-boundary data,positive source--absorber compatibilities,Lorentz-scaled proper boundary-clock event barriers,local flux-gauge completion,and a constant-flux threshold race that actualizes one definite event. The preparation-side ontic record\[\lambda_0=(v,\epsilon,\mathcal J,\Xi)\]is intervention-independent of the later local settings:\[P\!\left( \lambda_0 \mid \operatorname{do}(a), \operatorname{do}(b)\right)=P(\lambda_0).\]It is not postulated to be a complete Bell screening state.The completed event is the four-dimensional boundary object\[\Lambda^{\mathrm{comp}}_{ab}=\bigl( \lambda_0, \mathcal B_A(a), \mathcal B_B(b)\bigr).\]Thus PV-REC distinguishes preparation independence fromBell-Cauchy completeness:the complete event need not be encoded by a setting-independent variableon a hypersurface preceding the absorber boundary conditions. The central contention with the universalized reading of Bell's theorem istherefore precise.PV-REC accepts\[\BellCauchyScreen\LongrightarrowS\le2.\]It rejects the stronger inference\[\BoundaryLocal\Longrightarrow\BellCauchyScreen.\]Boundary-event locality is defined positively through:preparation intervention independence,region-local detector records,a source-star transaction graph with no primitive directed\(A\to B\) or \(B\to A\) edge,all-at-once source--absorber completion,Lorentz-covariant boundary laws,and accessible operational no-signalling.Bell screening is an additional past-Markov or Cauchy-screen condition,not a consequence of those boundary-locality principles. The balanced two-spin compatibility is derived natively in real boundarygeometry.Let \(n,m\in S^2\subset\R^3\) be analyzer directions and\(x,y\in\{\pm1\}\) local boundary-channel labels.Local balance,joint rotational covariance,separate affinity in the signed boundary vectors \(xn\) and \(ym\),neutrality of an unpolarized boundary,and source anti-alignment uniquely give\[m_{xy}(n,m)=\frac14\bigl(1-xy\,n\cdot m\bigr).\]No Hilbert-space state or amplitude is used in this derivation. The native interaction coordinate is the log cross-ratio\[\Theta(m)=\log\frac{m_{++}m_{--}} {m_{+-}m_{-+}}.\]For every strictly positive balanced binary table,\[E=\tanh\!\left(\frac{\Theta}{4}\right).\]PV acts as the projective homothety\[\Theta\longmapsto\Theta_\kappa=\kappa\Theta,\qquad\kappa(v)=\sqrt{1-\frac{v^2}{c^2}}.\]The unique positive entrywise transformation that scales everybinary interaction cross-ratio by \(\kappa\) is, up to a commonnormalization factor,\[m\longmapsto m^\kappa.\]Thus the power law is characterized intrinsically by positive-coneprojective geometry rather than imported from quantum mechanics. The same homothety is realized physically by proper boundary-clockevent-action accumulation.PV event action is a positive capacity-counting functional on microscopicboundary channels, not the ordinary Lorentz-invariant bulk field action.If \(n^\mu\) is the apparatus-boundary four-velocity and \(u^\mu\) theboundary-microstructure four-velocity, then\[\gamma_{\mathrm{rel}}=-\frac{n_\mu u^\mu}{c^2},\qquad\kappa=\gamma_{\mathrm{rel}}^{-1},\qquad\dd\tau_u=\kappa\,\dd t_n.\]For a uniform sector this gives\[I_{xy}^{\mathrm{proper}}=\kappa I_{xy},\qquadI_{xy}=-\log m_{xy}.\]A continuous nondegenerate memoryless proper-capacity law is exponential;equilibrium calibration at \(\kappa=1\) fixes its rate uniquely, yielding\[\chi_{xy}^{\PV}=e^{-I_{xy}^{\mathrm{proper}}}=m_{xy}^{\kappa}.\] Beyond the balanced sector,local boundary continuity fixes prescribed row and column fluxes.The unique positive table in the diagonal orbit of the raw PVsusceptibility with those fluxes is its Sinkhorn flux-gauge completion:\[Q_{xy}=\alpha_x\beta_y m_{xy}^{\kappa}.\]The local multipliers change only local flux potentials.They preserve the interaction cross-ratio:\[\Theta(Q)=\kappa\Theta(m).\]No-signalling completion is therefore a local gauge fixing of boundaryfluxes, not an alteration of the PV interaction invariant. The physical hazards are\[\widehat\Gamma_{xy}=\Gamma_{\mathrm{trial}}Q_{xy}.\]Independent exponential race seeds define\[T_{xy}=\frac{\xi_{xy}}{\widehat\Gamma_{xy}},\qquad(X,Y)=\argmin_{x,y}T_{xy}.\]One definite event pair is actualized almost surely.The total hazard is constant,local row and column hazards are fixed by local flux conservation,and accessible local outcome, count-rate, and timing distributions areindependent of the remote setting.The native mechanism therefore satisfies\[\PrepIndependent,\qquad\BoundaryLocal,\qquad\RealEvent,\qquad\NoSignalAcc,\]while failing\[\BellCauchyScreen.\] In the balanced sector,\[P_{\PV}(x,y\mid a,b,v)=\frac{ m_{xy}(a,b)^{\kappa(v)}}{ \sum_{x',y'} m_{x'y'}(a,b)^{\kappa(v)}},\]and\[E_{\PV}(a,b\mid v)=-\tanh\!\left[ \kappa(v) \artanh(\cos(a-b))\right].\]At \(v=0\), \(\kappa=1\), and the ordinary singlet curve is recovered.For every fixed \(v\neq0\), the law is nonlinear in\(n\cdot m\) and differs from standard two-spin quantum mechanics. The native same-source event map\[(X,Y)=K_{ab}(\lambda_0)\]has a positive-probability replay defect:at fixed \(b,\lambda_0\), Bob's actualized outcome can change under acounterfactual change of Alice's absorber setting, and conversely.Hence the context-indexed native variables do not glue into localcounterfactual columns\[A_0,A_1,B_0,B_1.\]This is a global-section obstruction, not a rejection of ordinaryprobability.PV-REC uses positive Kolmogorov probability for race seeds and realizedcontexts; it rejects the Bell-local overlap identities required to turncontext-indexed events into a setting-independent past-screened sheet. The companion selection theory identifies setting-dependent selection asa translation on the same interaction-rapidity line:\[\artanh E_{\mathrm{obs}}=\artanh E_{\mathrm{full}}+\xi,\qquad\xi=\frac14\log A,\]where \(A\) is the acceptance cross-ratio.PV is instead a dilation:\[\artanh[-E_{\PV}]=\kappa\artanh(\cos\Delta).\]Every local-hidden-variable correlation family casts a chainedsubadditivity shadow on the agreement defect\[d=\frac{1+E}{2}.\]The PV defect is\[d_\kappa(\Delta)=\frac{\tan^{2\kappa}(\Delta/2)} {1+\tan^{2\kappa}(\Delta/2)}\sim\left(\frac{\Delta}{2}\right)^{2\kappa}.\]A bounded rapidity translation can rescale a defect but cannot change itssmall-angle H\"older exponent.Consequently, for\[\kappa>\frac12,\]no finite-budget Bell-local selection model reproduces the full PV familyon a punctured neighborhood of coincidence.Matching it down to angular resolution \(\Delta\) requires\[\Xi\ge\frac{2\kappa-1}{4}\log\frac1\Delta-C,\]or an acceptance cross-ratio discrimination diverging at least as\[\max(A,A^{-1})\gtrsim\Delta^{-(2\kappa-1)}.\]Thus PV-REC is not a hidden post-selection construction:its native selection audit is trivial,\[\gamma_{\PV}\equiv1,\]while any external Bell-local selection counterfeit of thesuper-threshold continuum law incurs a divergent resolution cost. The optimized angular CHSH threshold is exactly\[\kappa>\frac12,\qquad\frac{|v|}{c}<\frac{\sqrt3}{2}.\]At \(\kappa=\tfrac12\),the agreement-defect exponent reaches the metric-additivity boundary,the optimized angular CHSH family first saturates,and the shape law becomes the unique nontrivial quadratic sector:\[f_{1/2}(z)=\tanh\!\left(\frac12\artanh z\right)=\frac{1-\sqrt{1-z^2}}{z}.\] The positive-cone geometry supplies an additional exact identity.The Hilbert-projective diameter of the powered balanced table is\[\Delta_{\mathrm H}=4\kappa\artanh|z|,\]and its optimal Birkhoff contraction coefficient is\[\tanh\!\left(\frac{\Delta_{\mathrm H}}{4}\right)=|E_\kappa(z)|.\]Here ``Hilbert'' denotes Hilbert's projective cross-ratio metric on apositive cone, not a quantum Hilbert space. Finally, the theory is separated explicitly from standard two-spinquantum mechanics.Any fixed two-qubit state measured by ordinary projective spin observableshas a bilinear correlation\[E(n,m)=n^{\mathsf T}Tm.\]Joint rotational covariance forces \(T=tI\), and perfect same-axisanti-correlation forces \(t=-1\), hence\[E(n,m)=-n\cdot m.\]Therefore no fixed \(v\neq0\) PV sector can be represented by a fixedrotationally invariant two-qubit state with standard projective spinmeasurements.The \(v=0\) sector is the unique equilibrium point at which the standardtwo-spin quantum curve and the PV-REC curve coincide. PV-REC is accordingly classified as\[\boxed{\PV\text{-}\REC_{\TS}\in\NonHilbert\cap\BoundaryComplete\cap\BoundaryLocal\cap\PrepIndependent\cap\RealEvent\cap\NoSignalAcc\cap\neg\BellCauchyScreen.}\]Bell's inequality remains valid within the Bell-screenable class.The point of contention is whether that class exhausts physically localreal-event mechanics.PV-REC answers in the negative by constructing a distinct,time-symmetric boundary-event category with its own ontology,geometry, dynamics, actualization law, and falsifiable non-equilibriumpredictions.
Phenomenological--Velocity Relational Event Completion((\PV\text{-}\REC))is formulated as a non-Hilbert, time-symmetric,source--absorber boundary-event mechanics.Its native ontology contains no wavefunction,no Hilbert-space state postulate,no unitary state evolution,no Born-rule actualization,and no collapse law.The theory instead uses real spin-boundary data,positive source--absorber compatibilities,Lorentz-scaled proper boundary-clock event barriers,local flux-gauge completion,and a constant-flux threshold race that actualizes one definite event. The preparation-side ontic record[\lambda_0=(v,\epsilon,\mathcal J,\Xi)]is intervention-independent of the later local settings:[P!\left( \lambda_0 \mid \operatorname{do}(a), \operatorname{do}(b)\right)=P(\lambda_0).]It is not postulated to be a complete Bell screening state.The completed event is the four-dimensional boundary object[\Lambda^{\mathrm{comp}}{ab}=\bigl( \lambda_0, \mathcal B_A(a), \mathcal B_B(b)\bigr).]Thus PV-REC distinguishes preparation independence fromBell-Cauchy completeness:the complete event need not be encoded by a setting-independent variableon a hypersurface preceding the absorber boundary conditions. The central contention with the universalized reading of Bell's theorem istherefore precise.PV-REC accepts[\BellCauchyScreen\LongrightarrowS\le2.]It rejects the stronger inference[\BoundaryLocal\Longrightarrow\BellCauchyScreen.]Boundary-event locality is defined positively through:preparation intervention independence,region-local detector records,a source-star transaction graph with no primitive directed(A\to B) or (B\to A) edge,all-at-once source--absorber completion,Lorentz-covariant boundary laws,and accessible operational no-signalling.Bell screening is an additional past-Markov or Cauchy-screen condition,not a consequence of those boundary-locality principles. The balanced two-spin compatibility is derived natively in real boundarygeometry.Let (n,m\in S^2\subset\R^3) be analyzer directions and(x,y\in{\pm1}) local boundary-channel labels.Local balance,joint rotational covariance,separate affinity in the signed boundary vectors (xn) and (ym),neutrality of an unpolarized boundary,and source anti-alignment uniquely give[m{xy}(n,m)=\frac14\bigl(1-xy,n\cdot m\bigr).]No Hilbert-space state or amplitude is used in this derivation. The native interaction coordinate is the log cross-ratio[\Theta(m)=\log\frac{m_{++}m_{--}} {m_{+-}m_{-+}}.]For every strictly positive balanced binary table,[E=\tanh!\left(\frac{\Theta}{4}\right).]PV acts as the projective homothety[\Theta\longmapsto\Theta_\kappa=\kappa\Theta,\qquad\kappa(v)=\sqrt{1-\frac{v^2}{c^2}}.]The unique positive entrywise transformation that scales everybinary interaction cross-ratio by (\kappa) is, up to a commonnormalization factor,[m\longmapsto m^\kappa.]Thus the power law is characterized intrinsically by positive-coneprojective geometry rather than imported from quantum mechanics. The same homothety is realized physically by proper boundary-clockevent-action accumulation.PV event action is a positive capacity-counting functional on microscopicboundary channels, not the ordinary Lorentz-invariant bulk field action.If (n^\mu) is the apparatus-boundary four-velocity and (u^\mu) theboundary-microstructure four-velocity, then[\gamma_{\mathrm{rel}}=-\frac{n_\mu u^\mu}{c^2},\qquad\kappa=\gamma_{\mathrm{rel}}^{-1},\qquad\dd\tau_u=\kappa,\dd t_n.]For a uniform sector this gives[I_{xy}^{\mathrm{proper}}=\kappa I_{xy},\qquadI_{xy}=-\log m_{xy}.]A continuous nondegenerate memoryless proper-capacity law is exponential;equilibrium calibration at (\kappa=1) fixes its rate uniquely, yielding[\chi_{xy}^{\PV}=e^{-I_{xy}^{\mathrm{proper}}}=m_{xy}^{\kappa}.] Beyond the balanced sector,local boundary continuity fixes prescribed row and column fluxes.The unique positive table in the diagonal orbit of the raw PVsusceptibility with those fluxes is its Sinkhorn flux-gauge completion:[Q_{xy}=\alpha_x\beta_y m_{xy}^{\kappa}.]The local multipliers change only local flux potentials.They preserve the interaction cross-ratio:[\Theta(Q)=\kappa\Theta(m).]No-signalling completion is therefore a local gauge fixing of boundaryfluxes, not an alteration of the PV interaction invariant. The physical hazards are[\widehat\Gamma_{xy}=\Gamma_{\mathrm{trial}}Q_{xy}.]Independent exponential race seeds define[T_{xy}=\frac{\xi_{xy}}{\widehat\Gamma_{xy}},\qquad(X,Y)=\argmin_{x,y}T_{xy}.]One definite event pair is actualized almost surely.The total hazard is constant,local row and column hazards are fixed by local flux conservation,and accessible local outcome, count-rate, and timing distributions areindependent of the remote setting.The native mechanism therefore satisfies[\PrepIndependent,\qquad\BoundaryLocal,\qquad\RealEvent,\qquad\NoSignalAcc,]while failing[\BellCauchyScreen.] In the balanced sector,[P_{\PV}(x,y\mid a,b,v)=\frac{ m_{xy}(a,b)^{\kappa(v)}}{ \sum_{x',y'} m_{x'y'}(a,b)^{\kappa(v)}},]and[E_{\PV}(a,b\mid v)=-\tanh!\left[ \kappa(v) \artanh(\cos(a-b))\right].]At (v=0), (\kappa=1), and the ordinary singlet curve is recovered.For every fixed (v\neq0), the law is nonlinear in(n\cdot m) and differs from standard two-spin quantum mechanics. The native same-source event map[(X,Y)=K_{ab}(\lambda_0)]has a positive-probability replay defect:at fixed (b,\lambda_0), Bob's actualized outcome can change under acounterfactual change of Alice's absorber setting, and conversely.Hence the context-indexed native variables do not glue into localcounterfactual columns[A_0,A_1,B_0,B_1.]This is a global-section obstruction, not a rejection of ordinaryprobability.PV-REC uses positive Kolmogorov probability for race seeds and realizedcontexts; it rejects the Bell-local overlap identities required to turncontext-indexed events into a setting-independent past-screened sheet. The companion selection theory identifies setting-dependent selection asa translation on the same interaction-rapidity line:[\artanh E_{\mathrm{obs}}=\artanh E_{\mathrm{full}}+\xi,\qquad\xi=\frac14\log A,]where (A) is the acceptance cross-ratio.PV is instead a dilation:[\artanh[-E_{\PV}]=\kappa\artanh(\cos\Delta).]Every local-hidden-variable correlation family casts a chainedsubadditivity shadow on the agreement defect[d=\frac{1+E}{2}.]The PV defect is[d_\kappa(\Delta)=\frac{\tan^{2\kappa}(\Delta/2)} {1+\tan^{2\kappa}(\Delta/2)}\sim\left(\frac{\Delta}{2}\right)^{2\kappa}.]A bounded rapidity translation can rescale a defect but cannot change itssmall-angle H"older exponent.Consequently, for[\kappa>\frac12,]no finite-budget Bell-local selection model reproduces the full PV familyon a punctured neighborhood of coincidence.Matching it down to angular resolution (\Delta) requires[\Xi\ge\frac{2\kappa-1}{4}\log\frac1\Delta-C,]or an acceptance cross-ratio discrimination diverging at least as[\max(A,A^{-1})\gtrsim\Delta^{-(2\kappa-1)}.]Thus PV-REC is not a hidden post-selection construction:its native selection audit is trivial,[\gamma_{\PV}\equiv1,]while any external Bell-local selection counterfeit of thesuper-threshold continuum law incurs a divergent resolution cost. The optimized angular CHSH threshold is exactly[\kappa>\frac12,\qquad\frac{|v|}{c}<\frac{\sqrt3}{2}.]At (\kappa=\tfrac12),the agreement-defect exponent reaches the metric-additivity boundary,the optimized angular CHSH family first saturates,and the shape law becomes the unique nontrivial quadratic sector:[f_{1/2}(z)=\tanh!\left(\frac12\artanh z\right)=\frac{1-\sqrt{1-z^2}}{z}.] The positive-cone geometry supplies an additional exact identity.The Hilbert-projective diameter of the powered balanced table is[\Delta_{\mathrm H}=4\kappa\artanh|z|,]and its optimal Birkhoff contraction coefficient is[\tanh!\left(\frac{\Delta_{\mathrm H}}{4}\right)=|E_\kappa(z)|.]Here ``Hilbert'' denotes Hilbert's projective cross-ratio metric on apositive cone, not a quantum Hilbert space. Finally, the theory is separated explicitly from standard two-spinquantum mechanics.Any fixed two-qubit state measured by ordinary projective spin observableshas a bilinear correlation[E(n,m)=n^{\mathsf T}Tm.]Joint rotational covariance forces (T=tI), and perfect same-axisanti-correlation forces (t=-1), hence[E(n,m)=-n\cdot m.]Therefore no fixed (v\neq0) PV sector can be represented by a fixedrotationally invariant two-qubit state with standard projective spinmeasurements.The (v=0) sector is the unique equilibrium point at which the standardtwo-spin quantum curve and the PV-REC curve coincide. PV-REC is accordingly classified as[\boxed{\PV\text{-}\REC_{\TS}\in\NonHilbert\cap\BoundaryComplete\cap\BoundaryLocal\cap\PrepIndependent\cap\RealEvent\cap\NoSignalAcc\cap\neg\BellCauchyScreen.}]Bell's inequality remains valid within the Bell-screenable class.The point of contention is whether that class exhausts physically localreal-event mechanics.PV-REC answers in the negative by constructing a distinct,time-symmetric boundary-event category with its own ontology,geometry, dynamics, actualization law, and falsifiable non-equilibriumpredictions.
Phenomenological--Velocity Relational Event Completion (\(\PV\text{-}\REC\)) is formulated here as a time-symmetric source--absorber boundary-event proposal. The preparation-side source record λ0=(v,ϵ,J,Ξ) is assumed to be statistically independent of the later detector settings. It is not, however, identified with the complete Bell screening variable. The completed four-dimensional event includes the later local absorber boundary conditions, Λabcomp=(λ0,BA(a),BB(b)). Thus the proposal retains preparation independence but rejects emission-time scalar screening-off. In the balanced singlet sector, the source--absorber compatibility of the candidate joint event (x,y) is the usual singlet projector weight mxy(a,b)=41(1−xycos(a−b)). PV-REC treats this as a linear boundary compatibility rather than directly as the final event probability. With Ixy=−logmxy,κ(v)=1−c2v2, the PV rule rescales the event-action barrier by Ixy↦κ(v)Ixy, giving raw susceptibility mxyκ(v). To prevent setting-dependent total count-rate or latency signals, the physical hazards are normalized to constant total flux, \[ \widehat\Gamma_{xy}^{\PV} = \Gamma_{\mathrm{trial}} \frac{m_{xy}^{\kappa(v)}}{\sum_{x',y'}m_{x'y'}^{\kappa(v)}}. \] With independent exponential race seeds this gives \[ P_{\PV}(x,y\mid a,b,v) = \frac{m_{xy}(a,b)^{\kappa(v)}} {\sum_{x',y'}m_{x'y'}(a,b)^{\kappa(v)}}. \] Consequently, \[ E_{\PV}(a,b\mid v) = -\tanh\!\left[ \kappa(v)\artanh(\cos(a-b)) \right]. \] At v=0, the ordinary singlet law E=−cos(a−b) is recovered; fixed sectors with v=0 predict a distinct non-Born angular deformation. The construction is not Bell-local in the factorization/local-causality sense. Same-source replay generally fails: (X,Y)=Kab(λ0) does not factor into separated scalar response functions X=XA(a,λ0),Y=YB(b,λ0). Thus the model is \emph{Bell-nonseparable}. In the standard Bell vocabulary this is Bell-nonlocality in the technical factorization sense. In the present ontology it is more accurately described as global boundary-contextuality: the completed event depends on the full source--absorber boundary context, not on a directed controllable signal from one wing to the other. Operational no-signalling is retained. Beyond the balanced singlet sector, no-signalling is imposed by marginal-preserving boundary flux balancing, implemented for finite bipartite outcome spaces by Sinkhorn scaling and, more generally, by entropic projection onto the no-signalling constraint set. The proposal is phenomenological. It is not claimed here that standard quantum field theory, the Born rule, or the singlet compatibility kernel has been derived from the direct-action embedding. The value of the construction is instead that it supplies a precise time-symmetric, Bell-nonseparable, operationally no-signalling event law with explicit empirical consequences. A visibility-independent diagnostic is provided by the shape function \[ f_\kappa(z)=\tanh(\kappa\artanh z), \] for which, in a fixed-v sector, fκ′(0)fκ′′′(0)=2(1−κ2)=2v2/c2. Dense angular correlation data can therefore either bound v close to zero or reveal a deviation from the ordinary singlet shape.
This is the data set for the published, peer reviewed paper: https://ijqf.org/archives/8027 "Phenomenological Velocity and Bell–CHSH: Exceptional-Locus Semantics, Selection Simulations of − cos, and a Microcausal Realization:" https://ijqf.org/wp-content/uploads/2026/02/IJQF2026v12n2p10.pdf Bell--CHSH is often read as a universal prohibition on \emph{local hidden variables}. Mathematically, however, the CHSH bound S≤2 is a theorem about a particular hypothesis class: a \emph{single} Kolmogorov probability space carrying all counterfactual outcomes, together with measurement independence, Bell locality (factorized response functions), and bounded outputs. On that class the proof is pointwise and unconditional, so no algebraic reformulation of a parameter (including a ``phenomenological velocity'' (PV)) can yield Tsirelson-scale singlet correlations E(a,b)=−cos(a−b) on a Tsirelson quartet. We then make precise two distinct ways in which PV can remain meaningful as a \emph{local} and in that sense ``hidden'' parameter while coexisting with Tsirelson-scale correlations, without contradicting Bell's theorem. First, we formalize ``exceptional-locus/defined-only'' conventions as setting-dependent acceptance (post-selection): reported correlators are conditional expectations under a setting-indexed family of accepted laws {νab}. Within this rung-1 semantics we give an explicit Bell-local deterministic base model whose accepted-sample joint law exactly matches the singlet law (unbiased marginals and correlator −cos(a−b)) for all a,b, via an explicit acceptance rule γ(a,b,λ)∈[0,1]. Second, we give a PV-indexed microcausal (operator-algebraic) realization: PV parameterizes a canonical SU(2) unitary that conjugates local observables inside commuting Alice/Bob subalgebras, and in the singlet state the unconditional correlator equals −cos(a−b) and attains Tsirelson's value 22. The results clarify a semantic and structural trichotomy: any ``PV-to-Tsirelson'' account must (i) change the effective ensemble by selection/context, or (ii) leave the Kolmogorov (commutative) model class (e.g.\ via noncommutativity or quasi-probability), or (iii) relax a Bell premise such as measurement independence or Bell locality. In this sense, Bell--CHSH constrains a specific formalization of hidden variables rather than ownership of the terms ``local'', ``real'', or ``hidden'' in physics discourse.
A Kernel-Role Determinacy Theorem for Bell-Witness Fixation, No-Signaling, and Tensor-Level Compatibility This manuscript develops a fixed-domain AASC analysis of Bell-nonlocal quantum entanglement as a theorem of reusable witness determinacy. It does not introduce hidden variables, modify quantum mechanics, derive Hilbert-space formalism from below, or propose a new signaling mechanism. Instead, it asks a prior structural question: once a Bell-type operational family is used as the same non-degenerate witness across settings, lawful transformations, retained records, comparison, and later classification, what admissible same-domain standing form is forced? The governing claim is that AASC is not an optional interpretive overlay on physics. Once an operational Bell family is asserted as a determinate reusable witness, admissibility, standing, reference, and irreversibility are already active as minimal necessity conditions of same-witness objecthood. Target instantiation therefore carries governance occupancy: an alleged determinate Bell witness with no admissible governance realization is not an empty alternative, but a refusal or retyping of the target. The manuscript defines finite operational Bell objects with explicit setting carriers, outcome families, channel/interface data, standing-token valuations, continuation and report/reuse structure, comparison profiles, and act-time status. It then defines object-indexed query sorts, dependent response families, exactly recoverable operational simulations, lawful Bell groupoids, six governance-effect functors, and target-active extensional governance equivalence. The result is a typed interface in which arbitrary native formalisms can be compared by their target-bearing governance effects rather than by their internal syntax. The central Bell-specific theorem states that every same-target candidate-admissible Bell-governance construction is forced component by component into the certified boundary-compatible standing form on the complete target-active practice. Candidate admissibility does not assume the canonical form at the outset. Instead, target occupancy, lawful transport, standing-flow exclusion, compatibility certification, coordinate-deviation exhaustion, and componentwise closure force exact target-bearing realization and governance equivalence to the canonical representative CP∗C^{*}_{P}CP∗. The target-induced governance quotient therefore has exactly one element. The Bell carrier is treated as a compatible empirical model on a finite measurement cover. Bell-local completion is identified with the existence of a global section, and Bell nonlocality is the obstruction to such a section. The manuscript separates operational marginal invariance, absence of controllable communication, and exclusion of same-domain standing transport. No-signaling is therefore preserved without retyping Bell nonlocality as superluminal communication, metric transport, distant repair, or hidden standing transfer. The finite-dimensional quantum specialization uses the standard bipartite tensor-product and local-POVM carrier. Every separable realization in that carrier is Bell-local; therefore, if the fixed Bell-nonlocal empirical model has such a finite-dimensional quantum realization, every realizing state in that carrier is entangled. Post-quantum Bell-nonlocal no-signaling models remain within the general witness theorem but outside the quantum-entanglement corollary. The accompanying Lean 4 development, release v0.8.0, directly verifies the finite Bell-side witness carrier: denominator-32 vector/Born weights, structural no-signaling, CHSH numerator 808080, all sixteen deterministic local extrema, finite denominator-32 Bell-local-envelope exclusion, and conditional endpoint routing. Its direct witness certificate is vectorBorn_flagship_spine, and its deepest released endpoint is reasonableDeepestFiniteEntanglement_spine. The manuscript-level typed governance interface, empirical lawful groupoid, compatibility equivalence, coordinate-deviation exhaustion, completed role collapse, candidate-to-canonical derivation, and finite-dimensional quantum specialization are not attributed to the released Lean layer. This paper is the Bell-witness member of the paired AASC quantum determinacy arc. The record-side manuscript, Admissible Record Construction in Physically Non-Selective Quantum Regimes, fixes when local physical record occurrences stand as the same evidential record objects. The present manuscript fixes when the Bell operational family sampled by such records stands as the same reusable witness. The paired endpoint is therefore: record-role fixation+Bell-witness compatibility.\text{record-role fixation} + \text{Bell-witness compatibility}.record-role fixation+Bell-witness compatibility. The manuscript is classificatory and eliminative rather than mechanistic. It does not claim that AASC generates the Bell probabilities or supplies a hidden physical influence. Its claim is that reusable Bell-witness determinacy strongly constrains how Bell-nonlocal correlations may be coherently classified on a fixed domain: not as independently complete local states later connected by a mechanism, not as Bell-local hidden-variable value assignments, not as superluminal signaling, and not as same-act repair, but as boundary-level compatibility carried by one standing-preserving Bell witness.
This paper develops a reconstruction of probability that does not begin by assuming a probability space, a probability simplex, or the Born rule. Instead, it asks a prior foundational question: under what physical and mathematical conditions are a set of alternatives entitled to carry positive, additive probabilities at all? The construction begins with lawful histories, future interfaces, and retained records. It distinguishes coherent alternatives from exclusive empirical events, preventing multiple interfering routes from being mistaken for separate probabilistic outcomes. A future-sufficient quotient identifies distinctions that no admitted future test can detect, while a record-qualification step determines when alternatives have become genuine empirical events. Dagger-compatible comparison and positive record factorization then produce a positive comparison kernel, from which normalized event probabilities arise only after the relevant record sectors are certified as exclusive. The framework is extended beyond individual measurements. Compatible finite record refinements generate a global probability measure under explicit realizability conditions; stochastic kernels provide a compositional calculus for carrying probabilistic information through unresolved future structure; and locally classical measurement contexts are assembled into a single positive state on a generally non-Boolean effect structure. This allows globally coherent probability to exist even when no context-independent deterministic assignment of outcomes exists. On a reflection-positive Hilbert-space branch, admissible future contexts are represented by positive effects and positive normalized states by density operators. The Born state-effect law then appears as the finite-dimensional representation of a probability structure that has already been earned through record qualification, positivity, exclusivity, refinement, and contextual consistency rather than introduced as a primitive postulate. The paper also separates two mechanisms that are often conflated: quadratic Born-type weighting arises from dagger self-comparison, whereas equal probabilities for symmetric alternatives arise from symmetry of the outcome structure. A further state-effect bridge analysis classifies residual coordinate freedom and identifies an exceptional rank-two adjugate transformation with Lorentzian signature, which is removed when positive self-dual identification is required. The central interpretation is that probability can function as a lawful form of global completion: it can carry future-relevant law through unresolved structure even when deterministic uniqueness is unavailable. The paper does not derive a mechanism selecting one actual outcome and does not claim that every physical sector satisfies the stated hypotheses. Instead, it provides an explicit dependency-controlled architecture connecting lawful completion, reflection positivity, event qualification, contextuality, stochastic continuation, Hilbert reconstruction, and the Born rule.
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.
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