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PV-REC as a Time-Symmetric Boundary-Event Mechanics

Parker Emmerson

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

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 \[ \lambda_0=(v,\epsilon,\mathcal J,\Xi) \] 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, \[ \Lambda_{ab}^{\mathrm{comp}} = (\lambda_0,\mathcal B_A(a),\mathcal B_B(b)). \] 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 \[ m_{xy}(a,b)=\frac14\left(1-xy\cos(a-b)\right). \] PV-REC treats this as a linear boundary compatibility rather than directly as the final event probability. With \[ I_{xy}=-\log m_{xy}, \qquad \kappa(v)=\sqrt{1-\frac{v^2}{c^2}}, \] the PV rule rescales the event-action barrier by \[ I_{xy}\mapsto\kappa(v)I_{xy}, \] giving raw susceptibility \(m_{xy}^{\kappa(v)}\). 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 \(v\neq0\) predict a distinct non-Born angular deformation. The construction is not Bell-local in the factorization/local-causality sense. Same-source replay generally fails: \[ (X,Y)=K_{ab}(\lambda_0) \] does not factor into separated scalar response functions \[ X=X_A(a,\lambda_0), \qquad Y=Y_B(b,\lambda_0). \] 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, \[ \frac{f_\kappa'''(0)}{f_\kappa'(0)} = 2(1-\kappa^2) = 2v^2/c^2. \] Dense angular correlation data can therefore either bound \(v\) close to zero or reveal a deviation from the ordinary singlet shape.

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Phenomenlogical Velocity Relational Event Completion - PV-REC as a Time-Symmetric Boundary-Event Mechanics

Parker Emmerson

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.

•Aug 6, 2026•0 citations

Artian's Completed-Event Bell Reconstruction: Real Source Geometry, Access, and Durable Records

Attar Ali

From a finite event record to Bell correlations and their loss under recording A pair is prepared together, read at two separated instruments, and allowed to leave a material record. This paper builds the correlation law from a real source geometry in Artian's Quantum Traction Theory. Its source-to-access equation is \[\boxed{p_K(r,s\mid\mathbf a,\mathbf b)=\frac{C\!\left[(\Pi_r(\mathbf a)\otimes\Pi_s(\mathbf b)\otimes I)K\Omega\right]}{C(K\Omega)}}\] Here C is an additive, basis-invariant event capacity, Omega is the normalized orientation-neutral two-channel source, K is a specified record contact, and the projectors select the two instrument outputs. A real quarter-turn and a qualified shared-reference composition generate the readout algebra. Capacity symmetry fixes its quadratic weight. A separately stated capacity-to-frequency condition connects that weight to laboratory counts. The correlation follows before the Bell comparison For a phase-compensated record contact with overlap g between its two normalized record profiles, the construction gives \[E_K(\mathbf a,\mathbf b)=-a_zb_z-g(a_xb_x+a_yb_y),\qquad 0\leq g\leq1.\] The intact-source limit is E = -a dot b. The same calculation yields the fixed-setting and optimized Bell responses: \[\boxed{S_{45}=\sqrt{2}(1+g),\qquad S_{\mathrm{opt}}=2\sqrt{1+g^2}.}\] Fresh independent record contacts multiply their overlaps. The paper derives the repeated-contact law, a capacity-based overlap calibration, uncertainty bounds, and a protocol that separates calibration from the Bell observations being judged. A source-first account with an explicit dependency map A1 supplies event-time labels and the clock-angle correspondence; A4 supplies the real dial; A5-X supplies completed-event support, with its inherited A6/A7 conditions. The two-channel preparation, balanced composition, capacity symmetry, frequency bridge, and record coupling are printed separately. Each theorem states exactly which conditions it uses. The Access uncertainty commutator is treated as a distinct inherited algebraic premise, rather than inferred from Bell correlations. When a receiver follows the source Version 2.0 constructs a matched receiver rather than assigning its transformation by name. Preserving every probability of the stated bilinear contact forces its transport to be the conjugate of the source transport, up to an overall phase: \[\boxed{|r^{\mathrm T}V^{\mathrm T}Us|^2=|r^{\mathrm T}s|^2\ \text{for all }r,s\quad\Longrightarrow\quad V=e^{i\alpha}\overline U.}\] A finite set of basis and sum probes certifies this relation. A robust extension converts imperfect probe probabilities into an operator-error bound. Finite clock and shift operations then produce the comparison loop \[\boxed{\mathcal L=M+\omega(I-M),\qquad \omega=e^{2\pi i/d}.}\] The matched sector cancels the comparison phase while the unmatched sector retains it. This is a joint comparison law; each local readout can remain incompatible. The paper identifies the binary loop's ambiguity, provides additional probes that distinguish it, and transfers the Bell correlations into a complete detector record when the original carrier is consumed. One contact, two observable roles A specified marker contact supplies both the Bell record overlap and a resolved weak-measurement instrument: \[\begin{aligned}\gamma&=\sin\theta,\\K_0&=\operatorname{diag}(\cos(\theta/2),\sin(\theta/2)),\\K_1&=\operatorname{diag}(\sin(\theta/2),\cos(\theta/2)).\end{aligned}\] An independent replay executes this instrument on all 40,032 tree records in the public Feng and colleagues archive, reproducing every stored decoder success/failure score. The main five-setting batch contains 32,000 records; the complete archive has six settings. The reproduced job-cluster residual probabilities are p = 0.0188753 and p = 0.0442603, respectively. These diagnose modest tension with the ideal instrument under the stated retrospective covariance. The matched QTT and quantum representations make the same archival predictions, so this replay is not a significance test between them. The package preserves the complete source-first Bell construction and adds finite receiver certificates, physical transport-error bounds, a common-contact derivation, five vector figures, a twelve-item achievements ledger, and offline reproducibility materials. The next operational question is precise: which independently qualified physical contact fixes both the record overlap and the receiver comparison response? Dependencies: QTT Main Book; finite address-capacity counting; real source histories and composition; A1-CHSH spinor character; Observation as Access.

•Sep 25, 2026•2 citations

Bell-Nonlocal Quantum Entanglement as Boundary-Level Compatibility

Amos Jay Maley Maley

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. 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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.

•Apr 3, 2026•0 citations

Phenomenological Velocity and Bell–CHSH: Exceptional-Locus Semantics, Selection Simulations of − cos, and a Microcausal Realization

Parker Emmerson

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 $S\le 2$ 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 $\{\nu_{ab}\}$. 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 $\gamma(a,b,\lambda)\in[0,1]$. 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 $2\sqrt2$. 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.

•Feb 22, 2026•0 citations

Fields and Waves from Completed Events: A QTT Source-to-Observation Reconstruction

Attar Ali

How finite source updates become interference, field equations and detector records \[ \boxed{\begin{gathered} s_{n+1}=U_A s_n,\qquad U_A^T U_A=I,\\ p_y(n)=\|K_{y,R}U_A^n s_0\|^2,\\ \sum_y K_{y,R}^T K_{y,R}=I. \end{gathered}} \] Localized completed-event transport reaches the receiver The source exponential now has a derived map into the local two-tick phase history. In the qualified free-source and local-history receiver class: \[ \begin{gathered} \delta\Phi_R(N)=B_R D_A U_A^N\delta s,\\ D_AU_A^N=L_A^N D_A,\\ \operatorname{supp}(D_A\delta s)\subseteq A,\\ d_{\rm addr}(A,R)>N\\ \Longrightarrow\quad \delta\Phi_R(N)=0,\\ p_y^+(N)=p_y^-(N). \end{gathered} \] Here D_A is derived from the inherited recurrence, L_A is its radius-one history update, and B_R reads a declared address window. The complete instrument includes missed events. The paper supplies an exact one-rail example with probabilities one and zero inside the preparation region, and a leakage bound for imperfect localization. It also proves the infinite multidimensional compact-band obstruction and retains the nonlocal canonical-receiver counterexample. This is a scoped causal readout theorem, not universal interacting-field microcausality or a new experimental signal. \[ \operatorname{TV}(p_R^+,p_R^-)\le \min\{1,2\kappa_{R,N}\epsilon_{R,N}\}. \] The preparation leakage and finite response norm must be independently fixed; they cannot be fitted to absorb detector results. All inherited scientific content remains, with 2,161 further deterministic checks and 16 adversarial mutations in a third reproducibility module. The QTT construction written explicitly The admitted free-source law can be written directly as counted real-phase evolution followed by physical receiver access: \[ \begin{gathered} \Phi_R(N\widetilde t_A)=C_R\mathcal U_A(N)s_0,\\ \mathcal U_A(N)=\exp[-2NJ\arcsin\sqrt{Q_A}],\\ Q_A=Q\oplus Q,\qquad 0\le Q\le I,\\ Q=(-\Delta_A/4)|_{\mathcal H_{A6}},\\ J^2=-I,\quad J^T=-J,\quad[J,Q_A]=0,\\ N\in\mathbb N_0. \end{gathered} \] The paper proves exact equivalence to its inherited finite block propagator at every nonnegative integer source tick, including both spectral endpoints. Q is the address operator restricted to the A6-admitted band; its direct sum acts on both real dial components. The receiver and completed-event source retain their physical qualification conditions. The construction is exact in the stated coherent massless free-packet sector, not an assertion that every interacting field has this free law. QTT constructs wave and field behaviour from a finite source: real phase dials, admitted updates and completed receiver records. The paper carries that source through an exact two-path interference calculation, normalized sequential instruments and a quantitative detector-error bound. A smooth continuum field is not an additional starting substance in this construction. When does a field image contain enough information? A projected field has an autonomous linear update exactly when the source information it omits cannot feed back into view: \[ \begin{gathered} C_RU_A=V_RC_R\\ \Longleftrightarrow\\ U_A\ker C_R\subseteq\ker C_R. \end{gathered} \] Two worked sources have the same present retained image and opposite future output ports. The paper then constructs finite electromagnetic Gauss, Faraday and work balances, connects the inherited exact wave propagator to its smooth limit, and gives an A2 gravitational source reading. The reconstruction states its source metric, transport, instrument and count-completion premises explicitly. It distinguishes finite addresses from finitely many possible phase states, and physical source selection from mathematical representation. The ideal matched quantum calculation also requires no outcome fitting. The reconstruction archive includes full source, the scientific illustration, the inherited numerical tables and 1,109 deterministic checks with 15 mutation checks, plus 664 additional explicit-construction checks and 10 mutations. These are reproducible mathematical fixtures, not new experimental observations. The fixed finite-wave residual retains its separate laboratory-access and sensitivity requirements. Stable paper DOI · Wave Dynamics · Completed-Event Transport · Main Book · Corpus Tree

•Oct 20, 2026•0 citations

The Bell Calculation: Singlet Correlations from Joint Readouts of One Complement Unit (Note v1 to "Spacetime as Register, Not Stage")

Władysław Kurowski

A supplementary calculational note to the interpretational program "Spacetime as Register, Not Stage". The note reconstructs the full structure of spin-1/2 singlet correlations — the joint distribution P = 1/4(1−αβ·a·b), the correlation E(a,b) = −a·b, and CHSH S = 2√2 — from the program's picture of the "entangled" pair as one indivisible complement unit read out locally at two sites by independently oriented detector settings. The note proves sufficiency, not derivation, and keeps an explicit ledger of its epistemic status: two rules are added and underived (a half-angle spinor readout structure, and the square law for amplitudes), two are derived conditionally (antisymmetry, relative to an explicit phase bridge; normalization, fixed by completeness with no free parameter), and every remaining gap is recorded in a debt register. The model exits the factorizable (Bell) class exactly where quantum mechanics does: conditioned on the full preparation state, local marginals remain 1/2 (parameter independence holds per preparation state), while outcome independence fails — with no superluminal transport claim derived from statistics, no superdeterminism (the preparation state carries no free parameter that could co-vary with later settings), and operational no-signaling verified numerically. A deterministic hidden-orientation contrast model is included, reproducing the linear correlation −1+2θ/π and S ≤ 2, to show that the cosine requires a genuine exit from the factorizable class rather than a change of vocabulary. The English text is a full translation of the author's Polish source version (both files included; frozen 28.07.2026, changes tracked in the program's correction register, Series 33).

•Jul 28, 2026•3 citations
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 [ \lambda_0=(v,\epsilon,\mathcal J,\Xi) ] 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, [ \Lambda_{ab}^{\mathrm{comp}} = (\lambda_0,\mathcal B_A(a),\mathcal B_B(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 [ m_{xy}(a,b)=\frac14\left(1-xy\cos(a-b)\right). ] PV-REC treats this as a linear boundary compatibility rather than directly as the final event probability. With [ I_{xy}=-\log m_{xy}, \qquad \kappa(v)=\sqrt{1-\frac{v^2}{c^2}}, ] the PV rule rescales the event-action barrier by [ I_{xy}\mapsto\kappa(v)I_{xy}, ] giving raw susceptibility (m_{xy}^{\kappa(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\neq0) 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)=K_{ab}(\lambda_0) ] does not factor into separated scalar response functions [ X=X_A(a,\lambda_0), \qquad Y=Y_B(b,\lambda_0). ] 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, [ \frac{f_\kappa'''(0)}{f_\kappa'(0)} = 2(1-\kappa^2) = 2v^2/c^2. ] Dense angular correlation data can therefore either bound (v) close to zero or reveal a deviation from the ordinary singlet shape.
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 recordλ0=(v,ϵ,J,Ξ)\lambda_0=(v,\epsilon,\mathcal J,\Xi)λ0​=(v,ϵ,J,Ξ)is intervention-independent of the later local settings:P ⁣(λ0∣do⁡(a),do⁡(b))=P(λ0).P\!\left( \lambda_0 \mid \operatorname{do}(a), \operatorname{do}(b)\right)=P(\lambda_0).P(λ0​∣do(a),do(b))=P(λ0​).It is not postulated to be a complete Bell screening state.The completed event is the four-dimensional boundary objectΛabcomp=(λ0,BA(a),BB(b)).\Lambda^{\mathrm{comp}}_{ab}=\bigl( \lambda_0, \mathcal B_A(a), \mathcal B_B(b)\bigr).Λabcomp​=(λ0​,BA​(a),BB​(b)).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 directedA→BA\to BA→B or B→AB\to AB→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∈S2⊂R3n,m\in S^2\subset\R^3n,m∈S2⊂R3 be analyzer directions andx,y∈{±1}x,y\in\{\pm1\}x,y∈{±1} local boundary-channel labels.Local balance,joint rotational covariance,separate affinity in the signed boundary vectors xnxnxn and ymymym,neutrality of an unpolarized boundary,and source anti-alignment uniquely givemxy(n,m)=14(1−xy n⋅m).m_{xy}(n,m)=\frac14\bigl(1-xy\,n\cdot m\bigr).mxy​(n,m)=41​(1−xyn⋅m).No Hilbert-space state or amplitude is used in this derivation. The native interaction coordinate is the log cross-ratioΘ(m)=log⁡m++m−−m+−m−+.\Theta(m)=\log\frac{m_{++}m_{--}} {m_{+-}m_{-+}}.Θ(m)=logm+−​m−+​m++​m−−​​.For every strictly positive balanced binary table,E=tanh⁡ ⁣(Θ4).E=\tanh\!\left(\frac{\Theta}{4}\right).E=tanh(4Θ​).PV acts as the projective homothetyΘ⟼Θκ=κΘ,κ(v)=1−v2c2.\Theta\longmapsto\Theta_\kappa=\kappa\Theta,\qquad\kappa(v)=\sqrt{1-\frac{v^2}{c^2}}.Θ⟼Θκ​=κΘ,κ(v)=1−c2v2​​.The unique positive entrywise transformation that scales everybinary interaction cross-ratio by κ\kappaκ is, up to a commonnormalization factor,m⟼mκ.m\longmapsto m^\kappa.m⟼mκ.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 nμn^\munμ is the apparatus-boundary four-velocity and uμu^\muuμ 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 κ=1\kappa=1κ=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:Qxy=αxβymxyκ.Q_{xy}=\alpha_x\beta_y m_{xy}^{\kappa}.Qxy​=αx​βy​mxyκ​.The local multipliers change only local flux potentials.They preserve the interaction cross-ratio:Θ(Q)=κΘ(m).\Theta(Q)=\kappa\Theta(m).Θ(Q)=κΘ(m).No-signalling completion is therefore a local gauge fixing of boundaryfluxes, not an alteration of the PV interaction invariant. The physical hazards areΓ^xy=ΓtrialQxy.\widehat\Gamma_{xy}=\Gamma_{\mathrm{trial}}Q_{xy}.Γxy​=Γtrial​Qxy​.Independent exponential race seeds defineTxy=ξxyΓ^xy,(X,Y)=arg min⁡x,yTxy.T_{xy}=\frac{\xi_{xy}}{\widehat\Gamma_{xy}},\qquad(X,Y)=\argmin_{x,y}T_{xy}.Txy​=Γxy​ξxy​​,(X,Y)=argminx,y​Txy​.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=0v=0v=0, κ=1\kappa=1κ=1, and the ordinary singlet curve is recovered.For every fixed v≠0v\neq0v=0, the law is nonlinear inn⋅mn\cdot mn⋅m and differs from standard two-spin quantum mechanics. The native same-source event map(X,Y)=Kab(λ0)(X,Y)=K_{ab}(\lambda_0)(X,Y)=Kab​(λ0​)has a positive-probability replay defect:at fixed b,λ0b,\lambda_0b,λ0​, 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 columnsA0,A1,B0,B1.A_0,A_1,B_0,B_1.A0​,A1​,B0​,B1​.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 AAA 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 defectd=1+E2.d=\frac{1+E}{2}.d=21+E​.The PV defect isdκ(Δ)=tan⁡2κ(Δ/2)1+tan⁡2κ(Δ/2)∼(Δ2)2κ.d_\kappa(\Delta)=\frac{\tan^{2\kappa}(\Delta/2)} {1+\tan^{2\kappa}(\Delta/2)}\sim\left(\frac{\Delta}{2}\right)^{2\kappa}.dκ​(Δ)=1+tan2κ(Δ/2)tan2κ(Δ/2)​∼(2Δ​)2κ.A bounded rapidity translation can rescale a defect but cannot change itssmall-angle H\"older exponent.Consequently, forκ>12,\kappa>\frac12,κ>21​,no finite-budget Bell-local selection model reproduces the full PV familyon a punctured neighborhood of coincidence.Matching it down to angular resolution Δ\DeltaΔ requiresΞ≥2κ−14log⁡1Δ−C,\Xi\ge\frac{2\kappa-1}{4}\log\frac1\Delta-C,Ξ≥42κ−1​logΔ1​−C,or an acceptance cross-ratio discrimination diverging at least asmax⁡(A,A−1)≳Δ−(2κ−1).\max(A,A^{-1})\gtrsim\Delta^{-(2\kappa-1)}.max(A,A−1)≳Δ−(2κ−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 exactlyκ>12,∣v∣c<32.\kappa>\frac12,\qquad\frac{|v|}{c}<\frac{\sqrt3}{2}.κ>21​,c∣v∣​<23​​.At κ=12\kappa=\tfrac12κ=21​,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 istanh⁡ ⁣(ΔH4)=∣Eκ(z)∣.\tanh\!\left(\frac{\Delta_{\mathrm H}}{4}\right)=|E_\kappa(z)|.tanh(4ΔH​​)=∣Eκ​(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 correlationE(n,m)=nTTm.E(n,m)=n^{\mathsf T}Tm.E(n,m)=nTTm.Joint rotational covariance forces T=tIT=tIT=tI, and perfect same-axisanti-correlation forces t=−1t=-1t=−1, henceE(n,m)=−n⋅m.E(n,m)=-n\cdot m.E(n,m)=−n⋅m.Therefore no fixed v≠0v\neq0v=0 PV sector can be represented by a fixedrotationally invariant two-qubit state with standard projective spinmeasurements.The v=0v=0v=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.
From a finite event record to Bell correlations and their loss under recording A pair is prepared together, read at two separated instruments, and allowed to leave a material record. This paper builds the correlation law from a real source geometry in Artian's Quantum Traction Theory. Its source-to-access equation is pK(r,s∣a,b)=C ⁣[(Πr(a)⊗Πs(b)⊗I)KΩ]C(KΩ)\boxed{p_K(r,s\mid\mathbf a,\mathbf b)=\frac{C\!\left[(\Pi_r(\mathbf a)\otimes\Pi_s(\mathbf b)\otimes I)K\Omega\right]}{C(K\Omega)}}pK​(r,s∣a,b)=C(KΩ)C[(Πr​(a)⊗Πs​(b)⊗I)KΩ]​​ Here C is an additive, basis-invariant event capacity, Omega is the normalized orientation-neutral two-channel source, K is a specified record contact, and the projectors select the two instrument outputs. A real quarter-turn and a qualified shared-reference composition generate the readout algebra. Capacity symmetry fixes its quadratic weight. A separately stated capacity-to-frequency condition connects that weight to laboratory counts. The correlation follows before the Bell comparison For a phase-compensated record contact with overlap g between its two normalized record profiles, the construction gives EK(a,b)=−azbz−g(axbx+ayby),0≤g≤1.E_K(\mathbf a,\mathbf b)=-a_zb_z-g(a_xb_x+a_yb_y),\qquad 0\leq g\leq1.EK​(a,b)=−az​bz​−g(ax​bx​+ay​by​),0≤g≤1. The intact-source limit is E = -a dot b. The same calculation yields the fixed-setting and optimized Bell responses: S45=2(1+g),Sopt=21+g2.\boxed{S_{45}=\sqrt{2}(1+g),\qquad S_{\mathrm{opt}}=2\sqrt{1+g^2}.}S45​=2​(1+g),Sopt​=21+g2​.​ Fresh independent record contacts multiply their overlaps. The paper derives the repeated-contact law, a capacity-based overlap calibration, uncertainty bounds, and a protocol that separates calibration from the Bell observations being judged. A source-first account with an explicit dependency map A1 supplies event-time labels and the clock-angle correspondence; A4 supplies the real dial; A5-X supplies completed-event support, with its inherited A6/A7 conditions. The two-channel preparation, balanced composition, capacity symmetry, frequency bridge, and record coupling are printed separately. Each theorem states exactly which conditions it uses. The Access uncertainty commutator is treated as a distinct inherited algebraic premise, rather than inferred from Bell correlations. When a receiver follows the source Version 2.0 constructs a matched receiver rather than assigning its transformation by name. Preserving every probability of the stated bilinear contact forces its transport to be the conjugate of the source transport, up to an overall phase: ∣rTVTUs∣2=∣rTs∣2 for all r,s⟹V=eiαU‾.\boxed{|r^{\mathrm T}V^{\mathrm T}Us|^2=|r^{\mathrm T}s|^2\ \text{for all }r,s\quad\Longrightarrow\quad V=e^{i\alpha}\overline U.}∣rTVTUs∣2=∣rTs∣2 for all r,s⟹V=eiαU.​ A finite set of basis and sum probes certifies this relation. A robust extension converts imperfect probe probabilities into an operator-error bound. Finite clock and shift operations then produce the comparison loop L=M+ω(I−M),ω=e2πi/d.\boxed{\mathcal L=M+\omega(I-M),\qquad \omega=e^{2\pi i/d}.}L=M+ω(I−M),ω=e2πi/d.​ The matched sector cancels the comparison phase while the unmatched sector retains it. This is a joint comparison law; each local readout can remain incompatible. The paper identifies the binary loop's ambiguity, provides additional probes that distinguish it, and transfers the Bell correlations into a complete detector record when the original carrier is consumed. One contact, two observable roles A specified marker contact supplies both the Bell record overlap and a resolved weak-measurement instrument: γ=sin⁡θ,K0=diag⁡(cos⁡(θ/2),sin⁡(θ/2)),K1=diag⁡(sin⁡(θ/2),cos⁡(θ/2)).\begin{aligned}\gamma&=\sin\theta,\\K_0&=\operatorname{diag}(\cos(\theta/2),\sin(\theta/2)),\\K_1&=\operatorname{diag}(\sin(\theta/2),\cos(\theta/2)).\end{aligned}γK0​K1​​=sinθ,=diag(cos(θ/2),sin(θ/2)),=diag(sin(θ/2),cos(θ/2)).​ An independent replay executes this instrument on all 40,032 tree records in the public Feng and colleagues archive, reproducing every stored decoder success/failure score. The main five-setting batch contains 32,000 records; the complete archive has six settings. The reproduced job-cluster residual probabilities are p = 0.0188753 and p = 0.0442603, respectively. These diagnose modest tension with the ideal instrument under the stated retrospective covariance. The matched QTT and quantum representations make the same archival predictions, so this replay is not a significance test between them. The package preserves the complete source-first Bell construction and adds finite receiver certificates, physical transport-error bounds, a common-contact derivation, five vector figures, a twelve-item achievements ledger, and offline reproducibility materials. The next operational question is precise: which independently qualified physical contact fixes both the record overlap and the receiver comparison response? Dependencies: QTT Main Book; finite address-capacity counting; real source histories and composition; A1-CHSH spinor character; Observation as Access.
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 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≤2S\le 2S≤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)E(a,b)=-\cos(a-b)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}\{\nu_{ab}\}{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)-\cos(a-b)−cos(a−b)) for all a,ba,ba,b, via an explicit acceptance rule γ(a,b,λ)∈[0,1]\gamma(a,b,\lambda)\in[0,1]γ(a,b,λ)∈[0,1]. Second, we give a PV-indexed microcausal (operator-algebraic) realization: PV parameterizes a canonical SU(2)SU(2)SU(2) unitary that conjugates local observables inside commuting Alice/Bob subalgebras, and in the singlet state the unconditional correlator equals −cos⁡(a−b)-\cos(a-b)−cos(a−b) and attains Tsirelson's value 222\sqrt222​. 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.
How finite source updates become interference, field equations and detector records sn+1=UAsn,UATUA=I,py(n)=∥Ky,RUAns0∥2,∑yKy,RTKy,R=I. \boxed{\begin{gathered} s_{n+1}=U_A s_n,\qquad U_A^T U_A=I,\\ p_y(n)=\|K_{y,R}U_A^n s_0\|^2,\\ \sum_y K_{y,R}^T K_{y,R}=I. \end{gathered}} sn+1​=UA​sn​,UAT​UA​=I,py​(n)=∥Ky,R​UAn​s0​∥2,y∑​Ky,RT​Ky,R​=I.​​ Localized completed-event transport reaches the receiver The source exponential now has a derived map into the local two-tick phase history. In the qualified free-source and local-history receiver class: δΦR(N)=BRDAUANδs,DAUAN=LANDA,supp⁡(DAδs)⊆A,daddr(A,R)>N⟹δΦR(N)=0,py+(N)=py−(N). \begin{gathered} \delta\Phi_R(N)=B_R D_A U_A^N\delta s,\\ D_AU_A^N=L_A^N D_A,\\ \operatorname{supp}(D_A\delta s)\subseteq A,\\ d_{\rm addr}(A,R)>N\\ \Longrightarrow\quad \delta\Phi_R(N)=0,\\ p_y^+(N)=p_y^-(N). \end{gathered} δΦR​(N)=BR​DA​UAN​δs,DA​UAN​=LAN​DA​,supp(DA​δs)⊆A,daddr​(A,R)>N⟹δΦR​(N)=0,py+​(N)=py−​(N).​ Here D_A is derived from the inherited recurrence, L_A is its radius-one history update, and B_R reads a declared address window. The complete instrument includes missed events. The paper supplies an exact one-rail example with probabilities one and zero inside the preparation region, and a leakage bound for imperfect localization. It also proves the infinite multidimensional compact-band obstruction and retains the nonlocal canonical-receiver counterexample. This is a scoped causal readout theorem, not universal interacting-field microcausality or a new experimental signal. TV⁡(pR+,pR−)≤min⁡{1,2κR,NϵR,N}. \operatorname{TV}(p_R^+,p_R^-)\le \min\{1,2\kappa_{R,N}\epsilon_{R,N}\}. TV(pR+​,pR−​)≤min{1,2κR,N​ϵR,N​}. The preparation leakage and finite response norm must be independently fixed; they cannot be fitted to absorb detector results. All inherited scientific content remains, with 2,161 further deterministic checks and 16 adversarial mutations in a third reproducibility module. The QTT construction written explicitly The admitted free-source law can be written directly as counted real-phase evolution followed by physical receiver access: ΦR(Nt~A)=CRUA(N)s0,UA(N)=exp⁡[−2NJarcsin⁡QA],QA=Q⊕Q,0≤Q≤I,Q=(−ΔA/4)∣HA6,J2=−I,JT=−J,[J,QA]=0,N∈N0. \begin{gathered} \Phi_R(N\widetilde t_A)=C_R\mathcal U_A(N)s_0,\\ \mathcal U_A(N)=\exp[-2NJ\arcsin\sqrt{Q_A}],\\ Q_A=Q\oplus Q,\qquad 0\le Q\le I,\\ Q=(-\Delta_A/4)|_{\mathcal H_{A6}},\\ J^2=-I,\quad J^T=-J,\quad[J,Q_A]=0,\\ N\in\mathbb N_0. \end{gathered} ΦR​(NtA​)=CR​UA​(N)s0​,UA​(N)=exp[−2NJarcsinQA​​],QA​=Q⊕Q,0≤Q≤I,Q=(−ΔA​/4)∣HA6​​,J2=−I,JT=−J,[J,QA​]=0,N∈N0​.​ The paper proves exact equivalence to its inherited finite block propagator at every nonnegative integer source tick, including both spectral endpoints. Q is the address operator restricted to the A6-admitted band; its direct sum acts on both real dial components. The receiver and completed-event source retain their physical qualification conditions. The construction is exact in the stated coherent massless free-packet sector, not an assertion that every interacting field has this free law. QTT constructs wave and field behaviour from a finite source: real phase dials, admitted updates and completed receiver records. The paper carries that source through an exact two-path interference calculation, normalized sequential instruments and a quantitative detector-error bound. A smooth continuum field is not an additional starting substance in this construction. When does a field image contain enough information? A projected field has an autonomous linear update exactly when the source information it omits cannot feed back into view: CRUA=VRCR⟺UAker⁡CR⊆ker⁡CR. \begin{gathered} C_RU_A=V_RC_R\\ \Longleftrightarrow\\ U_A\ker C_R\subseteq\ker C_R. \end{gathered} CR​UA​=VR​CR​⟺UA​kerCR​⊆kerCR​.​ Two worked sources have the same present retained image and opposite future output ports. The paper then constructs finite electromagnetic Gauss, Faraday and work balances, connects the inherited exact wave propagator to its smooth limit, and gives an A2 gravitational source reading. The reconstruction states its source metric, transport, instrument and count-completion premises explicitly. It distinguishes finite addresses from finitely many possible phase states, and physical source selection from mathematical representation. The ideal matched quantum calculation also requires no outcome fitting. The reconstruction archive includes full source, the scientific illustration, the inherited numerical tables and 1,109 deterministic checks with 15 mutation checks, plus 664 additional explicit-construction checks and 10 mutations. These are reproducible mathematical fixtures, not new experimental observations. The fixed finite-wave residual retains its separate laboratory-access and sensitivity requirements. Stable paper DOI · Wave Dynamics · Completed-Event Transport · Main Book · Corpus Tree
A supplementary calculational note to the interpretational program "Spacetime as Register, Not Stage". The note reconstructs the full structure of spin-1/2 singlet correlations — the joint distribution P = 1/4(1−αβ·a·b), the correlation E(a,b) = −a·b, and CHSH S = 2√2 — from the program's picture of the "entangled" pair as one indivisible complement unit read out locally at two sites by independently oriented detector settings. The note proves sufficiency, not derivation, and keeps an explicit ledger of its epistemic status: two rules are added and underived (a half-angle spinor readout structure, and the square law for amplitudes), two are derived conditionally (antisymmetry, relative to an explicit phase bridge; normalization, fixed by completeness with no free parameter), and every remaining gap is recorded in a debt register. The model exits the factorizable (Bell) class exactly where quantum mechanics does: conditioned on the full preparation state, local marginals remain 1/2 (parameter independence holds per preparation state), while outcome independence fails — with no superluminal transport claim derived from statistics, no superdeterminism (the preparation state carries no free parameter that could co-vary with later settings), and operational no-signaling verified numerically. A deterministic hidden-orientation contrast model is included, reproducing the linear correlation −1+2θ/π and S ≤ 2, to show that the cosine requires a genuine exit from the factorizable class rather than a change of vocabulary. The English text is a full translation of the author's Polish source version (both files included; frozen 28.07.2026, changes tracked in the program's correction register, Series 33).