The phenomenological-velocity radical αlsin β= L+κ L−/κ, with κ= 1−v2/c2, is usually analysed by squaring, which yields a v-free compatibility constraint and the conclusion that the equation says nothing about v. That conclusion holds only for the squared equation. Working on the principal branch, we show that the product of principal roots equals ε L+L− with a branch factor ε∈{±1}that depends on the signs of L± and on whether |v|<cor |v|>c, and that squaring erases it. On the solution locus, ε reduces to sgn(lα) for |v|<c and to +1 for |v|>c, giving a selection rule: solutions with |v|<c require sin β >0, solutions with |v|>c require sgn(lα) = sgn sin β, and for β ∈[0,π] the |v|>cregime is admitted if and only if lα>0. The light cone |v|= c is simultaneously the branch discontinuity, the locus where L−/κ is undefined, and the value returned by symbolic solvers. On the locus, εis a nonseparable function of sgn(lα) and the velocity regime: the Z/2 parity obstruction of a maximally CHSH-violating sign table, produced here by a branch cut with no probability. Its definedness locus coincides pointwise with a setting-dependent acceptance map under which Bell-local responses carry −cos(a−b) exactly on accepted data with unbiased marginals; what the model relinquishes is the settings-independence of the trial set, not locality, and we state the protocol under which the mechanism predicts no violation and is thereby refutable. Transporting the asymmetric decoration into an operator setting yields E=−cos(a−b+ 2ϑ(v)) with S = 2√2 |cos 2ϑ|. Keywords: principal branch; square-root identity; Riemann surface; Lorentz factor; light cone; phenomenological velocity; selection rule; Bell–CHSH; computer algebra.
The phenomenological-velocity radical αlsin β= L+κ L−/κ, with κ= 1−v2/c2, is usually analysed by squaring, which yields a v-free compatibility constraint and the conclusion that the equation says nothing about v. That conclusion holds only for the squared equation. Working on the principal branch, we show that the product of principal roots equals ε L+L− with a branch factor ε∈{±1}that depends on the signs of L± and on whether |v|<cor |v|>c, and that squaring erases it. On the solution locus, ε reduces to sgn(lα) for |v|<c and to +1 for |v|>c, giving a selection rule: solutions with |v|<c require sin β >0, solutions with |v|>c require sgn(lα) = sgn sin β, and for β ∈[0,π] the |v|>cregime is admitted if and only if lα>0. The light cone |v|= c is simultaneously the branch discontinuity, the locus where L−/κ is undefined, and the value returned by symbolic solvers. On the locus, εis a nonseparable function of sgn(lα) and the velocity regime: the Z/2 parity obstruction of a maximally CHSH-violating sign table, produced here by a branch cut with no probability. Its definedness locus coincides pointwise with a setting-dependent acceptance map under which Bell-local responses carry −cos(a−b) exactly on accepted data with unbiased marginals; what the model relinquishes is the settings-independence of the trial set, not locality, and we state the protocol under which the mechanism predicts no violation and is thereby refutable. Transporting the asymmetric decoration into an operator setting yields E=−cos(a−b+ 2ϑ(v)) with S = 2√2 |cos 2ϑ|. Keywords: principal branch; square-root identity; Riemann surface; Lorentz factor; light cone; phenomenological velocity; selection rule; Bell–CHSH; computer algebra.
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 recurring position in the recent Bell–CHSH literature holds that spin is a bivector, that the individual detection events of a bivector model are local and Boolean, that the phase relation among them appears only at the statistical level, and that Bell’s construction therefore does not apply to bivector spin. This paper gives that position an exact mathematical home and an exact boundary, working throughout in a multiplicative presence grammar: every object formed is a positive magnitude, every criterion is a coincidence of presences, and the sites classically described by vanishing are treated as non-formation sites of the domain or as coincidences with the unit. We prove that every positive binary compatibility table factors uniquely as mxy = C AxByDxy with all four factors positive; that local gauge freedom — the local, Boolean, wing-by-wing data — exhausts the scalar and wing factors; that the unique gauge invariant is the interaction presence D, canonically the coefficient of the wing bivector eA ∧eB under the character grading; and that the recorded correlation is the M¨obius emission E = (D2−1)/(D2+1) of exactly this invariant. Correlation is bivector-graded. Two commuting bivector actions organize the geometry: the rotor double cover of joint rotations, whose sign pair ±R realizes the mirror involution of the calculus, acting on the compact factor; and the phenomenological-velocity homothety D 7→ Dκ , the unique positive dilation of the interaction presence, acting on the non-compact factor. Their joint equivariance forces the algebraic, transcendental-free family Eκ = (1−c) κ−(1+c) κ / (1−c) κ+(1+c) κ , c = n·m, Born at κ = 1 and falsifiably non-Born below it; the hyperbolic form − tanh(κ artanh c) is its classical shadow. The class-relative CHSH bound is itself given a native formulation as a covering theorem about inscribed rosters, proved by inscription logic on the interior with no signed quantities, no zero criterion, and no evaluation at extremal assignments; the classical bound is recovered as the shadow of this theorem, while the textbook pointwise proof route, and the determinism step of the original 1964 argument, are shown to rest on non-formation sites. The bivector position’s two-level ontology and its inapplicability claim are thereby established at theorem grade, in a corrected type, and its spacetime-algebra reading is not. Its disproof claim is resolved by fixing the referent the literature has left unfixed: theorems are individuated by statement and proof, not by name; of the native bound no disproof is possible, while of the theorem as asserted in the founding paper — whose scope claim is refuted by exhibition and whose determinism lever is convicted at formation — the disproof stands.
Abstract We present a unified analysis showing that the dominant numerical and qualitative structure of Bell–CHSH correlations—including the ubiquitous appearance of √2, sinusoidal correlation curves, and violations of the CHSH bound—arises generically from Euclidean projection plus measurement-induced dimensional reduction. The argument is developed in three interlocking layers. A geometry-first layer uses macroscopic analogues (tuned guitar strings with threshold detectors) to make the projection mechanism inspectable: a shared relational state, local measurement axes, and a recording map that collapses continuous amplitudes to binary outcomes. A threshold-scan (“dial”) test shows that conditioned CHSH values rise smoothly above 2 as the acceptance criterion tightens. A Fourier-analytic layer demonstrates that the sign function—the operator that converts continuous amplitudes to ±1—injects an infinite series of odd harmonics into a pure fundamental signal. The classical triangle-wave correlation E(θ) = 2θ/π − 1 is the fundamental plus harmonic distortion; the quantum correlation E(θ) = −cos θ shares the same fundamental mode but without higher-harmonic content and with unit amplitude. A detection threshold acts as a conditioning map that progressively strips the distortion harmonics. At threshold τ ≈ 0.23, total harmonic distortion reaches near zero, the correlation matches −cos θ, and |S| = 2√2. A combinatorial layer introduces the Q/R/x ontological framework—where Q is the object, R the definite binary relationship, and x the genuinely ambiguous state—and connects it to the formula (n2 − n)/2. Binary measurement (n = 2) supports 1 pairwise relationship and a linear correlation. Acknowledging x as a third state (n = 3) yields 3 pairs—exactly the minimum for planar angular geometry and the cosine. These three layers converge: the gap between |S| = 2 and |S| = 2√2 is the total harmonic distortion introduced by binary quantisation of a continuous angular signal—equivalently, the information-theoretic cost of projecting a ternary relational reality onto a binary measurement space. We show that the hard-threshold model produces CHSH violations only through post-selection on coincidences; when no-detection events are properly accounted for, the model respects all Bell-type inequalities, including the Eberhard/CH inequality used in modern “loophole-free” experiments. This honestly limits the model’s scope but sharpens the open question: whether the trial-definition process upstream of the detectors (heralding, phase-matching) constitutes a form of selection beyond the hard threshold that could affect 1 full-accounting statistics. We propose the “dial test”—sweeping acceptance criteria and plotting violation strength versus acceptance fraction—as a concrete, falsifiable protocol for probing ensemble neutrality in any Bell experiment. Nothing in this work challenges the empirical validity of quantum mechanics or the logical structure of Bell’s theorem. Rather, it challenges the necessity of interpreting Bell-type correlations as evidence for an ontological division of reality into “quantum” and “classical” domains. The analysis suggests that what distinguishes the so-called quantum regime is not a change in physical law, but a change in how much of the underlying relational structure survives measurement and recording. If this reading is correct, its most consequential implication is that the correlations which motivated the inference of “action at a distance” have a local geometric origin: they are properties of a shared state established in the past, not signals transmitted in the present, and their statistical character is determined by how severely the observation compresses what was already there.
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).
Consolidated edition. This version merges the three previous papers of the programme into a single document, reordered by logical dependency rather than chronology so that the surviving core is separable from the speculation built on it. 63 pages, 26 sections in 7 parts, 8 appendices. Eloy René Becerra Daly What is established, in decreasing order of confidence [EXACT] Identities. The Λ-hierarchy reduces to τ = ℏ/E and ℓ = the reduced Compton wavelength. The accessible information satisfies IQ = (γ/τ)β exactly. A distinguished point at D/L = 1/2 carries four coincident properties — exactly one bit, the entanglement sudden-death point for C0 = 1, the maximum of D·C̄ at L²/4, and a Bures arc of π/3 — all invariant in β, τ and L. [OBS] Verification. D + C = 1 holds to machine precision across 650 circuits on two IBM backends and ~5.5×106 shots. [THM] Limits. A no-go theorem shows the temporal exponent of dephasing is independent of the pulse-sequence filter, which removes the derivation previously claimed for Axiom 3 and relabels it as an empirical postulate. [THM] The transferable result. Pollard's criterion used as a licence condition: β ≤ 1 certifies that the curve admits a positive rate density, β > 1 proves that none exists. In reliability engineering an infant-mortality exponent licenses reading the batch as heterogeneous, while a wear-out exponent forbids it under any partition. Six transfers are registered, each with the observation that would retire it. The inference this work does not license A fitted β < 1 is not evidence of quantum behaviour. The theorem establishes representability of the curve, not identification of the mechanism — the map from mechanism to curve is many-to-one — so what follows is weaker and sufficient: the data cannot be adduced as evidence of coherence, because a classical account of them exists. The decisive test, which is not performed here On a single sample, the anomalous-diffusion exponent and the stretched-relaxation exponent must satisfy β = (1 + αdiff)/2, with β ≥ 1/2 as a corollary. Both quantities are already tabulated in the NMR relaxometry and diffusion-imaging literature for overlapping sample sets. No new experiment is required to attempt this refutation. Errata to previous versions The critical qubit number is n* = 1.848, not 2.6. The LIGO ringdown significance, with the internal error scaled by √(χ²/ν) = 1.669 as the catalogue's own goodness of fit requires, is 12.5σ; we record that our first correction of it, 20.9σ, was itself wrong for neglecting that scaling, and that the originally published 13.8σ was closer to the defensible value than our correction. A reported 18 kHz emergence mode lies below the Rayleigh limit of its own data. The published comoving density of remnants corresponds to Ω = 0.063, not the 0.12 it is stated to reproduce. Endpoint, labelled CONJ throughout The programme closes with a dark-matter candidate: Planck-scale black-hole remnants at Mmin = mP/(2√π) = 6.140 µg, an exact identity rather than the approximation 0.28 mP under which it was published. Evaluating it shows the usual dismissal to be right for the wrong reason: the remnant carries 148.6 J at galactic velocity and one transit is expected per 0.86 km²·yr, so rarity is not the obstacle; but the mean free path in rock is 1023 light years and the probability of one interaction while crossing the Earth is 10-32. The candidate is cross-section-limited, not exposure-limited, and the two demand opposite experiments. Four falsifiers are given, two requiring no new instrument. Reproducibility Every numerical claim is recomputed from the equations by a dependency-free implementation: 354 regression tests and 37 published-number checks, over a registry of 138 equations each carrying source, evidence level and declared domain. Where a published number could not be reproduced it is recorded as an erratum rather than adjusted; where a correction of ours proved wrong it is recorded as such. The register documents 27 inconsistencies, of which 11 remain open, each with the observation that would close it.
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.