The EPR Paradox Is Not a Paradox: A Supplement to "Observation as Simultaneous Emission and Absorption" — Reading the Position–Momentum Pair as a Closure Identity
The EPR argument has three steps. (i) Measuring the position of one member A of a pair predicts with certainty the position of the other member B, and measuring the momentum predicts B's momentum (perfect correlation). (ii) The measurement at A does not disturb the distant B (locality). (iii) Hence B's position and momentum are simultaneously elements of reality, and the quantum-mechanical description is incomplete. This paper takes the readout rule of the preceding paper "Observation as Simultaneous Emission and Absorption" (Concept DOI 10.5281/zenodo.23857884) as its premise and re-reads the argument. The pair is a closure q_A + q_B = 0 that held once at the source, from which x_B = −x_A and p_B = −p_A follow simultaneously as identities. A single end has no value (the one-sided distribution is uniform), nothing happens at B when A is read, and the correlation depends neither on the order of the readouts nor on the distance between the two. The element of reality is the relation called the closure, not an attribute of one end. Step (ii) is correct, step (i) is a relational identity, and only the attribution in step (iii) is mistaken. That a destructive readout exists, drops the chosen variable to one of two values and severs the relation with the conjugate variable, is placed as a known fact (resonant absorption, photodetection, homodyne detection); deriving its mechanism is a separate task. "Instantaneous selection" is not written: what can be shown is only that no finite-speed influence exists. No new number appears. The claim is that, given the preceding paper, EPR is neither a paradox nor a mystery but a misreading of a relational identity. Earlier work reaching the same conclusion (Peres, Mermin, Smerlak and Rovelli, Griffiths, Jarrett, Spekkens, Wharton and Argaman, Cramer) is placed up front, and what belongs to this paper alone is limited to four points. Japanese original (authoritative) and English translation, with TeX sources, PDFs, two figures (Japanese and English labels), the figure program and the numerical identity checks.
The EPR argument has three steps. (i) Measuring the position of one member A of a pair predicts with certainty the position of the other member B, and measuring the momentum predicts B's momentum (perfect correlation). (ii) The measurement at A does not disturb the distant B (locality). (iii) Hence B's position and momentum are simultaneously elements of reality, and the quantum-mechanical description is incomplete. This paper takes the readout rule of the preceding paper "Observation as Simultaneous Emission and Absorption" (Concept DOI 10.5281/zenodo.23857884) as its premise and re-reads the argument. The pair is a closure q_A + q_B = 0 that held once at the source, from which x_B = −x_A and p_B = −p_A follow simultaneously as identities. A single end has no value (the one-sided distribution is uniform), nothing happens at B when A is read, and the correlation depends neither on the order of the readouts nor on the distance between the two. The element of reality is the relation called the closure, not an attribute of one end. Step (ii) is correct, step (i) is a relational identity, and only the attribution in step (iii) is mistaken. That a destructive readout exists, drops the chosen variable to one of two values and severs the relation with the conjugate variable, is placed as a known fact (resonant absorption, photodetection, homodyne detection); deriving its mechanism is a separate task. "Instantaneous selection" is not written: what can be shown is only that no finite-speed influence exists. No new number appears. The claim is that, given the preceding paper, EPR is neither a paradox nor a mystery but a misreading of a relational identity. Earlier work reaching the same conclusion (Peres, Mermin, Smerlak and Rovelli, Griffiths, Jarrett, Spekkens, Wharton and Argaman, Cramer) is placed up front, and what belongs to this paper alone is limited to four points. Japanese original (authoritative) and English translation, with TeX sources, PDFs, two figures (Japanese and English labels), the figure program and the numerical identity checks.
The EPR paradox is shown to be a result of the wrong definitions in quantum mechanics of such physical quantities as momentum and position. The corrected definitions show that momentum and position of particles can be precisely defined simultaneously and uncertainty relations do not prevent it. Therefore, with correct definition of physical quantities in quantum mechanics, the paradox disappears. The Bohm Aharonov version of the EPR paradox is illustrated by the example of measuring the polarization of photon pairs. The common view of the two photon state radiated by a source is presented. The common Bell’s inequality is discussed and the simplest version of it for a specific hidden variable is derived. A possibility of an experimental study of whether a photon and its polarization are preexistent before measurements is considered.
This paper reexamines the EPR paradox through a logical separation of “prediction” and “observation.” Even when a conditional prediction supplied by quantum theory reaches probability 1, this fact alone does not imply that an observation event has occurred at a remote location or that a particular ontology holds for an unobserved quantity. When this distinction is generalized into a minimal rule against generating ontology without warrant from an observational formalism, the inference that depends on EPR’s criterion of reality can be stopped without introducing new dynamics or a superluminal mechanism. At the same time, correlations that resist local causal factorization, as manifested in violations of Bell inequalities, remain. Thus, EPR-type “instantaneous remote determination” need not be identified with Bell nonlocality. The paper further reconsiders the possibility of reading state collapse as conditioning on an observation outcome rather than as transport through spacetime, its relation to the invariance of the speed of light, Bohr’s response to EPR and the logical ambiguity of the Copenhagen interpretation, and Einstein’s ontological concerns. Finally, the problem is considered as a “renewed Kantian overreach of reason,” in which an overreach that had been constrained by modern science and Kantian critique reappears through the mediation of precise quantum formalism. AI Assistance DisclosureGenerative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification. Keywords: EPR paradox, prediction, observation, state collapse, Bell nonlocality, Copenhagen interpretation, ontology, Kant
It is shown that there is one purely deterministic outcome when measurement is made on the state function chosen by EPR to describe the combined two-particle system – the distance between the two particles is preserved the same. Further, it is shown that, surprisingly, the Ψ-function designed according to QM leads to the following paradox – despite the fact that the two particles move in opposite directions, in time the distance between them becomes shorter and shorter. As is known, Einstein, Podolsky and Rosen (EPR) [1] observe two particles which have interacted in the past (for 0 Tt< <) but the interaction between them has ceased for times Tt>. Nevertheless, both particles, even at times Tt> EPR consider to be described by one common state-function, namely 1 2 ( ,)x xΨ. The authors have chosen to explore one particular form of that state-function, namely ( ) 1 2 01 2 (), i x x x pdpex x − −Ψ = ∫ =. (1) This function may be rewritten as ( ) 2 0 11 2 (), i ix x p x pdpe ex x − +Ψ = ∫ = = (2) and then 1 i x p e = can be considered as the orthonormal basis of an observable A pertaining to particle 1 (position of particle 1) while the exponents
What quantum mechanics calls "observation" has lumped together three interactions of different character: one that switches the discrete state of the partner, one that leaves the partner's state unswitched and shifts only the internal phases, and one that severs the mutual phase relation. This paper places the substance of observation in the second, the readout of the interference of a phase difference, and adopts only three premises. (P1) The distance between two parties is set by the ratio of a shared viewing angle to the partner's size; only ratios are readable and the absolute scale is not. (P2) An interaction is a single relation in which emission and absorption hold simultaneously; the speed of light appears on the readout side as a conversion factor between local clocks. (P3) The phase of a relation carries a harmonic order n that depends on the kind of wave. From these, the fact that a probability is the product of an amplitude and its conjugate (the form of the Born rule), the two-party correlation E = ∓cos nθ, the upper bound 2√2 of the CHSH quantity (the Tsirelson bound), and no-signaling follow in a few lines. The normalization that turns this into a probability is the division that discards the size ratio, i.e. the choice of the η = 0 section. As a consequence, the bound is 2√2 independently of the kind of wave, and the detector angle giving the maximal violation shrinks in proportion to 1/n (45° for spin-½, 22.5° for photons, 11.25° for gravitational waves). For a localized relation that is a sum of harmonics the correlation is the autocorrelation of the pulse shape; the bound holds for any weights, equality is attained only for a single harmonic, and a rectangular pulse reduces to a local hidden-variable model with S ≤ 2. Japanese original (authoritative) and English translation, with TeX sources, PDFs, five figures (Japanese and English labels) and the figure-generating program.
We argue that the so-called entangled states in quantum theory are not something exceptional, deserving a special attention in our efforts to understand conceptual foundations of quantum world. They appear by constructing the basis states of a compound system via the basis states of entering subsystems and describe it as a wholeness. While a system is considered as a wholeness, the individual members, forming the entangled state, have no physical meaning. In consequence, there is no physical ground for Einstein, Podolsky and Rosen (EPR) correlations arising in a process of decaying the system, being in the entangled state, into its constituent parts. The same regards to Bell’s introduced non-locality of quantum world. 1
Common introduction to all the records of this series. The content of this record follows below. Mathematical Materiality (MM) in brief. Why are the laws of physics the same everywhere and at every instant, even though nothing forces matter to obey them? MM proposes an answer: the laws are not rules imposed on matter from the outside; rather, matter itself is made of mathematical relations. And mathematics cannot change: a relation binds two distinct terms everywhere and always. In logic, nothing exists on its own: to exist means to be in relation to another logical entity, and outside this web of relations nothing can exist, not even other universes. This reading already meets experimental measurements at two points. First, the ratio of dark to ordinary matter is read as 3√3 = 5.196; when left free in the fits, Planck and DESI DR2 data give 5.21–5.22 (as registered in record 2.44, https://doi.org/10.5281/zenodo.23329080). Second, the acceleration scale of galaxies is read as a₀ = cH₀/6, the same scale that appears in Verlinde's emergent gravity: the computed value is 1.09–1.18 × 10⁻¹⁰ m/s² against the measured value of 1.20 (record 2.46.1, https://doi.org/10.5281/zenodo.23506655). Both readings were formulated with the measurements already known; the tests to check them against future data are registered in the same records. In the first logical nexus between space and time — which became homogeneous with Einstein — there is a non-local phase, made only of a pure logical distinction: space is not time. Between the two there is no relation of quantitative calculation, only the logical entity they have in common, which is needed to make them exist with respect to each other. This level is not distant: it is present in every phase of reality, in every place and at every instant. It shows in entanglement, where two distant systems remain correlated more strongly than any explanation based on separate properties and free of faster-than-light influences allows, without anything physically travelling between them. In this non-local phase the two distinct terms, space and time, share only one logical element, which we shall call their greatest common divisor (GCD), just as the numbers 3 and 5 have only 1 in common. The local world arises when a relation is counted through this GCD: its multiples and its fractions form a matrix. Seen by an observer inside the system like us, the multiples appear as time and the fractions as space. When that same multiple is directly contained in the matrix of the point of observation, it turns into mass — that is, into concrete matter as we usually understand it. There is therefore a close link between time and mass, in full agreement with Einstein's relativity, which ties mass to spacetime. The observed objects, while being logically autonomous entities in their local relations, enter into relation with the observer through a logical diffraction — that is, a decomposition into fractions — of the observer's matrix. This reading also gives a reason for the limit of the speed of light, which in Einstein's relativity is instead taken as a starting principle. The speed of light is the fixed exchange rate between the multiples and the fractions of the same GCD, meaning how much space a single step of time is worth: exceeding it would mean, for some observer, covering space without counting time, and thus leaving the local world. Logic always starts from a logical relative, and this exchange rate has nothing absolute in itself either: it is a necessary relative which, inside the system, becomes an absolute hypothesis, the same for everyone and impossible to exceed. Outside the system, by contrast, it has no justification at all, because outside there is only non-existence: justifying something requires another term to relate it to, and out there there is none. Asking for a reason from outside is like asking what lies north of the North Pole. The same theoretical framework that reads space, time, mass and the observed objects also admits quantum mechanics. The non-local phase and the mirror-like logical reciprocity between the two ends of the nexus give a reason for the probabilistic behaviour of matter. When the observed object is large enough, its mass fixes it in the matrix of the point of observation: for this reason we see it in one precise place and at one precise instant. When instead the object falls below a certain mass threshold, the principal GCD (that of the point of observation) and the GCD of the diffraction that generates the object are forced to become equal, reflecting each other as in a mirror. At that point the object no longer occupies a single point of the spacetime generated by the observer, but spreads over several possible positions, and we observe it as probability. It is like having to describe the number 10 using only the number 1: the only way is to distribute the 1 over ten positions, one for each unit of the 10. Today we know that even objects made of thousands of atoms behave in this way, and no experiment has yet found the boundary. The idea that the passage from quantum to classical behaviour depends on mass is close to that of Penrose and Diósi; in MM the threshold is the point where the two GCDs, the principal one and that of the local diffraction, become equal. For all these reasons the universe behaves like a simulation without really being one. It follows exact rules, made of counts, everywhere and at every instant, but no machine runs them and no programmer wrote them: a structure of mathematical relations does not need to be computed in order to exist. Tegmark (2008) had already noted that such a simulation need not be run, while Vazza (2025) estimated that running it physically on a machine would require resources that the very laws of the universe do not allow. This record develops one aspect of the theoretical framework. The work is shared openly so that it can be judged by as many readers as possible: anyone who knows someone interested can pass on this single link, valid for the whole framework (with an overview in English and Italian): https://doi.org/10.5281/zenodo.23535916. The entry page, with the complete list of all records, is https://doi.org/10.5281/zenodo.22667516. To cite the framework: Lanciano, U. (2026). The number 1 in front of the mirror: Mathematical Materiality from the nexus to the cosmos. Zenodo. https://doi.org/10.5281/zenodo.23535916. The records have been deposited since 5 August 2026, each with its own date, and are released under CC BY 4.0: anyone may use them by citing the source. Objections and errors found are welcome at riverantares@gmail.com, since Zenodo has no space for comments. Bell experiments without loopholes give CHSH values above the local bound 2, and high-precision measurements are compatible with Tsirelson's quantum bound 2√2 (Poh et al. 2015: 2.82759 ± 0.00051). This record asks what the author's framework, Mathematical Materiality (MM), must contain to respect both facts, and where it then stands among interpretations of quantum mechanics. No mathematical priority is claimed: the theorems used are published (Tsirelson; Barnum et al. 2010; Sorkin's quantum measures; Craig, Dowker et al. 2007; Aharonov's two-time formalism; Cramer; Argaman 2010; Wharton and Argaman 2020), and the record separates published results, derivations in the working notes and numerical conjectures. Predictions were registered with SHA-256 fingerprints before each test; failed predictions are reported. (1) Local readings and the Tsirelson bound. If each wing is read with the Born geometry (Euclidean state space with all its effects), no no-signalling joint event exceeds 2√2. Replacing the Euclidean ball with ℓ^p balls raises the maximum: 2^(2−1/p) for p ≥ 2 (the value of Ver Steeg and Wehner 2009), above 2√2 also for 1 < p < 2 (exact value conjectural). With marginals of length m the maximum is the quantum 2√(2 − m²). Two registered predictions failed: a hand formula, wrong for p < 2, and a three-dimensional search that stayed below the value already reached by the two-dimensional construction. (2) Two slits on the Feynman checkerboard (MM 2.40). In the 2.40 representation the imaginary turn weight −i sin θ produces destructive interference (dark fringes); with positive weights the two-slit pattern never falls below the sum of the single slits. The fringes fall at −143.2 and +140.4 lattice sites against ±142.1 predicted from the lattice dispersion. For the preparations tested, a real coin gives the same probabilities. A first registered run failed because its "central region" was badly defined; the second run was registered after seeing the first. (3) Can a future boundary give Born frequencies? In a minimal model, a future that selects the outcome reproduces Born frequencies only with some rules (weighting each future by the closure of its loop, or an exponential race), not with others. The measure over futures remains a postulate. (4) A locally mediated retrocausal singlet model. The future reading of one wing fixes a parameter carried back to the source and forward to the other wing, which reads it with the local Born rule. Singlet correlations are reproduced with every link along paths inside light cones. Within this family, no observable signalling holds only for the balanced (Born) weight, as already discussed by Argaman (2010); the angular law is assumed. (5) Self-consistency and the balanced weight. In the pair model the Born weight is put in by hand. With any other weight a bit could be signalled through the loop and, composed with a second link (Tolman's antitelephone, assumed), into the sender's past; a paradox protocol then makes consistent histories as improbable as one likes (0.02 with p = 0.51 and 10 001 pairs), while with the Born weight the channel carries nothing. This does not make non-Born weights impossible: the step from "arbitrarily improbable" to "cannot exist" is stated as the author's postulate. (