> This working draft (v0.1) explores a picture in which space and time are themselves information, so that information is the medium rather than something carried by one. Two entangled particles are treated as two points on a single ripple spreading from one source, and a measurement is treated as a "write": new information written on top of existing information by constructive and destructive interference.>> Three toy models are simulated (400,000 runs each, Python code included). A ripple that only carries copies of a phase set at the source reaches the local CHSH bound, |S| = 2. A single shared medium rewritten by the first measurement reaches |S| ≈ 2.83, matching the quantum value 2√2, while keeping each detector's own results at 50/50 (no signalling). A short interference argument gives the singlet correlation E(Δ) = −cos 2Δ.>> A proposed resolution of the "how fast" problem is given: distance is treated as the spreading time of information waves before they interact, so no distance yet exists between the entangled pair. The model reproduces standard quantum predictions but does not yet make a new testable prediction, and its key steps are stated as assumptions. Comments are welcome.
This working draft (v0.1) explores a picture in which space and time are themselves information, so that information is the medium rather than something carried by one. Two entangled particles are treated as two points on a single ripple spreading from one source, and a measurement is treated as a "write": new information written on top of existing information by constructive and destructive interference.>> Three toy models are simulated (400,000 runs each, Python code included). A ripple that only carries copies of a phase set at the source reaches the local CHSH bound, |S| = 2. A single shared medium rewritten by the first measurement reaches |S| ≈ 2.83, matching the quantum value 2√2, while keeping each detector's own results at 50/50 (no signalling). A short interference argument gives the singlet correlation E(Δ) = −cos 2Δ.>> A proposed resolution of the "how fast" problem is given: distance is treated as the spreading time of information waves before they interact, so no distance yet exists between the entangled pair. The model reproduces standard quantum predictions but does not yet make a new testable prediction, and its key steps are stated as assumptions. Comments are welcome.
Entanglement is usually met as a paradox: either something crosses space faster than light, or the outcomes were fixed in advance, and Bell's theorem forbids the second while relativity forbids the first. Within the informational framework, in which observable space is a projection of a correlation graph, the paradox rests on a hidden confusion between two distinct quantities. One is a coherence between the phase modes of the two particles, a factorizability F that runs from 1 for a classical pair to 0 for a maximally correlated one. The other is an informational distance, an arithmetic measure of how two elements differ, as real as the gap between two primes and needing no space. They are not the same number, and what binds a measured pair is the first, never the second. From the framework's phase structure, together with the standard two-level representation and the Born rule, the singlet correlation E(a,b) = -(1-F) cos(a-b) and the Bell parameter |S| = 2 sqrt2 (1-F) follow with no free parameter and no measured constant, so the violation |S| > 2 is a geometric consequence of small F. The full -cos, rather than the half of a product of marginal probabilities, is obtained from the joint amplitude, the direct signature of the projection being linear on amplitudes (Axiom 9). A local Haar-twirl lemma fixes the fully decohered fraction as the maximally mixed state, and a product of two locally isotropic (depolarizing) channels contracts the singlet's Pauli correlation tensor to a single factor, forcing the partially decohered singlet to be a Werner state with weight p = lambda_A lambda_B; the coincidence visibility then fixes p = 1-F, so the mixing weight is exactly the arithmetic factorizability; that is the bridge, derived rather than posited. Under decoherence F climbs toward 1 along an extension of the collective scaling law, F(N) = 1 - (1-F0)/sqrt(N) with N = 1 + n, and two distinct thresholds follow: the CHSH violation dies at a decorrelation count of order unity, while the entanglement itself survives to a count 9/2 times larger (a ratio of counts, not of times or distances). Throughout, the informational distance |Z_a-Z_b| does not move at all. Non-locality is therefore a property of coherence, not of a distance that shrinks to zero: the correlation never needed space to begin with. The informational space is not an interpretation added at the end; it is what the four opening questions require, and entanglement is the plainest evidence that it is there. All results are derived with zero free parameters and no measured constant; a self-contained standard-library verification script asserts every boxed result.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
Abstract. Bell's inequality established that no local hidden-variable theory can exceed a CHSH value of 2, andexperiment has repeatedly confirmed the quantum-mechanical prediction 2√2. Without modifying any measuredfact, this paper presents one structural picture of the origin of entanglement correlations — within a frameworkin which a particle is a bound triple of three wavons at mutual 120° phase: the two entangled systems do notcarry locally predetermined values, but share a single phase relation, and a measurement projects that sharedphase onto a direction angle. Within this picture the correlation function E(a,b) = ±cos 2(a−b) and CHSH = 2√2are reproduced, and no-signalling holds as a result rather than an assumption. For contrast, the value-sharinglocal translation is shown to fail, confined to CHSH ≤ 2, making explicit the criterion that separates the twopictures. At the present stage the picture is a restatement mathematically equivalent to the quantum formalism;no new prediction is claimed. Two forcing problems that must be proved before the restatement can be promotedto a derivation, and four points at which the picture is exposed to test and refutation, are stated explicitly.Keywords: Bell inequality · CHSH · quantum entanglement · no-signalling theorem · interpretation · phasesuperposition · wavon
Toner and Bacon showed that the singlet correlation E = -cos(Delta) can be reproduced exactly by two shared uniform random unit vectors and one bit of communication. We ask which unobserved quantities of a down-conversion source could supply the four uniform numbers that fix those vectors. For a collinear, nondegenerate source, the emission time and the signal-idler delay (which depends on the position of emission in the crystal and is shifted but not removed by compensation optics), together with the signal and idler optical phases as two clocks, give four such numbers. A Fourier argument then gives the departure of the correlation from -cos(Delta) in closed form: it vanishes at aligned and opposite settings, appears at intermediate angles, and is controlled by |sinc[(f_s - f_i) T]|, where T is the delay spread across the crystal. For type-I BBO at 763/849 nm the predicted deviation at a relative angle of 90 degrees reaches up to 14% at 1 mm and 42% at 0.3 mm, against 0.9% at 15.76 mm, while the CHSH value stays near 2.83. A thickness series with fixed optics would test this, and a null result at 0.5 mm would rule the construction out. The construction still needs either a communicated bit or setting choices that depend on the source. We give an exact measurement-dependent version and state what a physical cause would have to supply; we do not identify one.
In this chapter we shall deal with the most recent and challenging development of quantum theory, and also one of the most important ones for interpretational, foundational, and even technological issues. This field finds its roots in the observation that quantum states can be viewed as bricks of information in a way that is intrinsically different from classical information. As a consequence, the ability to manipulate quantum states translates immediately into a new form of information processing and exchange. What has been discovered during this conceptual passage is that this type of information processing is in many respects much richer than its classical analogue. This has contributed to the understanding of quantum states as an extension of the classical concept of state and not as a defective reality (see also Subsec. 2.3.4 and Sec. 15.5). The impossibility of knowing the value of all observables at the same time that had been seen as a strong limitation in the early days of quantum mechanics, now turns out to be a manifestation of a different – but not necessarily poorer – resource. On the contrary, we have increasingly discovered that superposition and entanglement are additional informational resources. These resources, for instance, allow for certain particular computations that are much faster on a quantum device than on its classical counterpart, and therefore the former is able to solve problems that cannot be practically solved using classical means.
Union Dipole Theory postulates two distinct channels of transfer: the particle moves through the permeable medium at a bounded speed, while a disturbance within that medium is stated to propagate at an unbounded one (Al-Mayahi, 2014). This paper asks what that structure implies for correlated pairs, and obtains the singlet joint distribution and the Tsirelson value from a single principle. The principle is single-valuedness of the internal phase — the same condition that produced the phase-closure result of the companion paper. Applied to a pair, it does two things at once. It indexes admissible transitions by integers, so that a shared pair quantum resolves into one channel and not partly into both. And, applied to the two pair channels as degrees of freedom of the medium, it fixes their action integrals, which for a linear medium are proportional to the stored energy. The channel weights are therefore not an added rule about rates: they are what closure gives. One condition supplies both the counting and the quadratic weighting. The consequences follow without further assumption. Writing the pair on its shared internal phase, the two channel amplitudes are 2cos(θ/2)cos φ and −2sin(θ/2)sin φ. Their closed action integrals stand in the ratio cos²(θ/2) to sin²(θ/2); a single shared quantum normalises them; and the correlation is E = −a·b with a CHSH maximum of exactly 2√2. An inversion symmetry then fixes the four joint outcome probabilities uniquely and makes the single-station marginals exactly one half at every angle, so no-signalling holds for the construction and not only for the architecture. Three negative controls are computed rather than asserted, and each is derived in closed form. Extracting outcomes by any local rule gives −(1 − 2θ/π) and |S| = 2 exactly. Sending the indivisible transition to the channel of larger instantaneous drive gives the same linear correlation and the same value 2, and no threshold weight repairs it. Weighting the channels by amplitude rather than by action gives S = 1.657 at the standard settings and, on optimisation over the four settings, exactly 2. All three controls therefore terminate at the Bell bound, and the distance between 2 and 2√2 is attributable to the closure measure and to nothing else in the construction. Two structural results complete the account. The correlation is independent of distance because it is a ratio: any common factor multiplying both channel weights — the inverse-square attenuation of the medium included — cancels identically between numerator and denominator, so the disturbance law is obeyed in full while the correlation does not register it. And a residual anisotropy of the medium leaves every prediction unchanged: perfect anticorrelation at zero angle survives at every order, and the CHSH maximum remains exactly 2√2 on the circular sections of the response ellipsoid. Finally, generalising the channel weighting to an arbitrary exponent shows that the correlation is an ordinary cosine for exactly one value, n = 2, and that the maximal CHSH value is strictly increasing in the exponent and passes through 2√2 at exactly that value. The position of the quantum ceiling is thereby tied to a constitutive property — the quadratic, that is linearly elastic, response of the medium — which is itself what the closure condition selects. The framework is not a local hidden-variable theory, does not evade Bell’s theorem, and lies outside the class the theorem constrains; this is stated as a classification and not as an achievement. What is read rather than derived is named, and what remains open is named: in particular, no dynamical account is given of how the shared medium delivers the single transition to one channel across two separated stations. Two appendices follow. The first compares the predicted angular form with Bell correlations reported on three physically unrelated platforms, spanning CHSH values from 2.07 to 2.71, all of which report the same angular form: strength varies, shape does not. The second sets out a two-coordinate hardware diagnostic which follows from the result and which is the subject of United Kingdom patent application GB 2619873.9, filed 26 August 2026. Companion papers: Al-Mayahi, A. (2026a). Internal Time as a Counted Quantity. https://doi.org/10.5281/zenodo.22110704 — Al-Mayahi, A. (2026b). Union-Dipole Continuum Electrodynamics. https://doi.org/10.5281/zenodo.22110528