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.
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.
For local hidden-variable accounts of the CHSH experiment that are faithful -- reproducing the observed singles and coincidence rates and unbiased marginals of a polarization singlet with symmetric detector efficiency ηeff -- I determine the minimal measurement dependence M (Hall's measure) needed to reach a CHSH value S. Linear programming over all local strategies yields M(S,ηeff)=max{0,ηeff((S+2)ηeff−4)/6}, whose edges reproduce Hall's bound, the Garg-Mermin threshold, and the postselection ceiling. The quantum point is certified exactly: M(22,9/10)=(272−33)/100, by primal and dual certificates in Q(2). I solve the detection profile under which a deterministic sign model reproduces the singlet exactly, and prove exact quantum correlations and angle-independent coincidence rates jointly impossible in this class. Mixtures of this profile with measurement dependence trade M against a cos4(a−b) modulation of the fourfold coincidence sum, up to 12.4% in relative amplitude, identical at the two CHSH angles; existing data exclude its full amplitude and bound its mixtures with flat detection to a detection share of order 15%; its sub-percent content is unbounded. A two-setting cross-ratio and a settings-torus log-Fourier protocol, calibrated against channel-efficiency imbalance, reach 5σ at 0.1% within hours; a flat result excludes the solved family to a detection share of order 1%. A local mechanism, moiré phase locking, with deterministic phase evolution and quenched randomness, attains the certified floor exactly at ηeff=1 and, with the solved profile, reaches M=0 at ηeff≈0.80. Its floor-attaining census shares a low-order Fourier skeleton across independent solutions and is not realizable by station-local offsets on the terrains tested.
Paper T14D investigates how a supplied Bell-correlated relational rule can be converted into two separately finalized detector records. The calculation uses the previously established Toner–Bacon one-bit communication protocol together with an angle and residual schedule passed from Paper T14C. Two shared random directions determine Alice’s ideal response and a one-bit message that is used in Bob’s response. Separate noisy first-passage detector processes then convert the ideal responses into records with variable finalization times, rare sign changes, finite-horizon handling, record hashing, and later trial matching. The record generator does not consult the target Bell correlation, a joint probability table, or a globally sampled outcome pair. An independent-seed hardening run generated and retained six million three hundred thousand primary records. The ideal responses produced a Bell value near the expected maximum, while the final records showed the small attenuation predicted by the measured pointer flip rates. The exact attenuation calculation reproduced the final record result to numerical precision. Communication analysis showed that the message carries one-half bit of information about a choice between orthogonal Alice settings when the shared directions are known. Removing T14C frame and state provenance did not change the generated outcomes, while removing the scheduled residual altered Bob’s response in the residual audit. Message removal and cross-trial message permutation damaged the Bell correlation, detector asymmetry produced measurable marginal bias, and no primary records were discarded. The finalized outcome marginals were consistent with the tested no-signalling equalities, although the complete protocol contains explicit communication from Alice to Bob. The result supports a communication-assisted first-passage record-encoding architecture. It does not derive the one-bit update from microscopic Holosphere dynamics, establish Bell locality, or provide a relativistically local model of spacelike measurement
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
> 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.