Observation as Simultaneous Emission and Absorption: Deriving the Born Rule, Bell Correlations and the Tsirelson Bound from a Viewing-Angle Readout Rule
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.
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.
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.
Evidence Paper VI of the Existence Equation series. From Born rule to Bell inequality: one observer to two. In the parent work (The Existence Equation), the Born rule P = |Ψ|² is not an axiom but a derived consequence: the condensation term α|Ψ|²Ψ projects the continuous phase of the deviation field onto discrete binary outcomes (±1), and time-averaging the projected phase recovers cos θ correlations — verified by simulation on a 256³ lattice (hydrogen s-orbital, 30,000 steps). The Born rule emerges from the measurement geometry of one observer. If the Born rule for one observer is a consequence of ED's condensation mechanism, then for two observers sharing a single conserved field (Ψ_A + Ψ_B = 0), the Tsirelson bound S = 2√2 follows as pure measurement geometry. The Bell inequality violation is not a separate mystery — it is the natural two-observer extension of the same mechanism that produces the Born rule. Input: 2 items. Nothing else. #InputDescription 1Continuous conserved fieldΨ_A + Ψ_B = 0 (single deviation field shared between two sites) 2Binary measurementDiscrete outcomes ±1 from nonlinear condensation α|Ψ|²Ψ (same mechanism that produces the Born rule) No Hilbert space. No C*-algebra. No Born rule postulate. No tensor product. The measurement geometry decides. The logical chain: StepWhat happensWhere demonstrated 1α|Ψ|²Ψ projects continuous phase onto ±1Main paper (axiom + simulation) 2Time-averaging → P = |Ψ|² (Born rule)Main paper (256³ lattice, 30k steps) 3One observer: cos θ projectionMain paper (consequence of steps 1–2) 4Two observers + conserved field → cos θ correlationThis paper 5cos θ + binary outcomes + unit vectors → S = 2√2This paper Output: S_max = 2 × √2 — decomposed, not just calculated. FactorValueOrigin 2Bob's constraintTwo unit measurement vectors: |b|² + |b'|² = 2 √2Alice's diagonalDiagonal of Alice's 2D choice space (2 settings) 2√2ProductGeometric ceiling of binary measurement Why this matters: The geometric ceiling 2√2 exists for all theories, including classical ones. The "Classical" configuration in the table below has |u| + |v| = 2√2 — the ceiling is fully available — yet achieves only S = 2. What differs is the correlation function. Separable hidden variables produce piecewise-linear correlations. Only cos θ correlations — which ED derives from field conservation, not from Hilbert space — can saturate the ceiling. Ceiling vs. Saturation — the key separation: Configuration|S||u||v||u|+|v| Optimal (quantum)2.82841.41421.41422.8284 Uniform2.59811.00001.73212.7321 Tight2.38900.76541.84782.6131 Classical2.00001.41421.41422.8284 The constraint |u|² + |v|² = 4 verified to 10⁻¹⁶ precision across 2D, 3D, and 10D measurement spaces — the bound is dimension-independent and reflects the structure of binary measurement, not of physical space. The structural reading of Bell violation: Standard quantum mechanics treats the Born rule as an axiom and calculates 2√2 through Hilbert space operator norms. The Existence Equation derives the Born rule from condensation, then explains 2√2 as the geometric consequence of applying that same mechanism to two observers sharing a conserved field. The Bell inequality violation is not evidence for non-locality — it is evidence for non-separability of a single conserved deviation field. The Bell inequality violation is not spooky. It is geometric. It measures the diagonal of the binary measurement lattice. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
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
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).
The Geometric Block Universe is a single completed geometric structure - a compact curved space carrying fields - in which every history compatible with the laws is present, together with the records that observers hold within it. Nothing in the block evolves. What an observer calls time is a relational ordering of records by internal coordinates, with a geometric direction, the logarithm of total volume, serving as the clock; where that clock stalls, a second clock built from accumulated volume carries the ordering through. Records are stable, mutually exclusive cells of a finite record space, the harmonics of low degree on a three-sphere, and an observer is a body that writes and reads such records. Three things that ordinary quantum mechanics must postulate are absent from this picture. There is no collapse: a measurement selects nothing, because every compatible outcome is already realised and a definite result is definite only relative to the observer's own record. There is no splitting of universes: the histories are labelled components of one structure, the number of them is not a primitive quantity, and nothing multiplies when a record is written. And there is no fundamental probability: the block contains only counts of qualified observer occurrences, and what an observer experiences as chance is self-location - the fraction of its own occurrences that carry a given record. Probability is an emergent appearance, not an ingredient. This paper shows that the quantum weights follow from counting. An earlier rigidity theorem of the programme showed that a positive linear incidence functional invariant under full-carrier heat resolution is proportional to volume, so that normalised incidence equals an inherited geometric density; the Born form of that density was an input, and this paper removes it. Three statements about counting- that the count of occurrences in a set of record points is the integral of a density depending on the amplitude at the point, that one attempt counts one, and that a non-interacting reader's counts cannot depend on a distant preparation, force the density to be the squared amplitude, with presence, unit counting, affinity and Born proportions for every sharp reader as consequences rather than assumptions. The Born weight is the conserved charge of the phase symmetry of each record, exact for the recorder the theory already contains, and its push-forward over states is the uniform reference distribution that earlier readers haad to be given by hand. The affine structure that makes counts quadratic is not free either: it is the positive first-order structure of the clock direction, derived for a clock that is a symmetry, and it survives the de Sitter trajectory the geometry selects with a bounce mixing 1/sinh²(πν) that is the same for every record mode. The recorder is realised on the canonical records of the five-rung carrier as a complete observer: it writes exactly distinguishable records at discrete pointer displacements, samples its pointer at the rate set by the evaluation overlap spectrum, and returns the branch weight with no threshold, no reference population and no preparation law; where the two records are not exactly distinguishable, the deviation from Born proportions is bounded and second order in their overlap. What remains assumed is stated once: that counting is local on the record manifold, that an attempt counts one, the selection of the record space, the standing of the recorder, and the identification of occurrence fraction with first-person credence. An unconditional derivation is not claimed. One consequence differs from quantum mechanics with continuous pointers and is recorded as a candidate signature: a recorded two-branch superposition decoheres, as a function of the write duration, with exact nulls, a partial revival of 21/91 at half the pointer period, full recurrence when the fibre closes, and a residual average coherence of 9/91.
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.