This paper presents a deterministic geometric derivation of the standard two-path interference law \(P(\Delta)=\cos^2(\Delta/2)\). The model is defined on the unit circle and uses deterministic preparation and projection rather than deriving the interference law from complex amplitude superposition together with the Born rule. A hidden directional variable is uniformly distributed before preparation and then biased according to geometric alignment with a preparation axis. Measurement is modeled deterministically by the sign of directional projection onto a measurement axis. The preparation-weighted geometric ensemble yields the correlation \(E(\Delta)=\cos\Delta\), which reproduces the standard two-path interference probability exactly. In this construction, the harmonic phase dependence appears as an overlap identity on \(S^1\) rather than as a consequence of probabilistic collapse or intrinsic complex amplitude addition. The construction is intentionally minimal. It does not attempt to reconstruct the full Hilbert space structure of quantum theory or provide a complete physical hidden-variable theory. It demonstrates only that the Born interference law for two coherent paths is consistent with deterministic geometric preparation and projection on a continuous hidden-direction space. This paper belongs to the Quantum Foundations area of the Coherence Geometry research corpus and is structurally paired with CGI-RSR-000021, "Geometric Substrate Models and Bell–CHSH Correlations: A Structural Analysis of Assumption Relaxation." Internal reference: CGI-RSR-000022.
This paper presents a deterministic geometric derivation of the standard two-path interference law (P(\Delta)=\cos^2(\Delta/2)). The model is defined on the unit circle and uses deterministic preparation and projection rather than deriving the interference law from complex amplitude superposition together with the Born rule. A hidden directional variable is uniformly distributed before preparation and then biased according to geometric alignment with a preparation axis. Measurement is modeled deterministically by the sign of directional projection onto a measurement axis. The preparation-weighted geometric ensemble yields the correlation (E(\Delta)=\cos\Delta), which reproduces the standard two-path interference probability exactly. In this construction, the harmonic phase dependence appears as an overlap identity on (S^1) rather than as a consequence of probabilistic collapse or intrinsic complex amplitude addition. The construction is intentionally minimal. It does not attempt to reconstruct the full Hilbert space structure of quantum theory or provide a complete physical hidden-variable theory. It demonstrates only that the Born interference law for two coherent paths is consistent with deterministic geometric preparation and projection on a continuous hidden-direction space. This paper belongs to the Quantum Foundations area of the Coherence Geometry research corpus and is structurally paired with CGI-RSR-000021, "Geometric Substrate Models and Bell–CHSH Correlations: A Structural Analysis of Assumption Relaxation." Internal reference: CGI-RSR-000022.
Abstract We develop a geometric framework in which quantum amplitudes, superposition, and probabilistic interpretation emerge from phase–channel geometry rather than from axiomatic postulates. The theory is formulated using generalized phase algebras Cₙ, with particular emphasis on the reduction chain C₆ → C₃ → C₂. At the C₆ level, the adjoint-defined norm selects a two-dimensional visible subspace spanned by antisymmetric phase channels, enforcing superposition as a direct consequence of higher phase symmetry. Projection to lower phase geometries collapses this superposition to a single effective channel, not through an external measurement postulate but via a curvature-driven stability mechanism. We show that distinct visible phase directions in C₆ project onto the same visible direction in C₃, with their geometric differences encoded solely in scalar amplitudes. These amplitude weights are fixed by phase geometry and lead naturally to a squared-norm probability rule. In this sense, the Born rule is obtained as a geometric projection law rather than assumed as a foundational axiom. The framework thus provides a unified geometric account of superposition, collapse, and probabilistic interpretation, and suggests a reinterpretation of quantum amplitudes as remnants of higher-dimensional phase geometry.
T62 establishes an exact complex amplitude-square representation for the interference structure derived in T61. Starting from the reduced interference-form expressionPtot=P1+P2+2P1P2sin(ΔΩ),The theorem shows that, after a phase shift transformation, the expression can be rewritten identically in the amplitude-square formPtot=P1+eiϕP22.The result, therefore, demonstrates that once the reduced antisymmetric transport structure generates an interference-form expectation law, a complex amplitude representation follows algebraically. The theorem identifies the emergence of complex amplitude structure as a consequence of the interference geometry rather than as an independently imposed assumption. T62 does not derive the Born rule, full Hilbert-space quantum mechanics, measurement theory, or wavefunction collapse. The theorem establishes only that the reduced interference structure admits an exact complex amplitude-square representation compatible with Born-type algebraic form. The resulting complex amplitude object is an emergent representation constructed from the interference expression rather than a fully axiomatized quantum state space. Status: solid for the exact algebraic amplitude-square representation derived from the interference-form structure of T61; conditional on the matched interference subclass and reduced-sector parameter correspondence; speculative for any identification with full quantum probability theory or physical wavefunction ontology.
How does a purely geometric projection — without Hilbert space, tensor products, or the Born rule — recover the exact quantum correlation E(a,b) = −cos θ and saturate Tsirelson's bound 2√2?We derive both results from the involution φ(x) = 9 − x on the digit alphabet {1, …, 9}, which generates four inversion pairs summing to 9 with barycenter μ = 4.5 and projection constant k_arc = 1/2. A hidden variable λ emitted by the source is projected sequentially onto detector axes a and b, each projection costing k_arc = 1/2. The product of two such projections yields E(a,b) = −cos θ, matching the quantum prediction exactly. The CHSH sum of four correlations at optimal angles gives S = 2√2, saturating Tsirelson's bound.The projection constant k_arc = 1/2 is established by seven independent paths: (I) global — barycenter over closure, 4.5/9; (II) local — half-span of the maximal pair, 3.5/7; (III) per-pair — every inversion pair individually encodes half-span/span = 1/2; (IV) positional — exact center of the sequence 1–8, with four elements on each side; (V) cadastral — the barycenters of the three row-pairs {2,4}, {5,7}, {8,1} average to 4.5; (VI) geometric — the interface mirror 11 divided by π = 22/7 gives 3.5, then 3.5/7 = 1/2; (VII) linker — 5.5/11 = 1/2, the barycenter of the phase cycle over the interface operator. Seven paths, sharing neither data nor method, converging to the same constant. This excludes numerical coincidence.No Hilbert space is assumed. No tensor product is constructed. No Born rule is invoked. The correlation is geometric, not probabilistic.New in v4.0: We add the complete arithmetic foundation — the Universal Fold Theorem proving that every multiplication table folds into inversion pairs summing to 9, the visual multiplication table mirrors, and the proof that base 10 is not arbitrary. We introduce entanglement as bipolar inversion: if x + φ(x) = 9, knowing x determines φ(x) instantaneously at any distance — not by causal signal but by structural constraint. This is quantum entanglement without mystery. Einstein's objection to "spooky action at a distance" is resolved: the correlation is not action, it is structure. The minimum of relation is 2 = DR(11), the interface operator — the smallest cost of knowing the other.Keywords: Tsirelson's bound, quantum correlations, Bell inequality, CHSH, geometric projection, hidden variables, digital roots, inversion symmetry, entanglement, EPR, sevenfold convergence.
The Born rule — that the probability of obtaining outcome |phi> in a measurement of a quantum system in state |psi> is P = |<phi|psi>|^2 — is one of the most empirically well-confirmed and conceptually contested statements in physics. Existing derivation attempts (Gleason’s theorem; Deutsch–Wallace decision theory; Zurek envariance; many-worlds branch counting) each establish it under their own assumptions, but none has achieved universal acceptance. We show that in the Three Time Dimensions (3+3) spacetime framework (de Haan 2026, book DOI 10.5281/zenodo.19633127), in which the third time dimension t_3 is compactified as a discrete two-sphere S^2 with 2^152 Planck-area cells, the Born rule emerges through three independent geometric routes that all converge on the same result P = |<phi|psi>|^2. Route 1 (Malus’s Law on S^2): the wavefunction is literally a direction on the t_3 S^2; the Bloch sphere IS this S^2; Malus’s 1809 law P = cos^2(theta) is identical to the Born rule for bosons, with SU(2) double cover giving cos^2(theta/2) for fermions. Route 2 (Gleason + S^2 non-contextuality): Gleason’s 1957 theorem uniquely selects |<phi|psi>|^2 given non-contextuality, which in (3+3) is a consequence of the S^2 Riemannian metric rather than an independent postulate. Route 3 (Lévy measure + t_2 isotropy): the t_2 expansion perturbs all t_3 directions isotropically; the unique continuous binary-outcome probability measure is the Lévy / Haar-invariant spherical measure, which gives P(+n_A) = cos^2(theta/2). The three routes use distinct mathematical tools (projection, Hilbert-space theorem, stochastic analysis) and distinct physical inputs (geometry, Riemannian metric, dynamical isotropy). They must converge if the geometric identification is correct; they would diverge if it were wrong. Malus’s 1809 law is the Born rule avant la lettre. The empirical optical law taught to undergraduates as a fact is, under this reading, a direct geometric consequence of photon polarisations being directions on S^2 — a statement about projection geometry that predates quantum mechanics by a century. The same geometric identification simultaneously resolves four foundational puzzles: (1) Schrödinger’s cat is never in superposition (the cat’s ~10^29 t_3 modes are entrained by environmental t_2 perturbation within femtoseconds of the first atomic interaction); (2) the three-polariser paradox dissolves (non-commuting measurements are non-commuting rotations on S^2); (3) the Bell/Tsirelson bound 2sqrt(2) is a geometric consequence of S^2 having dimension exactly 2 — empirical confirmation by loophole-free Bell tests (Hensen, Giustina, Shalm 2015) confirms the S^2 identification; (4) the cosmological decoherence floor tau_cosm = 1/H_0 ~ 14.5 Gyr is a structural upper bound on quantum coherence set by the t_2 cosmic expansion — distinguishing the framework from many-worlds interpretations. Three supporting elegance claims establish the framework’s quantum-foundational coherence: (A) the complex-valuedness of psi in C follows from t_2 being a rotation (the imaginary unit i is the generator of the t_2 precession rotation); (B) the Schrödinger equation i hbar dpsi/dt = H psi is the infinitesimal form of t_2 precession rotating the phase of the t_3 configuration; (C) the canonical commutation [x, p] = i hbar follows from the foam lattice structure. The paper is complementary to the companion Quantum Computing preprint [DOI 10.5281/zenodo.19651560] which uses the Bloch-sphere-as-S^2 identification for applied hardware-level proposals. The Born Rule paper provides the foundational underpinning for why that identification is correct at the level of quantum-mechanical first principles. The paper is honest about what remains open. Four specific items are explicitly acknowledged (§11): (1) the framework does not yet have a 6D Lagrangian formulation from which the Born rule would fall out of path-integral quantisation; (2) the uniqueness of the S^2 identification is only partially addressed — higher-dimensional spheres are ruled out by the Tsirelson bound, but tori and Calabi–Yau alternatives are not examined; (3) the path-integral reformulation of quantum mechanics within (3+3) has not been constructed; (4) detailed comparison with QBism, relational quantum mechanics, and consistent histories is outside the paper’s scope. Falsifiability. Six routes: Tsirelson-bound overshoot/undershoot beyond 2sqrt(2) + measurement precision; indefinite quantum coherence beyond cosmological timescales; real-valued QM experimental confirmation (already ruled out by Renou et al. 2021, reconfirmed by independent tests); detection of genuine contextuality; detection of discrete S^2 cell structure at near-Planck scales. The Born rule, in (3+3), is not a mysterious postulate; it is Malus’s Law on a compact sphere that nobody knew was there.
Deterministic Quantum Mechanics (DQM) is a research program investigating whether the probabilistic content of quantum mechanics can be understood as an emergent statistical description of an underlying deterministic physical theory. This paper consolidates the program’s cumulative results into a single account.The central proposal is that a complete microscopic state evolves deterministically, that the quantum state is a coarse-grained ensemble descriptor rather than the complete physical state, and that measurement outcomes are selected by which deterministic basin of an enlarged phase space a microscopic trajectory occupies. Under this proposal, Born probabilities are basin measures rather than fundamental stochastic laws.For a qubit, this program is realized exactly: a hidden variable λ on the unit sphere, a preparation-dependent density μ_r(λ) = [r·λ]₊/π, and the deterministic response A(n,λ) = sgn(n·λ) reproduce the qubit Born rule P(±|r,n) = (1 ± r·n)/2 exactly, together with sequential-measurement statistics and the associated expectation values. This preparation measure is itself shown to be the deterministic pushforward of a uniformly distributed microscopic preparation coordinate, removing any appeal to fundamental preparation-level stochasticity.Measurement is then developed as deterministic basin selection within a finite-time Hamiltonian architecture, with a finite-time stability theorem governing when microscopic perturbations cannot dislodge a trajectory from its basin. Entangled correlations are reproduced by an autonomous, rotationally covariant Bell Hamiltonian containing dynamical setting registers and a conserved-momentum phase shear; this mechanism reproduces the singlet correlation E(a,b) = −a·b and the CHSH value 2√2 in its ideal limit, while preserving exactly uniform local marginals at finite setting inertia and yielding an explicit, quantitatively bounded finite-inertia correction. A finite Hamiltonian bath (Model-C) supplies a deterministic amplification and record-retention mechanism; its analytic structure — including an exact completed-square cancellation of the static bath backreaction and an exact finite-bath spectral equation — is derived independently of, and is shown to survive, a documented numerical implementation failure (the C14 program), which is retained in the record as a diagnosed, non-Hamiltonian right-hand side rather than as evidence against the underlying construction.Extending the program beyond the qubit exposes a genuine mathematical obstruction: no deterministic basin defined directly on the qutrit pure-state manifold ℂP² can reproduce an arbitrary rank-one Born probability, because positivity forces any exact basin onto a measure-zero point. This is proved rigorously. The resolution is not the abandonment of determinism but the enlargement of the hidden state beyond the quantum ray itself, to a genuine mechanical phase space such as the cotangent bundle T*SU(3), on which the quantum state re-emerges as a coarse-grained moment and on which contextual, Kochen–Specker-compatible measurement dynamics can in principle be realized.Throughout, the paper enforces a strict evidentiary hierarchy — mathematical construction, analytic theorem, numerical validation, and experimental confirmation are never conflated — and closes with an explicit account of what has been proved, what remains open, and what would be required to render the program experimentally falsifiable.
FINDING: Born rule remains the probabilistic core of quantum mechanics; recent derivation attempts from measurement models; single-photon interference confirms wavefunction modulus-squared statistics. | MATH: Born rule: P(a)=∣⟨a∣ψ⟩∣2 (probability = squared modulus of amplitude). Single-photon interference: intensity I∝∣ψ1+ψ2∣2=∣ψ1∣2+∣ψ2∣2+2Re(ψ1∗ψ2) — cross-term encodes phase coherence. Derivation (arXiv:2408.06375) posits nondeterminism from measurement apparatus model, not from wavefunction collapse postulate alone. | CONNECTION: Interference cross-term 2∣ψ1∣∣ψ2∣cos(Δϕ) — phase difference Δϕ maps to geometric phase (Berry phase) on projective Hilbert space. For equal amplitudes, visibility V=2I1I2/(I1+I2) reaches 1.0, but any deviation from Born rule would appear as non-quadratic power law P∝∣ψ∣α with α=2. No evidence of \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com