The double-slit experiment combines three observational facts: interference fringes in the coherent regime, position-resolved detections, and continuous fringe suppression under which-path resolution. Standard quantum mechanics accommodates all three through postulates without deriving them from deeper structure. This paper shows that the standard far-field interference law and the coherence-to-which-path transition follow as structural corollaries of results established in Quantum Mechanics as the Boundary Algebra of a Non-Invertible Projection (doi:10.5281/zenodo.19535588) and its companion (doi:10.5281/zenodo.19571321). Two results suffice for the algebraic derivation: the Born rule as the canonical quadratic measure on Im(Π), and the kernel spinor as the Hilbert quadrature of the boundary field. A third result — measurement irreversibility as a kinematic consequence of the projection geometry — belongs to the physical interpretation of measurement and temporal ordering within the same framework. The paper derives the Fraunhofer intensity law I(x) ∝ |Ψ₁ + Ψ₂|² and the detector-overlap formula with fringe visibility V = |γ|, within the standard Fraunhofer approximation and without additional postulates. It does not claim a first-principles derivation of the full boundary propagator or of single-event arrival statistics. A companion numerical simulation implements all three coherence regimes. A self-contained appendix summarises the three imported results in compact form.
The double-slit experiment combines three observational facts: interference fringes in the coherent regime, position-resolved detections, and continuous fringe suppression under which-path resolution. Standard quantum mechanics accommodates all three through postulates without deriving them from deeper structure. This paper shows that the standard far-field interference law and the coherence-to-which-path transition follow as structural corollaries of results established in Quantum Mechanics as the Boundary Algebra of a Non-Invertible Projection (doi:10.5281/zenodo.19535588) and its companion (doi:10.5281/zenodo.19571321). Two results suffice for the algebraic derivation: the Born rule as the canonical quadratic measure on Im(Π), and the kernel spinor as the Hilbert quadrature of the boundary field. A third result — measurement irreversibility as a kinematic consequence of the projection geometry — belongs to the physical interpretation of measurement and temporal ordering within the same framework. The paper derives the Fraunhofer intensity law I(x) ∝ |Ψ₁ + Ψ₂|² and the detector-overlap formula with fringe visibility V = |γ|, within the standard Fraunhofer approximation and without additional postulates. It does not claim a first-principles derivation of the full boundary propagator or of single-event arrival statistics. A companion numerical simulation implements all three coherence regimes. A self-contained appendix summarises the three imported results in compact form.
This paper argues that quantum mechanics is not a separate postulate system to be reconciled with spacetime physics. It is the boundary algebra of a non-invertible projection: the algebraic structure that a globally coherent bulk field is forced to produce when it becomes observable. The bulk field is a Majorana spinor, Ξ = CΞ̄, on M^5 = R^4 × S^1(τ), governed by a single conservation law. The projection Π onto the observable boundary is not an external measurement device added afterward: it is the form the field must take in order to admit an observable image at all. What survives Π defines the observable sector; what enters ker Π is lost structurally, not thermodynamically. From this single mechanism, four boundary structures emerge: the Hilbert space of quantum mechanics, physical time, matter stability, and the Born rule. These are four boundary readings of one conservation law, K_τ = ξη = α^−1, resolved by the same tangential dynamics on the invariant surface Σ. Observable states are equivalence classes [Ξ] = Ξ + ker Π. The Born rule is the canonical quadratic probability measure induced by the projection geometry. Measurement irreversibility is the non-injectivity of Π, with non-vanishing protected by the transcendence of π. Physical time is not assumed in advance: it emerges as the output of chiral projection through a conformal map from the compact cyclic fibre to the irreversible real line. The Wheeler–DeWitt equation is not a paradox but the correct timeless description of the bulk. Spacetime emerges after quantum mechanics, not before it. The fine-structure constant is not a free parameter. It is the unique observable residue forced by the self-coherent projection structure. The double-slit interference pattern, the Born rule, and measurement irreversibility are derived as structural corollaries of this paper in the companion work doi:10.5281/zenodo.19654630
This volume establishes that the quantum propagator and double-slit interference pattern are derived consequences of the algebraic structure of H0(A3), not independent postulates. Starting from the presentation <a, b, c | a²=a, b²=b, c²=c, aba=bab, bcb=cbc, ac=ca>, we construct the transfer operator T = La + Lb + Lc as the adjacency operator of the Cayley graph, show that its spectrum {0, 1, 2, 3} with degeneracies (1, 11, 11, 1) is forced by idempotence, and derive the propagator K(d, s; t) = ⟨d|exp(−iTt)|s⟩ via spectral decomposition. Explicit numerical computation on the 24-element monoid verifies quantum interference: for certain detectors, the coherent probability |A1 + A2|² differs from the incoherent sum |A1|² + |A2|². We show that the commutation relation ac = ca in A3 suppresses interference at 20 of 24 detectors, while the fully braided algebra H0(Ã2) with C(3) = 0 produces maximal fringes. Measurement is not postulated but derived: the idempotent projection Ls² = Ls destroys coherence between paths, yielding the Born rule as a theorem. Keywords: 0-Hecke monoid, derived propagator, double-slit interference, transfer operator, idempotent projection, commutation deficit, Cayley graph Laplacian
The double‑slit experiment epitomizes the wave–particle duality of quantum mechanics. In standard treatments, the interference pattern follows from superposition, while measurement of “which path” destroys coherence. The precise mechanism that turns a coherent superposition into a localized detection event remains an open question. We propose an operator‑based model within the open‑quantum‑system framework that describes the entire process through seven effective Lindblad generators: fluctuating diffusion (initial wavepacket spreading), cyclic reset (quantum Zeno effect), spatial‑mode splitting (simultaneous excitation of both slits), free evolution (interference formation), liminal projection (threshold‑activated measurement), irreversible loss (detector inefficiency), and subspace mapping (projection onto a definite detection outcome). Unlike standard quantum mechanics, the model introduces a nonlinear decoherence rate that depends on the spatial spread of the wavefunction, capturing environment back‑action through a feedback‑controlled coupling. This dependence leads to a small but testable reduction of fringe visibility with increasing particle flux or wavepacket width, even in the absence of explicit which‑path measurement. The measurement activation is modeled as a smooth threshold function, giving a sharp transition in detection probability as the system reaches the screen. Analytical expressions for visibility and the critical Zeno frequency are derived, and numerical simulations are presented. The predictions are accessible with current experimental technology (weak measurements, controlled decoherence, cavity‑QED systems). The framework suggests that wave–particle duality can be understood as a dissipative phase transition, bridging foundational quantum theory with practical open‑system control.
This paper develops an interpretive framework for time, measurement, and probability within the Kittim Projection. Building on the geometric and thermodynamic foundations established in the companion foundational paper, the present work focuses on the structural consequences of a non-invertible projection from a depth-indexed coherence structure to observable three-dimensional space. A distinction is introduced between coordinate time, which parametrizes dynamical descriptions on observable space, and projection time, which encodes irreversible ordering of coherence propagation across depth layers. Within this framework, quantum measurement is interpreted as a structural consequence of non-invertibility: observable outcomes correspond to equivalence classes of depth pathways, and collapse arises as information loss under projection rather than as an additional dynamical postulate. A probability weighting scheme is constructed from depth-dependent visibility factors. Under equilibrium conditions, this scheme yields a quadratic weighting structurally analogous to the Born rule, without assuming wavefunctions or Hilbert spaces as primitive elements. Interference phenomena are likewise described as arising from indistinguishability of depth pathways under projection, rather than from wave evolution. The paper is explicitly interpretive and foundational in scope. It does not attempt to reconstruct the full formalism of quantum mechanics or to make new experimental predictions. Instead, it provides a geometric and structural perspective on time, measurement, and probability, suitable for foundations-of-physics and philosophy-of-physics contexts.
We propose a mathematical model for the double-slit experiment based on the block-alternating infinite products A(K,d) and B(K,d). The two products play strictly separated roles: B(K,d) encodes the background Minkowski spacetime (the stage), invariant under observation, while A(K,d) encodes the propagating matter wave (the actor), whose coherence parameter d acts as a phase switch. The coherence parameter d=1 corresponds to alternating phase (AC structure, full interference), while d=1/2 and d=infinity are algebraically identical and correspond to constant phase (DC structure, no interference). Observation---defined as physical interaction with a detector that makes path information available---transitions the observed path from d=1 to d=1/2. Unobserved paths remain at d=1 and continue to interfere. At the screen (K=1/2, the null boundary), A(1/2,d)=1 for all d, equalising the amplitudes. The interference pattern is then determined by d alone, yielding |psi|^2 = 4cos^2(pi*Delta/2) (unobserved) or |psi|^2 = 2 (observed). Combined with the envelope W[B] proportional to sinc^2(beta) arising from the Minkowski structure of B, the standard formula I proportional to sinc^2(beta)*cos^2(pi*Delta/2) is recovered as a mathematical consequence. The measurement problem is resolved algebraically: decoherence is not a collapse of the wave envelope but a transition of the phase parameter d within the A structure. The spacetime stage B is unaffected by the act of measurement.
This paper presents a detector-coupled interpretation of the Schrödinger equation within R-Theory (Resonance Theory), reframing quantum mechanics as the evolution of coherence in an underlying zero field rather than as the motion of fundamental particles. Instead of treating particles as primary entities whose positions are revealed by measurement, R-Theory models reality as coherent resonant modes. Particle-like events emerge only when coherence couples locally to a material environment, forming what are termed resonance knots. Detection is therefore not an abstract postulate but a physical conversion process. The work proposes a minimal extension of the standard non-relativistic Schrödinger equation by introducing a local knot-formation rate Γ(x), interpreted as a property of detector material and configuration. This leads to an “R-Schrödinger” equation in which coherence loss represents the conversion of extended coherence into localized detection events. The familiar Born rule |Ψ|² is correspondingly refined into an event statistics rule p_click(x,t) = Γ(x) |Ψ(x,t)|², making detector participation explicit. Key contributions of this paper include: A clear operational definition of the zero field as a coherence substrate A reinterpretation of the wavefunction as coherence amplitude rather than particle probability An explicit physical role for detectors in knot formation A continuity equation with a conversion (sink) term linking coherence flow to detection Experimentally testable implications emphasizing detector-dependent statistics in interference experiments Mathematically, the framework aligns with known open-system and dissipative approaches, but conceptually reverses the hierarchy: detector coupling is treated as the mechanism that produces particle-like events, not as an external add-on to an otherwise complete theory. This paper is part of the R-Theory Phase Series and is intended as a conservative bridge between standard quantum mechanics and a resonance-based, phase-first ontology, with direct relevance to double-slit experiments, detector physics, and foundational questions about measurement.