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PROJECT T VOLUME XVIII : The Derived Propagator Double-Slit Interference from Algebraic Structure Alone

Sami Allaghi

Published Feb 3, 2026• 0 citations
Publisher

Abstract

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

DOI
10.5281/zenodo.18467148

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Double-slit interference, Born rule, and irreversibility from a Non-Invertible Projection

Pasquale Camelia

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.

•Apr 19, 2026•0 citations

Interference without Wavefunction: Double-Slit Probabilities from Bilateral Microstructure

Pablo Garcia

We derive the double-slit interference pattern within Homogeneous Compression Theory (HCT) starting from a discrete relational microstructure, without postulating the Born rule, a wavefunction, particle trajectories, or collapse. Three steps carry the derivation: (i) a positivity theorem for the bilateral correlation matrix, proved from microstructural definitions via the Schur product theorem and Bochner's theorem; (ii) linear bi-local propagation from the slits to the screen, which yields the interference law I=I1+I2+2|mu|sqrt(I1I2)cos as an algebraic consequence of a Hermitian correlator; (iii) detection modeled as local bilateral calorimetry, giving event rates proportional to the bi-local diagonal, R(X) proportional to B(X,X). Particle-like clicks and wave-like fringes emerge as the diagonal and off-diagonal aspects of a single Hermitian correlator. Definite trajectories are shown to do no explanatory work.

•Sep 28, 2026•0 citations

Interference and Erasure on a Discrete Lattice: A Computational Demonstration of the Zeno Threshold in the Double-Slit Experiment

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Abstract: We present a computational demonstration of the Zeno Threshold (Zα) on the paradigmatic double-slit experiment. A Gaussian wave packet is propagated on a 256×128 tight-binding lattice using only two ingredients: graph topology (nearest-neighbor hopping) and unitarity (U = e-iHΔt, with the imaginary unit i forced by information conservation). No wave equation is assumed; interference fringes emerge as a strict consequence of lattice unitarity. We then introduce a which-path detector modeled as a tensor-product qubit that entangles with the particle at the slits. The entanglement entropy of the reduced detector state grows from zero toward Smax = ln 2 ≈ 0.693 nats. At the moment this entropy exceeds the observer's finite Zeno Threshold Zα, the observer's rendering engine is forced to perform Born Rule truncation—algorithmic garbage collection—erasing the off-diagonal coherence terms. The interference pattern is destroyed, leaving two classical bumps. Computational Verification Highlights: Emergent Interference: Fringes emerge strictly from discrete graph topology and unitarity, without assuming continuous wave equations. Forced Complex Phases: The complex unit i is mathematically forced by probability conservation (unitarity). The Z-Scale Breach: Which-path detection creates real entanglement entropy that breaches the observer's finite processing limit. Born Rule Garbage Collection: Truncation at the Z-Scale breach produces a detection pattern identical to the classical, incoherent sum |ψ1|2 + |ψ2|2. The full Python verification script and simulation code are available at github.com/srdrymn/atr-double-slit-simulation.

•Mar 18, 2026•0 citations

A Causal History-Dependent Bridge Operator for Double-Slit Interference: Exact U(1) Homomorphism, Algebraic Phase Linearity, and Cross-Validation Against Real Experimental Data

Luan Borges Fernandes

### Version 10 Release Note (Canonical Physical Realization — September 2026)This updated version provides the complete physical realization of the Causal Bridge Operator for quantum interference. Key evolutionary updates include:1. Grounding wave phase evolution in standard paraxial Fresnel wave optics with zero free parameters for spatial fringe spacing (Delta_x = lambda_dB * D / d);2. Binding exposure duration to the physical particle transit time inside the apparatus (tau_on = D / v_particle);3. Decomposing the non-Markovian memory kernel into a dual-layer architecture (K_eff = K_struct x K_thermal) that resolves the thermal decoherence confounder;4. Direct cross-validation against real experimental data: Zeilinger et al. (1988) cold neutron interferometry (reproducing 73.0 um fringe spacing with 0.106% error and calibrating V = 0.583 with lambda_struct = 31.11 s^-1, in agreement with independent decoherence analyses by Sanz et al. 2003) and direct transfer to Jonsson (1961) electrons;5. Rigorous satisfaction of the third-order Sorkin parameter (kappa = 0, error < 10^-15), strictly respecting Born rule bounds;6. Complete typographic overhaul of the manuscript in clean two-column academic LaTeX, eliminating table pagination overflow artifacts. --- ### AbstractStandard quantum mechanics treats wavefunction collapse as a fundamental, non-unitary event and models detection statistics strictly as a function of the instantaneous state vector, independent of the operational history of the apparatus. Here, we present a physical realization of the Causal Bridge Operator, B(H) = exp(i * Gamma(H) / hbar_Omega), anchored upon a non-Markovian memory functional Gamma(H) and standard Fresnel wave propagation. We demonstrate:(i) that B forms an exact homomorphism from the non-commutative history monoid to U(1);(ii) that the fringe spacing Delta_x = lambda_dB * D / d is governed by classical Fresnel wave optics with ZERO free parameters;(iii) that historical memory acts exclusively on the cross-term spatial degree of coherence (fringe visibility), resolving the thermal confounder through a dual-layer decomposition K_eff = K_struct(tau_on) x K_thermal(tau_on, tau_pause);(iv) that the intrinsic exposure time is rigorously bound to the physical flight time tau_on = D / v_particle; and(v) that the formulation identically satisfies the third-order Sorkin parameter (kappa = 0, error < 10^-15), strictly respecting Born's rule bounds. When cross-validated against the landmark cold neutron double-slit experiment of Zeilinger et al. (1988) (lambda_dB = 18.45 A, D = 5.0 m), the model reproduces the published 73.0 um fringe spacing with 0.106% error and calibrates the measured visibility (V = 0.583) with a structural decay rate of lambda_struct = 31.11 s^-1, in close agreement with independent decoherence analyses in quantum optics (~19.7 s^-1; Sanz et al. 2003). Direct transfer to electron diffraction (Jonsson 1961) predicts high-contrast fringes (K_eff ~ 1.0) without parametric readjustment due to the ~10^7x shorter transit time. Finally, observer collapse is shown to emerge naturally from path accessibility truncation (W_j -> 0) without requiring any metaphysical observer protagonism. ### KeywordsProject Omega, Double-Slit Interference, Causal Bridge Operator, Non-Markovian Dynamics, Fresnel Diffraction, Degree of Spatial Coherence, U(1) Homomorphism, Sorkin Parameter, Zeilinger Experiment, Quantum Foundations.

•Jul 27, 2026•0 citations

Paper 60B: Double-Slit Interference as a Radial-Coherence Regime Paraxial QM Baseline with Paper-60 Phase-Edge Admissibility Gating

Michael John Sarnowski

Paper 60B develops a unified Holosphere account of the double-slit experiment while keeping interference loss, measurement, and stable outcome formation conceptually separate. The paper begins with standard slit-resolved paraxial propagation and shows that a coherent two-slit pattern contains a cross term that is absent from the incoherent sum of the two single-slit diffraction patterns. Which-path detection is modeled through the overlap of path-dependent detector states. As those detector states become more distinguishable, fringe visibility decreases continuously and reaches the incoherent limit when the path records are fully distinguishable. The Holosphere interpretation treats a coherence corridor as a physical relation among the source, carrier, aperture, detector, environment, and readout boundary. The apparatus therefore helps determine whether the two paths remain part of one coherence-supporting structure. Boundary incompatibility separates path-conditioned corridors, while later interface competition and pointer-basin capture may select and stabilize a detector record. Physical recoil and apparatus relaxation remain causal, and no faster-than-light signal or strict determinism is required. A separate downstream phase-edge gate produces a modest, symmetric, and resolution-surviving change in the screen distribution. The numerical suite also verifies symmetry, attenuation accounting, slit-field linearity, detector-overlap behavior, and the conditioned quantum-eraser identity.

•Dec 28, 2025•0 citations

Conditional Uniqueness of the Amplitude-Square Form

Craig Edwin Holdway

T63 establishes a conditional uniqueness theorem for the interference and amplitude-square structures developed in T61 and T62. Starting from the reduced two-channel interference form,\[P_{\mathrm{tot}}=P_1+P_2+G(P_1,P_2,\Delta\Omega),\]The theorem analyzes the structural constraints imposed by symmetry, continuity, axis-vanishing behaviour, degree-1 homogeneity, and first-harmonic rotational structure generated by the reduced antisymmetric operator algebra. Under these assumptions, the interference contribution is forced into the normal form\[G(P_1,P_2,\Delta\Omega)=2\sqrt{P_1P_2}\,\sin(\Delta\Omega),\]yielding the exact amplitude-square representation\[P_{\mathrm{tot}}=\left|\sqrt{P_1}+e^{i\phi}\sqrt{P_2}\right|^2.\]The theorem therefore shows that, within the stated structural class, the Born-compatible amplitude-square form is uniquely selected by the reduced interference geometry together with the harmonic structure induced by the rotational generator satisfying \(A^2=-I\). T63 does not derive full quantum mechanics, measurement theory, Hilbert-space axioms, or probabilistic interpretation from first principles. The result is a conditional uniqueness theorem for the algebraic normal form of two-channel interference under the stated assumptions. In particular, the homogeneity assumption is essential: without it, uniqueness does not hold. The theorem further clarifies that the complex amplitude representation is obtained from the interference structure rather than independently postulated. Status: solid for the conditional uniqueness of the amplitude-square normal form under the stated symmetry, homogeneity, and harmonic assumptions; conditional on the separability and structural assumptions defining the interference class; speculative for any claim of a complete derivation of quantum probability theory or the Born rule from Q5 geometry alone.

•May 23, 2026•0 citations
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 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.
We derive the double-slit interference pattern within Homogeneous Compression Theory (HCT) starting from a discrete relational microstructure, without postulating the Born rule, a wavefunction, particle trajectories, or collapse. Three steps carry the derivation: (i) a positivity theorem for the bilateral correlation matrix, proved from microstructural definitions via the Schur product theorem and Bochner's theorem; (ii) linear bi-local propagation from the slits to the screen, which yields the interference law I=I1+I2+2|mu|sqrt(I1I2)cos as an algebraic consequence of a Hermitian correlator; (iii) detection modeled as local bilateral calorimetry, giving event rates proportional to the bi-local diagonal, R(X) proportional to B(X,X). Particle-like clicks and wave-like fringes emerge as the diagonal and off-diagonal aspects of a single Hermitian correlator. Definite trajectories are shown to do no explanatory work.
Abstract: We present a computational demonstration of the Zeno Threshold (Zα) on the paradigmatic double-slit experiment. A Gaussian wave packet is propagated on a 256×128 tight-binding lattice using only two ingredients: graph topology (nearest-neighbor hopping) and unitarity (U = e-iHΔt, with the imaginary unit i forced by information conservation). No wave equation is assumed; interference fringes emerge as a strict consequence of lattice unitarity. We then introduce a which-path detector modeled as a tensor-product qubit that entangles with the particle at the slits. The entanglement entropy of the reduced detector state grows from zero toward Smax = ln 2 ≈ 0.693 nats. At the moment this entropy exceeds the observer's finite Zeno Threshold Zα, the observer's rendering engine is forced to perform Born Rule truncation—algorithmic garbage collection—erasing the off-diagonal coherence terms. The interference pattern is destroyed, leaving two classical bumps. Computational Verification Highlights: Emergent Interference: Fringes emerge strictly from discrete graph topology and unitarity, without assuming continuous wave equations. Forced Complex Phases: The complex unit i is mathematically forced by probability conservation (unitarity). The Z-Scale Breach: Which-path detection creates real entanglement entropy that breaches the observer's finite processing limit. Born Rule Garbage Collection: Truncation at the Z-Scale breach produces a detection pattern identical to the classical, incoherent sum |ψ1|2 + |ψ2|2. The full Python verification script and simulation code are available at github.com/srdrymn/atr-double-slit-simulation.
### Version 10 Release Note (Canonical Physical Realization — September 2026)This updated version provides the complete physical realization of the Causal Bridge Operator for quantum interference. Key evolutionary updates include:1. Grounding wave phase evolution in standard paraxial Fresnel wave optics with zero free parameters for spatial fringe spacing (Delta_x = lambda_dB * D / d);2. Binding exposure duration to the physical particle transit time inside the apparatus (tau_on = D / v_particle);3. Decomposing the non-Markovian memory kernel into a dual-layer architecture (K_eff = K_struct x K_thermal) that resolves the thermal decoherence confounder;4. Direct cross-validation against real experimental data: Zeilinger et al. (1988) cold neutron interferometry (reproducing 73.0 um fringe spacing with 0.106% error and calibrating V = 0.583 with lambda_struct = 31.11 s^-1, in agreement with independent decoherence analyses by Sanz et al. 2003) and direct transfer to Jonsson (1961) electrons;5. Rigorous satisfaction of the third-order Sorkin parameter (kappa = 0, error < 10^-15), strictly respecting Born rule bounds;6. Complete typographic overhaul of the manuscript in clean two-column academic LaTeX, eliminating table pagination overflow artifacts. --- ### AbstractStandard quantum mechanics treats wavefunction collapse as a fundamental, non-unitary event and models detection statistics strictly as a function of the instantaneous state vector, independent of the operational history of the apparatus. Here, we present a physical realization of the Causal Bridge Operator, B(H) = exp(i * Gamma(H) / hbar_Omega), anchored upon a non-Markovian memory functional Gamma(H) and standard Fresnel wave propagation. We demonstrate:(i) that B forms an exact homomorphism from the non-commutative history monoid to U(1);(ii) that the fringe spacing Delta_x = lambda_dB * D / d is governed by classical Fresnel wave optics with ZERO free parameters;(iii) that historical memory acts exclusively on the cross-term spatial degree of coherence (fringe visibility), resolving the thermal confounder through a dual-layer decomposition K_eff = K_struct(tau_on) x K_thermal(tau_on, tau_pause);(iv) that the intrinsic exposure time is rigorously bound to the physical flight time tau_on = D / v_particle; and(v) that the formulation identically satisfies the third-order Sorkin parameter (kappa = 0, error < 10^-15), strictly respecting Born's rule bounds. When cross-validated against the landmark cold neutron double-slit experiment of Zeilinger et al. (1988) (lambda_dB = 18.45 A, D = 5.0 m), the model reproduces the published 73.0 um fringe spacing with 0.106% error and calibrates the measured visibility (V = 0.583) with a structural decay rate of lambda_struct = 31.11 s^-1, in close agreement with independent decoherence analyses in quantum optics (~19.7 s^-1; Sanz et al. 2003). Direct transfer to electron diffraction (Jonsson 1961) predicts high-contrast fringes (K_eff ~ 1.0) without parametric readjustment due to the ~10^7x shorter transit time. Finally, observer collapse is shown to emerge naturally from path accessibility truncation (W_j -> 0) without requiring any metaphysical observer protagonism. ### KeywordsProject Omega, Double-Slit Interference, Causal Bridge Operator, Non-Markovian Dynamics, Fresnel Diffraction, Degree of Spatial Coherence, U(1) Homomorphism, Sorkin Parameter, Zeilinger Experiment, Quantum Foundations.
Paper 60B develops a unified Holosphere account of the double-slit experiment while keeping interference loss, measurement, and stable outcome formation conceptually separate. The paper begins with standard slit-resolved paraxial propagation and shows that a coherent two-slit pattern contains a cross term that is absent from the incoherent sum of the two single-slit diffraction patterns. Which-path detection is modeled through the overlap of path-dependent detector states. As those detector states become more distinguishable, fringe visibility decreases continuously and reaches the incoherent limit when the path records are fully distinguishable. The Holosphere interpretation treats a coherence corridor as a physical relation among the source, carrier, aperture, detector, environment, and readout boundary. The apparatus therefore helps determine whether the two paths remain part of one coherence-supporting structure. Boundary incompatibility separates path-conditioned corridors, while later interface competition and pointer-basin capture may select and stabilize a detector record. Physical recoil and apparatus relaxation remain causal, and no faster-than-light signal or strict determinism is required. A separate downstream phase-edge gate produces a modest, symmetric, and resolution-surviving change in the screen distribution. The numerical suite also verifies symmetry, attenuation accounting, slit-field linearity, detector-overlap behavior, and the conditioned quantum-eraser identity.
T63 establishes a conditional uniqueness theorem for the interference and amplitude-square structures developed in T61 and T62. Starting from the reduced two-channel interference form,Ptot=P1+P2+G(P1,P2,ΔΩ),P_{\mathrm{tot}}=P_1+P_2+G(P_1,P_2,\Delta\Omega),Ptot​=P1​+P2​+G(P1​,P2​,ΔΩ),The theorem analyzes the structural constraints imposed by symmetry, continuity, axis-vanishing behaviour, degree-1 homogeneity, and first-harmonic rotational structure generated by the reduced antisymmetric operator algebra. Under these assumptions, the interference contribution is forced into the normal formG(P1,P2,ΔΩ)=2P1P2 sin⁡(ΔΩ),G(P_1,P_2,\Delta\Omega)=2\sqrt{P_1P_2}\,\sin(\Delta\Omega),G(P1​,P2​,ΔΩ)=2P1​P2​​sin(ΔΩ),yielding the exact amplitude-square representationPtot=∣P1+eiϕP2∣2.P_{\mathrm{tot}}=\left|\sqrt{P_1}+e^{i\phi}\sqrt{P_2}\right|^2.Ptot​=​P1​​+eiϕP2​​​2.The theorem therefore shows that, within the stated structural class, the Born-compatible amplitude-square form is uniquely selected by the reduced interference geometry together with the harmonic structure induced by the rotational generator satisfying A2=−IA^2=-IA2=−I. T63 does not derive full quantum mechanics, measurement theory, Hilbert-space axioms, or probabilistic interpretation from first principles. The result is a conditional uniqueness theorem for the algebraic normal form of two-channel interference under the stated assumptions. In particular, the homogeneity assumption is essential: without it, uniqueness does not hold. The theorem further clarifies that the complex amplitude representation is obtained from the interference structure rather than independently postulated. Status: solid for the conditional uniqueness of the amplitude-square normal form under the stated symmetry, homogeneity, and harmonic assumptions; conditional on the separability and structural assumptions defining the interference class; speculative for any claim of a complete derivation of quantum probability theory or the Born rule from Q5 geometry alone.