This record provides the full paper and reproducibility package for UDCT v4.5, which investigates the dynamical emergence of Born Rule statistics within the Double-Slit Born-Backreaction (DSBB) framework — originally introduced in v4.4 as Dynamic Self-Regulating Bidirectional Back-Reaction. In this work, we introduce a measurement-induced deformation of the Stability Landscape through the Detector Freezing mechanism. By constructing a Free Energy Functional combining DSBB feedback energy with a Shannon entropy term, we demonstrate that the system relaxes toward a biased bimodal probability distribution. A perturbative analysis yields an explicit zeroth-order solution that reproduces the main features of full two-dimensional simulations with a minimum L2 relative error of only 2.41%. This result suggests that Born Rule-like statistics can emerge dynamically from the interplay between unitary evolution and geometric back-reaction, rather than being introduced as an independent postulate. Future directions include developing a more rigorous first-principles derivation from functional Fokker-Planck dynamics and extending the scalar feedback field to a chiral restoring tensor in four-dimensional spacetime. The reproducibility package contains the full paper and a self-contained Python script that allows anyone to exactly reproduce the main quantitative results reported in the paper (including the 2.41% L2 error and optimized parameters). Researchers are encouraged to download the code and run Mode A, which reproduces the paper’s benchmark results using the original dataset. □ Related Work This work builds upon the DSBB framework originally introduced in: Paik, W. S. (2026). UDCT v4.4: DSBB Double-Slit Framework and Stability Landscape Dynamics. Zenodo. https://doi.org/10.5281/zenodo.20793605
This record provides the full paper and reproducibility package for UDCT v4.5, which investigates the dynamical emergence of Born Rule statistics within the Double-Slit Born-Backreaction (DSBB) framework — originally introduced in v4.4 as Dynamic Self-Regulating Bidirectional Back-Reaction. In this work, we introduce a measurement-induced deformation of the Stability Landscape through the Detector Freezing mechanism. By constructing a Free Energy Functional combining DSBB feedback energy with a Shannon entropy term, we demonstrate that the system relaxes toward a biased bimodal probability distribution. A perturbative analysis yields an explicit zeroth-order solution that reproduces the main features of full two-dimensional simulations with a minimum L2 relative error of only 2.41%. This result suggests that Born Rule-like statistics can emerge dynamically from the interplay between unitary evolution and geometric back-reaction, rather than being introduced as an independent postulate. Future directions include developing a more rigorous first-principles derivation from functional Fokker-Planck dynamics and extending the scalar feedback field to a chiral restoring tensor in four-dimensional spacetime. The reproducibility package contains the full paper and a self-contained Python script that allows anyone to exactly reproduce the main quantitative results reported in the paper (including the 2.41% L2 error and optimized parameters). Researchers are encouraged to download the code and run Mode A, which reproduces the paper’s benchmark results using the original dataset. □ Related Work This work builds upon the DSBB framework originally introduced in: Paik, W. S. (2026). UDCT v4.4: DSBB Double-Slit Framework and Stability Landscape Dynamics. Zenodo. https://doi.org/10.5281/zenodo.20793605
UDCT v4.7: Parametric Robustness of the Emergent Born Rule in the DSBB Stability Landscape This paper presents a comprehensive One-Factor-at-a-Time (OFAT) robustness verification of the dynamically emergent Born Rule within the Double-Slit Born-Backreaction (DSBB) framework of the Unified Dynamic Cohesion Theory (UDCT v4.7). Building on the geometric derivation established in UDCT v4.5, we subject the DSBB system to twelve independent parametric perturbations across four key parameters: wavepacket width σ (±10%), detector freezing time t_freeze (±0.1), coupling strength λ (±20%), and effective temperature k_eff (±20%). Results show that the L2 relative error between the full 2D microscopic simulation and the theoretical zeroth-order solution remains tightly bounded within ±0.44% of the baseline value (2.41%), while preserving 100% bimodal structural integrity in all cases. These findings provide strong quantitative evidence that the emergent Born-like probability distribution functions as a robust geometric attractor inherent to the DSBB Stability Landscape. A major goal of this work is full reproducibility. The accompanying reproducibility package implements a dual-mode execution system (Mode A: Paper Reproduction Mode; Mode B: Full Scratch Mode) together with a self-healing environment designed to be accessible even to undergraduate students. This work establishes a high-precision, reproducible benchmark for future investigations into the geometric origin of quantum probability.
We present a detailed numerical investigation of the internal encoding of path information in the double-slit experiment within the Unified Dimensional Cohesion Theory (UDCT) v4.4 framework. We replace the previous static geometric memory kernel with a Dynamic Self-Regulating Bidirectional Back-Reaction (DSBB) paradigm. In this framework, the spacetime geometry itself dynamically responds to curvature errors induced by the projected wavefunction through a self-regulating feedback field B(r,t), whose damping is controlled by Gamma(B) = (1/tau_0)(1 + beta |B|^2). This eliminates any ad-hoc memory history while enabling real-time bidirectional coupling between the probability density and the effective geometric potential V_DSBB. Using the split-step Fourier method on a 256×256 grid, we demonstrate in two-dimensional simulations with a physically realistic double-slit barrier that the DSBB field develops clear spatial asymmetry between the two paths, internally encoding which-path information without external measurement. Furthermore, we introduce a Detector Freezing Mechanism that overwhelms the self-regulating loop at mid-evolution, driving the system toward classical bimodal distributions. Our results show near-perfect unitarity (norm conservation to \~10^{-15}) and provide concrete evidence that dynamic self-regulation in spacetime geometry can resolve the measurement problem and path distinguishability in a physically transparent manner. The DSBB framework offers significant conceptual and numerical advantages over the static memory approach of v4.3.
This package provides an educational reproducibility tool for exploring the dynamical emergence of the Born Rule based on the Unified Dynamic Cohesion Theory (UDCT). It includes:- An educational manual explaining the theoretical background, Stability Landscape, and Detector Freezing mechanism- A Python reproducibility package with visualization features (Mode A) designed for learning and verification purposes This release (v4.7.1) focuses on educational usability, including self-healing features and clear explanations connecting theory with code implementation. The package is intended for undergraduate students, self-learners, and researchers interested in the foundations of quantum mechanics and alternative interpretations of the Born Rule.
Unified Dimensional Cohesion Theory (UDCT v4.2) proposes that quantum phenomena such as double-slit interference can arise from the deterministic interaction between localized energy and a dynamically responsive microscopic spacetime endowed with memory. In this work, we present extensive two-dimensional numerical simulations to investigate this hypothesis. We implemented causal memory kernels acting on the Minkowski metric and systematically explored particle dynamics in a double-slit geometry. Through comprehensive parameter studies, we identified three distinct dynamical regimes: a weak-coupling regime recovering classical trajectories, an over-coupling regime characterized by chaotic scattering, and an intermediate regime of controlled spreading induced by geometric memory. Among the memory kernels examined, a damped oscillatory kernel produced the strongest suppression of trajectory spreading, achieving a minimum standard deviation of approximately 8.4. These results demonstrate that microscopic metric memory can exert significant deterministic influence on particle trajectories. However, clear bimodal interference patterns have not yet been observed, and stronger spreading control was often accompanied by reduced detection efficiency. Our findings establish a quantitative foundation for exploring geometric mechanisms underlying quantum phenomena while highlighting key challenges that remain for reproducing full interference patterns through spacetime hysteresis alone.
This paper extends the Unified Dimensional Cohesion Theory (UDCT v4.0) to the quantum domain by replacing the probabilistic foundations of quantum mechanics with a deterministic, geometric framework. We argue that the long-standing difficulties of quantum mechanics, including the measurement problem and the double-slit paradox, arise not from the intrinsic nature of particles, but from the structural assumption that spacetime provides a static, inert background. By extending the dynamic compactification mechanism into the microscopic regime, we introduce a non-local temporal memory kernel and dynamic geometric back-reaction at the quantum scale. Within this framework, quantum behavior emerges as the deterministic response of localized energy densities interacting with a living, responsive higher-dimensional geometry. This model offers a new geometric paradigm that aims to establish a physically transparent description of quantum foundations.
This work provides a structural derivation of the quadratic form of the Born rule in quantum mechanics. Rather than postulating the probability rule as a foundational axiom, the paper proves that the exponent governing quantum probabilities is uniquely fixed by stability, contractivity, and duality requirements imposed on admissible physical laws. The analysis is carried out within a general operator-theoretic and law-space framework, in which physical laws are treated as fixed points of recursive generative dynamics. By combining strict contractive flow (Kappa Law), topological rigidity (Law of Endogenous Constraint), and algebra–geometry duality (Monad Duality), the paper demonstrates that only the Hilbert–Schmidt geometry is compatible with global stability of law evolution. As a consequence, the quadratic probability rule emerges as a rigid structural invariant rather than a free modeling choice. The result implies that any hypothetical modification of quantum theory based on non-quadratic probability rules would necessarily violate at least one of the fundamental structural requirements of admissible physical law, such as uniform contractivity, spectral stability, or duality consistency. Relation to Tier-1 dynamical formulation: This work develops a law-level (structural) resolution of the Born exponent. A complementary Tier-1 dynamical realization, formulated entirely within a constrained operator framework and independent of law-level recursion language, is developed in: J. Rodgers, Saturation Geometry and the Structural Emergence of Measurement and the Born Rule in a Capacity-Constrained Dirac–Λ System, Zenodo (2026).https://doi.org/10.5281/zenodo.18704783 The Tier-1 formulation may be read independently and derives the quadratic Born rule as a consequence of modular implementability and saturation geometry within the coupled Dirac–Λ system.