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| Hlavní autor: | |
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| Médium: | Recurso digital |
| Jazyk: | angličtina |
| Vydáno: |
Zenodo
2025
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| Témata: | |
| On-line přístup: | https://doi.org/10.5281/zenodo.17864384 |
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- <p>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.</p> <p>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 (<a href="https://doi.org/10.5281/zenodo.17851714">Kappa Law</a>), topological rigidity (<a href="https://doi.org/10.5281/zenodo.17823404">Law of Endogenous Constraint</a>), 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.</p> <p>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.</p> <p><strong>Relation to Tier-1 dynamical formulation:</strong></p> <p>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:</p> <p>J. Rodgers, <em>Saturation Geometry and the Structural Emergence of Measurement and the Born Rule in a Capacity-Constrained Dirac–Λ System</em>, Zenodo (2026).<br><a href="https://doi.org/10.5281/zenodo.18704783" target="_new" rel="noopener">https://doi.org/10.5281/zenodo.18704783</a></p> <p>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.</p>