Additivity Symmetry III: The Activation Theorem
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| Lenguaje: | inglés |
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2025
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| _version_ | 1866902046569922560 |
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| author | Treppiedi, Attilio |
| author_facet | Treppiedi, Attilio |
| contents | <p> </p> <div> <div> <div>Translator</div> <div> </div> <div> </div> </div> <div> <div> </div> <div> </div> </div> </div> <p> </p> <p> </p> <div> <div> <div>We establish the Activation Theorem, showing that the additive group (R, +), when realized<br>through its constraint representation, necessarily gives rise to three and only three admissible<br>regimes of physical description. The constraint representation is an intrinsic structure of the<br>additive algebra, characterized by four fundamental features (FR1FR4): nite resolution, con-<br>stants as multipliers, phase consistency, and typicality. These features are not postulated but<br>follow directly from the algebraic properties of admissible representations of (R, +).<br>We prove that FR1FR4 are sucient to activate the three regimes of physics: (i) the quan-<br>tum regime, where Hilbert space kinematics and the Born rule arise from Bochner's theorem,<br>the GNS construction, and invariant averaging; (ii) the statistical regime, where Kolmogorov<br>probability and the LévyKhintchine formula emerge from the convolution semigroup structure;<br>and (iii) the topological regime, where extremal states reduce to Dirac codes and invariant<br>averages yield deterministic 0/1 truth values.<br>Each regime is derived by rigorous constructive proofs based solely on harmonic analysis,<br>operator algebras, probability theory, and convex state spaces. The Activation Theorem thus<br>provides the foundational algebraic result from which the quantum, statistical, and topological<br>descriptions of nature arise as necessary and mutually consistent channels of the same additive<br>symmetry under finite resolution.<br><br></div> </div> <div> </div> </div> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17264768 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Additivity Symmetry III: The Activation Theorem Treppiedi, Attilio Additivity Symmetry Activation Theorem closure structural constraint euler-lagrange admissability boundary conditions gauge regularity sieve conjecture finite resolution probes quantum regimes stastical mechanics discrete regimes representation theory operator algebras topological quantum field theory (TQFT) <p> </p> <div> <div> <div>Translator</div> <div> </div> <div> </div> </div> <div> <div> </div> <div> </div> </div> </div> <p> </p> <p> </p> <div> <div> <div>We establish the Activation Theorem, showing that the additive group (R, +), when realized<br>through its constraint representation, necessarily gives rise to three and only three admissible<br>regimes of physical description. The constraint representation is an intrinsic structure of the<br>additive algebra, characterized by four fundamental features (FR1FR4): nite resolution, con-<br>stants as multipliers, phase consistency, and typicality. These features are not postulated but<br>follow directly from the algebraic properties of admissible representations of (R, +).<br>We prove that FR1FR4 are sucient to activate the three regimes of physics: (i) the quan-<br>tum regime, where Hilbert space kinematics and the Born rule arise from Bochner's theorem,<br>the GNS construction, and invariant averaging; (ii) the statistical regime, where Kolmogorov<br>probability and the LévyKhintchine formula emerge from the convolution semigroup structure;<br>and (iii) the topological regime, where extremal states reduce to Dirac codes and invariant<br>averages yield deterministic 0/1 truth values.<br>Each regime is derived by rigorous constructive proofs based solely on harmonic analysis,<br>operator algebras, probability theory, and convex state spaces. The Activation Theorem thus<br>provides the foundational algebraic result from which the quantum, statistical, and topological<br>descriptions of nature arise as necessary and mutually consistent channels of the same additive<br>symmetry under finite resolution.<br><br></div> </div> <div> </div> </div> <p> </p> |
| title | Additivity Symmetry III: The Activation Theorem |
| topic | Additivity Symmetry Activation Theorem closure structural constraint euler-lagrange admissability boundary conditions gauge regularity sieve conjecture finite resolution probes quantum regimes stastical mechanics discrete regimes representation theory operator algebras topological quantum field theory (TQFT) |
| url | https://doi.org/10.5281/zenodo.17264768 |