The Math of Fields: A Geometric Theory of Legitimate Entry, Faithful Traversal, Coherent Persistence, and Field-Local Wisdom
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2026
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| _version_ | 1866901683868532736 |
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| author | Figurelli, Rogério |
| author_facet | Figurelli, Rogério |
| contents | <p>Field concepts recur whenever object-only description becomes too thin to carry distributed structure, lawful propagation, admissible transformation, and stable return across heterogeneous regimes. Physics, geometry, information, cognition, and modern machine systems all exhibit this recurrence. Yet the mathematics of fields remains fragmented across domains: physical fields are one discourse, statistical manifolds another, symbolic or institutional regimes another, and machine-mediated environments yet another. Our work proposes a bounded unification. The thesis is not that these regimes are ontologically identical, nor that one master substance explains them all. The narrower and stronger claim is that they share a common transport grammar whenever signals, meanings, and actions must travel without unacceptable collapse. We formalize that grammar through three route conditions: legitimate entry, faithful traversal, and coherent persistence. These are compressed into the field-local intelligence equation I_F = ΛΠΞ and the field-local wisdom equation W_F = (I_F)^(C_F), while preserving the generic law W = I^C. Here I_F denotes field intelligence, C_F denotes field consciousness, and W_F denotes field wisdom. This distinction is central. Our work does not redefine intelligence or consciousness in generic terms. It declares their field-specific forms under explicit field conditions. The paper develops a geometric field object, defines receivability, admissibility, coupling, transportability, semantic curvature, continuity, and holonomy, and shows how these terms compose into a route discipline for complex problems. It further situates the proposal relative to Einsteinian geometry, variational mechanics, gauge structure, thermodynamics, information geometry, category theory, cybernetics, and ecological perception. Finally, it derives an operational program for diagnosis and governed machine use. The result is a mathematical framework in which many difficult failures can be re-read not merely as failures of solution, but as failures of field traversal.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19561741 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The Math of Fields: A Geometric Theory of Legitimate Entry, Faithful Traversal, Coherent Persistence, and Field-Local Wisdom Figurelli, Rogério fields geometry transport admissibility semantic curvature intelligence consciousness wisdom complex problems <p>Field concepts recur whenever object-only description becomes too thin to carry distributed structure, lawful propagation, admissible transformation, and stable return across heterogeneous regimes. Physics, geometry, information, cognition, and modern machine systems all exhibit this recurrence. Yet the mathematics of fields remains fragmented across domains: physical fields are one discourse, statistical manifolds another, symbolic or institutional regimes another, and machine-mediated environments yet another. Our work proposes a bounded unification. The thesis is not that these regimes are ontologically identical, nor that one master substance explains them all. The narrower and stronger claim is that they share a common transport grammar whenever signals, meanings, and actions must travel without unacceptable collapse. We formalize that grammar through three route conditions: legitimate entry, faithful traversal, and coherent persistence. These are compressed into the field-local intelligence equation I_F = ΛΠΞ and the field-local wisdom equation W_F = (I_F)^(C_F), while preserving the generic law W = I^C. Here I_F denotes field intelligence, C_F denotes field consciousness, and W_F denotes field wisdom. This distinction is central. Our work does not redefine intelligence or consciousness in generic terms. It declares their field-specific forms under explicit field conditions. The paper develops a geometric field object, defines receivability, admissibility, coupling, transportability, semantic curvature, continuity, and holonomy, and shows how these terms compose into a route discipline for complex problems. It further situates the proposal relative to Einsteinian geometry, variational mechanics, gauge structure, thermodynamics, information geometry, category theory, cybernetics, and ecological perception. Finally, it derives an operational program for diagnosis and governed machine use. The result is a mathematical framework in which many difficult failures can be re-read not merely as failures of solution, but as failures of field traversal.</p> |
| title | The Math of Fields: A Geometric Theory of Legitimate Entry, Faithful Traversal, Coherent Persistence, and Field-Local Wisdom |
| topic | fields geometry transport admissibility semantic curvature intelligence consciousness wisdom complex problems |
| url | https://doi.org/10.5281/zenodo.19561741 |