The Infinium (△₁ₓ₁): The Geometric Quantum that Generates Mathematics A Unified Review Article: From Measure and Smoothness to a Universal Categorical Scheme, Motivic Foundation, and Formal Logic (2)
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2026
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| _version_ | 1866902025517662208 |
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| author | Petrov, Alexey (KAMAZ) |
| author_facet | Petrov, Alexey (KAMAZ) |
| contents | <p>This work presents a comprehensive overview of the main results of △‑ontology — a new approach to the foundations of mathematics, in which the foundation is the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2). It is shown how the Lebesgue measure, smoothness, metric, nilpotence, as well as all known types of numbers and spaces grow out of this single geometric quantum. The resolution of the ∼/# conflict in Synthetic Differential Geometry (SDG) and the connection with the Collatz conjecture are demonstrated. A rigorous categorical formulation of the Theory of Relational Differentials (TRD) is presented, with universal closure, a monoidal structure of self‑similarity, and the spectral gap λ₁ = 1 – ½√2. A complete categorical scheme is given. Then the motivic foundation is constructed: the infinium as an elementary motive M(ℑ) = ℚ(0) ⊕ ℚ(1)[1] ⊕ ℚ(1)[√2], and it is shown how L‑functions, the BSD conjecture, the Riemann Hypothesis, and mirror symmetry grow out of this motive. A new logical unit replacing the point is formulated, and a formal axiomatics in the form of type theory is given. The article concludes with a logical closure: all results are structural truths forced by the existence of the infinium.</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20366543 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Infinium (△₁ₓ₁): The Geometric Quantum that Generates Mathematics A Unified Review Article: From Measure and Smoothness to a Universal Categorical Scheme, Motivic Foundation, and Formal Logic (2) Petrov, Alexey (KAMAZ) infinium, △‑ontology, SDG, Lebesgue measure, smoothness, metric, nilpotence, Collatz conjecture, Pythagorean theorem, categorical scheme, spectral gap, generation of spaces, forcing, semantic consequence. Mathematics, Category Theory, Differential Geometry, Foundations of Mathematics. --- <p>This work presents a comprehensive overview of the main results of △‑ontology — a new approach to the foundations of mathematics, in which the foundation is the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2). It is shown how the Lebesgue measure, smoothness, metric, nilpotence, as well as all known types of numbers and spaces grow out of this single geometric quantum. The resolution of the ∼/# conflict in Synthetic Differential Geometry (SDG) and the connection with the Collatz conjecture are demonstrated. A rigorous categorical formulation of the Theory of Relational Differentials (TRD) is presented, with universal closure, a monoidal structure of self‑similarity, and the spectral gap λ₁ = 1 – ½√2. A complete categorical scheme is given. Then the motivic foundation is constructed: the infinium as an elementary motive M(ℑ) = ℚ(0) ⊕ ℚ(1)[1] ⊕ ℚ(1)[√2], and it is shown how L‑functions, the BSD conjecture, the Riemann Hypothesis, and mirror symmetry grow out of this motive. A new logical unit replacing the point is formulated, and a formal axiomatics in the form of type theory is given. The article concludes with a logical closure: all results are structural truths forced by the existence of the infinium.</p> <p> </p> |
| title | The Infinium (△₁ₓ₁): The Geometric Quantum that Generates Mathematics A Unified Review Article: From Measure and Smoothness to a Universal Categorical Scheme, Motivic Foundation, and Formal Logic (2) |
| topic | infinium, △‑ontology, SDG, Lebesgue measure, smoothness, metric, nilpotence, Collatz conjecture, Pythagorean theorem, categorical scheme, spectral gap, generation of spaces, forcing, semantic consequence. Mathematics, Category Theory, Differential Geometry, Foundations of Mathematics. --- |
| url | https://doi.org/10.5281/zenodo.20366543 |