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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19967674 |
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| _version_ | 1866901212355362816 |
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| author | Washburn |
| author_facet | Washburn |
| contents | <p>The companion paper [1] proves a conditional rigidity theorem on the continuous positive-ratio comparison setting: under the four classical Aristotelian conditions on a comparison operator, plus continuity and finite pairwise polynomial closure of the combiner, the derived cost function satisfies the Recognition Composition Law and (under unit log-curvature calibration) equals the canonical reciprocal cost J(x) = ½(x + x⁻¹) − 1. Conditional on that rigidity theorem (and on the inductive-type, quotient, and completion machinery of the ambient meta-language), we exhibit the standard Peano structure as the canonical orbit of any non-trivial generator inside the multiplicative group ℝ₍>0₎, lift the construction through the integers, rationals, reals, and the complex carrier, and identify each layer up to a transport equivalence with the usual number tower. The Peano core carries no domain-specific axiomatic content; the upper layers reuse standard algebraic and completion machinery through explicit transport equivalences. The companion meta-theorem [2], the Universal Forcing Theorem, asserts that in every admissible realization of the Law of Logic the same arithmetic structure is canonically forced up to unique isomorphism; the present paper supplies the concrete continuous-ratio realization on which that meta-theorem rests. The downstream consequence is that the elementary number tower used by Recognition Science is forced by a single law, realised concretely here on continuous positive ratios and invariant across all admissible realizations of that law, rather than imported as a separate axiomatic commitment, conditional on the named hypotheses recorded in Section 1.1.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19967674 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Arithmetic from the Law of Logic: The Continuous Positive-Ratio Realization Washburn <p>The companion paper [1] proves a conditional rigidity theorem on the continuous positive-ratio comparison setting: under the four classical Aristotelian conditions on a comparison operator, plus continuity and finite pairwise polynomial closure of the combiner, the derived cost function satisfies the Recognition Composition Law and (under unit log-curvature calibration) equals the canonical reciprocal cost J(x) = ½(x + x⁻¹) − 1. Conditional on that rigidity theorem (and on the inductive-type, quotient, and completion machinery of the ambient meta-language), we exhibit the standard Peano structure as the canonical orbit of any non-trivial generator inside the multiplicative group ℝ₍>0₎, lift the construction through the integers, rationals, reals, and the complex carrier, and identify each layer up to a transport equivalence with the usual number tower. The Peano core carries no domain-specific axiomatic content; the upper layers reuse standard algebraic and completion machinery through explicit transport equivalences. The companion meta-theorem [2], the Universal Forcing Theorem, asserts that in every admissible realization of the Law of Logic the same arithmetic structure is canonically forced up to unique isomorphism; the present paper supplies the concrete continuous-ratio realization on which that meta-theorem rests. The downstream consequence is that the elementary number tower used by Recognition Science is forced by a single law, realised concretely here on continuous positive ratios and invariant across all admissible realizations of that law, rather than imported as a separate axiomatic commitment, conditional on the named hypotheses recorded in Section 1.1.</p> |
| title | Arithmetic from the Law of Logic: The Continuous Positive-Ratio Realization |
| url | https://doi.org/10.5281/zenodo.19967674 |