Three-Sign Cancellation Hypernumber Systems and Associator Curvature
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866911467074224128 |
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| author | Kim, Jaehwan |
| author_facet | Kim, Jaehwan |
| contents | We introduce and study a three-sign cancellation hypernumber system $H$ which extends the real field by adjoining a third sign $Λ$. The underlying set is $H=\{0\}\cup\{+,-,Λ\}\times\mathbb{R}_{>0}$, with a single-valued multiplication and a hyperaddition $\boxplus$ designed to encode cancellation phenomena between positive and negative reals. The classical real line embeds as a genuine subfield $\mathbb{R}_{\mathrm{cl}}\subset H$, and all field operations agree with the usual ones on $\mathbb{R}$. The additive structure of $H$ is almost associative but not a canonical hypergroup. We give an explicit description of where associativity fails and compute, for triples of the form $(+,a),(-,b),(Λ,c)$, a closed formula for the associativity defect $κ(a,b,c)=2\min(a,b)=a+b-|a-b|$, which coincides with the loss of absolute value when adding $a$ and $-b$ in $\mathbb{R}$. To explain this behaviour, we construct an ambient ''cancellation monoid'' $(K,\oplus)$ on $\mathbb{R}\times\mathbb{R}_{\ge 0}$ which is strictly associative and records both real sums and accumulated cancellation mass. We prove that $H$ cannot be recovered from $K$ by any simple projection, and formulate an ambient reconstruction problem. In addition, scalar multiplication by real numbers (defined via the embedded copy of $\mathbb{R}$) distributes over $\boxplus$, and the sign-layer admits a canonical hypergroup envelope governing the possible signs of hypersums. The results provide a controlled example of a nonassociative hyperaddition sitting over the real field and suggest several directions for multisign generalizations and connections with hyperfields and tropical geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21207 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Three-Sign Cancellation Hypernumber Systems and Associator Curvature Kim, Jaehwan Rings and Algebras Group Theory 16Y99, 20N20, 14T05 We introduce and study a three-sign cancellation hypernumber system $H$ which extends the real field by adjoining a third sign $Λ$. The underlying set is $H=\{0\}\cup\{+,-,Λ\}\times\mathbb{R}_{>0}$, with a single-valued multiplication and a hyperaddition $\boxplus$ designed to encode cancellation phenomena between positive and negative reals. The classical real line embeds as a genuine subfield $\mathbb{R}_{\mathrm{cl}}\subset H$, and all field operations agree with the usual ones on $\mathbb{R}$. The additive structure of $H$ is almost associative but not a canonical hypergroup. We give an explicit description of where associativity fails and compute, for triples of the form $(+,a),(-,b),(Λ,c)$, a closed formula for the associativity defect $κ(a,b,c)=2\min(a,b)=a+b-|a-b|$, which coincides with the loss of absolute value when adding $a$ and $-b$ in $\mathbb{R}$. To explain this behaviour, we construct an ambient ''cancellation monoid'' $(K,\oplus)$ on $\mathbb{R}\times\mathbb{R}_{\ge 0}$ which is strictly associative and records both real sums and accumulated cancellation mass. We prove that $H$ cannot be recovered from $K$ by any simple projection, and formulate an ambient reconstruction problem. In addition, scalar multiplication by real numbers (defined via the embedded copy of $\mathbb{R}$) distributes over $\boxplus$, and the sign-layer admits a canonical hypergroup envelope governing the possible signs of hypersums. The results provide a controlled example of a nonassociative hyperaddition sitting over the real field and suggest several directions for multisign generalizations and connections with hyperfields and tropical geometry. |
| title | Three-Sign Cancellation Hypernumber Systems and Associator Curvature |
| topic | Rings and Algebras Group Theory 16Y99, 20N20, 14T05 |
| url | https://arxiv.org/abs/2602.21207 |