Pairwise-comparison-valued cosurfaces: a projective framework for multi-scale relational structures
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914528430653440 |
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| author | Magnot, Jean-Pierre |
| author_facet | Magnot, Jean-Pierre |
| contents | We introduce cosurfaces with values in the group \(\PC_n(H)\) of \(H\)-valued reciprocal pairwise comparison matrices. The composition law is covariant on upper triangular coefficients and contravariant on lower triangular coefficients, which makes \(\PC_n(H)\) a natural target for oriented gluing constructions. Starting from a directed family of finite oriented discretizations, we define finite configuration spaces, coarse-graining maps induced by ordered refinements, and the associated universal projective limit. This yields a multi-scale organization of local comparative data in which global objects are reconstructed only through compatibility across scales. In the stochastic setting, projectively compatible probability laws define a cylindrical semantics on the limit space. We also introduce inconsistency observables, interpreted as discrete curvature-type defects measuring obstructions to global coherence. The resulting framework is simultaneously geometric, algebraic, and probabilistic, and suggests a foundational perspective on relational structures built from local comparisons rather than absolute observables. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02931 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pairwise-comparison-valued cosurfaces: a projective framework for multi-scale relational structures Magnot, Jean-Pierre General Physics Primary 18A30, Secondary 15B99, 28A33, 60B10, 81T13 We introduce cosurfaces with values in the group \(\PC_n(H)\) of \(H\)-valued reciprocal pairwise comparison matrices. The composition law is covariant on upper triangular coefficients and contravariant on lower triangular coefficients, which makes \(\PC_n(H)\) a natural target for oriented gluing constructions. Starting from a directed family of finite oriented discretizations, we define finite configuration spaces, coarse-graining maps induced by ordered refinements, and the associated universal projective limit. This yields a multi-scale organization of local comparative data in which global objects are reconstructed only through compatibility across scales. In the stochastic setting, projectively compatible probability laws define a cylindrical semantics on the limit space. We also introduce inconsistency observables, interpreted as discrete curvature-type defects measuring obstructions to global coherence. The resulting framework is simultaneously geometric, algebraic, and probabilistic, and suggests a foundational perspective on relational structures built from local comparisons rather than absolute observables. |
| title | Pairwise-comparison-valued cosurfaces: a projective framework for multi-scale relational structures |
| topic | General Physics Primary 18A30, Secondary 15B99, 28A33, 60B10, 81T13 |
| url | https://arxiv.org/abs/2605.02931 |