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arXiv · 2605.02931

Pairwise-comparison-valued cosurfaces: a projective framework for multi-scale relational structures

Jean-Pierre Magnot
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Abstract

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.

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Jean-Pierre Magnot. 2026-04-28. Pairwise-comparison-valued cosurfaces: a projective framework for multi-scale relational structures. https://arxiv.org/abs/2605.02931

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