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

TL;DR

This paper develops a multi-scale, geometric framework for relational structures using cosurfaces valued in reciprocal comparison matrices, integrating algebraic, probabilistic, and geometric perspectives.

Contribution

It introduces a novel projective framework for multi-scale relational data based on cosurfaces with values in reciprocal comparison matrices, incorporating consistency and inconsistency measures.

Findings

Defines a multi-scale organization of local comparative data

Constructs a projective limit of finite configuration spaces

Introduces inconsistency observables as curvature-type defects

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.