Role-Invariant Algebra

This Zenodo record contains the source materials for ROLE-INVARIANT ALGEBRA by Daniel Hendriks, together with the accompanying finite computational prototype. Role-Invariant Algebra is an algebraic framework for studying the global consistency of mathematical descriptions whose local roles depend on a chosen context. Its central object is a context atlas: a finite graph of commutative contexts, typically maximal abelian subalgebras of a finite-dimensional matrix algebra, equipped with specified admissible transports between them. Each context provides a local system of minimal variable labels. Transport along an edge identifies the labels of neighbouring contexts, while transport around a closed path may return to the original context with its labels permuted. The resulting role monodromy is invariant under local gauge relabelling and measures the obstruction to choosing one globally coherent labelling throughout the atlas. The associated role defect records the fraction of labels that are not fixed by all closed context loops. The paper develops this finite-dimensional theory from first principles. It introduces: commutative contexts and conditional-expectation decompositions; context graphs and admissible transports; role monodromy and gauge invariance; the obstruction theorem for global role labelling; the associated role cover and orbit structure; quantitative bounds connecting local transport errors to global role defect; perturbation results for certified transports and moving contexts; filtered context atlases and monotone defect profiles. After local labels have been chosen, the edge transports determine a non-abelian Sₙ-valued cocycle on the context graph. Its holonomy representation ρ: π₁(Γ, v₀) → Sₙ contains the finite Role-Invariant, up to simultaneous conjugation. A complete global role labelling exists precisely when this monodromy representation is trivial. More generally, the labels that remain globally meaningful are exactly the fixed points of the monodromy image.The article emphasizes that this invariant is not determined by the spectral data of individual operators or individual loop transports. Global role coherence depends on the joint action of all admissible context loops. Different atlases may have locally similar or even spectrally indistinguishable transports while producing different monodromy groups, orbit structures, and role defects. A central practical example concerns multi-view asset association. Several local maps, camera views, inspection records, or sensor observations may describe the same finite set of assets using different local identifiers. Accepted pairwise associations provide transports between these local label systems. The Role-Invariant then tests whether the accepted associations are globally cycle-consistent. Nontrivial monodromy identifies labels that cannot be interpreted as globally reliable asset identities, while the role defect quantifies the extent of the inconsistency. A spanning-tree analysis also isolates which additional loop-closing associations introduce the obstruction and provides a minimal finite repair diagnostic. This application is presented as a deterministic consistency layer. It does not replace probabilistic data association, statistical estimation, computer-vision matching, or SLAM algorithms. Instead, it supplies an exact algebraic certificate after a finite collection of local associations has been accepted. The framework is positioned as a focused mathematical layer over established ideas from operator algebras, group actions, graph holonomy, non-abelian cocycles, covering spaces, and gluing theory. It does not claim to replace set theory, category theory, type theory, sheaf theory, differential geometry, operator algebra, persistent homology, or any existing foundation of mathematics. Its purpose is to isolate one specific structural question: When can locally defined descriptive roles be transported consistently through every admissible change of context? The accompanying computational prototype implements the finite theory and a number of experimental extensions. It includes exact monodromy and defect calculations, permutation-group routines, gauge and cocycle canonicalization, numerical transport certification, filtered context atlases, persistent role-defect profiles, finite-prefix tower experiments, and diagnostic examples. For a filtered or sampled atlas, the computational route is: sampled or weighted context data -> filtered finite context atlas -> role monodromy and orbit structure -> fixed labels and role defect -> defect profile across scales These computations are finite and inspectable. They provide a reproducible research platform for studying robustness, refinement, constrained realizability, dynamic updates, and possible continuum extensions. The software does not claim to compute a completed invariant of arbitrary continuous or infinite-dimensional algebraic systems.