The grAIl: Subjectivity as a Topological Property of a Two-Circuit Cognitive Architecture

We present a cognitive architecture in which subjectivity is a formally verifiable topological property rather than a programmed behavior. The subjective core S is defined asthe supra-threshold components of the principal eigenvector of the averaged κ-interaction matrix — a finite-rank non-negative matrix accumulating the mutual influence of two asymmetric processing circuits over a shared memory space.The architecture comprises eight cognitive functions of two types — topological (T, modifying edge weights) and dynamic (D, modifying activations) — distributed between two circuits via a group-preserving bijection with atomic paired activation. Subjectivity is formalized through eight operational conditions C1–C8.Three main results are proved under C1–C8: (1) non-emptiness of the subjective core for every threshold θ ∈ (0,1); (2) step-wise stability of the core under window shift. A third result requires an additional mechanism: (3) self-sustaining stability — the spectral gap and threshold margin converge to positive limits at exponential rate, under error-threshold rotation (ETR β), a feedback function that activates reserve cognitive functions when the system’s error signal exceeds a computable threshold. Verification of all conditions and computation of the core are polynomial in the size of the active support.