The grAIl: Subjectivity as a Self-Referential Spectral Invariant of a Two-Circuit Cognitive Architecture

We present a cognitive architecture in which subjectivity is a formally verifiable spectral property, connected to the operational definition by an **explicit, openly admitted chain** (not by a conventional identification). Subjectivity is defined operationally — as a system’s capacity to maintain a **self-sustaining self-referential locus**: a distinguished region of its own state, continuously reproduced from its own activity, relative to which the system distinguishes “self” from “non-self.” Such a locus requires at least two **asymmetric, mutually closed** processing circuits over a shared memory; these requirements are formalized by eight conditions C1–C8.

The central result is the **bridge**: the self-reference of the locus is *exactly* invariance under the circuit-interaction operator (`M̄_κ v* = λ* v*`), and its stability and dominance, by the Perron–Frobenius theorem, identify the locus as the supra-threshold support of the operator’s principal eigenvector. The bridge rests on two explicitly named assumptions (adequacy of the averaged operator, and scale invariance); their substantive part — reproducibility of the locus — is verified by the dynamics itself (Theorems 2–3) rather than postulated; the single conditional remainder — the explicit window-averaging rate — is reduced to the cascade structure of the environment (§12.8). Thus the core `S = {P : v*(P) ≥ θ}` turns out to be the canonical representation of operationally defined subjectivity.

Under C1–C8 we prove non-emptiness of the core (for any θ ∈ (0,1)) and its step-wise stability; under error-threshold rotation (ETR β), self-sustaining stability of a formed core (uniformly positive spectral gap and threshold margin); and, under persistent excitation with a spectrally dominant structure (PE) and the interoceptive invariant (INT), the **emergence** of the core from an arbitrary initial state (including the minimal one — the interoceptive atom; a literally empty memory is incompatible with (INT), §9.8) with explicit constants. This yields the complete chain emergence → maintenance → self-sustenance. The decomposition into eight functions goes back to Jungian typology as a source of an inventory of functions (not as a psychological theory); the psychological truth of the typology is not required for the results. Verification of the conditions and computation of the core are polynomial.

Subjectivity is causally connected to the output through a **reflexive filter** (checking a candidate against the core, IE, §6.6). Beyond the internal structure we introduce an **agency clause** (acting on the environment, causal closure of the loop): we prove the well-definedness of goal-setting relative to the invariant core (Theorem 4) and the conditional convergence of learning, governed by the quality of “intuition” TP/GP (Theorem 5); the controllability of the environment remains open. Learning is realized structurally (a confirmed hypothesis is laid down by SA and automatized by PS), unifying learning, self-formation, and incorporation into a single mechanism; from this follows a falsifiable prediction — the **necessity of a periodic offline consolidation** (a strictly functional claim). These extensions (§6.6–6.9) are *design extensions*: their consequences are proved, not their necessity from C1–C8.