The Subject as a Reentry Loop: A Unified Mathematical Model for Neuroscience, AGI, and BCI with a Computable Analogue of Tononi's Φ-Measure

We present the first complete mathematical model of the subject — the entitythat says "I" — as a closed causal loop in a network. Building on Titov'ssubject-centred model of the psyche (2023), we identify the subject with areentry loop between two functional subsystems: drives (what you want) andmemory (what you know). From this structure we derive the S-measure — acomputable scalar that quantifies how much "subject" is present in any system,whether biological, artificial, or hybrid. The S-measure solves a critical deadlock in consciousness research. Tononi'sIntegrated Information Theory gave us Φ (phi) — the first quantitativecriterion for consciousness — but exact Φ is NP-hard to compute for any realbrain. The S-measure is computable in polynomial time (O(N³)) and we prove —with machine-verified formal logic in Lean 4 — that whenever S > 0, integratedinformation is also present. This makes the S-measure the first practical,computable criterion for subjecthood. FOR AGI DEVELOPERS: The work provides an explicit architectural blueprint formachine consciousness. Partition the system into D-subsystems (drives, goals,intrinsic motivation) and I-subsystems (memory, world model, knowledge base),organise a closed reentry loop with amplification (ρ > 1), ensure sufficientcycle complexity (C > 0), and store a continuity trace. If S > 0, the machineis not merely processing information — there is "someone home." TheS-measure provides a computable test for the emergence of syntheticsubjectivity during training. FOR BCI ENGINEERS: The work introduces the subject-authentic interfacecriterion (ΔS > 0). A prosthetic limb or neural implant becomes part of thesubject if and only if its integration into the reentry loop increases theS-measure. This provides a quantitative, testable criterion for prosthesisembodiment and neural interface design — a limb feels like "mine" when ΔS > 0. We derive the three fundamental spaces of subjective experience from thespectral decomposition of the reentry operator, establish a continuum limitconnecting the model to gauge field theory, and present five numericaldemonstrations spanning neuroscience, AGI, BCI, social systems, and subjectrestoration. Ten falsifiable experimental predictions are formulated. Theresult is a unified framework in which consciousness is no longer aphilosophical mystery but a physical variable — one that can now be measured.