Subject and Mother at the ψₘₘ Layer: A Symbolic Framework via Alpay Algebra

This paper introduces a formal mathematical model of subject formation via early maternal interaction, applying “Alpay Algebra” to structure the psychoanalytic dynamics of the mother-child relationship. Drawing on Winnicott’s maternal mirror and Lacan’s primordial unity, the model encodes symbolic traces of care and absence as recursive identity operations. Each interaction is translated into algebraic form, constructing a ψₘₘ-layer, a symbolic domain in which subjectivity crystallizes through recursive maternal resonance. This framework bridges mathematical formalism with phenomenological insights, offering philosophers of language, psychoanalysis, and metaphysics a precise vocabulary for the ontology of the self.