A Mathematical Theory of Finite Informational Entities

El Bouazzaoui TAYB

Published Jul 5, 2026• 0 citations

Abstract

We present a self-contained axiomatic framework in which a fundamental entity is defined not by intrinsic materiality but as a finite, quantized information node governed by an internal duality. The central postulate—the Axiom of Operational Parity—asserts that an individual entity is mathematically identical to an interacting pair of itself and its conjugate: S≡S×SˉS≡S×Sˉ. Working in a finite-dimensional Hilbert space H=CdH=Cd with d≥3d≥3, we introduce a strictly positive quantization operator QQ encoding geometric bounds, an antiunitary conjugation CC, and a command vector ∣v⟩∣v⟩ satisfying two orthogonality conditions. From these we construct a pair of positive rank-one entity operators (S,Sˉ)(S,Sˉ) with equal trace ττ and orthogonal support. The entity's global state is postulated to be the maximally entangled Bell state of the active and conjugate sectors. Temporal persistence is governed by a dissipative binary recurrence—a deterministic discrete-time linear evolution coupling S(n)S(n) and Sˉ(n)Sˉ(n). We prove that positivity is preserved if and only if the coupling parameters satisfy ∣A−B∣≤1−γ∣A−B∣≤1−γ, and that the trace decays as Tr⁡(S(n))=τ(1−γ)nTr(S(n))=τ(1−γ)n. An intrinsic collapse criterion based on trace threshold is introduced. The main result, the Finite Lifetime Theorem, establishes that every entity returns to the zero state in a finite number of steps, with the explicit formula Nmax⁡=⌈log⁡α/log⁡(1−γ)⌉Nmax=⌈logα/log(1−γ)⌉, depending only on the dissipation constant γγ and the collapse threshold αα. We provide a categorical framing via dagger compact closed categories, distinguish rigorously between proved results and postulates, and identify open problems including the dimensional obstruction to realizing the type axiom in finite-dimensional operator models.

DOI: 10.5281/zenodo.21198703

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