e as the Finite Number: Algebraic Derivation of the Natural Logarithm Base

This paper establishes that the natural logarithm base e is not a transcendental number in any physically meaningful sense, but a structurally finite algebraic invariant emerging from the unitary closure condition of SU(3), the symmetry group necessitated by M3(C) within the framework of Cognitional Mechanics (CM). We demonstrate that e is the minimum non-trivial eigenvalue of the exponential map at the unit operation t=1 of the M3(C) Lie algebra. The transcendence of e, established by Hermite (1873) within the Tier-3 framework of classical analysis, is reinterpreted as an artifact of infinite-precision representation in the projective layer. In CM, transcendence is a result internal to Tier-3 axiomatics, not a property of the Tier-1 structural invariant. This reinterpretation extends to π and the general principle that infinite representations are physically vacuous beyond the Planck-scale operational bound (approximately 61 decimal digits). By deriving e from the SU(3) determinant constraint, this work provides the structural basis for information conservation in CM-GUT. Together with the previously established derivation of α−1 and π, this work demonstrates that the fundamental constants of nature are co-generated from the single algebraic seed M3(C).