The Geometric Invariance of the Critical Exponent Half: Proof of Semicircular Energy Conservation under Pure Exponential Substitution

This paper establishes the Theorem of Invariant Exponent Preservation, proving that the critical half-line (γ = 0.5) is an absolute geometric property of the 90° orthogonal isometric vision space, independent of the Riemann zeta function or arithmetic number theory. Historically, the value 1/2 has been viewed strictly as an algebraic constraint governing the non-trivial roots of the transcendental zeta function. We dismantle this structural monopoly by eliminating the Riemann zeta function from the field equations and substituting it with a continuous, entire exponential wave mirror (e^z or e^{−z²}). Embedded within a non-singular, three-dimensional parallel manifold where the imaginary unit acts as a physical elevation axis, we rigorously prove that the critical exponent remains completely invariant under this substitution. The numerical value 0.5 is derived as the exact volumetric energy signature of the uncrushed π-radian Hankel semicircle under a 90° isometric rotation matrix. This framework unifies projective linear algebra and quantum spectral fields, transforming the half-line from an arithmetic hypothesis into a universal law of geometric space conservation.