Pontryagin Duality and Galperin Dynamics: On Continuity as the Spectral Limit of Discrete Group Actions

This treatise interrogates the ontological status of the continuum through the lens of abstract harmonic analysis and topological dynamics. By leveraging the isomorphism between the dual group of integers and the circle established by Pontryagin Duality, we posit that continuous geometry is not a primitive entity but an emergent property of discrete group actions in the ergodic limit. Utilizing the Galperin collision model as a physical archetype, we demonstrate that the transcendental constant π arises strictly from the phase space counting measure of an integer-driven system. Furthermore, we address the constructivist critique of the Cantor set by reifying it as a spectral entity characterized by persistent, non-decaying Fourier coefficients, violating the Riemann-Lebesgue lemma. We conclude that within the physical-informational horizon determined by the Bekenstein bound and Shannon-Nyquist sampling theorem, a constructive continuity based on integer dynamics constitutes a complete and effective theory. This framework renders non-computable reals ontologically superfluous, identifying them as spectrally indistinguishable from thermal noise.