Why Mathematics Is Reversible and Physics Is Not: Discretization, Unstable Symmetry, and Irreversibility

This paper identifies a generic instability introduced by discretization in dynamical systems that admit exactly invertible loops in the continuous limit. Using a canonical quadratic example, we show that a symmetric fixed point that is marginal in continuous time (with linearization eigenvalue equal to one) becomes unstable under discrete iteration, with an eigenvalue of two. This “instability doubling” arises purely from the structure of discretization and does not rely on dissipation, entropy production, or explicit time-reversal symmetry breaking. We interpret this result geometrically in terms of variance partition between complementary channels, where the equipartition configuration acts as a sharp basin boundary under discrete dynamics. The associated closure gap grows exponentially with iteration count, producing irreversible drift away from symmetric configurations in any finite-resolution implementation. The analysis suggests that discretization alone provides a generic mechanism for dynamical irreversibility in systems that are exactly reversible in the continuous limit. While no claim is made that this mechanism fully explains the thermodynamic arrow of time, it establishes a minimal geometric route by which temporal asymmetry can arise from finite resolution.