Entropy Has No Direction: A Mirror-State Paradox Against Universal Monotonic Entropy Increase and a First-Principles Proof that Constraints Reshape the Entropy Distribution

Abstract: We present a purely theoretical, self-contained argument that the Second Law of Thermodynamics cannot be a universal fundamental law in the form ``entropy does not decrease'' (whether asserted trajectory-wise or as a universal statistical principle) when the underlying microscopic dynamics are time-reversal invariant. The core is a mirror-state construction: for any microstate $A$ one constructs its time-reversed partner $B$ (momenta inverted). If a universal monotonicity statement is applied to both $A$ and $B$, it implies that $A$ is a local minimum of entropy at every moment, which forces entropy to be constant and destroys any entropic arrow of time. The consistent replacement is that entropy is a stochastic variable described by a probability distribution $P(S)$, whose shape depends on constraints and boundary conditions. We then prove from first principles that constraints necessarily reshape the long-time entropy distribution $P_{\infty}(S;λ)$ by altering the invariant measure through changes in the Hamiltonian and/or the accessible phase space. A sharp criterion is given: in the microcanonical setting, the \emph{only} way $P_{\infty}^{(E)}(S;λ)$ can remain the same up to translation is when all accessible macrostate volumes are scaled by a common factor; otherwise the distribution changes structurally.