Local Time-Reversal Asymmetry in Finite Reversible Symbolic Systems

We study a finite deterministic system whose microscopic dynamics is globally periodic and therefore reversible, while an observer has access only to a finite symbolic projection. We formalize observed length-k statistics as a circulation on the binary De Bruijn graph and define the orientational subspace as the antisymmetric component of the circulation space under word reversal. This yields an exact dimension formula, for k ≥ 2, r_k = [2^(k−1) + 2 − 2^⌈k/2⌉ − 2^⌈(k−1)/2⌉]/2. The first values are 0, 0, 1, 3, 9, 21, 49, 105, 225 for k=2,…,10. The paper also gives an explicit period-6 example whose one-, two-, and three-symbol statistics are time-reversal symmetric but whose four-symbol statistics are asymmetric. The threshold that binary symbolic irreversibility cannot occur below order four is already known in the literature; accordingly, no novelty claim is made for that threshold. The proposed contribution is the finite reversible-orbit formulation together with the De Bruijn circulation/reversal framework and the general dimension formula. We distinguish local statistical asymmetry from thermodynamic irreversibility, causal arrows, psychological time, and cosmological arrows.