Selection of the Group Z4 by the Gray Code and Achiral Lock on the Orientation of a Logical Loop

Abstract

This work constructs a logical substrate placed below spacetime, energy and physical information, and asks what structures can exist there. On such a substrate, reversibility does not rest on a thermodynamic limit but on the absence of an exterior: erasing a distinction means exporting it to an environment, and a substrate that is the whole has no environment to receive it. On the minimal space of two binary distinctions, a single combinatorial criterion, namely that exactly one bit changes at each step, selects from the twenty-four permutations one pair of mirror four-cycles, that is, the cyclic group of order four together with its two generators. The four operators of the microGlobal programme, Reproduction, Differentiation, Hierarchisation and Linkage, are then not descriptive axioms but the components of that group. The criterion is not postulated: the selected pair is the unique cyclic arrangement of least Hamming action, and this minimality is shown to be the uniqueness of the primitive orbits under the atomicity of distinction, not an optimisation borrowed from physics. Two loops therefore remain, mirror images of one another and, so far, indistinguishable. We establish a lock: no achiral quantity, entropy included, can separate them, because they are conjugate under inversion of the generator; at the level where spacetime emerges they even coincide as operators, by commutativity. No selected arrow of time is thus inscribed in the substrate. What separates the two loops can only be a sign, carried by the discrete geometric phase that the Gray orbit acquires on the Weyl phase plane, a time-odd and gauge-invariant quantity equal to the enclosed oriented area. Its realisation as a Berry phase, and the choice of sign through boundary conditions, belong to the level at which energy emerges and are stated as conjectures. Two further results follow from the same finite algebra. The imaginary unit is not introduced by hand: it is an eigenvalue of the generator, forced by the order of the cycle, and the relative phase between the two chiral eigenvectors is visible in the probabilities themselves, which places the substrate in the quantum rather than the classical regime by Hardy’s counting criterion. And the binary character of distinction yields the monogamy of entanglement as a count rather than a postulate. The claims are deliberately narrow and stated so that they may fail: three of the four proposed refutations are settled by finite calculation on two bits.