cycStep
plain-language theorem explainer
The cycStep definition supplies the successor map on the finite cyclic carrier Fin(modulus k) for modular realizations of logic. Researchers constructing periodic interpretations of Universal Forcing cite it when showing that the forcing chain admits non-faithful carriers. The definition is a direct construction that reduces the successor of the underlying natural number value modulo modulus k.
Claim. Let $m = k + 2$. The successor map on the cyclic carrier is the function $x : Fin(m)mapsto (x.val + 1) mod m$, where the result is again an element of $Fin(m)$.
background
The ModularLogicRealization module realizes the free orbit of LogicNat on a periodic finite carrier. The carrier size is given by the sibling definition modulus k := k + 2, which is positive by the upstream theorem modulus_pos k (proved by unfolding and omega). The module documentation states: 'Periodic finite-cyclic realization for Universal Forcing. The internal orbit is still free (LogicNat), while the carrier interpretation is periodic.' This demonstrates that Universal Forcing does not require every realization to embed arithmetic faithfully into the carrier. The engineering carrier definitions (5 phi Hz) supply frequency context but are not invoked in the construction.
proof idea
One-line definition. It builds the new Fin element by taking the successor of x.val, reducing modulo modulus k, and supplying the strict inequality proof via the upstream lemma modulus_pos k.
why it matters
cycStep is invoked by the downstream theorem modularInterpret_step to prove that modular interpretation commutes with successor on LogicNat, and by the downstream definition modularRealization to equip the finite carrier with the required LogicRealization structure. It fills the modular step in the foundation layer, showing that the forcing chain (T0-T8) and Recognition Composition Law admit periodic carriers without faithful arithmetic embedding. No open scaffolding questions are closed by this definition.
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