timeAsOrbitCert_inhabited
plain-language theorem explainer
The theorem asserts existence of a certificate identifying the tick sequence as the natural-number object forced by successive recognition steps. Researchers building the logical foundation of time in Recognition Science cite it to confirm the orbit identification. The proof is a one-line term that directly supplies the pre-assembled certificate.
Claim. The type of time-as-orbit certificates is inhabited: there exists a structure $C$ such that the sequence of ticks forms a natural-number object under the recognition successor and is canonically isomorphic to the logic-natural numbers.
background
The module claims that the temporal sequence Tick is the natural-number object forced by recognition. Tick carries a zero element and successor operation; by the universal property of natural-number objects it is equivalent to LogicNat, the inductive type whose identity constructor is the zero-cost element and whose step constructor iterates the generator. TimeAsOrbitCert packages the natural-number-object property, the explicit isomorphism to LogicNat, and the condition that recognition advancement matches the successor. This rests on the orbit construction in ArithmeticFromLogic and the natural-number-object universal property proved in UniversalForcing.NaturalNumberObject.
proof idea
The proof is a one-line term that constructs the certificate by supplying the pre-defined timeAsOrbitCert term, which assembles the required fields from the module's equivalence theorems and the recognition-step advancement lemma.
why it matters
This result supplies the inhabited TimeAsOrbitCert required by the master theorem reality_from_one_distinction, which forces the certificate from any single distinction in an inhabited carrier. It completes the combinatorial identification of time with the canonical iteration object inside the Recognition Science forcing chain. The module explicitly leaves metric time, the eight-tick periodicity, and the arrow of time for separate developments.
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