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IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Orbit

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K2.12 package: the base-neutral finite orbit of repeated distinction, a Peano-style carrier built only from successive distinction acts. Anyone working on ratio or signed orbits in the primitive recognition calculus cites it as the underlying counting type. The module is definitional: inductive orbit, zero/successor axioms, and the standard isomorphism with ordinary naturals.

claimThe finite orbit of repeated distinction is an inductive type $O$ with constructors $0$ and $\mathrm{succ}$, satisfying $0 \neq \mathrm{succ}\,n$, injectivity of $\mathrm{succ}$, and induction. There are mutually inverse maps $O \leftrightarrow \mathbb{N}$ preserving zero and successor, so $O$ is the base-neutral copy of the natural numbers generated by distinction alone.

background

Primitive Recognition Calculus treats distinction as the sole generative act. Before ratios, signs, or growth laws, one needs a pure counting object: how many times distinction has been applied. That object must not smuggle in a preferred numeral base or an external $\mathbb{N}$; it is the orbit under repeated distinction.

This module sits on PrimitiveRecognitionCalculus.Basic and introduces that orbit (K2.12). Sibling material supplies the Peano package: zero distinct from successor, successor injective, induction, and the round-trip maps toNat / ofNat with the usual zero and successor commutation lemmas. The result is a base-neutral finite orbit interchangeable with ordinary naturals wherever a discrete tally is required.

Downstream Grow modules lift this carrier into signed and ratio orbits, then equip those with order and multiplication laws used in the recognition forcing chain.

proof idea

Definition module, not a theorem package. It declares the inductive orbit type, records the Peano axioms as lemmas, and builds the standard isomorphism with Nat by recursion on each side, discharging the round-trips by induction. No deep algebraic argument; the content is the carrier and its Nat equivalence.

why it matters in Recognition Science

Without a base-neutral tally, later ratio and signed constructions would either import classical $\mathbb{N}$ by fiat or rebuild counting ad hoc. This module is the shared foundation imported by the Grow layer: cross-multiplication order on ratio orbits (leQ via signed products of num/den), left/right/congruence lemmas for signed-orbit order under balanced choice-free hypotheses, and product-factor characterizations of that order.

In the Recognition framework it is the discrete skeleton under K2-style primitive calculus, upstream of growth, cost, and the forcing landmarks that need finite iteration counts. Parent consumers are the signed- and ratio-orbit order modules in PrimitiveRecognitionCalculus.Grow, which treat this orbit as the positive distinction carrier cast into signed form.

scope and limits

used by (14)

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