toZ_one
The projection extracts the underlying integer from any element of the subgroup of ℤ generated by 1. Researchers building order isomorphisms for binary scales cite it when mapping between the δ=1 subgroup and the full integers. The definition is a direct field projection with no computation or lemmas required.
claimLet $Δ_1 = {x ∈ ℤ | ∃ n ∈ ℤ, x = n · 1}$ be the subgroup of ℤ generated by 1. For $p ∈ Δ_1$ the map returns the integer representative $p.val$.
background
The RecogSpec.Scales module constructs binary scales via subgroups of the integers. DeltaSub δ is the subtype {x : ℤ // ∃ n : ℤ, x = n * δ}, the set of all integer multiples of a fixed generator δ. Specializing to δ = 1 recovers the full group ℤ under the natural embedding. The same construction appears upstream in LedgerUnits and UnitMapping, where it serves as an abstract placeholder for later scale mappings.
proof idea
One-line definition that returns the .val field of the subtype p : DeltaSub 1. No lemmas are invoked and no tactics are used.
why it matters in Recognition Science
It supplies the forward map of the equivalence equiv_delta_one : DeltaSub 1 ≃ ℤ that appears in both LedgerUnits and RecogSpec.Scales. The equivalence underpins the integer rungs of the φ-ladder and the clean order isomorphism needed for binary scales. In the Recognition framework this projection closes the δ = 1 case before φ-exponential wrappers are applied.
scope and limits
- Does not apply to subgroups generated by δ ≠ 1.
- Does not treat real-valued or non-discrete generators.
- Does not establish bijectivity or inverse maps.
- Does not reference φ, the eight-tick octave, or Recognition constants.
formal statement (Lean)
118def toZ_one (p : DeltaSub 1) : ℤ := p.val
proof body
Definition body.
119