octaveAlg_id_left
plain-language theorem explainer
The left identity law for morphisms in the OctaveAlg category states that composing the identity on the codomain with any morphism recovers the original map. Algebraists modeling the eight-tick octave structure would cite this when verifying unit laws in the category whose objects are additive groups isomorphic to ZMod 8. The proof proceeds by function extensionality on the underlying additive map followed by reflexivity.
Claim. Let $A$ and $B$ be objects in the octave algebra category, each an additive group isomorphic to $Z/8Z$, and let $f: A → B$ be an additive homomorphism. Then the composition of the identity morphism on $B$ with $f$ equals $f$.
background
OctaveAlgObj consists of additive groups equipped with an explicit isomorphism to ZMod 8, making them finite additive groups of order 8. OctaveAlgHom consists of additive group homomorphisms between such objects. The local setting is the construction of a category whose morphisms are these homomorphisms, with composition defined by map composition and identity given by the identity additive monoid homomorphism. Upstream, octaveAlg_id supplies the identity map as AddMonoidHom.id on the carrier, while octaveAlg_comp supplies the composite map via AddMonoidHom.comp.
proof idea
One-line wrapper that applies function extensionality to the underlying map of the composite and concludes by reflexivity.
why it matters
This identity law supplies one of the two unit axioms required to confirm that OctaveAlg forms a category, directly supporting the algebraic encoding of the eight-tick octave (T7) in the forcing chain. It ensures identities act as units under the composition operation already defined in the same module. No downstream theorems are recorded yet, leaving open whether this structure will be used to model self-similar fixed points or the phi-ladder.
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