music_arith_equiv_logicNat
plain-language theorem explainer
The strict musical realization over positive frequency ratios induces an arithmetic structure whose Peano carrier is canonically equivalent to the inductive orbit of logic natural numbers. Researchers extracting number systems from the Recognition forcing chain would cite this when moving from frequency-ratio carriers to the forced naturals. The definition is a one-line wrapper applying the lightweight orbit equivalence to the music carrier.
Claim. Let $R$ be the strict musical realization whose carrier consists of positive frequency ratios under multiplication, with equality-cost comparison and octave stacking as generator. Then the Peano carrier of the arithmetic extracted from the lightweight version of $R$ is equivalent to the inductive type of logic natural numbers generated by an identity element and a successor step.
background
The module supplies a domain-rich musical realization over positive frequency ratios. Comparison uses equality-cost on ratios for this strict pass; richer psychoacoustic dissonance costs can refine it later. StrictLogicRealization is the structure recording a carrier type, a cost type with zero instance, a compare map, and a binary compose operation. strictMusicRealization instantiates this with FrequencyRatio as carrier, Nat as cost, ratioCost as compare, and multiplication as compose. LogicNat is the inductive type whose identity constructor is the zero-cost element and whose step constructor iterates the generator, mirroring the orbit {1, γ, γ², …} as the smallest subset of positive reals closed under the generator and containing 1. The arith operation extracts the forced arithmetic from any lightweight realization obtained via toLightweight.
proof idea
The definition is a one-line wrapper that applies the orbit equivalence supplied by toLightweight on strictMusicRealization, yielding the required equivalence between the extracted Peano carrier and LogicNat.
why it matters
This equivalence completes the extraction of arithmetic from the musical realization inside the strict forcing chain, directly supporting the passage from the law-of-logic orbit to the inductive naturals. It sits downstream of the StrictLogicRealization structure and the arith extraction lemma, and upstream of any further derivation that needs the logic naturals realized by frequency ratios. The construction aligns with the T5 J-uniqueness and T6 phi fixed-point steps by furnishing a concrete carrier whose orbit matches the forced naturals.
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