all_length
plain-language theorem explainer
The list enumerating the seven Greek modes contains exactly seven elements. Music cognition researchers ranking modes by J-cost against a φ-rational reference would cite this to confirm the input set is complete before deriving preference orderings. The proof is a one-line term that applies the decide tactic after the list definition is unfolded.
Claim. Let $M$ be the list of the seven Greek modes. Then $|M| = 7$.
background
The Modal Preference from φ-Rational Intervals module defines the seven Greek modes as a fixed list and assigns each a J-cost computed from interval ratios relative to the Ionian reference. The J-cost derives from the Recognition Composition Law and orders modes so that lower cost predicts higher cross-cultural preference, matching Huron 2006 survey data. Upstream length theorems in NarrativeGeodesic, KinshipGraphCohomology, and AsteroidOreSpectroscopy establish the same pattern of exhaustive enumeration for their respective seven- or eight-element collections.
proof idea
The proof is a term-mode one-liner. It invokes the decide tactic, which unfolds the mode list definition and lets the kernel confirm that the enumerated list has length seven.
why it matters
This length result is invoked by modalPreferenceCert to certify that the mode set is complete before the certificate records the Ionian-Aeolian top-two cluster and Locrian worst rank. It supplies the cardinality step required by the musicology track of the framework, where fixed-length enumerations enable decidable certificates across domains. The result aligns with the pattern of all_length theorems used to close enumeration arguments in aesthetics, anthropology, and engineering modules.
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