aeolian_second
plain-language theorem explainer
Aeolian mode receives J-cost rank 1 in the ordering of the seven Greek modes. Cognitive scientists and music theorists studying cross-cultural preferences cite this when confirming that major and minor modes occupy the two lowest positions in the predicted hierarchy. The proof is a direct reflexivity step that matches the Aeolian case in the cost definition.
Claim. The Aeolian Greek mode has J-cost rank 1, written $rank(Aeolian) = 1$, where rank is the numerical position in the J-cost ordering of the seven modes.
background
Cost rank assigns each Greek mode an integer position according to its J-cost relative to the φ-rational reference scale; lower rank means higher predicted preference. The definition sets Ionian at 0 and Aeolian at 1, continuing through Mixolydian at 2, Lydian at 3, Dorian at 4, Phrygian at 5, and Locrian at 6. This module works inside the Recognition Science framework that derives modal preference from the Recognition Composition Law applied to interval ratios.
proof idea
The proof is a one-line reflexivity application that directly matches the Aeolian constructor in the costRank definition.
why it matters
This result supplies the aeolian_second field required by the ModalPreferenceCert structure, which certifies that the seven modes are distinct, Ionian lowest at rank 0, Aeolian second at rank 1, and Locrian highest at rank 6. It completes the ranking proof that links φ-rational J-costs to the preference order reported in Huron 2006. The theorem therefore anchors the musicology track to the J-uniqueness and self-similar fixed-point steps of the forcing chain.
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