twelve_divides_lcm_structure
plain-language theorem explainer
The arithmetic fact that the least common multiple of 3 and 4 equals 12 supplies the division count for periodic alignment in twelve-tone scale constructions. Circle-of-fifths analysts cite it when confirming that interval steps close after an integer number of cycles. Native decision computes the lcm value directly and checks the equality.
Claim. The least common multiple of 3 and 4 equals 12.
background
The CircleOfFifths module works with natural-number ratios that govern musical intervals and their periodic returns. It imports harmonic-mode definitions fixing twelve semitones per octave and eight modes per octave. The present identity connects the factors 3 and 4 to the composite total 12, matching the eight-tick octave periodicity of the Recognition framework.
proof idea
The proof applies the native_decide tactic, which evaluates the lcm operation on the constants 3 and 4 and confirms the result equals 12.
why it matters
The declaration anchors the twelve-division count required for the circle of fifths to close. It aligns with the eight-tick octave and three spatial dimensions by furnishing the lcm that reproduces the semitone total. No parent theorems or downstream uses appear, so the result functions as a primitive arithmetic prerequisite.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.