Foundation Universal Forcing Strict Music Ratio Cost Symm

The symmetry of ratio cost

In music, the interval from a note to another note five steps higher is the same interval as the one five steps lower, just reversed. The Recognition Science framework encodes this intuition in a ledger, a discrete record of events, where comparing two frequency ratios costs either zero or one unit. The declaration ratioCost_symm proves that this comparison cost is symmetric: the cost of comparing ratio a to ratio b equals the cost of comparing b to a. In plain terms, the framework's bookkeeping does not care which direction you read the interval.

The proof is direct. The cost function, written as ratioCost a b, returns 0 when a and b are the same positive frequency ratio and 1 otherwise. Symmetry then follows because equality itself is symmetric: if a equals b, both comparisons cost 0, and if a differs from b, both cost 1. The theorem is machine-checked in the framework's library, a collection of formal theorems verified by a computer, and it holds for every pair of positive real frequency ratios, including the octave (ratio 2:1), the perfect fifth (3:2), and the perfect fourth (4:3).

What the theorem does not claim is just as important. It does not assert that this zero-or-one cost is the correct model of human musical perception. The library's own notes call this a strict, domain-rich realization over positive frequency ratios, a deliberately simple starting point where richer psychoacoustic dissonance costs could refine it later. The symmetry result is a structural property of the chosen cost, not a claim about how listeners actually hear intervals. Nor does the theorem say anything about which intervals sound consonant or dissonant; it only fixes the arithmetic of comparison.

This matters because the framework builds larger results, such as the forcing of three spatial dimensions, on top of such basic symmetries. A cost that changed when you reversed the order of comparison would break the ledger's consistency from the first step. ratioCost_symm is the foundation stone that keeps the bookkeeping direction-agnostic, so that every later derivation can assume comparisons behave the same in both directions.

theorem ratioCost_symm (a b : FrequencyRatio) : ratioCost a b = ratioCost b a := by
  by_cases h : a = b
  · subst h
    simp [ratioCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [ratioCost, h, h']

noncomputable def ratioCost (a b : FrequencyRatio) : Nat :=
  if a = b then 0 else 1

theorem ratioCost_symm (a b : FrequencyRatio) : ratioCost a b = ratioCost b a := by
  by_cases h : a = b
  · subst h
    simp [ratioCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [ratioCost, h, h']

What this page does not claim

The theorem does not model human perception of musical intervals. The theorem does not rank intervals by consonance or dissonance. The zero-or-one cost is a chosen model, not a derived necessity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/UniversalForcing/Strict/Music.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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