Foundation Universal Forcing Music Realization Music Cost Symm

The interval ledger

In music, an interval is the distance between two pitches. The framework's ledger, a discrete record of recognition events, models that distance as a count of steps along a scale. The declaration musicCost_symm proves that this count is symmetric: the cost of moving from step a to step b equals the cost of moving from b to a. The proof is mechanical, checking the two cases where the steps are equal or different.

The cost function itself is deliberately simple. It assigns 0 when two steps are the same, and 1 when they differ. This is a definitional choice, not a derived law. The theorem only establishes that this choice respects symmetry, a property any reasonable distance measure should have. It says nothing about how intervals sound, what makes music pleasing, or how the framework's deeper cost function J(x) = (x + 1/x)/2 - 1 arises.

In Recognition Science, this declaration is a small building block. It shows that the musical realization, which reads pitch-ratio stacking as interval composition, satisfies one of the five plain conditions that eventually force the golden ratio and the eight-tick cycle. But musicCost_symm itself is a single step, not the whole staircase. It proves symmetry for a toy model, not for the full framework's arithmetic.

What the theorem does not claim is just as important as what it proves. It does not claim that music is reducible to counting, that the framework explains musical aesthetics, or that this symmetry condition alone forces any deeper structure. The declaration is a lemma, a checked fact about a simple function, not a statement about the nature of music.

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

def musicCost (a b : MusicalIntervalStep) : Nat :=
  if a = b then 0 else 1

/-- Musical realization as interval-step comparison. -/
def musicRealization : LogicRealization where
  Carrier := MusicalIntervalStep
  Cost := Nat
  zeroCost := inferInstance
  compare := musicCost
  zero := 0
  step := Nat.succ
  Orbit := LogicNat
  orbitZero := LogicNat.zero
  orbitStep := LogicNat.succ
  interpret := musicInterpret
  interpret_zero := by rfl
  interpret_step := by
    intro n
    show LogicNat.toNat (LogicNat.succ n) = Nat.succ (LogicNat.toNat n)
    rfl
  orbit_no_confusion := by intro n h; exact LogicNat.zero_ne_succ n h
  orbit_step_injective := LogicNat.succ_injective
  orbit_induction := by
    intro P h0 hs n
    exact LogicNat.induction (motive := P) h0 hs n
  orbitEquivLogicNat := Equiv.refl LogicNat
  orbitEquiv_zero := rfl
  orbitEquiv_step := by intro n; rfl
  identity := musicCost_self
  nonContradiction := musicCost_symm
  excludedMiddle := True
  composition := True
  actionInvariant := True
  nontrivial := by
    refine ⟨1, ?_⟩
    simp [musicCost]

What this page does not claim

The theorem does not claim that music is reducible to counting steps. It does not claim that the framework explains musical aesthetics or emotional response. This symmetry condition alone does not force any deeper structure without the other four conditions.

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/MusicRealization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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