Acoustics Middle C Frequency Rs Middle Cfreq Rs

Middle C in the framework

Middle C is the musical note that most piano tuners use as a reference, and its standard frequency is 261.63 hertz. The number comes from a convention: the A above middle C is set to 440 hertz, and the twelve notes of the octave are spaced by equal ratios, so middle C lands at 261.63 Hz. That value is a measurement standard, not a law of nature.

In Recognition Science, a ledger (a discrete record of events) assigns a cost to comparing two quantities, and the framework's central result forces that cost to take one specific form. The declaration MiddleCFreqRS in the framework's machine-checked library of formal theorems packages three general facts about that cost function. It proves that the cost is zero when the two quantities are equal, that the cost is never negative for positive inputs, and that a certain threshold involving the golden ratio is positive. These are properties of the cost function itself, stated for any two real numbers.

The file's research note records an intended match: phi to the 14th power times 0.31 gives about 261.3 hertz, close to middle C. The note calls this a match, and the arithmetic is correct. But the note is not a theorem. The formal code defines the cost using a ratio of two numbers m and e without ever saying what m and e mean for a musical note. Nothing in the Lean declarations connects the cost function to sound, to frequencies, or to the number 261.63. The structure MiddleCFreqRS is inhabited, meaning the three general facts are consistent, but it says nothing about acoustics.

What the declaration does establish is a template. It shows that if you want to apply the framework's cost to middle C, you would need to define m and e in acoustical terms first. The current file leaves that step undone. The general cost facts are proved; the musical application is not. A reader should take the file as a placeholder with a research aspiration, not as a derivation of the note's frequency.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem cert_inhabited : Nonempty MiddleCFreqRS := ⟨cert⟩

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The file does not prove that middle C is 261.63 hertz. The file does not define what m and e mean for a musical note. The file does not derive the golden ratio from acoustics.

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/Acoustics/Middle_C_Frequency_RS.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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