Acoustics Musical Note A4 Exact Rs A4 Exact Rs

The A4 standard in the framework

The musical note A4 is the reference pitch used to tune most instruments, set by the ISO 16:1975 standard to exactly 440 hertz, meaning the sound wave completes 440 cycles per second. This is a convention chosen by international agreement in 1955, not a fact discovered in nature. In the Recognition Science framework, the declaration A4ExactRS is a machine-checked bundle of statements about a cost function, a mathematical measure of how far one quantity is from another. The bundle does not itself connect to the 440 hertz standard; it contains three general facts that hold for any positive real numbers.

The first fact is that the cost of comparing a quantity with itself is zero, provided the quantity is not zero. The second is that the cost is never negative when both quantities are positive, so the cost cannot go below zero. The third is that a certain threshold value, defined as the golden ratio minus 1.5, is positive. These three statements are proved in the framework's machine-checked library of formal theorems, and they are true for any positive inputs, not just for musical frequencies. The declaration simply packages these three proved facts into a single named object, so that other work can refer to them together.

The framework's research notes mention an approximation: 440 hertz is close to the golden ratio raised to the seventeenth power times 0.123, which gives about 439.2 hertz. This is a research note recording where the idea was meant to go, not a result. The formal proof does not define what the mass or energy terms in the cost function mean for a musical note, so it cannot claim to derive the 440 hertz standard. What the declaration actually establishes is only the three general properties of the cost function, which apply equally to any positive quantities, whether they are frequencies, lengths, or masses.

structure A4ExactRS where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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 canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The declaration does not prove that A4 equals 440 hertz or any other frequency. The declaration does not derive the golden ratio from the musical note standard. The declaration does not establish any property specific to acoustics or sound.

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/Musical_Note_A4_Exact_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.

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