Physics Sound Speed Ratio From Phi

Sound speed ratio and the golden ratio

In an isotropic solid, sound can travel as a longitudinal wave, where particles move along the direction of travel, or as a transverse wave, where they move perpendicular to it. The two speeds differ, and their ratio v_T/v_L is not arbitrary. For a given material it is fixed by Poisson's ratio ν, the number that describes how much a material thins sideways when you stretch it. The standard formula is v_T/v_L = √((1−2ν)/(2−2ν)). For common metals ν is around 0.3, which puts the ratio near 0.5 to 0.6.

Recognition Science, a framework that derives physical structure from a forced cost function, identifies Poisson's ratio with the reciprocal of the square of the golden ratio φ ≈ 1.618. That is, ν = 1/φ² ≈ 0.382. Plugging that into the formula gives v_T/v_L = √(0.236/1.236) ≈ 0.437, which is close to φ^(−2.7). This is a model: a definitional choice, not a derived theorem. The framework's claim is that the golden ratio, which already appears in the framework's cost function, also shows up in the elasticity of solids.

What does the machine-checked library actually establish about this? The module domainCost, a cost function that measures the mismatch between a measured and an expected value, is defined as J(measured/expected), where J(x) = (x + 1/x)/2 − 1. The library establishes three general facts about this cost: it is zero when measured equals expected, it is never negative for positive inputs, and the threshold φ − 3/2 is positive. These are true for any positive numbers, not specifically for sound speeds. The module does not define what "measured" and "expected" mean for sound, so it does not establish that the sound speed ratio is φ^(−2.7). That identification remains a research note, not a result.

The formal content is a certificate: a packaged record that the cost function has the three basic properties. It is a small but honest step. The gap between the formal facts and the physical claim is precisely the missing definition of the sound speeds in terms of the framework's quantities. Until that definition exists, the sound speed ratio from φ is a hypothesis, not a theorem.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (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 sound speed ratio from φ is not established by the module; it is a research note. The module does not derive Poisson's ratio from the framework; it assumes the identification ν = 1/φ². The framework does not claim that all solids have the same sound speed ratio; real materials vary.

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/Physics/SoundSpeedRatioFromPhi.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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