Materials Thermal Expansion Rs

Thermal expansion and its RS cost

Thermal expansion is the tendency of matter to change its dimensions in response to a change in temperature. For most solids, heating increases the average spacing between atoms, so the material expands. The coefficient of thermal expansion, usually written α, quantifies this: it is the fractional change in length per degree of temperature change. For metals, α typically falls between 10 and 25 millionths per kelvin (10-25 × 10⁻⁶ K⁻¹). This is a classical, well-measured property of materials, with values tabulated for every common engineering alloy.

Recognition Science (RS) starts from a different question: what does it cost to recognize a change? The framework models a ledger, a discrete record of events, and assigns a cost to each recognition event. The central RS result is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This function is zero when its input equals 1, and it grows as the input moves away from 1. The framework's library of machine-checked theorems proves these properties for J, and the thermal expansion module applies this general cost to a ratio of two quantities, m and e.

In the module, the cost is defined as J(m/e), where m and e are real numbers. The module proves three facts about this cost. First, the cost is zero when m equals e (provided e is not zero). Second, the cost is never negative when both m and e are positive. Third, the number φ - 3/2, called the canonical threshold, is positive, where φ is the golden ratio, about 1.618. These three facts are bundled into a structure called a thermal expansion certificate, and the module proves such a certificate exists.

What the module does not do is connect these abstract facts to any specific material. The docstring records a research note: the intended idea was that α_T = J(φ) × 10⁻⁵ K⁻¹ ≈ 1.18 × 10⁻⁶ K⁻¹, or that for metals α_T ≈ φ^k × 10⁻⁶ K⁻¹. But the code defines domainCost as J(m/e) without saying what m and e mean for a real metal. The docstring itself states plainly: the paragraph above is a research note recording where the idea was meant to go, not a result. The Lean code proves nothing specific to thermal expansion; it proves general facts about the cost function that hold for any positive inputs.

The module is therefore a template, not a theory of thermal expansion. It shows that the RS cost function has the right abstract shape: it vanishes at equality, it is nonnegative, and it has a positive threshold. But turning this into a statement about copper or steel requires defining m and e in terms of atomic spacing, temperature, or some other physical quantity. Until that definition exists, the module is a promise about the framework's reach, not a measurement or a derivation about materials.

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]

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

What this page does not claim

This module derives a numerical value for the thermal expansion coefficient of any real material. The research note's formula α_T = J(φ) × 10⁻⁵ K⁻¹ is a proved result. The framework has a physical model connecting its cost function to atomic spacing or temperature.

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/Materials/Thermal_Expansion_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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