Materials Young Modulus Steel Rs

The steel modulus note

Young's modulus measures a material's stiffness: the ratio of stress, the force per area, to strain, the fractional change in length. For steel, the accepted value is about 200 gigapascals (GPa), meaning a steel rod one meter long and one square centimeter in cross-section would need about 20,000 kilograms of hanging weight to stretch by one millimeter. Engineers use this number daily to predict how much beams and columns will deflect under load.

The golden ratio phi, roughly 1.618, appears throughout geometry and number theory. Its twenty-fifth power equals about 196,000 megapascals, or 196 GPa. That is within two percent of steel's modulus. The observation is that a number built from phi lands close to a measured material constant, with no fitted parameters.

In Recognition Science, the framework models cost as a function of a ratio between two quantities, written J(m/e), where J(x) = (x + 1/x)/2 - 1. The module under discussion defines a ledger for a domain using this formula, and proves three general facts: the cost is zero when the two quantities are equal, it is never negative for positive inputs, and the threshold phi minus 1.5 is positive. These are properties of the cost function itself, not of steel.

The docstring records the intended connection to steel's modulus, but the formal code does not define what m and e mean for a material. The theorem file proves facts about the abstract cost function, and the 196 GPa match sits in a research note, not in a machine-checked theorem. The framework's library, a machine-checked collection of formal theorems, establishes the general cost properties; the material-specific claim remains a numerical observation.

What the module does establish, cleanly, is that the cost function has the basic properties a measure of mismatch should have: it vanishes at agreement, stays nonnegative, and has a positive threshold tied to the golden ratio. The steel connection is a prompt for future work, not a proven result. A reader can take the general cost facts as verified and treat the 196 GPa coincidence as an open target for a real derivation.

structure YoungModSteelCert 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

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

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)

What this page does not claim

The steel modulus match is a proved theorem; it is a numerical observation in a research note. The cost function's properties are specific to steel; they hold for any positive ratio m/e. The module derives the value 200 GPa from first principles.

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/Young_Modulus_Steel_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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