Materials Fatigue Life From Phi Ladder

Fatigue life and the phi ladder

Materials fatigue is the progressive, localized damage that occurs when a material is subjected to repeated loading. The standard engineering description is the S-N curve, also called the Wöhler curve after August Wöhler, who documented it in the 1860s. It plots stress amplitude S against the number of cycles to failure N_f. A common empirical form is N_f = N_0 × (σ_u/σ)^m, where σ_u is the ultimate strength, σ is the applied stress, and m is the fatigue exponent. For many metals, m lies near 3, but values vary widely with material and loading.

In Recognition Science, the framework models this exponent as a consequence of its cost function. The framework's central object is a cost, a measure of mismatch between a measured value and an expected value. The framework proves that any cost satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. From this, the framework derives the golden ratio φ = (1 + √5)/2 ≈ 1.618 as the unique self-similar scaling. The fatigue exponent m is then identified with φ² ≈ 2.618, a value within the scatter of many empirical S-N exponents.

The module named FatigueLifeFromPhiLadder in the framework's library does not prove this material claim. Its formal content is deliberately general. It defines a cost function domainCost(m, e) = J(m/e) and proves three facts: the cost is zero when m equals e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. The library's own docstring states plainly that these results are universal and say nothing specific to fatigue. What would make the module a theorem about fatigue is a definition of m and e in material terms, which the module does not provide.

The honest summary is that the phi ladder offers a suggestive identification, not a proved material law. The formal machinery guarantees the shape of the cost function and the positivity of a threshold, but the bridge from recognition geometry to Wöhler exponents remains a research note. A reader should take the φ² exponent as a hypothesis to test against measured S-N data, not as a derived consequence.

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 fatigue exponent m = φ² is a proved consequence of the framework's axioms. The module establishes any material-specific result about fatigue life. The S-N curve itself is derived from recognition geometry.

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/FatigueLifeFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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