Materials Fatigue Threshold From Jcost Endurance Threshold Band

The endurance threshold

Materials fatigue is the failure of metals and polymers under repeated loading at stresses far below the static yield strength. A steel bridge or turbine blade can snap after millions of cycles even though a single heavy load would not bend it. Engineers describe this with Wöhler's S-N curve, Basquin's law, and the Coffin-Manson relation. The endurance limit, about half the yield stress for many steels, is the stress below which a material can in principle survive infinite cycles.

In Recognition Science, the framework models fatigue through a cost function on the stress ratio r, defined as observed stress divided by yield stress. The cost is zero at r = 1, meaning no perturbation from the yield baseline, and positive for any other ratio. The framework's central result, J(x) = (x + 1/x)/2 - 1, applies here. The endurance limit corresponds to the golden-section value J(φ), which falls in the band between 0.11 and 0.13.

The theorem endurance_threshold_band, checked in the machine-checked library of formal theorems, proves that 0.11 < EnduranceThreshold < 0.13. Two regimes follow: IsInfiniteLife holds when the fatigue cost is below this threshold, and IsFatigueFailing holds at or above it. The theorem regimes_exclusive proves these two conditions cannot both hold for the same stress ratio. The framework also proves the cost is nonnegative, symmetric under taking the reciprocal stress ratio, and positive away from yield.

What the declaration does not claim is equally important. It does not prove that real steel actually fails at this exact threshold; it establishes a structural band within the framework's model. The comparison to the measured endurance limit of about 0.5 times yield stress for many steels is an empirical observation, not a theorem. The framework places fatigue alongside plaque vulnerability, ignition, and magnetic reconnection as sharing one universal cost quantum, but that unification across fields is a modeled analogy, not a derived physical law.

/-- Endurance threshold lies in the canonical band. -/
theorem endurance_threshold_band :
    0.11 < EnduranceThreshold ∧ EnduranceThreshold < 0.13 := by
  unfold EnduranceThreshold
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  rw [Cost.Jcost_eq_sq hphi_ne]
  have h_lo : (1.61 : ℝ) < phi := Constants.phi_gt_onePointSixOne
  have h_hi : phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
  have hpos : (0 : ℝ) < 2 * phi := by
    have : (0 : ℝ) < phi := Constants.phi_pos
    linarith
  refine ⟨?lo, ?hi⟩
  · rw [lt_div_iff₀ hpos]
    nlinarith [h_lo, h_hi]
  · rw [div_lt_iff₀ hpos]
    nlinarith [h_lo, h_hi]

theorem fatigueCost_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ fatigueCost r :=
  Cost.Jcost_nonneg hr

theorem fatigueCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    fatigueCost r = fatigueCost r⁻¹ := Cost.Jcost_symm hr

theorem fatigueCost_pos_off_yield {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < fatigueCost r := Cost.Jcost_pos_of_ne_one r hr hne

/-- Infinite-life and fatigue-failing are mutually exclusive. -/
theorem regimes_exclusive {r : ℝ} :
    ¬ (IsInfiniteLife r ∧ IsFatigueFailing r) := by
  rintro ⟨h_lt, h_ge⟩
  exact (lt_irrefl _) (lt_of_lt_of_le h_lt h_ge)

What this page does not claim

The theorem does not prove that physical steel fails exactly at this threshold. The measured endurance limit of about 0.5 times yield stress is an empirical observation, not a derived result. The framework's analogy across fatigue, combustion, and pathology is a modeled unification, not a proven physical law.

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

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