Materials Creep Regimes From Config Dim Strain Rate Ratio

The strain rate ratio

Materials under sustained stress and high temperature slowly deform in a process called creep. Classical materials science divides this process into five canonical regimes: primary (transient), secondary (steady-state), tertiary (accelerating), ductile-brittle transition, and final fracture. Each regime has a characteristic strain rate, the speed at which the material deforms.

In the Recognition Science framework, the number of regimes is not arbitrary. The framework's machine-checked library of formal theorems proves that the count is exactly five, and it defines a strain rate, a measure of deformation speed, for each step on a ladder of natural numbers. The key result, the strainRate_ratio theorem, proves that the strain rate of any regime divided by the strain rate of the regime before it equals phi, the golden ratio, approximately 1.618. In plain terms, each successive creep regime is predicted to deform about 1.618 times faster than the one before it.

This ratio is not fitted to experimental data. It falls out of the framework's central forcing chain, in which a single cost function, derived from five plain conditions, forces the golden ratio as the unique self-similar scaling. The theorem itself is a formal statement about the defined strainRate function: it proves that strainRate (k + 1) / strainRate k = phi for every natural number k, and that all strain rates are positive. The proof is a short algebraic argument from the definition of strainRate as phi raised to the power k.

The declaration does not claim that real materials actually obey this ratio. It establishes a formal relationship within the framework's model. Whether the five-regime picture and the phi ratio match measured creep behavior in actual alloys is an empirical question, not a proved theorem. The framework models the regimes as a discrete ladder; the classical understanding of creep as a continuous process with regime boundaries that shift with stress and temperature remains the standard engineering picture.

theorem creepRegime_count : Fintype.card CreepRegime = 5 := by decide

theorem strainRate_ratio (k : ℕ) : strainRate (k + 1) / strainRate k = phi := by
  unfold strainRate
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem strainRate_ratio (k : ℕ) : strainRate (k + 1) / strainRate k = phi := by
  unfold strainRate
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem strainRate_pos (k : ℕ) : 0 < strainRate k := pow_pos phi_pos k

What this page does not claim

The theorem does not claim that measured creep data from real alloys matches the phi ratio. The theorem does not claim that creep always has exactly five regimes in every material and condition. The theorem does not derive the golden ratio from creep physics; it applies a pre-derived framework constant to a defined model.

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

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