Materials Creep Regimes From Config Dim Strain Rate Pos

Strain rate positivity

Materials creep is the slow, time-dependent deformation of a solid under constant stress. It proceeds through five canonical regimes: primary (transient), secondary (steady-state), tertiary (accelerating), ductile-brittle transition, and final fracture. In Recognition Science, the framework's account, these regimes are indexed by a natural number k, and the strain rate at regime k is defined as phi raised to the power k, where phi is the golden ratio, approximately 1.618. The declaration strainRate_pos proves that this strain rate is always greater than zero for every natural number k.

The proof is immediate from the definition. Since phi is positive, any positive power of phi is positive. The theorem states: for all k, 0 < strainRate k. This is a formal theorem, checked by a machine, with no axioms beyond the standard three. It is part of a larger structure: the framework also proves that adjacent regimes have a strain rate ratio of exactly phi, and that there are exactly five regimes. The positivity theorem is the simplest of the three, but it is the one that guarantees the rates are physically meaningful as rates.

What the declaration does not claim is more interesting than what it does. It does not claim that these rates match any measured creep data. It does not claim that the golden ratio is the actual rate ratio in real materials. It does not claim that the five regimes are the only possible regimes. The theorem is purely structural: within the framework's definition, the rates are positive. The comparison to real-world creep measurements is an empirical check, not a theorem. The framework's library proves the structure; the measurements are a separate matter.

The consequence for a reader is this: the positivity theorem is a small but necessary piece of the framework's account of creep. It ensures that the ladder of rates, each rung a factor of phi above the last, never dips to zero or below. Without positivity, the ratio theorem would be meaningless, since division by zero would be possible. With it, the ladder is a well-defined sequence of positive real numbers, ready for comparison against experiment.

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

noncomputable def strainRate (k : ℕ) : ℝ := phi ^ k

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

What this page does not claim

The strain rate values match any measured creep data. The golden ratio is the empirically observed rate ratio in real materials. The five regimes are the only possible classification of creep stages.

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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