Materials Creep Regimes From Config Dim

The five creep regimes

Materials creep is the slow, time-dependent deformation of a solid under constant stress. It is a central concern in engineering because turbine blades, jet engine components, and nuclear reactor parts operate at temperatures where even a strong metal will gradually stretch and eventually fail. Standard materials science divides this process into five canonical regimes: primary (transient) creep, where the strain rate decreases as the material hardens; secondary (steady-state) creep, where the rate is roughly constant; tertiary (accelerating) creep, where the rate increases as damage accumulates; the ductile-brittle transition, where the failure mode changes; and final fracture. This five-stage picture is the classical description taught in every metallurgy course.

The Recognition Science framework begins with a simple observation: reality keeps a ledger, a discrete record of recognition events, and the cost of each recognition is forced by a proved functional equation. From that single equation, the framework derives a chain of consequences, including the golden ratio phi as the unique self-similar scaling. The materials module applies this chain to creep. It defines the five regimes as an inductive type, a data structure with exactly five named cases, and proves that the count is five. It then defines the characteristic strain rate of each regime as a power of phi and proves that the ratio of strain rates between adjacent regimes is exactly phi. In plain terms, the framework claims that each stage of creep runs at a rate that is a fixed multiple of the previous stage, and that multiple is the golden ratio.

The module does not claim to measure these rates or to explain the physical mechanisms that cause primary hardening or tertiary damage. It establishes a structural relationship: if the five regimes exist, and if the framework's forcing chain applies, then the rates must be φ-spaced. The proof is machine-checked, meaning a computer verified every step of the arithmetic and the logic, with no unproven assumptions. The practical consequence is a prediction: for a given material, the ratio of the strain rate in the tertiary regime to that in the secondary regime should be φ, and similarly for each adjacent pair. This is a falsifiable claim that a laboratory creep test could confirm or reject.

The value of this result is that it connects a practical engineering taxonomy to a deep mathematical structure. It suggests that the five regimes are not arbitrary; they are the natural number of stages that fall out of the framework's counting argument. Whether real materials obey the φ-spacing is an empirical question, not a theorem. The module provides the formal scaffold, and the experiment provides the verdict.

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_pos (k : ℕ) : 0 < strainRate k := pow_pos phi_pos k

What this page does not claim

This module does not measure any real material's creep rates. The φ-spacing is a prediction, not a verified empirical law. The framework does not explain the microstructural causes of primary, secondary, or tertiary creep.

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)

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.

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