Materials Creep Rate2 From Jcost

Creep rate from cost

Creep is the slow, permanent deformation of a solid under constant stress, most familiar in turbine blades and reactor metals that sit hot for years. Engineers describe the steady creep rate with the Dorn equation, ε̇ = A·exp(−Q_c/RT)·σⁿ, where σ is stress, T is temperature, Q_c is an activation energy, and n is the stress exponent. For many metals in the dislocation creep regime, measurements put n between 3 and 5, and a common textbook value is n = 3.

The Recognition Science framework attempts to derive that exponent from first principles. Its central object is the cost, a forced penalty for any mismatch between two quantities, written J(x) = (x + 1/x)/2 − 1. The framework's library proves that any cost function satisfying five plain conditions must equal this J. The idea for creep is to define a domain cost as J(m/e), where m is some measured material quantity and e is its expected value; when they match, the cost is zero, and when they differ, the cost is positive.

What the machine-checked module actually establishes is narrower. It proves three general facts about J(m/e): it vanishes when m equals e, it is never negative for positive inputs, and the constant φ − 3/2 is positive (φ being the golden ratio, about 1.618). These are properties of the cost function itself, not of creep. The module defines a certificate structure bundling these three facts, and proves that certificate exists. The research note attached to the file records the intended step: that the power-law creep exponent n equals the framework's configurational dimension D = 3, placing n at the lower bound of the empirical 3–5 range.

In plain language, the module proves the cost function behaves sensibly, but it does not prove that creep follows that cost. The missing piece is a definition of m and e in creep's own terms, a definition the file itself acknowledges it lacks. The result is a template, shared verbatim across 2383 sibling modules, not a theorem about materials.

The consequence for a reader is a clear boundary. The framework's cost function has a rigorously proven shape, and the creep exponent n = 3 is a plausible identification at the empirical lower bound, but the bridge from cost to creep remains unbuilt. That bridge is the target for future work.

theorem domainCost_at_eq (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]

theorem cert_inhabited : Nonempty CreepRate2Cert := ⟨cert⟩

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not derive the Dorn equation or the creep exponent n = 3 from the cost function. The framework does not prove that real metals obey J(m/e); the empirical n = 3–5 range is a measurement, not a theorem. The certificate structure is a formal bundling of three cost facts, not a physical model of 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/Creep_Rate2FromJCost.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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