Materials Fracture Mechanics From Jcost Paris Law Exponent Pos

The Paris exponent

The Paris law describes how a crack in a material grows under repeated loading. Engineers write it as da/dN = C(ΔK)^m, where da/dN is the crack growth per cycle, ΔK is the range of the stress intensity factor, and C and m are material constants. The exponent m is the quantity that matters most: a larger m means the crack accelerates more sharply as the stress range increases. For many metals, m sits near 3 to 4, and the value 4 appears often in handbooks.

The Recognition Science framework, a body of formal mathematics built from a single cost function, contains a small declaration about this exponent. The declaration parisLawExponent_pos establishes that the framework's Paris law exponent, which it sets to the natural number 4, is greater than zero. The proof is immediate: the exponent is defined as 4, and 4 is positive. This is a theorem in the framework's machine-checked library of formal theorems, meaning the statement is verified by a computer from the framework's axioms.

The framework derives the value 4 from a structural argument: it identifies the exponent as the configuration dimension of space (3) plus 1, coming from a four-point symmetry of the stress intensity field. This derivation is a prediction, not a measurement. The framework's claim is that the exponent should be 4 for a three-dimensional material, and it offers a falsifier: a precision fracture toughness measurement on a class of materials showing the critical strain energy release rate systematically outside the framework's predicted band by more than 50%.

What the declaration itself does not claim is equally clear. It does not prove that real materials have exponent 4; that is an empirical question. It does not establish the Paris law itself, nor does it derive the exponent from first principles within the formal system. The declaration is a single, narrow fact: the framework's chosen exponent is positive. The broader claims about why the exponent is 4 and how it connects to the framework's cost function remain at the level of prediction and definition, not formal proof.

/-- Paris law exponent is positive. -/
theorem parisLawExponent_pos : 0 < parisLawExponent := by
  rw [parisLawExponent_eq]; norm_num

/-- Paris law exponent: m = 4 = configDim + 1. -/
def parisLawExponent : ℕ := 4

What this page does not claim

This declaration does not prove that real materials have a Paris exponent of 4. This declaration does not derive the Paris law itself or the exponent from the framework's cost function. This declaration does not establish the value of the exponent as anything other than the framework's chosen definition.

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/FractureMechanicsFromJCost.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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