IndisputableMonolith.Materials.FractureToughnessFromJCost
The Materials.FractureToughnessFromJCost module applies the Canonical J-Cost Band template to the materials domain. Materials physicists working in Recognition Science cite it to obtain fracture toughness measures from J-cost ratios on the phi-ladder. The module follows the six-clause reusable structure imported from CanonicalJBand, establishing the matched-zero and nonnegativity clauses for the domain cert.
claimThe module supplies $FractureToughnessCert$ and $FractureRegime$ such that the J-cost function satisfies $J(1)=0$ and $J(x)≥0$ for $x>0$, together with the remaining four clauses of the Canonical J-Band template.
background
The upstream CanonicalJBand module defines the reusable six-clause J-cost-on-ratio template used for all domain certs in the master chain. Each domain cert must prove matched-zero at $J(1)=0$ and nonnegativity $J(x)≥0$ for $x>0$, plus four additional clauses on the ratio variable. Recognition Science defines $J(x)=(x+x^{-1})/2-1$, which equals zero at unity and is nonnegative for positive arguments. The present module sits in the Materials domain and imports only Mathlib plus this template.
proof idea
This is a definition module that imports the CanonicalJBand template and instantiates it via the sibling declarations FractureRegime, fractureRegimeCount, FractureToughnessCert, fractureToughnessCert, FractureToughCert and cert. No independent proofs appear; the structure simply supplies the domain-specific names and the required clause instances.
why it matters in Recognition Science
The module supplies one of the Plan v7 domain certs in the B-tier whole-science openings of the master cert chain. It feeds the materials instantiation of the J-cost band into the larger Recognition Science derivation of physical constants and mass ladders. No downstream declarations are recorded yet.
scope and limits
- Does not derive numerical values for fracture toughness.
- Does not address time-dependent or dynamic fracture.
- Does not connect to the eight-tick octave or spatial dimension forcing.
- Does not prove the full six clauses; relies on the imported template.