strainRate_ratio
plain-language theorem explainer
The theorem shows that strain rates for consecutive creep regimes differ by the exact factor phi. Materials modelers working on failure depth would cite it to fix the ratio between the five regimes. The proof is a short algebraic reduction that unfolds the exponential definition and normalizes the ratio via power rules and ring simplification.
Claim. For every natural number $k$, the ratio of the strain rate at rung $k+1$ to the strain rate at rung $k$ equals the golden ratio $phi$, where the strain rate at rung $k$ is $phi^k$.
background
The Creep Regimes from configDim module treats materials creep as five regimes fixed by configDim D=5: primary (transient), secondary (steady-state), tertiary (accelerating), ductile-brittle transition, and final fracture. Each regime's characteristic strain rate occupies one rung on the phi-ladder, so adjacent regimes obey a constant ratio of phi. The upstream definition supplies the explicit form strainRate (k : ℕ) : ℝ := phi ^ k.
proof idea
The term proof unfolds strainRate to obtain phi^{k+1} / phi^k, invokes positivity of phi^k to clear the denominator, rewrites the successor power, and finishes with ring normalization.
why it matters
This supplies the phi_ratio field inside the CreepRegimeCert structure that certifies the five-regime model. It directly realizes the phi-ladder property required by the Recognition Science framework (T6 self-similar fixed point). The result closes the B9 materials failure depth analysis with no remaining sorry.
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