cert
plain-language theorem explainer
The definition cert re-exports the canonical J-band certificate into the ultrasound therapy module. Acoustics modelers working on HIFU and LIPUS thresholds would cite it to anchor intensity ratios to the phi-ladder. It is a one-line alias to the six-clause structure from CanonicalJBand that confirms J(1)=0, reciprocity, and the phi-band bounds.
Claim. Let $J$ be the J-cost function. The certificate cert asserts $J(1)=0$, $J(x)=J(1/x)$ for $x≠0$, $J(φ)>0$ with $0.11<J(φ)<0.13$, and $J(1/φ²)>0$.
background
Recognition Science defines the J-cost via T5 as $J(x)=(x+x^{-1})/2-1$, obeying the Recognition Composition Law. The module UltrasoundTherapyThresholdFromJCost applies this to acoustic bioeffects, identifying therapeutic thresholds with the canonical golden-section quantum on intensity ratios and mechanical-index steps of order φ². The upstream CanonicalCert structure supplies the six clauses that certify the J-band properties reused across domains.
proof idea
This is a one-line wrapper that directly assigns the cert from Common.CanonicalJBand, inheriting the structure without further reduction.
why it matters
The declaration embeds the ultrasound therapy model inside the canonical J-band, connecting acoustic thresholds to T5 J-uniqueness and T6 phi fixed point. It supports the module claim that mechanical-index ratios track φ² transitions from diagnostic to ablative regimes. The module doc treats the empirical correspondence as a hypothesis while the band identity itself is theorem-level.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.