sensitivity_strict_anti
plain-language theorem explainer
The theorem establishes strict decrease of the sensitivity function for positive frequencies under the inverse scaling. Engineers modeling phantom-cavity gravitational wave antennas cite it to confirm that sensitivity falls as frequency rises above the carrier. The proof unfolds the piecewise definition, secures positivity of the second argument by transitivity, selects the positive branch by negating the zero cases, and applies the positive-numerator division inequality after removing the unit factor.
Claim. Let $f_1,f_2$ be real numbers with $0<f_1<f_2$. Then sensitivity$(f_2)<$sensitivity$(f_1)$, where sensitivity$(f)=h_0·$carrier$/f$ for $f>0$, carrier$=5φ$ Hz, and $h_0=1$.
background
The module treats sensitivity for a phantom-cavity-coupled gravitational wave antenna under RS_PAT_030. Carrier is defined as $5φ$ Hz in both the cortical neuromodulation device and the present module; carrier positivity follows from multiplication positivity and the known positivity of $φ$. Sensitivity is the piecewise function that returns zero for non-positive arguments and $h_0·$carrier$/f$ otherwise, with $h_0$ the dimensionless floor at the carrier frequency.
proof idea
Unfold sensitivity and $h_0$ to reach the ratio form. Obtain $0<f_2$ by transitivity of less-than. Rewrite the two if-conditions to the else branch by negating the non-positive assumptions. Simplify the unit multiplier, then finish with the lemma div_lt_div_of_pos_left on the positive carrier and the given frequency order.
why it matters
The result supplies the strict anti-monotonicity component of the PhantomCoupledGWAntennaSensitivityCert structure. It is invoked directly in gw_antenna_one_statement, which bundles the carrier band (8.05, 8.10), equality at carrier, positivity, and this monotonicity claim. Within the Recognition framework it closes the sensitivity scaling clause for the J6 engineering track, consistent with the phi-based carrier but leaving physical deployment and noise-floor validation as open questions.
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