Experimentally Measurable Quantification of Quantum HS° Resource Consumption (v1.0)

Quantum HS$^\circ$ models temporal (phase) alignment without an external clock as a $U(1)$-asymmetry resource theory: free states are time-translation invariant and free operations are time-translation covariant channels. This note gives a compact, measurement-facing route to quantify \emph{resource consumption} during an alignment attempt. The key point is that applying a group shift $\rho\mapsto U(\alpha)\rho U(\alpha)^\dagger$ does not consume HS$^\circ$ resource; consumption arises from the estimation--correction loop and residual uncertainty, which induces an effective phase-diffusion channel (a mixture of shifts). For a finite-dimensional clock state, we derive (i) an analytic baseline $\AHS(\rho_{\mathrm{in}})=\log M$, (ii) an identity $\Delta\AHS=S(\rho_{\mathrm{out}})$ for the maximally asymmetric input family under phase diffusion, and (iii) experimentally accessible proxies and lower bounds using coherence visibility and purity, enabling reproducible reporting of HS$^\circ$ resource expenditure under stated assumptions.