Model-level synthetic-flux control of hyperchaos order and matched-resource sensing in dissipative optomechanics

Within a normalised six-dimensional model of dissipative optomechanics (one cavity + two mechanical resonators), a synthetic-flux phase $Φ_{\rm syn}$ selects the \emph{order} of a drive- and coupling-gated hyperchaos transition---up to four simultaneously unstable Lyapunov directions, beyond any reported single-mode benchmark---while the same matched-resource force-sensing protocol yields no flux-induced enhancement on the chaotic attractor. Building on the topology of Muthukumar \emph{et al.}~[PR Applied \textbf{24}, 014053 (2025)], a phase-consistent Floquet--Lyapunov protocol (cross-checked by monodromy multipliers, dissipative volume balance and a 180-run three-seed audit) localises a Neimark--Sacker bifurcation at $E^{*}=\num{1.060}$ ($θ=0$). At a weakly coupled reference the matched Fisher gain reaches at most $\num{1.32}\times$ (flux-off) and $\num{1.16}\times$ (single-mode), with Monte-Carlo median $\num{1.039}\times$ (90\,\% CI $[\num{1.025},\num{1.053}]$); on the chaotic attractor the identical protocol returns a null result ($\mathcal{G}_{A/B}=\num{1.039}\pm\num{0.014}$). Truncated-Fock and truncated-Wigner checks support the mean-field description at selected points. Both the hyperchaos classification and the sensing result remain strictly model-level: the strong-coupling sector explored here lies $\num{2542}\times$ beyond anchored silicon optomechanical couplings. Closing that gap requires a measured inter-resonator hopping $J_m$ and fixed bath temperatures.