BMVPhaseRateNative
plain-language theorem explainer
The BMV entangling phase rate in RS-native units is $(G m_1 m_2/\hbar)\cdot g$ for masses $m_1,m_2$ and the four branch distances. Anyone citing the quantum-channel gravity bridge (T4) uses this as the closed $\varphi$-rational rate before SI conversion. The body is a direct product of the RS-native $G/\hbar$ with the T3 geometry factor.
Claim. For real masses $m_1,m_2$ and branch distances $r_{LL},r_{LR},r_{RL},r_{RR}$, the RS-native BMV entangling phase rate equals $\frac{G\, m_1 m_2}{\hbar}\cdot g(r_{LL},r_{LR},r_{RL},r_{RR})$, where $g$ is the geometry-dependent inverse-distance factor from the branch-phase invariant of T3.
background
Module Gravity IV builds the unit bridge from the dimensionless RS coupling $\kappa_{rs}=8\varphi^5$ to the dimensionful BMV entangling phase rate (Theorem 4 of Gravity from Recognition IV: The Quantum Channel). In RS-native units one has $\hbar=\varphi^{-5}$ and $G=\varphi^5/\pi$, so $G/\hbar=\varphi^{10}/\pi$ is fixed by $\varphi$ alone.
Upstream, Constants.G is the RS-native gravitational coupling $G=\lambda_{\mathrm{rec}}^2 c^3/(\pi\hbar)$ (not an SI prediction), and Constants.hbar is the native action quantum $\hbar=E_{\mathrm{coh}}\tau_0=\varphi^{-5}$. The geometry factor $g$ is the inverse-distance combination appearing in the T3 branch-phase invariant, divided by the duration $T$ to obtain a per-time rate.
By T3 the entangling invariant is $(G m_1 m_2 T/\hbar)\cdot g$; stripping $T$ yields the rate $(G m_1 m_2/\hbar)\cdot g$ defined here.
proof idea
Definition, not a theorem. The body is the one-line product $(G\cdot m_1\cdot m_2/\hbar)\cdot g(r_{LL},r_{LR},r_{RL},r_{RR})$, with $G$ and $\hbar$ the RS-native constants and $g$ the sibling geometry factor. No tactics or lemmas are applied; later theorems unfold this def and substitute $G/\hbar=\varphi^{10}/\pi$.
why it matters
This is the RS-native rate object that T4 converts to SI. Downstream, bmv_phase_rate_native_eq unfolds it to the closed form $(\varphi^{10}/\pi),m_1 m_2, g$. The SI rate bmvPhaseRateSI multiplies by an external calibration factor $U_{\mathrm{conv}}$, and bmvPhaseRateSI_eq_kappa_alpha_factored rewrites that as $U_{\mathrm{conv}}\cdot\kappa_{rs}\cdot\alpha_{RS}\cdot m_1 m_2\cdot g$ (the T4 master closed form).
Framework landmarks: RS-native $G=\varphi^5/\pi$ and $\hbar=\varphi^{-5}$ from the primer constants; the eight-tick and $\varphi$ fixed-point chain enter only through those constants. The dimensional bridge remains conditional on an ExternalCalibration anchor until SIBridgeClosure supplies one.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.