bmv_phase_rate_native_eq
plain-language theorem explainer
The RS-native BMV entangling phase rate equals (φ¹⁰/π) m₁ m₂ g for masses m₁, m₂ and geometry factor g from the four branch distances. Anyone deriving tabletop BMV predictions from Recognition Science constants would cite this closed form. The proof unfolds the rate definition and substitutes the native identity G/ℏ = φ¹⁰/π.
Claim. For all 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 $\bigl(\varphi^5/\pi\cdot\varphi^5\bigr)\,m_1 m_2\,g(r_{LL},r_{LR},r_{RL},r_{RR})$, where $g$ is the geometry factor built from the four inverse-distance combinations. Equivalently the rate is $(\varphi^{10}/\pi)\,m_1 m_2\,g$.
background
In Recognition Science native units one has $c=1$, $\hbar=\varphi^{-5}$, and $G=\varphi^5/\pi$, so the ratio $G/\hbar=\varphi^{10}/\pi$ is fixed by $\varphi$ alone. The BMV (Bose–Marletto–Vedral) entangling phase measures gravitationally mediated entanglement between two masses. By the quantum-channel analysis (T3), the per-time rate is $(G m_1 m_2/\hbar)\cdot g$, with $g$ the geometry-dependent combination of inverse distances among the four branch endpoints.
This module (Gravity IV, Unit Bridge) converts the dimensionless RS coupling $\kappa_{rs}=8\varphi^5$ into a dimensionful BMV phase rate. The native rate is the intermediate closed form before any SI calibration is applied. Upstream, $G$ is the RS-native gravitational coupling $G=\lambda_{\mathrm{rec}}^2 c^3/(\pi\hbar)$ (not a CODATA prediction), and $\hbar$ is the native action quantum defined as $\varphi^{-5}$.
proof idea
Unfold the definition of the native phase rate, which is $(G m_1 m_2/\hbar)\cdot g$. Algebraically regroup $G m_1 m_2/\hbar$ as $(G/\hbar),m_1 m_2$ by a one-line ring identity, then rewrite with the sibling lemma that $G/\hbar=\varphi^5/\pi\cdot\varphi^5$ in RS-native units. The geometry factor is left untouched. The entire argument is a two-step rewrite after unfolding.
why it matters
This is the closed-form identity named in the module doc as one of the three proved objects of Gravity IV Theorem 4 (the unit bridge from $\kappa_{rs}$ to the SI BMV phase rate). It pins the native rate to $\varphi^{10}/\pi$ times masses and geometry, so the $\kappa_{rs}$ band $(85.6,90.4)$ can propagate linearly into a native rate band via the sibling band theorem. Framework landmarks: $G=\varphi^5/\pi$ and $\hbar=\varphi^{-5}$ from the RS constant table (primer: $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$); $\varphi$ itself is forced at T6. No downstream dependents are recorded yet; siblings such as the SI phase-rate bridge and the $\kappa_{rs}$-band propagation build on the same native expression. The SI conversion remains conditional on an external calibration anchor.
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