G_constraint
plain-language theorem explainer
The G-constraint is the dimensional matching condition that equates the CODATA SI value of Newton's constant to the RS-native value scaled by the bridge conversion factors. Anyone proving uniqueness of the SI calibration map cites it as one of the three closing equations. It is a pure propositional definition: the equality itself, not a derived theorem.
Claim. For a bridge $b$ with positive factors $a_T$ (seconds per tick), $a_L$ (metres per voxel), and $a_M$ (kilograms per coherence-mass), the G-constraint is the proposition $G_{\mathrm{SI}} = G_{\mathrm{RS}}\cdot a_L^3/(a_M\cdot a_T^2)$, where $G_{\mathrm{RS}}=\varphi^5/\pi$ and $G_{\mathrm{SI}}$ is the CODATA anchor.
background
The SI Bridge Closure module fixes the unique calibration from Recognition Science native units to SI. In the native gauge one has $c_{\mathrm{RS}}=1$, $\hbar_{\mathrm{RS}}=\varphi^{-5}$, and $G_{\mathrm{RS}}=\varphi^5/\pi$, together with the recognition/Planck identity $G\cdot\pi\cdot\hbar=\lambda_{\mathrm{rec}}^2\cdot c^3$ at $\lambda_{\mathrm{rec}}=1$.
An SI bridge is a structure of three strictly positive conversion factors: $a_T$ (sec/tick), $a_L$ (m/voxel), $a_M$ (kg/coherence-mass). Matching the three dimensionful constants against SI values produces three constraints. Under SI-2019, $c_{\mathrm{SI}}$ and $\hbar_{\mathrm{SI}}$ are exact definitions; $G_{\mathrm{SI}}$ is the remaining CODATA measurement that anchors this bridge.
Upstream, $G_{\mathrm{RS}}$ is defined by $G=\lambda_{\mathrm{rec}}^2 c^3/(\pi\hbar)$ with unit $\lambda_{\mathrm{rec}}$ and $c$, and $\hbar=\varphi^{-5}$, giving $\varphi^5/\pi$. The external anchor $G_{\mathrm{SI}}=6.67430\times 10^{-11},\mathrm{m}^3/(\mathrm{kg},\mathrm{s}^2)$ is the CODATA 2022 value.
proof idea
Definitional only. The predicate on a bridge $b$ is exactly the equality $G_{\mathrm{SI}}=G_{\mathrm{RS}}\cdot(a_L^3)/(a_M,a_T^2)$. No tactics, no lemmas, no proof obligations.
why it matters
This predicate is the third conjunct of the closed-bridge condition (all three matching equations hold). Downstream, the ratio theorem from the $c$ and $G$ constraints alone yields $a_T/a_M=G_{\mathrm{SI}}/(G_{\mathrm{RS}},c_{\mathrm{SI}}^3)$. With the $\hbar$-constraint these close the calibration and deliver the module's main structural result $a_T^2=\pi,\hbar_{\mathrm{SI}},G_{\mathrm{SI}}/c_{\mathrm{SI}}^5$, i.e. $\tau_0=\sqrt{\pi},\tau_{\mathrm{Planck}}$.
In the framework this is the SI-side image of the native prediction $G=\varphi^5/\pi$ (primer constants). The module treats $G_{\mathrm{SI}}$ as an external anchor; it does not predict that number. Closure of the conversion map is the named frontier this definition helps finish.
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