si_bridge_closed_under_three_constraints
plain-language theorem explainer
Under the three SI matching constraints (c, ℏ, G), every closed bridge has tick conversion equal to √π times the Planck time. Anyone citing the unique RS-to-SI calibration map needs this master certificate. The proof is a one-line appeal to the closed-form identity already proved for τ₀.
Claim. For every SI bridge $b=(a_T,a_L,a_M)$ of positive conversion factors (seconds per tick, metres per voxel, kilograms per coherence-mass), if $b$ satisfies the three matching constraints $c_{\mathrm{SI}}=c_{\mathrm{RS}}\,a_L/a_T$, $\hbar_{\mathrm{SI}}=\hbar_{\mathrm{RS}}\,a_M a_L^2/a_T$, and $G_{\mathrm{SI}}=G_{\mathrm{RS}}\,a_L^3/(a_M a_T^2)$, then $a_T=\sqrt{\pi}\,\tau_{\mathrm{Planck}}$.
background
Recognition Science fixes the dimensionless triple in native units: $c_{\mathrm{RS}}=1$, $\hbar_{\mathrm{RS}}=\varphi^{-5}$, $G_{\mathrm{RS}}=\varphi^5/\pi$, together with the recognition/Planck bridge $G\cdot\pi\cdot\hbar=\lambda_{\mathrm{rec}}^2 c^3$ at $\lambda_{\mathrm{rec}}=\ell_0=1$. Laboratory display needs a dimensional bridge: three positive factors $a_T$ (sec/tick), $a_L$ (m/voxel), $a_M$ (kg/cohmass).
Matching the SI values of $c$, $\hbar$, and $G$ yields three algebraic constraints. Under SI-2019, $c_{\mathrm{SI}}$ and $\hbar_{\mathrm{SI}}$ are exact definitions; $G_{\mathrm{SI}}$ is the CODATA external anchor. A bridge is closed when all three constraints hold simultaneously.
The module's main algebraic result is $a_T^2=\pi,\hbar_{\mathrm{SI}} G_{\mathrm{SI}}/c_{\mathrm{SI}}^5$, equivalently $\tau_0=\sqrt{\pi},\tau_{\mathrm{Planck}}$. The native $G$ definition is the projection $G=\lambda_{\mathrm{rec}}^2 c^3/(\pi\hbar)$, not a prediction of the SI value of Newton's constant.
proof idea
Term-mode one-liner: the statement is definitionally the universal quantification already established by the closed-form identity tau0_eq_sqrt_pi_planck_time. No new algebra is performed here; the master certificate simply packages that identity as the claim that every closed bridge has $a_T=\sqrt{\pi},\tau_{\mathrm{Planck}}$. Upstream work derives $a_L=c_{\mathrm{SI}} a_T$ from the $c$-constraint, substitutes into the $\hbar$ and $G$ constraints with the RS-native values plugged in, and simplifies to the $\pi,\hbar G/c^5$ formula for $a_T^2$.
why it matters
This is the module's master SI-bridge calibration certificate. It closes the named open frontier of the dimensional bridge: once the single dimensional anchor $G_{\mathrm{SI}}$ is supplied, the tick (and by the same constraints the voxel and coherence-mass) conversion factors are uniquely fixed, with zero free dimensionless parameters.
Framework landmarks in play are the RS-native constants $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$ from the primer, and the recognition/Planck identity that forces the $\pi$ in $\tau_0=\sqrt{\pi},\tau_{\mathrm{Planck}}$. The five-clause certificate also records positivity of Planck time and of calibrated $\tau_0$, and the single-anchor character of any pure-number theory mapped to SI.
No downstream dependents are recorded yet; the declaration is the terminal structural theorem of SIBridgeClosure (status: 0 sorry, 0 RS-internal axiom). It does not feed mass-ladder or $\alpha$-band results directly; it only certifies unit conversion.
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