tau0_predicted_seconds
plain-language theorem explainer
Defines the calibrated RS tick duration in SI seconds as τ₀ = √π · τ_Planck. Anyone citing the SI bridge uniqueness result or the closed-form a_T uses this constant. It is a one-line abbreviation of the main algebraic identity a_T² = π · ℏ_SI · G_SI / c_SI⁵.
Claim. The predicted tick duration in seconds is $\tau_0 := \sqrt{\pi}\,\tau_{\mathrm{Planck}}$, where $\tau_{\mathrm{Planck}} = \sqrt{\hbar_{\mathrm{SI}} G_{\mathrm{SI}} / c_{\mathrm{SI}}^5}$ is the Planck time built from the SI fixings of $c$ and $\hbar$ and the measured $G$.
background
The SI Bridge Closure module fixes the unique conversion map from RS-native units to SI once a dimensional anchor is supplied. In RS-native units the framework predicts $c_{\mathrm{RS}}=1$, $\hbar_{\mathrm{RS}}=\varphi^{-5}$, $G_{\mathrm{RS}}=\varphi^5/\pi$, together with the recognition/Planck identity $G\cdot\pi\cdot\hbar=\lambda_{\mathrm{rec}}^2 c^3$ at $\lambda_{\mathrm{rec}}=\ell_0=1$.
The bridge is three positive factors $a_T$ (sec/tick), $a_L$ (m/voxel), $a_M$ (kg/cohmass) constrained by matching $c$, $\hbar$, and $G$. The main algebraic result is $a_T^2 = \pi,\hbar_{\mathrm{SI}} G_{\mathrm{SI}}/c_{\mathrm{SI}}^5$, i.e. $a_T=\sqrt{\pi},\tau_{\mathrm{Planck}}$.
Upstream, tau_Planck is defined as $\sqrt{\hbar_{\mathrm{SI}} G_{\mathrm{SI}}/c_{\mathrm{SI}}^5}$, the standard Planck time from SI-2019 $c$, $\hbar$ and CODATA $G$.
proof idea
Pure definitional abbreviation: multiply the already-defined Planck time by $\sqrt{\pi}$. No tactics or lemmas; the mathematical content lives in the uniqueness theorem that forces $a_T^2=\pi,\hbar_{\mathrm{SI}} G_{\mathrm{SI}}/c_{\mathrm{SI}}^5$.
why it matters
This is the closed-form SI value of the fundamental tick that the uniqueness theorem delivers. The certificate structure SIBridgeClosureCert packages it as clause 2 ("Closed form: $\tau_0=\sqrt{\pi},\tau_{\mathrm{Planck}}$") and clause 4 (positivity of the calibrated $\tau_0$). Downstream positivity is immediate from positivity of $\pi$ and of Planck time.
In the broader framework this pins the time unit of the eight-tick octave (T7) once the SI anchor is chosen. It does not invent a new prediction for $G$; it converts the RS-native $G_{\mathrm{RS}}=\varphi^5/\pi$ into a unique $a_T$ given measured $G_{\mathrm{SI}}$. Numerically $\tau_0\approx 9.55\times 10^{-44},\mathrm{s}$.
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