planck_length_eq
plain-language theorem explainer
In RS-native (Family A) units the Planck length equals $\sqrt{1/\pi}$. Constant-derivation and forcing-chain authors cite it to pin $\ell_P$ once $c$, $\hbar$, and $G$ are fixed algebraically in $\varphi$. The proof unfolds the definition, sets $c=1$, and rewrites via the $G\cdot\hbar=1/\pi$ product.
Claim. In RS-native units the Planck length equals $\sqrt{1/\pi}$, i.e. $\ell_P = \sqrt{\hbar G/c^3}$ reduces to $\sqrt{1/\pi}$ once $c=1$ and $G\cdot\hbar=1/\pi$.
background
Module ConstantDerivations derives $c$, $\hbar$, $G$, and $\alpha$ as ratios of RS-native quantities rather than free inputs. The chain runs Composition Law $\to$ unique $J$-cost $\to$ $\varphi$ and $D=3$ $\to$ eight-tick period $\to$ Level-4 constants.
The Planck length is defined by the usual combination $\ell_P=\sqrt{\hbar G/c^3}$. In Family A one has $c_{\mathrm{rs}}=1$ (causal coherence), $\hbar_{\mathrm{rs}}=\varphi^{-5}$ (IR gate), and $G_{\mathrm{rs}}=\varphi^5/\pi$ (curvature extremum with holographic $\pi$), so $\hbar G=1/\pi$ and $\ell_P^2=1/\pi$.
Upstream, c_rs_eq_one discharges $c=1$ by unfolding length/time units; the product identity $G\cdot\hbar=1/\pi$ is the algebraic content used after that simplification.
proof idea
Term-style rewrite proof. Unfold $\ell_P$ to $\sqrt{\hbar_{\mathrm{rs}} G_{\mathrm{rs}}/c_{\mathrm{rs}}^3}$. Replace $c_{\mathrm{rs}}$ by $1$ via c_rs_eq_one, then simplify $1^3$ and the resulting division. Commute the product and rewrite with G_ℏ_product (the identity $G_{\mathrm{rs}}\cdot\hbar_{\mathrm{rs}}=1/\pi$) to obtain $\sqrt{1/\pi}$.
why it matters
Closes the Planck-length slot in the Family-A constant package. Downstream, all_constants_from_phi packages $c=1$, $\hbar=\varphi^{-5}$, and $G\cdot\pi=\varphi^5$ as the canonical RS values; this lemma supplies the matching geometric length $\ell_P=\pi^{-1/2}$.
It is also consumed by t6_to_phi_constants_canonical_bridge_holds in the unified forcing chain: once T6 forces $\varphi$, the bridge asserts the fixed (not merely existential) exponents for the constants, with this identity confirming the Planck scale is pure $\pi$-geometry in RS units.
Primer landmarks: T5 $J$-uniqueness and T6 $\varphi$-forcing feed the algebraic values; $G=\varphi^5/\pi$ and $\hbar=\varphi^{-5}$ are the RS-native constants quoted in the framework primer.
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