constants_from_phi_canonical
plain-language theorem explainer
In RS-native units the fundamental constants are fixed by φ alone: c = 1, ℏ = φ^{-5}, G = φ^5/π, with Planck length √(1/π) and Planck mass √π·φ^{-5}. Cite this when packaging the Recognition Science constant set or the complete inevitability chain after T6. The proof is a short term projection of the six canonical equalities out of the T6-to-constants bridge.
Claim. In RS-native units, $c = 1$, $\hbar = \varphi^{-5}$, $G\cdot\pi = \varphi^{5}$ (equivalently $G = \varphi^{5}/\pi$), $G\cdot\hbar = 1/\pi$, the Planck length equals $\sqrt{1/\pi}$, and the Planck mass equals $\sqrt{\pi}\,\varphi^{-5}$, where $\varphi$ is the self-similar fixed point forced at T6.
background
The Unified Forcing Chain module shows that T0–T8 are forced from the Recognition Composition Law together with normalization $F(1)=0$ and calibration $F''(1)=1$. T6 forces $\varphi$ as the self-similar fixed point of the discrete ledger; once $\varphi$ is fixed, the constant package is no longer free.
In Family-A RS-native units the speed of light is the ratio of fundamental length to fundamental time. With both set to 1 one has $c=1$ by unit coherence, not as a free parameter. The reduced Planck constant is set to $\varphi^{-5}$. Newton's $G$ arises from the curvature formula $G=\lambda_{\mathrm{rec}}^{2}c^{3}/(\pi\hbar)$ with $\lambda_{\mathrm{rec}}=c=1$, hence $G=\varphi^{5}/\pi$. The Planck length and mass are the usual combinations $\sqrt{\hbar G/c^{3}}$ and $\sqrt{\hbar c/G}$, which collapse to $\sqrt{1/\pi}$ and $\sqrt{\pi},\varphi^{-5}$.
Upstream, the T6 bridge packages these identities once T6 is known to hold. This theorem records the six equalities as one conjunction with no existentials.
proof idea
Term-mode projection, not a fresh derivation. Instantiate the canonical bridge lemma that turns the standing T6 fact into a structure of six field equalities. Unpack that structure into the conjunction: $c=1$, $\hbar=\varphi^{-5}$, $G\cdot\pi=\varphi^{5}$, $G\cdot\hbar=1/\pi$, Planck length $\sqrt{1/\pi}$, Planck mass $\sqrt{\pi},\varphi^{-5}$. All algebraic content lives in the bridge and in the ConstantDerivations definitions; this declaration only assembles the canonical package.
why it matters
This is the constants-from-$\varphi$ step named in the module's stronger claim: after T6 forces $\varphi$, the values $c$, $\hbar$, and $G$ (and the derived Planck scales) are fixed rather than postulated. It sits on the T6 landmark of the forcing chain and matches the primer package $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^{5}/\pi$.
In the same module it supports the complete-inevitability packaging that follows the T0–T8 spine (the ultimate root theorem is stated as unconditional at the mathematical level, with physical axiom bundles pushed downstream). It does not pin the fine-structure constant; $\alpha^{-1}$ remains in the separate band $(137.030,137.039)$. The historical nearby bundle formerly called Gödel-dissolved is deprecated and is not the real consumer of this content.
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