constants_from_phi
plain-language theorem explainer
In RS-native units the speed of light is fixed at 1, and both ħ and G·π equal integer powers of φ. Anyone citing the forcing-chain claim that c, ħ, G are φ-derived uses this legacy existential form. The proof is a three-field projection of the spine-to-extras bridge under T0, T5, and T6.
Claim. In Recognition-Science native units, $c = 1$, and there exist integers $n$ such that $\hbar = \varphi^{n}$ and $G \cdot \pi = \varphi^{n}$.
background
The Unified Forcing Chain module shows T0–T8 are forced from the Recognition Composition Law plus normalization and calibration. T5 pins the unique cost $J(x) = (x+x^{-1})/2-1$; T6 forces $\varphi$ as the self-similar fixed point of the discrete ledger.
Once $\varphi$ is forced, the constants package is stated in RS-native units: $c$ is set to the unit speed, $\hbar$ and $G$ are required to lie on the $\varphi$-ladder (powers of $\varphi$). The gravitational coupling is the RS projection $G = \lambda_{\mathrm{rec}}^{2} c^{3}/(\pi\hbar)$; with $\lambda_{\mathrm{rec}}=c=1$ this collapses to $G=\varphi^{5}/\pi$ at the canonical values.
This declaration is the legacy existential interface. The companion constants_from_phi_canonical drops the existentials and records the concrete exponents $\hbar=\varphi^{-5}$, $G\cdot\pi=\varphi^{5}$.
proof idea
Term-mode unpacking. Instantiate the spine-to-extras bridge with the already-proved T0, T5, and T6 holds lemmas. The bridge structure carries three fields: unit speed of light, $\hbar$ on the $\varphi$-ladder, and $G\cdot\pi$ on the $\varphi$-ladder. Package those three fields as the conjunctive goal. No new algebra is done here; the work lives in the bridge.
why it matters
Closes the module-doc claim that "constants derived: $c$, $\hbar$, $G$, $\alpha$ all from $\varphi$" at the existential level of the complete inevitability chain. Downstream, the continuum-limit bridge and the variational-to-Born-rule canonical bridge both consume this φ-derived constants package when they move from the discrete ledger into continuum or measurement layers.
Framework landmarks: T5 (unique $J$), T6 ($\varphi$ forced), and the RS-native dictionary $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^{5}/\pi$. The existential form keeps a stable legacy API; the canonical companion pins the exact rungs used by Planck-length and Planck-mass identities ($\ell_{P}=\sqrt{1/\pi}$, $m_{P}=\sqrt{\pi},\varphi^{-5}$).
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.