T6_To_PhiConstants_Canonical_Bridge
plain-language theorem explainer
Bridge certificate from T6 (φ forced as the unique positive root of x²=x+1) to the fixed RS-native constants: c=1, ℏ=φ^{-5}, G·π=φ^5, the duality G·ℏ=1/π, and the matching Planck length/mass forms. Anyone assembling the complete forcing chain or the constants-from-φ extras cites it. It is a propositional structure; the companion holding theorem fills each field from ConstantDerivations.
Claim. Given a T6 witness that $\varphi$ is the unique positive solution of $x^2=x+1$, the bridge asserts: that uniqueness statement is available; in RS units $c=1$, $\hbar=\varphi^{-5}$, $G\cdot\pi=\varphi^5$; the duality $G\cdot\hbar=1/\pi$ holds; the Planck length is $\sqrt{1/\pi}$ and the Planck mass is $\sqrt{\pi}\,\varphi^{-5}$; and both $\hbar$ and $G$ are positive.
background
The module UnifiedForcingChain assembles the full inevitability spine from the absolute floor through T0–T8, all forced from the Recognition Composition Law plus normalization and calibration. T6 is the φ-forcing node: in a discrete ledger with self-similar cost, the only positive scaling ratio is $\varphi=(1+\sqrt{5})/2$, characterized by $\varphi^2=\varphi+1$ and uniqueness among positive roots.
RS-native units fix the tick $\tau_0=1$ and set $c=1$ as the length/time tick ratio. The constant layer then pins $\hbar=\varphi^{-5}$ and $G=\varphi^5/\pi$, so that the product duality $G\cdot\hbar=1/\pi$ is immediate, and the Planck length/mass collapse to $\sqrt{1/\pi}$ and $\sqrt{\pi},\varphi^{-5}$. Upstream positivity of $\hbar$ is already recorded as theorem C-004.2.
This structure does not re-derive those equalities; it packages them as a named bridge certificate parameterized by a T6 witness, so downstream spine-to-extras and complete-chain assemblies can demand fixed exponents rather than existential placeholders.
proof idea
Definitional structure (Prop), not a proved theorem. Each field is a named conjunct: φ-uniqueness copied from the T6 witness; the six canonical constant identities and two positivity facts drawn from ConstantDerivations ($c_{\mathrm{rs}}=1$, $\hbar_{\mathrm{rs}}=\varphi^{-5}$, $G_{\mathrm{rs}}\cdot\pi=\varphi^5$, duality, Planck length/mass, positivity).
The companion theorem t6_to_phi_constants_canonical_bridge_holds is the one-line filler: it applies h6.phi_unique and the ConstantDerivations lemmas (c_rs_eq_one, ℏ_rs_eq, G_pi_eq_phi5, and siblings). A Subsingleton instance records that any two certificates for a fixed T6 witness are propositionally equal.
why it matters
Closes the constants-from-φ step of the forcing chain: once T6 forces φ, every RS-native constant is fixed with explicit exponents, not merely shown to exist. Downstream, Spine_To_Extras_Bridge sources its constants extras from this canonical bridge so exponents stay fixed; CompleteForcingChain and ultimate_inevitability consume it as part of the unconditional root package.
Framework landmarks: T6 φ-forcing, and the primer identities $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$. The duality $G\cdot\hbar=1/\pi$ is the clean algebraic signature of that pair. This is the bridge the module doc means by "Constants derived: c, ℏ, G, α all from φ" at the T6 stage (α itself lives in the calibration layer).
No open scaffold here: claim_status is definition; the holding theorem already discharges inhabitance.
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