t6_to_phi_constants_canonical_bridge_holds
plain-language theorem explainer
Given T6 (φ forced as the unique positive root of x² = x + 1), the canonical RS unit constants are fixed: c = 1, ℏ = φ^{-5}, G·π = φ^5, with G·ℏ = 1/π and the matching Planck length and mass. Anyone citing the forcing chain's derived-constants surface or ultimate inevitability uses this bridge. The proof is a structure witness that plugs in the ConstantDerivations equalities and positivity lemmas.
Claim. If T6 holds (φ satisfies $\varphi^2 = \varphi + 1$, $\varphi > 0$, and is the unique positive solution of that equation), then the canonical $\varphi$-constants bridge holds: $c_{\mathrm{RS}} = 1$, $\hbar_{\mathrm{RS}} = \varphi^{-5}$, $G_{\mathrm{RS}}\,\pi = \varphi^5$, $G_{\mathrm{RS}}\,\hbar_{\mathrm{RS}} = 1/\pi$, with the corresponding Planck length and mass formulae, and with $\hbar_{\mathrm{RS}} > 0$ and $G_{\mathrm{RS}} > 0$. The exponents are fixed, not existential.
background
The Unified Forcing Chain module shows T0–T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the φ-forcing step: in a discrete ledger with self-similar cost, the only positive scaling ratio is $\varphi = (1+\sqrt{5})/2$, unique among positive solutions of $x^2 = x + 1$.
RS-native units set $c = 1$ (length/time tick ratio), $\hbar = \varphi^{-5}$, and $G = \varphi^5/\pi$. The bridge structure packages these as a certificate under a T6 hypothesis: uniqueness of φ is available, the three unit equalities hold, the duality $G\cdot\hbar = 1/\pi$ holds, and the Planck length/mass take their canonical closed forms.
A nearby comment records that α is not on this bridge. A former α formula was deleted as inconsistent with the repository band $(137.030, 137.039)$; α remains a free U(1) kinetic normalization that no κ-blind closure can pin.
proof idea
Term-mode structure construction. The hypothesis h6 : T6_Phi_Forced supplies phi_unique_available directly from h6.phi_unique.
Each remaining field is a one-line citation of a ConstantDerivations lemma: $c_{\mathrm{RS}} = 1$, $\hbar_{\mathrm{RS}} = \varphi^{-5}$, $G_{\mathrm{RS}},\pi = \varphi^5$, the product identity $G\cdot\hbar = 1/\pi$, the Planck length and mass equalities, and positivity of $\hbar$ and $G$. No new algebra is done here; the bridge is a typed assembly of already-proved constant identities under T6.
why it matters
This is the T6 → constants link in the complete inevitability chain. Downstream, constants_from_phi_canonical restates the same fixed equalities as a bare conjunction; spine_to_extras_bridge_holds sources its t6_* extras from this witness; and both ultimate_inevitability_canonical and ultimate_inevitability_extended replace legacy existential constant surfaces with these fixed exponents.
Framework landmarks: T6 (φ forced), and the RS-native package $c = 1$, $\hbar = \varphi^{-5}$, $G = \varphi^5/\pi$. The module doc's claim that constants derive from φ is discharged here for the gravitational/quantum unit surface. α is explicitly out of scope (κ-blind no-go), so the forcing chain does not overclaim fine-structure pinning. The mass-ladder bridge that follows in the file is separate: this declaration only certifies the unit constants.
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