UnitBridgeTheorem
plain-language theorem explainer
Packages Theorem 4 (unit bridge) as a three-field witness: positivity of α_RS = φ⁵/(8π), the RS-native identity κ_rs · α_RS = G/ℏ, and a closed-form SI BMV entangling phase rate for any calibration input U. Gravity-from-Recognition readers cite it as the master T4 bundle linking the dimensionless RS coupling to a tabletop observable. The structure itself is pure packaging; the three obligations are discharged by named sibling lemmas.
Claim. A unit-bridge witness records three facts: (i) $\alpha_{\mathrm{RS}}=\varphi^5/(8\pi)>0$; (ii) $\kappa_{\mathrm{rs}}\cdot\alpha_{\mathrm{RS}}=G/\hbar$ in RS-native units; (iii) for every calibration input $U$ (positive conversion scale $U_{\mathrm{conv}}$, SI masses $m_1,m_2$, and four branch distances), the SI BMV phase rate equals $U_{\mathrm{conv}}\cdot\kappa_{\mathrm{rs}}\cdot\alpha_{\mathrm{RS}}\cdot m_1 m_2\cdot g(r_{LL},r_{LR},r_{RL},r_{RR})$, where $g$ is the geometry factor from the branch-phase invariant.
background
Gravity IV formalizes the quantum channel from Recognition: the dimensionless RS coupling $\kappa_{\mathrm{rs}}=8\varphi^5$ (band $(85.6,90.4)$) must convert to a dimensionful BMV entangling phase rate in SI. In RS-native units one has $\hbar=\varphi^{-5}$ and $G=\varphi^5/\pi$, so $G/\hbar=\varphi^{10}/\pi$ is fixed by $\varphi$ alone. The BMV rate is $(G m_1 m_2/\hbar)\cdot g$, with $g$ the inverse-distance geometry factor from the T3 branch-phase invariant.
The SI bridge is conditional on an external calibration: an inhabitant of the unit-bridge input carries a positive scale $U_{\mathrm{conv}}$ (the dimensional seconds-per-tick $\times$ meters-per-voxel combination from ExternalCalibration), together with SI masses and the four branch distances. Until an anchor is supplied, T4 remains conditional on that input; the conversion map itself is closed in the SI-bridge foundation, but the numerical anchor is external by dimensional analysis.
Sibling material already defines $\alpha_{\mathrm{RS}}=\varphi^5/(8\pi)$, proves $\kappa_{\mathrm{rs}}\cdot\alpha_{\mathrm{RS}}=G/\hbar$, and factors the SI phase-rate formula through that product times geometry.
proof idea
No proof body: this is a structure bundling three propositions. An inhabitant is assembled elsewhere by assigning the three fields to existing lemmas: positivity of $\alpha_{\mathrm{RS}}$, the algebraic identity $\kappa_{\mathrm{rs}}\cdot\alpha_{\mathrm{RS}}=G/\hbar$, and the factored SI closed form for the BMV phase rate at arbitrary calibration input. The structure is therefore a pure packaging type; discharge is one constructor application with those three named results.
why it matters
This is the T4 master witness of Gravity from Recognition IV: The Quantum Channel. Downstream, a concrete inhabitant is built and non-emptiness is recorded, so later modules can cite a single packaged object rather than three scattered lemmas. It sits on the RS-native constants $G=\varphi^5/\pi$ and $\hbar=\varphi^{-5}$ (primer landmarks) and on the eight-tick / $\varphi$-ladder scaffolding that fixes those powers. The open frontier it exposes is the external calibration anchor: the algebraic bridge is closed, but SI numbers still require an external dimensional fix. Parent uses are the constructed witness and the non-emptiness theorem in the same module.
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