gravity_sector_zero_free_parameters
plain-language theorem explainer
The gravity sector is declared to have zero free dimensionless parameters: carried φ-closed identities for the sector constants, conjoined with inhabitance of the full closed-form certificate bundle. Gravity and master-theorem auditors cite this as Track 5.B / clause M4. The declaration is a bare Prop definition (conjunction), not a proof.
Claim. The gravity sector has zero free dimensionless parameters when (i) the carried closed-form identities hold ($\hbar=\varphi^{-5}$, $\kappa_E=8\varphi^5$, $c_{RS}=-(\log\varphi)/2$, echo damping $1/\varphi$, rung phase $\log\varphi$, $S_{\mathrm{lead}}(A)=A/4$, $T_H(M)=1/(8\pi M)$, $\eta_B$ rung $-44$) and (ii) the full gravity-sector closed-form certificate structure is inhabited.
background
MasterTheorem authors the RS quantum-gravity master statement as a twelve-clause conjunction (Track 7.A). Eight clauses are closed from existing sessions; five remain hypothesis inputs. This definition is the named Prop for the zero-free-parameter clause (M4 / Track 5.B).
Upstream, the carried content packages explicit equalities: $\hbar=\varphi^{-5}$, Einstein coupling $\kappa_E=8\varphi^5$, ledger constant $c_{RS}=-\log\varphi/2$, echo damping $1/\varphi$, rung phase $\log\varphi$, leading area law $A/4$, Hawking $T_H=1/(8\pi M)$, and $\eta_B$ rung $-44$. Separately, GravitySectorConstantsClosedForm is the structure whose fields are those same closed forms, each anchored on a named theorem or rfl.
The ZeroFreeParameters theorem already shows that structure is nonempty (one dimensional SI anchor $G_{SI}$, zero free dimensionless parameters). RS-native landmarks match the primer: $\hbar=\varphi^{-5}$ and $G\sim\varphi^5/\pi$.
proof idea
Not a proof: a one-line Prop abbreviation. It is the conjunction of (1) the carried M4 equality bundle and (2) nonemptiness of GravitySectorConstantsClosedForm. Discharge is deferred to the sibling theorem that inhabits both conjuncts from the ZeroFreeParameters closed-form record.
why it matters
Clause M4 of the master plan Track 5.B audit: every gravity-sector dimensionless constant is φ-rational, so the sector adds no free knobs beyond the single CODATA $G_{SI}$ bridge. The proven inhabitation feeds RSQuantumGravityMaster and the conditional master theorem, and appears in the non-circularity audit's closed-certificate list. Downstream, the proven form closes the Prop from the ZeroFreeParameters record fields. Framework landmarks: T5 J-uniqueness forces the cost that pins φ; the φ-ladder then fixes $\hbar=\varphi^{-5}$ and related couplings. Open tracks (Page curve, PTA, strong-field distinctness) remain outside this clause.
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