gravity_constants_audit_one_statement
plain-language theorem explainer
A single conjunction packages every gravity-sector constant as a closed-form function of φ: ħ = φ^{-5}, Einstein κ = 8φ^5, BMV α_RS = φ^5/(8π), BH entropy log-prefactor, echo damping 1/φ, Hawking T = 1/(8πM), area-law S = A/4, and η_B rung −44. Gravity auditors cite it as the Track 5.B one-statement audit. The proof is a pure term conjunction of named closed-form lemmas and definitional equalities.
Claim. In RS-native units one has simultaneously $\hbar = \varphi^{-5}$, $\kappa_E = 8\varphi^5$, $\kappa_{\mathrm{rs}} = 8\varphi^5$, $\alpha_{\mathrm{RS}} = \varphi^5/(8\pi)$, $c_{\mathrm{RS}} = -(\log\varphi)/2$, echo damping ratio $1/\varphi$, rung phase delay $\log\varphi$, Hawking temperature $T(M) = 1/(8\pi M)$ for all $M$, leading black-hole entropy $S(A) = A/4$ for all $A$, and baryon asymmetry rung $\eta_B = -44$.
background
Track 5.B of the quantum-gravity master plan asks for a Lean certificate that every gravity-sector constant is a closed-form function of the golden ratio φ, with a single dimensional SI anchor (G_SI). This module closes that audit: zero free dimensionless parameters in the gravity sector.
RS-native units fix c = 1 and set the action quantum ħ := E_coh · τ_0. The identity ħ = φ^{-5} is the native unit choice; the nontrivial content is the exponent 5 = D+2 forced by the configuration chain. Einstein’s coupling is κ = 8πG/c^4; with G = φ^5/π one obtains κ = 8φ^5. The same closed form appears as κ_rs in the zero-parameter gravity sector.
Upstream lemmas pin the pieces: hbar_eq_phi_inv_fifth (THEOREM C-004.1), kappa_einstein_eq, and kappa_rs_closed_form. Black-hole side constants (ledger entropy prefactor, echo damping, rung phase, Hawking temperature, area law) and the cosmology rung η_B = −44 are definitional or already proved equalities imported into the conjunction.
proof idea
Pure term-mode conjunction. The ten conjuncts are discharged in order by:
Constants.hbar_eq_phi_inv_fifthConstants.kappa_einstein_eqZeroParameterGravity.kappa_rs_closed_form4–7. fourrflsteps (α_RS, c_RS, echoDampingRatio, rungPhaseDelay are definitionally the stated φ-expressions)HawkingTemperatureFromRung.T_hawking_def(pointwise in M)BlackHoleEntropyFromLedger.S_lead_eq_BH(pointwise in A)rflfor η_B rung value −44.
No tactics, no new algebra: the audit is the packaging of already-closed identities.
why it matters
This is the human-readable one-statement form of Track 5.B. The sibling master certificate GravitySectorConstantsClosedForm / gravity_sector_zero_free_parameters bundles the same closed forms plus the SI-bridge anchor on CODATA G_SI; the present theorem is the pure φ-rational conjunction without the dimensional record.
Framework landmarks hit directly: ħ = φ^{-5} and G ∼ φ^5/π (primer RS-native constants), Einstein κ = 8φ^5, and the eight-tick / rung structure underlying η_B = −44 and the BH echo phase log φ. Together they underwrite the claim of zero free dimensionless gravity parameters.
No downstream dependents are recorded yet; the declaration is a terminal audit artifact for referees and the master-plan checklist, not an intermediate lemma in a longer derivation.
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