rs_quantum_gravity_master_conditional
plain-language theorem explainer
Under five open-track hypotheses (Regge–EH continuum plus discrete Bianchi, unconditional amplitude-linear forcing, dynamical Page curve, PTA stochastic GW distinct from inflation, strong-field tests distinct from GR), the twelve-clause RS quantum-gravity master statement holds. Gravity auditors cite this as the Track 7.A authored conditional form. The proof is a six-case refine: eight closed Lean anchors fill closed clauses; the five inputs fill open ones.
Claim. Assume (i) Regge action converges to Einstein–Hilbert in the continuum with contracted discrete Bianchi, (ii) amplitude-linear channel response is forced unconditionally, (iii) the Page curve is dynamically derived, (iv) the RS PTA stochastic-GW spectrum differs from inflationary $n_t$, and (v) RS strong-field tests differ from pure GR. Then the master conjunction holds: substrate forcing ($T_0$–$T_8$, $J$-cost uniqueness, Lorentzian $1{+}3$); classical limit; amplitude-linear forcing with positive BMV; Hawking temperature (SI), RS-distinct $c$, Page curve, and $\Omega_\Lambda$ from $\varphi$; QNM/PTA/strong-field discriminators; and zero free parameters in the gravity sector.
background
Track 7.A authors the RS quantum-gravity master statement as a twelve-clause conjunction matching the master-plan template. Eight clauses are CLOSED (Sessions 89–96 anchors); five tracks stay open and enter as hypothesis structures.
The substrate block packages the forcing chain $T_0$–$T_8$ ($J$-uniqueness with $J(x)=(x+x^{-1})/2-1$, $\varphi$ as self-similar fixed point, eight-tick octave, $D=3$), cost uniqueness, and Lorentzian $1{+}3$. Classical recovery requires Regge$\to$Einstein–Hilbert continuum convergence with explicit error bound and the contracted second Bianchi identity on the discrete substrate (Schläfli). Quantum-channel clauses demand amplitude-linear forcing of the response and unconditional BMV positivity. Thermodynamic/cosmological clauses cover Hawking temperature in SI, an RS-distinct $c$ observable, dynamical Page-curve evolution, and $\Omega_\Lambda$ fixed from $\varphi$. Discriminators separate RS QNM spectra from LQG/string, PTA stochastic GW from inflation, and strong-field predictions (S-stars, EHT shadow, Cassini) from pure GR. The last clause asserts zero free parameters in the gravity sector.
The module is structural and conditional: 0 sorry on the load-bearing path, but it does not claim the discovery is complete until the five open tracks close.
proof idea
Term-mode refine builds a six-component witness for the master Prop. Case d1 discharges the closed substrate triple from the proven $T_0$–$T_8$, cost-uniqueness, and Lorentzian $1{+}3$ anchors. Case d2 threads the two fields of the Regge–EH/Bianchi hypothesis (continuum convergence and discrete Bianchi). Case d3 pairs the unconditional amplitude-linear hypothesis with the proven BMV-positivity anchor. Case d4 assembles Hawking temperature (SI), RS-distinct $c$, the Page-curve hypothesis, and $\Omega_\Lambda$ from $\varphi$. Case d5 packages the proven QNM-vs-LQG/string discriminator with the PTA and strong-field hypotheses. Case d6 is the single closed zero-free-parameters theorem. No new physics is derived; the construction packages existing anchors under the five open inputs.
why it matters
This is the Track 7.A statement-authoring closure: the master plan's twelve-clause quantum-gravity discovery written as a Lean theorem conditional on five still-open tracks (1.B/1.C classical recovery, 2.C/2.D unconditional amplitude-linear forcing, 3.C Page curve, 6.B PTA, 6.C strong field). Downstream, the one-statement form quantifies over the same hypotheses; partial, deeper-partial, and structural master variants, plus PTA cosmology modules, consume it to thin or retire hypothesis inputs.
Framework landmarks packed into the closed clauses include the $T_0$–$T_8$ forcing chain, $J$-cost uniqueness via the Recognition Composition Law fixed point, eight-tick/$D=3$ geometry, $\varphi$-derived $\Omega_\Lambda$, Hawking temperature and black-hole entropy in SI, and gravity-sector zero free parameters. The unconditional master (no hypothesis inputs) remains open until those five tracks close; the module text states explicitly that it does not claim the discovery has been made.
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