QuantumGravityCert
plain-language theorem explainer
The structure certifies that Recognition Science subsumes five quantum gravity approaches while the bounce radius at each recognition rung is positive and strictly increasing with rung number and the echo delay remains positive. A physicist modeling black hole echoes or deriving quantum gravity from the Recognition functional equation would cite the certificate to confirm structural consistency with phi-ladder scaling. It is defined as a direct bundling of four properties whose verifications are supplied by sibling lemmas.
Claim. A certificate structure whose fields assert that the set of subsumed quantum gravity approaches has cardinality 5, that the bounce radius satisfies $0 < r(N)$ and $r(N) < r(N+1)$ for every natural number $N$, and that the echo delay satisfies $0 < t(N)$ for every $N$, where $r(N) = phi^N$ and $t(N) = 2 r(N) log phi$.
background
The module develops quantum gravity implications of Recognition Science in the strong-field regime. The bounce radius is defined as $r(N) = phi^N$ in RS-native units and the echo delay as $t(N) = 2 r(N) log phi$, with phi the self-similar fixed point from the forcing chain. The inductive type enumerating the five approaches consists of canonical quantum gravity, spin foam, causal sets, causal dynamical triangulations, and loop quantum gravity. Upstream results from BHEchoesLIGOCatalog and BlackHoleEchoesFromBounce establish the explicit forms of the bounce radius and echo delay, confirming the geometric round-trip multiplier $2 log phi$.
proof idea
This declaration is a structure definition with no proof body. It collects the four properties whose individual verifications are supplied by the cardinality computation for the approach type and the positivity and monotonicity lemmas for the bounce radius and echo delay.
why it matters
This structure supplies the bundled certificate instantiated in the quantumGravityCert definition, which serves as the top-level claim for the quantum gravity derivation from Recognition Science. It directly supports the module assertion that the five canonical approaches are subsumed under the RS framework with configDim equal to 5. The positivity and monotonicity conditions tie into the phi-ladder scaling and the eight-tick octave structure of the forcing chain, ensuring consistency with the Recognition Composition Law in the strong-field limit.
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