quantumGravityCert
plain-language theorem explainer
quantumGravityCert assembles the structural certificate confirming that Recognition Science subsumes exactly five quantum gravity approaches while the bounce radius remains positive and strictly increasing with rung number and the echo delay stays positive. Physicists modeling Planck-scale bounces or LIGO echo signatures would cite this record to verify the required positivity and monotonicity conditions hold in the RS framework. The definition is a direct record construction that assigns the approach count and delegates the three positivity or
Claim. The quantum gravity certificate is the structure asserting that the cardinality of the set of quantum gravity approaches equals five, that the bounce radius satisfies $r_ {min}(N)>0$ for every natural number $N$, that $r_{min}(N)<r_{min}(N+1)$, and that the echo delay satisfies $Delta t(N)>0$ for every $N$.
background
In the Recognition Science A4 strong-field sector the bounce radius is given by $r_{min}(N)=ell_P phi^{N/2}$ with $phi$ the self-similar fixed point and the echo delay by $Delta t(N)=2 r_{min}(N) log phi$. The module supplies the structural backing that exactly five canonical approaches (canonical QG, spin foam, causal sets, CDT, loop QG) are covered by configuration dimension five. Upstream theorems establish the needed properties: bounceRadius_pos unfolds the power to obtain strict positivity from $phi>1$; echoDelay_pos combines the logarithm positivity for $phi>1$ with the radius positivity; bounceRadius_mono uses the recurrence $phi^{N+1}=phi cdot phi^N$ together with positivity to obtain the strict increase.
proof idea
The definition is a direct record construction. It sets five_approaches to the constant qgApproachCount, bounce_pos to the theorem bounceRadius_pos, bounce_mono to bounceRadius_mono, and echo_pos to echoDelay_pos. No additional tactics or reductions are applied beyond these field assignments.
why it matters
This definition supplies the top-level certificate that the RS derivation of quantum gravity from the bounce and echo properties is complete for the A4 sector. It closes the structural requirements listed in the module documentation and relies on the phi-ladder and rung indexing. The certificate stands as a verified interface for downstream gravitational-wave echo modeling; no open questions are addressed here.
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