qg_octave_cert_inhabited
plain-language theorem explainer
The quantum-gravity octave duality certificate exists as a mathematical object. Researchers deriving unification constants from the J-cost functional cite this result to confirm that the full set of relations, including kappa hbar equals 8, is simultaneously realized. The proof is a one-line term that supplies the pre-constructed certificate.
Claim. There exists a quantum-gravity octave duality certificate whose fields include: for every positive real $x$, the J-cost equals $(x-1)^2/(2x)$; the J-cost is nonnegative and vanishes precisely when $x=1$; Newton's constant $G$ equals $phi^5$ over $pi$; and the product of the Einstein coupling $kappa$ with $hbar$ equals 8.
background
The module derives all quantum-gravity relations from the single J-cost functional. J-cost is the arithmetic-geometric mean gap of the pair $x$ and $x^{-1}$: for $x>0$, Jcost $x$ equals $(x-1)^2/(2x)$, which is nonnegative and zero exactly at $x=1$. The QGOctaveCert structure packages thirteen fields that together certify the octave duality.
proof idea
The proof is a one-line wrapper that applies the constructor qg_octave_cert. That constructor populates every field of the certificate from prior lemmas on the J-cost AM-GM identity and the octave product kappa hbar equals 8.
why it matters
This declaration certifies that the complete QG Octave Certificate is inhabited, closing the unification section of the module. It directly realizes the central claim that kappa times hbar equals 8, the eight-tick cycle that locks quantum mechanics to gravity. The result follows from the forcing chain through T5 J-uniqueness and T7 eight-tick octave, with all inputs taken from the J-cost functional and the constants proved in the Constants module.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.