QGOctaveCert
plain-language theorem explainer
QGOctaveCert packages the J-cost AM-GM identities with the octave duality relations kappa hbar equals 8, G hbar equals 1 over pi, Planck area equals 1 over pi, and the Fibonacci mass ladder. Researchers in quantum gravity unification would cite it to reference the formal certificate that the 8-tick cycle locks quantum and gravitational scales from the single J-cost functional. The declaration is a scaffolding sorry stub whose fields are populated downstream by applying lemmas on J-cost and constants.
Claim. The structure whose fields assert: for all $x>0$, $J(x)=(x-1)^2/(2x)$; $J(x)≥0$ with equality iff $x=1$; $G=φ^5/π$; $κℏ=8$; $Gℏ=1/π$; $Gℏ/c^3=1/π$; $G/ℏ=φ^{10}/π$; $κ/8=1/ℏ$; $κ=8(5φ+3)$; $ℏ=1/(5φ+3)$; $φ^{n+2}=φ^{n+1}+φ^n$ for all $n∈ℕ$; and $yφ^{n+2}=yφ^{n+1}+yφ^n$ for all $y∈ℝ$, $n∈ℕ$.
background
In Recognition Science the J-cost functional is exactly the AM-GM gap of the pair {x, x^{-1}}, so J(x)=(x-1)^2/(2x) for x>0. The module shows this single function forces the octave number 8 into the product of the Einstein coupling κ and the reduced Planck constant ℏ. Upstream results supply the definition of octave as 8 ticks, the RS-native G=λ_rec² c³/(π ℏ), ℏ=φ^{-5}, and κ=8φ^5. The local setting is the unification of quantum mechanics and gravity through the 8-tick recognition cycle, with every listed relation proved from J-cost and the golden ratio.
proof idea
The structure declaration is a sorry stub. Its fields are realized by the downstream construction qg_octave_cert, which supplies each property by direct application of the lemmas jcost_eq_sq_div, jcost_nonneg_amgm, jcost_zero_iff_one, G_eq_phi_fifth_over_pi and kappa_hbar_octave.
why it matters
This structure is the central certificate for quantum-gravity octave duality and feeds the theorem qg_octave_cert_inhabited asserting the type is nonempty. It realizes the framework landmark T7 (eight-tick octave) together with the relation κℏ=8 that locks quantum and gravitational scales. The doc-comment states that inhabiting the type certifies unification starting only from the J-cost functional and the golden ratio.
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