planck_area_pos
plain-language theorem explainer
The theorem establishes strict positivity of the Planck area in Recognition Science units. Researchers unifying quantum mechanics with gravity via the J-cost functional would cite it when fixing the recognition scale relative to spacetime discreteness. The proof is a one-line rewrite of the area expression to its inverse-pi form followed by a positivity check.
Claim. In RS-native units the Planck area satisfies $0 < G hbar / c^3$, where $G = lambda_rec^2 c^3 / (pi hbar)$ and $hbar = phi^{-5}$.
background
The module develops quantum-gravity octave duality from the single J-cost functional equation. J-cost is defined as the AM-GM gap: for $x > 0$, Jcost $x = (x-1)^2/(2x)$, which equals AM$(x,x^{-1})$ minus GM$(x,x^{-1})=1$. Constants are fixed in RS units by the phi-ladder: $hbar = phi^{-5}$ and $G = phi^5 / pi$, so their product is $1/pi$. The local setting is the eight-tick octave that forces $kappa hbar = 8$ and places the Planck scale at the recognition scale divided by sqrt(pi). Upstream results supply the explicit forms of $G$ and $hbar$ used in the rewrite.
proof idea
The proof is a one-line wrapper. It rewrites the target inequality via the lemma planck_area_eq_inv_pi, which substitutes the RS expression $G hbar / c^3 = 1/pi$, after which the positivity tactic on the reals confirms $1/pi > 0$.
why it matters
The result supplies the positivity half of QG-003 (Planck area equals $1/pi$ in RS) inside the octave-duality development. It feeds the Quantum Gravity Octave Certificate and the claim that Planck scale equals recognition scale over sqrt(pi). The parent chain is the T5 J-uniqueness and T7 eight-tick octave that fix the constants; no open scaffolding remains.
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