spiralRadius_pos
plain-language theorem explainer
The theorem establishes that the spiral radius remains strictly positive for any positive base radius r₀, regardless of the integer tier parameter kappa or real phase theta. Engineers modeling Nautilus-class access paths for Class C powers would cite this to confirm geometric admissibility in the technological stack. The proof is a direct term reduction that unfolds the radius definition and applies the standard positivity rules for multiplication and real exponentiation with positive base.
Claim. Let $r_0 > 0$. For any integer $k$ and real $t$, the spiral radius satisfies $0 < r_0 · ϕ^{f(k,t)}$ where $ϕ > 0$ is the golden ratio and $f$ is the exponent determined by the tier and phase.
background
The Nautilus power tier model in the Superhuman module encodes geometric parameters for Class C technological access, with power tiers treated as structural definitions and engineering constants drawn from the NTL provisional patent stack. The spiral radius is constructed as a base length scaled by a power of the golden ratio phi, whose positivity follows from the self-similar fixed point in the Recognition Science forcing chain. Upstream results supply the fundamental tick as the RS time quantum and the Tick structure as the discrete atomic unit of temporal progression, though the radius statement itself uses only real-analysis facts about phi.
proof idea
The term proof unfolds the definition of spiralRadius, then applies mul_pos to the hypothesis 0 < r₀ together with the positivity of the remaining phi-powered factor. The second step invokes Real.rpow_pos_of_pos on the established fact that phi is positive.
why it matters
This result secures a basic positivity sanity check inside the Nautilus technological access path, ensuring geometric parameters stay consistent with the positive-definite requirements of the Recognition Science framework. It supports the engineering parameters from the NTL patent stack for Class C powers and connects to the phi-ladder and eight-tick octave structure. Although currently unused downstream, it closes an elementary prerequisite for tier power range definitions.
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