Gspher
plain-language theorem explainer
Gspher defines the spherical candidate for the cost function in log coordinates as G(t) equals cosine of t minus one. Researchers examining the curvature gate cite this definition to exhibit the periodic solution excluded by non-negativity. The definition is a direct transcription of the spherical ODE solution with constant positive curvature.
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background
The Curvature Gate module analyzes the cost metric in log coordinates, where G(t) equals F of e to the t and the line element is ds squared equals G double prime of t times dt squared. Constant curvature yields three solutions: flat with zero curvature as t squared over two, hyperbolic with curvature minus one as cosh t minus one from the recognition composition law, and spherical with curvature plus one as cos t minus one. The spherical form is introduced explicitly here. Upstream results supply the reparametrization G from the functional equation module and the J-cost expression from gravity.
proof idea
The definition is a direct transcription of the spherical solution to the ODE G double prime equals minus G minus one.
why it matters
This definition supplies the spherical case used by curvature_gate_main and curvature_gate_summary to contrast solutions. It is shown to violate non-negativity in Gspher_nonpositive and Gspher_negative_at_pi, thereby excluding spherical geometry. The result supports selection of the hyperbolic solution corresponding to the recognition composition law, consistent with the eight-tick octave and three spatial dimensions in the framework.
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