Scalarstack Geometry in a 45◦ Cone: Exact Cone/Sphere Operators, Rational Void, Silver Gears, and a Heuristic Near-7/8 Note

Inside a $45^\circ$ cone we stack tangent spheres two ways: a chain sliding down the axis, and a nested sequence shrinking inward at the center. The first mechanism is a translating cascade governed by the silver-ratio contraction $3-2\sqrt2$; the second is a concentric sphere-cube recursion governed by the contraction $1/\sqrt3$.

For the axial chain in a general cone we derive an exact occupied volume fraction, specializing in the $45^\circ$ case to $2/7$ occupied and $5/7$ empty. For the trapped-tip cascade beneath the primary tangent sphere, the packed cascade volume carries $\sqrt2$, yet the remaining void cancels to the rational value $\pi^4/84$, exactly $1/28$ of the full cone volume. The same geometry also yields a $1/\sqrt2$ bicone fill fraction, a Pell-type curvature ladder in $\Z[\sqrt2]$, and invariant fill ratios for the inward sphere-cube operator: $2/(\pi\sqrt3)$, $\pi/6$, and $2/\pi$.

The paper is split deliberately. Part I contains exact results in cone and sphere geometry. Part II records a speculative numerical observation: a near-$7/8$ rotational transient associated with $\rho = 1/\ln(\pi)$. That constant is not derived in this paper; it is imported from the broader scalar framework as a quantity to test against the geometry. The near-$7/8$ language is therefore heuristic, not a theorem.