Scalarstack Geometry in a 45◦ Cone: Rational Void, Silver Gears, Concentric Compression, and a Near 7/8 Resonance

We study tangent-sphere geometry inside a right circular cone, emphasizing the distinguished $45^\circ$ case in which the cone height equals the base radius. Several exact structures emerge. First, the infinite axial tangent chain admits a closed-form volume law; in the $45^\circ$ cone the occupied fraction is exactly $2/7$, leaving $5/7$ empty. Second, a trapped-tip cascade beneath the primary tangent sphere is governed by the silver-ratio contraction $3-2\sqrt{2}$, and its residual void volume is exactly $\pi^4/84$, i.e. $1/28$ of the full cone volume. Third, in curvature-space the axial bend ladder lies naturally in the quadratic ring $\mathbb{Z}[\sqrt{2}]$, with Pell-type sequences arising from powers of $1+\sqrt{2}$. 

We then introduce a second, orthogonal geometric recursion internal to the primary sphere: the inward concentric operator $S_n \to C_n \to S_{n+1}$, where each cube is inscribed in the current sphere and each next sphere is inscribed in that cube. This yields the exact radius contraction $r_{n+1}=r_n/\sqrt{3}$, universal alternating fill fractions $2/(\pi\sqrt{3})$ and $\pi/6$, and the invariant area lock $A(C_n)/A(S_n)=2/\pi$. The primary sphere is further shown to mediate a precise $L_1 \supset L_2 \supset L_\infty$ metric bridge between an outer octahedral model and an inner cubic core. Finally, we record a near-$7/8$ rotational transient associated with $\rho=1/\ln(\pi)$, interpreted cautiously as a near-rational phase-locking phenomenon with slow linear dephasing rather than as a foundational acoustic law.

The resulting picture is a two-operator geometry: a downward silver-ratio translating cascade and an inward $\sqrt{3}$-based concentric compression, both embedded in a rigid $45^\circ$ cone architecture.