Method and limitations.

Core posture

Two classical readings, one cube. Divisor bands are arithmetic. Sphere shells are metric. The tool shows both without identifying them.

Native operations

• GCD shell identity q=n/gcd(n,kx,ky,kz).
• Jordan shell count J3(q) and divisor-band conservation.
• Ideal cube frame with insphere, midsphere, and circumsphere radii 1, sqrt(2), sqrt(3).
• Ray target, snapped address, and snap error.

Quiet guardrails

• Arithmetic band and metric shell are separate addresses of the same point.
• EXACT means the target is available in the active grid. SNAP means the tool selected the nearest available rational address.
• Coordinate origin is additive 0. The count J3(1)=1 is multiplicative empty-product behavior. These are different operation rooms.
• This v1 does not include density, perfect-number, Chamber Lift, Picone, or physical overlays.