A rotational source ledger pins every inertia, speed, and energy value back to a cited source, chaining the inputs so the energy budget closes from named quantities. The bookkeeping is exact only for the stated inertia, angular speed, and source values in the ideal rigid-body model. Real systems add distributed-mass geometry, slipping, friction, bearing losses, axis alignment, sensor calibration, finite precision, and measurement uncertainty.
The final lesson keeps each source claim tied to a readable diagram. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The energy source ledger shows the spin square
The checked rotational-energy ledger stores inertia 8, spin 3, and energy 36.
8,3,36
The rolling source ledger keeps both energy parts
The rolling ledger keeps 9 joules rotational, 9 joules translational, and 18 joules total.