PRH | Essay | 7.20 • Fifty–Plus Exercises with Solutions on Two-Channel Arithmetic

We present a compact exercise set ( $60+$ problems with solutions) that trains the twochannel view of arithmetic-an additive channel and a multiplicative channel-equipped with blurs ( $B_\tau^{+}, B_{\varepsilon}^{\times}$), a bridge $U$, and honest switches ( $S_{\times \rightarrow+}, S_{+\rightarrow \times}$ ). Part I develops the $2 \times 2$ block-operator calculus. Part II practices the mixed-expression word notation and its blur budget. Part III casts the same moves as quantum-style amplitudes on a two-component state. Between the preliminaries and Part I we insert a short section on the sharp/blur notation $x^{\sharp}, x^{\text {b }}$ and a generic blur operator $\mathcal{B}_\sigma$, so that an entire equation (or whole proof) can be declared to live at a given blur "lens". The final mini-drills touch the Collatz accelerated step and show how the per-step budget $\rho$ drops out immediately from the count of switches.