Input and output penalties on history states

Expand the history in the orthogonal legal clock basis. In $H_{in}$ the projector $\mathrm{\mid }0⟩⟨0{\mathrm{\mid }}_{C_1}$ is one exactly on $c_0$ and zero on $c_t$ for $t\ge 1$. At $c_0$, the sum of ancilla projectors has expectation $\mathbb{E}[N_{bad}]$ in $\varphi$; since $N_{bad}\ge 1$ on the dirty subspace, this expectation is at least $p_{bad}$. The history clock weight is $1\mathrm{/}N$, proving the input identity and inequality.

The output projector $\mathrm{\mid }1⟩⟨1{\mathrm{\mid }}_{C_T}$ is one only on legal clock word $c_T$. The history weight there is $1\mathrm{/}N$; multiplying by the explicit coefficient $N$ cancels it. The remaining computation state is $U_T\cdots U_1\varphi$, whose reject-bit probability is $p_{rej}$ by [F2]. Thus the output identity holds. The kernel of $H_{in}+H_{out}$ imposes a zero-ancilla projector only at clock $c_0$ and the accept projector only at clock $c_T$; these clock sectors are orthogonal, so the removed squared norm is $(p_{bad}+p_{rej})\mathrm{/}N$. For a clean ancilla $p_{bad}=0$ and every input penalty vanishes; acceptance at least $1-ϵ$ gives $p_{rej}\le ϵ$.