An explicit scalar phase completes Clifford+T to full unitary density

The number $\sqrt{2}$ is irrational. Otherwise write $\sqrt{2}=a\mathrm{/}b$ with positive coprime integers $a,b$. Then $a^2=2b^2$ makes $a$ even, so $a=2c$ and $b^2=2c^2$ makes $b$ even, contradicting coprimality. Thus [F2] applies to the scalar phases $e^{2\pi in\sqrt{2}}$.

The gadget matrix is $(e^{\pm 2\pi i\sqrt{2}}{I}_2)\otimes I_{2^{m-1}}=e^{\pm 2\pi i\sqrt{2}}{I}_{2^m}$. Consequently it intertwines $V$ with $P$ or $P^{\ast }$ exactly, including the scalar phase. Its channel is the identity on the whole physical operator space, also with an external reference. The schedule has one supplied gate or idle on each of the $m$ distinct coordinates, so [F5] gives size $m$ and depth one. Its literal unitary propagation changes no operator support.

Fix $U\in U(2)$ and $\epsilon >0$. Write $\pi :U(2)\to PU(2)$ for the quotient. The set $\pi (B(U,\epsilon \mathrm{/}2))$ is open, because its inverse image is ${\bigcup }_{\mathrm{\mid }\lambda \mathrm{\mid }=1}\lambda B(U,\epsilon \mathrm{/}2)$, a union of open operator-norm balls. By [F1], some matrix word $W$ in $H,T$ and their adjoints has projective class in that open set, so $\mathrm{\parallel }U-\lambda W{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}<\epsilon \mathrm{/}2$ for some $\mathrm{\mid }\lambda \mathrm{\mid }=1$. By step 1.1 and [F2], choose an integer $n$ with $\mathrm{\mid }{e}^{2\pi in\sqrt{2}}-\lambda \mathrm{\mid }<\epsilon \mathrm{/}2$. Since $W$ is unitary, [F3] gives $\mathrm{\parallel }W{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=1$ and homogeneity and the triangle inequality from [F6] yield $\mathrm{\parallel }U-P^{n}W{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}<\epsilon$. This proves density in the full matrix group $U(2)$.

A union of at most $q$ faulty locations in this layer touches at most $q$ coordinates per block. Expand the incoming Kraus operators from [F4] as finite sums of per-block supported terms. Each product of a fault Kraus operator and such a term has per-block support in the union of their supports, of cardinality at most $s+q$. The ideal scalar gates commute with all of them. Their composite is CPTP and hence is a joint error channel of the type in [F4]; it need not factor across blocks. This proves joint $(s+q)$-deviation and the spread-one fault budget with the external reference unchanged. In particular, $s=q=1$ gives the computation-stage budget two needed when the declared capacity is two.

Replace a phase location by the same phase on its first child and idle locations on its other $m-1$ children; replace an idle by $m$ idles. At each replacement exactly one descendant carries the phase, so the full matrix after any number of replacements remains $e^{\pm 2\pi i\sqrt{2}}I$. Every elementary operation used is again $P,P^{\ast }$ or an idle. Thus this part of the finite alphabet is closed under recursive replacement, with at most $m$ children per phase or idle gadget and the exact level-one size and depth already proved. No controlled phase is identified with an identity gate: the claim concerns the displayed scalar matrices themselves, while their unconditioned channels are identity.