Topological Quantum Algorithms and Braid Compilation

Fix $d\in \{2,5,8\}$ and put $SU(d)=\{T\in M_d(\mathbb{C}):T^{\ast }T=I_d,\det T=1\}\mathrm{.}$ A certified $SU(d)$ matrix input is an exact target $T\in SU(d)$ together with a deterministic Cauchy-name procedure which, on unary input $1^b$ for $b\ge 1$, returns a rational complex matrix $Q_b$ such that each real and imaginary coordinate of each entry differs from the corresponding coordinate of $T$ by at most $2^{-b}$. The interface also specifies a worst-case bit-time bound $t(b)$ for the query. A polynomial-time name has $t(b)\le (b+1{)}^c$ for some fixed integer $c\ge 1$. The target is the matrix specified by the entire Cauchy name; if two targets satisfy these bounds for the same sequence $(Q_b{)}_{b\ge 1}$, every coordinate difference is at most $2^{1-b}$ for all $b$, hence is zero. No individual $Q_b$ is required to be unitary or special unitary. A requested tolerance is a positive binary rational and its numerator/denominator encoding length is recorded separately. For $d=2$ this specializes the unary-name convention of Effective Unitary Matrix Input. For $d=5,8$ it is the same entrywise interface in the fixed block dimensions; no dimension is input to an algorithm as an unbounded parameter.

Let $U_1,\dots ,U_m$ and $V_1,\dots ,V_m$ be unitary operators on the same finite-dimensional Hilbert space, and suppose $\mathrm{\parallel }{U}_j-V_j{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {ϵ}_j$ with ${ϵ}_j\ge 0$. With empty products equal to the identity and empty sums equal to zero, $\mathrm{\parallel }{U}_m\cdots U_1-V_m\cdots V_1{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {\sum }_{j=1}^m{ϵ}_j\mathrm{.}$

Let $U,V$ be unitary operators on a finite-dimensional Hilbert space, let $\rho$ be any density operator, and let $\mathsf{M}$ be any finite POVM. If $\mathrm{\parallel }U-V{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le ϵ$, then the outcome distributions obtained by measuring $U\rho U^{\ast }$ and $V\rho V^{\ast }$ have total-variation distance at most $ϵ$.

Work in a fixed unitary braided fusion category $\mathcal{C}$ with the representative set $\mathrm{Irr}(\mathcal{C})$ of Multiplicity-aware fusion-tree basis. Fix ordered inputs $a_1,\dots ,a_n\in \mathrm{Irr}(\mathcal{C})$, $n\ge 1$, a parenthesized fusion tree, and a total charge $c\in \mathrm{Irr}(\mathcal{C})$ for which the fusion space is nonzero. Simple charges, multiplicity bases, and admissible fusion-tree vectors have the conventions of Multiplicity-aware fusion-tree basis. An encoded input subspace is specified by a fixed-charge isometry as in Fixed-charge encoding and code projector. Each task instance supplies either an initial unit vector in that subspace as an abstract quantum input, or a finite state-preparation procedure whose specified output is that vector and whose preparation operations and ancillas are recorded. The instance also gives a nonnegative rational error threshold and specifies the correctness relation and metric to which it applies.

A finite anyon decision task is a promise pair $(Y,N)$ of disjoint sets of binary strings with domain $D=Y\cup N$, together with a finite encoding of admissible fusion-tree input labels; a solver must answer yes on $Y$ and no on $N$, with no obligation outside $D$. A finite anyon search task is a domain $D$ and a finite-string witness relation, with a valid witness required only for encoded inputs in $D$. The input record states the number and order of strands, the finite alphabet of positive and inverse Artin generators available to the solver, any prescribed finite braid word, and the finite set of adjacent fusion-measurement locations. For distinct input charges, an adjacent exchange has the typed unitary F/R action from the current ordered-list fusion space to the space for the list with those charges exchanged; track that list after every generator. These colored maps and their Artin relations are supplied by Gauge covariance of fusion-space braid actions. If a single-space $B_n$ representation is required for the physical-interface record Physical braid implementation interface, use the finite orthogonal direct sum of fusion spaces over all distinct orderings of the fixed input multiset, with the same total $c$. Each generator maps each summand to its exchanged-list summand by the typed unitary; its inverse reverses that map. The colored Artin relations on each summand give the relations on the direct sum, hence a unitary $B_n$ action. Record the occupied summand and interpret measurements in its declared fusion-tree basis. Identical charges give the usual single fixed-space action. The physical-interface record specifies the mathematical action and separately states its physical realization hypotheses. The state-preparation field is an abstract task input and does not assert that a physical device prepares the state or realizes the braid. Input length and correctness-on-promises follow QA task and promise domain.

In the unitary braided fusion category $\mathcal{C}$ fixed by Finite anyon task and promise interface, use its fixed representative set $\mathrm{Irr}(\mathcal{C})$. Fix $n\ge 1$, ordered inputs $a_1,\dots ,a_n\in \mathrm{Irr}(\mathcal{C})$, a total charge $c\in \mathrm{Irr}(\mathcal{C})$ with nonzero fusion space, and one parenthesized binary tree. Every charge label below belongs to this same representative set. An encoded fusion input consists of the finite tree shape, a simple-charge label on each internal edge, and a multiplicity-basis index at each trivalent vertex. A label is admissible exactly when every chosen vertex index belongs to the specified basis of its multiplicity space $V_{xy}^z$; the allowed local vertex data and resulting orthonormal fusion-tree vectors are those of Multiplicity-aware fusion-tree basis. The input encoding uses fixed self-delimiting binary codes for charge labels and nonnegative multiplicity indices, with out-of-range labels rejected as invalid encodings. It also records the total charge and an identifier for the selected computational subspace isometry required by Finite anyon task and promise interface, as in Fixed-charge encoding and code projector. For a fixed task family, a declared codebook assigns each such finite identifier a computational dimension and a particular isometry into the specified fusion space. The codebook and its isometries are fixed background model data; unknown identifiers are invalid. If isometries instead vary as part of the classical input, the task must declare a finite description language and its exact interpretation as isometries, and charge the full descriptions as input bits. This does not claim that every arbitrary complex isometry has such a finite description; an arbitrary abstract isometry may remain background model data only. The tree labels and isometry identifiers are finite classical data; they do not themselves specify or prepare the initial encoded quantum state, which is the separate state/preparation field in the task record.

For a finite anyon solver instance and one run, record separately the tuple $(L_{\mathrm{i}\mathrm{n}},N_{\mathrm{a}\mathrm{n}\mathrm{y}},\mathrm{\mid }b\mathrm{\mid },N_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}},N_{\mathrm{a}\mathrm{n}\mathrm{c}},N_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}},B_{\mathrm{o}\mathrm{u}\mathrm{t}},b_{\max },t_{\mathrm{c}\mathrm{l}},T_H,{ϵ}_{\mathrm{o}\mathrm{p}},{ϵ}_{\mathrm{t}\mathrm{r}},{ϵ}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}},{\mathrm{\ell }}_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}},p_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}},N_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}},t_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}},{\mathsf{C}}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}},N_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{l}\mathrm{e}})\mathrm{.}$ Here $L_{\mathrm{i}\mathrm{n}}$ is the complete finite classical input-description bit length, including the declared task, state/preparation descriptions when classically encoded, and tolerance descriptions; a directly supplied quantum input is not assumed to have a finite classical coordinate description; $N_{\mathrm{a}\mathrm{n}\mathrm{y}}$ is the number of ordered simple-charge strand labels in the fusion-space instance; and $\mathrm{\mid }b\mathrm{\mid }$ is the number of braid-generator occurrences (inverse generators count as one). $N_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}$ counts elementary operations in the declared preparation procedure, while $N_{\mathrm{a}\mathrm{n}\mathrm{c}}$ counts the ancillary anyons or other physical ancilla systems used by preparation or computation. If the initial state is supplied directly as an abstract quantum input with no preparation interface, $N_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}$ is “not applicable”, not silently set to zero; $N_{\mathrm{a}\mathrm{n}\mathrm{c}}$ still records any declared solver ancillas (zero if none, or “not applicable” if no physical ancilla model is supplied). $N_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$ is the number of fusion measurements, and $B_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the total bit length of their classical outcome record under its declared finite encoding (zero when no measurement occurs). The largest unary precision query to a certified matrix name is $b_{\max }$; $t_{\mathrm{c}\mathrm{l}}$ is the classical bit time used by search and postprocessing. $T_H$ is the total Hamiltonian evolution time over the run in the declared physical time unit; it is “not applicable” for an abstract braid-only model without Hamiltonian evolution. The error coordinates ${ϵ}_{\mathrm{o}\mathrm{p}},{ϵ}_{\mathrm{t}\mathrm{r}},{ϵ}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}},{\mathrm{\ell }}_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}},p_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}$ are respectively a stated operator-norm approximation bound, an independent in-band transport bound, an implementation-to-ideal-measurement total-variation bound, and leakage amplitude and probability bounds. $N_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}},t_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}$ count phase corrections and their classical control cost. ${\mathsf{C}}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}$ is the complete applicable circuit cost record of Circuit-family and computation cost model, including description length $L(C)$, gate count $g(C)$, minimum-layer depth $d(C)$, wire count $w(C)$, declared clean/dirty ancilla counts, generator and acceptance costs, each named oracle's query count, and tolerance encoding. $N_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{l}\mathrm{e}}$ counts all declared elementary nonoracle gates, including preparation and computation, as in Nonoracle quantum gate cost. For an adaptive solver, declare the branch convention and report worst-case branch costs; for a single circuit use that circuit's record. These coordinates may overlap $N_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}}$ and $N_{\mathrm{a}\mathrm{n}\mathrm{c}}$; they are separate reported resources, not disjoint summands to be added. Without a circuit implementation interface, ${\mathsf{C}}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}$ and $N_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{l}\mathrm{e}}$ are “not applicable”. Classical input bit cost follows Classical bit-operation cost and finite nonoracle gates follow Nonoracle quantum gate cost. An inapplicable coordinate is marked “not applicable”; it is not assigned value zero. Unless an interface explicitly states a bound, no braid length, preparation operation, ancilla, matrix precision, measurement, Hamiltonian time, search step, or phase correction is free. The tuple is a bookkeeping convention; it does not assert any of its error or runtime bounds.

Let $H$ be a fixed finite fusion space with an orthonormal fusion-tree basis in the category and simple-representative domain of Fusion-tree task input encoding. For a colored task, $H$ is the currently occupied ordered-list summand tracked by the task record. For an adjacent pair of leaves or a declared adjacent pair of subtrees, recouple the tree by its specified unitary $F$-moves so that the pair fuses first. For each allowed simple total charge $a$ of that pair, let $P_a$ be the orthogonal projection onto the span of basis vectors whose pair label is $a$, transported back by the inverse recoupling unitary. Distinct channel subspaces are orthogonal and their direct sum is $H$, so $(P_a{)}_a$ is a finite projective measurement in the sense of Finite projective measurements. Its outcome is $a$, its unnormalized branch vector on input $\psi$ is $P_a\psi$, and the recoupling follows the inverse-precomposition F convention of F-symbol as a recoupling matrix, whose unitarity is proved in F-symbol unitarity. The branch vector is unnormalized until its probability is checked.

Let $n\ge 1$ and $c\in \{1,\tau \}$ with $V({\tau }^n\to c)\mathrm{\ne }0$, using the left-associated tree for both fixed-charge spaces. In the ordered list of $3n$ Fibonacci $\tau$ charges, group consecutive leaves into $n$ triples; within each triple fuse its first two charges and then its third, and fuse the $n$ triple outputs by the left-associated $n$-leaf tree to total $c$. Restrict each triple output to $\tau$; its two local paths are indexed by the charge $x\in \{1,\tau \}$ after the first two leaves. Fix a rebracketing sequence of associators from this grouped tree to the fixed left-associated $3n$-leaf tree, and let $R_{n,c}$ be the induced composite unitary. For each tensor basis vector $e_{x_1}\otimes \cdots \otimes e_{x_n}\otimes v_{\lambda }$, where $e_1,e_{\tau }$ are the local triple paths and $v_{\lambda }$ is a basis vector of $V({\tau }^n\to c)$, map it to its grouped fusion-tree basis vector, then apply $R_{n,c}$. Extend linearly and call the resulting map $J_{n,c}$. Its image is the code ${\mathcal{C}}_{n,c}$ and it is an isometry; set $P_{n,c}=J_{n,c}{J}_{n,c}^{\ast }$ for the code projector. The factor $V({\tau }^n\to c)$ retains all spectator fusion paths and is not a physical preparation at zero cost.

For the fixed Fibonacci convention, write $\rho (b)$ for the physical braid matrix and $B(b)$ for the normalized word matrix obtained by removing the common exchange scalar $R_1$ from each Artin generator, as in The braid phase is a common scalar character. If $e(b)$ is exponent sum, then $\rho (b)=R_1^{e(b)}B(b)\mathrm{.}$ On the three-anyon total-$\tau$ block the SK alphabet instead uses the special-unitary lifts $z{B}_j$ with $z=e^{3\pi i\mathrm{/}10}$, so a word of exponent sum $k$ satisfies $\rho (b)=(R_1{z}^{-1}{)}^k\widehat{B}(b),\widehat{B}(b)=z^{k}B(b)\mathrm{.}$ A phase reference record stores the exact generator scalar, exponent sum, which normalized convention was compiled, the resulting scalar on each coherently used branch, and any correction operation and its declared cost under Circuit-family and computation cost model. A standalone fixed-sector computation may report its output projectively using the metric of Projective braid gate and operator metric. In coherent control the active branch scalar is relative to the inactive branch and must remain in the record. No phase correction is assigned zero cost unless its interface says so. This ledger distinguishes the common $R_1^{e(b)}$ phase from the additional $z^{-e(b)}$ needed when the compiled alphabet is the SU(2) lift.

Let $n\ge 1$, $c\in \{1,\tau \}$, and consider the fixed fusion space $V({\tau }^{3n}\to c)$ with its grouped tree of $n$ consecutive triples and outer triple-charge tree. Its orthogonal decomposition is indexed by local triple outputs $a_i\in \{1,\tau \}$ and has factors $({⨂}_{i=1}^{n}V({\tau }^3\to a_i))\otimes V(a_1\otimes \cdots \otimes a_n\to c),$ with zero-dimensional summands omitted. A braid word supported on the internal three strands of the $i$th triple acts on this summand as ${\rho }_{a_i}(b)$ on $V({\tau }^3\to a_i)$ and the identity on every other local triple factor and on the outer spectator factor. Reassociating to the fixed left-associated $3n$-leaf tree conjugates this operator by a unitary; its operator norm and the stated local-factor action are preserved.

Let $(P_a{)}_{a\in \mathcal{A}}$ be the finite fusion-channel projectors of Fusion-channel measurement access on a finite-dimensional fusion space $H$, and let $\psi \in H$ be a unit vector. The channel probabilities $p_a=\mathrm{\parallel }{P}_a\psi {\mathrm{\parallel }}^2$ are nonnegative and sum to one. If $p_a>0$, the conditional branch is the unit vector $P_a\psi \mathrm{/}\sqrt{p_a}$. If $p_a=0$, then $P_a\psi =0$ and no normalized conditional state is defined.

Inputs: certified Cauchy names for targets $T_5\in SU(5)$ and $T_8\in SU(8)$; a positive binary-rational tolerance $\epsilon$; and the fixed normalized six-strand Fibonacci B6 generators in both total-charge blocks. Use the ordered alphabet $({\sigma }_1,\dots ,{\sigma }_5,{\sigma }_1^{-1},\dots ,{\sigma }_5^{-1})$ for $B_6$. For $d\in \{5,8\}$, form the exact generators $B_i^{(d)}=rI+(1-r)P_i^{(d)}$ from the Fibonacci path-projector formula Fibonacci B6 path projectors satisfy the braid relations, and use $(B_i^{(d)}{)}^{\ast }$ for inverse letters. To produce a rational approximant to a word block at precision $p$, write $b=a_1\cdots a_{\mathrm{\ell }}$ in Artin product order and multiply its exact letter matrices as $B_{a_1}^{(d)}\cdots B_{a_{\mathrm{\ell }}}^{(d)}$ (the rightmost factor acts first). Write $q=\sqrt{5}>0$, $\varphi =(1+q)\mathrm{/}2$, $s=\sqrt{\varphi }>0$, $h=\sqrt{10+2q}>0$, and $r=-(2\varphi {)}^{-1}-ih\mathrm{/}4$, the fixed Fibonacci data of Fibonacci fusion data and chiral convention. Start with dyadic root intervals $[2,3]$ for $q$, $[1,2]$ for $s$, and $[3,4]$ for $h$. At each pass $t=1,2,\dots$, bisect each interval once, keeping the half certified by its defining square equation and exact rational comparisons. To compare a rational midpoint square for $s$ or $h$, refine the $q$ interval until the resulting rational enclosure of its radicand lies strictly on one side; these radicands are irrational, so each comparison terminates. For real intervals $I=[a,b]$ and $J=[c,d]$, use $I+J=[a+c,b+d]$, $-I=[-b,-a]$, $IJ=[\min (ac,ad,bc,bd),\max (ac,ad,bc,bd)]$, and $I\mathrm{/}J=I[1\mathrm{/}d,1\mathrm{/}c]$ when $0<c\le d$. Represent a complex enclosure by a pair of real intervals and multiply $(I_1,I_2)(J_1,J_2)$ as $(I_1{J}_1-I_2{J}_2,I_1{J}_2+I_2{J}_1)$; conjugation sends $(I_1,I_2)$ to $(I_1,-I_2)$. Use these rational endpoint operations to evaluate each exact matrix-entry expression. Stop when every real and imaginary coordinate interval has width at most $2^{1-p}$, and return its midpoint; the coordinate error is then at most $2^{-p}$. These intervals converge to the exact entries, and all divisions are by positive powers of $\varphi >1$, so this loop terminates. At stage $s_0=1,2,\dots$, inspect word lengths $\mathrm{\ell }=0,\dots ,s_0$ in lexicographic order and precisions $1\le p\le s_0$ in increasing order, skipping pairs already inspected. For each new pair $(b,p)$, produce rational complex matrix approximants $P_{5,b,p},P_{8,b,p}$ by the interval procedure and request rational target approximants $Q_{5,p},Q_{8,p}$. For $d\in \{5,8\}$ compute the rational number $s_{d,b,p}={\sum }_{i,j}(\mathrm{\mid }\mathrm{\Re }(P_{d,b,p}-Q_{d,p}{)}_{ij}\mathrm{\mid }+\mathrm{\mid }\mathrm{\Im }(P_{d,b,p}-Q_{d,p}{)}_{ij}\mathrm{\mid })\mathrm{.}$ Return the first word for which both rational tests $s_{d,b,p}+4d{2}^{-p}<\epsilon (d=5,8)$ hold. Output the word, its exponent sum, $p$, and for each block the rational value $s_{d,b,p}$ and the verified strict test, together with measured search and precision costs in the ledger of Anyon compiler resource ledger. The returned matrices are normalized by the common $R_1$ scalar; their physical phase is recorded by Braid phase reference and correction ledger. This is a dovetailed partial search definition: no cutoff or useful word-length/time bound is asserted here. A separate theorem in this pair proves termination under certified target access.

For this compiler subroutine's ledger record, $N_{\mathrm{a}\mathrm{n}\mathrm{y}}=6$, ${ϵ}_{\mathrm{o}\mathrm{p}}=\epsilon$, $N_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=B_{\mathrm{o}\mathrm{u}\mathrm{t}}=0$, and $N_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}=t_{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}=0$ because it performs no measurement or phase correction. Preparation, ancilla, Hamiltonian-time, transport, measurement- implementation and physical-leakage coordinates are “not applicable” because this mathematical search supplies no physical realization. $L_{\mathrm{i}\mathrm{n}}$, $\mathrm{\mid }b\mathrm{\mid }$, $b_{\max }$, and $t_{\mathrm{c}\mathrm{l}}$ record the encoded target/compiler inputs, returned word, greatest precision request, and measured search/postprocessing bit time.

A finite adiabatic braid input consists of a finite-dimensional complex Hilbert space $V$, a $C^2$ Hermitian path $H(s)$ on $0\le s\le 1$, a norm-continuous rank-$m$ orthogonal spectral projection $P(s)$, with $m\ge 1$, onto a coherently followed band with uniform positive full gap, a slow-time schedule $T>0$, and an explicit initial isometry $A_0:{\mathbb{C}}^m\to \mathrm{ran}P(0)$. These are the path and propagator hypotheses of Finite adiabatic path model; the output propagator $U_T$ solves $i{U}_T^{\mathrm{\prime }}(s)=TH(s)U_T(s)$. Also supply an endpoint isometry $A_1:{\mathbb{C}}^m\to \mathrm{ran}P(1)$ and a specified Fibonacci braid word $b$ with categorical action $\rho (b)$ on a specified $m$-dimensional fusion space $V_{\mathrm{c}\mathrm{a}\mathrm{t}}$, in the fixed convention of Fibonacci fusion data and chiral convention, and a unitary target ${\rho }_{\mathrm{l}\mathrm{o}\mathrm{g}}(b)\in U({\mathbb{C}}^m)$ in the declared logical input/output frames. Supply unitary coordinate maps $Q_j:{\mathbb{C}}^m\to V_{\mathrm{c}\mathrm{a}\mathrm{t}}$ for $j=0,1$ and set ${\rho }_{\mathrm{l}\mathrm{o}\mathrm{g}}(b)=Q_1^{\ast }\rho (b)Q_0$. The band identifications are then ${\iota }_j=A_j{Q}_j^{\ast }$, as in Physical braid implementation interface. The target is the selected categorical $F\mathrm{/}R$ action written in those frames, including the phase convention recorded by Braid phase reference and correction ledger; specifying it does not assert that the physical path realizes it. The in-band transport condition is the separate hypothesis $\mathrm{\parallel }P(1)U_T(1)A_0-A_1{\rho }_{\mathrm{l}\mathrm{o}\mathrm{g}}(b){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {ϵ}_{\mathrm{t}\mathrm{r}}\mathrm{.}$ This condition is not a consequence of a projector-leakage estimate. The model also specifies a finite output measurement and, if needed, its own implementation-variation bound. The JRS expression and leakage consequence are handled as a separate result.

For every Fibonacci braid word $b$ with exponent sum $k$, its physical operator differs from the normalized operator by the common scalar $R_1^k$ on every fixed-charge block. On the three-anyon SU(2)-lift alphabet $z{B}_j$, the corresponding scalar is $(R_1{z}^{-1}{)}^k$. The scalar leaves uncontrolled fixed-sector measurement probabilities unchanged; under a coherent control it is a relative phase. If the scalar is not $1$, it cannot be discarded without a phase-reference correction; if it is $1$, the controlled maps coincide and no correction is needed.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. For every certified Cauchy input pair $(T_5,T_8)\in SU(5)\times SU(8)$ and every positive rational $\epsilon$, the exhaustive compiler of Exhaustive certified Fibonacci B6 block-pair search halts with one six-strand Fibonacci braid word whose normalized total-charge-$1$ and total-charge-$\tau$ matrices are each within operator norm $\epsilon$ of $T_5,T_8$, respectively. Density proves termination, but this result gives no useful joint-block word-length or search-time bound.

For a finite adiabatic braid input with rank-$m$ spectral band and schedule $T>0$, define $A_m(s)=\frac{1}{T}[\frac{m\mathrm{\parallel }{H}^{\mathrm{\prime }}(0)\mathrm{\parallel }}{g(0{)}^2}+\frac{m\mathrm{\parallel }{H}^{\mathrm{\prime }}(s)\mathrm{\parallel }}{g(s{)}^2}+{\int }_0^s(\frac{m\mathrm{\parallel }{H}^{\mathrm{\prime }\mathrm{\prime }}(r)\mathrm{\parallel }}{g(r{)}^2}+\frac{7m\sqrt{m}\mathrm{\parallel }{H}^{\mathrm{\prime }}(r){\mathrm{\parallel }}^2}{g(r{)}^3})dr]\mathrm{.}$ Then $\mathrm{\parallel }{U}_T(s)P(0)U_T(s{)}^{\ast }-P(s)\mathrm{\parallel }\le A_m(s)$, and every unit initial band vector has leakage probability at most $A_m(s{)}^2$. In particular, $\mathrm{\parallel }[I-P(1)]U_T(1)A_0{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le A_m(1)$ and leakage probability is at most $A_m(1{)}^2$. This result does not prove the separate in-band transport condition or identify band transport with Fibonacci $F\mathrm{/}R$ data.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and the fixed Fibonacci convention $B_1=\mathrm{diag}(1,r)$, $B_2=F{B}_{1}F$ with $r=e^{-3\pi i\mathrm{/}5}$. Put $z=e^{3\pi i\mathrm{/}10}$ and let $\mathrm{\Lambda }=⟨z{B}_1,z{B}_2⟩\le SU(2)$. Then $\mathrm{\Lambda }$ is dense in $SU(2)$ in operator norm.

Let $A_0$ and $A_1$ be the initial and endpoint isometries in a finite adiabatic braid input, let $V$ be the ideal logical unitary, and let $W_c$ be a compiled logical unitary. Suppose $\mathrm{\parallel }{W}_c-V{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {ϵ}_c,\mathrm{\parallel }P(1)U_T(1)A_0-A_{1}V{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {ϵ}_{\mathrm{t}\mathrm{r}},$ and let the JRS leakage amplitude be bounded by $A_m(1)$. Fix a finite POVM $\mathsf{M}$ on the physical output space. If the implemented measurement distribution differs from the distribution of this same $\mathsf{M}$ on $U_T(1)A_0\rho A_0^{\ast }{U}_T(1{)}^{\ast }$ by at most ${ϵ}_m$ in total variation, then for every input density operator $\rho$ the total measured-output variation from the ideal compiled output $A_1{W}_c\rho W_c^{\ast }{A}_1^{\ast }$ is at most ${ϵ}_c+{ϵ}_{\mathrm{t}\mathrm{r}}+A_m(1)+{ϵ}_m\mathrm{.}$ The transport mismatch is an explicit independent premise; the JRS estimate bounds leakage only.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and use the six-anyon triple-triple encoding Six-anyon triple-encoded subspace and leakage complement. For any $U\in U(4)$ put $T_1=U\oplus (\det U{)}^{-1}\in SU(5),T_{\tau }=U\oplus \mathrm{diag}((\det U{)}^{-1},1,1,1)\in SU(8)\mathrm{.}$ For every $\delta >0$, one normalized six-strand braid word $b$ has both block errors $\mathrm{\parallel }{B}_1(b)-T_1{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}<\delta$ and $\mathrm{\parallel }{B}_{\tau }(b)-T_{\tau }{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}<\delta$. In either total-charge block, the normalized braid's logical block error and leakage amplitude relative to the encoded action of $U$ are at most $\delta$. The physical categorical braid is $\zeta B(b)$ with $\zeta =R_1^{e(b)}$; its logical block error is at most $\delta$ relative to the phase-matched target $\zeta U$, and its leakage amplitude is unchanged. This common scalar is recorded under Braid phase reference and correction ledger and must be handled under coherent control. When the two determinant-corrected targets are supplied as certified Cauchy inputs, the dovetailed B6 compiler finds such a word. No useful joint-block word-length or search-time bound is asserted.