Exhaustive certified Fibonacci B6 block-pair search

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.