Exceptional parabolic-orbit length certificates for E6, E7, E8, F4, H3 and H4

Data and dual matrices. For each of the six rows, [F6] identifies the recorded diagram with the standard diagram of the type and the recorded parabolic with the stated type, and the deleted nodes of the statement are exactly the certificate's deleted-node entries. By [F7] every edge scalar is exact in the recorded coefficient field; [F2] then gives the exact dual simple-reflection matrices in dual simple-root coordinates, while nonedges have scalar $2\cos (\pi \mathrm{/}2)=0$. The script applies these matrices to the seed coordinate vector, performs exact breadth-first search, and records each state, each generator image, an attaining word and its graph distance. The certificate matrix is for the dual action. If $R_t$ is a canonical simple-reflection matrix and $G$ is the Gram matrix, then $R_t^{\mathsf{T}}G{R}_t=G$ and $R_t^2=I$ by [F1], so its dual matrix $R_t^{\mathsf{T}}=G{R}_t{G}^{-1}$. Multiplying in the same recorded order shows that the dual-action Coxeter matrix is similar to the canonical Coxeter matrix; thus their characteristic polynomials agree. The canonical characteristic polynomial is the exact trace result of [F8], and the certificate also checks its cyclotomic factorization. The seven certificate flags (orbit closure, word upper bounds, edge-distance lower bounds, reflection involutions, the Poincare product identity, the cyclotomic spectrum identity, and the final characteristic recurrence) are true; rerunning the script reproduces the certificate byte for byte. Thus these are exact coordinate computations, with no floating-point approximation.

The recorded lengths are the distances. Fix one of the six types. The recorded state set contains the seed and is closed under every generator, and every recorded state is reached from the seed by its recorded generator sequence; hence it is exactly the orbit $W\cdot f_0$. Let $d^{\mathrm{\prime }}$ be the recorded distance. Each recorded word has length $d^{\mathrm{\prime }}$, so the graph distance $d$ is at most $d^{\mathrm{\prime }}$. Conversely, the certificate verifies that $d^{\mathrm{\prime }}$ changes by at most one along every generator edge and vanishes at the seed; summing this bound along any edge path gives $d^{\mathrm{\prime }}\le d$. Therefore $d^{\mathrm{\prime }}=d$, which by [F3] is the minimal left-coset length. The recorded orbit sizes and distance polynomials are thus correct, and [F3] gives $P_W=P_{W_T}{\sum }_v{t}^{d(v)}$.

Degree comparison and induction. The exact coefficient check of the Poincare product identity agrees with the degree multisets of [F4]. The six identities reduce to $[4{]}_t(1+t^4+t^8)=[12{]}_t$, $[5{]}_t(1+t^5)=[10{]}_t$, $[9{]}_t(1+t^9)=[18{]}_t$, $[6{]}_t(1+t^6)=[12{]}_t$, $[10{]}_t(1+t^{10})=[20{]}_t$, and $[12{]}_t(1+t^{12})=[24{]}_t$, where $[n{]}_t=1+t+\cdots +t^{n-1}$; multiplying by the corresponding parabolic degree products gives exactly the six ambient degree products in the statement. Using the quotient factorization established in step 2.1, each identity converts a known $P_{W_T}$ into $P_W$. For $F_4$, [F5] gives $P_{B_3}=[2{]}_t[4{]}_t[6{]}_t$, and the first relation gives $P_{F_4}=[2{]}_t[6{]}_t[8{]}_t[12{]}_t$. For $H_3$, [F5] gives $P_{I_2(5)}=[2{]}_t[5{]}_t$, and the second relation gives $P_{H_3}=[2{]}_t[6{]}_t[10{]}_t$; the fourth and fifth relations then give $P_{H_4}=[2{]}_t[12{]}_t[20{]}_t[30{]}_t$. For $E_6$, [F5] gives $P_{D_5}=[2{]}_t[4{]}_t[5{]}_t[6{]}_t[8{]}_t$, and the first relation gives $P_{E_6}=[2{]}_t[5{]}_t[6{]}_t[8{]}_t[9{]}_t[12{]}_t$; the second and third relations give $P_{E_7}=[2{]}_t[6{]}_t[8{]}_t[10{]}_t[12{]}_t[14{]}_t[18{]}_t$; the fourth, fifth and sixth relations give $P_{E_8}=[2{]}_t[8{]}_t[12{]}_t[14{]}_t[18{]}_t[20{]}_t[24{]}_t[30{]}_t$. Each result is ${\prod }_{d\in D(W)}[d{]}_t={\prod }_{i=1}^{\mathrm{\mid }S\mathrm{\mid }}(1+t+\cdots +t^{d_i-1})$. Choice enters only through the degree determinations of [F4]; the certificate's orbit and distance verification is choice-free.