Exceptional Brunnian Words: Undecidability and One-Relator Boundaries

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1. Definition 2.1
   PF:2026.000009/v1/DEF-2.1

   The common-representative source-literal and quotient-killing Brunnian subgroups are \[ B_X^{\src}(G;N) = q\!\left(\bigcap_{i=1}^n\delta_i^{-1}(N)\right), \qquad B_X^{\quo}(G) = \bigcap_{i=1}^nR_i. \] The literal full-support subgroup $L_X(G)$ is generated by all finite binary-bracket commutator values whose leaves are $q(x_i)^{\pm1}$, with repetitions allowed and every label required to occur. The subgroup $C_X^{\mathrm{fat}}(G)$ is defined by allowing a leaf labelled $i$ to be any element of $R_i$.

2. Proposition 2.2: The two numerator semantics
   PF:2026.000009/v1/PROP-2.2

   For every $i$, the quotient map $G\to G/R_i$ is universal among homomorphisms killing $q(x_i)$, and its kernel is $R_i$. Hence \[ B_X^{\quo}(G) = \{g\in G:g\text{ dies after each relation }q(x_i)=1\text{ is imposed}\}. \] Moreover, \[ B_X^{\src}(G;N) =q(H_N), \qquad H_N:=\bigcap_{i=1}^n\delta_i^{-1}(N), \] and B_X^{\quo}(G) = q\!\left(\bigcap_{i=1}^nR_i^FN\right), \qquad B_X^{\src}(G;N)\leq B_X^{\quo}(G). These predicates are genuinely different. For \[ F=F(x,y),\qquad N=\langle\!\langle xy\rangle\!\rangle_F,\qquad G=F/N, \] consider the following conditions on $w\in F$: \item $\delta_x(w),\delta_y(w)\in N$; \item $q(w)$ has a representative $v$ satisfying $\delta_x(v),\delta_y(v)\in N$; \item $q(w)\in\langl…

3. Theorem 2.3: Full-support denominator and the sphere benchmark
   PF:2026.000009/v1/THM-2.3

   Let $G$ be generated by a finite indexed tuple $X=(x_1,\ldots,x_n)$, and let $R_i=\langle\!\langle x_i\rangle\!\rangle_G$. Then L_X(G) = C_X^{\mathrm{fat}}(G) = \Fat(R_1,\ldots,R_n) = \Sym(R_1,\ldots,R_n) \leq B_X^{\quo}(G) = \bigcap_{i=1}^nR_i. For $d\geq1$, let \[ F_d=F(x_0,\ldots,x_d),\qquad r_d=x_0x_1\cdots x_d,\qquad G_d=F_d/\langle\!\langle r_d\rangle\!\rangle, \] and let $q_d\colon F_d\to G_d$ be the quotient map. Then B_{X_d}^{\src}(G_d;\langle\!\langle r_d\rangle\!\rangle) = L_{X_d}(G_d), whereas \frac{B_{X_d}^{\quo}(G_d)}{L_{X_d}(G_d)} \cong\pi_{d+1}(S^2). For $d=1$, G_1\cong\Z,\qquad B_{X_1}^{\src}=L_{X_1}=\{1\},\qquad B_{X_1}^{\quo}=G_1. For $d=2$, the element \[ [q_d(x_1),q_d(x_2)] \] lies in $B…

4. Theorem 3.1: Supplied-word reductions for source semantics
   PF:2026.000009/v1/THM-3.1

   Supplied-word recognition for \[ B_X^{\src}(G;N)\setminus L_X(G) \] (Definition 2.1) is uniformly undecidable for arbitrary finite marked presentations. The reduction may be arranged so that the supplied word itself satisfies \delta_i(w)\in N\qquad(1\leq i\leq n). The conclusion remains true under the promise that all displayed marks are nonidentity and pairwise distinct. More precisely, from \[ H=\langle y_1,\ldots,y_m\mid s_1,\ldots,s_\ell\rangle \] and a word $u$, one can effectively construct a marked presentation and a word $W$ such that (equation reference) holds and \in B_X^{\src}(G;N)\setminus L_X(G) \quad\Longleftrightarrow\quad u\neq1\text{ in }H.

5. Theorem 3.2: Source existence and its exact arithmetical boundary
   PF:2026.000009/v1/THM-3.2

   Fix effective encodings of finite marked presentations and words. \item From every pair $(H,w)$ with $H$ displayed on $m\geq1$ generators, one can effectively construct a marked presentation \[ \cP_{\mathrm{gen}}(H,w) =\langle y_1,\ldots,y_n\mid t_1,\ldots,t_q\rangle, \qquad n=2m+6, \] and a word $V_{\mathrm{gen}}(H,w)$ such that all marks are nonidentity and pairwise distinct and w=1\text{ in }H \Longleftrightarrow B_{\mathrm{gen}}^{\src}=\{1\} \Longleftrightarrow B_{\mathrm{gen}}^{\src}\setminus L_{\mathrm{gen}}=\varnothing. If $w\neq1$, then the single supplied word satisfies every literal deletion relation and represents an element of $B_{\mathrm{gen}}^{\src}\setminus L_{\mathrm{gen}}$. \item For arbitra…

6. Theorem 4.1: A minimal three-label source example
   PF:2026.000009/v1/THM-4.1

   Let \[ F=F(x,y,z),\qquad N=\langle\!\langle y^2z^2,\ xz^2,\ xy^2\rangle\!\rangle_F, \qquad G=F/N, \] and put $w=xy^2z^2$. Then \[ \delta_x(w)=y^2z^2,\qquad \delta_y(w)=xz^2,\qquad \delta_z(w)=xy^2, \] and q(w)\in B_X^{\src}(G;N)\setminus L_X(G), \qquad q(w)\neq1. The marked elements $q(x),q(y),q(z)$ are pairwise distinct. Three labels are minimal for a source-literal exceptional example. Every finite marked one-relator presentation has empty source-literal exceptional set, including the case of an identity relator. Therefore at least two defining relators are necessary. The example above uses three relators; no claim is made that three relators are minimal.

7. Theorem 5.1: A genuine-marking supplied-word gadget
   PF:2026.000009/v1/THM-5.1

   Let \[ H=\langle y_1,\ldots,y_m\mid S\rangle,\qquad m\geq1, \] and let $u$ be a word in the $y_j^{\pm1}$. For $v=v(y_1,\ldots,y_m)$, put \[ \widehat v=v(d_1c_1^{-2},\ldots,d_mc_m^{-2}). \] Let \[ X=\{a,c_1,d_1,\ldots,c_m,d_m,s,z\}, \] and define \[ D=[a,c_1,d_1,\ldots,c_m,d_m], \qquad W=[D,s]. \] Set G(H,u)= \left\langle X\ \middle|\ \widehat r=1\ (r\in S),\quad W=\widehat u\,z \right\rangle. Then: \item \[ G(H,u)\cong H*F(a,c_1,\ldots,c_m,s), \quad d_j=y_jc_j^2,\quad z=u^{-1}W; \] \item all elements of $X$ are nonidentity and pairwise distinct, and they generate $G(H,u)$; \item $W$ lies in the normal closure of every element of $X\setminus\{z\}$; \item W\in\langle\!\langle z\rangle\!\rangle \quad\Longleftri…

8. Theorem 5.2: A whole-quotient central-involution switch
   PF:2026.000009/v1/THM-5.2

   Let \[ H=\langle y_1,\ldots,y_m\mid s_1,\ldots,s_\ell\rangle, \qquad m\geq1, \] and let $u$ be a word. Put \[ \widehat v=v(d_1c_1^{-2},\ldots,d_mc_m^{-2}), \qquad E=\{r,c_1,d_1,\ldots,c_m,d_m\}, \] introduce stable letters $q_e$ for $e\in E$ and further symbols $a,z,q$, and set \[ k=a^{-1}[\widehat u,r]. \] Let \[ X=\{z,a,q\}\cup E\cup\{q_e:e\in E\} \] and define G(H,u)= \left\langle X\ \middle|\ {ll} \widehat{s_j}=1 &(1\leq j\leq\ell),\\ z^2=1,\ [z,x]=1 &(x\in X\setminus\{z\}),\\ q_e^{-1}eq_e=ez &(e\in E),\\ q^{-1}kq=kz.& \right\rangle. Then every mark is nonidentity, distinct marks represent distinct elements, and $X$ generates $G(H,u)$. Moreover, \[ R_z=\langle z\rangle\cong C_2, \] and B_X^{\quo}(G(H,u))…

9. Theorem 6.1: Arithmetical boundary for quotient killing
   PF:2026.000009/v1/THM-6.1

   For either \[ B_X^{\quo}(G)\setminus L_X(G) \quad\text{or}\quad B_X^{\quo}(G)\setminus C_X^{\mathrm{fat}}(G), \] the following hold for arbitrary finite marked presentations. \item The uniform supplied-word relation is two-computably-enumerable and $\Delta^0_2$, but is not computably enumerable. No convergent characteristic approximation starting at $0$ can change at most once. \item The witness-existence index set belongs to \[ \Sigma^0_2\setminus\Sigma^0_1. \] No $\Sigma^0_2$-hardness or completeness assertion is made. \item No positive-only computable witness algorithm exists. Relative to the halting oracle $\mathbf 0'$, positive witness production is possible.

10. Theorem 7.1: Integrated computability matrix
    PF:2026.000009/v1/THM-7.1

    Fix effective encodings of finite marked presentations and words. Use the two numerators and two denominators of Definition 2.1. Supplied-word recognition means a terminating decision algorithm. Positive-only witness search outputs only valid exceptional words and halts on every positive presentation. Total witness-or-correct-nonexistence always terminates with either a valid witness or a correct negative report. An arbitrary marking carries no promise on the displayed values. A genuine-set marking is promised to consist of pairwise distinct nonidentity elements. Then the ordinary-computability matrix is {c|c|c|c|c} \text{semantics} &\text{marking} &\text{recognition} &\text{positive-only witness} &\text{tot…