\section[Hopf-pairing maps and single-S boundary]
{Hopf-pairing images, primitive caps, and the
single-\texorpdfstring{\(S\)}{S} boundary}
\label{p4:chap:hopf-riesz-single-s}

This section constructs the permitted pairing-image and cap outputs, then
tests which scalar agreements admit a state-level comparison.

Figure~\ref{p4b:fig:one-holed-torus-running} is the genus-one instance to
keep in view: the maps in this section act on the reduced root-of-unity fiber, and
the primitive caps contract that fiber only after the pairing-induced map has
been typed.  Neither operation reverses the finite-stage arrows in the
figure.

This section separates four objects which have similar finite-root formulas
but different mathematical types:
\[
\begin{array}{c}
\text{a strict common rootwise/Rees state},\\
\text{an honest line-valued root-product state},\\
\text{a scalar obtained only after link-functional contraction},\\
\text{a generic boundary state over a fraction field}.
\end{array}
\]
Only the first three occur here, and they are never identified.  The source
constructions are organized below by the operation they support: transport,
closed contraction, or a boundary-level single-\(S\) test.

Half-Smith transport preserves the coefficient vector while keeping the
normalization line separate.  The maximal Newton-compatible pullback makes
these typed maps available on closed images.  The two primitive caps then
select coefficient zero and identify the common scalar graph, into which the
Habiro ring supplies finite-stage representatives with compatible formal and
root/cap readings.  The normalized single-S color is tested after
contraction, while the honest line-valued root state is kept separate.  The
final no-go results and the exact AS0 radical mark which state-level
comparisons fail and which finite quotient statement remains available.

\subsection{Coefficient-preserving Hopf-pairing frames}

Fix an odd prime \(p\ge5\), put \(d_p=(p-1)/2\), and write
\[
 \mathcal O_p=\Z[A_p],\qquad \lambda_p=1+A_p,\qquad
 \Lambda_p=\operatorname{diag}(1,\lambda_p,\ldots,\lambda_p^{d_p-1}).
\]
The exact Smith factorization in Newton coordinates and Hopf square are
\[
 H_pC_p=L_p\Lambda_p^2,\qquad
 C_p,L_p\in\operatorname{GL}_{d_p}(\mathcal O_p),
 \qquad H_p^2=\delta_p I.
\]
Let \(\mathbf d_p\) generate the normalization line with
\(\mathbf d_p^2=\delta_p\), and put
\(\mathbf d_p^\vee=\mathbf d_p^{-1}\).  Define
\[
 w_{n,p}=\lambda_p^{-n}C_pe_n,\qquad
 g_{n,p}=\lambda_p^nL_pe_n,\qquad
 f_{n,p}=\mathbf d_p^\vee g_{n,p}.
\]
The first frame spans the saturated source \(\mathcal S_p^+\); the other
two span the lowered-Hopf and raised, line-valued output lattices.
Here \(C_p,L_p\) are the explicit polynomial matrices of
Section~\ref{p4:chap:smith-newton}, and their GM comparison is
\eqref{p4:eq:selected-frame-transition}.  The division by
\(\lambda_p^n\) belongs to the invariant saturation.  The factor
\(\mathbf d_p^\vee\) belongs to the inverse dimension line and is
retained separately.

\begin{lemma}[Half-Smith transport]
\label{p4:lem:half-smith-transport}
\[
 H_pw_{n,p}=g_{n,p},\qquad
 \boldsymbol S_p^+w_{n,p}=f_{n,p},\qquad
 \boldsymbol S_p^+=\mathbf d_p^\vee H_p.
\]
Thus both arrows preserve the coefficient vector in their displayed typed
frames.  The oppositely oriented cylinder
\(\boldsymbol S_p^-:f_{n,p}\mapsto w_{n,p}\) is the inverse after line
contraction.
\end{lemma}

\begin{proof}
 The Smith factorization is used as a frame change: the inverse diagonal
factor clears the source frame, the Hopf matrix produces the integral target
frame, and the normalization line supplies the oppositely typed inverse.
Thus coefficient preservation is a statement in the displayed frames, not an
untyped identification of the two cylinders.

Multiply \(H_pC_p=L_p\Lambda_p^2\) on the right by
\(\Lambda_p^{-1}\).  Its \(n\)-th column is the first identity; the second
is obtained by the separately typed line generator.  Finally,
\((\mathbf d_p^\vee)^2H_p^2=1\).
\end{proof}

Let \(\mathscr C_W^T\) be the maximal Newton-compatible pullback of
Theorem~\ref{p4:thm:maximal-newton-pullback}.  On its formal coefficient
copy the following arrows preserve coefficients; on its \(p\)-th root
coordinate they apply \(H_p\) and \(\boldsymbol S_p^+\).  Denote the paired
maps by \(\mathcal H_{\mathrm{pair}}\) and
\(\mathcal S_{\mathrm{pair}}^+\), and define
\[
 \mathscr C_H^T=\mathcal H_{\mathrm{pair}}(\mathscr C_W^T),
 \qquad
 \mathscr C_S^T=\mathcal S_{\mathrm{pair}}^+(\mathscr C_W^T).
\]

\begin{theorem}[Strict images of the Hopf-pairing maps]
\label{p4:thm:strict-hopf-riesz-carrier}
The carriers \(\mathscr C_H^T\) and \(\mathscr C_S^T\) are closed,
complete, and Hausdorff, and
\[
 \mathcal H_{\mathrm{pair}}:\mathscr C_W^T\xrightarrow{\sim}\mathscr C_H^T,
 \qquad
 \mathcal S_{\mathrm{pair}}^+:\mathscr C_W^T\xrightarrow{\sim}\mathscr C_S^T
\]
are inverse bicontinuous isomorphisms onto their images.  The second inverse
is the oppositely oriented typed cylinder, not an untyped reuse of the same
matrix.

Moreover, with \(T_{\mathrm{pair}}\) from the preceding section,
\[
 \mathcal S_{\mathrm{mix}}
 =\mathcal S_{\mathrm{pair}}^+T_{\mathrm{pair}}^{-1}:
 \mathscr C_v^{\mathrm{base}}\xrightarrow{\sim}\mathscr C_S^T
\]
is bicontinuous.  The compatible triple graph
\[
 \{(u,T_{\mathrm{pair}}u,\mathcal S_{\mathrm{pair}}^+u):
 u\in\mathscr C_W^T\}
\]
is maximal among graphs with source in the full strict Newton carrier and
monomial coordinate in \(\mathscr C_v^{\mathrm{base}}\).
\end{theorem}

\begin{proof}
 The two Hopf-pairing maps are first checked on the displayed integral frames,
where the half-Smith factors cancel coefficientwise.  Their ambient product
maps then transport the closed Newton-compatible pullback to closed images.
The mixed map is the same comparison composed with the previously fixed
Newton identification; no new rootwise choice is made at this stage.

The root maps send an integral basis to an integral basis by
Lemma~\ref{p4:lem:half-smith-transport}; the formal map is the identity
between typed coefficient copies.  Their ambient product maps are therefore
homeomorphisms.  They transport the closed, complete pullback
\(\mathscr C_W^T\) to closed, complete images.  The mixed arrow is the
composition with the already proved Newton isomorphism.  A source point has
an admissible monomial coordinate exactly when it belongs to
\(T_{\mathrm{pair}}^{-1}(\mathscr C_v^{\mathrm{base}})
=\mathscr C_W^T\), which proves maximality.
\end{proof}

\subsection{Primitive caps and the vacuum pair}

Let \(y_p=e_0^{\mathsf T}\), define the complementary cap
\(h_p=y_pH_p\), and define the identity-double cap by the line-valued row
\(\widehat y_p=\mathbf d_py_p\).  Let
\(R^{\mathrm{all}}\subset R^{\mathrm{ev}}\) be the subring of fractions
whose reduced denominators are units at every declared admissible reduced root.  Let
\[
 \mathscr C_{\mathrm{sc}}
 =\overline{\{(\operatorname{Taylor}_\lambda c,
       (\operatorname{Ev}_p c)_p):c\in R^{\mathrm{all}}\}}
 \subset\mathcal R_\lambda\times\prod_{p\ge5}\mathcal O_p
\]
be the rank-one graph of compatible formal and root/cap readings.  It is not the evaluation graph of
all \(\mathcal R_\lambda\).

\begin{theorem}[Primitive two-cap vacuum]
\label{p4:thm:primitive-two-cap-vacuum}
For all active \(n\),
\[
 h_p(w_{n,p})=\delta_{n0},\qquad
 \widehat y_p(f_{n,p})=\delta_{n0}.
\]
The paired caps extend continuously to
\[
 h_W:\mathscr C_W^T\to\mathscr C_{\mathrm{sc}},\qquad
 \widehat y_S:\mathscr C_S^T\to\mathscr C_{\mathrm{sc}},
\]
and obey
\[
 \widehat y_S\mathcal S_{\mathrm{pair}}^+=h_W,\qquad
 h_W\mathcal S_{\mathrm{pair}}^-=\widehat y_S.
\]
The vacua
\[
 \Omega_p^+=w_{0,p}=e_0,\qquad
 \Omega_p^-=\mathbf d_p^\vee H_pe_0
\]
are exchanged by the typed \(S\)-cylinder and both declared caps have value
one.  After positive root-fiber trivialization,
\[
 S_{00,p}=D_p^{-1},\qquad (D_py_p)(\Omega_p^+)=D_p.
\]
The second displayed value remains normalization-line-valued.
\end{theorem}

\begin{proof}
 The two caps both select the zeroth vector in their respective typed frames.
The graph of compatible formal and root/cap readings makes that coefficient compatible between the
formal and finite-root coordinates, and continuity extends the finite
calculation to the closed carriers.  The scalar \(S_{00}\) value appears only
after the positive root-fiber trivialization of the normalization line.

The constructed zeroth column \(C_pe_0=e_0\) and row
\(e_0^TL_p=e_0^T\) fix
\(y_pL_p\Lambda_p=e_0^{\mathsf T}\).  Hence
\[
 y_pH_pC_p\Lambda_p^{-1}=e_0^{\mathsf T},
 \qquad
 \mathbf d_py_p(\mathbf d_p^\vee L_p\Lambda_p)=e_0^{\mathsf T}.
\]
On formal coefficients the caps select label zero.  On finite-stage representatives
with compatible formal and root/cap readings that coefficient lies in the all-incidence-safe ring, so the
values land in \(\mathscr C_{\mathrm{sc}}\); continuity extends the caps to
the closures.  Since \(H_{p,00}=1\), the \(S_{00}\) formulas follow.
\end{proof}

In the monomial chart the cap has a particularly useful closed formula.

\begin{theorem}[Common monomial cap]
\label{p4:thm:common-monomial-cap}
\leavevmode\newline
For a finite monomial vector \(Y=\sum_kc_kv^k\),
\[
 \varepsilon_v(Y)=\sum_kc_k\left(\frac{\lambda}{A}\right)^k
\]
is the zeroth Newton coefficient of \(T_{\mathrm{pair}}^{-1}Y\).  It extends
uniquely to a continuous map
\(\mathscr C_v^{\mathrm{base}}\to\mathscr C_{\mathrm{sc}}\).  Formally it is
the convergent series above; physically it evaluates the already formed
finite quotient class at
\[
 v=\rho_{0,p}=1-q_p^{-1}=\frac{\lambda_p}{A_p}.
\]
No completed formal series is evaluated termwise at a root.
\end{theorem}

\begin{proof}
 The row-zero inverse divided difference supplies the powers of
\(\lambda/A\), and their positive \(\lambda\)-order controls convergence on
the truncation-dense coefficient module.  At an admissible reduced root,
evaluation is performed only on the finite quotient.  Agreement on the dense
finite common core then extends the formal and root-specialized cap maps to
the common closure.

The row-zero inverse divided difference is
\(\eta_0^k\), where
\(\eta_0=(A+A^{-1}+2)/(1+A)=\lambda/A\).  Its positive
\(\lambda\)-order gives uniform convergence and continuity on \(c_0\).
At an admissible reduced root, \(\rho_{0,p}\) is a root of the quotient polynomial,
so evaluation is well defined on the finite quotient.  The two maps agree
on the dense finite common core and therefore extend uniquely to its
closure.
\end{proof}

\subsection{The Habiro ring inside the scalar graph}

Let
\[
 \widehat{\Z[q]}_{\mathrm H}
 =\varprojlim_N\Z[q]/((q)_N),\qquad
 (q)_N=\prod_{j=1}^N(1-q^j),
\]
with \(q=-A\) and \(\lambda=1-q\).

\begin{theorem}[Habiro-to-common scalar embedding]
\label{p4:thm:habiro-common-embedding}
The map
\[
 j_{\mathrm H}(f)=
 \left(\iota_\lambda(f),
       (\operatorname{Ev}_{q_p}f)_{p\ge5}\right)
\]
is a continuous injective ring homomorphism
\(\widehat{\Z[q]}_{\mathrm H}\to\mathscr C_{\mathrm{sc}}\).  For every
requested Taylor order and finite set of primes, one polynomial
representative supplies both approximations.
\end{theorem}

\begin{proof}
Modulo \((q)_m\), choose a polynomial representative \(a_m(q)\).  Since
\((q)_m=\lambda^m u_m(\lambda)\), its Taylor error has order at least \(m\).
If a root has order at most \(m\), one factor of \((q)_m\) vanishes there,
so the same representative is exact at that root.  Taking \(m\) larger than
the requested Taylor order and all selected root orders gives the required
finite-stage representative.  Habiro's injective Taylor theorem makes \(j_{\mathrm H}\)
injective.
\end{proof}

For \(N\ge1\), define the shell polynomial
\[
 g_N(q)=\frac{\prod_{j=N+1}^{2N+1}(q^j-1)}{q-1}.
\]
It has exact \(\lambda\)-order \(N\), is divisible by \((q)_N\), and
vanishes at a primitive \(p\)-th root once \(N\ge(p-1)/2\).  The exact
leading term is
\[
 g_N(1-\lambda)=(-1)^N\frac{(2N+1)!}{N!}\lambda^N
 +O(\lambda^{N+1}).
\]

The literal finite approximants used below arise from Habiro's uniform
positive twist color on a surgery link.  We therefore isolate the class of
integral homology spheres admitting an algebraically split all-$(-1)$-framed
presentation before stating the approximation theorem.

\begin{definition}[All-negative Hoste sector]
\label{p4:def:all-negative-hoste-sector}
An all-negative admissible presentation is an algebraically split link
\(L^-\subset S^3\) with every framing \(-1\).  Its linking matrix is
\(-I\), and its surgery manifold is an integral homology sphere.  This
sector consists only of manifolds admitting such a presentation.
\end{definition}

Let \(\omega_{+,<N}\) be Habiro's positive twist color truncated before
label \(N\), and set
\[
 \mathscr B_N(L^-)=J_{L^0}(\omega_{+,<N},\ldots,\omega_{+,<N}).
\]

\begin{theorem}[Negative Hoste scalar and literal approximants]
\label{p4:thm:negative-hoste-scalar}
For an all-negative presentation of \(M\), the polynomials \(\mathscr B_N(L^-)\)
converge in the Habiro ring to \(J_M\), and
\[
 \mathscr C_-(M)=j_{\mathrm H}(J_M)\in\mathscr C_{\mathrm{sc}}
\]
is independent of that presentation.  The same \(\mathscr B_N\) obey
\[
 \iota_\lambda(J_M-\mathscr B_N)\in\lambda^N\Z[[\lambda]]
\]
and is exactly equal to the \(p\)-th root value once
\(N\ge(p-1)/2\).  Its two coordinates are the Ohtsuki series and the odd
WRT values.
\end{theorem}

\begin{proof}
 The Hoste truncation is controlled by its successive shell difference.  Link
divisibility places each new term in a shell that has both the required
Taylor order and the required finite-root vanishing; the twist formula then
identifies the limit with the presentation-independent Habiro scalar.

Multilinearity writes \(\mathscr B_{N+1}-\mathscr B_N\) as a sum of colored-link terms whose
largest filtration label is \(N\).  Habiro's colored-link divisibility\newline
puts every term in \(g_N(q)\Z[q,q^{-1}]\).  The shell facts above make the
sequence cyclotomic Cauchy, give exact Taylor order \(N\), and kill every
omitted shell at the \(p\)-th root after \(N\ge(p-1)/2\).  Habiro's twist
formula identifies the limit with his surgery definition of \(J_M\);
presentation independence and the Ohtsuki/WRT coordinate statements are
his unified-invariant theorems.
\end{proof}

\subsection{Normalized single-\texorpdfstring{\(S\)}{S} colors and scalar
contraction}

Keep the bracket variable \(\widetilde A\), its square \(a\), and Habiro's
variable \(q_{\mathrm H}\) distinct:
\[
 v=a=\widetilde A^2,\qquad q_{\mathrm H}=\widetilde A^4=v^2.
\]
Write \(\{s\}_v=v^s-v^{-s}\) and
\(\{n\}_v!=\prod_{s=1}^n\{s\}_v\).
At order \(\kappa\ge3\), freeze
\[
 \begin{gathered}
 \widetilde A_{\kappa}=-e^{\pi i/(2\kappa)},\qquad
 \alpha_{\kappa}=e^{\pi i/(2\kappa)}=-\widetilde A_{\kappa},\qquad\\
 q_{\mathrm H,\kappa}=e^{2\pi i/\kappa},\qquad
 \xi_{\kappa}=e^{3\pi i(\kappa-2)/(4\kappa)}.
 \end{gathered}
\]
The sign ensures that Habiro's twist specializes to the frozen \(T_{\kappa}\).
Let \(S_{\kappa}=D_{\kappa}^{-1}\Pi_{\kappa}\), let \(h_{\kappa}\) be the signed zero-framed cap, and
let
\[
 F_{\epsilon,\kappa}=\sum_j\Delta_{j,\kappa}^2\theta_{j,\kappa}^{\epsilon}
 =D_{\kappa}\xi_{\kappa}^\epsilon.
\]

\begin{proposition}[Anomaly-corrected single-\(S\) color]
\label{p4:prop:normalized-single-s-colour}
For \(\epsilon=\pm1\),
\[
 \boldsymbol\Omega_{\kappa,\epsilon}
 =\frac{T_{\kappa}^{-\epsilon}\boldsymbol\Omega_{\kappa}}{F_{-\epsilon,\kappa}}
 =\xi_{\kappa}^\epsilon T_{\kappa}^{-\epsilon}S_{\kappa}e_0,
 \qquad
 h_{\kappa}\boldsymbol\Omega_{\kappa,\epsilon}=1.
\]
The anomaly factor \(\xi_{\kappa}^\epsilon\) remains root-specific; only the total
dimension has canceled.
\end{proposition}

\begin{proof}
 The signed dimension column identifies the unnormalized modular color as
\(D_\kappa S_\kappa e_0\).  The Gauss-sum ratio supplies the root-specific
anomaly factor, while the signed cap identity fixes the normalization so that
the resulting color contracts to one.

The first column of \(\Pi_{\kappa}\) is the signed dimension vector, so
\(\boldsymbol\Omega_{\kappa}=D_{\kappa}S_{\kappa}e_0\).  The Gauss identities give
\(D_{\kappa}/F_{-\epsilon,\kappa}=\xi_{\kappa}^\epsilon\).  The cap identity is the definition
of the normalizing Gauss denominator.
\end{proof}

Put
\[
 E_{\kappa}=\Q(\alpha_{\kappa},\zeta_8),\qquad
 W_{\kappa}=\bigoplus_{j=0}^{\kappa-2}E_{\kappa}f_j,
\]
and let \(\operatorname{Ev}_{\kappa}\) denote Habiro's regular root evaluation
\(q_{\mathrm H}^{1/4}\mapsto\alpha_{\kappa}\), followed by scalar extension to
\(E_{\kappa}\).  A \emph{regular test color} is a finite \(E_{\kappa}\)-linear
combination of rep\-resentation-ring colors whose lifted scalar
denominators do not vanish at \(\alpha_{\kappa}\).  On one distinguished
component of an algebraically split, zero-framed link, intersect the
kernels of the functionals obtained by putting arbitrary regular test
colors on all remaining components.  Denote the resulting \(E_{\kappa}\)-linear
radical and quotient by
\[
 \mathcal N_{\kappa}^{\mathrm{AS}},\qquad
 \mathcal Q_{\kappa}^{\mathrm{AS}}=W_{\kappa}/\mathcal N_{\kappa}^{\mathrm{AS}}.
\]

\begin{theorem}[Functional single-\(S\) equality]
\label{p4:thm:functional-single-s-equality}
Habiro's truncated twist color \(\overline\omega_{\kappa,\epsilon}\) and the
normalized modular color have the same class:
\[
 [\overline\omega_{\kappa,\epsilon}]_{\mathrm{AS}}
 =[\xi_{\kappa}^\epsilon T_{\kappa}^{-\epsilon}S_{\kappa}e_0]_{\mathrm{AS}}.
\]
They are not equal as vectors for any \(\kappa\ge3\).
\end{theorem}

\begin{proof}
 The equality is tested only after applying all declared algebraically split
link functionals, so it is a quotient statement.  Habiro's comparison lemma
puts the difference in the resulting radical, while the last supported basis
coefficient separates the two vectors before quotienting.  This is the local
model for the later distinction between scalar contraction and state equality.

Habiro's algebraically split comparison lemma says that every declared
link functional has the same value on the two colors, which is exactly
membership of their difference in \(\mathcal N_{\kappa}^{\mathrm{AS}}\).  Vector
equality fails by support: the truncated color is supported through
\(\lfloor(\kappa-2)/2\rfloor\), whereas the normalized full color has a nonzero
coefficient at \(f_{\kappa-2}\).
\end{proof}

For an odd prime \(p\), let
\(\sigma_p(\zeta_p)=\zeta_p^{(p+1)/2}=q_p^{\mathrm{com}}\).

\begin{theorem}[Contracted anomaly-corrected single-\(S\) formula]
\label{p4:thm:contracted-single-s-formula}
If \(M=S^3_{L^-}\) is in the all-negative Hoste sector, then
\[
 \sigma_p^{-1}\!\left(\operatorname{pr}_p\mathscr C_-(M)\right)
 =J_{L^0}^{(p)}
 \bigl(\xi_pT_p^{-1}S_pe_0,\ldots,\xi_pT_p^{-1}S_pe_0\bigr)
 =\tau_{\zeta_p}(M).
\]
Equivalently, \(\sigma_p\) may be applied after the complete scalar
contraction to recover the root-specialized common-root coordinate.  It is not
applied to an uncontracted state.
\end{theorem}

\begin{proof}
 First contract the integral Habiro scalar, where Galois naturality is
available.  At an admissible reduced root, Habiro divisibility truncates each twist
color; the functional single-\(S\) equality can then be applied component by
component, and the finite-root surgery formula identifies the final scalar.
The Galois operation is consequently post-contraction, not a transport of an
uncontracted boundary state.

The common scalar coordinate is
\(\operatorname{Ev}_{q_p^{\mathrm{com}}}J_M\).  Integral coefficients give
Galois naturality of the already contracted Habiro evaluation.  At
\(\zeta_p\), Habiro divisibility truncates every twist color.  Replace the
components successively using
Theorem~\ref{p4:thm:functional-single-s-equality}; after every replacement
the remaining colors are still valid test colors.  Habiro's finite-root
surgery formula identifies the final contraction with \(\tau_{\zeta_p}(M)\).
\end{proof}

\subsection{The honest line-valued root state}

For an odd prime \(p\ge5\), put
\[
 \widetilde A_p=-e^{\pi i/(2p)},\qquad
 E_p^{\mathrm{sk}}=\Q(\widetilde A_p)=\Q(\zeta_{4p}),\qquad
 E_p^{\mathrm{an}}=E_p^{\mathrm{sk}}(\zeta_8),\qquad
 \xi_p=\zeta_8^{-1}\widetilde A_p^{-3}.
\]
Let
\[
 W_p^{\mathrm F}=\bigoplus_{j=0}^{p-2}E_p^{\mathrm{sk}}f_j,\qquad
 V_p^{\mathrm E}=\bigoplus_{a=0}^{(p-3)/2}E_p^{\mathrm{sk}}e_a,
 \qquad e_a=f_{2a}.
\]
Let
\[
 \mathcal N_p^{\mathrm F}=E_p^{\mathrm{sk}}d_p^{\mathrm F},\qquad
 \mathcal N_p^{\mathrm E}=E_p^{\mathrm{sk}}d_p^{\mathrm E}
\]
be the full and even total-dimension lines.  Their chosen quadratic
trivializations satisfy
\((D_p^{\mathrm F})^2=2(D_p^{\mathrm E})^2\).  Introduce the square-root
line
\(\mathcal R_p=E_p^{\mathrm{sk}}r_p\), with
\(r_p^{\otimes2}\mapsto2\), and the anomaly line
\(\mathcal A_p=E_p^{\mathrm{sk}}\eta_p\), whose base change to
\(E_p^{\mathrm{an}}\) sends \(\eta_p\mapsto\xi_p\).  The total-dimension
identity is retained as the line isomorphism
\[
 \phi_{N,p}:\mathcal N_p^{\mathrm F}
 \xrightarrow{\sim}\mathcal R_p\otimes\mathcal N_p^{\mathrm E},
 \qquad
 d_p^{\mathrm F}\longmapsto r_p\otimes d_p^{\mathrm E}.
\]

Define the split parity map
\[
 \iota_p:V_p^{\mathrm E}\longrightarrow W_p^{\mathrm F},\qquad
 \iota_p(e_a)=f_{2a}+i^{-p}f_{p-2-2a}.
\]
Even-coordinate projection is its left inverse.  Dualizing
\(\phi_{N,p}\), define the typed split injection
\[
 \mathcal J_p
 =\operatorname{id}_{\mathcal A_p}\otimes\phi_{N,p}^{\vee}\otimes\iota_p:
 \mathcal A_p\otimes\mathcal R_p^\vee
 \otimes(\mathcal N_p^{\mathrm E})^\vee\otimes V_p^{\mathrm E}
 \longrightarrow
 \mathcal A_p\otimes(\mathcal N_p^{\mathrm F})^\vee
 \otimes W_p^{\mathrm F}.
\]

\begin{theorem}[Exact rootwise parity bridge]
\label{p4:thm:line-valued-parity-bridge}
Let
\[
 u_p^{\mathrm E,+}
 =\eta_p\otimes r_p^\vee\otimes(d_p^{\mathrm E})^\vee
   \otimes(T_p^{\mathrm E})^{-1}H_pe_0
\]
and
\[
 u_p^{\mathrm F,+}
 =\eta_p\otimes(d_p^{\mathrm F})^\vee
   \otimes(T_p^{\mathrm F})^{-1}\Pi_pe_0.
\]
Then
\[
 \mathcal J_p(u_p^{\mathrm E,+})=u_p^{\mathrm F,+}.
\]
After line trivialization, the full state is
\(\xi_p(T_p^{\mathrm F})^{-1}S_p^{\mathrm F}e_0\).
The product \(\prod_{p\ge5}u_p^{\mathrm F,+}\) is an honest point of the
complete Hausdorff product of finite line-valued root spaces.
\end{theorem}

\begin{proof}
 The parity injection is checked on the simple-current pair of labels, whose
sign and phase give the second coefficient in the split vector.  The
normalization-line isomorphism supplies the missing square-root factor, and
the continuous left inverses make the product of finite root states a genuine
point of the product space.

For the simple current \(J(j)=p-2-j\),
\[
 \Delta_{J(j)}=-\Delta_j,\qquad
 \frac{\theta_{J(j)}^{-1}\Delta_{J(j)}}
      {\theta_j^{-1}\Delta_j}=(-1)^ji^{-p}.
\]
For \(j=2a\) this is exactly the second coefficient in \(\iota_p(e_a)\).
The normalization-line factors agree because the full total dimension is
\(\sqrt2\) times the even total dimension.  Products of the split maps have
continuous left inverses, so the product state is well defined and has
closed image.
\end{proof}

The primitive \(SO(3)\) cap of the even state has residual value
\(\xi_p/\sqrt2\) in its anomaly/square-root line.  The scalar-primitive full
link-functional cap uses the blow-up Gauss contraction.  They are different
typed caps.

\subsection{Three exact state-level no-gos}

\begin{theorem}[No state-natural scalar Galois transport]
\label{p4:thm:galois-state-no-go}
There is no all-prime family of invertible
\(\sigma_p\)-semilinear intertwiners between the standard and frozen
common-root even twists, even up to a central scalar.
\end{theorem}

\begin{proof}
 It is enough to compare the smallest declared prime: scalar Galois transport
changes the unordered eigenvalue ratios, while a central factor changes all
eigenvalues together and cannot change those ratios.  The first-prime spectral
contradiction therefore rules out an all-prime state intertwiner.

At \(p=5\), the spectrum after scalar Galois transport is
\(\{1,\zeta_5\}\), whereas the frozen twist spectrum is
\(\{1,\zeta_5^2\}\).  Their characteristic polynomials differ.  Even after
central rescaling, the unordered eigenvalue-ratio sets
\(\{\zeta_5^{\pm1}\}\) and \(\{\zeta_5^{\pm2}\}\) are disjoint.
\end{proof}

\begin{theorem}[No strict common longitude-twist vacuum]
\label{p4:thm:single-s-c0-no-go}
For either longitude-twist sign and every family of central root units
\(c_p\), the root family \(c_p\omega_{p,\pm}\) is not the root component of
a point of the strict genus-one rootwise/Rees module.  Consequently the
anomaly-corrected even single-\(S\) state has no lift to the maximal
image module of the Hopf-pairing map.
\end{theorem}

\begin{proof}
 The strict rootwise/Rees module requires eventual divisibility in the formal Newton
coordinate, but the last active root-admissible label has a rootwise unit
coefficient.  Comparing those two statements at the maximal active label
gives the contradiction, and central root units cannot change divisibility.

In the saturated Newton frame, the coefficient of the last active label
\(n=(p-3)/2\) is a \(\lambda_p\)-adic unit for either twist sign: modulo \(\lambda_p\),
the GM and Newton bases differ by diagonal units, and every twist coefficient
has denominator congruent to \(-2\).  If a strict common lift existed, its
formal \(c_0\) coordinate would be divisible by \(\lambda\) at all sufficiently
large labels.  The exact compatible finite-stage representative criterion would then make the
last active root-fiber coefficient divisible by \(\lambda_p\), contradicting its
unit value.  Central root units cannot alter this contradiction.
\end{proof}

\begin{corollary}[Hereditary arbitrary-genus obstruction]
\label{p4:cor:single-s-hereditary-no-go}
Fix \(g\ge2\).  Any proposed common carrier admitting a continuous
tensor-local primitive-cap reduction to the existing genus-one target
cannot contain
\[
 u_p^{\mathrm E,+}\otimes\mathbf1_p^{\otimes(g-1)}
\]
as an all-prime common state.
\end{corollary}

\begin{proof}
Contract the last \(g-1\) primitive vacua.  Their caps are one, so a
hypothetical lift would give the forbidden genus-one lift of
Theorem~\ref{p4:thm:single-s-c0-no-go}.
\end{proof}

This is a no-go for the present strict \(c_0\) architecture, not for
arbitrary genus itself.

\subsection{The exact AS0 radical}

Let \(p_n=\operatorname{Ev}_{\kappa}(P'_n)\), \(k=\kappa-2\), and
\(d_{\kappa}=\lfloor k/2\rfloor\).  For \(0\le n\le k\), the factorial
denominator in \(P'_n\) is nonzero at \(\alpha_{\kappa}\), so \(p_n\in W_{\kappa}\) is
well defined.

\begin{lemma}[Specialized \(P'\)-basis]
\label{p4:lem:specialized-p-prime-basis}
The vectors \(p_0,\ldots,p_k\) form an \(E_{\kappa}\)-basis of \(W_{\kappa}\).
\end{lemma}

\begin{proof}
Both \(P_n\) and \(V_n\) are monic of degree \(n\) in \(V_1\), so their
transition matrix is triangular with diagonal one.  Dividing \(P_n\) by
the nonzero specialized factorial preserves triangularity and leaves a
nonzero diagonal.
\end{proof}

\begin{theorem}[Exact AS0 radical and failure of MCG descent]
\label{p4:thm:exact-as0-radical}
For every \(\kappa\ge3\),
\[
 \mathcal N_{\kappa}^{\mathrm{AS}}
 =\operatorname{Span}_{E_{\kappa}}\{p_{d_{\kappa}+1},\ldots,p_k\},
 \qquad
 \dim\mathcal Q_{\kappa}^{\mathrm{AS}}=\left\lfloor\frac{\kappa}{2}\right\rfloor.
\]
Equivalently, in the representation gauge,
\[
 \mathcal N_{\kappa}^{\mathrm{AS}}
 =\operatorname{Span}_{E_{\kappa}}\{V_{k-j}-V_j:0\le j<k-j\}.
\]
This radical is preserved by neither the full modular \(S\) matrix nor the
full twist \(T\).  Hence \(\mathcal Q_{\kappa}^{\mathrm{AS}}\) is a contracted
link-functional quotient, not a torus state quotient.
\end{theorem}

\paragraph{Divisibility, Borromean detection, and gauge symmetry.}
The radical computation has two directions.  Habiro divisibility kills the
high specialized \(P'\)-basis vectors under every regular test color, while
Borromean tests detect the low coordinates.  Symmetry under
\(j\mapsto\kappa-j\) gives the representation-gauge generators, and explicit
\(S\)- and \(T\)-images supply witnesses that the contracted radical is not a
torus-state subspace.

\begin{proof}
\emph{High indices vanish.}
Fix \(n>d_{\kappa}\).  Expand each regular test color generically as a finite
linear combination of \(V_m\)'s, then use Habiro's generic identity
\[
 V_m=\sum_{a=0}^m
 \begin{bmatrix}m+a+1\\2a+1\end{bmatrix}_{v}\{a\}_v!\,P'_a .
\]
Habiro's colored-link divisibility applies to the resulting integral
generic colored-Jones expression: up to a Laurent unit it is divisible by
\[
 G_h(q)=\frac{\prod_{s=h+1}^{2h+1}(q^s-1)}{q-1},
\]
where \(h\ge n\) is the maximum \(P'\)-index.  If \(h<\kappa\), then
\(h+1\le \kappa\le2h+1\); if \(h\ge \kappa\), the interval of \(h+1\) consecutive
integers contains a multiple of \(\kappa\).  Thus
\(\operatorname{Ev}_{\kappa}(G_h)=0\).  The original test denominators are root
units, so every declared functional kills \(p_n\).  This argument
specializes the divisible integral expression, not any individual high
\(P'_a\), whose factorial denominator may vanish.

\emph{Low coordinates are detected.}
Conversely, the Borromean rings with the regular colors \(p_i,p_i\) give
\[
 J_{\mathrm{Bor}}(P'_i,P'_i,P'_n)=\delta_{in}\lambda_i,
 \qquad
 \lambda_i=(-1)^i
 \frac{\prod_{s=i+1}^{2i+1}(v^s-v^{-s})}{v-v^{-1}},
\]
and \(\operatorname{Ev}_{\kappa}(\lambda_i)\ne0\) for \(i\le d_{\kappa}\).  Expanding a
radical vector in the basis of
Lemma~\ref{p4:lem:specialized-p-prime-basis}, these tests force all its
coefficients through \(d_{\kappa}\) to vanish.  This proves the first equality and
the dimension formula.

\emph{Representation-gauge generators.}
The specialized coefficients of \(P'_i\) in \(V_j\) are invariant under
\(j\mapsto k-j\) for \(i\le d_{\kappa}\), since
\(\{\kappa-r\}=\{r\}\).  Hence every \(V_{k-j}-V_j\) belongs to the radical;
one relation per nontrivial orbit is independent, and their number equals
the radical dimension, proving the second equality.

\emph{Failure of \(S\)- and \(T\)-stability.}
For failure of \(S\)-stability, put \(r_0=V_k-V_0\).  Up to a nonzero
common scalar,
\[
 (Sr_0)_i=\bigl((-1)^i-1\bigr)
 \sin\!\left(\frac{(i+1)\pi}{\kappa}\right).
\]
The orbit containing \(V_1\) gives a nonzero quotient coordinate for odd
\(\kappa\), for even \(\kappa\ge6\), and at \(\kappa=4\); the same formula gives a
nonzero quotient class at \(\kappa=3\).  Thus \(Sr_0\) is not radical.  For
\(T\),
\[
 [T(V_{k-j}-V_j)]=(\theta_{k-j}-\theta_j)[V_j],
 \qquad
 \frac{\theta_{k-j}}{\theta_j}
 =\exp\!\left(\frac{\pi i(\kappa-2-2j)}2\right).
\]
Take \(j=0\), except when \(\kappa\equiv2\pmod4\), where \(j=1\) is an allowed
nontrivial orbit for \(\kappa\ge6\) and the ratio is \(-1\).  The quotient class
is nonzero in every case, so the radical is not \(T\)-stable.
\end{proof}

The resulting quotient records exactly the closed link functionals used to
test the colors.  Its failure to be preserved by $S$ and $T$ is therefore a
failure of torus-state transport, not a failure of the contracted scalar
functional; the latter remains a separate closed observable with its own cap.

\begin{scopefirewall}[Regular-color specialization]
\label{p4:firewall:regular-colour-specialization}
The inclusion proof above does not specialize arbitrary high
\(P'_a\) coefficientwise: its factorial denominator can vanish at a
\(\kappa\)-th root.  It first forms Habiro's integral generic colored-Jones
expression, uses its \(G_h(q)\)-divisibility, and only then specializes.
This is why the other link components are restricted to regular test
colors.
\end{scopefirewall}

\begin{scopefirewall}[State, scalar, and generic targets]
\label{p4:firewall:state-scalar-generic}
The line-valued root-product single-\(S\) state exists.  Its contracted
all-negative Hoste scalar exists and lies in the common Habiro graph.  The
same state does not lie in the strict rootwise/Rees \(c_0\) module, and scalar
Galois transport is not state-natural.  None of these statements constructs
or obstructs the later fraction-field generic stated-skein state: that is a
different carrier with a cap defined only after geometric closure.
\end{scopefirewall}