CK-degree-zero-descent-publication.tex

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  pdftitle={An isomorphism between six-root quantum gluing modules},
  pdfauthor={FPRD Lab · AI-assisted research manuscript},
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\title{An isomorphism between six-root quantum gluing modules}
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\subtitle{Degree-zero descent and Chen-Kricker Conjecture 5.27}
\author{FPRD Lab · AI-assisted research manuscript}
\date{8 September 2026 · Publication edition 1}

\begin{document}
\maketitle

{
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\hypertarget{abstract}{%
\subsection{Abstract}\label{abstract}}

Two constructions of the three-dimensional quantum trace assign a
quantum gluing module to an ideally triangulated cusped three-manifold.
Chen and Kricker construct a surjective comparison between their
explicit six-root presentation and the Panitch-Park target, which is
defined as an image inside a larger quotient. They conjecture that this
comparison is injective. We prove the missing intersection equality for
the standard mixed-sided presentation used in their comparison, thereby
establishing Conjecture 5.27 of arXiv:2606.13268v1 in that presentation.
The argument combines a reversible local right-normalization, refining a
calculation already in Chen-Kricker, with projection onto the zero
component of a quantum-torus lattice grading. It works over their
coefficient ring without generic-parameter denominators. Their existing
trace-comparison theorems then identify the named extended traces in
their common conventions. Opposite roots are not identified in the
theorem. Neither an original six-root peripheral-action theorem nor
Panitch-Romo's Dehn-filled quantum-trace conjecture is proved.

\textbf{Origin and review status.} This is AI-assisted work, not
independently reviewed mathematics. No external referee, source author,
or institution has approved the proof. The fresh audit reconstructs the
argument and uses a separately implemented exact verifier; it is
independent of the earlier calculation in method, not independent human
expert review. No claim of first priority is made.

\hypertarget{what-is-being-compared}{%
\subsection{1. What is being compared?}\label{what-is-being-compared}}

An ideal triangulation builds a cusped three-manifold from tetrahedra
whose vertices have been removed. Classically, shape coordinates
describe how the tetrahedra glue. Quantum shape coordinates obey scalar
commutation rules instead of commuting. A quantum trace turns a framed
link, considered modulo skein relations, into an element of a module
built from those coordinates.

There are two different-looking ways to construct the target. One
imposes a visible list of vertex, tetrahedron and edge relations
directly on the shape quantum torus. The other first works with a larger
quantum torus attached to face pieces, imposes relations there, and then
takes the image of the shape coordinates. Those procedures need not
agree: a relation using the larger algebra could create a previously
invisible relation among the shape coordinates.

Write \(S\subset U\) for the shape and ambient tori, respectively. If
\(J\subset S\) is the visible relation module and
\(\mathcal R\subset U\) is the ambient relation module, the comparison
has the form

\[
 S/J\longrightarrow \operatorname{im}(S\longrightarrow U/\mathcal R)
                 =S/(S\cap\mathcal R).
 \tag{1}
\]

Showing that the visible relations hold proves
\(J\subset S\cap\mathcal R\), hence surjectivity. Injectivity requires
the reverse inclusion. This is the precise issue in Chen-Kricker {[}CK,
Proposition 5.25, Remark 5.26 and Conjecture 5.27{]}. The result below
computes that full intersection; it does not redefine the image target
by the presentation one hopes to prove.

\hypertarget{thm-main}{%
\subsubsection{Theorem 1.1. Comparison isomorphism}\label{thm-main}}

Let \(Y\) be an oriented cusped three-manifold without actual manifold
boundary, with a finite ideal triangulation \(\mathcal T\), as in {[}CK,
Section 5{]}. Let \(R\) be a commutative coefficient ring with
distinguished units \(t\) and \(v_0\), put \(A=t^2\), and require
\(v_0^2=-A^2\). Use the Laurent face-cone tori and the mixed-sided
ambient relations specified in Section 2 below. Then the
generator-matching comparison of {[}CK, Proposition 5.25{]} is an
isomorphism of \(R\)-modules:

\[
 \mu:Q_{CK}=S/J\overset{\sim}{\longrightarrow}
 Q_{PP}^{\mathrm{ext}}=\operatorname{im}(S\longrightarrow U/\mathcal R).
 \tag{2}
\]

Equivalently, \(S\cap\mathcal R=J\). This proves {[}CK, Conjecture
5.27{]} for the intended right-tetrahedron/left-edge presentation used
in its comparison. The exponent ``ext'' here records that all six roots
per tetrahedron are retained; it is not a change to CK's target.

The quotient in (2) is a module quotient, not generally an algebra
quotient. ``Isomorphism'' does not assert that the quantum trace itself
is injective, that the target is nonzero, or that the resulting
coordinates are independent of triangulation.

\hypertarget{source-reading-convention-that-is-part-of-the-statement}{%
\subsubsection{Source-reading convention that is part of the
statement}\label{source-reading-convention-that-is-part-of-the-statement}}

There is a genuine inconsistency in {[}CK, Definition 5.7(i){]}: the
local tetrahedron ideal is defined on the right, but its global
extension is called a left ideal once. The theorem uses its
\textbf{right-generated} extension. This is the reading consistent with
CK's Definition 5.11, its explicit right-multiplication proof on
pp.~78-79, and Panitch-Park's independent ambient presentation {[}PP1,
Section 5.2 and Remark 5.9{]}. Equation (5.6) is read with that right
ideal. An isolated literal left-ideal reading would change the target,
not merely its notation, and is not covered here. We have not obtained
author confirmation of an erratum.

A second shorthand is less consequential: CK Definition 5.22(ii) lists
the six generators without explicitly listing their inverses. We use the
Laurent quantum torus of its immediately following Remark 5.23 and
equation (5.7), as required by the source comparison itself. Thus
neither issue is silently resolved by changing an algebra in the proof.

\hypertarget{the-complete-algebraic-setup}{%
\subsection{2. The complete algebraic
setup}\label{the-complete-algebraic-setup}}

\hypertarget{quantum-tori-and-weyl-monomials}{%
\subsubsection{2.1. Quantum tori and Weyl
monomials}\label{quantum-tori-and-weyl-monomials}}

For a free lattice \(\Lambda\) and an integral skew-symmetric form
\(B\), define the quantum torus \(R_B[\Lambda]\) to have the \(R\)-basis
\(\{W(a):a\in\Lambda\}\), with

\[
 W(a)W(b)=t^{B(a,b)}W(a+b),\qquad W(0)=1.
 \tag{3}
\]

Bilinearity makes this multiplication associative. Every \(W(a)\) is a
unit, with inverse \(W(-a)\). In coordinate generators \(u_j=W(e_j)\),
the relations are \(u_ju_k=A^{B_{jk}}u_ku_j\). The notation
\([u_1^{a_1}\cdots u_m^{a_m}]\) means \(W(a)\), so

\[
 W(a)=t^{-\sum_{j<k}B_{jk}a_ja_k}
       u_1^{a_1}\cdots u_m^{a_m}.
 \tag{4}
\]

This basis remains a basis if \(B\) has a radical, if \(R\) has zero
divisors, or if \(A\) is specialized to a root of unity. A degenerate
commutation form creates central monomials, not linear identifications
between different monomials. This elementary distinction is important to
the proof.

\hypertarget{ambient-variables-and-the-six-root-inclusion}{%
\subsubsection{2.2. Ambient variables and the six-root
inclusion}\label{ambient-variables-and-the-six-root-inclusion}}

A \emph{bare face} is an occurrence before face pairings are made. Do
not identify its variables with those in the face paired to it. In a
tetrahedron, use the labels

\[
\begin{array}{c|ccc}
h_1&z&z'&y''\\
h_2&z'&z''&y\\
h_3&z''&z&y'\\
h_4&y&y'&y''.
\end{array}
\tag{5}
\]

For each entry \(a\) in row \(h\), introduce an invertible half-edge
variable \(u_{a,h}\). Consecutive entries in every cyclic row of (5)
satisfy \(u_{a,h}u_{b,h}=A u_{b,h}u_{a,h}\). Variables in different rows
commute. These are CK's face-cone tori, written using the cyclic order
that makes the displayed commutators positive.

Let \(U\) be the tensor product of these tori over all bare faces. With
\(N\) tetrahedra, its monomial lattice has rank \(12N\). Regrouping the
tensor factors by paired faces gives exactly the ambient tensor product
of face-suspension modules in {[}CK, equation (5.6){]}. There is no
additional face-identification quotient in \(U\).

Each bare tetrahedron edge \(e\) belongs to two distinct bare face
occurrences \(h,h'\). Define its shape generator by

\[
 X_e=u_{e,h}u_{e,h'}.
 \tag{6}
\]

The factors commute. Let \(S\) be the subtorus generated by all
\(X_e^{\pm1}\). In local calculations, abbreviate these six generators
as \(z,z',z'',y,y',y''\). Opposite edges are different generators. Their
skew matrix, in that order, is

\[
 \Omega_6=\begin{pmatrix}\Omega&\Omega\\\Omega&\Omega\end{pmatrix},
 \qquad
 \Omega=\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix}.
 \tag{7}
\]

For example, \(zz'=Az'z\), while \(z\) and \(y\) commute. Equation (6)
identifies the abstract CK shape torus with this actual subtorus:
different shape exponents use disjoint pairs of half-edge coordinates.
Section 3 proves the stronger degree-zero assertion needed later.

\hypertarget{visible-shape-relations}{%
\subsubsection{2.3. Visible shape
relations}\label{visible-shape-relations}}

For a tetrahedron \(T\), define four vertex monomials and one Lagrangian
polynomial:

\[
\begin{aligned}
v_T&=[zz'z''],&v_{T,1}&=[zy'y''],\\
v_{T,2}&=[yz'y''],&v_{T,3}&=[yy'z''],\\
f_T&=z^2+(z'')^{-2}-1.
\end{aligned}
\tag{8}
\]

All four vertex monomials are central in \(S\). Let \(V_T\) be the right
ideal of \(S\) generated by their differences from \(v_0\), and let
\(L_T=f_TS\). Here ``right ideal'' means that the relation is on the
left of its arbitrary multiplier.

For every manifold edge \(e\), let \(e_1,\ldots,e_k\) be its bare-edge
occurrences and set

\[
 H_e=[X_{e_1}\cdots X_{e_k}],\qquad b_e=H_e+A^2.
 \tag{9}
\]

Let \(E=\sum_e Sb_e\), a left ideal. Edge relations are therefore on the
right of arbitrary multipliers. The CK relation module is

\[
 J=\sum_T(V_T+L_T)+E.
 \tag{10}
\]

All edges are kept. No independent-row choice, geometric solution, rank
hypothesis or regularity assumption is used. The presentation agrees
with {[}CK, Definitions 3.1 and 3.7, Remark 3.8{]}.

\hypertarget{the-raw-ambient-list}{%
\subsubsection{2.4. The raw ambient list}\label{the-raw-ambient-list}}

Put \(c=v_0^{-1}\). For edges \(a,b\) sharing the unique face \(h\),
write

\[
 G_{ab}=c[u_{a,h}u_{b,h}].
 \tag{11}
\]

These are the triangle-generator images in {[}CK, Lemma 5.18{]}. For the
vertex incident to \(z,z',z''\), the two ambient relations are

\[
 k=G_{zz'}G_{z'z''}G_{z''z}-v_0^{-2},
 \tag{12}
\]

\[
\begin{split}
 r={}&[G_{zy'}^{-1}G_{y'z''}]\\
     &-[G_{z''y}^{-1}G_{yz'}]G_{zz'}
      -G_{z'z''}^{-1}[G_{z'y''}^{-1}G_{y''z}].
\end{split}
\tag{13}
\]

There is one pair for each of the four vertices, transported with the
oriented tetrahedral labeling. Denote them \(k_{T,w},r_{T,w}\).

For an edge, the bigon-generator images multiply to an ambient monomial
\(B_e\); each image is the product of the corresponding half-edge
variables on the two sides of a paired face, with coefficient 1. Include
both \(B_e+A^2\) and \(B_e^{-1}+A^{-2}\). The full relation module is

\[
\begin{split}
\mathcal R={}&\sum_{T,w}\bigl(k_{T,w}U+r_{T,w}U\bigr)\\
 &+\sum_e\bigl(U(B_e+A^2)+U(B_e^{-1}+A^{-2})\bigr).
\end{split}
\tag{14}
\]

This is the right/left extension of the entire source generator lists,
not just their intersection with the shape torus. If a right ideal is
first formed in the triangle algebra and then its image is extended to
\(U\), the result is precisely the right ideal generated by the images
of its generators; the image of every source multiplier is already an
allowed \(U\)-multiplier. The analogous observation applies on the left
to the bigon algebra. Thus (14) is the ambient denominator of (1) with
the source-reading convention stated above.

\hypertarget{a-projection-that-detects-every-hidden-relation}{%
\subsection{3. A projection that detects every hidden
relation}\label{a-projection-that-detects-every-hidden-relation}}

Let \(a_{e,h}\) denote the lattice basis element of a half-edge
variable. The exponent lattice of \(S\) inside that of \(U\) is

\[
 L=\bigoplus_e\mathbb Z(a_{e,h}+a_{e,h'}).
 \tag{15}
\]

Choose either ordering of each pair \((h,h')\), and define

\[
 d:\Lambda\longrightarrow\mathbb Z^{6N},\qquad
 d(a)_e=a_{e,h}-a_{e,h'}.
 \tag{16}
\]

Because the coordinate pairs are disjoint, \(\ker d=L\), and \(d\) is
onto. In particular, \(\Lambda/L\) is free; no lattice saturation or
fractional monomial has been assumed.

Group the basis monomials of \(U\) by their degree in \(\Lambda/L\).
Equation (3) gives

\[
 U=\bigoplus_{g\in\Lambda/L}U_g,\qquad U_gU_h\subset U_{g+h},
 \qquad U_0=S.
 \tag{17}
\]

Define \(P_0:U\to S\) by retaining only degree-zero monomials. It is an
\(S\)-bimodule retraction:

\[
 P_0(su)=sP_0(u),\qquad P_0(us)=P_0(u)s
 \quad(s\in S,u\in U).
 \tag{18}
\]

Multiplication by \(s\) does not change degree. No averaging or division
is involved.

\hypertarget{lem-intersection}{%
\subsubsection{Lemma 3.1. Mixed-sided
intersection}\label{lem-intersection}}

If \(a_i,b_j\in S\), then

\[
 S\cap\left(\sum_i a_iU+\sum_j Ub_j\right)
       =\sum_i a_iS+\sum_j Sb_j.
 \tag{19}
\]

\textbf{Proof.} One inclusion is immediate. Conversely, write an element
\(x\) of the left side as a finite sum \(x=\sum_i a_iu_i+\sum_jv_jb_j\).
Since \(x\in S\), apply \(P_0\) and (18):

\[
 x=\sum_i a_iP_0(u_i)+\sum_jP_0(v_j)b_j.
\]

This belongs to the right side. No summand has been moved across its
multiplier. \(\square\)

For a simple picture of the projection, take two half-edge variables
\(u,v\) paired into \(X=uv\). Then \(u^av^b\) has degree \(a-b\), so
\(P_0(u^av^b)\) vanishes unless \(a=b\). In particular,
\(P_0(u)=P_0(u^{-1})=0\), but \(P_0(uu^{-1})=1\). The retraction is
\textbf{not} an algebra homomorphism. The bimodule property, which is
weaker, is exactly what (19) needs.

We cannot yet apply the lemma to (14): the raw trinomial \(r\) is not in
\(S\). A reversible change of generators is essential.

\hypertarget{reversible-right-normalization}{%
\subsection{4. Reversible
right-normalization}\label{reversible-right-normalization}}

All calculations in this section take place in the twelve-variable
ambient torus of one tetrahedron, before imposing any relation. Write

\[
 v=[zz'z''],\qquad f=z^2+(z'')^{-2}-1,
 \qquad h=u_{z'',h_3}^{-1}u_{z,h_3}.
 \tag{20}
\]

The last product is \textbf{ordered} as written, not Weyl ordered. It is
a unit of \(U\).

\hypertarget{lem-normalization}{%
\subsubsection{Lemma 4.1. Exact local
identities}\label{lem-normalization}}

For (12)-(13), one has

\[
 k=c^3(v-v_0)
 \tag{21}
\]

and

\[
 rh=-t^{-1}f+t^{-1}(v-v_0)
        \bigl(v^{-1}z^2-v_0^{-1}(z'')^{-2}\bigr).
 \tag{22}
\]

Consequently,

\[
 kU+rU=(v-v_0)U+fU.
 \tag{23}
\]

\textbf{Calculation.} The three factors of (12) lie in three different
face factors. Collecting their half-edge exponents gives
\(c^3v-c^2=c^3(v-v_0)\). For the trinomial, the commutation rules in (5)
give

\[
 rh=t^{-1}-c(z'')^{-1}z'z-c^{-1}(z'')^{-1}(z')^{-1}z.
 \tag{24}
\]

The remaining products of shape generators satisfy

\[
 (z'')^{-1}z'z=t^{-1}v(z'')^{-2},\qquad
 (z'')^{-1}(z')^{-1}z=t^{-1}v^{-1}z^2.
 \tag{25}
\]

Substitute (25) into (24) and expand the right side of (22). Its
coefficient of \((z'')^{-2}\) is \(-t^{-1}v/v_0\), its coefficient of
\(z^2\) is \(-t^{-1}v_0/v\), and its constant is \(t^{-1}\), agreeing
exactly. Equations (24)-(25) also provide a direct three-term
calculation that can be checked by hand.

Finally, \(rU=(rh)U\). Since \(c\) and \(t\) are units, (21)-(22) give
both inclusions in (23). More explicitly, if

\[
 C_v=(v-v_0)\bigl(v^{-1}z^2-v_0^{-1}(z'')^{-2}\bigr),
\]

then \(f=-t\,rh+C_v\) and \(r=t^{-1}(-f+C_v)h^{-1}\). Every multiplying
factor is on the right of its relation, as required. \(\square\)

The local multiplication through (24) is already present in {[}CK,
Proposition 5.25, p.~79{]}, where it proves the visible relations
vanish. The point of (21)-(23) is to retain the exact correction term
and reverse the argument: the \textbf{entire generated ambient right
ideal} is an extension of shape relations.

\hypertarget{why-this-handles-the-whole-tetrahedron-list}{%
\subsubsection{4.2. Why this handles the whole tetrahedron
list}\label{why-this-handles-the-whole-tetrahedron-list}}

Label the tetrahedron's vertices \(0,1,2,3\), with edges

\[
 z=01,\ z'=02,\ z''=03,\ y=23,\ y'=13,\ y''=12.
 \tag{26}
\]

An even permutation of the four vertices permutes half-edge occurrences
and preserves their skew form. Hence it defines an automorphism of the
ambient quantum torus. The identity and the three double transpositions
take vertex 0 to all four vertices. Applying them to (21)-(23) supplies
the four required generator-pair equalities. No orientation-reversing
symmetry, inversion of \(A\), or unrecorded scalar is used.

If the local cyclic choice at a vertex is changed, the resulting
Lagrangian has the same right ideal modulo vertex relations. For
example, put \(f'=(z')^2+z^{-2}-1\). Direct calculation gives

\[
 fz^{-2}+f'=(v^2-v_0^2)[z^{-2}(z'')^{-2}].
 \tag{27}
\]

This uses \(v_0^2=-A^2\). The right side belongs to the vertex ideal
because \(v\) is central in \(S\) and \(v^2-v_0^2=(v-v_0)(v+v_0)\).
Cyclic permutations give the other two relations. Since \(z^{-2}\) is a
unit, (27) works in both directions.

To compare opposite choices, put

\[
 r_1=yz^{-1},\qquad r_2=y'(z')^{-1},\qquad r_3=y''(z'')^{-1}.
 \tag{28}
\]

These are central in \(S\). Dividing the other vertex monomials by \(v\)
gives \(r_2r_3,r_1r_3,r_1r_2\). Thus, in the algebra quotient by the
central vertex relations,

\[
 r_2r_3=r_1r_3=r_1r_2=1.
 \tag{29}
\]

In particular \(r_1=r_2=r_3\) and \(r_1^2=1\), without division by 2.
For an explicit ideal witness, set \(q_1=r_2r_3-1\), \(q_2=r_1r_3-1\),
\(q_3=r_1r_2-1\). Then

\[
 r_1-r_2=(q_2-q_1)r_3^{-1},\qquad
 r_1^2-1=r_1(q_2-q_1)r_3^{-1}+q_3.
 \tag{30}
\]

Also \(q_j=v^{-1}(v_{T,j}-v_T)\) with the evident ordering of the three
vertices. Hence opposite squares agree modulo the vertex ideal. This
proves that the four normalized vertex Lagrangians, with any allowed
cyclic choices, add no relation beyond \(V_T+L_T\). We have established

\[
 \sum_w(k_{T,w}U+r_{T,w}U)=(V_T+L_T)U.
 \tag{31}
\]

Only centrality inside \(S\) was used. It would be incorrect to commute
\(v\) or the ratios (28) through arbitrary half-edge variables in \(U\).

\hypertarget{edge-relations-and-the-proof-of-theorem-1.1}{%
\subsection{5. Edge relations and the proof of Theorem
1.1}\label{edge-relations-and-the-proof-of-theorem-1.1}}

Around a manifold edge, the bigon images are products of the two
corresponding half-edge variables in each paired face. Such images
commute. For two different bigons in the same face suspension, the
commutation exponent on one side of the face is the negative of the
exponent on the other side; their sum is zero. Bigons in different face
suspensions commute by the tensor-product presentation.

This argument still works if an edge visits a face suspension more than
once. A bare half-edge occurrence is a distinct coordinate; revisiting
the same paired face can use other commuting bigon generators but does
not duplicate or merge its coordinates. The product uses exactly the
half-edge occurrences paired in (6). Consequently,

\[
 B_e=H_e=[X_{e_1}\cdots X_{e_k}].
 \tag{32}
\]

There is no residual Weyl scalar: the factors have zero mutual pairings.
This is also the edge-image computation in {[}CK, Proposition 5.25,
p.~78{]}. The negative-power generator is redundant on the specified
side:

\[
 H_e^{-1}+A^{-2}=A^{-2}H_e^{-1}(H_e+A^2).
 \tag{33}
\]

It follows from (14), (31), (32) and (33) that

\[
 \mathcal R=\sum_T(V_T+L_T)U+\sum_e U(H_e+A^2).
 \tag{34}
\]

The equality is crucial: proving only that the right side lies in the
left would recover the known surjection but would not prove injectivity.

\textbf{Proof of Theorem 1.1.} Every displayed generator on the right of
(34) belongs to \(S=U_0\). Apply Lemma 3.1 and the definition (10) to
obtain

\[
 S\cap\mathcal R=\sum_T(V_T+L_T)+\sum_e S(H_e+A^2)=J.
 \tag{35}
\]

The actual comparison (1) is therefore bijective. \(\square\)

An explicit inverse sends the class of a shape representative \(s\) in
the ambient image to its class in \(S/J\). More generally, if \(u\in U\)
represents an element of that image, send it to \([P_0(u)]\). Indeed,
\(u-s\in\mathcal R\) for some \(s\in S\), and (34) implies
\(P_0(u-s)\in J\), so this formula is independent of all
representatives.

The theorem uses only \(t^{\pm1}\) and \(v_0^{\pm1}\). It therefore
holds directly in the stated ring, not by an unsupported specialization
argument from a generic field. It remains valid if some or all edge
relations are omitted. That is an algebraic observation; no new
boundary-marked trace theorem is being inferred from it. Multiple
pairings or self-pairings of distinct bare face occurrences do not
change the lattice argument.

\hypertarget{consequences-for-the-named-traces}{%
\subsection{6. Consequences for the named
traces}\label{consequences-for-the-named-traces}}

The new theorem concerns targets. The trace identity comes from an
additional, already established input. In their common conventions,
{[}CK, Theorems 4.9 and 5.34{]} prove

\[
 \tau_{GY}^{CK}=\tau_{CK},\qquad
 \tau_{PP}^{CK}=\mu\circ\tau_{CK}.
 \tag{36}
\]

Here the superscript records CK's orientation, labeling, product and
state conventions. In particular their GY trace is the extended six-root
version, not the opposite-root-identified target silently read in
unchanged coordinates.

\hypertarget{cor-traces}{%
\subsubsection{Corollary 6.1. Convention-adjusted extended trace
comparison}\label{cor-traces}}

With the hypotheses of Theorem 1.1 and the trace conventions of (36),
the isomorphism \(\mu\) intertwines the named extended GY and normalized
PP traces. Equivalently,

\[
 \mu^{-1}\tau_{PP}^{CK}=\tau_{GY}^{CK}.
 \tag{37}
\]

\textbf{Proof.} Substitute the first identity of (36) into the second,
then use Theorem 1.1. \(\square\)

This corollary credits CK for the named-trace theorem; matching algebra
generators alone would not prove it. We have checked the relevant
theorem statements and their conventions, not independently re-proved
the full geometric constructions of CK, GY or PP.

\hypertarget{normalization-and-the-original-pp-branch}{%
\subsubsection{6.2. Normalization and the original PP
branch}\label{normalization-and-the-original-pp-branch}}

The CK triangle and bigon constants are \((c_T,c_B)=(v_0^{-1},1)\). Over
\(k=\mathbb Q(i,t)\), choose \(v_0=iA\), so this is \((-iA^{-1},1)\).
With PP2's \(q=-A^2\), it is the first normalized branch
\((q^{-1/2},1)\) in {[}PP2, Theorem 2.20 and Remark 2.21{]}, not the raw
PP1 constants \((A^{-1},-i)\). The comparison (37) must not be
advertised as an unexplained equality of raw coordinate formulas.

There is nevertheless an explicit original-to-normalized transport. Give
the dual four-regular multigraph of \(\mathcal T\) an Eulerian
orientation, treating the two ends of a loop as separate incidences.
Every tetrahedron has two incoming and two outgoing faces. Assign
\(\tau_h=1\) to outgoing face occurrences and \(\tau_h=-1\) to incoming
ones. Paired occurrences have opposite signs.

Over the auxiliary extension \(k(s)\), \(s^2=-i\), scale every original
PP half-edge generator on \(h\) by \(s\tau_h\). The triangle and bigon
coefficients become

\[
 A^{-1}(s\tau_h)^2=-iA^{-1},\qquad
 (-i)(s\tau_h)(s\tau_{\bar h})=1.
 \tag{38}
\]

These are identities on the face trace's defining bimodule generators,
so they conjugate its map. Tensoring and passing through the same
splitting construction conjugates the global named trace as well. The
induced shape map already has coefficients in \(k\):

\[
 \Phi(x_e)=i\epsilon_eX_e,\qquad
 \epsilon_e=-\tau_h\tau_{h'}\in\{1,-1\}.
 \tag{39}
\]

The product of the four \(\tau_h\)'s at a tetrahedron is 1. Thus
opposite edges have the same \(\epsilon_e\), the product around a
tetrahedral vertex is -1, and the product around a manifold edge is 1.
The original vertex value \(-A\) and edge value \(-i^kA^2\) transform
correctly; the original Lagrangian transforms to the negative of the
corresponding normalized one. Therefore \(\Phi\) induces an isomorphism
of original and normalized six-root targets, and

\[
 \Phi\tau_{PP1}=\tau_{PP}^{CK}.
 \tag{40}
\]

The auxiliary extension is faithfully flat, so this identity descends to
\(k\): both the shape map and the trace images are defined there. We do
not assume that \(\sqrt{-i}\) lies in \(k\). This transport uses an
Eulerian choice and is not claimed to be a canonical or Pachner-natural
identification of all target elements. It is ancillary to Theorem 1.1,
whose integral coefficient-ring statement needs no such choice or
extension.

\hypertarget{opposite-roots-and-the-boundary-of-the-result}{%
\subsection{7. Opposite roots and the boundary of the
result}\label{opposite-roots-and-the-boundary-of-the-result}}

\hypertarget{the-identified-target-is-a-further-quotient}{%
\subsubsection{7.1. The identified target is a further
quotient}\label{the-identified-target-is-a-further-quotient}}

Equations (28)-(30) leave one central involution \(\sigma_T\) per
tetrahedron:

\[
 y=\sigma_Tz,\quad y'=\sigma_Tz',\quad y''=\sigma_Tz'',
 \quad\sigma_T^2=1.
 \tag{41}
\]

These central shape operators preserve the mixed-sided relation module,
so they act on both targets of (2). Since \(\mu\) matches all six roots,
quotienting both sides by \(\sigma_T-1\) gives an isomorphism of the
identified targets and intertwines the quotiented traces. This quotient
is taken \textbf{on the shape modules}, not as a two-sided quotient of
the full half-edge torus.

One cannot reverse that step. Over \(k\), the local module before global
edge relations has nonzero plus and minus sectors. To see this, first
impose only the central vertex relations. The resulting algebra has an
explicit presentation

\[
 k\langle z^{\pm1},(z'')^{\pm1}\mid zz''=A^{-1}z''z\rangle
 \otimes_k k[\sigma]/(\sigma^2-1).
 \tag{42}
\]

Indeed, the eliminated generator is \(z'=t\,v_0z^{-1}(z'')^{-1}\), and
the opposite generators are given by (41); substitution verifies both
directions of this presentation. The remaining right relation is
\(f=z^2+(z'')^{-2}-1\), independent of \(\sigma\).

In a quantum torus over a field, the Newton polytope of a nonzero
product is the Minkowski sum of the factor polytopes. For any exposing
linear functional, the unique extreme monomial of each factor
contributes a nonzero extreme product coefficient, so the extremal terms
cannot cancel. Thus a Laurent polynomial with more than one exponent in
its support cannot be a unit or have a right inverse. The right ideal
\(fS\) in the rank-two factor of (42) is proper. Each sign sector is
therefore nonzero. In the minus sector, the class of \(z-y=2z\) is
nonzero: otherwise right multiplication by \(z^{-1}/2\) would put 1 in
that proper ideal.

This is a local obstruction to a \textbf{shape-matching} inverse of
opposite-root identification. It does not exclude arbitrary linear
splittings and does not assert that every sign sector survives the
global edge quotient.

\hypertarget{exact-peripheral-factors-do-not-identify-named-insertions}{%
\subsubsection{7.2. Exact peripheral factors do not identify named
insertions}\label{exact-peripheral-factors-do-not-identify-named-insertions}}

For a six-root exponent vector
\(a=(a_z,a_{z'},a_{z''},b_y,b_{y'},b_{y''})\), set

\[
 r=a_z+b_y,\quad s=a_{z'}+b_{y'},\quad t_*=a_{z''}+b_{y''},
 \quad\beta=b_y+b_{y'}+b_{y''}.
\]

In the central-vertex quotient, Weyl elimination gives

\[
 W_6(a)=\sigma_T^{\beta}v_0^s
 A^{(r-s)(t_*-s)/2}z^{r-s}(z'')^{t_*-s}.
 \tag{43}
\]

This follows by collapsing opposite exponents and extracting the central
Weyl monomial \([zz'z'']^s=v_0^s\). Formula (4) supplies the final
ordered factor. The symbol \(t_*\) is an exponent coordinate, distinct
from the coefficient-ring unit \(t=A^{1/2}\).

For example, replacing the row for \(z\) by the row for \(y\) leaves
\((r,s,t_*)\) unchanged but multiplies the result by \(\sigma_T\). Thus
an identified peripheral formula cannot detect every six-root factor.
Over several tetrahedra the invisible factor can be a product of the
\(\sigma_T\)'s.

If two candidate peripheral rows \(a,b\) have skew pairing \(2\kappa\),
the precise Weyl factor is

\[
 W(a+b)=A^{-\kappa}W(a)W(b).
 \tag{44}
\]

Right-multiplication operators reverse composition: \(R_MR_L=R_{LM}\)
when \(R_M(x)=xM\). These are algebra identities. They do not establish
that the chosen rows represent the actual named meridian, longitude and
sum-of-slopes insertions.

The theorem on peripheral insertions in {[}GYI, Theorem 5.7{]} uses a
target in which opposite edges have the same variables {[}GYI, Section
3.3{]}. To import it into the original six-root PP setting requires an
additive signed lift and an exact translation of collar, state, product
and label conventions. Neither is supplied by the isomorphism of targets
alone. The new theorem removes the CK comparison-kernel obstruction; it
does not remove this separate peripheral obstruction.

In particular, {[}PR, Conjecture 7.1{]} requires a nontrivial compatible
Dehn-filled quotient. Theorem 1.1 proves no such quotient or its
nonvanishing. That conjecture remains open in this work.

\hypertarget{credit-significance-and-reproducibility}{%
\subsection{8. Credit, significance and
reproducibility}\label{credit-significance-and-reproducibility}}

\hypertarget{what-is-new-and-how-large-is-it}{%
\subsubsection{8.1. What is new, and how large is
it?}\label{what-is-new-and-how-large-is-it}}

The quantum trace constructions belong to Garoufalidis-Yu and
Panitch-Park. The third construction, comparison map, surjectivity and
trace intertwining belong to Chen-Kricker. The key local multiplication
is substantially already in CK. Projection to graded components is a
standard algebraic method; indeed PP1's Proposition 5.7 uses a related
parity-grading intersection argument for a different image.

The contribution here is the reversible equal-right-ideal formulation of
the full tetrahedron list, combined with the specific paired-half-edge
grading that proves the missing ambient intersection equality for the
\textbf{six-root shape image}. Neither the desired equality nor
comparison injectivity is assumed at an intermediate step.

Under the FPRD publication calibration, this meets the ``recognized open
problem'' criterion because it addresses the precise assertion of a
numbered conjecture in a primary mathematical paper. It is a focused,
potentially publishable result in quantum topology: it completes a
comparison between target modules. It is not evidence of a major
breakthrough in Dehn filling, quantum invariants or geometric topology
as a whole. Its recent date and short proof neither disqualify the
result nor inflate its magnitude. Internal completion and external
acceptance remain distinct.

\hypertarget{bounded-priority-check}{%
\subsubsection{8.2. Bounded priority
check}\label{bounded-priority-check}}

A fresh check on 8 September 2026 inspected CK and PP2 submission
histories, the published PP1 and GY papers, the current author-hosted
GY-index revision, the latest relevant Panitch-Romo primary paper, and
the research lists of Samuel Panitch and Sunghyuk Park. Targeted
primary-domain searches looked for the title, conjecture number,
comparison injectivity and quantum-gluing grading arguments. No
primary-source resolution of CK Conjecture 5.27 was located. The current
CK record exposed only v1 and still states the assertion as a
conjecture.

This is a bounded check, not proof of novelty or first discovery.
Unindexed papers, forthcoming revisions, theses, talks and private work
may have been missed. The source-and-priority ledger records the scope
and URLs. No author contact was made.

\hypertarget{exact-verification-and-its-limits}{%
\subsubsection{8.3. Exact verification and its
limits}\label{exact-verification-and-its-limits}}

The release includes \texttt{verify-ck-pbw.py}, a new standard-library
Python verifier using ordinary ordered monomials and explicit reordering
factors. It does not import the earlier ATTACK verifier, which instead
multiplies in a Weyl basis. The new verifier checks the universal ring
\(\mathbb Z[i,t^{\pm1}]\); writing \(i=v_0/A\) shows this is the
universal ring with \(v_0^2=-A^2\). Its identities therefore map to
every ring in Theorem 1.1.

It checks the base calculation and its reverse witnesses, all twelve
orientation-preserving tetrahedral symmetries, cyclic right-ideal
witnesses, central-ratio eliminations, degree-zero behavior, paired-face
edge commutator cancellation, the inverse-edge relation, and a
deliberate changed-sidedness countercheck. It uses no floating-point
approximations and searches no manifolds, triangulations, slopes or
skein configurations.

These computations certify the finite algebraic identities. The proof
for arbitrary triangulations is the lattice argument and Lemma 3.1, not
a finite sample. The source trace theorems remain cited mathematical
dependencies. There is no Lean or other proof-assistant formalization,
no exhaustive priority certification, and no external peer review.

\hypertarget{references}{%
\subsection{References}\label{references}}

\textbf{{[}CK{]}} Qingjing Chen and Andrew Kricker, \emph{On 3d Quantum
Trace Maps}, arXiv:2606.13268v1, 11 June 2026.
\href{https://arxiv.org/html/2606.13268v1}{Primary text}. Especially
Definitions 3.1, 3.7, 5.7, 5.11, 5.17 and 5.22; Lemma 5.18; equation
(5.6); Proposition 5.25; Remark 5.26; Conjecture 5.27; Theorems 4.9 and
5.34.

\textbf{{[}PP1{]}} Samuel Panitch and Sunghyuk Park, \emph{3d quantum
trace map}, Algebraic \& Geometric Topology 26 (2026), 2255-2314. DOI
10.2140/agt.2026.26.2255.
\href{https://msp.org/agt/2026/26-6/agt-v26-n6-p09-p.pdf}{Published
primary PDF}. Especially Section 5.2, Proposition 5.7, Proposition 5.8,
Remark 5.9 and Definition 5.13.

\textbf{{[}PP2{]}} Samuel Panitch and Sunghyuk Park, \emph{Compatibility
of quantum trace and UV-IR maps}, arXiv:2509.09100v1, 11 September 2025.
\href{https://arxiv.org/abs/2509.09100v1}{Primary record}. Especially
Theorem 2.20, Remark 2.21 and Definition 2.22 with its opposite-root
quotient footnote.

\textbf{{[}GY{]}} Stavros Garoufalidis and Tao Yu, \emph{A quantum trace
map for 3-manifolds}, Advances in Mathematics 486 (2026), 110735. DOI
10.1016/j.aim.2025.110735.
\href{https://people.mpim-bonn.mpg.de/stavros/publications/printed/quantum_trace_map.pdf}{Published
primary PDF}. Especially Section 4.10, Theorem 4.25 and Remark 5.1.

\textbf{{[}GYI{]}} Stavros Garoufalidis and Tao Yu, \emph{The 3d-index
of the 3d-skein module via the quantum trace map}, arXiv:2406.04918;
author revision dated 6 August 2025.
\href{https://people.mpim-bonn.mpg.de/stavros/publications/3d-index-skein-module.pdf}{Inspected
primary revision}. Especially Section 3.3, Remark 3.1, Theorem 5.7 and
Appendix A.

\textbf{{[}PR{]}} Samuel Panitch and Mauricio Romo, \emph{On the length
conjecture for twist knots}, arXiv:2609.02706v1.
\href{https://arxiv.org/html/2609.02706v1}{Primary text}. Section 7,
Conjecture 7.1 is the larger, unresolved Dehn-filling objective, not a
theorem input here.

\input{fprd-afterword.tex}
\end{document}