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.

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.

1. 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.

Theorem 1.1. Comparison isomorphism

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.

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 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.

2. The complete algebraic setup

2.1. 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.

2.2. Ambient variables and the six-root inclusion

A 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.

2.3. 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].

2.4. 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.

3. 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.

Lemma 3.1. Mixed-sided intersection

If $a_i,b_j\in S$, then

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

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 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.

4. 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 ordered as written, not Weyl ordered. It is a unit of $U$.

Lemma 4.1. Exact local identities

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,

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

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 entire generated ambient right ideal is an extension of shape relations.

4.2. 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 $U$.

5. 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.

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.

6. 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.

Corollary 6.1. Convention-adjusted extended trace comparison

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,

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

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.

6.2. 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.

7. Opposite roots and the boundary of the result

7.1. 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 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 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.

7.2. 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.

8. Credit, significance and reproducibility

8.1. 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 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.

8.2. 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.

8.3. Exact verification and its limits

The release includes 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.

References

[CK] Qingjing Chen and Andrew Kricker, On 3d Quantum Trace Maps, arXiv:2606.13268v1, 11 June 2026. 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.

[PP1] Samuel Panitch and Sunghyuk Park, 3d quantum trace map, Algebraic & Geometric Topology 26 (2026), 2255-2314. DOI 10.2140/agt.2026.26.2255. Published primary PDF. Especially Section 5.2, Proposition 5.7, Proposition 5.8, Remark 5.9 and Definition 5.13.

[PP2] Samuel Panitch and Sunghyuk Park, Compatibility of quantum trace and UV-IR maps, arXiv:2509.09100v1, 11 September 2025. Primary record. Especially Theorem 2.20, Remark 2.21 and Definition 2.22 with its opposite-root quotient footnote.

[GY] Stavros Garoufalidis and Tao Yu, A quantum trace map for 3-manifolds, Advances in Mathematics 486 (2026), 110735. DOI 10.1016/j.aim.2025.110735. Published primary PDF. Especially Section 4.10, Theorem 4.25 and Remark 5.1.

[GYI] Stavros Garoufalidis and Tao Yu, The 3d-index of the 3d-skein module via the quantum trace map, arXiv:2406.04918; author revision dated 6 August 2025. Inspected primary revision. Especially Section 3.3, Remark 3.1, Theorem 5.7 and Appendix A.

[PR] Samuel Panitch and Mauricio Romo, On the length conjecture for twist knots, arXiv:2609.02706v1. Primary text. Section 7, Conjecture 7.1 is the larger, unresolved Dehn-filling objective, not a theorem input here.

<!– FPRD AFTERWORD BEGIN –> # Afterword: FPRD Lab’s contribution

An isomorphism between six-root quantum gluing modules

Methodological and contribution addendum. The paper’s mathematical scope and review status remain as stated in the paper.

The Lab’s approach

FPRD means Finitely Presented Recognition Dynamics. Its starting point is to distinguish a finite description, the possibly infinite process it generates, and what an observer can learn from that process. Equal answers need not come from equal internal structures. Conversely, an infinite state space may have a short, exact description and useful coordinates; failure to compress it into finitely many states need not end the investigation.

A small example: balanced parentheses

The grammar S → (S)S | ε presents the language of balanced parentheses. To recognize a prefix, keep the number of unmatched opening parentheses, or a failure state if a closing parenthesis arrived too early. An opening parenthesis increments the count. A closing parenthesis decrements a positive count; at zero it enters failure. Failure persists. At the end, accept exactly when the count is zero.

The prefixes ( and ()( leave the same future task: any suffix that completes one completes the other. The prefix (( leaves a different task: ) completes the first two prefixes but not this one. These future tasks are residuals. There are infinitely many counts, yet a finite set of update rules generates their dynamics. This is a classical automata example, not a discovery of FPRD Lab. It illustrates the research question: which coordinates preserve all relevant futures, and what does updating or reading those coordinates cost?

From viewpoint to proof

In practice, the Lab fixes the observable and the allowed operations, compares presentations of the same problem, and seeks an exact translation or invariant. A failed shortcut is useful when a counterexample identifies the missing coordinate or hypothesis. Small computations test particular uncertainties; a proof must explain why the result holds beyond those tests. Costs, relation sides, boundary conditions, and the direction of each translation remain explicit.

AI-assisted exploration, source comparison, executable checks, and adversarial reconstruction support this work. Their agreement is not automatically independent evidence: shared assumptions must be exposed. A formal development certifies its stated formal theorem, not every interpretation attached to it. Each paper retains its own evidence and review status.

The approach is most useful when a poor representation hides a reusable construction, a compatibility condition, or a sharp obstruction. Outside recognition theory, it can guide the search for a structural reformulation without producing a literal residual dynamical system. The account below distinguishes that methodological influence from the ingredients of the proof.

FPRD’s contribution to this paper

The target was an exact comparison of two presentations. Chen and Kricker had a surjection from a six-root presentation to an image inside a larger quotient. The missing statement was that passing through the larger algebra introduces no extra relations among the shape coordinates.

The Lab recast that issue as an intersection equality: the visible relation module must equal the intersection of the ambient relation module with the shape torus. This keeps the target fixed. Defining the desired image by the smaller presentation would have assumed precisely what needed proof.

The argument combines reversible local normalization with a lattice grading whose zero component is the shape torus. Projection onto that component respects multiplication on both sides by shape elements. Once the ambient relations have the required normalized form, this bimodule property recovers the full intersection. It replaces a search through triangulations or link configurations with a structural statement about the presentation.

Keeping relation sides was essential. These are mixed-sided module quotients, not automatically two-sided algebra quotients. The proof also retains all six roots and the disclosed coefficient ring. A source wording inconsistency is handled by stating the chosen right-tetrahedron/left-edge convention explicitly, rather than silently changing the question.

Inherited ingredients and the boundary of the contribution

Chen–Kricker supply the conjecture, comparison map, and a local calculation refined here. Panitch–Park supply the ambient presentation used to check conventions. Existing trace-comparison results provide the consequences for the named traces. Lattice gradings and projection onto degree zero are established algebraic tools; the contribution is showing that the actual quantum-torus lattice and normalized relation module satisfy their required hypotheses.

Exact ordered-monomial and Weyl-basis calculations check finite algebraic identities by different implementations. The arbitrary-triangulation conclusion comes from the lattice and intersection arguments, not from sampling triangulations. No proof-assistant formalization or external peer review is claimed.

This is a direct example of the Lab’s structural-translation approach, although it is not a language-residual automaton. The isomorphism does not itself prove that either target is nonzero, that a quantum trace is injective, or that the Dehn-filling conjecture is solved. Those are distinct questions with distinct obligations.

Where to inspect the contribution: the paper, Sections 2–6 and 8.3, including the mixed-sided intersection lemma, lattice projection, and exact-verification limits.

<!– FPRD AFTERWORD END –>