Degree-zero descent for six-root quantum gluing modules

FPRD Lab candidate-proof note — 8 September 2026

Status and exact claim

This note proves the injectivity assertion in Chen–Kricker, Conjecture 5.27, arXiv:2606.13268v1, for the quantum gluing modules defined in that paper. The map is the actual generator-matching map into the ambient image Panitch–Park target, not a replacement target defined by the desired presentation. The proof identifies the full intersection of the ambient relation module with the six-root shape subtorus.

The argument has two parts: an exact local right normalization of the tetrahedron relations, and projection onto the zero component of a lattice grading. It uses neither triangulation enumeration nor specializations, flatness, positivity, or a conjectural injectivity input.

Combining this result with CK’s already proved named-trace comparison gives an isomorphism intertwining their convention-adjusted extended GY trace and their normalized PP trace. Opposite-root identification can then be performed on both sides. A separate explicit Eulerian scaling transports this comparison to the original PP constants.

This does not prove the original six-root extension of GY-index Theorem 5.7, nor Panitch–Romo Conjecture 7.1. We give exact peripheral Weyl and central-factor formulas, but distinguish them from the still unproved assertion that specified six-root lifts are the named peripheral insertion operators. In particular, the theorem below must not be used to silently combine the identified GY boundary theorem with CK’s extended comparison.

This is a candidate-proof packet ready for a PUBLISH-mode preparation and review pass, not an already reviewed Publication. External novelty and priority remain unassessed. A limited fresh primary-literature check is recorded in the source ledger.

1. Sourced inputs and notation

CK means Chen–Kricker, On 3d Quantum Trace Maps, arXiv:2606.13268v1. We use their Definitions 3.1, 3.7, 5.7, 5.17, and 5.22, equation (5.6), and Lemma 5.18 as definitions and presentation inputs. Their Proposition 5.25 supplies the surjective comparison; their Theorems 4.9 and 5.34 supply named-trace compatibility. Their Remark 5.26 explicitly leaves the reverse inclusion of relation modules unresolved.

PP1 is Panitch–Park, 3d quantum trace map, Algebraic & Geometric Topology 26 (2026), 2255–2314. Section 5.2 and Proposition 5.8 independently confirm the complete ambient relation module and its sidedness. PP2 is Panitch–Park, arXiv:2509.09100v1, Theorem 2.20, Remark 2.21, and Definition 2.22.

Let the coefficient ring (R) contain units (A^{1/2}) and (v_0), with

$$v_0^2=-A^2, \qquad c=v_0^{-1}. \tag{1.1}$$

The comparison proof uses no other denominators. For the sign-sector and generic peripheral discussion, use (k=Q(i,A^{1/2})), with

$$v_0=iA, \qquad c=-iA^{-1}. \tag{1.2}$$

For a cusped manifold with an ideal triangulation $\mathcal T$, put

$$U=\bigotimes_{f\in\mathcal T^{(2)}}\mathbf S f =\bigotimes_{h\in\mathbf f(\mathcal T)}\mathbf C h. \tag{1.3}$$

Here $h$ is a bare face occurrence, not just an identified face. There are three quantum-torus generators (u_{e,h}) for each occurrence. Each bare tetrahedral edge (e) belongs to exactly two face occurrences (h,h’); its six-root shape generator is

$$X_e=u_{e,h}u_{e,h'}. \tag{1.4}$$

These two factors commute because they belong to different tensor factors. Let (SU) be the subtorus generated by all (X_e^{}). No opposite roots are identified in (S).

In CK’s common conventions, write the ambient relation module as

$$\mathcal R=\sum_T\rho(\widetilde I_T)U +U\eta(\widetilde I_E). \tag{1.5}$$

The tetrahedron relations generate right ideals; edge relations generate a left ideal. This means tetrahedron relations occur on the left of arbitrary multipliers and edge relations occur on the right. The actual PP target is

$$Q_{PP}^{\rm ext}=\operatorname{im}(S\longrightarrow U/\mathcal R) =S/(S\cap\mathcal R). \tag{1.6}$$

The CK target is (Q_{CK}=S/J), where

$$J=\sum_T(V_T+L_T)+E; \tag{1.7}$$

(V_T,L_T) are the right ideals in (S) generated by tetrahedral vertex and Lagrangian relations, and (E) is the left ideal generated by all edge relations. We retain all edges; independence of edge rows is not needed.

2. The zero-degree retraction

The monomial lattice of (U) is the free lattice

$$\Lambda=\bigoplus_{(e,h)}\mathbb Z\,a_{e,h}.$$

The shape lattice is

$$L=\bigoplus_e\mathbb Z(a_{e,h}+a_{e,h'}). \tag{2.1}$$

The pairs of coordinates in (2.1) are disjoint. In particular the map from the abstract six-root lattice into $\Lambda$ is injective. Choose an ordering of each pair and define

$$d_e(a)=a_{e,h}-a_{e,h'}. \tag{2.2}$$

Then $\ker d=L$ and (/LZ^{6N}). The orderings only change signs of grading coordinates, not the degree-zero part. The Weyl-monomial basis of a quantum torus therefore gives a direct sum

$$U=\bigoplus_{g\in\Lambda/L} U_g, \qquad U_0=S. \tag{2.3}$$

The projection (P_0:UU_0=S) is an (S)-bimodule map:

$$P_0(su)=sP_0(u),\qquad P_0(us)=P_0(u)s. \tag{2.4}$$

It is generally not an algebra homomorphism. No averaging, division by a group order, or choice of a splitting of the quantum torus is involved.

Lemma 2.1 — mixed-sided intersection

If (r_i,e_jS), then

$$S\cap\left(\sum_i r_iU+\sum_j Ue_j\right) =\sum_i r_iS+\sum_j Se_j. \tag{2.5}$$

Proof. The right side is visibly contained in the left. If (xS) belongs to the ambient sum, write (x=_i r_i u_i+_j v_j e_j). Applying (P_0) and using (2.4) gives (x=_i r_iP_0(u_i)+_jP_0(v_j)e_j), as required. ∎

One cannot apply this lemma to the unnormalized tetrahedron trinomials: those are not themselves in (S). The next calculation is the necessary bridge.

3. Exact tetrahedron normalization

Use CK’s labels (z,z’,z’‘,y,y’,y’’), with face occurrences

$$\begin{array}{c|ccc} h_1&z&z'&y''\\ h_2&z'&z''&y\\ h_3&z''&z&y'\\ h_4&y&y'&y''. \end{array} \tag{3.1}$$

In each displayed cyclic row, consecutive half-edge variables satisfy (u_a u_b=A u_bu_a). This yields the shape skew matrix

$$\Omega_6=\begin{pmatrix}\Omega&\Omega\\\Omega&\Omega\end{pmatrix}, \qquad \Omega=\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix}. \tag{3.2}$$

For exponent vectors (a,b), our precise Weyl convention is

$$W(a)W(b)=A^{\Omega(a,b)/2}W(a+b). \tag{3.3}$$

Write

$$v=[zz'z''],\qquad f=z^2+(z'')^{-2}-1. \tag{3.4}$$

For two edges (a,b) sharing face (h), put

$$G_{ab}=c[u_{a,h}u_{b,h}]. \tag{3.5}$$

These are exactly the images under CK Lemma 5.18 of the triangle generators. The image of the vertex relation at ((z,z’,z’’)) is

$$G_{zz'}G_{z'z''}G_{z''z}-v_0^{-2} =c^3(v-v_0). \tag{3.6}$$

The image of the other relation at that vertex is

$$r=[G_{zy'}^{-1}G_{y'z''}] -[G_{z''y}^{-1}G_{yz'}]G_{zz'} -G_{z'z''}^{-1}[G_{z'y''}^{-1}G_{y''z}]. \tag{3.7}$$

Let the right normalizing unit be

$$h=u_{z'',h_3}^{-1}u_{z,h_3}. \tag{3.8}$$

It is an ordered product, not an implicit Weyl product. Direct calculation gives the following identity in (U), before any quotient:

$$\boxed{\quad rh=-A^{-1/2}f+ A^{-1/2}(v-v_0)\bigl(v^{-1}z^2-v_0^{-1}(z'')^{-2}\bigr). \quad} \tag{3.9}$$

For transparency, an intermediate form is

$$rh=A^{-1/2}-c(z'')^{-1}z'z-c^{-1}(z'')^{-1}(z')^{-1}z, \tag{3.10}$$

and the two ordered products satisfy

$$(z'')^{-1}z'z=A^{-1/2}v(z'')^{-2},\qquad (z'')^{-1}(z')^{-1}z=A^{-1/2}v^{-1}z^2. \tag{3.11}$$

Equations (3.10)–(3.11) imply (3.9), using only (1.1). The companion standard-library script verifies (3.6) and (3.9) as exact Laurent polynomial identities in twelve noncommuting half-edge variables. Thus the check does not assume the sought intersection theorem.

Since (hU^), one has (rU=(rh)U). Equation (3.6) puts (v-v_0) into the same right ideal, so (3.9) proves equality of the right ideals generated by the two ambient relations and by (v-v_0,f). Repeating by the orientation-preserving symmetries taking the chosen vertex to any vertex handles the entire CK tetrahedron relation list. This is one symbolic local identity with tetrahedral symmetry, not a search over configurations.

The other vertex Lagrangians generate no extra right ideal in (S). This is stated in CK Definition 3.1 and follows directly from the vertex relations. For example, with (f’= (z’)^2+z{-2}-1), one has

$$fz^{-2}+f'=(v^2-v_0^2)[z^{-2}(z'')^{-2}], \tag{3.12}$$

and cyclic permutations give the remaining Lagrangians. Opposite squares are equal modulo the four vertex relations, as proved in Section 6 below. Every multiplier in this reduction is on the right, or is central in (S). No vertex monomial is assumed central in the larger torus (U).

Consequently

$$\sum_T\rho(\widetilde I_T)U =\sum_T(V_T+L_T)U. \tag{3.13}$$

4. Edge relations and the comparison isomorphism

For an edge with bare-edge occurrences (e_1,,e_k), set

$$E_e=[X_{e_1}\cdots X_{e_k}]. \tag{4.1}$$

The images under CK’s bigon map of the factors around the edge are (u_{e_{j-1},h_1^{{(j)}}u_{e_j,h_2}{(j)}}). These factors commute in (U): commutation contributions on the two sides of an identified face cancel. Their product is the Weyl monomial (4.1). Thus the edge generator is (E_e+A^2S), with no unrecorded (i)- or (A)-factor. This is also CK Proposition 5.25’s explicit edge calculation. Negative-power edge generators give the same left ideal since

$$E_e^{-1}+A^{-2}=A^{-2}E_e^{-1}(E_e+A^2). \tag{4.2}$$

Combining (3.13) with the edge description gives the equality, not just an inclusion,

$$\mathcal R= \sum_T(V_T+L_T)U+UE. \tag{4.3}$$

Here (4.3) means the right extensions of the displayed tetrahedron generators and the left extensions of the displayed edge generators; the order must not be reversed.

Theorem 4.1 — CK comparison injectivity

For every triangulated cusped manifold in the scope of CK Section 5, the comparison of CK Proposition 5.25 is an isomorphism

$$\mu:Q_{CK}\xrightarrow{\ \cong\ }Q_{PP}^{\rm ext}. \tag{4.4}$$

In particular, CK Conjecture 5.27 holds for its stated target modules.

Proof. Apply Lemma 2.1 to (4.3). Its right side intersects (S) in exactly (J), hence (SR=J). The comparison in CK is the map induced by the inclusion (SU); therefore (S/JS/(SR)) is an isomorphism. ∎

An explicit inverse sends the class of (sS) in the ambient image to the class of the same (s) in (S/J). More generally, if an arbitrary representative (uU) represents an element of that image, its inverse class is ([P_0(u)]S/J). The proof shows this is independent of the representative.

The proof remains valid over (R) as in (1.1). In particular it does not require characteristic zero, a generic parameter, or a nonzero target. It applies locally as well when some or all edge relations are omitted. Self-pairings of distinct face occurrences and multiple pairings cause no change to (2.1); face occurrences must never be merged in the ambient lattice.

5. What this proves about the named traces

CK’s Theorems 4.9 and 5.34 already establish, in their explicitly common orientation, labeling and stated-skein conventions,

$$\tau_{GY}^{CK}=\tau_{CK},\qquad \tau_{PP}^{CK}=\mu\tau_{CK}. \tag{5.1}$$

The theorem proved here upgrades the second comparison to an isomorphism of targets. Equation (5.1) is a named-trace equality from a primary source, not an inference from matching target presentations.

The exact face coefficients used by CK are

$$(c_T,c_B)=(v_0^{-1},1)=(-iA^{-1},1). \tag{5.2}$$

With PP2’s (q=-A^2) and the chosen square root (q^{1/2}=iA), this is PP2’s first normalized branch ((q^{-1/2},1)). CK Lemma 5.18 explicitly notes its use of PP2 conventions; its notation should not be read as the raw constants of PP1.

Because $\mu$ matches all six generators, it intertwines the central opposite-root quotient maps. Therefore it induces an isomorphism of identified targets and intertwines the respective quotiented named traces. This proves the identified comparison in CK’s common conventions without assuming comparison injectivity after quotienting. Here identification is a central quotient of the shape image module, not a two-sided quotient of the full face-cone torus $U$; opposite-root ratios are generally not central in $U$.

Two limitations remain important:

1. CK’s extended GY trace is not already the identified target of GY-index Theorem 5.7. The quotient is a separate, noninvertible map.
2. CK’s conventions are not the original numerical coordinate conventions of GY-index. CK Section 4.1 reverses a lantern product; its introduction and Section 4.3 also flag different labels and orientations. No equation in this note identifies a raw GY-index monomial with a raw PP monomial at fixed (A) by ignoring those changes. In particular the present named comparison does not certify the still missing peripheral-coordinate dictionary needed to import GY-index Theorem 5.7.

Original PP constants: exact global scaling

For completeness, the named normalized trace can be related to the original PP1 constants ((A^{-1},-i)), keeping all six roots. Give the 4-regular dual multigraph an Eulerian orientation, counting face occurrences; each tetrahedron has two incoming and two outgoing faces. Set $\tau_h=1$ on outgoing occurrences and $\tau_h=-1$ on incoming ones. Over (k(s)), with (s^2=-i), scale each PP1 half generator by (s_h). Then

$$A^{-1}(s\tau_h)^2=-iA^{-1},\qquad (-i)(s\tau_h)(s\tau_{\bar h})=1. \tag{5.3}$$

These are exactly the triangle and bigon coefficients in (5.2), so the face bimodule trace formulas are conjugated, not merely the relations. The induced shape map is already defined over (k):

$$\Phi(x_e)=i\epsilon_eX_e, \qquad \epsilon_e=-\tau_h\tau_{h'}\in\{\pm1\}. \tag{5.4}$$

The four $\tau_h$ at a tetrahedron multiply to (1). Hence opposite edges have equal $\epsilon_e$, each vertex triple has product (-1), and the product of $\epsilon_e$ around a manifold edge is (1). In particular (5.4) takes the original central values to the right ones:

$$[x_1x_2x_3]=-A, \qquad [x_{e_1}\cdots x_{e_k}]=-i^kA^2 \tag{5.5}$$

map to (i[X_1X_2X_3]=-A) and (i^kE_e=-ikA^2). The original Lagrangian trinomial becomes the negative of the appropriate normalized Lagrangian. Thus $\Phi$ is an isomorphism of original and normalized six-root targets, and

$$\Phi\tau_{PP1}=\tau_{PP}^{CK}. \tag{5.6}$$

Although the face-level demonstration adjoins (s), (5.4)–(5.6) descend to (k), since the extension is faithful and the trace images are in the shape subtorus. This descent must be stated: () need not belong to (Q(i,A^{1/2})). Different Eulerian choices need not give the same map of the whole target. No Pachner-natural choice is asserted.

6. Central sectors are retained, not erased

The opposite differences lie in the radical of (3.2). Consequently

$$r=y/z,\qquad r'=y'/z',\qquad r''=y''/z'' \tag{6.1}$$

are central in (S). Divide the three other vertex monomials by the base vertex (v). Their ratios are (r’r’‘,rr’‘,rr’). Setting all four vertices equal to (v_0) gives

$$r=r'=r''=\sigma,\qquad\sigma^2=1. \tag{6.2}$$

Thus opposite squares agree without identifying the roots. Over (k), the commuting central idempotents ((1_T)/2) split the module into sign sectors; after global edge relations, some sectors can be zero. We do not claim that all (2^N) global sectors survive.

The identified quotient is the all-(+) quotient (sigma_T=1). Locally, before global edge relations, the minus sector is genuinely nonzero. After eliminating the central vertex, the local algebra is a rank-two quantum torus tensored with (k[]/(^2-1)); the tetrahedron relation is a nonmonomial Laurent polynomial in the rank-two torus and is independent of $\sigma$. Its generated right ideal is proper. Indeed, a product’s Newton polytope is the Minkowski sum of the factor polytopes in a quantum torus over a field (exposed leading monomials cannot cancel); a nonmonomial Laurent polynomial cannot have a right inverse. In the minus sector, the vector (z-y=2z) is therefore nonzero.

This supplies a decisive local obstruction to a shape-matching lift of the identified target into the six-root target: the two opposite standard diagonal arcs have the same image after identification but different images before it. CK Lemma 4.6 and the local face-cone comparison identify these with the actual diagonal-arc trace values. The obstruction does not rule out a correctly refined six-root boundary theorem. It rules out recovering that theorem by inverting the opposite-root quotient.

7. Exact peripheral factors, and the remaining named-action gate

This section fixes all algebraic factors while explicitly withholding the unproved named-action assertion.

For one tetrahedron, let a six-root exponent vector be

$$a=(a_z,a_{z'},a_{z''},b_y,b_{y'},b_{y''}).$$

Put

$$r=a_z+b_y,\quad s=a_{z'}+b_{y'},\quad t=a_{z''}+b_{y''}, \quad\beta=b_y+b_{y'}+b_{y''}. \tag{7.1}$$

In the central-vertex quotient, exact Weyl elimination gives

$$\boxed{\quad W_6(a)=\sigma^{\beta}(iA)^s A^{(r-s)(t-s)/2}z^{r-s}(z'')^{t-s}. \quad} \tag{7.2}$$

For multiple tetrahedra multiply these expressions; different tetrahedra commute. The factor $\sigma^\beta$ is invisible in the identified target. It is not an optional sign convention. For the unscaled original PP1 roots in the same CK edge ordering, the analogous formula uses the vertex value ((-A)^s) in place of ((iA)^s), with the same Weyl factor.

For any lifted exponent vector (a), write (d(a)=_e a_e) and (^a=_e_e^{a_e}). The original-root expression corresponding under (5.4) to the normalized monomial (W_X(a)) is exactly

$$\mathcal M(a)=i^{-d(a)}\epsilon^a W_x(a). \tag{7.3}$$

Consequently its symmetric trace expression is

$$\mathcal M(a)+\mathcal M(a)^{-1} =i^{-d(a)}\epsilon^aW_x(a) +i^{d(a)}\epsilon^aW_x(-a). \tag{7.4}$$

Now take candidate meridian and longitude lifts (a,bZ^{6N}) whose collapsed peripheral rows have pairing (2), where $\kappa\in\{1,-1\}$ records the chosen collar orientation. Set (M=W(a)), (L=W(b)). The exact three candidate operators are

$$\begin{array}{c|c} \mu & M+M^{-1}\\ \lambda & L+L^{-1}\\ \mu+\lambda & W(a+b)+W(-a-b) =A^{-\kappa}(ML+M^{-1}L^{-1}). \end{array} \tag{7.5}$$

Here (ML=A^{2}LM). If these act by right multiplication, the composition of the associated linear operators reverses the product: (R_MR_L=R_{LM}). This reversal must be retained in any boundary algebra convention. For the prior handoff’s left-torus convention, $\kappa=1$; the opposite algebra has $\kappa=-1$.

Substituting (7.3) into (7.5) gives every corresponding original PP1 Weyl and scalar factor. These monomials are well-defined normalizer operators whenever their collapsed rows are peripheral: the six-root skew form factors through collapse, so they commute with the edge monomials; right multiplication automatically preserves the tetrahedron right ideals.

What has not been proved: that a specified lift (a), (b), and the additive lift (a+b) in (7.5), after the CK/GY collar and state-convention translation, give the insertions of the actual three named peripheral skeins in the original six-root trace. GY-index Theorem 5.7 proves this only after opposite-root identification.

The remaining ambiguity can be expressed exactly. If two six-root rows have the same collapsed three-root row, their Weyl monomials differ by

$$\prod_T\sigma_T^{\nu_T}, \tag{7.6}$$

with no extra Weyl factor, since the difference is radical. A geometric sum-of-slopes representative could therefore have the same identified monomial as (W(a+b)) while still carrying (7.6). The identified boundary theorem cannot detect it. If rows agree only after central-vertex elimination, the additional ((iA)^s) factors in (7.2) must also be kept.

We inspected both the original GY-index appendix and the author-hosted revision dated 6 August 2025. Their off-diagonal cancellations are an intrinsic promising route, but the exact six-root additive lift and the stated/collar convention dictionary have not been proved here. Replaying selected skein configurations would not settle that structural gate. No such configuration search was undertaken.

8. Hostile internal audit and stopping boundary

The comparison proof survived the following checks:

• Correct target: (Q_{PP}^{}) remains the image in the full ambient quotient (1.6). It is not redefined to make $\mu$ injective.
• Full relation list: all vertex and Lagrangian generators are handled by (3.6)–(3.13) and tetrahedral symmetry; all edge generators, including inverses, are handled by (4.1)–(4.2).
• Sidedness: the trinomial is normalized on the right. Projection is an (S)-bimodule map, so left-edge and right-tetrahedron relations are both retained. No two-sided closure of a Lagrangian ideal is taken.
• Six-root lattice: degree zero is the full six-root torus, not the three-root identified torus, an even-exponent torus, or the image of the skein module. Identification occurs only after Theorem 4.1.
• Named traces: source CK Theorem 5.34 supplies the actual intertwining; the new proof supplies its invertible target map. GY-index Theorem 5.7 is not an input to Theorem 4.1 or (5.1).
• Denominators: (3.9) uses only units already in CK’s ring. Division by (2) is restricted to the separate sign-sector discussion, and the original-scaling argument explicitly descends from the auxiliary square-root extension.

There is a minor source-text inconsistency worth recording: CK Definition 5.7(i) first calls $\widetilde I_T$ a right ideal and one sentence later calls its global extension a left ideal. Equation (5.6), their subsequent right-multiplication proof, their Definition 5.11(i), and PP1’s explicit ambient presentation consistently use the right-generated tetrahedron relation module. Theorem 4.1 concerns that stated target. Taking the isolated word “left” as an instruction to impose a new two-sided closure would define a different object and invalidate the argument.

Next mode: PUBLISH — candidate-proof preparation and review, focused on CK Conjecture 5.27. The recognized conjecture is explicitly identified and a complete proof is included; the packet is not yet a public Publication. The six-root peripheral ATTACK stops at the signed additive-lift/convention gate in Section 7, not at an alleged counterexample. Future work should construct that intrinsic lift or prove an obstruction to it, not enumerate manifolds, slopes, triangulations, or skein configurations.

9. Credit and significance

Garoufalidis–Yu and Panitch–Park constructed the quantum traces. CK constructed the common face-cone comparison, established surjectivity and named-trace intertwining, and isolated the injectivity conjecture. The local normalization in Section 3 is a sharpened exact form of the calculation already in CK Proposition 5.25; it is not claimed as a new quantum-trace formula.

The Lab contribution is the degree-zero (S)-bimodule retraction and its use, after proving equality of the normalized sided ambient relation modules, to establish the missing reverse inclusion and hence injectivity. The original-branch Eulerian scaling restates the preceding Lab packet with its necessary auxiliary coefficient extension made explicit. The central-factor audit and stopping boundary prevent these results from being overstated as a Dehn-filling or six-root peripheral theorem.

External novelty is unassessed; a search failing to locate another proof does not establish priority. No author was contacted, and no material in this packet has been published by this research agent.