A source-native six-root peripheral lift

Two named insertions determine the additive peripheral action

FPRD Lab, 8 September 2026. ATTACK research note. AI-assisted work, without external expert review.

Retained first checkpoint. Its native theorem and algebraic formulas were re-audited in the continuation. The local gate in Section 5.4 is addressed in FPRD-local-collar-square-2026-09-08.md; read that note and local-collar-hostile-audit.md for the current status. The text below preserves the earlier claim boundary, not the latest handoff.

Result and boundary of the result

The peripheral-action theorem of Garoufalidis and Yu can be extended to their full six-root target, at generic parameter, by retaining bare-edge labels in their prequotient diagrammatic calculation. An additive lift is then obtained from two actual peripheral curves. The torus skein commutator determines the named sum-of-slopes insertion, including its central sign and Weyl factor.

This is a source-native extension of the GY argument, not an inverse to opposite-root identification. The proof substantially imports GY’s diagrammatic work. The new algebraic step is a two-generator argument that makes a separately guessed sum-of-slopes sign unnecessary. Its novelty is unassessed. This is a supporting research note, not a new Publication.

An exact algebraic comparison with the CK convention is also given below. It is not yet proved to be the comparison of the raw named traces: endpoint duality, ribbon framing and the collar identification must still be made into a commuting local diagram. CK’s existing convention-adjusted trace comparison and the Lab’s comparison isomorphism remain available, but do not supply that missing raw dictionary by themselves.

Panitch-Romo Conjecture 7.1 remains open here. No new nonvanishing or unrestricted Dehn-filling theorem is claimed.

1. Keep the two presentations separate

Work over

$$k=\mathbb Q(i,A^{1/2}),\qquad i^2=-1,$$

with indeterminate $A^{1/2}$. GY-index’s parameter satisfies $q^{1/8}=A^{1/2}$, so $q^{1/4}=A$. Let $S_6$ be the tensor product of the six-root quantum tori of an ideal triangulation. In one tetrahedron, order the roots as

$$(z,z',z'',y,y',y''),\qquad B_6=\begin{pmatrix}\Omega&\Omega\\\Omega&\Omega\end{pmatrix},\qquad \Omega=\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix}. \tag{1}$$

Our Weyl notation is

$$W(a)W(b)=A^{B_6(a,b)/2}W(a+b). \tag{2}$$

The four central vertex monomials are

$$v_0=W(e_z+e_{z'}+e_{z''}),\quad v_1=W(e_z+e_{y'}+e_{y''}),\quad v_2=W(e_y+e_{z'}+e_{y''}),\quad v_3=W(e_y+e_{y'}+e_{z''}). \tag{3}$$

Here $v_j$ denotes a monomial, not the scalar used under that name in the previous publication. All four are set to $iA$. For a manifold edge $e$, let $h_e$ be its vector of actual bare-edge occurrences, and let $H_e=W(h_e)$.

The original GY-sided full target is

$$Q_G^6=S_6/J_G,\qquad J_G=\sum_{T,j}S_6(v_{T,j}-iA) +\sum_T S_6 f_T^- +\sum_e(H_e+A^2)S_6, \quad f_T^-=z_T^{-2}+(z_T'')^2-1. \tag{4}$$

Thus tetrahedron relations have arbitrary multipliers on the left, and edge relations have arbitrary multipliers on the right. This is the extension in [GY, Theorem 4.25 and Remark 5.1], after the parameter substitution above. Denote its named trace by

$$\tau_G^6:\mathcal S(M)\longrightarrow Q_G^6.$$

It is not the CK presentation. In CK’s convention, tetrahedron relations have multipliers on the right, edges have multipliers on the left, and the Lagrangian is

$$f_T^+=z_T^2+(z_T'')^{-2}-1. \tag{5}$$

Central vertex elimination in either presentation gives

$$y=\sigma_Tz,\quad y'=\sigma_Tz',\quad y''=\sigma_Tz'',\qquad \sigma_T^2=1. \tag{6}$$

The identified GY-index target is the further quotient $Q_G^3=Q_G^6/(\sigma_T-1)_TQ_G^6$. The local minus sector is nonzero, as proved in the preceding packet. That quotient is never inverted below.

2. A general two-generator lemma

Let $\mathcal S(T^2)$ have the usual parameter $A$, with ordered homology basis $(\lambda,\mu)$ in the GY convention. Write $K_a$ for its Chebyshev-normalized slope elements. The product-to-sum identities for the two basis curves are

$$\begin{aligned} K_\lambda K_\mu&=A K_{\lambda+\mu}+A^{-1}K_{\lambda-\mu},\\ K_\mu K_\lambda&=A^{-1}K_{\lambda+\mu}+A K_{\lambda-\mu}. \end{aligned} \tag{7}$$

Consequently, over $k$,

$$K_{\lambda+\mu}= \frac{A K_\lambda K_\mu-A^{-1}K_\mu K_\lambda}{A^2-A^{-2}}. \tag{8}$$

The three elements $K_\lambda,K_\mu,K_{\lambda+\mu}$ generate the torus skein algebra [GYI, Appendix A, citing Bullock-Przytycki]. Equation (8) therefore makes the first two alone a generating set after the indicated localization.

Lemma 2.1. Two insertions suffice

Let $V$ be a left $\mathcal S(T^2)$-module, let $Q$ be a vector space over $k$, and let $\tau:V\to Q$ be linear. Suppose invertible operators $L,M$ on $Q$ satisfy

$$LM=A^2ML \tag{9}$$

and, for every $x\in V$,

$$\tau(K_\lambda x)=(L+L^{-1})\tau(x),\qquad \tau(K_\mu x)=(M+M^{-1})\tau(x). \tag{10}$$

Define

$$E_{p\lambda+q\mu}=A^{-pq}L^pM^q. \tag{11}$$

Then the torus skein action on $Q$ given by $K_\gamma\mapsto E_\gamma+E_\gamma^{-1}$ intertwines $\tau$ for every $\gamma$.

Proof. The operators in (11) obey

$$E_aE_b=A^{\det(a,b)}E_{a+b},\qquad E_{-a}=E_a^{-1}. \tag{12}$$

The symmetric combinations therefore obey the Frohman-Gelca product-to-sum law. This defines the desired torus skein representation. It agrees with the action on $V$ through $\tau$ on the two generators by (10), hence on their products and linear combinations. Alternatively, substitution in (8) explicitly gives

$$\tau(K_{\lambda+\mu}x)= \bigl(A^{-1}LM+A^{-1}L^{-1}M^{-1}\bigr)\tau(x). \tag{13}$$

The second summand is $(A^{-1}LM)^{-1}$, not $A L^{-1}M^{-1}$. This is the exact Weyl factor. $\square$

There is no surjectivity or faithfulness hypothesis on $\tau$. In particular, the lemma is about the actual action on every input, not just the trace of the empty link. It does not assert that a directly drawn normal representative of $\lambda+\mu$ has an exponent vector literally equal to the sum of the two drawn basis vectors.

The localization matters. This argument alone proves no version at $A^4=1$, and it cannot be moved back to an integral coefficient ring without a separate argument.

3. Lift the actual two-state calculation before taking a quotient

Choose an admissible diagrammatic basis of peripheral curves as in [GYI, Appendix A.1]. This is the basis obtained from their fundamental-domain construction, not an enumeration of slopes or triangulations. The topological construction and the prequotient local identities in that appendix are formal dependencies here.

For an oriented curve $\gamma$ in that setup, its lantern diagram has standard segments and substandard segments. A standard segment is assigned the actual root of its bare edge; an opposite bare edge is assigned its own $y$-root. Define

$$\widetilde C_\gamma= \sum_{\text{small counterclockwise segments}} e_{\text{bare edge}} -\sum_{\text{big clockwise segments}}e_{\text{bare edge}}. \tag{14}$$

Equivalently, this is the exponent vector of the all-minus contribution after each substandard segment is replaced by its parallel standard segment with reversed state. This equivalent definition fixes the sign without relying on a prime-label convention. Standard $(\epsilon,\epsilon)$ arcs have image $z^{-\epsilon}$ in the native GY convention. The all-plus contribution has the opposite vector.

When matching a separately fixed oriented cusp-row convention, select between these two opposite vectors by requiring $\pi\widetilde C_\gamma=C_\gamma$ for that source row. GYI’s identification of its two constant-state monomials as $m_\gamma^{\pm1}$ supplies precisely these two choices. This calibrates the orientation of the eigenmonomial; it introduces no extra central sign and does not change the symmetric insertion formula. In the rest of the note, $\widetilde C_\gamma$ denotes the calibrated choice.

Proposition 3.1. Source-native six-root insertion formula

For either of the two admissible basis curves and every $\alpha\in\mathcal S(M)$,

$$\tau_G^6(K_\gamma\alpha)= \bigl(W(\widetilde C_\gamma)+W(-\widetilde C_\gamma)\bigr)\tau_G^6(\alpha). \tag{15}$$

Proof and quotient audit. We use the calculation in [GYI, Appendix A], but track the stage at which every assertion is valid.

1. The height-exchange identities of Lemma A.1 and the commutation statements of Corollary A.2 are identities in the stated skein algebra. They do not use a shape quotient.

2. Lemma A.3 is an identity in the corner-reduced lantern module. It turns a diagonal substandard arc into the parallel standard arc with the opposite state, modulo products containing a standard off-diagonal arc. The parallel arc is at the same bare edge, not its opposite. No $z=y$ substitution occurs.

3. Standard arcs span that corner-reduced module [GY, Lemma 4.22]. A product with an off-diagonal standard factor vanishes in the full local target [GY, Section 4.9 and Theorem 4.25], before the extra opposite-root quotient. Thus a state cycle with a transition $(+,-)$ vanishes by the same height-exchange argument as in GYI. Only the two constant state assignments remain. This argument works with an arbitrary additional skein $\alpha$, as required for an action theorem.

4. The all-equal-state commutation and Weyl-cancellation claims in GYI’s proof are assertions about the original arcs and their standard replacements in the lantern skein algebra. The only root commutation form used after applying the local trace is (1). Opposite-root identification does not change that form: if $\pi(a,b)=a+b$ collapses opposite exponents, then

   $$B_6(u,v)=\Omega(\pi u,\pi v). \tag{16}$$

   Hence their cancellation works with the six distinct labels. The exact Weyl monomials left by the two state choices are those in (15).

5. In their Type II closing configuration, the annulus counit contributes $q^{-1/4}=A^{-1}$. Its compensation is the unmatched endpoint-height contribution to Weyl ordering. Both are prequotient statements. The annular buffer has no shape root and introduces no opposite-root identification. The same correction must be retained; simply omitting it would give a wrong factor.

Every vanishing or replacement used in the proof is therefore valid in the full target (4). We have not inferred (15) from its image in $Q_G^3$. $\square$

This proposition is an extension by inspection and reuse of the source proof, not an independently invented substitute for that proof. The geometric basis construction and the detailed local height-exchange identities are still sourced mathematical inputs.

4. The additive signed lift

Let $a=\widetilde C_\lambda$ and $b=\widetilde C_\mu$ for the admissible ordered basis, oriented so that

$$B_6(a,b)=2. \tag{17}$$

The collapsed vectors are the source’s peripheral rows. The Neumann-Zagier relations cited in [GYI, Section 5.2] give

$$B_6(a,h_e)=B_6(b,h_e)=0 \tag{18}$$

for every edge. The same argument gives zero pairing between different boundary components.

Left multiplication by $W(a)^{\pm1}$ preserves the left-generated tetrahedron relations. Equation (18) makes it preserve the right-generated edge relations as well. It therefore defines an invertible operator on the whole $Q_G^6$. The same holds for $b$. No centrality in the larger face-cone torus is being assumed.

Theorem 4.1. Generic native six-root peripheral linearity

In the setup of [GYI, Theorem 5.7 and its Appendix A], the extended named trace $\tau_G^6$ admits a peripheral quantum-torus action on its full six-root target such that

$$\widetilde C_{p\lambda+q\mu}=pa+qb,\qquad \widetilde m_{p\lambda+q\mu}=W(pa+qb) =A^{-pq}W(a)^pW(b)^q, \tag{19}$$

and, for every slope element and every skein $\alpha$,

$$\boxed{\quad \tau_G^6(K_\gamma\alpha)= \bigl(\widetilde m_\gamma+\widetilde m_\gamma^{-1}\bigr) \tau_G^6(\alpha). \quad} \tag{20}$$

Opposite-root identification takes this action to the action in [GYI, Theorem 5.7]. The construction applies to all torus ends simultaneously.

Proof. Equations (17)-(18) supply the operators and their exact commutation law. Proposition 3.1 supplies the two named insertions. Apply Lemma 2.1. Equation (16) and the definition of the source’s additive rows identify the result after the opposite-root quotient. Different ends commute by the same pairing identity. $\square$

A prescribed meridian and longitude need not be the initial diagrammatic basis. Express their homology classes in that basis and use (19). Equation (20) concerns their actual skein insertions, so this change of basis does not rename the question.

No canonical action on every element of $Q_G^6$, independent of all choices of initial diagrammatic basis, is asserted. The resulting actions agree on the trace image because they describe the same named insertions. Agreement on that image is not silently promoted to equality on the full target.

Exact central signs

For a local vector $u=(u_z,u_{z'},u_{z''},u_y,u_{y'},u_{y''})$, put

$$r=u_z+u_y,\quad s=u_{z'}+u_{y'},\quad t_*=u_{z''}+u_{y''},\quad \beta=u_y+u_{y'}+u_{y''}.$$

The central-vertex reduction is

$$W(u)=\sigma_T^\beta(iA)^s A^{(r-s)(t_*-s)/2}z^{r-s}(z'')^{t_*-s}. \tag{21}$$

In (19), $\beta_T(pa+qb)=p\beta_T(a)+q\beta_T(b)$. Thus the sum-of-slopes sign is the product of the two basis signs, with no undetermined third sign. These factors remain present in every surviving minus sector.

5. What can be proved about the raw convention dictionary

5.1. A fixed-parameter algebraic dictionary

Let $P$ interchange $z\leftrightarrow z''$ and $y\leftrightarrow y''$, fixing the prime roots. Then

$$P^tB_6P=-B_6. \tag{22}$$

With every bare-edge incidence relabeled by $P$,

$$D_P(W(u))=W(Pu),\qquad D_P(A^{1/2})=A^{1/2},\quad D_P(i)=i \tag{23}$$

is an anti-isomorphism of the full quantum tori. It sends $f^-$ to $f^+$, permutes the four vertex relations, and sends each edge relation to its relabeled edge relation. It consequently identifies the GY-sided and CK-sided module presentations. In particular,

$$D_P(Mx)=D_P(x)D_P(M). \tag{24}$$

The source-native left action is transformed algebraically into right multiplication. For $R_X(x)=xX$, composition satisfies $R_XR_Y=R_{YX}$. In the ordered GY basis, the transformed monomials themselves have the opposite commutation sign; the right operators recover the original sign.

More explicitly, with $L_C=D_P(L)$ and $M_C=D_P(M)$, one has $M_CL_C=A^2L_CM_C$, and the sum monomial is $A^{-1}M_CL_C$. This is the order used by the earlier CK-sided filling packet; it is not $A^{-1}L_CM_C$.

On diagonal standard-arc values the necessary state dictionary is

$$\text{GY state }\epsilon\ \longleftrightarrow\ \text{CK state }-\epsilon,\qquad z_e^{-\epsilon}\longmapsto X_{Pe}^{-\epsilon}. \tag{25}$$

This is an exact algebraic check, not a proof that (25) extends to the actual collar map on all stated tangles.

5.2. A second, semilinear dictionary

There is also an anti-isomorphism with

$$D_{\mathrm{inv}}(A^{1/2})=A^{-1/2},\quad D_{\mathrm{inv}}(i)=-i,\quad D_{\mathrm{inv}}(W(u))=W(-u). \tag{26}$$

It sends $f^-$ to $f^+$. A vertex relation goes to $v^{-1}+iA^{-1}$, which is generated by $v-iA$, and an edge relation goes to

$$H_e^{-1}+A^{-2}=A^{-2}H_e^{-1}(H_e+A^2). \tag{27}$$

Its coefficient conjugation is essential. It is related algebraically to the GY chirality formula [GYI, Section 5.6, (97)], but is not a same-parameter identification of the two raw named traces.

5.3. A genuine obstruction to the naive dictionary

There is no fixed-coefficient anti-homomorphism of the unquotiented generic shape tori sending every root to the inverse of the identically labeled root. If $zz'=Az'z$, the proposed anti-image would require

$$(z')^{-1}z^{-1}=A z^{-1}(z')^{-1},$$

whereas the actual relation gives the factor $A^{-1}$. Since these are units, the proposal would force $A^2=1$. This is a torus-level obstruction, not a claim that every global quotient is nonzero.

Thus it is not legitimate to combine “reverse the product,” “keep the parameter,” and “send the GY minus-state root to the CK minus-state inverse” without further changes. The fixed-parameter map (23) uses an odd label permutation and a diagonal state flip instead.

5.4. Why the remaining named comparison does not follow

The local named comparison still needs a map of the underlying stated/corner-reduced constructions which:

• implements the actual reversal of lantern collar and marking direction;
• fixes the duality factors for mixed endpoint states, not only the diagonal values in (25);
• translates GY’s vector-framing diagrams to PP/CK’s symmetric ribbon convention, including the distinguished half-twist;
• commutes with splitting and annular counit before passing to the shape target.

The scalar check is nontrivial. In the usual GY boundary convention, the two nonzero cup coefficients are $A^{-1/2}$ and $-A^{-5/2}$; in the corresponding symmetric PP/CK convention they are $iA$ and $-iA^{-1}$. Their common ratio is $iA^{3/2}$, the chosen half-twist factor. A correct geometric dictionary must account for this, rather than equating the displayed cup diagrams. This calculation identifies a required normalization, not a completed collar map.

The exact minimal outstanding test is a prequotient local intertwining diagram on the elementary stated arcs and their cup/cap dualities, with inverse duality factors cancelling across each paired cut. Once that diagram is proved, the source-native action (20), (23)-(24), CK’s named comparison and the original-to-normalized PP transport can be composed. A new independent calculation of the named sum slope will not be needed.

6. Original PP branch factors, kept explicit

The preceding Lab note constructs an Eulerian normalization with

$$\Phi(x_e)=i\epsilon_eX_e,\qquad \epsilon_e\in\{\pm1\},\qquad \epsilon_e=\epsilon_{e^{\mathrm{opp}}}. \tag{28}$$

Here $x$ is the original PP1 root and $X$ the CK/first-normalized-PP root. For every integer vector $u$,

$$\Phi^{-1}(W_X(u))=i^{-\sum_eu_e}\epsilon^u W_x(u). \tag{29}$$

Consequently the algebraic image of a native lifted monomial is

$$\Phi^{-1}D_P(W(u))=i^{-\sum_eu_e}\epsilon^{Pu}W_x(Pu). \tag{30}$$

The inverse term has the inverse scalar. For example, the image of $W(u)+W(-u)$ is

$$\epsilon^{Pu}\left(i^{-\sum u_e}W_x(Pu)+i^{\sum u_e}W_x(-Pu)\right), \tag{31}$$

not generally a common scalar times $W_x(Pu)+W_x(-Pu)$. Opposite-root signs are unchanged by (28), and must still be retained.

Equations (30)-(31) are exact algebraic transport formulas. Calling them the original PP named peripheral action remains conditional on the local dictionary in Section 5.4.

7. An exact underdetermination test

Let $C$ be the central group generated by the $\sigma_T$’s. For any character

$$\chi:H_1(T^2;\mathbb Z)\longrightarrow C,$$

the operators

$$E'_\gamma=\chi(\gamma)E_\gamma \tag{32}$$

have the same quantum-torus multiplication law, and the same image after all $\sigma_T=1$. Their symmetric combinations define another algebraic torus skein action. These actions need not agree before the quotient and need not intertwine the raw named trace.

This proves exactly why the identified theorem and an algebraic comparison of targets cannot select the raw named lift. The ambiguity is already parameterized by the two values of $\chi$ on a basis. It is not an independently free sign for every slope. The source-native calculation (15) fixes it for $\tau_G^6$; the unresolved collar square is needed to transport that fixing to the original PP named trace.

8. Status, significance and next question

Internally proved, using stated source dependencies: the generic native six-root peripheral-action extension, the two-generator interpolation lemma, exact central/Weyl formulas, the two algebraic anti-isomorphisms, and the naive-dictionary obstruction.

Not proved: the raw GY-index-to-CK/PP named collar dictionary; an integral or root-of-unity version of the two-generator argument; triangulation-natural equality of the entire target actions; arbitrary-filling nonvanishing; Panitch-Romo Conjecture 7.1.

The source’s original theorem, its diagrammatic lemmas, and the Frohman-Gelca torus embedding are substantial formal dependencies. Most of the geometric work is theirs. The advance is useful internal progress on the precise obstruction, not a major quantum-topological result. A bounded primary-literature check did not locate a separately stated six-root version; that is not evidence of priority. Novelty remains unknown.

Next mode: ATTACK. Prove or disprove the local endpoint-duality/collar square in Section 5.4. Work with its intrinsic stated-skein operations, not with a list of manifolds or normal-curve configurations. Do not restart the sum-of-slopes sign problem: over the generic field it follows from the two named basis insertions by (8).

References

[GY] Stavros Garoufalidis and Tao Yu, A quantum trace map for 3-manifolds, Advances in Mathematics 486 (2026), 110735. Published primary PDF. Sections 4.6, 4.9-4.10; Theorem 4.25; Remark 5.1. Preprint: arXiv:2403.12424.

[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. Sections 2.2-2.3, 3.3, 5.2, 5.5-5.6; Theorem 5.7; Appendix A. The appendix supplies the geometric influence and the prequotient identities used here.

[CK] Qingjing Chen and Andrew Kricker, On 3d Quantum Trace Maps, arXiv:2606.13268v1. Section 4.1 and Remark 4.2; Section 4.2; Theorems 4.9 and 5.34.

[PP1] Samuel Panitch and Sunghyuk Park, 3d quantum trace map, Algebraic & Geometric Topology 26 (2026), 2255-2314. Published primary PDF. Preprint: arXiv:2403.12850v1. Stated ribbon relations and original face normalization.

[PP2] Samuel Panitch and Sunghyuk Park, Compatibility of quantum trace and UV-IR maps, arXiv:2509.09100v1. Section 2.1; Theorem 2.20; Remark 2.21; Definition 2.22 and its opposite-root footnote.

[FG] Charles Frohman and Razvan Gelca, Skein modules and the noncommutative torus, Transactions of the American Mathematical Society 352 (2000), 4877-4888. The torus embedding/product-to-sum theorem is the input recalled in [GYI, (94)].

[BP] Doug Bullock and Jozef Przytycki, Multiplicative structure of Kauffman bracket skein module quantizations, Proceedings of the American Mathematical Society 128 (2000), 923-931. Three-generation of the torus algebra, as invoked in [GYI, Appendix A].

[PR] Samuel Panitch and Mauricio Romo, On the length conjecture for twist knots, arXiv:2609.02706v1. Conjecture 7.1 remains the larger unresolved target.

[FPRD-CK] FPRD Lab, An isomorphism between six-root quantum gluing modules: Degree-zero descent and Chen-Kricker Conjecture 5.27, 8 September 2026. Publication. Prior target comparison and original-PP Eulerian normalization; not a proof of the raw named dictionary.