The local collar square and signed six-root peripheral transport

FPRD Lab, 8 September 2026. Supporting ATTACK research note. AI-assisted and internally audited; not externally reviewed or author-endorsed. This is not a new Publication or a solution of Panitch–Romo Conjecture 7.1.

Result and the precise new input

The preserved native six-root peripheral argument survives a fresh audit against Garoufalidis–Yu’s prequotient construction. Its two named basis insertions determine the additive peripheral action at generic parameter. The outstanding local convention problem can be separated into four operations: conversion of vector-framing pictures to ribbon pictures, reversal of the marking with unweighted state flip, reversal of the collar product, and reversal of the cyclic bare-edge labels.

The relevant geometric operation is already supplied by Panitch–Park’s published Remarks 3.4–3.5. Their later weighted self-duality in Proposition 3.10 is a different operation. Substituting that weighted map on both sides of a same-state cut fails the splitting and counit squares. Keeping these operations separate gives the local comparison below, and hence the named peripheral action on CK’s and the original PP six-root targets.

The geometric ingredients are not newly invented here: PP supply marking reversal and the ribbon-picture conversion; GY supply corner reduction, the standard-arc spanning and annular counit, and the peripheral calculation; CK specify the opposite collar and prove their convention-adjusted named comparison. The contribution is the explicit composition, the choice of the unweighted rather than weighted operation, and an audit that retains the full six-root signs and all sides of multiplication. Novelty and publication significance remain unassessed. This is a narrow program bridge, not a new general nonvanishing theorem.

1. Conventions fixed before making a comparison

Work over

$$k=\mathbb Q(i,t),\qquad t=A^{1/2},\quad v=iA=it^2,\quad h=iA^{3/2}=it^3. \tag{1}$$

Thus $v^2=-A^2$, $h^2=-A^3$. GY-index’s $q^{1/8}$ is $t$, whereas the parameter called $q$ in the original GY paper is $A$. No coefficient conjugation occurs in the dictionary of this note.

The six-root order is $(z,z',z'',y,y',y'')$. Put

$$B_6=\begin{pmatrix}\Omega&\Omega\\\Omega&\Omega\end{pmatrix},\qquad \Omega=\begin{pmatrix}0&1&-1\\-1&0&1\\1&-1&0\end{pmatrix},\qquad W(u)W(w)=A^{B_6(u,w)/2}W(u+w). \tag{2}$$

A label always denotes an actual bare-edge occurrence. Opposite edges are distinct. Source and target letters below are in their respective conventions, not an assertion that identically printed letters label the same embedded edge.

The native GY relation module is

$$J_G=\sum_{T,j}S(v_{T,j}-v)+\sum_T S\bigl(z_T^{-2}+(z_T'')^2-1\bigr) +\sum_e(H_e+A^2)S. \tag{3}$$

The four $v_{T,j}$ are the Weyl products at the four tetrahedral vertices. CK has the opposite sides and the polynomial $z_T^2+(z_T'')^{-2}-1$. These are module quotients, not two-sided algebra quotients.

We use CK’s actual opposite-collar convention: the interval coordinate increases in the direction of its boundary marking, opposite to the GY lantern collar. The embedded three-dimensional neighborhood and ambient orientation are kept fixed. Reversing its interval direction reverses the induced surface orientation as well. It does not reflect the ambient three-manifold or send $A$ to $A^{-1}$. This is the convention explicitly described in CK, Section 4.1 and Remark 4.2.

To anchor the coordinate dictionary, use a common ideal vertex and common prime-labeled incident edge in the two diagrams. Their cyclic orders are opposite. Fix the remaining label anchor so that

$$P:(z_G,z'_G,z''_G,y_G,y'_G,y''_G) \longmapsto(z''_C,z'_C,z_C,y''_C,y'_C,y_C). \tag{4}$$

This is a dictionary between the labels of the same embedded arcs, not an orientation-reversing move of a tetrahedron. The two coherent cyclic orders are related by an odd permutation; after cyclic choices the displayed transposition is an allowed anchor. If independently stored diagrams use different anchors, first record their actual edge bijection and conjugate (4) by those cyclic relabelings. Relabel every face incidence and every edge row, not just the letters in a polynomial. GY’s oriented vertex convention and standard-arc labeling are in GY, Sections 2.1–2.2 and Figure 13; CK’s are in its Section 1 and Figure 1.1.

The algebraic check is

$$P^tB_6P=-B_6,\qquad D_P(W(u))=W_C(Pu),\qquad D_P(xy)=D_P(y)D_P(x). \tag{5}$$

Equation (5) alone is not the named comparison. Sections 2–5 identify its geometric origin.

2. Cup and cap: the ribbon conversion in both directions

Let $V$ have state basis $(e_+,e_-)$. Use the boundary-slot order in which the GY trivial-arc coefficients are $A^{-1/2}$ for $(+,-)$ and $-A^{-5/2}$ for $(-,+)$. In these slots the evaluation and coevaluation matrices are

$$E_G=\begin{pmatrix}0&t^{-1}\\-t^{-5}&0\end{pmatrix},\qquad C_G=\begin{pmatrix}0&-t^5\\t&0\end{pmatrix}. \tag{6}$$

The symmetric PP/CK ribbon pictures use

$$E_R=C_R=\begin{pmatrix}0&v\\v^{-1}&0\end{pmatrix}. \tag{7}$$

These formulae include both mixed states; the equal-state entries vanish. The exact picture conversion is

$$E_G=h^{-1}E_R,\qquad C_G=hC_R. \tag{8}$$

In particular both snake identities survive:

$$E_GC_G=C_GE_G=I=E_RC_R=C_RE_R. \tag{9}$$

The first equality in (8) is the cup ratio found in the preceding checkpoint. Its partner, the inverse factor on the cap, is indispensable. Multiplying both cup and cap by $h$ would leave $h^2=-A^3$ at a cancellation, changing the framing.

These are not arbitrary normalization choices. PP1, Remark 3.5, printed p. 2265 explicitly draws both conversions from the ordinary vertically framed pictures, including the negative half twist on the cup and the positive half twist on the cap. It also explains that the half-twist relation is central. GY-index’s relations (8)–(9) are exactly (6) after (1). The local conversion therefore respects the complete boundary relations, as well as the unchanged crossing and loop relations. A choice of a framed ribbon is retained; we do not identify an arbitrary extra half twist with zero.

For a standard lantern arc the vertical framing line is the collar line. Reversing the collar changes its chosen vector from $n$ to $-n$, but not the ribbon line $\mathbb R n$. Consequently this arc remains the untwisted standard ribbon arc used in CK Section 4.2. There is no additional power of $h$ on its diagonal matrix entries. The factors in (8) convert the different displayed cup/cap pictures; they are not per-endpoint weights on every standard arc.

3. Marking reversal and the actual splitting/counit square

Let $\Gamma_G$ be the marking in a GY collar and $\Gamma_C=-\Gamma_G$. In the symmetric ribbon description define

$$F_\Gamma([L,s])=[L,-s]. \tag{10}$$

The underlying embedded ribbon and its framing are unchanged. PP1, Remark 3.4 states that simultaneously reversing the marking and all endpoint states preserves the defining skein relations. Thus (10) is an actual map of the marked skein modules, not just an assignment of diagonal trace values. It is involutive. With the collar coordinates of Section 1 it reverses stacking order. This is a comparison between two marked/cooriented collars, not the false assertion that fixing every closed curve gives an anti-automorphism of a single fixed oriented surface skein algebra.

In the state basis, $F=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. Reversing the marking reverses the boundary-slot reading, so its evaluation matrix is $E_{R,-}=F^tE_RF$, and similarly for the cap. Hence

$$F^t E_{R,-}F=E_R,\qquad F C_R F^t=C_{R,-}. \tag{11}$$

Equations (8) and (11) are the mixed-state endpoint square. Their geometric content comes from PP1 Remarks 3.4–3.5; the matrix equations expose the scalar bookkeeping rather than replace those source statements.

Lemma 3.1. Splitting and the annular counit commute with the unweighted map

Compose the ordinary-to-ribbon picture conversion with (10), on every piece obtained by a GY splitting. The resulting maps satisfy

$$(F_{\Gamma_1}\otimes F_{\Gamma_2})\Theta_G =\Theta_C F_\Gamma, \tag{12}$$

and for the standard annulus,

$$\epsilon_C F_{\mathcal A}=\epsilon_G. \tag{13}$$

Proof. Choose the same physical cut and the same physical endpoint heights, expressed in the two opposite collar coordinates. For one intersection the splitting tensor is

$$\delta_{\rm cut}=e_+\otimes e_++e_-\otimes e_-. \tag{14}$$

This is the matching-state tensor, not the cup or cap in (6)–(7). Applying $F\otimes F$ to it returns (14). More generally the source state assignment $s$ corresponds bijectively to the target assignment $-s$, with the same ribbon pieces and coefficient 1. The conversion at a paired cut is a change of endpoint picture on the same framed ribbon; its inverse picture change occurs on the paired side. Equation (9) says their duality factors cancel. Thus (12) is equality term by term, for arbitrarily many cut intersections and with any additional tangle present. It is not a finite search over possible diagrams.

For the annulus let $a_{\mu\nu}$ be its untwisted standard matrix-element arc, using the same standard arc on both sides. GY’s annular counit has $\epsilon_G(a_{\mu\nu})=\delta_{\mu\nu}$ [GY, Theorem 4.18 and Corollary 4.19]. The target arc is $a^C_{-\mu,-\nu}$, so its counit is the same scalar. The assertion extends to the standard parallel-arc spanning set, on which the counit is the product of these Kronecker deltas, and hence to the whole corner-reduced annulus. This is also the usual matrix-element counit of the bigon algebra used in CK Section 4.2. Equivalently, (12) makes the map a coalgebra map and the displayed counit is the unique counit. No new choice of counit is made to force commutation. $\square$

The annular correction in GY-index’s Type II peripheral calculation is still $A^{-1}$, not 1. The lemma preserves that calculation; it does not discard the nonstandard height exchange used to evaluate that particular annular diagram. Only the standard spanning diagrams have the bare Kronecker-delta formula.

3.2. A useful failed replacement: weighted self-duality

PP1 Proposition 3.10, expressed on unrescaled generators, is

$$d([L,s])=v^{-\sum s}[L,-s]. \tag{15}$$

It is suited to the balanced action relations in PP’s three-dimensional splitting theorem. It is not (10). Applying (15) to both sides of a same-state cut multiplies the term of state $s$ by $v^{-2s}$, namely $-A^{-2}$ or $-A^2$. On the standard annular arc it gives

$$\epsilon(d(a_{++}))=v^{-2}\ne1=\epsilon(a_{++}). \tag{16}$$

The weighted map therefore fails (12)–(13) if inserted without changing the cut contraction. This is not a criticism of PP’s balanced splitting theorem, which uses its matching rescaled actions. A nontrivial endpoint weight can be used if the inverse weight is put on the paired side. If $F_w(e_s)=w_s e_{-s}$, the elementary criterion is $w_s^{(1)}w_s^{(2)}=1$. Using the same arbitrary weight on both sides does not satisfy it. The exact checker verifies the criterion with a formal Laurent variable, not sample values.

There is also a limited uniqueness check: for a uniform endpoint rescaling used on both sides, preservation of (14) forces $w_+^2=w_-^2=1$. Preservation of either nonzero mixed cup forces $w_+w_-=1$. Over $k$, this leaves a common sign only, invisible on every tangle’s even total number of endpoints. Within this restricted uniform ansatz the unweighted square has no continuous normalization freedom. We make no classification claim for arbitrary boundary-dependent or nonlocal transformations.

4. The twisted corner quotient, with sides retained

For homogeneous tangles write the source product as

$$x\cdot_G y=A^{-b_G(d(x),d(y))/2}\,x\cup_G y. \tag{17}$$

The target state degree is $-Pd(x)$, where here $P$ denotes the induced reversal of cyclic boundary slots. Reversing the surface orientation in the same ambient oriented collar reverses the cyclic triangle form:

$$b_C(Pa,Pb)=-b_G(a,b). \tag{18}$$

Both state signs have changed, so their two minus signs cancel inside a bilinear form. Combining (18) with reversal of stacking gives

$$F(x\cdot_G y)=F(y)\cdot_C F(x). \tag{19}$$

For clarity, the target exponent on the right is $-b_C(-Pd(y),-Pd(x))/2=-b_G(d(x),d(y))/2$, exactly the source exponent. Treating either degree reversal or triangle-order reversal as optional would be wrong. Equations (17)–(18) are the two cocycles in GY’s (31) and CK Section 4.1, written in common physical slots. The elementary matrix identity for reversal of $\Omega$ certifies (18) for all degrees at once.

The corner relations now match as actual stated relations. In the GY corner the diagonal values are

$$\text{state }--:v^{-1},\qquad \text{state }++:v, \tag{20}$$

with the admissible mixed corner equal to 1 and the bad corner equal to 0. These are GY’s (32), or GY-index’s (13). The state flip takes them to CK’s $++:v^{-1}$, $--:v$, good mixed corner 1 and bad corner 0. CK Definition 4.1 lists $++:v^{-1}$ and the good mixed corner; the inverse diagonal relation is equivalent in the reduced triangle algebra. The reversal of the marking also reverses the viewpoint defining the bad arc, so the bad arc is sent to the bad arc, not to its nonzero mixed partner.

Equation (19) sends a source relation with arbitrary multiplier on the left to the corresponding target relation with arbitrary multiplier on the right. It therefore descends through the corner ideal on its actual side. Similarly, the standard off-diagonal submodule

$$\sum_e\bigl(a^e_{+-}\,\mathcal S_G^{\rm cr} +a^e_{-+}\,\mathcal S_G^{\rm cr}\bigr) \tag{21}$$

goes to the target submodule with those factors on the right. This is exactly the image of CK’s left-generated off-diagonal ideal in Section 4.2. Neither corner quotient is being treated as an algebra. The relevant standard-arc subalgebra survives as the six-root quantum torus before the final corner relations.

On a standard diagonal arc the source root is its $--$ state, whereas the target root is its $++$ state. The untwisted-ribbon observation following (9) therefore gives

$$F(a^e_{\mu\nu})=a^{Pe,C}_{-\mu,-\nu},\qquad D_P(z_e)=X_{Pe},\qquad D_P(W_G(u))=W_C(Pu). \tag{22}$$

All four state choices, not just the two diagonal ones, were used before the off-diagonal quotient. By GY’s standard-arc spanning and CK’s Section 4.2, (22) identifies the local quantum traces on a spanning module. It is not inferred from the corresponding three-root quotient.

Algebraically, (5) sends $z^{-2}+(z'')^2-1$ exactly to $z_C^2+(z_C'')^{-2}-1$, permutes the four vertex monomials, and sends an edge monomial to its actual relabeled edge monomial. Hence it descends to a bijection

$$\overline D_P:Q_G^6\overset{\sim}{\longrightarrow}Q_C^6. \tag{23}$$

The bijection is of modules; its notation does not assert multiplication of arbitrary quotient classes. Its inverse is the same anchored reversal. The algebraic statements in this paragraph were independently checked in the preserved verifier and re-run in this continuation.

5. Named closed traces: the completed comparison square

Theorem 5.1. Same-parameter named collar dictionary

For an ideally triangulated cusped oriented three-manifold in the GY/CK setup, choose the common bare-edge anchors (4). With the source-reading convention for CK’s mixed-sided Laurent target retained, one has

$$\boxed{\quad\overline D_P\,\tau_G^6=\tau_{GY}^{CK}=\tau_{CK}.\quad} \tag{24}$$

The underlying closed framed link is unchanged; no mirror, chirality conjugation, central sign character or independently transported unnamed trace is substituted.

Proof. Represent a closed framed link on the embedded dual surface, as in GY’s construction. The ordinary-to-ribbon conversion represents the same framing. Reversing the collar changes its normal vector to the negative vector but leaves its ribbon line unchanged. There are no states to flip on the closed link. Thus the top left map in the comparison is the identity on the actual closed skein in the ambient manifold.

Split along the same physical face circles. Lemma 3.1 gives equality of the two state sums in the tensor product of the marked local modules, with every new state flipped and every physical height expressed in the reversed collar. Section 4 takes this equality through the corner and off-diagonal quotients, and (22) identifies the local trace maps on the standard-arc spanning modules. The edge rows and their sides are carried by (5), so the equality descends to the global target. This proves the first equality in (24) using the actual construction, not just the target presentation. CK Theorem 4.9 supplies the second equality. The surjection from the dual-surface skein onto the three-manifold skein is GY’s source input and CK Remark 4.4; because the two constructed maps agree before that surjection and descend, the equality holds for every link in its domain. $\square$

There is a separate semilinear inverse-root anti-map, $A\mapsto A^{-1}$, $i\mapsto-i$, $W(u)\mapsto W(-u)$. It remains algebraically valid but is not the map used in (24). Fixed-parameter inversion of identically labeled roots is not an anti-map: it already has a nonzero defect in the unquotiented quantum torus.

6. Audit and use of the native signed additive lift

The earlier note is retained as the native six-root proof. Its Section 5’s outstanding local gate is superseded by Sections 2–5 above, not by an assertion that its original algebraic check was sufficient.

The fresh audit re-read GY Sections 4.6–4.10 and Remark 5.1 and the entire GY-index Appendix A in the retained author revision dated 6 August 2025. The vulnerable steps are as follows.

1. The substandard arc is replaced by the parallel standard arc at the same bare edge, not its opposite. GY-index Lemma A.3 is a corner-reduced prequotient identity, modulo standard off-diagonal terms.
2. Products containing a standard off-diagonal factor vanish in the full local GY target defined by (60), before opposite-root identification. Standard arcs span the corner-reduced module by GY Lemma 4.22. The arbitrary extra skein in Appendix A is retained.
3. The height-exchange and Weyl-cancellation statements in Appendix A are prequotient statements. The six-root form is the pullback of the collapsed form: $B_6(u,w)=\Omega(\pi u,\pi w)$, $\pi(a,b)=a+b$. Retaining bare-edge labels therefore introduces no new commutation factor.
4. The Type II closing annulus contributes $A^{-1}$, cancelled by the residual height-order factor. It is retained. A standard-counit formula cannot be applied before standardizing that diagram.
5. The source’s admissible diagrammatic basis and its local lemmas remain substantial dependencies. We have audited the quotient stages; we do not claim an independent replacement proof of GY’s geometric lemmas.

Let $a,b$ be the two native basis rows, calibrated so their collapsed rows are the GY oriented peripheral rows, with $B_6(a,b)=2$. Put $L=W_G(a)$, $M=W_G(b)$. They commute with every edge row and define invertible left operators on all of $Q_G^6$, with $LM=A^2ML$. Their named insertion identities hold on every input. The exact skein identity

$$K_{\lambda+\mu}= \frac{A K_\lambda K_\mu-A^{-1}K_\mu K_\lambda}{A^2-A^{-2}} \tag{25}$$

and the source three-generator theorem then give

$$u_{p\lambda+q\mu}=pa+qb,\qquad E^G_\gamma=W_G(u_\gamma)=A^{-pq}L^pM^q, \tag{26}$$

$$\tau_G^6(K_\gamma\alpha)=(E^G_\gamma+(E^G_\gamma)^{-1})\tau_G^6(\alpha). \tag{27}$$

This is a generic statement; (25) uses $(A^2-A^{-2})^{-1}$. No integral or $A^4=1$ all-slope statement follows from this argument. No equality on the entire target between the actions built from different initial diagrammatic bases is asserted; they agree on the trace image. A trace need not be surjective for (25)–(27).

Corollary 6.1. Actual CK peripheral action

Define $E^C_\gamma=W_C(Pu_\gamma)$. Then right multiplication preserves the CK relation module and

$$\boxed{\quad\tau_{CK}(K_\gamma\alpha) =\tau_{CK}(\alpha)(E^C_\gamma+(E^C_\gamma)^{-1}).\quad} \tag{28}$$

Proof. Apply (24) to (27) and use reversal of multiplication in (5). Preservation of the quotient follows either by conjugating the native operator through (23), or directly because the monomial commutes with all target edge rows. $\square$

Put $L_C=D_P(L)$, $M_C=D_P(M)$. Then

$$M_CL_C=A^2L_CM_C,\qquad E^C_{p\lambda+q\mu}=A^{-pq}M_C^qL_C^p, \tag{29}$$

and the named sum monomial is $A^{-1}M_CL_C$, not $A^{-1}L_CM_C$. Right operators reverse composition, so their operator commutator has the original GY sign.

7. Original PP branch and central signs

Use the existing Eulerian branch transport from the CK degree-zero-descent publication, Section 6.2. It provides

$$\Phi(x_e)=i\epsilon_eX_e,\qquad \epsilon_e\in\{\pm1\},\quad \epsilon_e=\epsilon_{e^{\rm opp}},\qquad \Phi\tau_{PP1}=\mu\tau_{CK}. \tag{30}$$

Here $\mu$ is CK’s generator-matching map, now invertible by the published intersection theorem. The original PP1 constants are $(A^{-1},-i)$, while CK uses $(-iA^{-1},1)$. The Eulerian face construction adjoins a square root of $-i$; the shape map and trace identity descend to $k$. This prior result is a dependency, not re-proved by the present local tensor verifier.

Set

$$E^{PP}_\gamma =i^{-\sum_e(u_\gamma)_e}\epsilon^{Pu_\gamma} W_x(Pu_\gamma). \tag{31}$$

Equations (24), (28) and (30) give the original named PP1 formula

$$\boxed{\quad\tau_{PP1}(K_\gamma\alpha) =\tau_{PP1}(\alpha)(E^{PP}_\gamma+(E^{PP}_\gamma)^{-1}).\quad} \tag{32}$$

The inverse summand in (32) carries the reciprocal scalar. Explicitly its symmetric factor is

$$\epsilon^{Pu_\gamma} \left(i^{-\sum u_\gamma}W_x(Pu_\gamma) +i^{\sum u_\gamma}W_x(-Pu_\gamma)\right). \tag{33}$$

One may not extract a common power of $i$ from both summands without an additional parity justification.

For $u=(u_z,u_{z'},u_{z''},u_y,u_{y'},u_{y''})$, define

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

The exact central-vertex reduction is

$$W(u)=\sigma_T^\beta v^s A^{(r-s)(t_*-s)/2}z^{r-s}(z'')^{t_*-s},\qquad\sigma_T^2=1. \tag{34}$$

Since $\beta(pa+qb)=p\beta(a)+q\beta(b)$, the signed lift is additive. Neither (24) nor (32) discards the factors $\sigma_T$. A central $\sigma$-valued character could still be multiplied into an abstract torus action without affecting the identified target, but it would change (22) on the native standard-arc generators. The actual local square, rather than agreement after $\sigma_T=1$, fixes that ambiguity here.

8. What this changes for Dehn filling, and what it does not

The previous generic boundary-reconstruction theorem applies to any trace with a genuine torus-skein action. Thus (32) supplies its missing named-action hypothesis for the original PP six-root target, with (31) fixing the branch and collar conventions. The resulting relative targets and relative trace use the prior GJV/GJS generic category and excision inputs; this note does not replace them by open face-suspension monomial formulae.

For a filling meridian operator $m$, in the collar convention of that earlier theorem, its intrinsic filled target is isomorphic to $\operatorname{coker}(m+A^2)$. Equations (23), (30) intertwine these operators and identify the corresponding filled constructions. This gives an exact named filling construction; it does not show that this cokernel is always nonzero.

There is a conditional transferred nonvanishing consequence. The native six-root target maps onto its identified GY target, compatibly with (26). Tensoring for filling is right exact, so its filled target maps onto the identified filled target. Whenever the prior strictly positive geometric-filling theorem supplies a nonzero identified filled vacuum, the full six-root filled vacuum is also nonzero. The bijections (23), (30) carry this to the original PP target. This consequence retains every positivity, actual logarithmic-surgery, parameter and vacuum hypothesis in the positive-geometric proof note. Those hypotheses have not been re-established for arbitrary triangulations or fillings.

In particular, the local minus sector being nonzero does not prove that every global sign sector or every filled sector survives. The comparison theorem and the action theorem cannot turn a surjective operator into a nonsurjective one without a new argument.

Next precise question: for the actual six-root target and the named filling monomial (31), what intrinsic condition forces $m+A^2$ to be nonsurjective, beyond the regular-intersection or strictly positive geometric chamber? A universal detecting functional, a genuinely nonvanishing deformation-theoretic construction, or a decisive structural obstruction would address the remaining issue. Enumerating manifolds, slopes or triangulations would not answer that question.

Panitch–Romo Conjecture 7.1 remains open in this work. The present bridge is internally proved using the listed source inputs; no external referee has verified it. Chen and Kricker have been contacted separately, with Joshua Gay copied, concerning the earlier comparison publication; no response, confirmation or endorsement is inferred, and no new contact was made in this continuation.

References and reproducibility

• GY: Garoufalidis–Yu, A quantum trace map for 3-manifolds, Advances in Mathematics 486 (2026), 110735, published PDF. Sections 2.1–2.2, 4.3–4.10, Theorem 4.25, Remark 5.1.
• GYI: Garoufalidis–Yu, The 3d-index of the 3d-skein module via the quantum trace map, author revision dated 6 August 2025. Sections 2.2–2.4, 3.3, 5.2, 5.5; full Appendix A. The torus product-to-sum and three-generator inputs are credited there to Frohman–Gelca and Bullock–Przytycki.
• PP1: Panitch–Park, 3d quantum trace map, Algebraic & Geometric Topology 26 (2026), 2255–2314, published PDF. Definition 3.3, Remarks 3.4–3.5 and 3.9, Proposition 3.10, Theorem 3.21. The newly used local conversion and marking symmetry are theirs.
• CK: Chen–Kricker, On 3d Quantum Trace Maps, arXiv:2606.13268v1. Definition 2.2, Sections 4.1–4.3, especially Remark 4.2, Lemmas 4.3 and 4.6, Theorem 4.9; Theorem 5.34.
• PP2: Panitch–Park, Compatibility of quantum trace and UV–IR maps, arXiv:2509.09100v1. Definition 2.2 and footnote 5, Theorem 2.20, Remark 2.21.
• PR: Panitch–Romo, On the length conjecture for twist knots, arXiv:2609.02706v1, Conjecture 7.1.

Run python3 verify-six-root-peripheral.py and python3 verify-local-collar.py. Their outputs are retained separately. They certify exact algebra and tensor identities only. The geometric statements used from the cited papers remain explicit proof dependencies. The current hostile audit and source ledger specify the claim boundaries and inspected source pages.