The support conjecture for rank-two quantum triangular bases

Public text index for scholarly discovery. It mirrors theorem, lemma, proposition and related statements from this immutable version and does not imply human verification.

1. Lemma 3.1
   PF:2026.000012/v1/LEM-3.1

   Suppose \({\mathscr F}_{\nu,w}\ne\varnothing\), \(u\in{\operatorname{Dom}}(w)\) is uniform, and \(u\leq\nu\). With the equivariant structures fixed above, there are graded representation isomorphisms for the cyclic group \(\langle\sigma\rangle\) generated by \(\sigma\): \[ V^j_{\nu,u;w}\cong V^j_{\nu-u,0;w\ominus u}. \] Consequently \(\rho_{\nu,u;w}=\rho_{\nu-u,0;w\ominus u}\).

2. Proposition 4.1
   PF:2026.000012/v1/PROP-4.1

   For a uniform \(\nu=(x,y)\) satisfying (equation reference), \sum_{\substack{u\in{\operatorname{Dom}}(w),\,u\leq\nu\\u\text{ uniform}}} \rho_{\nu,u;w}(z)P_{u,w}(z) = z^{d_0-\widetilde d} \genfrac{[}{]}{0pt}{}{a'}{x}_{z^m}\genfrac{[}{]}{0pt}{}{d'+cx}{y}_{z^n}.

3. Proposition 4.2
   PF:2026.000012/v1/PROP-4.2

   The element \(G_w\) is the triangular basis element represented by the standard monomial \(M'[a-a',d'-d]\) in the adjacent seed \((X_1,X_0)\). In particular, it depends only on \(a-a'\) and \(d'-d\).

4. Corollary 4.3
   PF:2026.000012/v1/COR-4.3

   For any four-parameter frame, \[ x>[a'-a]_+\quad\Longrightarrow\quad \rho_{(x,y),0;w}=0. \]

5. Lemma 5.1
   PF:2026.000012/v1/LEM-5.1

   For \(0\ne u=(s,t)\in{\operatorname{Dom}}(w)\), set \[ F_u&=-Q(s,t)+mAs-n(cA-B)t,\\ F_u^{\mathrm{sw}}&=-Q(s,t)-mAs+n(cA-B)t,\\ J_u&=-Q(s,t)-2mbst+mAs+n(cA-B)t. \] Then \deg P_{u,w}\leq\min(-1,F_u,F_u^{\mathrm{sw}},J_u).

6. Lemma 5.2
   PF:2026.000012/v1/LEM-5.2

   For every frame and every admissible \(\nu\), \[ \deg\rho_{\nu,0;w}\leq F_w(\nu),\qquad F_w(\nu)<0\ \Longrightarrow\ \rho_{\nu,0;w}=0. \] For the special frame (equation reference), a nonzero \(\rho_{\nu,u;w}\), \(u=(s,t)\), also satisfies x-s\leq[A-bt]_+.

7. Lemma 6.1
   PF:2026.000012/v1/LEM-6.1

   Suppose \(A,B>0\) and \(cA-B>0\), and assume either \(bc>4\) and \(cA^2-bcAB+bB^2<0\), or \(bc=4\) and \(B=cA/2\). Let \(x,y,s,t\) be real numbers satisfying \(F_w(x,y)\geq0\) and 0<s\leq x\leq A,\qquad 0<t\leq y,\qquad s\leq bt,\quad t\leq cs,\quad t\leq cA-B,\\ x-s\leq[A-bt]_+. Define \Delta&=Q(s,t)+m(A-2x)s+n(2cx-2y-cA+B)t,\\ L_2&=2[Q(s,t)-mxs+n(cx-y)t],\\ L_3&=2[Q(s,t)+m(A-x)s+n(cx-y-cA+B)t],\\ L_4&=2[ms^2+nt^2-mxs+n(cx-y-cA+B)t]. Then at least one of \[ \Delta\geq0,\qquad L_2>0,\qquad L_3>0,\qquad L_4>0 \] holds.

8. Proposition 6.2
   PF:2026.000012/v1/PROP-6.2

   For the frame (equation reference), suppose \(F_w(\nu)\geq0\). Every nonzero uniform rank \(u\) satisfies \[ \deg\bigl(\rho_{\nu,u;w}P_{u,w}\bigr)<F_w(\nu). \]

9. Theorem 7.1
   PF:2026.000012/v1/THM-7.1

   Let \(b,c,A,B\) be positive integers satisfying \[ cA^2-bcAB+bB^2\leq0. \] For the coefficients in (equation reference), set \[ D(p,q)=cAq+bBp-bp^2-bcpq-cq^2. \] Then \[ e(p,q)\ne0 \quad\Longleftrightarrow\quad 0\leq p\leq B,\quad 0\leq q\leq A,\quad D(p,q)\geq0. \] At every point in this support, \[ \deg_{{\mathbf v}}e(p,q)=D(p,q),\qquad [{\mathbf v}^{D(p,q)}]e(p,q)=1. \] The coefficients are bar invariant. In particular, \(e(p,q)=1\) whenever \(D(p,q)=0\) in the stated rectangle.