Physical pair zero exclusion and its allocation polynomial infrastructure.

Abstract

Physical pair zero exclusion and its allocation polynomial infrastructure.

For a finite vertex type, write kappa(d)=d^2(d^2-1)/24, c_d=(d^2-3)/(4(d-1)) when d differs from one and c_1=0. Put L_1=13/48 and, for d at least two, L_d=d(d^2-1)(d-3)/96+d(d+1)(2d-3)/(16(d-1)). The form H is one half sum_v d_v^2 t_v^2 plus one half sum_uv a_uv t_u t_v minus z/4 times sum_v d_v(d_v^2-1)t_v plus z^2 sum_v kappa(d_v). The form Q is the pairQuadratic of the source, with one half the ordered distinct-vertex sum of d_u d_v(d_u d_v-a_uv)(z+eta_u)(z+eta_v)/9 and diagonal term kappa(d_v)(z^2+2z eta_v). The Cayley map is c(w)=(1-w)/(1+w), and eta(d,X)_v=c((-1)^d_v X^d_v).

The complexification is bilinear, with no conjugates. The graph identity retains the leaf constraint t_v=2z/3 and the exact 13/48 compensation per leaf. The internal degree-2 remainder is 5/16. The finite index types have no cardinality bound. The real parts of the physical Cayley variables are positive. Leaves obey eta_v=z in real and imaginary parts. The positive real quadratic value, the real sum-of-squares certificate and the bilinear lemma imply nonvanishing for every complex X of norm less than one; independent arbitrary leaf weights are outside this statement.

Lemma 1.1 (Rank one minus squares nonzero).

$$\begin{array}{r}J\text{finite},w_j\ge 0,K\ge 0,l,t_j\in \mathbb{C}\\ (\Re l{)}^2/2-\sum _j{w}_j(\Re t_j{)}^2>0⟹l^2/2-\sum _j{w}_j{t}_j^2+K\ne 0\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.rank_one_minus_squares_nonzero (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Lemma 1.2 (Graph sum of squares).

$$\begin{array}{r}V\text{finite},R\text{a field of characteristic zero},d_v\ge 1,a_{uv}=a_{vu},\sum _v{a}_{uv}=d_u,d_v=1⟹t_v=2z/3\\ H(d,a,z,t)=\sum _v{d}_v(d_v-1)(t_v-c_{d_v}z{)}^2/2+\sum _{u,v}{a}_{uv}(t_u+t_v-z/2{)}^2/4+\sum _v{L}_{d_v}{z}^2\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.graph_sum_of_squares (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Lemma 1.3 (Pairquadratic rank one).

$$\begin{array}{r}V\text{finite},R\text{a field of characteristic zero},d_v\in \mathbb{N},a_{vv}=0,z,{\mathrm{eta}}_v\in R,t_v=d_v(z+{\mathrm{eta}}_v)/3\\ \mathcal{Q}(d,a,z,\mathrm{eta})=(\sum _v{d}_v{t}_v{)}^2/2-H(d,a,z,t)\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.pairQuadratic_rank_one (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Theorem 1.4 (Physical pairquadratic nonzero).

$$\begin{array}{r}V\text{finite},d_v\ge 1,0\le a_{uv}\le 1,a_{uv}=a_{vu},a_{vv}=0,\sum _v{a}_{uv}=d_u,\exists v{d}_v\ge 2\\ \forall X\in \mathbb{C},K\in \mathbb{R},|X|<1\wedge K\ge 0⟹\mathcal{Q}(d,a,c(-X),\mathrm{eta}(d,X))+K\ne 0\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.physical_pairQuadratic_nonzero (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Write F(d,r;X)=1+(-1)^d X^(d+r) and P(X)=sum_alpha w_alpha product_v F(d_v,r_alpha,v;X). Let h count the vertices of odd d_v. Define the quotient vertex factor as the geometric sum from i=0 to d+r-1 when d is odd, and as F(d,r;X) when d is even; Q is the same weighted allocation sum with these quotient factors. The notation p_C means extension of real coefficients to the complex numbers, ord_1 means root multiplicity at one, and H_N(p)(u,v)=sum over k at most N of p_k u^k v^(N-k) is canonical homogenization. Natural degree uses zero as the natural degree of the zero polynomial.

The hypotheses quantify over arbitrary finite allocation types, positive degrees and weights, a common total mass and even degree sum. No coefficient cancellation is assumed. Every odd-degree vertex contributes one factor 1-X; the quotient has strictly positive value at one. Reciprocity transfers open-disk exclusion to the exterior. Homogenization uses the canonical Mathlib polynomial construction.

Lemma 1.5 (Allocation degree and leading).

$$\begin{array}{r}V,A\text{finite},A\ne \varnothing ,d_v>0,w_{\alpha }>0,r_{\alpha v}\in \mathbb{N},\sum _v(d_v+r_{\alpha v})=N,\sum _v{d}_v\text{even}\\ \mathrm{natDegree}(P)=N\wedge \mathrm{leadingCoeff}(P)=\sum _{\alpha }{w}_{\alpha }\wedge P\ne 0\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.allocation_degree_and_leading (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Lemma 1.6 (Allocation multiplicity one).

$$\begin{array}{r}V,A\text{finite},A\ne \varnothing ,d_v>0,r_{\alpha v}\in \mathbb{N},w_{\alpha }>0⟹{\mathrm{ord}}_1(P)=h\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.allocation_multiplicity_one (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Lemma 1.7 (Allocation roots on circle).

$$\begin{array}{r}V,A\text{finite},d_v,r_{\alpha v}\in \mathbb{N},w_{\alpha }\in \mathbb{R},\sum _v(d_v+r_{\alpha v})=N,\sum _v{d}_v\text{even}\\ (\forall z\in \mathbb{C},|z|<1⟹P_{\mathbb{C}}(z)\ne 0)⟹(\forall z\in \mathbb{C},P_{\mathbb{C}}(z)=0⟹|z|=1)\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.allocation_roots_on_circle (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Lemma 1.8 (Allocation quotient degree).

$$\begin{array}{r}V,A\text{finite},A\ne \varnothing ,d_v>0,r_{\alpha v}\in \mathbb{N},w_{\alpha }>0,\mathrm{natDegree}(P)=N⟹\mathrm{natDegree}(Q)=N-h\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.allocation_quotient_degree (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Lemma 1.9 (Homogenize complex eval).

$$\begin{array}{r}\forall p\in \mathbb{R}[X],N\in \mathbb{N},(u,v)\in {\mathbb{C}}^2,\mathrm{natDegree}(p)\le N\wedge v\ne 0⟹H_N(p)(u,v)=p_{\mathbb{C}}(u/v)v^N\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.homogenize_complex_eval (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

Theorem 1.10 (Homogenize complex infinity).

$$\begin{array}{r}\forall p\in \mathbb{R}[X],N\in \mathbb{N},u\in \mathbb{C},H_N(p)(u,0)=p_N{u}^N\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/LeeYang/TreePairStability.homogenize_complex_infinity (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. B. Fiol; J. Martínez-Montoya; A. Rios Fukelman (2020). The planar limit of N=2 superconformal quiver theories. DOI: 10.1007/JHEP08(2020)161. URL: https://arxiv.org/abs/2006.06379v2.

Commentary.

Natural-number subtraction is truncated at zero. All parameters range universally over the indicated domains. The displayed conditions are the hypotheses of the relation that follows.

References

• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.allocation_degree_and_leading
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.allocation_multiplicity_one
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.allocation_quotient_degree
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.allocation_roots_on_circle
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.graph_sum_of_squares
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.homogenize_complex_eval
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.homogenize_complex_infinity
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.pairQuadratic_rank_one
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.physical_pairQuadratic_nonzero
• Truth anchor: D5/S3/Zeros/LeeYang/TreePairStability.rank_one_minus_squares_nonzero