A Finite Certificate for the Positive $n=11$ Vasc Inequality

1.1 Convention

• The Vasc expression is order-dependent, but is invariant under cyclic rotation and common positive scaling.Source: §1.1 · p. 1

1.2 Related work

• Prior certificate approaches include successive difference substitutions, structured column-stochastic substitutions, Pólya-related methods, and AM–GM-based certificates.Source: §1.2 · pp. 1–2
• Earlier work covers the positive n=9 case, while a reported n=11 result concerns only a small cone; this paper partitions all normalized rank words.Source: §1.2 · p. 2

1.3 Statement

• Theorem 1 states the Vasc cyclic inequality for every strictly positive eleven-tuple, and every constant positive tuple attains equality.Source: §1.3 · p. 2

1.4 Proof strategy

• The proof reduces the inequality to homogeneous polynomials indexed by normalized rank words, then uses nonnegative coefficients or midpoint decompositions on subdivided gap cones.A finite certificate assembles these cones before recovering the rational inequality.Source: §1.4 · p. 2

2.1 Clearing the positive denominators

• Multiplying by the positive denominator product yields an integer polynomial whose sign is equivalent to that of the rational expression.The polynomial is homogeneous of degree eleven.Source: §2.1 · p. 3

2.2 Normalized rank words

• After rotating a distinct-coordinate tuple so its minimum is first, its rank word records the ranks at the original cyclic positions; there are 10! such words.Source: §2.2 · p. 3
• Cumulative gaps represent weakly increasing positive lists, with nonnegative gaps allowing ties and zero values in the polynomial certificates.The gap-to-list construction and its inverse are given by adjacent differences and telescoping sums.Source: §2.2 · p. 3
• For each rank word, substituting its linear forms defines a polynomial on the strictly positive gap domain.The passage also notes that some possible degree-eleven monomial coefficients are zero.Source: §2.2 · p. 4

• Each leaf certifies its defining polynomial on a transformed orthant, using g = Tz for a path-reconstructed nonnegative integer matrix T.Source: §3 · p. 4
• The transformed variables z lie in R11, and the leaf certificate concerns nonnegative values on the transformed orthant.Source: §3 · p. 4
• Monomials use the convention 0^0 = 1.Source: §3 · p. 4

3.1.1 Weighted midpoint circuits

• A weighted midpoint circuit uses positive integer weights and exponent vectors satisfying the stated midpoint relation and degree condition.In a degree-(11 + k) leaf, all three exponents have degree 11 + k.Source: §3.1.1 · p. 4
• Every circuit satisfying the midpoint relation and d^2 ≤ 4uv is nonnegative on R11.Source: §3.1.1 · p. 4

3.1.2 The exact leaf test

• A leaf is accepted either when every transformed-polynomial coefficient is nonnegative or when every residual coefficient is nonnegative after circuit subtraction.The residual is formed for k in {0, 1, 2, 3}; its coefficients are obtained by exact expansion.Source: §3.1.2 · p. 5
• An accepted leaf proves the transformed polynomial is nonnegative throughout the nonnegative orthant.Source: §3.1.2 · p. 5

3.2 Finite covers of gap cones

• A cover begins at the identity matrix, and every internal node must have the prescribed children and complete accepted subtrees.Child images may overlap on their boundaries.Source: §3.2 · p. 5, §3.2.2 · p. 6

3.2.1 Binary comparison nodes

• Binary comparison splits on either ordering of two coordinates, using substitutions that produce two children covering the parent orthant.The child matrices are computed from the parent matrix by adding one selected column to another.Source: §3.2.1 · p. 5

3.2.2 Partial sorting nodes

• Partial sorting orders a selected group of 2–6 coordinates and creates one child for each permutation, covering the parent orthant.The substitutions use successive coordinate differences; ties can place a point in multiple children.Source: §3.2.2 · p. 6
• If every leaf is accepted, induction over the finite tree proves the root polynomial nonnegative for every nonnegative gap vector.The induction combines leaf soundness with the coverage of each binary or partial sorting node.Source: §3.2.2 · p. 6

3.3.1 The rank-word partition

• Every rank word belongs to one of four disjoint classes: coefficientwise-nonnegative roots, ordinary midpoint leaves, binary trees, or trees with partial sorting nodes.The missing set, outside-exception set, and pairwise intersections are all empty.Source: §3.3.1 · pp. 6–7

3.3.2 Independent polynomial reconstruction

• The checker reconstructs each cleared polynomial from its linear factors and checks every root with a complete degree-eleven coefficient array.Shared permutation prefixes reduce repeated work; the recurrence is also compared with direct expansion and exact point evaluations.Source: §3.3.2 · p. 7

3.3.3 Arithmetic safety and recorded workload

• The untransformed polynomial coefficient norm is bounded by 2,921,591,507,056,640, below 2^52, supporting signed 64-bit classification arithmetic.Transformed-tree verification uses signed 128-bit arithmetic with safety checks, while the circuit inequality is checked with arbitrary-precision integers.Source: §3.3.3 · pp. 7–8
• The ordinary batches check 81,961,459 circuits and 123,084,490,482 residual coefficient positions; sort-containing batches cover 36 roots with 4,575 leaves.The sort-containing batches contain 1, 29, and 6 roots, with 120, 3,533, and 922 leaves, respectively.Source: §3.3.3 · p. 8

3.4 Proof of the main theorem

• For pairwise-distinct coordinates, cyclic rotation places the minimum first, yielding a unique rank word and strictly positive gaps.The cyclic expression is unchanged by rotation.Source: §3.4 · p. 8
• The accepted class of each rank word proves its gap polynomial nonnegative, and the positive denominator then yields the inequality for every distinct positive tuple.The proof uses coefficientwise nonnegativity for one class, accepted identity-matrix leaves for another, and cover soundness for the tree classes.Source: §3.4 · p. 8
• Distinct positive tuples are dense, so continuity extends the inequality to tuples with ties; when all coordinates equal a positive value, every numerator is zero.The denominator remains positive on the domain.Source: §3.4 · p. 8

4 Discussion

• Cumulative sums encode ordered values as nonnegative parameters, while midpoint AM–GM circuits supply nonnegative polynomial summands.Exact expansion turns the certificate conditions into integer coefficient inequalities.Source: §4 · p. 8
• The disjoint rank-word partition connects local decompositions on gap cones to every normalized rank word.The certificate conditions reduce to exact integer coefficient inequalities.Source: §4 · pp. 8–9
• Binary comparison resolves one ordering relation at a time, while partial sorting handles a selected group at once; larger sorting steps are concentrated on 36 roots.Direct coefficients and ordinary midpoint leaves settle more than 99.9% of the root family.Source: §4 · p. 9