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[2] ai.viXra.org:2609.0038 [pdf] submitted on 2026-09-17 22:36:50
Authors: Wen PeiLin
Comments: 81 Pages.
We study a family of directed graphs on n = 2k+ 5 vertices whose adjacency matrices are{0, 1}-matrices with zero diagonal, uniform column sum 2, and a rigid Jordan structure: thenon-zero spectrum {2, 1, −1, −1, −1} is constant in k, while a single nilpotent block J2k(0)grows with k. We call this the Wen class Uk and define it through five axioms D1—D5. The main result of the paper settles the existence question. We construct an explicit infinite subfamily {Am}m≥1 with ord(Am) = 2m + 7, obtained by iterating a chain-insertionoperator T on a fixed 7 × 7 core C. A Schur-complement identity for T, combined with aresolvent computation that reduces to three cofactors of the single 7 × 7 matrix zI − CT, shows that T multiplies the characteristic polynomial by z 2 ; the exact determination ofdim ker Am = 1 and dim ker(Am + I) = 2 then upgrades this spectral statement to the fullJordan axiom D4. Consequently Uk 6= ∅ for every integer k ≥ 1: the class is infinite, andno order is empty. Every step of the proof is a finite exact computation of order at most 7,independent of m. We prove the closed-walk counting formula NL = 2L + 1 + 3(−1)L, the rank formula rank(Am) = max(n − m, 5), the minimal polynomial x 2k (x − 2)(x − 1)(x + 1)2 of degree n − 1, the commutant dimension 7 + 2k, the Newton-identity recurrence of order 5,and the doubler-chain power-of-two property. A computer search at seven orders (k =1, . . . , 7, i.e. n = 7, . . . , 19) produced 233 pairwise non-isomorphic members, distributed as 34, 50, 39, 22, 68, 13, 7; these numbers are lower bounds for the number of isomorphism classes at each order, and ten representatives of each order are listed in Appendix 18. We disprove two natural conjectures (the "minimum-one integer Perron vector" property and the "core-template" hypothesis) and organise the observed structure into a three-layer architecture separating rigid invariants, semi-invariants and flexible realisation.
Category: Combinatorics and Graph Theory
[1] ai.viXra.org:2609.0017 [pdf] submitted on 2026-09-08 16:35:15
Authors: Marian Opial
Comments: 13 pages. Licensed under CC BY 4.0. AI-assisted.
For a finite set S of integers, its integer Turnpike data record the multiplicity of every positive difference. In the dense frequency-vector encoding, realizability reduces to parity of self-reciprocal irreducible factors and is therefore decidable in polynomial time by ordinary polynomial factorization. Sparse binary-exponent encoding is different: the associated reciprocal polynomial can have exponentially large degree. We construct an infinite family of sparse, coefficientwise nonnegative, Fourier-nonnegative, arithmetically admissible Turnpike instances with the fixed point count n = 107,520 that are nevertheless nonrealizable. Each instance contains, to odd multiplicity, a primitive noncyclotomic self-reciprocal irreducible of degree 2^k, while the sparse input length is Theta(k). Thus Fourier positivity, elementary count identities, and visible monomial-composition tests do not imply any polynomial bound on the degree of the remaining obstruction. Several complementary results locate the residual difficulty. Under the Turnpike scalar identities, an integral Hermitian sum of squares collapses to one binary rank-one factor; a parity-safe multiplier cannot raise the square-root-total/center ratio; and a strictly Fourier-positive integer NO instance can lie in the midpoint of two YES instances. Every odd self-reciprocal factor has infinitely many reciprocal quadratic shadows modulo primes, with a density lower bound in terms of permutation rank, but cyclotomic examples rule out naive small-prime enumeration. Finally, a ramified quadratic trace gadget converts a Legendre symbol into the desired characteristic-zero reciprocal-parity bit while preserving Fourier positivity. A local-to-global counterexample shows why this gadget does not yet yield a hardness reduction.
Category: Combinatorics and Graph Theory