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[3] ai.viXra.org:2602.0094 [pdf] submitted on 2026-02-19 17:15:36
Authors: Vinicius F. S. Santos
Comments: 45 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We develop the spectral theory of operators whose eigenvalue structure is governed by the golden ratio φ = (1 +√5)/2. The foundation is the golden resolvent factorisation: for any real symmetric matrix A, λ2I − λA − A2 = (λI − φA) (λI + A/φ). This identity controls the spectrum of golden companion operators, decomposing eigenvalues into transparent pairs (μφ and −μ/φ) and coupled modes (roots of an explicit secular equation), governed by the Galois conjugation φ 7→ −φ−1 of Q(√5)/Q. We establish six main results: (1) the Golden Amplification Theorem, producing the transparent eigenvalue pairs and their eigenvectors; (2) the Secular Equation, a closed-form characteristic polynomial for coupled modes; (3) the Secular Sensitivity Theorem, identifying the secular weight with the coupled eigenvector norm and establishing Lipschitz continuity of coupled eigenvalues in the coupling vector; (4) the Positive Boost Inequality, bounding submatrix spectral radii via nodal-domain restrictions; (5) the Galois Transfer Principle, exactly classifying partition transfer for conjugate eigenvector pairs via the Transfer/Pareto Trichotomy—with automatic spectral dominance for transparent modes and unavoidable Pareto regimes for coupled modes; (6) the Chebyshev Ladder, generalising the amplification ratio to 2 cos(π/(2p+3)) through the cyclotomic fields Q(ζ2p+3)+. The spectral spread of the golden pair equals the generator of the different ideal of Z[φ], connecting the framework to the arithmetic of Q(√5).
Category: Combinatorics and Graph Theory
[2] ai.viXra.org:2602.0026 [pdf] submitted on 2026-02-08 00:53:04
Authors: Vinicius F S Santos
Comments: 17 Pages. (Note by viXra Admin: Parts of the texts are cut off!)
The Erdős—Lovász Tihany Conjecture (1968) asserts that every graph G with χ(G) ≥ s + t − 1 > ω(G) admits a vertex partition into parts with chromatic numbers ≥ s and ≥ t, respectively. We prove the conjecture for the infinite family of pairs (3, k−2) on Mycielski graphs Mk for all k ≥ 5. Our approach is spectral, centred on the golden ratio φ = (1+√5)/2. The pentagon C5—the minimal graph with χ > ω—has adjacency eigenvalues {2, φ−1, φ−1,−φ,−φ}, and this golden spectral structure propagates through the Mycielski construction: the golden ratio’s defining equation μ2 − μ − 1 = 0 arises exactly from the Mycielski eigenvalue relation (Lemma 2.1). We prove the C5-Peeling Existence Theorem: for every Mk (k ≥ 4), a direction in the golden eigenspace peels off a C5 from Mk. The proof is constructive via spectral interferometry: two Mycielski lift paths span a four-dimensional subspace whose layer-control matrix has det = √5, enabling independent phase steering to select a diagonal lift C5—the reverse cycle through alternating address layers. Computationally, the Hoffman margin of this partition is F = φ−3 = √5 − 2 exactly, verified for all k ≤ 12. The key advance is the Golden Sub-Induction (Theorem 1.2): for k ≥ 6, the 1/φ shadow attenuation forces the peeled C5 into the original vertex block of Mk, so the remainder Mk Pk contains Mycielski(Mk−1 Pk−1) as a subgraph, where (Pk)k≥5 is a coherent family of peeled pentagons. Since χ(Mycielski(G)) = χ(G) + 1 for any graph with an edge, this yields the inductive bound χ(Mk Pk) ≥ k − 2. Combined with χ(C5) = 3, this settles the Tihany conjecture for the pair (3, k−2) on Mk for every k ≥ 5—an infinite family of previously open cases.
Category: Combinatorics and Graph Theory
[1] ai.viXra.org:2602.0007 [pdf] submitted on 2026-02-02 19:31:23
Authors: Lucas Aloisio
Comments: 5 Pages.
We revisit the classical Unit Distance Problem posed by Erdős in 1946. While the upper bound of O(n4/3) established by Spencer, Szemerédi, and Trotter (1984) is tight for systems of pseudo-circles, it fails to account for the algebraic rigidity inherent to the Euclidean metric. By integrating structural rigidity decompositionwith the theory of Cayley-Menger varieties, we demonstrate that unit distance graphsexceeding a critical density must contain rigid bipartite subgraphs. We prove a "FlatnessLemma," supported by symbolic computation of the elimination ideal, showing that the configuration variety of a unit-distance K3,3 (and by extension K4,4) in R2 is algebraically singular and collapses to a lower-dimensional locus. This dimensional reduction precludes the existence of the amorphous, high-incidence structures required to sustain the n4/3 scaling, effectively improving the upper bound for non-degenerate Euclidean configurations.
Category: Combinatorics and Graph Theory