The Budargham Constant λ(E5): Minimal Polynomial, 26×26 Incidence Matrix M, and 1,000-Digit Expansion

Abstract

The Budargham Constant ; λE5 (BE5):λE5 = 3.1415932235776478077659351649002756071087111553047325144046... This record formally documents the mathematical constant λE5 (the EOA Limiting Ratio under encoding E5). It is defined as the unique strictly dominant real root (Perron root) of the monic, irreducible degree-13 integer polynomial: G2(x) = x13 - 9x12 + 30x11 - 38x10 - 26x9 + 152x8 - 210x7 + 116x6 + 34x5 - 98x4 + 74x3 - 45x2 + 21x - 4 Key Properties:• Spectrum of M: λE5 is the spectral radius of the released 26×26 non-negative integer incidence matrix M (matrix_M_E5.csv). Over Q, the characteristic polynomial factors with an equal-degree split: charpoly(M) = (x - 2) · x3 · (x - 1)9 · G2(x). • Galois Invariant: G2(x) is irreducible with full symmetric Galois group S13 (order 13! = 6,227,020,800). • Certified Precision: Full 1,000-digit expansion is certified in lambda_E5_1000_digits.txt (verified by checksums.sha256). • Proximity to π: |λ_E5 - π| ≈ 5.70 × 10^-7 (treated as a numerical coincidence). • Independent Verification: verify_e5.py recomputes the characteristic polynomial factorization and all 1,000 digits directly from M and G2(x). Staged Disclosure:The matrix M, polynomial G2(x), and 1,000 digits are fully public. The phonetic letter-name strings and recursive β-operator that generated M are withheld under staged disclosure. Access for collaboration: bdarghamneurolabs@gmail.com.