The Budargham Constant λ(E5): Minimal Polynomial, 26x26 Incidence Matrix M, and 1,000-Digit Expansion

The Budargham Constant ; λ_E5 (B_E5):
λ_E5 = 3.1415932235776478077659351649002756071087111553047325144046...

This record formally documents the mathematical constant λ_E5 (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: λ_E5 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).

• Reciprocal Constant: 1/λ_E5 ≈0.318309886... marks the exact radius of convergence of the generating function.

• 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: [e-mail].

Access the full matrix, 1,000-digit file, and verification scripts on Zenodo: doi.org/10.5281/zenodo.23395237