This paper establishes a rigorous algebraic encoding of the Ramanujan constant 1103—the spectral weight in Ramanujan's celebrated series for 1/π—within the spectral theory of the Fibonacci matrix. The starting point is the canonical Lucas–Zeckendorf decomposition of 1103: 1103 = L₁₄ + L₁₁ + L₈ + L₅ + L₂ The five indices 14, 11, 8, 5, 2 form an arithmetic progression with common difference −3, and every index satisfies n ≡ 2 (mod 3). This purely combinatorial observation becomes algebraically significant when combined with the classical identity Tr (Fⁿ) = Lₙ, where F is the Fibonacci matrix. It follows immediately that 1103 is the trace of a sum of Fibonacci matrix powers: 1103 = Tr (F² + F⁵ + F⁸ + F¹¹ + F¹⁴) This forces the definition of a one-parameter family of 2×2 integer matrices: Sₘ = ∑ₖ₌₀ᵐ F³ᵏ⁺² for which S₄ encodes the Ramanujan constant. The paper computes the complete spectrum of Sₘ in closed form: Trace: Tr (Sₘ) = (L₃ₘ₊₄ − 1) /2 Eigenvalue coefficients: aₘ = ∑ᵢ F₃ᵢ₊₂, bₘ = ∑ᵢ F₃ᵢ₊₁ Determinant: det (Sₘ) = (L₃₍ₘ₊₁₎ − 1 − (−1) ᵐ⁺¹) /4 All spectral invariants are expressed purely in Lucas and Fibonacci numbers, with no free parameters. The arithmetic progression structure of the Zeckendorf indices propagates faithfully into the mod-3 separation of the eigenvalue components. The entire construction follows uniquely from the Zeckendorf decomposition of 1103; no parameter is tuned, no arbitrary choice is made. The paper does not claim to explain why 1103 occurs in Ramanujan's series—that question belongs to complex multiplication theory—but establishes a precise, invertible algebraic encoding of this constant within Fibonacci matrix algebra. This is the third installment of a series. The first paper reported the arithmetic progression structure in the Zeckendorf decomposition of 1103. The second extended the verification to five discriminants and identified the boundaries of the phenomenon. The present paper provides the rigorous spectral-theoretic foundation, upgrading the earlier numerical observations to closed-form theorems.
Li Yunlong (Sat,) studied this question.