The Golden Ratio in Infinite Products: A Cyclotomic Parallel to the Rogers–Ramanujan Continued Fraction

We introduce and study the two-parameter family of block-alternating infinite products M^(d)(a,b), where the exponent imposes a block-alternating sign pattern of period 2d. For d=1 we establish a closed-form identity that transforms the classical sine-ratio product into a tangent ratio, yielding algebraic values in Q(√2), Q(√3), and Q(√5). For d=2 at (a,b)=(2/5,4/5), we prove the rational value M^(2)(2/5,4/5)=5/2 via a double √5-cancellation rooted in the Gauss multiplication formula and the arithmetic of Q(ζ₂₀). As d→∞, limiting values for 1/5-multiple parameters lie in the golden-ratio field Q(√5), with φ=(1+√5)/2 appearing explicitly. The paper concludes by drawing a structural parallel with the Rogers–Ramanujan continued fraction R(q), showing that the same field Q(√5) arises through the arithmetic of cyclotomic fields rather than modular geometry: the engine is the same; the vehicle is different.