Metallic Means and the Pythagorean Angle Lattice: A Complete Classification of Metallic Values at Every Depth of the Acute-Angle Difference

Every metallic value appearing at any depth of the acute-angle difference of a primitive Pythagorean triple is completely classified: they arise only from odd Gaussian powers of consecutive-leg seeds and lie in the Pell orbit of 1 + √2. For a primitive Pythagorean triple T = (a, b, c) with 0 < a < b < c, let Δ(T) be the difference of its two acute angles. We prove that the quarter-difference coordinate cot(Δ/4) = (a + b + c√2)/(b − a) lies, for every triple, in the single field ℚ(√2) with norm −1, and that it is an integral-index metallic mean M_n exactly when the legs are consecutive, in which case cot(Δ/4) = M_{2(a+b)} = (a + b) + c√2 = (1 + √2)^β for an odd β ≥ 3. The passage from divided angles to divided differences is a projective involution in PGL₂, and every metallic value occurring at any depth 4d lies in the Pell orbit {(1 + √2)^β}. We introduce an odd Gaussian-power action T_d on primitive triples, which preserves primitivity, satisfies T_e ∘ T_d = T_{ed} and multiplies Δ by d in the unfolded chamber. The main theorem classifies completely the solutions of cot(Δ(T)/(4d)) = M_n: they exist exactly for odd d, odd β ≥ 3 with d · 4 arctan((1 + √2)^{−β}) < π/2, and T = T_d(T_β), where T_β is the consecutive-leg seed; the hypotenuse is P_β^d, with P_β a Pell number. Each metallic mean thus has a finite, explicit depth spectrum, for example {1, 3, 5} for M_{14}, realised by (3, 4, 5), (44, 117, 125) and (237, 3116, 3125). We further determine the minimal polynomial of the divided difference coordinates of the seeds, of degree 2^d, using Kummer theory and the theorem of Pethő and Cohn on perfect powers in the Pell sequence; we prove a fibre-rigidity theorem for the lattice coordinates; and we record the consequences for the squareclass-return problem, including a genus-one curve with j-invariant 1728 attached to every primitive Pythagorean source.