The Pythagorean Angle Lattice in Quantum Mechanics: Exact Spin Rotations, Gate Synthesis Obstructions and Arithmetic No-Go Theorems for Quantum Computation

The Pythagorean Angle Lattice produces exact spinor-norm and continuity no-go theorems for single-qubit gate synthesis, free and dense groups of difference gates, and exponent walls at positive distance from the Clifford group. Let θ be the smaller acute angle of a primitive Pythagorean triangle (a, b, c) and z = eiθ = (b + ia)/c. The rational norm-one phases form the Pythagorean angle lattice P ⊂ ℚ(i). We study the exact arithmetic of the associated single-qubit rotations and of the difference angle Δ = π/2 − 2θ, whose spin rotation is U_n(Δ) = (A I − i B n · σ)/(√2 c) with A = a + b, B = b − a. Within the sector of rational rotation matrices, Gaussian rational eigenphases occur exactly for rational Bloch axes, and the obstruction to an exact rational spin lift is the spinor squareclass, which on these phases is the halving class [N(1 + z)]. Every Pythagorean difference gate has spinor norm [2] and factors exactly as a quarter-turn times the rational spin gate defined by the triple. This gives a spinor-norm obstruction to exact synthesis with forced length parity, an obstruction to synthesising the rotation by θ from difference and Clifford gates unless c is a perfect square, and an exact criterion for rational fourth roots; difference gates about orthogonal axes generate a free group, and about non-parallel axes a dense group with approximating words of logarithmic length. For pairs of gates we prove exact commutator identities, including a relation between the commutator scalar and the Born overlap of PAL-internal axis states; nested commutators obey an exact logistic recurrence for their transverse intensity, with a transition at θ = π/12 and a 2-adic filtration of their spectral discriminants. In odd prime-power dimensions, suitable Weil operators admit exponent intertwiners U ↦ U⁻² at projective distance at least 1/(2√2) and diamond distance at least √(1 − p⁻ᵏ) from the Clifford group, with Weyl spreading at least 1/360. Finally, density of the dyadic classes gives a continuity obstruction: without an a priori height bound, no fixed finite-copy continuous protocol separates them with a uniform margin. All results concern exact arithmetic structures inside ordinary quantum mechanics; no modification of its postulates is proposed.