Pythagorean Angle Lattice: Rational Division, Degree Rigidity, Source–Division Duality, and Global Reconstruction

Degree–return rigidity on the Pythagorean Angle Lattice identifies algebraic degree with minimal return depth in the quotient T_PAL, yielding a source–depth–branch normal form and a complete global reconstruction of the lattice and its profinite division boundary. The Pythagorean Angle Lattice (PAL) is studied from an intrinsic division-theoretic point of view. The classical group of rational points of the unit circle supplies a free abelian Gaussian source lattice L_PAL; we pass to its rationalization V_PAL = L_PAL ⊗_ℤ ℚ and to the quotient T_PAL = V_PAL / L_PAL, which measures the finite obstruction to rational angle division. The numerical PAL is recovered from the Pythagorean angle character by rational division followed by the cotangent map. Our main result is a degree–return rigidity theorem. For an oriented PAL value carrying a nonzero Gaussian source, the algebraic degree over ℚ equals the minimal Pythagorean return depth, equivalently the order of the corresponding class in T_PAL. Consequently a value of degree D satisfies [ℚ([n]_PAL X) : ℚ] = D / gcd(D, n). The hypothesis under which this holds is identified exactly: the rational branch shift must be subordinate to the lattice denominator, which is precisely the condition that the value lie in the image of some archimedean realization of the angle character. We show by an explicit example that the statement fails without this hypothesis, and that the failure is a genuine Kummer-theoretic phenomenon rather than a technical restriction. From rigidity we deduce a source–depth–branch normal form: a source-bearing PAL value determines, and is determined by, its primitive source, source multiplicity, division depth and division branch. For a finite configuration X = (X_1, …, X_k) inside one realization we introduce relation groups R_X ⊆ K_X ⊆ ℤ^k recording source cancellation and integral return; they give an exact sequence whose terminal quotient is a finite division group, whose order function is the degree function, and whose Pontryagin dual is the finite branch group. These finite data assemble functorially: direct limits reconstruct L_PAL ⊂ V_PAL, and the inverse limit of the finite duals reconstructs the profinite division boundary ∂_f PAL = T_PAL^∨, for which the coherent extensions of the angle character form a torsor.