Springborn Span Transport and the Hidden-Scale Euler Family

Fix q ∈ (0, 1). For a finite q-rational circle [x], let

Uₓ = [x]♭_q, Vₓ = [x]♯_q, Lₓ := Vₓ − Uₓ, hₓ := log Lₓ.

The period of the normalized two-pole oper attached to [x] is

Πₓ(λ) = Lₓ B(2λ + 1, 1 − 2λ).

The interior oper branch therefore carries an exact even-zeta expansion, while the regularized boundary of the same Euler family carries an exact Mellin/Hurwitz-zeta tower. The only missing transport law was the behavior of the hidden span L_z under Springborn composition.

For a finite inner regular pair (x, y), write

z := x ⊕ₛ y.

Using the unique orientation-preserving hyperbolic involution exchanging the two q-rational circles, we prove the exact endpoint formulas

V_z = (L_y Vₓ + Lₓ U_y) / (Lₓ + L_y),

U_z = (L_y Vₓ(u∞ − Uₓ) + Lₓ U_y(u∞ − V_y)) / (L_y(u∞ − Vₓ) + Lₓ(u∞ − U_y)), u∞ := 1 / (1 − q),

and the exact span law

L_z = ((Vₓ − V_z)(V_z − V_y)) / (u∞ − V_z)
= LₓL_y(Vₓ − U_y)(Uₓ − V_y) / ((Lₓ + L_y)²(u∞ − V_z)).

Thus the hidden scale closes under Springborn transport. Consequently the complete oper period, the regularized boundary potential, the full Mellin stack, the endpoint zeta jets, and the finite Yamamoto truncations all transport by the same explicit scalar factor L_z.