证明
Lemma 45 ([Coefficient bounds and unary removal]). For the even source let U be…Lemma 44 ([Stabilized minimum depth]). Assume the uniform arithmetic lower…观点
We first check integrality of the generators. Diamonds act by scalars on B_U and on each unary term. For, the only apparent denominator is removed by ⟨u⟩H_U=ψ_U(u)H_U+ 12∑_J B_U[J] (ψ_U(u)-η(u)χ_ -4 (u)^i(-J/u))Θ_J; the difference of the two characters is zero or twice a unit. The same formula applies after multiplication by theta. Good Hecke operators commute with diamonds, and their coefficient formula preserves integrality. Thus all generators of M are integral. Finitely many Fourier coefficients inject the ambient finite-dimensional space into a finite-dimensional coefficient space, so M embeds in O^r for some finite r . It is consequently a finite, closed O -module. Continuity and simultaneous Chebotarev approximation on the finitely many eigensystems approximate t(u) and d(u) by T_q and q⟨q⟩ at every prescribed precision. These operators preserve M , so their limits do as well. They act by scalars on each good eigensystem and by the identity on its old multiplicity space, which retains the asserted commutation with level operators. The formula (T_qf)[n]=f[qn]+q(⟨q⟩f)[n/q] has no second term when q n . Successive distinct fresh primes therefore give the coefficient assertion. On a single character system, determinants reduce to 1 because all characters have 2 -power order and the cyclotomic character is odd. On a subtracted vector the displayed diamond formula leaves unary corrections. A fresh trace kills each correction modulo the maximal ideal: the tested index nq cannot have its prescribed squarefree part J when q nJ . This continues to hold after further fresh tests. Thus d(u)t(v)=t(v) on the reduced trace orbit. Apply the matrix identity tr(A)tr(B) =tr(AB)+(B)tr(AB^ -1 ) to each characteristic-zero system and use that determinant identity. Finally t(1)=2 before reduction.