Eventual Scalar Recurrences for Constrained Orthogonal Polynomials

Let \(S\) be a finite-codimensional subspace of \(\mathbb R[x]\) with a positive definite moment inner product, and let \(q_n\) be its degree-ordered monic orthogonal basis. For a fixed nonzero polynomial \(a\) of degree \(d\), an eventual scalar band recurrence for \(a q_n\) exists exactly when multiplication by \(a\) preserves \(S+S^\perp\), where the complement is taken inside the polynomial space. The width can then be chosen to be \(d\). Some nonconstant multiplier works exactly when this sum contains a nonzero polynomial ideal.

For the inner product on \([0,1]\) and the single constraint \(\int_2^3 p(x)\,dx=0\), every nonconstant multiplier sends infinitely many \(q_n\) outside \(S\). A mean-zero Legendre example shows why stabilizing \(S\) itself is not necessary for a tail recurrence. These elementary results refine an established stabilizer approach; they do not classify differential, coupled, or compressed-operator recurrences, or settle the broader AIM problem. This AI-assisted note is self-audited and unrefereed; absolute priority and independent review are not claimed.