  We consider the extension of techniques for bounding higher-dimension
operators in quantum effective field theories to higher-point operators.
Working in the context of theories polynomial in $X=(\partial \phi)^2$, we
examine how the techniques of bounding such operators based on causality,
analyticity of scattering amplitudes, and unitarity of the spectral
representation are all modified for operators beyond $(\partial \phi)^4$. Under
weak-coupling assumptions that we clarify, we show using all three methods that
in theories in which the coefficient $\lambda_n$ of the $X^n$ term for some $n$
is larger than the other terms in units of the cutoff, $\lambda_n$ must be
positive (respectively, negative) for $n$ even (odd), in mostly-plus metric
signature. Along the way, we present a first-principles derivation of the
propagator numerator for all massive higher-spin bosons in arbitrary dimension.
We remark on subtleties and challenges of bounding $P(X)$ theories in greater
generality. Finally, we examine the connections among energy conditions,
causality, stability, and the involution condition on the Legendre transform
relating the Lagrangian and Hamiltonian.
