In the field of transverse modes, the intrinsic relationship between orbital angular momentum (OAM, $\ell$), which encodes azimuthal twisting, and the radial quantum number $p$, which governs the transverse energy-radius distribution, has long lacked a unified, coordinate-free characterization. In the fixed-order degenerate subspace, it remains an open theoretical question how these degrees of freedom are jointly constrained by a common structure, and how one can distinguish and read out $(p,|\ell \left|\right)$ with minimal information in practice. Starting from symmetry, we prove that while second-order intensity moments are strictly degenerate, fourth-order moments are the first and minimal tier that lifts the $(p,|\ell \left|\right)$ degeneracy. This yields a basis-invariant dimensionless observer $Q$ which turns a structural statement into a single-frame, calibration-free readout. On the engineering side, we exploit the intrinsic order-charge coupling to construct Euler-Hermite-Gaussian (EHG) modes: equal-amplitude, staircase-phase superpositions on the fixed-order ladder whose astigmatic evolution realizes a Dirichlet kernel with strict notches. In a single-parameter “$m$ measurement,” the EHG selector implements a low-crosstalk, near-identity transfer in the $m$ basis; the observer $Q$ then provides a basis-agnostic verification of $(p,|\ell \left|\right)$. This coupling-guided construction yields zero sampling leakage at the designed lattice sites, supports cross-platform, reproducible thresholds under aberrations and apertures, and forms an end-to-end standard operating procedure for state preparation, demultiplexing, and quality control in high-dimensional photonic and quantum links.

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• Angular momentum of light
• Quantum information processing