A Recursion-Driven Structural Transition in Gas Enshrouded Early AGN: Cross-Scale Invariants in Little Red Dots from z ≈ 3–10

This paper presents the first operator-level analysis of structural transitions in early active galactic nuclei (AGN), using the recently observed population of gas-enshrouded “little red dots” (LRDs) at z ≈ 9 as an empirical test case. Building on the recursion operator Ψ = ∂S/∂R, which measures the sensitivity of structural rules to recurrent activation, we show that LRD spectra exhibit a four-step precursor sequence preceding the rise of Ψ: 1. Phase tightening 2. Modal persistence 3. Curvature shift in S 4. Irreversible structural drift Using Balmer-break morphology, broad-line formation, and continuum deformation in the CAPERS-LRD-z9 source, we demonstrate that this sequence appears in the same internal order as in other recursive physical systems (e.g., seismic drift, pulsar recovery, LIGO ringdowns). This cross-scale stability suggests that the transition belongs to recursion itself rather than to the astrophysical domain. We show that early AGN near the reionization epoch conform to the same operator law governing recursion-driven structural reconfiguration, with the onset of Ψ marking the point at which recurrent excitation can no longer be absorbed by the existing structural rule-class. The result positions LRDs as a natural astrophysical realization of a universal recursion-induced transition, extending the empirical validity of the operator across more than forty orders of magnitude in energy.