A structural degeneracy explains reionization tensions and limits dark matter constraints

Over the past decade, reionization studies have yielded persistent factor-of-two-to-five disagreements in the inferred ionizing escape fraction $f_{\mathrm{esc}}$ and peak star formation efficiency $f_{*,0}$, compounded by JWST's discovery of unexpectedly bright $z>10$ galaxies. We show that this discrepancy arises from an algebraically exact structural degeneracy: the ionizing photon rate $\dot{n}_{\mathrm{ion}} \propto f_{\mathrm{esc}} \times f_{*,0}$ renders all reionization-history probes, including Thomson optical depth, neutral hydrogen fraction, UV luminosity function, and quasar proximity zones, sensitive only to their product, leading to an intrinsically non-invertible mapping between model parameters and observations. We demonstrate the robustness of this degeneracy using a large suite of N-body simulations of self-interacting dark matter haloes spanning $10^9$-$10^{11} M_\odot$. Despite substantial changes to galaxy-scale structure, observables remain indistinguishable once the effective ionizing emissivity is matched, severely limiting reionization-based dark matter probes. We identify that only observables sensitive to the spatial topology of ionized regions can break this degeneracy. Our results provide a unified explanation for the scatter among published constraints and establish a framework for interpreting reionization observations and their implications for early galaxy formation and dark matter.