Confidently Wrong: Why Ignoring Binaries Biases IMF Inference at Large Sample Sizes

The stellar initial mass function (IMF) high-mass slope $\alpha$ is routinely measured by fitting single-star models to photometric samples that contain 20-90% unresolved binaries. This practice introduces a systematic negative bias on $\alpha$ that is constant with sample size $N$. Because posterior credible intervals shrink as $1/\sqrt{N}$, at sufficiently large $N$ the bias exceeds the reported uncertainty and the true value falls outside the credible interval - a regime we call "confidently wrong." We bracket this bias between two limiting observation operators: mass-addition $(m_\text{obs} = m_1 + m_2)$, a formal upper bound on unresolved-system mass overestimation, and luminosity-addition $(m_\text{obs} = L^{-1}(L_1 + L_2))$, an idealized lower-bias photometric case based on the ZAMS mass-luminosity relation. Across four astrophysical environments spanning $\alpha = 1.60-2.30$, we find: (1) mass-addition bias of $0.054-0.086$ with crossover to confidently wrong at $N_\text{cross} \sim 5{,}000-10{,}000$; (2) luminosity-addition bias of $0.011-0.021$ with $N_\text{cross} \sim 75{,}000-150{,}000$; and (3) a binary-aware mixture likelihood that marginalizes over the Moe & Di Stefano (2017) binary population model recovers the true slope in the synthetic tests presented here. Published single-star IMF slopes can therefore plausibly carry systematic errors of order $0.01-0.09$ if unresolved binaries are not modeled, comparable to or exceeding reported uncertainties in some regimes. Since current and upcoming surveys (Gaia, JWST, Roman, LSST) will deliver $N = 10^4-10^6$ resolved stars per rich cluster, binary-aware inference is likely necessary to avoid binary-driven systematic bias in the large-$N$ single-star-fitting regime.