We present multi-epoch mid-infrared (IR) photometry and the optical discovery observations of the impostor supernova (SN) 2010da in NGC 300 using new and archival Spitzer Space Telescope images and ground-based observatories. The mid-IR counterpart of SN 2010da was detected as SPIRITS 14bme in the SPitzer InfraRed Intensive Transient Survey (SPIRITS), an ongoing systematic search for IR transients. A sharp increase in the 3.6 $mu$m flux followed by a rapid decrease measured ~150 d before and ~80 d after the initial outburst, respectively, reveal a mid-IR counterpart to the coincident optical and high luminosity X-ray outbursts. At late times after the outburst (~2000 d), the 3.6 and 4.5 $mu$m emission increased to over a factor of 2 times the progenitor flux. We attribute the re-brightening mid-IR emission to continued dust production and increasing luminosity of the surviving system associated with SN 2010da. We analyze the evolution of the dust temperature, mass, luminosity, and equilibrium temperature radius in order to resolve the nature of SN 2010da. We address the leading interpretation of SN 2010da as an eruption from a luminous blue variable (LBV) high-mass X-ray binary (HMXB) system. We propose that SN 2010da is instead a supergiant (sg)B[e]-HMXB based on similar luminosities and dust masses exhibited by two other known sgB[e]-HMXB systems. Additionally, the SN 2010da progenitor occupies a similar region on a mid-IR color-magnitude diagram (CMD) with known sgB[e] stars in the Large Magellanic Cloud. The lower limit estimated for the orbital eccentricity of the sgB[e]-HMXB (e>0.82) from X-ray luminosity measurements is high compared to known sgHMXBs and supports the claim that SN 2010da may be associated with a newly formed HMXB system.
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