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I did further research and made some discoveries about some of the constants in HeartGold/SoulSilver (well, technically just SoulSilver, but I imagine HeartGold is the same). Constant 304 is not specifically the status of roaming Latias/Latios, but rather the status for Steven Stone. 0 is "Not Appeared" in PKHeX, which is the default value before anything happens. 1 is "Ready to Roam" in PKHeX, which I've confirmed causes Steven to talk to the player in Vermilion City. This is likely set to 1 when the player obtains Copycat's doll from the man in the Pokemon Fan Club house in Vermilion, since Steven talks to the player on their first step anywhere in Vermilion City after this constant is set to 1. 2 is "Roamed" in PKHeX, which I've confirmed is set immediately after Steven talks to the player in Vermilion City (regardless of whether Latias/Latios is actively roaming). 3 seems like it is set once the player has defeated Red and Steven Stone is in Silph Co in Saffron City, ready to give the player a Hoenn starter. 4 is set once the player obtains the Hoenn starter (confirmed from my own save file). 5 is set when the player talks to Steven in the Pewter City Museum (which I've also confirmed). Constant 75 (currently undocumented) is the current location of roaming Latias/Latios. I confirmed this by moving between Vermilion and the route north of it and observing this constant change. I'm sure there's a map somewhere of the values of this constant to the locations to which they refer, but setting this constant to anything deliberate is pointless because Latias/Latios roams to a new location when you load the game. The only value I can confirm is 3 = Route 6. Here's the big one: constant 72 (currently undocumented) is the result of the last encounter with roaming Latias/Latios (where it didn't flee). A value of 1 means the player caught Latios/Latias, and a value of 2 means the player caused Latios/Latias to faint (I confirmed both of these values.) Constant 72 does not have to be reset to 0 in order for Latias/Latios to roam. If you reset constant 304 to 1 and talk to Steven in Vermilion, Latias/Latios will start roaming regardless of constant 72. One you capture or defeat Latias/Latios, constant 72 is set to 1 or 2, respectively. If you enter the Hall of Fame when constant 72 is set to 2, when the game saves before the credits, this constant is reset to 0, and Latias/Latios will be roaming when you reload the game. So after researching/learning all of this, I've determined, unfortunately, that I have indeed caught roaming Latios on my SoulSilver, and I must have traded it away at some point. Hopefully this research helps someone in a similar predicament!

@theSLAYER Oh wow. Thank you so much. I didn't know any of this. I really appreciate the help! I'll try your suggestions and report back! @theSLAYER OMG it worked. Thank you so much!