Partly. It's a decent starting point and a bad place to stop.
For anybody who hasn't heard it: multiply the age of your system by the cost of the repair. Over $5,000, replace. Under, repair. A 10-year-old unit with a $300 repair is $3,000, so you fix it.
The logic isn't wrong. It's trying to weigh two real things at once — how much life the system has left, and how bad the current failure is. I'd rather a homeowner have that in their head than nothing.
Here's where it falls apart.
It doesn't ask what broke. A $400 capacitor and contactor job on a 15-year-old unit comes out to $6,000, and the rule says replace. But a capacitor is a wear part. It's supposed to fail. Replacing one doesn't tell you anything about the compressor, the coil, or how much life is left. Meanwhile a $400 partial repair on a failing compressor is a completely different situation, and the formula treats them the same.
The threshold hasn't kept up. That number was set when a changeout cost a fraction of what it does now. If a new system runs ten or eleven thousand dollars, triggering that decision at $5,000 is asking people to spend a lot of money on the strength of some arithmetic.
And notice who publishes it. You'll find the $5,000 rule on manufacturer websites. I'm not saying anybody's being dishonest, but a formula that outputs "buy a new system" is a comfortable formula for a company that sells them.
What I actually look at: what failed and whether it's a wear part or a core component. What else on the system is close to going. Whether it's an R-22 system, because refrigerant on those is expensive enough now that a leak changes the math by itself. How long you plan to be in the house. And what you can spend, which is a real factor whether or not it fits in a formula.
Sometimes I do the math, land over $5,000, and still tell somebody to fix it — because the thing that broke was a part, not the system. And sometimes I land under and tell them not to, because I can see what's coming.