OK Karl, let's shift discussion of your Mazda here. We get going in other thread's and create a disjointed story...
So does the Mazda's Idle RPM stay high even after driving w/ hot engine? It should be dropping
well under 1000 when hot, could be a vac. leak, could be the IAC staying open. Every engine w/ modern injection I've seen idle drops down at hot re-start. It's noticeable during ECU re-learn drive sequence, at each park-shut down-restart, RPM goes high then immediately drops down, it's an indication the engine is warmed up. Something maybe missed could be temp sensors- intake air temp, coolant temp, input sensors to computer, there will be spec's for those available, check OBD readings. Does OBD show closed-loop operation, check correlation between FT's, temp inputs, O2 voltages, load, TPS, exp. mechanic can watch OBD2 readings real-time and spot anomalies. Does engine reach op. temp like over 180 deg.? Can check if input temps to ECU match temps checked w/ infrared thermometer at rad. and thermostat housing. Need INfrared thermometer, scanguage/scantool, digital voltmeter to get readings.
Another thing to check is exhaust gas how does tailpipe smell, is there moisture in exhaust. Moisture is good sign of good combustion, if it's dry AND smelly it may mean rich mix too much fuel going in and/or bad cat. convert. Coolant in exhaust from previous head gasket leak can kill cat, look at downstream O2 voltage, I think if cat is good then downstream O2 should stay around .5V w/ hot engine/cat's. Check cat's w/ infrared thermometer cat's "light off" around 600 deg.F. A 5 gas smog machine can identify if HC or CO is too high, and the nose is useful too. When something is wrong there will be evidence somewhere. Codes, other symptoms too. Vac. leaks can also be in evap. system, gas cap, charcoal canister, hoses/tubes or the control solonoid valve. EVAP solonoid can go bad, or stick open. Can cause vac. leak not on engine. Worth checking.
What about fuel press. regulator? Vac. leak from vac. hose or leaky internal diaphragm. Check fuel press. compare to spec's. If tailpipe is stinky or black then I start asking/ looking at fuel system, vac. leaks, cat's even ignition, and see if it's running smoothly. If it runs smooth all the time, clean tailpipe smell, brown inside tip, no burps hesitation or other weirdness, then it makes me think there is a problem that ECU is compensating for. See if temp. inputs to ECU look OK. Does it warm up normally, or take a long time? Is t-stat AND rad. cap OK? IF you and friend can't find anything, could be worthwhile to let pro diagnostic shop look it over. Gotta be such a shop there somewhere there's one here in Reno. You drive lotta miles burn lotsa gas, lotsa $$! that dang Mazda is draggin' anchor somewhere!!

Keep looking Inspector Karl!!!
Tracy G