I'm going to have to call you out on that one (not just because you seem to have a collection of red herrings). In a sport where your contact with the asphalt is less surface area than half your shoe and your largest contact point with your equipment isn't much larger than a cell phone -- it makes A LOT of sense that core strength can help in a variety of ways.
To briefly name a few:
Anything that involves taking one or both hands off the bar
Anything involving higher speeds
Avoiding damaged bits of road
And certainly not least of all; limiting wasted movement in other parts of your body (e.g. isolating your legs)
Core strength and stability has been a staple of mine for many years due to various physical hobbies and I have seen time and time again how it's benefitted me when compared to riding partners who do not have such a base.
If you're calling me out then you just don't understand physics. What you wrote was anecdotal drivel. There are dozens of red herrings in cycling propagated only by 'tradition' and this is one of them. NM of force are NM of force. If your max pedal torque is say 70NM/leg and you weigh 70kg, the force of gravity alone produces a higher torque required to climb a stair. You can find this
In terms of studies Levin et. al 2008 showed a decrease in performance from an experimental group performing resistance training over a control group performing cycle
Also, there was a good QA analysis of a 4hr pro 1 road race that concluded: "The highest AEPF he generated during the 4 h race was ~500 N, whereas eyeballing things it appears that the Y intercept of his AEPF-CPV relationship (IOW, his maximal strenght) is 900-1000N. IOW, it looks as if he never needed to use more than 50-60% of his maximal force generating capacity (a.k.a., strength) while pedaling." So, never once did the person exert the force you'd need to run up a set of stairs. Core muscles are postural and thus slow twitch and do not somehow "fatigue" from sitting on a bike and pedaling the way they might if the pelvis weren't supported. In weightlifting terms, the above rider never used more than 60% of the force of a 1 rep max single legged squat and that was during the peak sprint. This would represent a load lower than bodyweight that one could sustain for very long, continuous periods of time without fatiguing. When we used to do rep max tests, 60% was usually 30 reps or so, but the more aerobic athletes could bang out upwards of 50. So, his force for 10s was equal to a load someone could squat with one leg for over a minute. Not high.
Average core stability is more than enough for cycling unless you have pelvic tilt/alignment problems. I think you also clearly don't understand that to active the core as a stabilization chain, the load has be extremely high. Coasting through a corner at 40mph might incur a lot of g forces, but the body does not need to brace the pelvis more to do this if the weighting of the bike is correct.
Seems Andy Coggan is in agreement in places in this thread, which features everyone endlessly debating with no evidence:http://forum.slowtwitch.com/cgi-bin/gforum.cgi?post=1161585
I would link the many wattage debates, but won't help if you're not a member
The big caveat to all of this is overall pelvic alignment and stability. Many people work desk jobs and pros spend an ungodly amount of time sitting whether on the bike or on the couch recovering and thus become fairly unfunctional. If you have functionality problems then yes, it is of course beneficial. Better functionality usually leads to better athletic performance over time, but core strength in and of itself is not essential for cycling. Its simply not a limiter in many sports, especially ones where your core is fully supported. If your friends have "core" problem chances are their bike fit is terrible.
Don't take me too seriously. The only person that doesn't hate