Luckily JR is still around...and caveat I am not a health professional, but I have paid close attention to this for decades. That said I have actually been gaining weight recently since I blew out my knee and reduced my physical activity level without adjusting my diet down to compensate... (so this is a do what I say, not what I am doing at the moment) :-(
I would advise against any short term weight loss plan... Life is actually too long to expect short term fixes to stick. What do you think will happen after the brief quick weight loss? If you return to anything resembling your previous pattern that weight will come back on, maybe more because now you are burning less baseline calories to carry around the spare tire.
This is a huge subject so I will touch on the broad strokes.
Starvation (actually called caloric restriction or CR by the people who do this on purpose). There is a great deal of science surrounding the effects of continuous CR. There are true believers who restrict their caloric intake to slow down aging (hoping to live longer). There have been promising experiments on animals and even primates, taking decades, where the caloric restricted primates did age more gracefully. However there are consequences to CR. First you have to plan very carefully to get adequate nutrition when eating less food. A second phenomenon when severely restricted your body will "remodel" your body will literally consume or eat it self to keep the brain fed with fuel. We will not run out of fat for many months but the body will still remodel and atrophy little used muscles. The body is not smart enough know how much fat it has in the gas tank (actually the fat mass does generate some hormones like a pre-evolutionary part of our endocrine system), but we do not use this information. Instead the body operates on a strict short term sugar economy, We have a small amount of circulating sugar in our blood and perhaps 2000-3000kCal of glycogen stored in muscles and organs, a little more than a typical day's total energy budget. If we starve ourselves by fasting or restricting carbohydrates until our glycogen is depleted, the body will shift to Ketosis where fat is converted to ketones, a sugar replacement that the brain can run on. Muscles can burn fat, but the brain needs it's sugar (or ketones), or it will shut down (a bad thing).
Nutrition... The body generally stores fat soluble vitamins (like E) but most water soluble vitamins (like C) need to be constantly replenished. Even Vit D commonly extracted from sunlight is stored in the skin fat. When severely restricting calories, you need to eat smarter to get full nutrition.
"Energy Balance"... yes, weight gain or loss is pretty much as simple as balancing a check book, but there can be some confounding factors. A major fraction of our energy consumed daily is the bodies overhead. Sugar to keep the brain happy, energy to keep us warm, energy to carry our body up and around. When we starve ourselves or go into restriction, the body goes into survival mode and reduces that baseline consumption. CR practitioner do this on purpose, but anyone trying to lose weight wants to avoid this slow down in baseline metabolism.
I don't know how much Keith weighs now but 1000-1500 kCal + cycling daily sounds too low for a healthy sustainable pattern. Another tidbit about weight don't be too focussed on every pound. Our water weight can fluctuate quite a bit and the weight is not the important part, it is how you feel. In fact muscle is heavier than fat so converting fat to muscle could actually gain weight or negate fat losses.
My advice would be to target some final destination weight, and activity level that you would like to maintain. Based in that weight, and activity level you can compute a nominal caloric intake to maintain that weight. Then adjust your regular diet for nutrition so you can meet nutrition goals with that caloric budget. If you are carrying 35# more than that destination target, eating maintenance levels for that target weight will result in a modest energy deficit and weight will come off slowly, as you consume that extra weight your diet is not supporting.
Instead of starving yourself and performing excessive workouts to lose weight, just start now on that final maintenance diet, and the excess weight will slowly and healthfully melt off.
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Too sudden weight loss can release toxins that accumulate in our stored body fat. I am not aware of people actually getting sick from this, but it sounds scary enough to avoid.
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Your perception of weakness may be low blood sugar. While physical exercise should improve the bodies ability to regulate blood sugar. Our muscles work with insulin to convert sugar to glycogen for storage so simple exercise can mitigate against adult onset diabetes. Also on a low protein diet if the leg muscles need more raw material to repair (muscles tear and rebuild when exercised). If there is inadequate protein in the diet, the body will consume little used muscles elsewhere to get the needed raw material. Did you ever see a marathon runner with big biceps, no.. QED.
As i get older it is more painful to get on the weight machine to work out. I can tell that I have lost strength in recent years due to this. If you don't use it you lose it. Getting old sucks. I would suggest mixing in some upper body strength work to complement the bike workouts. This can be as simple as push-ups and/or pull-ups.
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There are tons of resources on the WWW to help manage nutrition levels, The CR group has a website CRSOC and while they are IMO too extreme on the caloric restriction side, they have great advice about nutrition.
There are also tons of BS and snake oil on the WWW... The diet industry is a Bazillion dollar business and growing as it is basic human nature to pursue easy fixes for hard problems... Avoid anything that promises fast results as that is unhealthy by definition.
So to recap... instead of a high impact short term program, consider looking at this as a permanent long term lifestyle adjustment. Make small incremental changes that you can live with. Whenever possible turn these changes into good habits. The weight will come off at it's own pace.
JR
PS: During weight loss our blood chemistry (HDL, LDL, etc) improves dramatically. It seems like a good idea to prolong this period of unusually good blood chemistry by losing the excess weight as slowly as possible.
PPS: OK I just saw you comments about avoiding anaerobic work. Perhaps you can think of some aerobic workout to support upper body strength. In bicycle sprints the upper body is involved somewhat, i have an exercise bike (Schwin air-dyne) with arm cranks (I need to use it more).