When all other variables are accounted for, calories drive fat-loss and weight-gain.

The first law of thermodynamics is an expression of the principle of conservation of energy. It states that energy can be transformed (changed from one form to another), but cannot be created or destroyed.
Our bodies take in energy (food) to perform work (measured in joules or converted to calories spent).

But as you can see, not all food calories are used to perform physical work.  Sometimes the kcal from your food is:

1. stored as energy
2. lost as heat
3. used to fuel or support metabolic reactions
4. used to fuel digestion
5. not completely digested

And then sometimes...you burn less calories for the same amount of work as you've done in the past!  Sometimes you feel like you need to put in twice as much work for the same results!

Does this mean that for some reason the first law is being broken?  That the energy going in is out of balance with the energy coming out?

No.  What it means is that YOUR BODY IS NOT A CLOSED SYSTEM, and it RESPONDS AND ADAPTS to its environment and the stresses placed on it.


A VERY IMPORTANT LESSON:
"Fat burning" and "increased metabolism" are somewhat opposing terms.  I know you've heard ads for products that claim to "stoke your metabolic fire" so you can "burn off fat".  My advice: don't waste your money.  Here's why:

1. Losing fat (and by consequence, bodymass), effectively decreases your metabolism and increases hunger cues

Your bodymass directly influences your resting metabolic rate.  The more mass you have, the more calories your body needs to function.  When you lose weight, your requirement for calories decreases.  The other side of this coin is that the higher the proportion of muscle mass to fat mass you have, the higher your metabolism.  What does this tell us?

Any weight-loss program that fails to include progressive weight-training is destined to fail long-term by not attempting to combat the reduction in metabolism due to reduction in body-mass (other social strategies and meal planning options are necessary to combat this as well).  Weight-training places necessary demand on the muscles to grow, or at least maintain mass.  One of the most crucial elements of exercise physiology that goes ignored by endurance athletes who rely solely on endurance exercise to manage their weight is that the aerobic energy system actually uses 3 fuel sources (simplified):

1. Fatty Acids (low intensity or rest)
2. Glycogen/Glucose (higher intensity)
3. Protein from deaminated muscle tissue (low or high intensity, amplified in the absence of stored carbohydrate)

Aerobics oxidize fat, but they also eat up muscle tissue.  This is generally not an issue if you eat a moderate to "high" protein diet, along with refraining from training in fasted states (glycogen depleted), and if you incorporate progressive resistance training at sufficient intensities to stimulate muscle protein synthesis (MPS).  Unfortunately, many endurance athletes I know rarely meet their minimum protein requirements, let alone that needed for athletes (1.2-1.8 g/kg bw/day).  Their A) calories are often too low and B) carb intake to high to justify the a) distances they cover and b) how low their average intensity is.  But that's a story for another time.

Bottom-line: the "fat-burning zone" is meaningless if you are overeating calories.  If your goal is to lose weight you must create a caloric deficit by consuming less calories than your body needed that day.  You can do this by eating less, or exercising more, or both.  

What you DON'T want to do is train more and more and more while eating less and less and less.  This is one form of what I call "spinning your wheels".  It's kind of the parallel equivalent of training all the time but eating whatever you want - both result in running yourself into the ground and getting nowhere fast.  Doing the former will provoke your body to adapt by lowering metabolism to deal with the excess work performed in the absence of adequate energy - a survival mechanism. 

Support your body, don't starve it.

"Energy Rules! Energy Conversion and the Laws of Thermodynamics - More About the First and Second Laws". Uwsp.edu. Retrieved 2010-09-12.

Klok, M. D., Jakobsdottir, S. and Drent, M. L. (2007), The role of leptin and ghrelin in the regulation of food intake and body weight in humans: a review. Obesity Reviews, 8: 21–34. doi:10.1111/j.1467-789X.2006.00270.x

Antonio, J., Lowery, L. M. (2012) Dietary Protein and Resistance Exercise. CRC Press, Taylor & Francis Group. 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL

Nesheim, M., Nestle, M., (2012) Why Calories Count: From Science To Politics. University of California Press Ltd. Berkeley and Los Angeles, California.

When it comes to protein brownies, the popular choice has always been to use beans to impart the chewy, richness we think of when it comes to brownies.  I've never liked any of the bean-based brownie recipes I've tried.  If you want to make delicious brownies that not only taste amazing but are also packed with protein and fiber - you should use hemp protein!  I've always thought that there was a missed opportunity having such an affordable, nutrient rich protein powder without having recipes that didn't taste distinctly...green.  Your search is over.

1. Nutrient Dense Food

There are 4 criteria that we must satisfy when eating food for fitness related reasons:

• Fuel
  
• Nutrition
• Health
• Building Muscle

So how do these 4 things differ?

Fuel: we must eat enough to fuel our ability to do things we enjoy

Nutrition: we must consider the specific raw components the body requires in response to activity and where the balance lies (E.g. Endurance exercise vs. bodybuilding)

Health: is "the optimal interplay of the body's organs".  What and how much vitamins and nutrients do I need to avoid deficiencies?  What special considerations do I have (i.e. vegan, Celiac, dairy sensitivity, etc.)?

Building Muscle: you need a minimum amount of protein to avoid muscle wasting - a health related goal.  You need an optimal amount of protein to build lean muscle - a fitness related goal.

These are all slightly different criteria, but none of them can be achieved without nutrient dense food.  There are no magic foods that provide a blueprint to the perfect body, but a perfect body cannot be built out of gummy bears ;)

As adapted from Tom Venuto's book The Body Fat Solution

In his best-selling book The Body Fat Solution, Tom Venuto espouses simplicity over novelty and sustainable, real-world solutions to weight-loss and weight-management.  Many diets that restrict certain foods, or focus too closely on isolated nutrients, hormones, "magic bullets" are cast aside in favour of simple lifestyle changes.  The big picture is "synergy", and your success depends upon a "total-life approach that addresses all of the true root causes of the body fat problem in every area of your life - physical, mental, emotional, and social."
The causes of obesity make it a complex issue, but the physiology of body-fat is actually quite simple.  It is nothing more than energy balance.  Fat metabolism is an intricate area of study, but "you don't need to understand electricity to light your home, you only need to flip the switch".  It's amazing to me that most people I meet know intuitively how to lose weight; they just start eating less.  They are correct in their conclusion, but seldom successful in execution.  I know I need lumber and nails to build a home (a permit helps too), but that knowledge is useless without the blueprints.  I'm here to tell you that you don't need to be an expert to understand these blueprints, you just have to be willing to follow them.

In the introduction to his book, Tom remarks on how he built his own physique following a short list of crucial principles.

1. Nutrient-dense food
   
2. Carefully controlled calorie intake
   
3. Progressive resistance training
   
4. The right dose of cardio
   
5. A mind focused on well-formed goals
   
6. Strong emotional drive
   
7. A great support system of friends, training partners, and mentors
   

I will attempt to expand on these points on a weekly basis.