NOx = various oxides of nitrogen, a required precursor to smog
HC = hydrocarbons, the other ingredient to smog, also carcinogenic
CO = carbon monoxide, poisonous
CO2 = carbon dioxide, inert, byproduct of combustion
H2O = water, inert
O2 = oxygen, we breathe, necessary for life, consumed in combustion
N2 = nitrogen, inert, composes ~70% of air breathed
You need to distinguish between "engine out" emissions and "catalyst out" emissions. NOx is a result of combustion temps, the higher the temps the more NOx is created as a result of N2 + O2 <---> NOx equilibrium reaction. After the temps drop from the expansion of the gas in the power stroke, the equilibrium that was reached at high temps gets "locked in". The NOx can't auto reduce back to the N2 natural state that composes 70% of our air.
The modern 3-way catalyst has a two stage process. First it oxidizes the NO to a NO2 formula and also oxidizes the CO to CO2. This preparatory step is required to allow the NOx reduction reaction to take place on the next catalyst stage. CO oxidization is the easiest part of catalytic reduction and is usually not the most concerning factor developing an emissions approved engine.
Rich burn is preferred not just for quenching the reaction to reduce initial engine out NOx, the excess HCs from rich combustion are used by the catalyst as part of the NOx reduction reaction.
The function of EGR is also to reduce NOx production by introducing an inert gas that absorb some of the combustion temperature. This works because the piston still draws in the same mass of air, but requires less fuel (less energy, less heat) while maintaining rich burn.
Water injection can effectively achieve the same thing, but care must be taken not to wash oil from the cylinder walls causing massive increases in cooling system heat rejection and cylinder and piston deterioration due to loss of lubricant. Or dilute the oil lubrication needed for bearings and moving parts. Or cause too much reduction of combustion temp and form soot in the chamber, piston ring pack, and oil. I do not know what affect additional water would have on the catalyst; it might not be a problem since its a natural byproduct of combustion.
So that on the surface of the catalyst you get this reaction:
NOx + HC --> N2 + H2O + CO2 (+ heat)
The right proportion of HC:NOx is required to maintain a good reduction and prevent fouling of the catalyst.
The only good way to develop this is to use an engine dyno with emissions datalogging with the full engine control parameters: fuel injection timing, pressure, quantity, ignition timing, and valve actuation control, variable EGR controls, etc. Only this equipment can provide a full map for cold start, hot start, high altitude, high humidity, low humidity, low octane fuel, high octane fuel, ethanol cut fuel, a new phase of the moon, etc...
Reducing NOx is very important, I like having blue skies and fresh air. Thanks for your effort and good luck.