For specific values, you'll need the performance date for your aircraft. You'l need weight, temperature, thrust, and altitude, for a still-air computation, along with aircraft configuration, to include any performance-limiting statements that are true for that particular takeoff (bleeds, flaps, configuration deviation items such as missing inspection covers, gap seals, etc).

Your original post asked about a publication explaining what the climb segments are, and the criteria that defines each segment. Yes, such publications exist, multiple sources were cited, and additional explanations given.

You asked "what happens if V2 is significantly exceeded during the second segment". You were told.

You asked if there were any mathematical formulas or flight tests done to prove it. You were also told: yes, there are. Yes, they were used for the certification of the airplane, and the result of the design of the airplane, and the extensive flight testing that followed, is the operating parameters published in the performance data for that aircraft. Your performance is met when you follow the correct procedure. Your performance will NOT be met if you do not follow that procedure. By how much will you fail to meet the published climb data, if you don't fly on speed? Depends on numerous factors, the combination of which is specific to your aircraft in that location on that day at that power setting at that weight at that temperature at that power setting at that elevation in that configuration and condition.

You appear to be trying to make a case to others that their practice of not flying the published speeds, will have a negative effect. You appear to be saying you want the math to prove it. You have better access to the performance data for your aircraft than the rest of us will especially given that we don't know what aircraft you are flying.

What is correct, regardless of which aircraft you're flying, or any of the particulars, is that unless you fly the established parameters and profile for your departure, you will not get the performance desired, or predicted or required.

With all engines operating, the airplane may well exceed minimum performance gradient by a substantial margin, but don't forget that the minimum climb gradient in climb segments isn't an all-engine proposition, You may well exceed minimum climb gradient criteria at V2+50, with all engines operating. The same may not be true with loss of thrust on one (or more) engines. We conduct the departure with plans for the potential loss of thrust that may accompany an engine failure, or the need to reduce thrust following a problem. We conduct a runway or departure analysis for accelerate and stop distance, and the subsequent climb away from that runway, using a specific set of values. Speeds less or greater than those values will compromise the predicted performance. How much those values are compromised depends on numerous variables specific to your aircraft and environmental considerations. The math has been done by the manufacturer, and flight tests have verified the predicated data, which has then been published for your aircraft. If your aircraft does not include climb data in each segment for V2+50, then you may have to stop and think why. You may lack that information from the manufacturer that did the math and provided the performance numbers you do need, and it falls upon you to use and follow those numbers, if you want the performance predicted.

You could level off and clean up, and you may get a higher climb rate, but the segments of the initial climb are about gradient; distance forward vs altitude gained, not gross climb rate. Therefore, if you're looking for a math formula, you're going to need to know the forward progress of the aircraft over the ground, as well as the altitude increase in time, relative to that forward gain, and the impacts of your takeoff weigtt and CG, as well as ambient temp, elevation, etc. Density altitude matters. If you change any of the published parameters, you push outside those parameters. By climbing at V2+50, you're increasing forward speed, and altering your climb gradient. If you're experiencing sufficient rate of climb at that time and location, with all engines, perhaps you're still well above the desired minimum gradient. Lose thrust on one or more engines and all bets are off. How much are they off? Depends on numerous operational variables, and variables that you've added to the problem. At that point you may have a real problem that you've created, and you may not have published data to account for it. You certainly won't have had that data available to do your takeoff or runway analysis, which means you're guessing at your performance, which is never a good idea (especially given that you already have data guaranteeing your takeoff performance, if followed).

"What happens if I violate the parameters of the guaranteed climb segment performance predictions?" you may ask. You don't get the performance. By how much? Depends. If you choose to attempt to recreate the test program for the aircraft, without access to any of the design and prediction data used for that test and evaluation program, that's all on you. "I guess it will climb better at V2+50" isn't much of a test card, but at that point, the manufacturer data becomes irrelevant, and you're playing test pilot. Save your data.