Sorry, this will be a long post.
If your nitrite were too high, it should read at the top of the scale. Do you have plants in this tank?
It is possible to calculate the amount of nitrite and nitrate a given level of ammonia should produce when using an total ion kit like API uses. If one were using the nitrogen scale, the only thing that matters in NH3/NH4, N)2 and NO3 is the N aka nitrogen. So, if ome has a test that uses the Nitrogen scale, 1 ppm of ammonia = 1 ppm nitrite = 1 ppm nitrate. This is because all that is being measured is the N. But when using the total ion scale one is not only measuring the N they are also measuring the 3 or 4H in the ammonia, the 2 O int the nitrite and the 3 O in the nitrate. If we look at the atomic weights of all these things here is what we have with which to work.
The atomic weight of N is 14.0067, O is 15.9994 and H is 1.00794. And with this info we can calculate the atomic weight for each ion:
Ammonia as NH3 = 14.0064 + 1.00794 x 3 = 3.02370 which results in an atomic weight of 17.03010
Ammonia as NH4 = 14.0064 + 1.00794 x 4 = 4.03176 which results in an atomic weight of 18.03816
But the API kits measure Total ammonia which is NH3 + NH4. Depending on the pH and temp of the water, one could calculate the exact values in a tank. However, for this discussion we can consider TA would average an atomic weight of about17.80 since it is mostly NH4.
Nitrite as NO2 = 14.0064 + 15.9994 x 2 = 31.99880 which results in an atomic weight of 46.0052.
Nitrate as NO3 = 14.0064 + 15.9994 x 3 = 47.99820 which results in an atomic weight of 62.0046.
From all the above we can now say that 1 ppm of TA on the ion scale has an atomic weight of 17.8. If that is converted completely to nitrite the result would be an atomic weight of 46.0052. And from this we can calculate that 17.8 becomes 46.0052 and that means there is 2.58 time as much nitrite as TA (46.0052 divided by17.8). 1 ppm of total ammonia would turn into a maximum of 2.58 ppm of nitrite. Continuing in this vein, 1 ppm ot TA if it is completely converted to nitrite which in turn is completely converted to nitrate would mean that 17.8 becomes 62.0046 and that means there is 3.48 times as much nitrate as T (62.0046/17.8). Moreover we can see that 46.0052 of nitrite becomes 62.0046 of nitrate, or 1.35 times as much nitrate as nitrite (62.0046/46.0052).
These numbers are a guide to the relative values for TA, NO2 and NO3 that are the maximum that can be in in the water. However, things can mitigate the numbers. Ammonia can evaporate or be consumed by plants or algae. Some amount of nitrite and nitrate can become an acid. So the process is not as exact as the numbers indicate, However, we can use them to ballpark what is going on.
In this case the 4 ppm of TA in the tank on day one was 1.5 ppm on day 4. that means 2.5 ppm of ammonia was converted to something else. If it were all turned to nitrite theRE would have been 2.5 x 2.58 = 6.45 ppm of nitrite. This should have PRODUCED a reading for nitrite , not 0. Further, that nitrite would have become a maximum of 8.71 ppm of nitrate. Bearing in mind there is not a 100% throughput, the actual nitrite and nitrate levels would be a bit lower than the absolute math indicates. Evemn so there sould be a nitrite reading for sure.
Since we know that that the ammonia bacteria multiplies faster than the ones for nitrite conversion, and since the nitrite bacteria do not start to reproduce until there is nitrite present. There most certainly shoul be a nitrite reading on day 4. So i am thinking there must be testing error or an expired kit. The nitrate kit is the least accurate of the three and it is most inaccurate between 0 and 20 ppm. The problem there is that the nitrate kit works by turning the nitrate into nitrite and then measuring that. During a fishless cycle oif there is nitrite present, it will get counted as nitrate on the test for that.
It seems more likely to me that there is nitrite but not nitrate. The nitrate reading could actually be nitrite. In order to get an idea of what is actually going on, I wonder what the tap water parameters are is in this case. There is a;so a possible issue with the use of dechlor. Most of these today are able to detoxify chloramine as well as chlorine. When the former is broken down, it becomes chlorine and ammonia. If one has dechlor which also detoxifies ammonia, an ammonia test can be inaccurate unless it is done very soon after the dechlor has been added.
More info is needed to figure out what is going on. Test your tap parameters, Be sure to leave the water out overnight or else bubble it in a glass with an air stone for 15 minutes before testing for pH. Test for ammonia, pH and nitrate in the tap. If you have the kit, test for KH. In addition to the test restuls, tell us what decor and substrate are in the tank as well as about any live plants or obvious algae. Also,what dechlor do you use and do you add anything else to the water?