A partition breaks a whole number into an unordered sum. For example, three can be written as three, two plus one, or one plus one plus one. An overpartition adds one choice: the first occurrence of each distinct part may be marked. That gives eight overpartitions of three.

As the numbers grow, these counts become enormous. Yet some leave no remainder when divided by a particular prime. This collection gives computer-assisted proofs of several such patterns proposed by Wei, for the primes seven, seventeen, nineteen and twenty-three. Every pattern has precise restrictions on which numbers are counted.

The key is not to calculate endlessly. Modular forms turn an infinite statement into a finite, mathematically justified checking task. In the modulo-seven paper, a sharper level reduces the required coefficient bound from over one point eight million to eight thousand and sixty-four.

The modulo-nineteen case needs more care. Its starting function has poles. The proof first changes the function without changing its coefficients modulo nineteen, then selects the desired terms. Exact arithmetic checks the remaining cusp cancellations before the final coefficient test can be used.

A third paper explains why selected patterns continue when their arguments are repeatedly multiplied by four. The correction terms are eigenforms: the operation changes their scale but preserves their relevant zero coefficients. A further result says exactly which levels make one entire modulo-thirteen class vanish: the level must leave remainder three on division by four.

These are specialised number-theory results, not demonstrated practical applications or a general classification of all congruences. The three papers share some base cases, which are counted only once. The written arguments and reproducible exact code are linked on Evidence Press. This unrefereed collection is dated the twelfth of October, twenty twenty-six. This synthetic briefing is not additional scientific evidence.