The flagship matrix performs the CRC division from the exact message and generator bits. Every row is recomputed by XORing or skipping at the current leading bit.
highlighted = computed this step
Why the matrix is the proof
The CRC remainder is not typed separately from the division. The matrix shows each XOR placement and the running state down to the final bits.
dividend=11010011101100000
Divide by the generator
The dividend 11010011101100000 is divided by generator 1011. The engine recomputes the rows from those bits.
11010011101100000÷1011
XOR or skip
When the leading bit is 1, XOR the generator at that position; when it is 0, skip. This example has 9 XOR steps.
xor steps=9
The final bits
After the last aligned step, the remainder is 100.
remainder=100
Summary
CRC division is ordinary-looking long division, except subtraction is XOR and the final remainder is the CRC. Polynomial arithmetic over the exact bits; timing/throughput is not modeled here.