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## 3) What exactly is derived from congruences (mod arguments)

You write that I claims about common prime factors from congruences. But your text contains multiple “mod” steps and uses them to force contradictions. The critique is not “congruences are useless.” The critique is:

- When you infer a strong conclusion (divisibility, coprimality, or existence/nonexistence of a shared prime) from a congruence, you need an explicit justification: which lemma is being applied and why its assumptions hold in your setting.

So again, the concrete request is: **please identify one representative congruence step where you believe the proof forces a contradiction, and provide the exact reasoning chain** (including any needed lemma). Then we can evaluate that step precisely.

## Bottom line

If you think my critique is “completely different from your paper,” the fastest way to show that is not to assert “I never wrote that,” but to **quote the exact passage** (equation number / paragraph) and explain the intended inference. I am happy to correct my statement once the disputed steps are pinned to specific lines.