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1)The Chilluk has everything to do with heicha d’mifrach kal vachomer. When the kal vachomer will be mufrach and will leave you in a situation where the halacha is more kal than the kal case then we discard the technicality of dayo.
2)Of course it makes sense that something more chamur should be more chamur elsewhere, but it also makes sense that it should be equal elsewhere – which is in fact how it is in almost every kal vachomer.
3)There is always a possibility of an unknown pircha but we assume that when the Gemara makes and accepts a kal vachomer they cleared the standard. In fact there is either a Tosafos or a Maharsha in Kiddushin which asks why the gemara doesn’t make a certain kal vachomer and answers that the Gemara must have had some pircha that we don’t know of. (If I have time maybe I’ll locate the exact source.)
So I don’t think my question is based on questionable assumptions. We agree that the Rabbanan and R’ Tarfon are arguing about the guidelines. I think that the guideline that they are arguing about is whether we uphold the technicality of dayo when it would force us to discard a kal vachomer and therefore have a halacha that doesn’t make sense. What do you think they are arguing about?