okay, let's form a conglomerate. Deposit the best, most explicit New Church heresies you have.

Let's try to have class though. Let's try and have the source for the quote and the corresponding dogma contradicted.

I will start.

I don't deny the apparent Heresies. In fact, here's a good one:

From Evangeliuum Gaudium- http://w2.vatican.va/content/francesco/en/apost_exhortations/docuмents/papa-francesco_esortazione-ap_20131124_evangelii-gaudium.html

"We hold the Jєωιѕн people in special regard because their covenant with God has never been revoked, for “the gifts and the call of God are irrevocable” (Rom 11:29). The Church, which shares with Jєωs an important part of the sacred Scriptures, looks upon the people of the covenant and their faith as one of the sacred roots of her own Christian identity (cf. Rom 11:16-18). As Christians, we cannot consider Judaism as a foreign religion; nor do we include the Jєωs among those called to turn from idols and to serve the true God (cf. 1 Thes 1:9). With them, we believe in the one God who acts in history, and with them we accept his revealed word."

This is just flat out heresy. Flat out.

Denzinger 712, 714 Old Numbering, Sources of Catholic Dogma: http://patristica.net/denzinger/#n700

712 "It firmly believes, professes, and teaches that the matter pertaining to the law of the Old Testament, of the Mosiac law, which are divided into ceremonies, sacred rites, sacrifices, and sacraments, because they were established to signify something in the future, although they were suited to the divine worship at that time, after our Lord's coming had been signified by them, ceased, and the sacraments of the New Testament began;"

And again,

714 "It firmly believes, professes, and proclaims that those not living within the Catholic Church, not only pagans, but also Jєωs and heretics and schismatics cannot become participants in eternal life, but will depart "into everlasting fire which was prepared for the devil and his angels" [Matt. 25:41], unless before the end of life the same have been added to the flock; and that the unity of the ecclesiastical body is so strong that only to those remaining in it are the sacraments of the Church of benefit for salvation, and do fastings, almsgiving, and other functions of piety and exercises of Christian service produce eternal reward, and that no one, whatever almsgiving he has practiced, even if he has shed blood for the name of Christ, can be saved, unless he has remained in the bosom and unity of the Catholic Church."

Must it be something formally taught (such as the Encyclical you just linked)?  Or can it be casually but pervasively taught (and believed by the modern faithful)?

Sure, just cite it and what dogmas it trespasses and where we can find the dogma.