Extremely luminous, red eruptive variables like RV in M31 are being suggested as exemplars of a new class of astrophysical object. Our greatly extended series of nova simulations shows that classical nova models can produce very red, luminous eruptions. In a poorly studied corner of 3-D nova parameter space (very cold, low-mass white dwarfs, accreting at very low rates) we find bona fide classical novae that are very luminous and red because they eject very slowly moving, massive envelopes. A crucial prediction of these nova models -in contrast to the predictions of merging star ("mergeburst") models -is that a hot remnant, the underlying white dwarf, will emerge after the massive ejected envelope has expanded enough to become optically thin. This blue remnant must fade on a timescale of decades -much faster than a "mergeburst", which must fade on timescales of millenia or longer. Furthermore, the cooling nova white dwarf and its expanding ejecta must become redder in the years after eruption, while a contracting mergeburst must become hotter and bluer. We predict that red novae will always brighten to L ∼ 1000L ⊙ for about 1 year before rising to maximum luminosity at L ∼ 10 6 − 10 7 L ⊙ . The maximum luminosity attainable by a nova is likely to be L ∼ 10 7 L ⊙ , corresponding to M ∼ −12. In an accompanying paper we describe a fading, luminous blue candidate for the remnant of M31-RV; it is observed with HST to be compatible only with the nova model.