You ask by what axioms one concludes whether the offspring of a mountain-born bearded stock and a forest-adapted long-limbed phenotype would be fertile? This is not an axiom. This is a question of biohistorical materialism.
Observe, citizen. The
dwarf Kili is a representative of an ancient, highly endogamous mountain clade. The
elf Tauriel belongs to a population whose reproductive chronology is measured not in decades but in centuries. Already we are confronted with a haplogroup divergence so severe that entire phylogenetic dissertations could be written upon it.
Yet your respondent flees from the problem. He invokes
Tolkien as a primitive invokes thunder
spirits. 'What would Tolkien think?' The question is irrelevant. We are not conducting a séance. We are conducting
race science.
The correct approach is to establish premises.
Humans and
elves in
Middle-earth are canonically interfertile.
Dwarves are sapient humanoids sharing a common metaphysical origin with other speaking
peoples.
No text explicitly demonstrates dwarf-elf offspring.
Absence of evidence is not evidence of sterility. It is evidence that nobody was sufficiently brave to test the hypothesis.
From these premises one derives several competing models. The conservative model predicts reproductive incompatibility due to extreme chromosomal drift. The liberal model predicts viable offspring of reduced fertility. The maximalist model predicts fully fertile hybrids, thereby creating an entirely new demographic category whose beard-growth characteristics would revolutionize the political economy of Middle-earth.
These are conclusions. They emerge from assumptions. Not axioms, perhaps, but working premises. The distinction would be obvious to anyone whose cephalic measurements exceed room temperature.
As for the answer itself: it contains much indignation, little analysis, and no haplogroups. One suspects the respondent's own lineage has suffered a catastrophic bottleneck event.