The Steppe-Origin Transmission Model, Reconsidered: Why the Narrow Feasible Corner Does Not Survive Formal Axiomatization

This is the third paper in a sequence whose purpose, taken as a whole, is to establish theaxiomatic truth of the claims made in Paper 1 by subjecting them to the same rigor anadversarial reviewer would apply -- including independent effective-population-sizereconstruction (hapROH) of both the steppe source and the Swat Valley samples themselves --rather than to defend or restate them. It examines the demographic feasibility of the strongSteppe-origin transmission model (SOTM) for Indo-Aryan ancestry and language transmission intoSouth Asia. The first paper Kadambi2026a argued, via a qualitative proof bycontradiction, that the SOTM's feasible set is empty. The second Kadambi2026bstress-tested that claim computationally, finding mechanisms that survive numerical testing atheadline rates as high as 23. 2\% before a series of corrections converged toward 1--2\%. Thispaper extends that testing to every mechanism proposed across the sequence -- a directsteppe-to-Indus model motivated by the documented absence of BMAC-core ancestry in South Asia, a founder-regional model under two independently-derived transit assumptions, and the ``Indusattractor'' alternative named in Paper 1's own title -- and checks each against the real, documented scale of the Indus population (500, 000--5, 000, 000), realistic mountain-pastoraltransit pace, actual cattle-droving logistics, and the sparse, outlier-status archaeologicalsignature the ancient-DNA record actually shows. Every mechanism tested either collapses towardzero or requires a cohort size that directly contradicts the evidence for how small andundetected the migrating population must have been; the one mechanism structurally differentenough to avoid this (a slow demographic trickle rather than a bulk movement) failsindependently, by depleting the steppe source faster than it can regenerate within the actualdated contact window. We show these two failure modes are not coincidental but jointlyexhaustive: any point on the spectrum from a single migration to an arbitrarily slow tricklemust trigger one constraint or the other, which we state and prove formally as two theoremsderived from primitive logistical admissibility axioms whose numerical bounds are calibratedby ethnographic, archaeological, and genetic evidence already accepted in the field. Separately, we show that the same demographic mechanism required to make theancestry claim survivable makes the accompanying language-transmission claim mathematicallyworse, not comparably uncertain, since language has no equivalent to ancestry's indefinitelow-frequency detectability. We conclude that, given the field's own accepted premises, anempty feasible set is not a probabilistic likelihood but the only mathematically validconclusion available, and state precisely what would need to be found -- a specificarchaeological signature, a specific isotopic result -- to overturn it.