Bhāradvāja Vimāna: The Axis of Convergence

For every p, integer or not, there is an elementary map that multiplies your correct digits by exactly p at every step. So the speed is something you control. Before you build anything you say how fast you want it to converge, and the map that converges exactly that fast is one line. Here it is. Send the pair (x, y) to the even and the odd half of the binomial expansion of (x+y)p, and call it Vp, the Vimāna: a thing measured out, and in the classical sense a vehicle that crosses distance at the speed of thought. Both of those are meant. In the mūla-bheda coordinate μ = (x−y) / (x+y) this map is nothing but μ ↦ μp. So the digits obey Dn = pn D0, and that is an equality. No error constant, no asymptotic regime, no restriction on p. It preserves x²−y², and it composes: Vp ∘ Vr = Vpr. So the order of convergence is not a thing you work out after the method is built. It is a coordinate on a group, and you set it first. It is also free: at integer p, reaching D digits costs 2 log₂D multiplications whichever integer you pick. Choosing p decides how much work sits inside one step, not how much work there is, because the quartic algorithm is the quadratic composed with itself. The axis is stratified: polynomial at integer p, algebraic at rational p, transcendental everywhere else, and that is why the classical record sits on the integers. For π the axis is exactly ℚ. Some of those integers are already taken. Brahmagupta's bhāvanā of 628 CE is Vp over ℤ[√D], the oldest one we know of. Newton's square-root iteration is p = 2 and Halley's is p = 3. On the unit hyperbola it is the Chebyshev polynomials, whose composition rule is the group law itself; on the unit circle it is de Moivre. The arithmetic-geometric mean and Landen's transformation are the same power map read in the nome, which is what puts the Borwein algorithms at 2, 3, 4, 5, 7, 9. Every one of them is one fixed point on an axis you can now move along. One thing to be clear about: the speed here is convergence per step, not execution time on a machine. Multiplying bigger numbers costs more and none of that is being claimed away. The claim is about how much convergence a single iteration buys, which is what all of these were measured by in the first place. Companion code (MIT): github.com/bhardvaja/vimana