52. A finite approximation with an explicit price
On this page · Permalink · Source line 2079
A curved probability domain can sometimes be replaced by a finite one after changing the dynamics. That change must be quantified, not described as exact refinement.
Let \(P_n\) be the regular \(n\)-gon inscribed in the unit disk, with a vertex at \((1,0)\) and \(n\geq3\). Replace the rotate branch by
while retaining the same probe branches. The spatial coordinates are attenuated, but total probability mass remains unchanged.
Proposition 47: Polyhedral approximation of the rotation model
If
then the cone over \(P_n\) is invariant under all modified operators and contains \(z_0\). Consequently the modified model admits a finite positive lift. The original and modified models are both valid on the larger cone \(C_{\mathrm L}\), and for every common policy their length-\(T\) laws satisfy
The angle-independent sufficient condition can be improved for a fixed polygon and rotation. For the rational diamond \(|x|+|y|\leq t\), the present rotation is invariant after attenuation whenever \(\alpha\leq5/7\). Thus \(\alpha=2/3\) gives an entirely rational four-generator certificate.
Read proof · Recorded Lean scope · Permalink · Source line 2091
Proof
The regular polygon contains the disk of radius \(\cos(\pi/n)\) and lies in the unit disk. Every \(\alpha Rv\) with \(v\in P_n\) has Euclidean norm at most \(\alpha\), hence lies in \(P_n\) under the stated condition. The probe effects remain nonnegative because \(P_n\) lies in the disk, and their normalized images are the retained vertex \((1,0)\).
The base norm of \(C_{\mathrm L}\) is
Indeed the convex hull of the unit disks in the planes \(t=1\) and \(t=-1\) is the cylinder with this unit ball. The rotate-branch discrepancy on a normalized state is at most \(1-\alpha\) in this norm. Probe discrepancies are zero. Proposition 43 gives the law bound. For the diamond, the induced \(\ell^1\) norm of \(R\) is \(7/5\), so \(\alpha\|R\|_1\leq1\) proves invariance. \(\square\)
For a target attenuation loss \(0<\eta\leq1\), choosing \(\alpha=1-\eta\) and sufficiently many polygon vertices satisfies the certificate. The inequality \(1-\cos u\leq u^2/2\) shows that \(n\geq\pi/\sqrt{2\eta}\) is sufficient, with an integer ceiling and \(n\geq3\). This is a constructive upper bound on certificate size, not a lower bound or a proof of optimality.
The distinction is exact. At \(\alpha=1\) there is no finite polyhedral certificate for this model. At \(\alpha<1\) a sufficiently fine polygon works, at the cost of a declared change in the observation law. The reference figures show that tradeoff rather than labeling the approximation an identical model.