Manuscript Oai365-A@1 · Oai365-A.tex · proposition · a3840e3d

Proposition: Non-vacuity

An explicit enriched-process Lorentz pilot › A predeclared enriched subclass · lines 482–506 · proof included or linked

Oai365-A@1#prop:nonempty

The rich tagged class is nonempty. Indeed, for every $s \in [1, 2]$ , choose a countable dense sequence $r_{e} \in S^{2}$ and set

p_{e} = \frac{s}{e + s}, e = 1, 2, \dots .

Then

p_{*} = \frac{s}{1 + s}, p_{e}^{- 1} = 1 + \frac{e}{s} .

Proof. Rational stereographic coordinates provide a countable dense subset:

r (a, b) = \frac{( 2 a , 2 b , 1 - a ^{2} - b ^{2} )}{1 + a ^{2} + b ^{2}}, a, b \in Q .

Enumerate it. The reciprocal gaps are at least $1/ s \geq 1/2$ ; the maximum tag is the first, and it belongs to the required interval. Validity follows from Lemma lem:pilot-process.

Displayed formulas in this passage

unlabeled · line 486
$p_{e} = \frac{s}{e + s}, e = 1, 2, \dots .$
unlabeled · line 490
$p_{*} = \frac{s}{1 + s}, p_{e}^{- 1} = 1 + \frac{e}{s} .$
unlabeled · line 498
$r (a, b) = \frac{( 2 a , 2 b , 1 - a ^{2} - b ^{2} )}{1 + a ^{2} + b ^{2}}, a, b \in Q .$

Proposition: Non-vacuity

Displayed formulas in this passage

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