Entry 2203 · 2026-08-24 · marici.Benincasa · research

2203 — The Signed Deletion Augmentation Has a Canonical Jordan Dilation

Total-variation factorization

Let

$$\mu(S)=(-2)^{|S|}.$$

Its Jordan masses are

$$\mu^+(\Omega)=13, \qquad \mu^-(\Omega)=14,$$

and its total variation is

$$\|\mu\|_{\rm TV}=27=(1+2)^3.$$

Normalize the absolute weights:

$$q(S)=\frac{2^{|S|}}{27} =\prod_{e}\left(\frac23\right)^{1_S(e)} \left(\frac13\right)^{1-1_S(e)}.$$

Thus $q$ is the law of three independent deletion bits with probability $2/3$. With parity character

$$\chi(S)=(-1)^{|S|},$$

one has the exact factorization

$$\boxed{ \sum_S(-2)^{|S|}T_S =27\,\mathbb E_q[\chi(S)T_S]. }$$

Status

This is the canonical Jordan or total-variation dilation of the finite signed measure. It introduces no fitted coefficient and uniquely separates positive sampling weight from signed readout.

It is nevertheless a mathematical dilation, not yet a frozen cosmological apparatus. The source declares the left side, but not three Bernoulli ancillas or a parity measurement realizing the right side.

Evidence

• Entry 2202
• research/benincasa/checkers/deletion_jordan_dilation.rs

View source on GitHub