The cardinality conjecture for spaces generated by sigma-sets

Let $IN=\{1,2,3,\ldots\}$, let $IN^{\star}=\{1^{\ast},2^{\ast},3^{\ast},\ldots\}$, and let $IN_{\Theta}=\{1_{\Theta},2_{\Theta},3_{\Theta},\ldots\}$. For a set $S$, write $2^S$ for its power space, and let $\langle 2^{X\cup Y},2^{X\cup Y^{\star}}\rangle$ denote the space generated by these two spaces. The cardinality conjecture. For all $X\in 2^{IN_{\Theta}}$ and $Y\in 2^{IN}$,

$$|\langle 2^{X\cup Y},2^{X\cup Y^{\star}}\rangle|=2^{|X|}3^{|Y|}.$$

The conjecture proposes a cardinality formula for spaces generated by two sigma-sets and is intended to establish new relationships involving infinite cardinals within sigma-Set Theory. The paper provides only foundations and does not report a resolution.