Properties of Boolean functions/hard

Hard properties can be assigned to a BF, without referencing its arity.

Weight is the quotient of true and all places of the truth table. It is usually shown as an integer for a given arity (soft weight).
This triangle shows the numbers of BF by weight.

Nonlinearity is the extent to which a BF is not linear. It is usually shown as an integer for a given arity (soft nonlinearity).

Atoms or atomvals are the relevant arguments of the BF. They correspond to the circles of an Euler diagram. The number of atoms is the valency.
I this project the letters A, B, C... correspond to atoms 0, 1, 2...   E.g. ${\displaystyle B\land C}$ has atoms 1 and 2.

The root of a BF created by replacing its atoms with the set {0, ..., valency−1}.
When it is its own root, the BF is dense.

based on input negation and permutation: family (negation), faction (permutation), clan (both)   (Typically represented by Smallest Zhegalkin index.)

extended by complement: super-family, super-faction, super-clan   (Families and clans can be self-complementary. Factions can not.)

(Further extension by half-complement leads to ultra-famlies and -clans, which are soft properties.)

The prefect is a way to assign each BF to a linear BF. A linear is assigned to itself.    (3-ary images)

calculating prefect from Zhegalkin index
			The possible results are the linear functions.

These properties assign every almost every BF to a finite set of integers. The result for the contradiction is infinite. Soft equivalents have been defined to avoid this problem.
The cardinalities of the results are gravity and depth.