La maison d'Hades

Since there are, in principle, an infinite number of ways to deform classical tensor products via an uncountable set of cubes and an indefinite number of ways to decompose a given deformed tensor product, discovering methods that provide such decompositions seems, at first glance, to be a particularly difficult challenge to overcome.

To try to bring some order to this process, I propose a classification of the methods that could be developed

The term "intrinsic" refers here to any approach considering the system (S) already encountered during the premises of the discussion regarding "trivial decompositions" (reminder):

| ⊗_A(a, b) > = [P]. | b > + | z > ∈ E*(D, K)

The classic statement

Let there be a set of D linear combinations written as a function of the D components of the target. This set constitutes a system (S) such that:

[M]. |b> = |y>

The classic problem considers the pair ([M], y) from M(D, K) x E(D, K) as known and seeks to uncover an unknown b in E(D, K):

b = ?

The formal analogy between a decomposition and a classical system is striking. Since there is always at least one trivial decomposition, any given decomposition can be written as:

{_AΦ(a) - [P]}. | b > = | z >

By posing:

[M] = _AΦ(a) - [P]         y = z

It is possible to have the illusion of being faced with a classic statement.
This appearance is deceptive.

The so-called (E) question
 
Within a discussion examining the decompositions of a deformed tensor product, there is no question of discovering the components of the target.

The known data are: the deforming cube A, the projectile a, and the target b.

The unknowns are the principal part [P] and the residual part z of the decomposition, the existence of which is presumed.

Thus, here, it is appropriate to read the system (S) with a different perspective that the colors attempt to materialize:

| ⊗_A(a, b) > = [P]. | b > + | z > ∈ E*(D, K)

Due to the systematic existence of at least one trivial decomposition, the system (S) can always be rewritten as:

{_AΦ(a) - [P]}. | b > = | z >

The formal resemblance to a classical system allows for only one thing, namely understanding the importance of the polynomial. :

Λ(a) = |_AΦ(a) - [P]|

Definition – Intrinsic method for decomposing a deformed tensor product.

Any mathematical method capable of discovering the pairs ([P], z) using only the known data: A, a, and b.