ADDENDUM:
In NKS terms, what the basis/target machinery changes is this: it takes two of Wolfram's central qualitative moves; behavioral classification by inspection, and the observer-dependence of complexity and turns them into linear algebra with numbers attached.
Equivalence claims become measurable. NKS is full of statements like "rule 18 contains rule 90" or "rule X behaves like rule Y on this background," each established by a cleverly found encoding. The decomposition calculus gives that search a computational counterpart: expand rule X's diagonal in rule Y's diagonal basis over Z_m and read off three numbers — coefficient count, stability under window growth, residual rank. "Emulates" goes from an existence claim about some encoding to a matrix property you can scan for across all 256×256 rule pairs. NKS establishes resemblance by inspection; the change-of-basis viewpoint adds a way to measure its rank.
The observer becomes a formal object. One of NKS's deeper theses is that complexity is partly in the eye of the analyzer — a system looks random relative to the perceptual and computational tools brought to it. The basis choice makes that precise for one sharply-defined observer class: linear observers over Z_m. Each basis rule is an observer, and a target diagonal's coefficient growth in that basis measures how much structure that observer can see. Rule 30 having a dense, growing spectrum in every additive basis becomes a formal, checkable version of "rule 30 is random to all linear eyes" — falsifiable, since a single basis where it goes sparse would overturn it. The PCE intuition that rule 30 achieves maximal sophistication gets, along this one axis, an invariance statement to prove or refute rather than a principle to take on faith.
It also sharpens a distinction NKS leaves informal. On any finite window, an underdetermined linear system will "decompose" almost anything, which can look like a universality-flavored discovery until you check stability — and realize you've only shown that a simple-seed rule diagonal is p-power periodic on that window. That's a statement about the target's reachable class, not an elevation of the basis. Additive rules are closed-form predictable, so nothing expressible in their basis with stable finite coefficients can carry irreducibility. The framework turns a distinction NKS gestures at — emulation vs. universality — into something checkable.
And it complements the book's direction of reading. NKS reads patterns forward; run the rule, look at what appears. The decomposition reads them backward: given the appearance, project it onto every algebra simple enough to be fully understood, and define what remains; the part orthogonal to all additive bases at all moduli, the coefficient growth invariant under every linear change of observer; as the genuinely irreducible content. Computational irreducibility becomes a residue you compute: it's whatever survives every basis you throw at it.
