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The Routing Axis

The Routing axis is the framework's name for the codomain side of any transformation step — the choice of where a step sends its transformed result. It carries two of the four directional primitives: Loop-back (route the result back into the system) and Propagation (emit the result outward). Where the companion Selection axis governs what a step admits or excludes, Routing governs what happens to the result once the step has run. The two axes together are how the framework explains why the primitives number exactly four.

The carving idea

The framework insists the primitive set is not merely listed at four but carved at four. The carving claim: a recursive transformation step has two formal sides — a domain side (Selection) and a codomain side (Routing) — and each side is a binary choice. The product {toward, away} × {loop-back, propagation} then yields the four primitives. Routing supplies the second factor. This is the point where the contestability gradient becomes visible, because the count's two halves are held with different strength.

A note on names: the seed called the routing pair inward and outward (symbols ↺ ↻); the rename to loop-back and propagation is exact and structure-preserving — terminology over an unchanged architecture.

Why no fewer than four

The two axes are independent: Routing cannot collapse into Selection, and Selection cannot collapse into Routing. The independence is shown behaviorally — the four combinations admit-and-retain, admit-and-emit, exclude-and-retain, exclude-and-emit are genuinely distinct. Because neither axis can be dropped without losing distinctions, the primitive set cannot fall below four. This no-fewer claim is held as frame-internal: it follows by construction, contestable only by declining the carving. Routing's independence is what makes loop-back and propagation irreducible to the Selection moves rather than special cases of them.

The Selection/Routing decomposition is kept throughout because it is load-bearing twice over: it generates both the emotion set (via predicate binding) and the Force set.

Why "no more" is only contested

The framework's argument that the two axes are exhaustive — that there is no third directional side — is strong as a mapping. Magnitude, timing, medium, gain, threshold, and channel all describe how a step occurs, not a new direction it can take; and the hold/maintain stress test resolves cleanly (hold is identity; maintain is identity or active loop-back at zero current error). Even so, the 2026 adversarial run surfaced candidate motions — a memory or temporal operation, for instance — that the two-axis carving does not obviously contain. So the exactly-four / no-more claim is held as a carving / count claim, contestable by counter-instance, and the benchmark treats four-primitive sufficiency as an open proof-burden.

The upshot: the four-primitive set is a contestable carving — the most accurate carving available, not a sealed closure. A carving can be the best one going without being final, exactly as the framework's no-possession condition predicts.

Common misreadings

Routing is not a hierarchy and not a challenge-proof closure. It must not be confused with the content of what is routed — loop-back and propagation name only the direction (home vs. outward), not the value carried. And it must be kept distinct from the higher-order aims it feeds: the bare propagation operator is not the same as the Telos's outward continuation. Finally, Routing does not collapse into Selection; the four combinations are behaviorally distinct.

Formal status

Formal status. Epistemic: Derived. The no-fewer claim is frame-internal — the four combinations are behaviorally distinct by construction, contestable only by declining the carving. The exactly-four / no-more claim is carving/count, contestable by counter-instance, and the benchmark treats four-primitive sufficiency as an open proof-burden. Alethic: the decomposition aspires to map the real structure of a transformation step and maps it well — the surrender is of closure, not of accuracy. Provenance: the carving is canonical (the four semantics are authority-fixed); its exhaustiveness is treatise-side demoted to a contestable carving on a benchmark-flagged open burden; the seed's inward/outward underlies the rename.