Exploration — do the partial-level ladders decompose into dimensions?

Status: POST-RELEASE. An open question, not a proposal to land. Nothing here has been accepted, and nothing in data/ or compute/ has been changed on its account — the scored corpus and the ladders it is scored against are unchanged. The papers carry a footnote saying the question is open; that footnote is the whole of what has shipped.

The question

Each primitive's partial levels are written as a ladder: one number, ordered. Several primitives look like they have more than one independent internal axis, which would make an intermediate rung a projection — it ranks whichever axis the ladder happened to order and says nothing about the others.

Tree is the clearest instance, and Paper 6 §"Tree: Three Dimensions" already states the evidence without drawing the conclusion:

"Plan 9 achieved depth and scope but not hash-based bindings. Git achieved depth and bindings but fixed namespace scope."

Binding (a path resolving to a content hash over an immutable store) and composition (assembling a namespace from parts its owner does not control) vary independently, and both off-diagonal corners have occupants. One T number cannot say which a system has.

Why it is not urgent

The endpoints converge. At the maximal setting the axes are all at maximum together, so Full T, Full I and the rest name the same position under either reading. Every load-bearing claim in the corpus about the entity system itself is stated at the maximum and is therefore unaffected. The ambiguity lives strictly in the intermediate rungs, which is where the corpus is doing comparative description rather than making structural claims.

That is the reason this is a footnote and not a rewrite.

What a future pass would have to decide

1. Whether the axes are sub-axes of one primitive or primitives in their own right. Identity is already documented as one primitive with three internal dimensions (Paper 6 §"Identity: Three Dimensions"), so there is precedent for the first answer. Nothing yet distinguishes the two cases.

2. How to score, if a ladder is kept. Either keep one number and state in prose which axis it ranks, or score the axes separately. The second costs a schema change and would have to be general across all 26 domains, not special-cased to this one.

3. Whether the same thing is true of the other five primitives. Not established. It looked that way on a quick pass, and a quick pass is exactly what should not settle it.

   ⚠ Sharpen this before spending a session on it — the question is narrower than it reads. Paper 6 §Internal Substructure already names internal axes for all six primitives (E 3, I 3, T 3, M 3, X 4, P 4), and has for a long time. So "do the other five have axes" is answered in the affirmative by the corpus itself. What is not established is independence — whether a given primitive's axes vary separately across real systems, which is what makes a rung a projection rather than a summary. Tree has that evidence (Plan 9 and Git occupy opposite off-diagonal corners) and Identity has it (the three-corner table in §"Identity: Three Dimensions"). The other four have named axes and no occupancy evidence cited either way. That is the gap to close, and it is the mn column's job, not the definitions'.

What the corpus reach actually is — measured, so nobody re-scopes this by grep

Counting substrate rung tokens ([EIXTMP][0-4]) per paper: Paper 6 146, Paper 0 80, Paper 5 6 (X0–X3, biology mapping, already hedged), Papers 3 and 10 2 each (I1, both anchored to assigned identity, not content-derived and both stated against Full I), and zero in the other ten. Paper 4 has none — its gradient is the compilation gradient. Papers 11, 12 and 13 use partial level in the methodology's domain-general sense.

So a ladder rewrite's paper-side blast radius is Papers 0 and 6, plus one sentence each in 3 and 10. That is the papers only. It does not contradict the warning below — the integers still travel into data/ and the arrangement cascade, and that is where the four-directory surprise lives.

Method note for whoever picks this up

The honest way to answer (1) and (3) is not to re-adjudicate rung by rung. It is to name the underlying capability predicates per primitive, score the corpus on those predicates, and then measure pairwise independence — how many systems have A without B and B without A. Independence is a claim about real systems and should be counted, not asserted. That is a general instrument and it would apply to every domain, which is the bar AGENTS.md sets for any new capability.

One practical warning for whoever attempts it

The rung labels are integers, and they travel further than the concept does. A reworked ladder does not stop at the primitive's own definition: the same integers appear in the scored manifestations, in the walk endpoints, in the trajectory snapshots, and in the arrangement's load-bearing cascade, where a level is named as a threshold rather than described. A search for the concept will report that nothing outside the papers depends on it, and that report will be wrong.

Grep the number, not the noun. A trial pass estimated this change as papers-only on exactly that reasoning, and the estimate was wrong by four directories.