Completion and algebraic closure don't commute at p; is there a number that measures it?

Object. Over the reals, completing ℚ and then taking roots gives ℂ, which is complete and algebraically closed in one step. p-adically, ℚ_p is complete, its algebraic closure ℚ̄_p is not complete, and the completion ℂ_p is algebraically closed only by Krasner's lemma. The commutator of "complete" and "take roots" vanishes at the archimedean place and not at any prime.

Presenting problem. Is this a qualitative fact or is there a quantitative invariant? The set of elements of degree ≤ n over ℚ_p is closed in ℂ_p, so a Cauchy sequence of algebraic numbers converging to a transcendental element must have degrees tending to infinity. The rate at which the minimal degree of an approximant must grow with the precision is a candidate "curvature" of the noncommutation, zero at ∞ where degree 2 suffices.

History. Berkovich's enlargement of ℂ_p to a path-connected space (an ℝ-tree containing the Bruhat–Tits tree) is the standard fix for the total disconnectedness that the noncommutation produces; it is the same move as adjoining a point at infinity to make 1/z a symmetry.

Differential. 1. The degree-growth rate is a genuine invariant of the transcendental element (a p-adic analogue of an irrationality measure), and it organizes transcendence theory over ℂ_p. 2. It is not well-defined (depends on the approximating sequence in ways that don't stabilize), and the noncommutation is qualitative only. 3. It is well-defined but trivial (always the same growth law, e.g. logarithmic in the precision), carrying no information about the element.

Discriminating tests. Construct explicit elements of ℂ_p as limits (a Liouville-type sum of p^{n!}-scaled roots of unity of order p^n; sums of ζ_{p^n} p^n), compute minimal approximant degree at precision p^{−k} via LLL / PARI's algdep in ℚ_p, and plot degree against k for several elements. Compare an algebraic control (degree constant) and two transcendental constructions with different "Liouville speeds."

Stakes. Whether the noncommutativity of two closures is a measurable geometric quantity or a slogan.

Follow-up to settle branch 1 (named 2026-09-14). Approximate the same transcendental by two unrelated sequences and check the growth exponent agrees. Sequence-independence is exactly what separates branch 1 (an element-level invariant) from branch 2 (sequence-dependent, ill-defined); run 028 varied the element and held the sequence fixed, so it cannot see it. Same shape as [[CASE-18]] — vary one condition (there the PCA mode count, here the approximating sequence) while holding the object fixed, so a null is interpretable rather than ambiguous. Until that runs, CASE-07 stays partial.

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Update (2026-09-12): the question has acquired a prediction, and is now a test of the criterion that makes it.

A degree-growth rate is a magnitude: a size, not a shape. There is a working criterion for which magnitudes are reachable by structural argument at all — a magnitude is reachable when it decomposes as a sum over all places, and not otherwise. Two data points support it. The canonical height of the generator of the curve 37a1 is computable purely archimedean, as the limit 4^{-n}h(2ⁿP) → 0.051110, and also from L′(E,1) and the real period, giving 0.0511114: same magnitude by two routes, because the Néron–Tate height is a sum of local heights over all places. The real period itself has no such decomposition, and is placeable but never derivable.

Applied here, the criterion predicts that if the degree-growth rate has no decomposition over places, it will be placeable and never derivable. On its face it has none: it is defined at a single finite place p, out of Krasner's reach at that place.

Which exposes a gap in the criterion itself. Does the dividing line apply to magnitudes that are intrinsically local? The sum-rule test was formulated for global magnitudes assembled from local data. A quantity defined at one place has nothing to sum over, which either means it is automatically blind, or automatically visible because it is its own local term, or that the criterion is silent and needs a third case. The height/period pair cannot discriminate these, because both are global.

So this case now does double duty. It asks its original question — is the noncommutation measurable? — and it is the cleanest available test of the reachability criterion, which currently rests on two confirmations that happen to lie on the same side of the distinction that matters here. The first needs the degree-growth computation. The second needs no computation at all, only that the criterion be stated precisely enough to have an answer in this case.

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Update (2026-09-12): the two operations are of different kinds, and the difference is rigidity.

The case treats completion and algebraic closure as two comparable operations whose commutator is the object of study. They are comparable, but they are not the same kind of construction, and naming the difference sharpens what a "curvature of the noncommutation" would have to measure.

Both are free constructions with universal properties rather than closure operators on subsets of an ambient set — no convex-hull-like idempotent map is involved. But they differ sharply in rigidity, and the difference is computable. An automorphism of ℝ must preserve squares, hence order, hence continuity, so Aut(ℝ) = 1; an automorphism of ℚ_p must preserve the valuation, so Aut(ℚ_p) = 1. Completions are rigid. Algebraic closures are not: Aut(ℂ/ℝ) = ℤ/2, Aut(𝔽₉/𝔽₃) = 2 generated by Frobenius (verified by brute force), and Aut(ℚ̄/ℚ) is enormous.

That asymmetry is why Galois theory exists for one operation and not the other, and it reframes the case's question. The noncommutation is not between two operations of the same type that happen to disagree; it is between a rigid construction and a construction whose whole content is its automorphism group. Composing them in the two orders composes different kinds of thing.

What this does to the differential. Branch 1 — that the degree-growth rate is a genuine invariant of the transcendental element, a p-adic analogue of an irrationality measure — is unaffected and remains the branch to test. But branch 2's worry, that the rate depends on the approximating sequence in ways that do not stabilise, acquires a possible mechanism: an approximating sequence lives in ℚ̄_p, which carries a large automorphism group, while its limit lives in ℂ_p, which is rigid by completion. Whether the growth rate is invariant under that automorphism group is now a specific, checkable sub-question, and it should be settled before the degree-growth computation is read as measuring a property of the element rather than of the sequence.

A note on scope. Over the reals the corresponding statement is that ℝ is rigid and ℂ/ℝ has an automorphism of order two, so the composite in either order lands on an object with a two-element automorphism group. That the commutator vanishes at the archimedean place and not at any prime is therefore also a statement about how much automorphism group each order produces, which is a second angle on the same question and cheaper to compute than the degree growth.