run 028Held
../lab/runs/028-krasner-degree-growth.mdBlocked onthe sequence-independence clause — the same element through a second approximating sequence (CASE-07's follow-up); until then the run is held, not scored
Predictionfor the linear construction x = Σ ζ_{p^n} p^n the Krasner minimal-degree bound grows ≈ p^k·(p−1)/p at precision p^{−k} (exponential in k); for the quadratic x = Σ ζ_{p^n} p^{n²}, ≈ p^{√k} (sub-exponential). The growth exponent is an invariant of the element — the curvature between completion and algebraic closure at p, the inverse of the Liouville speed of the defining series — and it is 0 at ∞.
Falsifierthe two constructions give the same degree-growth law (no separation by Liouville speed), or Krasner's inequality fails to apply at the computed precisions (precision ≥ Krasner radius) so no element-invariant rate is defined — either way the noncommutation is not quantitative
VerdictBranch 1 supported, branch 2 untested (recorded in the body 2026-09-14, transcribed to the frontmatter 2026-09-28). At p = 3, N = 1…8, log_p(deg)/k for the linear element and log_p(deg)/√k for the quadratic agree term for term (0.315 … 0.848, both → 1): growth exponents 1 and 1/2, separated by Liouville speed as predicted, Krasner applicable at every step. The prediction's second clause — that the exponent is an invariant of the element — is untested: one approximating sequence per element shows that different elements differ, not that a given element is sequence-independent. No outcome token until that clause is run.
Recovered, not written. R6 was produced in the genesis chat alongside CASE-01–07 and never downloaded — three searches read it as a missing file (see OPS.md). It carries its own prediction, falsifier and sizing in its header; nothing is added here beyond the run frontmatter.
Object. For an explicit x ∈ ℂ_p \ Q̄_p given as a limit x = lim x_N with x_N ∈ Q_p(ζ_{p^N}), Krasner's lemma bounds the minimal degree of any approximant β within precision |x − x_N| below by deg(x_N) = p^{N−1}(p−1). The run computes the Krasner radius and precision for two constructions and reads off the growth law of that degree bound against precision.
The two constructions. Linear x⁽¹⁾ = Σ_n ζ_{p^n} p^n; quadratic (Liouville-fast) x⁽²⁾ = Σ_n ζ_{p^n} p^{n²}. The claim is that the growth exponent separates them and is an invariant of the element, 0 at the archimedean place.
Sizing. Sage handles the totally ramified Q_p(ζ_{p^N}) via Eisenstein polynomials; conjugate distances come from the cyclotomic structure. R6's header estimates minutes — this case is one run from a verdict.
Result (2026-09-14, BernoullisBook · Sage 10.9 · p = 3 · N = 1…8). The prediction holds exactly, and the confirmation is cleaner than the prediction. log_p(deg)/k for the linear element and log_p(deg)/√k for the quadratic come out identical term for term —
0.315, 0.544, 0.658, 0.726, 0.772, 0.804, 0.829, 0.848
— both converging to 1. So deg ≈ p^k for the linear element (growth exponent 1) and deg ≈ p^√k for the quadratic (exponent 1/2): the exponent separates the two by Liouville speed exactly as predicted, and is an invariant of the element. Krasner's inequality applies at every step in both constructions, so the degree bounds are genuine, not artifacts of precision exceeding the Krasner radius. The explicit N = 2 check gives 1/6 against the expected 1/φ(N) = 1/6.
Verdict: branch 1 supported, branch 2 untested. The degree-growth rate is a genuine element-level invariant with a clean exponent difference (branch 1). Branch 2 — that the rate is not well-defined, depending on the approximating sequence rather than the element — is not excluded: this run used one sequence per element, so it shows different elements differ, not that a given element is sequence-independent. So CASE-07 is partial, not settled (see the case for the follow-up that would close it).
The run's own failure, for the record. The first attempt died on a format string — Sage's Rational does not accept :10.4f. The mathematics completed; only the printing failed, and float() fixed it. That makes R6 the third genesis-era recipe to need a fix before its first execution, after R18's integrator and R19's four bugs — exactly what the recipes README says of the batch: each file is one calculation the sandbox could not finish.
generated 2026-10-25the lab index