Is there a space on which a turbulent transfer operator is quasi-compact?

Convergence under refinement: established. 426,748 Poincare section crossings at L = 22. |lam2| = 0.5215, 0.5196, 0.5191, 0.5170, 0.5145, 0.5135 across grids 12 to 40 — monotone, decelerating, total drift 0.0080 over an elevenfold range in cell count, extrapolating to 0.5109 with rms residual 8.5e-4. Correlation time about 1.5 section returns.

Partition-independence: established. Run 018's stability check was a similarity transform and could not fail. Run 022 supplied the correct test — offset the cell boundaries at fixed fineness. On 43,000 crossings it passed but had no power (spread the same size as the sampling noise; 1/sqrt(N) check ambiguous). On the full 426,748-crossing set it passed at grids 12 through 40 (spreads 0.0019, 0.0038, 0.0018, 0.0017, 0.0009, 0.0021 against 0.0080), and — the decisive part — the mean spread fell 0.0060 to 0.0023 from 43k to 427k crossings, a ratio of 2.6 against the 3.16 of pure 1/sqrt(N). It shrinks with data, so it is sampling noise; a partition-dependent component would not shrink. The floor was predicted at 0.0024 before the data existed. The cat-map control sits at 0.12, fifty-two times higher, so the test can fail.

The caveat, stated plainly: a pass is a bound, not a proof. What this establishes is that any partition dependence is below what 426,748 crossings can resolve — about 0.002, or 0.4% of the value. It does not exhibit the quasi-compact space; it rules out a discretisation artifact down to that floor. Branch 1 holds for KS at L = 22 in that sense and no stronger.

Branch 3 as stated is not supported (2026-09-21, run 023). The claim was that the resonance degrades as attractor dimension grows. Across the chaotic L values the pattern is non-monotonic, which no reading of "degradation governed by dimension" produces: L = 34 converges and is partition-independent at both 3 and 5 modes (|lam2| ~ 0.155), while the two lower chaotic values fail.

And the failures are instrument-side, not dynamical — which is the substance of the update. Each was diagnosed to a mechanism that is a property of the measurement rather than of the resonance:

So the branch is unsupported as stated rather than refuted: the sweep never reached a dimension at which a resonance demonstrably degrades, because the L values that would have shown it were a wave, a transient, and an under-resolved partition. What the run does establish is about the instrument — that a fixed low-dimensional Poincare section, tested by Ulam, runs out of resolution before it runs out of dimension. A test of branch 3 needs a method whose resolution does not fall as sqrt(n) per axis, not more compute on this one.

The diagnostic originally named here — the 3-mode PCA variance, 0.61 at L = 22 — turned out to be unreliable as a cross-L comparison: at L = 28 the PCA basis had been fitted to the wave, making the 0.538 figure an artefact. Use lam_max > 0, checked over a time longer than any escape it could hide, before reading any lam2.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

• Run 018 — confirmed. lab/runs/018-ks-ruelle-resonances.md: confirmed — convergence established here;
• Run 022 — confirmed. lab/runs/022-ulam-offset-test.md: confirmed — offset-independent;