WP-100 established the geometry: hexagon at 78.5°N with axis radius R₆ = 13,260 km and λ₆ = 13,886 km; decagon at 63.0°S with R₁₀ = 29,641 km and λ₁₀ = 18,624 km. COMPUTED It also recorded, in its §7, the objection that undercuts every selection-law story: the discovery paper’s own shallow-water runs relax to whatever wavenumber they are seeded with. If the jet inherits its count from the forcing, no law is picking it.
The natural next move is empirical rather than theoretical — go to the archive and watch the number. This note is the feasibility calculation that move requires, and it returns a smaller answer than the one proposed.
WP-97 established that a single field carrying both a sixfold and a tenfold symmetry is invariant under C₃₀, because ⟨1/6, 1/10⟩ = ⟨1/30⟩ in ℚ/ℤ, and that nobody has photographed a triacontagon. CHECKED That result is true and stays true. It is also a theorem in the periodic category, and the distinction is worth stating because it is the whole content of the quasicrystal literature.
A lattice admitting an n-fold rotation exists in dimension φ(n) and no lower — the companion matrix of the n-th cyclotomic polynomial acting on ℤφ(n).
So thirty was doing the work, and in the quasiperiodic category thirty’s price is eight dimensions rather than a constant. COMPUTED The realised quasicrystal orders in nature — 5, 8, 10 and 12 — are exactly the n with φ(n) = 4, which is why decagonal and dodecagonal solids exist and 7-, 9-, 11- and 13-fold ones do not.
Every statement above is about lattices. A zonal wave has m = 2πR/λ and admits any integer; nothing in fluid dynamics knows about φ(n). That 12 is the only other order available to a crystal at the same embedding cost as 10 says nothing whatever about which wavenumber a jet will adopt. This is the line tools/verify-polar/README.md draws, and it is drawn here for the same reason.
The decagon drifts eastward at 2.5 m s⁻¹ and its vertices oscillate in longitude with a 32-day period. CHECKED On the WP-100 radius those are two independent clocks:
A denominator of thirteen before the ratio comes within half a per cent of a rational is not a resonance. Two clocks with no low-order commensurability is motion on a 2-torus that never closes — a quasiperiodic signature read out of the data rather than assumed. COMPUTED
The published uncertainties on the 2.5 m s⁻¹ drift and the 32-day libration are not in hand here. If the drift bar admits ±0.5 m s⁻¹, then 8/3 = 2.667 sits comfortably inside, the clocks lock, and the quasiperiodic reading dies on the spot. OPEN
Saturn currently shows a warm pole and a cold pole simultaneously, which invites a comparison. With LD = NH/f and the Coriolis difference removed:
Let the south warm to the north’s value with the jet held at 63°S and m: 10 → 11.72. COMPUTED Against the requirement table — m = 11 needs −9.1%, m = 12 needs −16.7% — the measured −14.7% sits between them, and the 2% gap is smaller than any error bar available. Eleven and twelve are not separated by this.
It assumes a selection law exists, which is precisely what WP-100 §7 reports the discovery paper’s own model denying. It computes the product NH rather than N — a warmer pole has a larger scale height, which pushes λ the other way, and the split is not determined here. It identifies λ with LD when the fastest-growing mode is some multiple of LD that may differ between the two jets. It treats a hexagon stable for forty-four years as a free instability when its persistence suggests a trapped mode. And it borrows the north’s stratification for the south. Five assumptions, two data points, no error bars. OPEN
The Hubble OPAL programme added Saturn in 2018, after Cassini’s de-orbit. A MAST query on the programme’s map target returns an unbroken annual cadence:
The discovery paper’s record begins in 2023. That appears to leave five archived years, 2018–2022, untouched and public — a self-contained archival project. CHECKED
It does not, because for most of that window Saturn’s south pole was edge-on from Earth. Northern summer solstice was 2017; the south only opened as the planet approached the May 2025 equinox. Taking JPL Horizons’ sub-observer latitude and asking what emission angle 63°S presents — a point is visible when its surface normal lies within 90° of the line of sight, which makes this a planetographic condition on an oblate body, not a spherical one:
At a μ ≥ 0.20 cut, the first usable epoch is 2022. The untouched, usable archival window before the paper’s 2023 baseline is one year, not five. COMPUTED 2018–2020 sit at μ = 0.016–0.101: above the limb, but edge-on, with the wave smeared along it. 2021 is marginal at 0.169. The paper beginning at 2023 is very nearly when the geometry became usable, and is not an oversight.
One correction belongs here because it was made in the course of the work. A first pass used the spherical relation 90° − |sub-observer latitude| with Horizons’ planetodetic value and concluded that 63°S was behind the limb in 2018 and 2019 by 4.0° and 1.9°. That was wrong in sign: on the oblate figure the correct condition is planetographic, and 63°S clears the limb by 0.94° in 2018. The conclusion survives — those years are unusable — but for the reason μ gives rather than the one the first calculation gave. A gap argued from the wrong quantity is a gap a referee dissolves in one line.
§4’s worst assumption is the borrowed northern stratification. It can be removed, because Cassini observed the south pole in its own summer: arrival 2004, southern equinox August 2009, giving five years of CIRS coverage at exactly these latitudes in the opposite season. CHECKED
So the measurement is available without borrowing anything: take NH at 60–65°S from CIRS in southern summer, take the wavenumber history from OPAL 2022 onward, and constrain what the same pole should adopt approaching southern summer solstice. Same latitude band, two seasons, no hexagon-as-equilibrium assumption and no cross-hemisphere transfer. Both halves are already on disk.
If wavenumber is set by a stratification-dependent length rather than
inherited from the forcing, the southern count must change as the pole warms toward southern summer
solstice in 2032, when μ at 63°S reaches 0.806 and the viewing geometry is the
best it will be this century. The direction is upward; the magnitude is not resolved between 11 and 12
by anything in §4.
Refutation: if the decagon is still ten-sided at southern summer solstice, this line is
wrong. No reinterpretation is offered in advance and none should be accepted afterwards.
The archival half tests the same fork retrospectively, at one epoch. If 2022 already shows m = 10, the wave predates the published record and the inheritance reading gains. If 2022 shows a different count, a parameter is tracking and the selection-law reading gains. If 2022 shows no wave, the interesting question becomes what happened between 2022 and 2023.
It proposes no mechanism for wavenumber selection and offers none. OPEN §3 is conditional on error bars not in hand; §4 is conditional on a premise the discovery paper’s own model argues against, and is carried as a check rather than as support. The φ(n) material in §2 is crystallography and does not bear on an atmosphere. Nothing here is a statement about the contact-geometric framework elsewhere in this corpus — the decagon does not become dm³ because it has ten sides, and ChladniPolygon.lean and WP-77 already say why that bridge is unavailable.
The μ ≥ 0.20 cut is a convention, not a derivation. A team willing to work at μ = 0.17 gets 2021 as well, and the honest form of §5 is that the window is one epoch at this threshold and two at a looser one.
This note exists because a five-year archival gap, 2018–2022, was proposed and described as definitive. The programme dates behind it are correct and check out against MAST. The gap is not, because the calculation that was never run is the one that decides it: for four of those five years the target latitude was presented edge-on. COMPUTED
WP-97 corrected a symmetry claim by naming an observation. WP-100 took the measurement and asked what the pair is. This note is the third of that kind and the least interesting to write: it asks whether the instrument could have seen the thing at all, and finds that mostly it could not. That question costs one afternoon and is the first one a referee asks.
[1] Sánchez-Lavega, Simon, Wong, Fletcher et al., Sci. Adv. 12, eaee4251 (2026).
[2] Fletcher et al., “Seasonal evolution of Saturn’s polar temperatures and
composition”, Icarus 250, 131 (2015); arXiv:1412.6416.
[3] Simon et al., Hubble Outer Planet Atmospheres Legacy (OPAL), MAST; programmes 15262, 15502,
15929, 16266, 16790, 16995, 17294, 17843.
[4] JPL Horizons, target 699, location 500@399, quantity 14.
[5] IAU 2015 Saturn reference spheroid, a = 60,268 km, c = 54,364 km.
[6] Bindi et al., “Decagonite”, Am. Mineral. 100, 2340 (2015) — cited
for §2 only, and for crystals only.