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Round 38.A — The drumhead / Bessel-disk 15th-rung candidate: a contrast rung that sharpens the spine as SO(3)-specific

Dispatched 2026-05-26 on the rolling draft PR #690 (open-queue item: "15th Reading-D rung candidate — a drumhead/Bessel-disk rung, Class-L off S², on a 2D domain"). Generating code: verify_round38_drumhead_disk_contrast_rung.py. Tested per [[feedback_dont_pre_commit_spike_query_operators]].

Result — a CONTRAST / control rung, not another 2ℓ+1 rung

Every prior rung realized the Class-L spine (§11.9.22) as the S²/SO(3) ladder: eigenvalue ℓ(ℓ+1), degeneracy 2ℓ+1 = {1,3,5,7,…}, first-three = the k=3 triad {1,3,5}. The drumhead tests the spine off S², on a 2D disk (Dirichlet Laplacian):

  • modes J_m(j_{m,n} r/a)·{cos mθ, sin mθ} (Morse–Feshbach; Courant–Hilbert), ordered by Bessel zeros j_{m,n} (verified: j_{0,1}=2.4048 < j_{1,1}=3.8317 < j_{2,1}=5.1356 < j_{0,2}=5.5201 < …);
  • m=0 non-degenerate, m≥1 doubly degenerate (cos/sin) → angular degeneracy ladder {1, 2, 2, 2, …} — not {1,3,5,7,…};
  • because the disk's symmetry group is O(2) (irreps: 1D trivial at m=0 + a 2D irrep per m≥1), not SO(3) (irreps dim 2ℓ+1).

So the disk's "first-three" is {1,2,2}, not {1,3,5}. The Class-L mechanism persists off S² (Laplacian eigenspaces; degeneracy = irrep-dim of the domain's symmetry group), but the specific ladder changes with the symmetry group.

Verdict per Spike #229 tiers

🟢 (a)-bit-exact contrast + (b)-interpretive sharpening (honest NEGATIVE on "another 2ℓ+1 rung"). The drumhead is not a 15th 2ℓ+1 rung — it is a control that sharpens what the spine is: the 2ℓ+1 odd ladder and the {1,3,5} k=3 triad are specifically an S²/SO(3) phenomenon, not universal. Class-L's universal content is "Laplacian eigenspaces, degeneracy = irrep-dimension of the domain's symmetry group" (SO(3) → {1,3,5,…}; O(2) → {1,2,2,…}). The Reading-D "9 contiguous S² rungs" count is unchanged. Cascade A ∘ L (O(2) irreps) ∘ C (cos/sin reflection-parity doublet) ∘ K (Dirichlet boundary truncation). New candidate stance [[user_stance_classL_spine_is_symmetry_group_relative]].

HONEST SCOPE: (a)-bit-exact for the Bessel-zero ordering + the {1,2,2}-vs-{1,3,5} degeneracy contrast + O(2)/SO(3) irrep dims (standard math, Explore-verified); (b)-interpretive for the "spine is symmetry-group-relative" reading; no new physics. Newly-revealed coupled item (for roadmap): this invites a deliberate small sub-thread — re-reading each prior ladder rung's symmetry group (are they all genuinely SO(3)/S², or do any secretly live on a different group whose irrep-dims happen to coincide?). Parked as a roadmap candidate, not auto-dispatched.

Discipline

  • Honest NEGATIVE (it is a contrast, not a 2ℓ+1 rung); the ladder count is held fixed, not inflated.
  • Bessel zeros DLMF Table 10.21; membrane modes Morse–Feshbach 1953 / Courant–Hilbert; O(2)/SO(3) irreps Arfken / Dresselhaus (Explore-verified).
  • Lands on rolling PR #690 (Round 38.A); unsolved-maths §11.9.31. (Domain-geometry contrast, not metric-field — no MFO section, consistent with the R37 recursive-check discipline.)