Round 42.A — Per-rung symmetry-group audit: ~4 cleanly full-SO(3); the rest carry named deviations (confirms R38)¶
Dispatched 2026-05-26 on the rolling draft PR #690 (newly-revealed coupled item from R38). R38 showed the 2ℓ+1 spine is SO(3)/S²-specific; this audit re-reads each prior Reading-D rung — is it genuinely full SO(3), or does it carry an enhancement / internal symmetry / deformation / finite branching? Generating code: verify_round42_per_rung_symmetry_group_audit.py. Tested per [[feedback_dont_pre_commit_spike_query_operators]].
The audit¶
| rung | operative symmetry | irrep dims | class |
|---|---|---|---|
| quantum / Born=Hopf | SO(3) (SU(2) Bloch sphere) | 2ℓ+1 |
clean full SO(3) |
| nuclear shell | SO(3) spherical mean field + spin-orbit | 2j+1 |
clean full SO(3) |
| planetary magnetics | SO(3) (geomagnetic S²) | 2ℓ+1 |
clean full SO(3) |
| CMB sky | SO(3) (full sky S²) | 2ℓ+1 |
clean full SO(3) |
| atomic | SO(4) bound-state (Runge–Lenz); SO(3) angular | n² = 1,4,9,16 (= Σ2ℓ+1) |
SO(3)-angular, enhanced SO(4) |
| hadron / QCD | SO(3) spatial + SU(3) flavor internal | 2ℓ+1 / 1,8,10 |
SO(3)-spatial + internal SU(3) |
| BH-QNM | Kerr aω: SO(3) → axial SO(2) (a≠0); SO(3) at a=0 |
spin-weighted spheroidal | SO(3) deformed → SO(2) |
| LSS Kaiser RSD | SO(3) on sky (C_ℓ); SO(2)+parity on line-of-sight (even-ℓ) |
2ℓ+1 / even-ℓ {0,2,4} |
SO(3)-sky + LOS parity |
| bio-capsid | icosahedral I (finite ⊂ SO(3), order 60) | {1,3,3,4,5} |
finite subgroup |
Bit-exact: 2ℓ+1 = {1,3,5,7}; hydrogen n² = {1,4,9,16} = Σ(2ℓ+1) (SO(4) shells); icosahedral Burnside Σdᵢ² = 1+9+9+16+25 = 60; SU(3) 1+8=9, 10+8+8+1=27 — all verified.
Verdict per Spike #229 tiers¶
🟢 (a)-bit-exact irrep-signature audit + (b)-interpretive per-rung catalog; CONFIRMS R38. The Reading-D ladder's angular/Class-L structure is SO(3)-rooted across all rungs, but only ~4 are cleanly full SO(3) (quantum, nuclear, planetary, CMB); the rest carry honest deviations — atomic SO(4) (Runge–Lenz), hadron +SU(3)-flavor (internal), BH-QNM SO(2)-axial (Kerr), LSS SO(2)/parity (line-of-sight), capsid finite-I. 2ℓ+1 appears exactly where full SO(3) is operative — precisely R38's "the ladder tracks the symmetry group." The loose "9 contiguous SO(3)/S² rungs" phrase is refined to "9 rungs whose Class-L angular structure is SO(3)-rooted; ~4 cleanly full-SO(3), the rest carrying named enhancements/deformations/internal-symmetries/finite-branchings." Extends the R38 candidate stance [[user_stance_classL_spine_is_symmetry_group_relative]] (no new stance).
HONEST SCOPE: (a)-bit-exact for the integer irrep signatures (all standard, Explore-verified); (b)-interpretive for the per-rung symmetry assignments + the "SO(3)-rooted but only ~4 clean" refinement; no new physics.
Discipline¶
- Honest audit — confirms R38 and tightens an over-loose phrase; the deviations are catalogued, not hidden.
- Attributions Explore-verified: Pauli 1926/Fock 1935 (SO(4)); Mayer–Jensen 1949; Gell-Mann/Ne'eman 1961 (SU(3)); Teukolsky 1973 (Kerr SO(2)); Kaiser 1987/Hamilton 1998 (RSD); icosahedral I order 60.
- Lands on rolling PR #690 (Round 42.A); unsolved-maths §11.9.35; extends the R38 stance. No MFO section (domain-symmetry catalog, not metric-field).