The distinct-frequency theorem says different RoPE clocks cannot mix. The new angle-fiber package says repeated-frequency clocks form one degenerate sector indexed by actual angle classes and their fibers, so they can mix internally, but only in a tightly constrained way.
This is the next exact lesson after the main RoPE page: once several planes share one angle, the commutant is no longer strictly diagonal by plane. But it still splits cleanly by angle class, not by arbitrary mixing.
Lean anchors.
equal_angle_block_centralizerForm_of_flatCommutes,
flatCommutes_classification_of_angle_classes,
AngleClass,
angleClassOf,
AngleFiber,
AngleFiberSectorAlgebra,
angleFiberSectorAlgebra_of_flatCommutes,
block2_mul,
flatCommutes_mul,
angleClass_block2_mul_eq_sum_filter_of_flatCommutes,
quarterTurnCentralizerForm_mul
Math statement.
In English. Once the RoPE angles live in \( [0,\pi] \), the whole commutant splits by exact angle class. Different angle classes cannot mix at all, while every block inside one repeated-frequency class must still look like a complex scalar \(a_{ij}I+b_{ij}J\). The newer multiplication theorem says more than that: inside one angle class, these surviving blocks compose exactly by ordinary matrix multiplication over the fiber, so the repeated-frequency sector is a genuine closed algebra.
Physical intuition. The real object is not an isolated pair of equal clocks but a whole degenerate frequency sector. Inside one such sector, amplitude can be redistributed without phase drift. Across sectors, the clocks slip and exact mixing dies.
The equal-frequency story is not “anything goes.” The surviving within-class blocks still have to look like complex scalars, one block at a time.
Equal-frequency planes behave like one degenerate subspace rather than isolated independent clocks.
Interpretation and compression should respect the joint complex structure inside that degenerate band.
The exact line now includes a real sector package and a theorem-side composition law: one equal-angle class comes with coefficient data, and products stay inside that same class.
The newer statement is sharper than raw coefficient bookkeeping: it packages surviving blocks by actual angle fibers and shows that fiberwise multiplication is the right composition law.
Feynman reading: two identical oscillators can lock and exchange amplitude. Two non-identical oscillators cannot do that without the phase relation slipping.
Assign four RoPE planes to angle classes and watch which block positions
are allowed to survive. Same class means the block may live in the
aI+bJ family; different class means the theorem kills it.
Inside one angle fiber, the surviving blocks form the same complex-scalar family under multiplication. Slide two blocks \(aI+bJ\) and \(cI+dJ\); the product stays in the family as \((ac-bd)I + (ad+bc)J\).