Key finding: both Goormaghtigh collisions have exact rho=3.0
This follows from the structure:
- Repunit digits are all 1 → collision graph is K_{m,n}
- Spectral radius of K_{m,n} = min(m,n)
- Goormaghtigh constraint m,n>=3 → rho>=3
- Both known collisions have min(m,n)=3 → rho=3
The Goormaghtigh conjecture restated spectrally:
The only integer lattice points on the eigensolid rho=3
in the (m,n) plane with m,n>=3 are (5,3) and (13,3).
Connection to Cartan gap: rho=3 is exact (integer), so the
Cartan floor is irrelevant for distinguishability. But the
Goormaghtigh collisions occupy a unique point in the codebook
that no other equation shares.
