Method Selector

The ELC boundary is determined by the algorithm you pick, not the quantity you compute. Same number, different routing, different stratum. Pick a problem; see which method stays inside ELC.

Eigenvalues of an n × n matrix

The quantity is the same — the algorithm decides whether you stay inside ELC.

iterative

Lanczos / power method

ELC-interiorevery step stays in the expression lattice

CostO(kn²) per iteration; ~30 iters to 64-bit precision

Each step is a matrix-vector product and a normalization — every operation is +, ×, /, sqrt on positives. Fully ELC-interior. The limit is an eigenvalue that may lie outside ELC, but you never wrote it as a closed-form expression.

closed-form

Characteristic-polynomial roots

ELC-boundaryroutes through a boundary op (abs, max, ⁿ√ for n ≥ 5)

Costimpossible for generic n ≥ 5 (Abel-Ruffini)

For n ≤ 4 you can write each eigenvalue as a radical expression (ELC-interior). For n ≥ 5 the generic characteristic polynomial has Galois group S₅, unsolvable — no closed-form exists.