Why the Generator Becomes Block-Diagonal

The previous lesson showed that transport never leaves its own sector.

If nothing moves between sectors, the generator describing that transport cannot contain connections between them.

SUPPOSE exact sector preservation holds, i.e. P_a U(t) P_b = 0 for a ≠ b.

THEN, under the stated continuous-generator assumptions, the generator is block diagonal, H₀ = ⊕_a H_a, shown schematically as the 2×2 matrix below.

U(t) = e^−iH₀t uses natural units with ħ = 1 throughout.

Total Relational State Space

Illustration

Sector V₁

Amplitude (constant)

Phase

Sector V₂

Amplitude (constant)

Phase

Reconstructed generator H₀

Exact sector preservation removes the off-diagonal generator blocks.

Select a region of the generator to see what it governs.

Because transport preserves each sector, the reconstructed generator contains no terms that move states from one sector into another.

Internal evolution✓

Cross-sector evolution✗

“Transport remains inside each sector.”

“Still an open mathematical question.”

Established conditionally

Still Open

What does a block-diagonal generator prevent?

Under the QM3 sector-preservation and generator assumptions, the generator takes block-diagonal form.

Internal evolution is allowed.

Connections between sectors remain absent.

The next question is whether those missing connections can ever be reconstructed.