Quantum Foundations III · Lesson 4

Can Sectors Interact?

Suppose several invariant sectors exist.
Could transport carry a configuration from one into another?

Let's try.

Tap any configuration marker to begin.

Show Paths

Try as many times as you like. Drag a configuration marker towards a neighbouring sector — it slows, the edge glows and ripples, and it slides around the inside.

What to notice

✓Configuration markers move freely inside their own sector.
✓Each sector stays intact while its phase evolves.
✓Under exact sector-preserving transport, coupling is unavailable.

If transport preserves each sector exactly, then coupling between sectors is unavailable under this definition and these assumptions. QM3 identifies an obstruction rather than proving interaction impossible under every possible future formulation. The obstruction is relative to the candidate definition of a relational sector, and the adequacy of exact invariance remains open.

The Logical Chain

Candidate definition: sector = exactly invariant under transportDEFINED HERE
Transport preserves each sectorDERIVED CONDITIONALLY
Cross-sector transport terms vanishDERIVED CONDITIONALLY
Block-diagonal structureDERIVED CONDITIONALLY

Block-Diagonal Transport

DERIVED CONDITIONALLY

Where compatible embeddings and the required linear representation are available, transport takes this schematic form:

U =
[ U₁   0    0
   0   U₂   0
   0    0   U₃ ]

Exact sector preservation removes the off-diagonal transport terms. This conclusion depends on the sector definition and the required representation/embedding structure, and is not an unconditional property of nature.

To represent several sectors together in a common linear description, the relevant sector structures must admit compatible embeddings into that description. QM3 therefore treats the block-matrix analysis conditionally on those embeddings being available.

The Coupling Obstruction

IDENTIFIED OBSTRUCTION

If multiple sectors are defined to be exactly invariant under transport, then transport acts independently within those sectors. In the corresponding block representation, the off-diagonal coupling terms vanish. Therefore the same exact invariance that defines the sectors also prevents ordinary cross-sector coupling.

Is exact invariance therefore too strong?

If exact invariance leaves no route between sectors…
where could coupling come from?

This is called the Coupling Obstruction.

Continue to Lesson 5