Chapter 9 — Architecture Value and Separation
Part III · The Separation Bench · Week 8
K→𝒢→Q→Ξ
The book’s flagship theorem: value separates into what the state is worth, what the existing pathways are worth, and what the capacity to make pathways is worth. Architecture value is compound-option value on an unrolled architecture — exactly — but the unrolling is exponentially expensive.
Learning objectives
- LOS 9.2 — Recompute Chapter 1’s premium as a compound option on the unrolled architecture.
- LOS 9.3 — Prove the exponential lower bound on value-faithful unrollings: \(n\) latent pathways force \(2^n\) meta-states (Lemma 9.3).
- LOS 9.4 — Define aware value and the unawareness premium.
The laboratory module
Module 9 — The Separation Bench. An unroller, a compression meter, an awareness bench, and a calculus bench.
The miniature’s premium unrolled. Chapter 1’s architecture premium \(\Pi = 5.5\) is exactly the compound-option value on the unrolled product graph, dying at \(\kappa^* = 7.5\). The unrolled architecture has a vertex set that cannot be smaller than the meta-state lattice: \(n\) independent latent pathways generate \(2^n\) meta-states.
| pathways \(n\) | meta-states \(2^n\) |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
This is the combinatorial heart of the Separation Theorem’s counting argument — and why \(\Xi\) is, at minimum, an exponentially compressed object.
Guided experiments
- Reproduce the miniature’s 5.5 on the unroller, then break it by raising \(\kappa\) past 7.5.
- Find the smallest knife-edge tie that halves the compression meter’s bound.
- Construct an awareness set under which the enumerable premium is zero and the unawareness premium is the whole premium.
ch09 · 8/8 PASS