Chapter 5 — The Transformation Network
Part II · Reachability · Week 4
K→𝒢→Q→Ξ
The map is the asset. This chapter founds the signature network — a mid-market firm’s capital-state classes and the admissible transformations between them — and asks what a position can reach. Reachability turns out to be a lattice of fixed points, and the interval between its extremes is a measurable economic object.
Learning objectives
- LOS 5.2 — Build the transformation network and compute stratified reachability.
- LOS 5.5–5.6 — Classify positions by the Type I/II/III taxonomy and compute transformation depth.
- LOS 5.8 — Run the two Kleene iterations for the viable-reachability fixed point and read the gap.
The laboratory module
Module 5 — The Network Explorer. Instruments on the (editable) signature network: a strata explorer, a fixed-point stepper (Kleene iterations from bottom and top), a type classifier, and a matrix differ for depth.
The six classes are [Private], [Sponsor-owned], [Listed], [Securitized], [JV], [Wound-down]. From [Private] at the ledger parameters (\(\ell=3\), \(\lambda=\tfrac12\)):
| Stratum | Question | Reaches |
|---|---|---|
| \(\mathfrak{R}_M\) | engineer: any route? | all six |
| \(\mathfrak{R}_L\) | lawyer: gates satisfiable? | all but [JV] |
| \(\mathfrak{R}_V^+\) | growth banker: value-consistent financing? | all but [JV] |
| \(\mathfrak{R}_F^-\) | credit banker: grounded financing? | all but [JV], [Securitized] |
The viability gap \(\mathfrak{R}_V^+ \setminus \mathfrak{R}_F^- = \{[\text{Securitized}]\}\): the securitization step (cost 13) is affordable when the lender pledges the program-inclusive value (capacity \(3 + \tfrac12\cdot 22 = 14\)) but not the standalone value (capacity \(3 + \tfrac12\cdot 18 = 12\)). A doctrine shock removes the gap class with no change to any edge, gate, or law — the germ of Chapter 13’s crisis mechanics.
Guided experiments
- Reproduce the strata table on the signature network.
- Find the smallest \(\lambda\) at which [Securitized] enters the least fixed point, and the smallest at which it enters the greatest; interpret the interval.
- Build a two-class bootstrap with a gap, then destroy it by switching the pledging rule.
- Produce one Type II and one Type III edit and record the condensations.
ch05 · 8/8 PASS
Exercises
B · Computations
- 5.7 Run both Kleene iterations on the signature network and identify the gap. Hint: the gap is the set of classes reachable optimistically but not grounded — subtract the two fixed points.
Statements and hints surfaced here; full solutions in the Instructor’s Manual.