Chapter 7 — The Geometry of Transformation

Part II · Distance, Access, Liquidity · Week 6

K→𝒢→Q→Ξ

Distance before price. Equip the network with a cost gauge and the transformation cost becomes a directed quasi-metric — asymmetric, because ownership is a ratchet. Valuation on a frozen map is then a geometry problem, solved by a network Hopf–Lax theorem.

Learning objectives

  • LOS 7.1 — Define cost gauges and the transformation quasi-metric; compute distance matrices at a stratum and gauge.
  • LOS 7.4 — State and prove the network Hopf–Lax theorem and use it to compute frozen-architecture value.
  • LOS 7.5 — Read liquidity as local geometry.

The laboratory module

Module 7 — The Geometer. Instruments on the signature network: a radial distance map, forward/backward distance matrices, a Hopf–Lax intercept race, and a spectrum-and-pruning bench.

The directed distances from [Private] at the default gauge (money weight 1, time weight 1):

To Geodesic \(d_T\)
[Sponsor-owned] sponsor sale 2
[Wound-down] wind-down 5
[Securitized] mint, exercise 14
[Listed] sale, sponsor IPO 18
[JV] (partner gate) \(\infty\)

Two readings. The ratchet: [Private] reaches [Sponsor-owned] at 2, but [Sponsor-owned] reaches [Private] only at 25 — the geometry prices the ratchet’s teeth. The detour: [Listed] at 18 is not a ticket but an itinerary — the direct gate is frozen shut, so the geodesic runs \(2 + 16\) through sponsor ownership. The accessibility spectrum from [Private] is \(\{0, 2, 5, 14, 18\}\); a covenant forbidding the mint drops the 14 entry.

Guided experiments

  1. Find gauge weights under which the listing route overtakes the sponsor route in the intercept race.
  2. Measure the [Private]⇄[Sponsor-owned] round trip at three gauges and decompose each.
  3. Construct a gate change that moves [Listed] from distance 18 to 12 and record the effect on program value.
  4. Exhibit the depth/distance divergence: two targets ordered oppositely by \(D_s\) and by \(d_T\).

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ch07 · 8/8 PASS

Exercises

B · Computations

  • 7.5 Price a covenant that prunes the securitization route both in feasibility and in value. Hint: feasibility loss is certain (a class drops from the spectrum); value loss is state-contingent (zero if the pruned route is off the active geodesic).

Statements and hints surfaced here; full solutions in the Instructor’s Manual.