Chapter 3 — The Capital State

Part I · Representation and the Impossibility of a Scalar · Week 2

K→𝒢→Q→Ξ

The chapter that makes the four-coordinate representation forced, not chosen: no scalar functional of a capital system can be simultaneously monotone in the state, sensitive to the architecture, invariant under pure claim repackaging, and consistent under composition. The proof is a tie-chain propagated across an institutional gate.

Learning objectives

  • LOS 3.1 — State the capital system as a fibered state and name each coordinate of the quadruple.
  • LOS 3.3 — Prove the Scalar Impossibility Theorem, including the tie lemma (3.16) and the ladder lemma (3.17).
  • LOS 3.4 — Identify, given a gate configuration, which scalar readings RM1–RM4 are licensed on which cells (Proposition 3.24).

The laboratory module

Module 3 — Representation and Projections. Two instruments on a two-attribute state plane: a tie-chain animator (choose a candidate scalar, seed a tie pair, watch the ladder march across a gate — the contradiction as a visible event) and a license-map explorer (place gates, tile the plane into cells, toggle the RM1–RM4 licenses).

The tie-chain, worked. Take the linear scalar \(\varphi = 2k_1 + 3k_2\). A tie pair \(p \sim p+(s,-\delta)\) requires \(2s = 3\delta\), so \(s/\delta = 3/2\). Starting from \(p = (1,1)\) with \(\delta = 0.3\) (hence \(s = 0.45\)), the ladder identity

\[p + (0, m\delta) \;\sim\; p + (ms, 0)\]

marches the right endpoint across the gate \(\theta = 2\) on coordinate 1 while the left endpoint stays outside. At \(m = 3\) the right endpoint reaches \(k_1 = 2.35\) (inside), the left stays at \(k_1 = 1\) (outside), and both share \(\varphi = 7.7\) — a genuine tie. Two systems with the same measure, different admissible operations: the sufficiency premise fails at its own clause.

Guided experiments

  1. Run the animation for all three scalar families (linear, Cobb–Douglas, hand-drawn spline); record where each level set meets the gate.
  2. State why no monotone re-drawing of the level sets can avoid the crossing.
  3. License-map explorer — build a two-gate configuration, identify the cell a firm sits in, and state which scalar readings are licensed and which threshold event expires them.

📓 Notebook · 📊 Workbook · 🎞 Slides (.pptx) · Open the Laboratory →

ch03 · 8/8 PASS

Exercises

A · Concept checks

  • 3.1 Name the four coordinates of the quadruple and give, for each, one capital event that moves it and nothing else.

B · Computations

  • 3.7 Run both acts of the worked example: (a) show \(\varphi = k_1^2 k_2\) fails sufficiency with a tie pair; (b) for \(\varphi = 2k_1 + 3k_2\), derive the tie ratio \(3/2\) and tabulate the ladder to a gate at \(\theta = 2\). Hint: the multiplicative form’s images diverge under enrichment; the linear form’s don’t — but the linear form’s ladder still crosses the gate.

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