MTC Chapter 1 — The Miniature Capital System¶

Part I · Week 1 · Seed 20260101

The miniature is the smallest model in which the book's central phenomenon occurs: two systems that agree on every scalar reading yet differ in value. This notebook reproduces the decomposition $90+10+5.5$, the threshold $\kappa^*=7.5$, and the two-pathway additivity $6.5=5.5+1$.

In [ ]:
import sys, os
sys.path.insert(0, r"/home/claude/mtc/mtc-course/downloads/engine_lib")
sys.path.insert(0, r"/home/claude/mtc/nbbuild")
from mtcplot import apply_style, MTC_RED, BRASS, SLATE, MTC_RED_LITE, MTC_RED_DARK
import matplotlib.pyplot as plt
import numpy as np
apply_style()

The decomposition waterfall¶

State value, path increment, and the architecture premium — the three enlargements of the admissible operation set.

In [ ]:
from engines import mtc_engine_ch01 as e
w = e.waterfall()
labels = ["V_state", "path\nincrement", "architecture\npremium"]
vals = [w["state_value"], w["path_increment"], w["architecture_premium"]]
cum = [0, w["state_value"], w["state_value"]+w["path_increment"]]
fig, ax = plt.subplots()
colors = [SLATE, BRASS, MTC_RED]
for i,(l,v,c) in enumerate(zip(labels, vals, cum)):
    ax.bar(i, vals[i], bottom=c, color=colors[i], edgecolor="white", width=0.62)
    ax.text(i, c+vals[i]+2, f"+{vals[i]:g}" if i else f"{vals[i]:g}", ha="center", fontweight="bold", color=MTC_RED_DARK)
ax.axhline(w["V_B"], color=MTC_RED_DARK, ls="--", lw=1)
ax.text(2.4, w["V_B"]+1, f"V_B = {w['V_B']:g}", color=MTC_RED_DARK, fontweight="bold")
ax.set_xticks(range(3)); ax.set_xticklabels(labels)
ax.set_ylabel("value"); ax.set_title("System B: the architecture premium decomposition")
ax.set_ylim(0, 120)
plt.tight_layout(); plt.show()
print("V_state =", w["state_value"], " path increment =", w["path_increment"], " premium =", w["architecture_premium"], " V_B =", w["V_B"])

The premium as a function of minting cost¶

The premium is $(7.5-\kappa)^+$ — a kinked line dying at $\kappa^*=7.5$. Everything Chapter 9 says about meta-capacity has this shape as its two-period shadow.

In [ ]:
kappas = np.linspace(0, 10, 200)
prem = [max(e.architecture_premium(kappa=k), None) if False else e.architecture_premium(kappa=k) for k in kappas]
prem = [max(p,  -1) for p in prem]
fig, ax = plt.subplots()
ax.plot(kappas, [e.architecture_premium(kappa=k) for k in kappas], color=MTC_RED, lw=2.2)
ax.axhline(0, color=SLATE, lw=0.8)
ax.axvline(7.5, color=BRASS, ls="--", lw=1.5)
ax.text(7.6, 4, "$\\kappa^* = 7.5$", color=BRASS, fontweight="bold")
ax.scatter([2],[5.5], color=MTC_RED_DARK, zorder=5)
ax.text(2.2, 5.7, "book: 5.5 at $\\kappa=2$", color=MTC_RED_DARK)
ax.set_xlabel("minting cost $\\kappa$"); ax.set_ylabel("architecture premium")
ax.set_title("The price of capacity: premium $(7.5-\\kappa)^+$")
plt.tight_layout(); plt.show()
print("kappa* =", e.kappa_star())

Two-pathway additivity¶

Add a second (expansion) pathway: the premia add, $6.5 = 5.5 + 1$, because the pathways are exercise-independent.

In [ ]:
jp = e.joint_premium()
fig, ax = plt.subplots(figsize=(7,4))
parts = ["securitization\n5.5", "expansion\n1.0", "joint\n6.5"]
vals = [jp["premium_securitization"], jp["premium_expansion"], jp["joint_premium"]]
ax.bar(parts, vals, color=[MTC_RED, BRASS, MTC_RED_DARK], edgecolor="white", width=0.6)
for i,v in enumerate(vals): ax.text(i, v+0.1, f"{v:g}", ha="center", fontweight="bold", color=MTC_RED_DARK)
ax.set_ylabel("premium"); ax.set_title("Additivity of independent pathways")
plt.tight_layout(); plt.show()
print(jp)

Validation¶

Every number above is reproduced by the seeded engine; the checks below must all pass.

In [ ]:
from engines import mtc_engine_ch01 as _e
v = _e.validation_checks(); v.pop("_values", None)
for k, val in v.items():
    print(f"  {k}: {val}")
assert v["ALL_PASS"], "validation failed"
print("\nALL_PASS — matches notebook + workbook + API")