MTC Chapter 3 — The Impossibility of a Scalar

Part I · Week 2 · Seed 20260301

The tie-chain: a monotone scalar $\varphi=2k_1+3k_2$ is propagated across a gate by the ladder identity, until the right endpoint crosses the threshold while the left stays outside — a witnessed contradiction.

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 ladder marching across the gate

Each rung is a genuine tie ($\varphi$ equal on both endpoints), yet at $m=3$ the right endpoint enters the gate ($k_1\ge 2$) while the left stays out.

from engines import mtc_engine_ch03 as e
L = e.ladder()
fig, ax = plt.subplots()
for row in L["rows"][:7]:
    lx, ly = row["left"]; rx, ry = row["right"]
    color = MTC_RED if row["m"]==L["crossing_row"] else SLATE
    ax.plot([lx, rx], [ly, ry], "-o", color=color, alpha=0.9 if row["m"]==L["crossing_row"] else 0.5, lw=2 if row["m"]==L["crossing_row"] else 1)
ax.axvline(L["gate_theta"], color=BRASS, ls="--", lw=1.8)
ax.text(L["gate_theta"]+0.03, 2.4, "gate $\\theta=2$", color=BRASS, fontweight="bold", rotation=90, va="top")
ax.set_xlabel("$k_1$ (liquid resources)"); ax.set_ylabel("$k_2$ (productive plant)")
ax.set_title("The tie-chain crosses the gate (crossing at m=3, in red)")
plt.tight_layout(); plt.show()
c = L["crossing_row"]; row = L["rows"][c]
print(f"crossing at m={c}: left k1={row['left'][0]} (out), right k1={row['right'][0]:.2f} (in), phi both = {row['phi_left']:.1f}")

Validation

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

from engines import mtc_engine_ch03 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")