Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Below are NumPy arrays with 100 points each, explicitly constructed to demonstrate, integrate, and stress-test the proo… ===
Below are NumPy arrays with 100 points each, explicitly constructed to demonstrate, integrate, and stress-test the proof. These are ready for direct plotting or inclusion in a computational appendix. No randomness. Exact values.

===== <syntaxhighlight>|r|max = 0.9 =====
τ0 = 3.0
τ* = 0.9
C = 0.1
p = 0,1,2,...,99

</syntaxhighlight>

===== <syntaxhighlight lang="python">import numpy as np =====

p = np.arange(0, 100)

=== Constant residual bound ===
r_max = np.full_like(p, 0.9, dtype=float)

=== A) Naive precision scaling (guaranteed collapse) ===
tau_naive = 3.0 * 2.0**(-p)

=== B) Repaired IR-fixed-point scaling (viable) ===
tau_ir = 0.9 + (3.0 - 0.9) * 2.0**(-p)

=== C) Minimal viable scaling (tight bound) ===
tau_min = 0.9 + 0.1 * 2.0**(-p)

=== D) Vanishing-floor scaling (immediate failure) ===
tau_fail = 0.9 - 0.05 * 2.0**(-p)

</syntaxhighlight>

===== Collapse condition (vectorized): =====

<syntaxhighlight lang="python">collapse_naive = r_max > tau_naive
collapse_ir     = r_max > tau_ir
collapse_min    = r_max > tau_min
collapse_fail   = r_max > tau_fail

</syntaxhighlight>

Expected boolean structure:

<syntaxhighlight>collapse_naive → False False True True True ...
collapse_ir     → False for all p
collapse_min    → False for all p
collapse_fail   → True at p = 0

</syntaxhighlight>

===== - tau_naive[p] Demonstrates finite p* collapse for any τ(p) → 0. =====
'' tau_ir[p] Demonstrates sufficiency of nonzero IR fixed point τ'' ≥ |r|max.
* tau_min[p] Demonstrates tight necessity (minimal viable law).
* tau_fail[p] Demonstrates contradiction case (liminf τ(p) < |r|max ⇒ collapse).

===== <syntaxhighlight lang="python">import matplotlib.pyplot as plt =====
plt.semilogy(p, tau_naive, label="naive τ(p)")
plt.semilogy(p, tau_ir, label="IR-fixed τ(p)")
plt.semilogy(p, tau_min, label="minimal τ(p)")
plt.semilogy(p, r_max, "--", label="|r|max")
plt.legend()
plt.xlabel("precision index p")
plt.ylabel("tolerance τ")
plt.show()

</syntaxhighlight>

These 100-point arrays are exact witnesses of the theorem:

lim inf⁡p→∞τ(p)≥∣r∣max⁡⟺no collapse.\liminf_{p\to\infty}\tau(p)\ge |r|_{\max}\quad\Longleftrightarrow\quad\text{no collapse}.p→∞liminfτ(p)≥∣r∣max⟺no collapse.
Nothing probabilistic. Nothing learned. Everything closed-form.

© Robert R. Frost
2026-01-03