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

=== Assistant: Below are explicit numerical point sets you can plot or tabulate to demonstrate and interrogate the proof. ===
Below are explicit numerical point sets you can plot (or tabulate) to demonstrate and interrogate the proof. They are chosen to isolate each logical claim: naive collapse, repaired viability, and boundary tightness. No graphics included—only coordinates.

==== Residual bound (constant): ====

∣r∣max⁡=0.9|r|_{\max}=0.9∣r∣max=0.9
Precision index:

p∈N0p\in\mathbb{N}_0p∈N0

==== Tolerance law: ====

τnaive(p)=τ0 2−p,τ0=3.0\tau_{\text{naive}}(p)=\tau_0\,2^{-p},\quad \tau_0=3.0τnaive(p)=τ02−p,τ0=3.0
Decision rule:

COLLAPSE if ∣r∣max⁡>τ(p)\text{COLLAPSE if } |r|_{\max}>\tau(p)COLLAPSE if ∣r∣max>τ(p)
Points (p, τ(p))

<syntaxhighlight>(0, 3.0000)
(1, 1.5000)
(2, 0.7500)   ← first violation (0.9 > 0.75)
(3, 0.3750)
(4, 0.1875)

</syntaxhighlight>

Overlay line (constant):

<syntaxhighlight>|r|max = 0.9

</syntaxhighlight>

Inference demonstrated by points:
Finite p'' exists (here p''=2). Collapse is inevitable.

==== Tolerance law: ====

τIR(p)=τ\''+(τ0−τ\'')2−p,τ\''=0.9, τ0=3.0\tau_{\text{IR}}(p)=\tau^\''+(\tau_0-\tau^\'')2^{-p},\quad \tau^\''=0.9,\ \tau_0=3.0τIR(p)=τ\''+(τ0−τ\'')2−p,τ\*=0.9, τ0=3.0
Points (p, τ(p))

<syntaxhighlight>(0, 3.0000)
(1, 1.9500)
(2, 1.4250)
(3, 1.1625)
(4, 1.03125)
(5, 0.965625)
(6, 0.9328125)
(∞, 0.9000)

</syntaxhighlight>

Overlay line (constant):

<syntaxhighlight>|r|max = 0.9

</syntaxhighlight>

Inference demonstrated by points:
For all finite p, τ(p) ≥ |r|max. No collapse ever occurs.

==== Tolerance law (minimal floor): ====

τmin(p)=∣r∣max⁡+C 2−p,C=0.1\tau_{\text{min}}(p)=|r|_{\max}+C\,2^{-p},\quad C=0.1τmin(p)=∣r∣max+C2−p,C=0.1
Points (p, τ(p))

<syntaxhighlight>(0, 1.0000)
(1, 0.9500)
(2, 0.9250)
(3, 0.9125)
(4, 0.90625)
(5, 0.903125)
(∞, 0.9000)

</syntaxhighlight>

Inference demonstrated by points:
This is the tightest viable scaling. Any smaller floor collapses.

==== Tolerance law: ====

τ(p)=0.9−0.05⋅2−p\tau(p)=0.9-0.05\cdot 2^{-p}τ(p)=0.9−0.05⋅2−p
Points (p, τ(p))

<syntaxhighlight>(0, 0.8500)   ← immediate violation
(1, 0.8750)
(2, 0.8875)

</syntaxhighlight>

Inference demonstrated by points:
If liminf τ(p) < |r|max, collapse occurs (often immediately).

==== Using: ====

p∗=⌈log⁡2 ⁣(τ0∣r∣max⁡)⌉p^*=\left\lceil\log_2\!\left(\frac{\tau_0}{|r|_{\max}}\right)\right\rceilp∗=⌈log2(∣r∣maxτ0)⌉
''Points (τ0, p'')*

<syntaxhighlight>(1.0, 1)
(1.8, 1)
(2.0, 2)
(3.0, 2)
(5.0, 3)
(10.0, 4)

</syntaxhighlight>

Inference demonstrated by points:
Higher τ0 delays collapse but never prevents it if τ(p)→0.

==== - Sets A & D instantiate the necessity direction: if τ(p)→0 or liminf τ(p)<|r|max ⇒ finite p*. ====
'' Sets B & C instantiate the sufficiency direction: nonzero IR floor τ''≥|r|max ⇒ ∀p STAND.
* Set E shows the collapse boundary explicitly as a function of τ0 (diagnostic power).

If you want, I can export these point sets as CSV, NumPy arrays, or LaTeX tables for direct inclusion.
© Robert R. Frost
2026-01-03