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

=== Assistant: Below are clean, compile-ready tables numeric only intended to be ingested into a single composite graph in the next re… ===
Below are clean, compile-ready tables (numeric only) intended to be ingested into a single composite graph in the next response.
They are mutually aligned on the same x-axis (p) and require no interpretation layer.
All values are exact, deterministic, and closed-form.

==== Column: p ====
Domain: 0 … 999

<syntaxhighlight>p
0
1
2
3
4
5
…
995
996
997
998
999

</syntaxhighlight>

(Use as the shared x-axis for all subsequent tables.)

==== Column: r_max ====

Definition:

rmax⁡(p)=0.9∀pr_{\max}(p) = 0.9 \quad \forall prmax(p)=0.9∀p
<syntaxhighlight>r_max
0.9
0.9
0.9
0.9
0.9
0.9
…
0.9
0.9
0.9
0.9
0.9

</syntaxhighlight>

Purpose in composite graph: constant horizontal line.

==== Column: tau_naive ====

Definition:

τnaive(p)=3.0⋅2−p\tau_{\text{naive}}(p)=3.0\cdot 2^{-p}τnaive(p)=3.0⋅2−p
First values:

<syntaxhighlight>p    tau_naive
0    3.000000000000000
1    1.500000000000000
2    0.750000000000000
3    0.375000000000000
4    0.187500000000000
5    0.093750000000000

</syntaxhighlight>

Asymptotic tail:

<syntaxhighlight>p    tau_naive
10   0.002929687500000
20   0.000002861022949
50   2.664535259100e-15
100  2.367159491070e-30
999  ~0.0 (underflow)

</syntaxhighlight>

Purpose: descending curve crossing <code>r_max</code> at p = 2.

==== Column: tau_ir ====

Definition:

τIR(p)=0.9+(3.0−0.9)⋅2−p\tau_{\text{IR}}(p)=0.9+(3.0-0.9)\cdot 2^{-p}τIR(p)=0.9+(3.0−0.9)⋅2−p
First values:

<syntaxhighlight>p    tau_ir
0    3.000000000000000
1    1.950000000000000
2    1.425000000000000
3    1.162500000000000
4    1.031250000000000
5    0.965625000000000

</syntaxhighlight>

Asymptotic tail:

<syntaxhighlight>p    tau_ir
10   0.902050781250000
20   0.900002861022949
50   0.900000000000003
100  0.900000000000000
999  0.900000000000000

</syntaxhighlight>

Purpose: monotone curve saturating from above to <code>r_max</code>.

==== Column: tau_min ====

Definition:

τmin⁡(p)=0.9+0.1⋅2−p\tau_{\min}(p)=0.9+0.1\cdot 2^{-p}τmin(p)=0.9+0.1⋅2−p
First values:

<syntaxhighlight>p    tau_min
0    1.000000000000000
1    0.950000000000000
2    0.925000000000000
3    0.912500000000000
4    0.906250000000000
5    0.903125000000000

</syntaxhighlight>

Asymptotic tail:

<syntaxhighlight>p    tau_min
10   0.900097656250000
20   0.900000095367432
50   0.900000000000000
999  0.900000000000000

</syntaxhighlight>

Purpose: tight envelope demonstrating necessity.

==== Column: tau_fail ====

Definition:

τfail(p)=0.9−0.05⋅2−p\tau_{\text{fail}}(p)=0.9-0.05\cdot 2^{-p}τfail(p)=0.9−0.05⋅2−p
First values:

<syntaxhighlight>p    tau_fail
0    0.850000000000000
1    0.875000000000000
2    0.887500000000000
3    0.893750000000000
4    0.896875000000000
5    0.898437500000000

</syntaxhighlight>

Asymptotic tail:

<syntaxhighlight>p    tau_fail
10   0.899951171875000
20   0.899999904632568
50   0.900000000000000 (approach from below)

</syntaxhighlight>

Purpose: curve strictly below <code>r_max</code> → collapse region.

==== Column: collapse_naive ====

Definition:

collapse_naive(p)=1[ 0.9>τnaive(p) ]\text{collapse\_naive}(p)=\mathbf{1}[\,0.9>\tau_{\text{naive}}(p)\,]collapse_naive(p)=1[0.9>τnaive(p)]
<syntaxhighlight>p    collapse_naive
0    0
1    0
2    1
3    1
4    1
…
999  1

</syntaxhighlight>

Purpose: step-function overlay marking collapse onset.

==== All tables share the same p index. ====
A single composite graph should include:
* X-axis: p
* Y-axis: tolerance / residual magnitude
* Curves: - tau_naive - tau_ir - tau_min - tau_fail
* Horizontal line: - r_max
* Optional binary shading: - collapse_naive

No further transformation is required.

When you say “next”, I will:
* assemble these into a single composite plot specification (exact axes, legends, layering), or
* emit plot-ready code (NumPy → Matplotlib / LaTeX PGFPlots) with no ambiguity.

© Robert R. Frost
2026-01-03