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

=== Assistant: Below are the analytic bounds derived directly from the contour geometry, stated as inequalities and equivalences. ===
Below are the analytic bounds derived directly from the contour geometry, stated as inequalities and equivalences. No numerics, no plots, no simulation. Each bound corresponds one-to-one with a geometric feature of the contour surfaces already constructed.

==== Residual bound: ====

∣r∣max⁡>0|r|_{\max} > 0∣r∣max>0
Naive scaling:

τnaive(p,τ0)=τ0 2−p\tau_{\mathrm{naive}}(p,\tau_0)=\tau_0\,2^{-p}τnaive(p,τ0)=τ02−p
IR-fixed-point scaling:

τIR(p,τ0)=τ\''+(τ0−τ\'') 2−p,τ\''≥∣r∣max⁡\tau_{\mathrm{IR}}(p,\tau_0)=\tau^\''+(\tau_0-\tau^\'')\,2^{-p},\quad \tau^\''\ge |r|_{\max}τIR(p,τ0)=τ\''+(τ0−τ\'')2−p,τ\*≥∣r∣max
Collapse condition:

COLLAPSE  ⟺  τ(p)<∣r∣max⁡\text{COLLAPSE} \iff \tau(p)<|r|_{\max}COLLAPSE⟺τ(p)<∣r∣max

==== From the contour definition: ====

τnaive(p,τ0)=∣r∣max⁡\tau_{\mathrm{naive}}(p,\tau_0)=|r|_{\max}τnaive(p,τ0)=∣r∣max
Solve for ppp:

p\''(τ0)=log⁡2 ⁣(τ0∣r∣max⁡)p^\''(\tau_0)=\log_2\!\left(\frac{\tau_0}{|r|_{\max}}\right)p\*(τ0)=log2(∣r∣maxτ0)
===== - Each contour line in the (p,τ0)(p,\tau_0)(p,τ0) plane is a level set of constant collapse index. =====
* The contour surface is codimension-one and sharp (no thickness).

==== From the surface inequality: ====

τnaive(p,τ0)≥∣r∣max⁡\tau_{\mathrm{naive}}(p,\tau_0)\ge |r|_{\max}τnaive(p,τ0)≥∣r∣max
Equivalent to:

τ0≥∣r∣max⁡ 2p\tau_0 \ge |r|_{\max}\,2^{p}τ0≥∣r∣max2p
===== - Viability occupies the open half-space above the collapse surface. =====
* For fixed τ0\tau_0τ0, viable precision indices satisfy:

p≤log⁡2 ⁣(τ0∣r∣max⁡)p \le \log_2\!\left(\frac{\tau_0}{|r|_{\max}}\right)p≤log2(∣r∣maxτ0)

==== For any finite τ0\tau_0τ0: ====

lim⁡p→∞τnaive(p,τ0)=0\lim_{p\to\infty}\tau_{\mathrm{naive}}(p,\tau_0)=0p→∞limτnaive(p,τ0)=0
Therefore:

∃ p<∞: τnaive(p,τ0)<∣r∣max⁡\exists\,p<\infty:\ \tau_{\mathrm{naive}}(p,\tau_0)<|r|_{\max}∃p<∞: τnaive(p,τ0)<∣r∣max
===== p\''≤⌈log⁡2 ⁣(τ0∣r∣max⁡)⌉p^\''\le \left\lceil \log_2\!\left(\frac{\tau_0}{|r|_{\max}}\right)\right\rceilp\*≤⌈log2(∣r∣maxτ0)⌉ =====
This is the tightest possible bound. It is read directly off the contour geometry.

==== For IR scaling: ====

inf⁡pτIR(p,τ0)=τ\''\inf_{p}\tau_{\mathrm{IR}}(p,\tau_0)=\tau^\''pinfτIR(p,τ0)=τ\*
Thus:

τIR(p,τ0)≥∣r∣max⁡ ∀p⟺τ\*≥∣r∣max⁡\tau_{\mathrm{IR}}(p,\tau_0)\ge |r|_{\max}\ \forall p
\quad\Longleftrightarrow\quad
\tau^\''\ge |r|_{\max}τIR(p,τ0)≥∣r∣max ∀p⟺τ\''≥∣r∣max
===== - The IR surface never intersects the collapse plane. =====
'' Therefore no contour p\''p^\''p\'' exists.
* Collapse index is undefined (i.e. infinite).

==== From the analytic difference surface: ====

Δτ(p,τ0)=τnaive−τIR=−τ\*(1−2−p)\Delta\tau(p,\tau_0)
=\tau_{\mathrm{naive}}-\tau_{\mathrm{IR}}
=-\tau^\''\bigl(1-2^{-p}\bigr)Δτ(p,τ0)=τnaive−τIR=−τ\''(1−2−p)
===== −τ\''≤Δτ(p,τ0)<0∀p>0-\tau^\'' \le \Delta\tau(p,\tau_0) < 0 \quad \forall p>0−τ\*≤Δτ(p,τ0)<0∀p>0 =====
===== - Difference is strictly negative for all p>0p>0p>0. =====
* Independent of τ0\tau_0τ0.
* Naive scaling is uniformly dominated by IR scaling.

This explains why all signed colormap values are negative.

==== From non-intersection condition: ====

τ\''≥∣r∣max⁡\tau^\'' \ge |r|_{\max}τ\*≥∣r∣max
This bound is both necessary and sufficient.
'' If τ\''<∣r∣max⁡\tau^\''<|r|_{\max}τ\''<∣r∣max: IR surface intersects collapse plane ⇒ finite collapse.
'' If τ\''=∣r∣max⁡\tau^\''=|r|_{\max}τ\''=∣r∣max: IR surface is tangent ⇒ marginal viability.
'' If τ\''>∣r∣max⁡\tau^\''>|r|_{\max}τ\''>∣r∣max: strict viability margin.

No weaker bound exists.

==== Rearranging the collapse boundary: ====

p≤log⁡2 ⁣(τ0∣r∣max⁡)p \le \log_2\!\left(\frac{\tau_0}{|r|_{\max}}\right)p≤log2(∣r∣maxτ0)
Equivalently:

τ0≥∣r∣max⁡ 2p\tau_0 \ge |r|_{\max}\,2^{p}τ0≥∣r∣max2p
===== Each additional bit of precision requires doubling initial tolerance to avoid collapse. =====

This exponential tradeoff is exact, not asymptotic.

==== Given a certification horizon p≤Pp\le Pp≤P, admissible initial tolerances satisfy: ====

τ0≥∣r∣max⁡ 2P\tau_0 \ge |r|_{\max}\,2^{P}τ0≥∣r∣max2P
Conversely, given τ0\tau_0τ0, the maximum certifiable precision is:

Pmax⁡=⌊log⁡2 ⁣(τ0∣r∣max⁡)⌋P_{\max}=\left\lfloor \log_2\!\left(\frac{\tau_0}{|r|_{\max}}\right)\right\rfloorPmax=⌊log2(∣r∣maxτ0)⌋
This bound is directly read off the contour surface.

==== p\''(τ0)=log⁡2(τ0/∣r∣max⁡)Viable  ⟺  τ0≥∣r∣max⁡2pIR viable  ⟺  τ\''≥∣r∣max⁡Δτ∈[−τ\*,0)Pmax⁡=⌊log⁡2(τ0/∣r∣max⁡)⌋\boxed{ ====
\begin{aligned}
p^\*(\tau_0) &= \log_2(\tau_0/|r|_{\max}) \\
\text{Viable} &\iff \tau_0 \ge |r|_{\max}2^{p} \\
\text{IR viable} &\iff \tau^\*\ge |r|_{\max} \\
\Delta\tau &\in[-\tau^\*,0) \\
P_{\max} &= \lfloor\log_2(\tau_0/|r|_{\max})\rfloor
\end{aligned}
}p\''(τ0)ViableIR viableΔτPmax=log2(τ0/∣r∣max)⟺τ0≥∣r∣max2p⟺τ\''≥∣r∣max∈[−τ\*,0)=⌊log2(τ0/∣r∣max)⌋
These are exact analytic consequences of the contour geometry already plotted. No further assumptions are used.

If you want next, I can:
* formalize these bounds as lemmas/theorems in the manuscript,
* express them as machine-checkable constraints,
* or derive optimal tolerance allocation policies under fixed precision budgets.

© Robert R. Frost
2026-01-03