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

=== Assistant: Below is the exact scaling law for τ as a function of precision p required to restore asymptotic viability under the AN… ===
Below is the exact scaling law for τ as a function of precision p required to restore asymptotic viability under the ANF semantics. This is a closed-form condition. No simulation. No approximation.

DEFINITION OF ASYMPTOTIC VIABILITY

Asymptotic viability requires:

∀ p ∈ ℕ₀ : |r|max ≤ τ(p)

where
τ(p) is the precision-dependent structural tolerance,
|r|max is the worst-case residual under the admissible disturbance class.

In your run:

|r|max = |y₀ − θ − b| + ε_max

This quantity is constant in p.

CURRENT (FAILING) SCALING LAW

You used:

τ(p) = τ₀ / 2^p

This implies:

lim_{p→∞} τ(p) = 0

Therefore:

∃ finite p'' such that τ(p'') < |r|max
⇒ inevitable collapse

This scaling cannot be asymptotically viable unless |r|max = 0.

NECESSARY AND SUFFICIENT CONDITION FOR VIABILITY

To avoid collapse for all p, we require:

inf_p τ(p) ≥ |r|max

Equivalently:

τ(p) must not decay to zero faster than a constant lower bound.

This yields the fundamental constraint:

τ(p) ≥ |r|max  ∀ p

This is exact.

MINIMAL ASYMPTOTICALLY VIABLE SCALING LAW

The weakest (tightest) viable scaling is:

τ(p) = |r|max + ε with ε > 0

This gives:

lim_{p→∞} τ(p) = |r|max + ε
⇒ strict survival margin

This is the minimal constant-floor scaling.

GENERAL CLASS OF VIABLE SCALINGS

Any τ(p) satisfying:

liminf_{p→∞} τ(p) ≥ |r|max

is asymptotically viable.

Examples (all valid):
# Constant floor τ(p) = τ₀ ≥ |r|max
# Exponential decay with floor τ(p) = |r|max + C·2^{-p}
# Polynomial decay with floor τ(p) = |r|max + C/(p+1)^α (α > 0)
# Logarithmic decay with floor τ(p) = |r|max + C/log(p+2)

Invalid (guaranteed collapse):

τ(p) = τ₀·2^{-p}
τ(p) = τ₀/(p+1)^α
τ(p) → 0

REQUIRED SCALING IN YOUR NUMERICAL CASE

From the run:

|r|max = 0.9

Therefore, necessary and sufficient condition:

τ(p) ≥ 0.9 ∀ p

Minimal viable choice:

τ(p) = 0.9 + ε (ε arbitrarily small, >0)

Example viable law:

τ(p) = 0.9 + 0.1·2^{-p}

Check:

p=0 → τ=1.0
p→∞ → τ→0.9
Always τ(p) ≥ |r|max

⇒ NO COLLAPSE, EVER.

INTERPRETATION IN THEORY TERMS (PRECISE)

• Precision scaling may increase resolution
• But structural tolerance must encode physical irreducible spread
• That spread is exactly |r|max

Asymptotic viability therefore requires:

: 

This is equivalent to asserting that exact measurements do not exist.

ONE-LINE FORMAL RESULT

: 

This is the tight bound. Nothing weaker works.

If you want, the next exact step would be to:
• rewrite this as a formal theorem in the Methods section, or
• connect τ(p) to a renormalization-group fixed point interpretation.

© Robert R. Frost
2026-01-03