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

=== Assistant: Below is the strict meaning of the results, stated at the level of theory semantics and algorithmic consequence. ===
Below is the strict meaning of the results, stated at the level of theory semantics and algorithmic consequence. No narrative interpretation beyond what is logically implied.

MEANING OF THE RESULTS
# Separation of regimes is real and absolute

The run demonstrates a hard separation between two logically distinct regimes:

• Structural (finite-resolution) regime
• Asymptotic (infinite-precision) regime

In the structural regime, the system survives.
In the asymptotic regime, the same system fails deterministically.

This is not a numerical artifact. It is a property of the decision logic.
# Bounded robustness is genuine but conditional

With bounded adversarial noise:

|r|max = |y₀ − θ − b| + ε_max = 0.9
τ_struct = 3.0

Since 0.9 < 3.0, the system is provably robust to all admissible disturbances in the declared noise class.

Meaning:
• No amount of bounded disturbance can falsify the model
• Robustness is binary, not probabilistic
• This is a true certification margin

This corresponds to a hard robustness guarantee, not statistical confidence.
# Precision scaling is a stronger adversary than noise

Precision scaling introduces a qualitatively different stressor:

τ_p = τ_struct / 2^p

As p increases, the allowed residual shrinks without bound.
Eventually:

τ_p < |r|max

At p = 2, collapse occurs.

Meaning:
• The model relies on finite resolution for survival
• It does not admit a consistent infinite-precision limit
• The collapse is inevitable unless τ_struct → ∞

This is not “overfitting” or “instability” in the ML sense.
It is resolution dependence.
# δ (linear scaling rate) becomes irrelevant under bounded noise

Once linear drift is replaced by bounded noise:

• δ no longer contributes to collapse
• Time index k disappears from the failure condition
• The system becomes time-invariant

Meaning:
• Failure is no longer “eventual”
• It is purely structural vs asymptotic
• This isolates robustness cleanly from temporal drift

This confirms that earlier linear-scaling collapse was not stochastic—it was geometric.
# What the final verdict actually says

The final verdict was COLLAPSE, triggered by PrecisionFailure, not by structure, feasibility, or evidence.

Meaning in precise terms:

• The theory is empirically adequate at finite resolution
• The theory is not resolution-invariant
• Therefore, it cannot be certified as a fixed-point theory in the limit p → ∞

This is a stronger and more specific conclusion than generic falsification.
# What this does ''not'' mean

It does NOT mean:

• The model is “wrong” at current experimental scales
• The model lacks robustness to real-world noise
• The model fails statistically or probabilistically
• The model is unstable under bounded perturbations

All of those were explicitly ruled out.
# What this means operationally

In operational terms:

• The theory is valid as an effective infrared theory
• It is invalid as a scale-free, asymptotically exact theory
• Its domain of authority is finite-resolution physics
• Its failure mode is mathematically clean and diagnosable

This is exactly the distinction your framework is designed to expose.
# One-line formal meaning

: 

If you want, the next logically meaningful step would be to formalize this as a theorem:

: 

or to compute how τ must scale with p to restore asymptotic viability.

© Robert R. Frost
2026-01-03