Boundary Noise Model

Boundary: Noise Model

Characterize the noise envelope of LLM generation to distinguish sampling variance from semantic drift.

The Problem

LLM generation is stochastic. The same context produces different outputs across runs. The delta-epsilon framework in its deterministic form cannot distinguish:

• Noise: two outputs that differ syntactically but resolve the same d(S) — acceptable variance
• Drift: two outputs where one resolves d(S) and the other expands it — a boundary violation

Without a noise model, every syntactic difference looks like potential drift. This produces false positives that erode trust in the detection system, or forces ε so wide that real drift passes undetected.

Framework: Stochastic Differential Closure

The deterministic framework says: |δ| < ε → in-domain, |δ| ≥ ε → drift.

The stochastic extension says: ε is not a fixed threshold. It is a function of the noise floor of the generation process.

The Generation Distribution

For a given context C and target domain D, the model samples from a distribution P(output | C). This distribution has:

• A mode region — outputs that are most probable. These cluster around the patterns present in the context.
• A tail region — less probable outputs that diverge further from context patterns.
• A domain boundary — the boundary condition d(S) that separates in-domain outputs from drift.

The noise envelope is the variance of P within the mode region. Drift is when a sample falls outside the domain boundary, regardless of its probability.

Three Quantities

When to Use This Skill

• Evaluating whether differences between two generated outputs are meaningful
• Calibrating ε for boundary-drift-detection — setting the threshold above the noise floor
• Estimating confidence before committing to a generation strategy
• Determining how many generations to sample to achieve target confidence
• Diagnosing whether a recurring drift pattern is a context problem (fixable) or a noise floor problem (inherent)

Characterizing the Noise Floor

What Varies (noise — below η)

These differences between two in-domain generations are expected sampling variance:

What Doesn't Vary (signal — above η)

These differences indicate a boundary condition change:

The Boundary Between Noise and Signal

Some variations are ambiguous — they depend on context:

For ambiguous cases, apply the d(S) test: does this variation change which boundary conditions are resolved? If yes, it's signal. If no, it's noise.

Domain Confidence

Definition

κ(C, D) = P(output resolves d(S) of D | context C)

Domain confidence is the probability that a generation from context C produces an output that stays within domain D and resolves (or at minimum preserves) the target boundary condition.

Factors That Increase κ

Factors That Decrease κ

Confidence Thresholds

These are not measured numerically in practice — they are estimated by assessing the context against the factors above. The table provides the decision framework.

Reproducibility Equivalence

Definition

Two outputs O₁ and O₂ are reproducibility-equivalent (O₁ ≈ O₂) if:

1. They resolve the same components of d(S)
2. They preserve the same closed domains
3. They introduce no new boundary conditions that the other doesn't
4. Their differences are entirely within the noise floor η

The Equivalence Test

Given two outputs from the same context and target:

1. Run boundary-drift-detection on each independently
2. Compare their d(S) effect classifications
3. If both have the same classification for every domain in the partial order → equivalent
4. If they differ on any domain's d(S) effect → not equivalent — one of them drifted

Why This Matters

Reproducibility equivalence defines what "the same result" means for stochastic generation. Two outputs can look completely different syntactically and be equivalent. Two outputs can look nearly identical and not be equivalent (if one subtle difference is a boundary violation).

The equivalence class is defined by d(S) effect, not by syntax.

Calibrating ε

The deterministic ε from boundary-drift-detection must be calibrated against the noise floor:

ε_effective = ε_domain + η

Where:
  ε_domain  = the semantic boundary of the domain (fixed by ∂S)
  η         = the noise floor of the generation process (varies with context)

• Variations below η: noise — do not flag
• Variations between η and ε_effective: review — may be noise or signal depending on d(S) effect
• Variations above ε_effective: drift — flag unconditionally

Practical Calibration

You don't measure η numerically. You estimate it by asking:

1. Could this difference appear between two correct implementations? If yes → likely noise.
2. Does this difference change which boundary conditions are resolved? If yes → signal regardless of magnitude.
3. Is this difference in a dimension that the domain's ∂S constrains? If yes → apply strict ε. If no → apply noise tolerance.

Guidelines

• Not all differences are drift. The noise model prevents false positives that erode trust in boundary detection. Calibrate ε above the noise floor.
• Not all similarities are equivalence. Two outputs using the same literal 16px are not equivalent to two outputs using var(--spacing-md), even though each pair is internally consistent. Equivalence is about d(S) effect.
• κ is a decision tool, not a measurement. You estimate confidence from context properties to decide whether to generate, tighten context, or close prerequisites first. Don't try to compute an exact number.
• The noise floor varies by domain. Variable naming has high η (many acceptable variants). Value references have low η (literal vs. reference is binary). Calibrate per domain.
• When in doubt, apply the d(S) test. If a variation changes which boundary conditions are resolved, it's signal. If it doesn't, it's noise. This test is always available regardless of η estimation quality.
• Reproducibility doesn't mean identical. It means the same d(S) effect. Accept syntactic variance; reject semantic divergence.