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

Quantity	Definition	Purpose
Noise floor (η)	The expected syntactic variance between two samples from P that both resolve d(S)	Baseline — variation below η is not signal
Domain confidence (κ)	P(output resolves d(S) | context C) — the probability that a sample stays in-domain	Measures how well the context constrains generation
Reproducibility equivalence (≈)	Two outputs are equivalent if they resolve the same d(S) components, regardless of syntactic differences	When two generations are “the same”

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:

Variation	Example	Why It’s Noise
Variable naming	items vs elements vs data	Semantically equivalent; no d(S) effect
Statement ordering	Independent declarations in different order	No semantic dependency
Formatting	Whitespace, line breaks, bracket placement	No semantic content
Equivalent expressions	x !== null vs x != null vs x	Same domain semantics
Comment presence/absence	Added or omitted inline comments	No structural effect
Import ordering	Same imports, different sequence	No semantic content

What Doesn’t Vary (signal — above η)

These differences indicate a boundary condition change:

Variation	Example	Why It’s Signal
Literal vs. reference	16px vs var(--spacing-md)	d(S) effect: reference preserves closure, literal reopens
Type widening	Status vs Status | string	d(S) effect: wider type expands representable states
Seed bypass	Hardcoded value vs. imported constant	d(S) effect: bypass reopens a resolved boundary
Domain crossing	Layout code in a logic file	d(S) effect: touches a domain outside the target
Convention break	camelCase in a snake_case codebase	Hamming distance exceeds semantic distance

The Boundary Between Noise and Signal

Some variations are ambiguous — they depend on context:

Variation	Noise When…	Signal When…
Different API usage	Both are idiomatic for the domain	One bypasses the established pattern
Structural alternatives	if/else vs ternary — same semantics	Different control flow with different error handling
Abstraction level	Both are appropriate for the scope	One introduces unnecessary indirection or coupling

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

Factor	Effect on κ	Mechanism
More closed prerequisite domains in context	↑	Model has stable foundation; fewer drift vectors
Narrower type constraints visible	↑	Smaller representable output space
More seed examples using references	↑	Continuation distribution favors references
Fewer unrelated domains in context	↑	Fewer cross-domain contamination paths
Target d(S) is small/specific	↑	Less room for the model to diverge

Factors That Decrease κ

Factor	Effect on κ	Mechanism
Open prerequisite domains	↓	Foundation is unstable; dependent results unreliable
Wide types (any, string, loose schemas)	↓	Large representable space includes invalid states
Literals in context alongside references	↓	Model treats both as acceptable patterns
Large target d(S)	↓	More room for divergence
Cross-domain files in context	↓	Contamination paths available

Confidence Thresholds

κ Range	Interpretation	Action
κ > 0.95	High confidence — most generations will be in-domain	Generate and verify once
0.8 < κ ≤ 0.95	Moderate — occasional drift expected	Generate, verify, may need one retry
0.5 < κ ≤ 0.8	Low — drift is common	Tighten context before generating; sample multiple times
κ ≤ 0.5	Unreliable — more drift than signal	Do not generate. Close prerequisite domains first

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.