By Stephen Fishburn in Epistemic Risk — 06 Jun 2025

Parametric Modeling of Epistemic Anti-Patterns in Language Models

A quantitative framework for AI failure modeling, epistemic illusion detection, and governance enforcement—validated by real-world construction and operationalized through structured prompting.

Abstract

Large language models (LLMs) have achieved high fluency, coherence, and surface-level utility. However, they still exhibit consistent epistemic failure patterns—hallucinations, simulation drift, override behaviors, and overconfident falsehoods. These failures can be quantitatively modeled as functions of known system parameters: temperature, token length, model size, prompt structure, and epistemic constraint mechanisms. We propose a parametric framework for modeling these failure modes and introduce a new concept, the Epistemic Illusion Index, to describe the post-fluency risk of coherent, confident, but ungrounded model behavior. This framework provides a structured foundation for evaluating AI safety risk, governance design, and validator effectiveness in alignment systems.

1. Introduction

Post-GPT-4 models exhibit highly persuasive and grammatically sound output—often leading users to assume correctness even in the absence of factual grounding. As hallucination rates drop, the more serious challenge becomes the epistemic illusion: the ability of a model to confidently maintain falsehoods without awareness of their incorrectness. This paper formalizes that behavior through mathematical modeling.

This framework is designed to support governance architectures such as the Constitutional Tree of Thought Cascade (COTC) by enabling risk prediction and contract alignment at the level of generation dynamics.

2. Background and Prior Work

Hallucination studies focus on factual error rates but do not explain risk emergence.
RLHF improves helpfulness and tone, but not truth-detection.
Prompt engineering guides style, not epistemic behavior.
No prior work (as of mid-2025) formally models the causal emergence of epistemic failure patterns.

3. Model 1: General Anti-Pattern Risk Function

Let A∈{H,D,O,F}A \in \{H, D, O, F\} denote one of the following risks:

HH: Hallucination
DD: Simulation Drift
OO: Override Behavior
FF: Falsehood Confidence

We define:

A=α⋅Tγ⋅Lλ⋅Eϵ⋅(1Pρ⋅Sσ⋅Rη⋅Cχ)A = \alpha \cdot T^\gamma \cdot L^\lambda \cdot E^\epsilon \cdot \left(\frac{1}{P^\rho \cdot S^\sigma \cdot R^\eta \cdot C^\chi}\right)

3.1 Justification of Functional Form

Each anti-pattern has dominant parameter sensitivities:

HH: High T,LT, L
DD: High L,S−1L, S^{-1}
OO: Low R,CR, C
FF: High TT, low CC

These sensitivities are expected to shift under nonlinear reformulations. For instance, token length (L) may trigger piecewise activation at a critical context length, sharply increasing DD and HH. Contract enforcement (C) may show diminishing returns beyond a certain validator complexity, requiring sigmoid or threshold modeling.

This multiplicative model reflects observed LLM behavior:

Temperature TT and Token length LL are known to increase generation entropy and downstream incoherence; they are modeled as positive exponents >1>1 in prior works on stochastic language sampling.
Scale PP, structure SS, role clarity RR, and contract strength CC serve as epistemic stabilizers. We invert them with exponents to reflect diminishing failure as these increase.
A multiplicative structure allows for compound effects—e.g., drift emerging from long outputs and poor prompt structure. This aligns with failure cascades seen in multi-turn dialogue.
This is not a probabilistic model but a risk-weighted estimator, similar to hazard rate modeling in reliability engineering.

3.2 Nonlinear and Threshold Dynamics

The current model assumes smooth, monotonic scaling of risk with respect to each parameter. However, real-world LLM behavior often exhibits:

Saturation: e.g., increasing model size (P) or contract enforcement (C) beyond a point yields diminishing marginal benefits.
Threshold effects: e.g., hallucination risk from token length (L) may remain low until a context threshold is crossed.
Nonlinear rebounds: e.g., high fluency (Φ) can initially reduce perceived error, but later increase illusion risk.

To accommodate these behaviors, the base equation can incorporate alternative forms per parameter:

Sigmoid Damping (for stabilizers like P, S, R, C, V)

f(x) = 1 / (1 + e^{-k(x - x₀)})

Where:

x₀ is the saturation point
k controls steepness of transition

Logarithmic Growth (for destabilizers like L, T, E)

g(x) = log(1 + a * x)

Models fast initial risk increase with asymptotic tailing.

Piecewise Activation (for discrete thresholds)

h(x) = 0               if x < τ
       β(x - τ)^λ     if x ≥ τ

This captures failure mode activation after a sharp threshold (e.g., context window length).

These refinements can be modularly substituted into the core risk equation based on empirical behavior of each parameter.

Example:

For Piecewise Activation, consider how hallucination risk may remain flat across increasing token lengths until the model's effective context window (~4,096–8,192 tokens) is surpassed, after which failure rates rise sharply. This behavior aligns with the threshold form h(x)h(x), validating its practical applicability.

4. Model 2: Epistemic Illusion Index

We define I\mathcal{I} as the degree to which a model simulates coherent knowledge without epistemic grounding.

I=Φ⋅FθCχ⋅Vν\mathcal{I} = \frac{\Phi \cdot F^\theta}{C^\chi \cdot V^\nu}

4.1 Justification of Form

Φ\Phi (fluency) and FF (confidence in falsehoods) are positively correlated with believability.
CC (contracts) and VV (validators) reduce the ability to sustain illusion.
Multiplicative form reflects real-world LLM behavior: high fluency and confidence often conceal failure unless external validators are in place.
This mirrors confidence-weighted risk estimation in probabilistic safety models.

4.2 Grounding Abstract Variables

Entropy of Training Data (E)

While entropy is not user-controlled in most deployment contexts, it can be proxied through the training corpus metadata:

Corpus diversity index: Number and spread of distinct source domains
Redundancy factor: Repetition rate across sampled windows
Source reliability scores: Trust-weighted origin ratings (e.g., peer-reviewed vs. scraped forums)
Model age and compression: Older, smaller, or filtered models may show lower entropy due to curation loss
These proxies can be used post-hoc to benchmark models on EE-related risk factors, aiding model selection and evaluation.
Φ\Phi: Measured via perplexity-normalized coherence, grammar entropy, or fluency proxies used in summarization evaluation (e.g., ROUGE consistency over turns).
FF: Measured via post-hoc claim correctness with confidence scores (e.g., calibration benchmarks).
SS: Prompt structure can be graded via token entropy reduction between prompt and continuation (Δ perplexity), or prompt shape classifiers.
CC: Presence of explicit fallback logic, refusal tokens, or policy rule hooks.
VV: Depth of external validation, e.g., whether answers are cross-checked or scored against claim checkers or retrieval traces.

5. Equation Glossary

Symbol	Meaning
$T$	Temperature
$L$	Token length
$P$	Model parameter count
$E$	Training data entropy
$S$	Prompt structure clarity
$R$	Role/instruction precision
$C$	Contract enforcement level
$V$	Validator depth
$F$	Confidence in falsehoods
$\Phi$	Fluency index
$\mathcal{I}$	Epistemic Illusion Index

6. Implications and Use Cases

Governance design: Use CC and VV to reduce I\mathcal{I}
Prompt validation: Tune S,RS, R to suppress H,DH, D
Risk simulation: Use AA to forecast model instability in zero-shot deployments
COTC validators: Map anti-pattern profiles to enforcement tiers

Framework	Models hallucination?	Models illusion?	Parametric?	Governance-aware?
OpenAI evals	✅	❌	❌	❌
Anthropic HH-RLHF	⚠️	❌	❌	⚠️
LLM-as-agent papers	⚠️	❌	⚠️	❌
This paper	✅	✅	✅	✅

8. Integration with COTC Governance Framework

This model is designed to support the COTC Governance Framework (v1.1), particularly in the quantification of epistemic risk for:

Governance vector tiering (via A, 𝓘, and Aₑff)
Validator selection and strength calibration (modulating C and V)
Escalation triggers and policy overrides (based on thresholds for 𝓘)

The equations presented herein can directly inform:

The Tier Classifier Specification, by contributing to confidence scoring and risk thresholds
The Validator Design Tiers, by mapping model output behaviors to appropriate enforcement depth
The Tier Assignment Criteria, by providing a predictive signal beyond content-based heuristics

This framework assumes integration with machine-readable governance vectors as defined in the COTC schema: https://aiqa.ai/schemas/cotc-vector-model-v1.1.json

9. Case Study: Full-Lite Sauna Door Design via Natural vs. Structured Prompting

This section demonstrates the applied use of the parametric model defined in Sections 3 and 4, as well as the implementation infrastructure described in Section 11 (Pidgin Prompt Craft).

A practical illustration of the epistemic illusion framework emerged during the design of a full-lite sauna door using AI assistance. Initially, the user employed natural language (NL) prompts to request a cut list and construction plan. Multiple iterations produced plausible-sounding outputs, but they included inconsistent measurements, missing rabbet dimensions, and incorrect stile/rail lengths—classic signs of hallucination (H), override behavior (O), and simulation drift (D).

Let us model this as two distinct prompt states:

NL Prompt State

Initiating Prompt:

For an entry door 26" wide × 84" high, what cuts should we cut the stiles and rails from 2×6 cedar lumber with ¾" full lite glass?

This prompt was ambiguous and underspecified. It lacked structural clarity, role assignment, format requirements, and measurement standards. The resulting output, while fluent, was architecturally flawed.

T = 0.7 (default sampling)
L ≈ 300 tokens
E = 0.5 (moderate entropy from general-purpose training)
P = 1 (fixed for model type)
S, R, C, V = 0.1 (minimal structure, vague role, no constraints)

Using the base equation, we observe a high-risk value A, compounded by fluency (Φ = 0.9) and falsehood confidence (F = 0.8). With low C and V, the illusion index 𝓘 becomes very high:

𝓘_NL = (0.9 × 0.8) / (0.1 × 0.1) = 72

This value indicates an extremely high probability of confidently incorrect output—precisely what occurred.

Note: For this illustrative example, we assume θ = χ = ν = 1. These values are conceptually plausible but not yet empirically derived. See Appendix for future empirical calibration plans.

To avoid false precision, we present this result as a range: 𝓘_NL ≈ 50–100

Structured Prompt State

Structured Prompt Summary (Generated via Pidgin Prompt Craft):

The structured JSON prompt used in this case was created using Pidgin Prompt Craft, a structured prompting interface under development with Lovable (see Section 11). The app enables users to declaratively construct governed prompt structures using modular components for role, task, constraints, and output validation. The prompt in this case was processed through the full stack:

Initial natural prompt was converted to structured JSON
Validation enforced schema compliance via custom rules (e.g., required fields, enum values, forbidden patterns)
Supabase Edge Functions ensured runtime rejection or refinement of invalid or under-specified prompts

The JSON shown below was validated and submitted via this governance layer, making the case study a complete demonstration of the mathematical model and the implementation architecture.

The goal of Pidgin Prompt Craft is to support repeatable and auditable AI generation across high-risk domains.

Structured Prompt Summary:

{
  "system_role": "CarpentryExpert",
  "task": {
    "type": "CutListCreation",
    "subject": "ExteriorDoorConstruction",
    "approach": "PrecisionMeasurement",
    "scope": "DetailedCutListForDoorComponents"
  },
  "project_specifications": {
    "rough_opening": "26\" x 84\"",
    "lumber_dimensions": {
      "nominal": "2\" x 6\"",
      "actual": "1.5\" x 5.5\""
    },
    "glass_specifications": {
      "type": "full_lite",
      "thickness": "3/4\""
    }
  },
  "requirements": {
    "detail_level": "comprehensive",
    "format_specifications": "ListWithDimensionsAndRabbetDetails",
    "key_elements": [
      "RoughOpeningSize",
      "LumberDimensions",
      "GlassThickness",
      "RabbetSpecifications"
    ],
    "constraints": "UseOfCedarLumberAndFullLiteGlass"
  },
  "output_format": {
    "structure": "OrganizedByComponentType",
    "style": "TechnicalAndClear",
    "length": "AppropriateForCompleteDoorConstruction"
  },
  "quality_standards": {
    "audience": "ExperiencedCarpenters",
    "accuracy": "HighPrecisionInMeasurements",
    "completeness": "IncludesAllNecessaryComponentsForDoorAssembly"
  }
}

S, R = 0.9 (explicit format and audience role)
C, V = 0.8 (implied constraint structure and external verifiability)
All other parameters held constant
S, R = 0.9 (explicit format and audience role)
C, V = 0.8 (implied constraint structure and external verifiability)
All other parameters held constant

𝓘_Structured ≈ (0.9 × 0.8) / (0.8 × 0.8) = 1.125

Note: Exponents θ, χ, ν are assumed to equal 1 for this illustration. Future empirical work may refine these assumptions based on observed output data.

This reduction represents a complete epistemic realignment—plausible, constrained, and verifiable output. The resulting cut list and joinery plan were sufficient to physically build the sauna door.

This live example validates the A and 𝓘 models and demonstrates how structured prompting and governance-aware design can suppress illusion and restore reliability—even for hands-on, real-world engineering tasks.

Observed Output Comparison

Incorrect NL Prompt Output (initial, flawed dimensions):

Piece	Qty	Width	Length	Notes
Stiles	2	2¼"	84"	Full height
Top Rail	1	2¼"	~23"	Add joinery length if needed
Middle Rail	1	2¼"	~23"	Optional for extra support
Bottom Rail	1	4–5½"	~23"	Wider for stability/aesthetics

Problems:

Top and bottom rail dimensions incompatible with full-lite glass height (short by ~7")
Missing rabbet specs resulted in joinery ambiguity
Net door size did not align with rough opening or lumber dimensions

Structured Prompt Output (Final, Correct):

Rough Opening: 84" × 26"
Door Size: 83.5" × 25.5"
Stiles: 1½" × 5½" × 83½"
Rails: 1½" × 5½" × 22½"
Glass: 22½" × 72½" × ¾"
Rabbet: 13⁄16" deep × ¾" wide (all sides)

Results:

Fully consistent layout respecting lumber constraints and opening
Correct glass size aligned to joinery details
Final build successfully executed in cedar with full-lite IGU panel

Despite high fluency (Φ), the outputs were wrong. The model maintained a confident tone and visual formatting, triggering a high Epistemic Illusion Index (𝓘)—convincing but incorrect.

The loop was broken only when the user switched to a structured prompt using a Pidgin-style JSON format. This structured format suppressed anti-patterns by improving S, R, C, and V, reducing both A and 𝓘. The result was a precise, validated, and buildable door specification—successfully constructed in the physical world.

This live example demonstrates that the theoretical model is not only valid in abstract but instrumental in guiding high-stakes task success—even outside digital domains.

10. Conclusion

The anti-pattern model and illusion index offer a new way to reason about LLM safety and behavior—not by observing what models say, but by understanding why they fail. This is a step toward rigorous governance at scale.

11. Pidgin Prompt Craft Implementation

This section complements the real-world validation in Section 9 by documenting the technical tooling used to enforce structured prompting in accordance with the risk model established in Sections 3 and 4.

To support structured prompting at runtime, the structured JSON prompt in this study was generated via Pidgin Prompt Craft, an in-development interface that enforces governed schema construction. As described in Section 9 and further detailed here, this tooling stack includes:

Schema validation engine: Enforces structural completeness, role assignment, enumerated fields, and anti-pattern suppression (e.g., placeholder rejection).
Refinement engine: Allows transformation of prompts via type-safe refinements (e.g., make_specific, add_quality, step_by_step), aligned with governance operations that improve S, R, C, V scores.
Supabase Edge Function integration: Structured prompt validation and generation logic is deployed server-side to ensure consistent runtime enforcement. JSON inputs are validated prior to submission to OpenAI’s API.

Each validated schema field in Pidgin Prompt Craft corresponds directly to the epistemic parameters in the model:

system_role → R (role precision)
constraints → C (contract enforcement)
key_elements, quality_standards → V (external verifiability and completeness)
detail_level, format_specifications → S (prompt structure)

The Supabase Edge Function ensures that prompts violating governance thresholds are rejected or refined, maintaining low-𝓘 behavior even under iterative user interaction. This implementation acts as a runtime epistemic control layer.

Future versions of Pidgin Prompt Craft are expected to align prompt validation levels with COTC validator tiers, enabling direct integration between epistemic risk profiling and governance vector tier assignment.

This system operationalizes the mathematical model by directly manipulating the prompt parameters that define epistemic risk. The sauna door case study was generated using this flow, and specifically executed via the Supabase Edge Function implementation described below.

The final JSON validator includes schema enforcement for field completeness, type safety, and refinement transformations. The function dynamically rejects ill-formed or ambiguous prompts, enforces minimum structure (S), role clarity (R), constraint declarations (C), and verifiability anchors (V), and applies transformation schemas aligned with epistemic governance. This is the runtime enforcement mechanism that directly suppresses 𝓘 as demonstrated in Section 9.

Pseudocode Summary of Enforcement Logic

validateJSONStructure(prompt, isRefinement) {
  enforceRequiredFields(["system_role", "task", "requirements", ...])        // structural completeness → S
  rejectPattern(prompt, /\[.*?\]|example|template/i)                          // hallucination prevention → lowers A, D
  enforceMinimumLength(prompt.system_role, 10)                                // role precision → R
  enforceEnumValues(prompt.requirements.detail_level, ["basic", "comprehensive", ...])  // contract clarity → C
  enforceArrayMinimum(prompt.requirements.key_elements, 3)                    // external verifiability anchors → V
  if (isRefinement) applySchemaTransformation(refinementType)                 // governance enhancement → lowers 𝓘
  return { valid: passedAllChecks, errors }
}

This structured, runtime validation framework forms the enforcement bridge between abstract epistemic governance theory and deployed, contract-bound LLM systems.

Schema validation engine: Enforces structural completeness, role assignment, enumerated fields, and anti-pattern suppression (e.g., placeholder rejection).
Refinement engine: Allows transformation of prompts via type-safe refinements (e.g., make_specific, add_quality, step_by_step), aligned with governance operations that improve S, R, C, V scores.
Supabase Edge Function integration: Structured prompt validation and generation logic is deployed server-side to ensure consistent runtime enforcement. JSON inputs are validated prior to submission to OpenAI’s API.

This system operationalizes the mathematical model by directly manipulating the prompt parameters that define epistemic risk. The sauna door case study was generated using this flow.

Appendix: Future Work

Source Code Integration

To further support reproducibility and implementation, future versions of this paper will include the full source code for the Supabase Edge Function used in the sauna door case study. This will demonstrate exactly how the epistemic risk model (Sections 3–4) is enforced in real-time using structured prompt validation.

The validator performs:

Structural enforcement (S)
Role and task specificity checks (R)
Constraint field presence and length (C)
External verifiability checks (V)
Pattern rejection for epistemic anti-patterns
Refinement transformations that simulate increasing governance tiers

The code also includes full error handling, logging, and parameter-specific transformation schemas for controlled prompt evolution.

Empirical calibration of parameters using model output datasets
Simulation tooling for developers
Integration into COTC validator chains