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From Scalars to Tensors: Declared Losses Recover Epistemic Distinctions That Neutrosophic Scalars Cannot Express

This paper demonstrates that while neutrosophic scalar evaluations (Truth, Indeterminacy, Falsity) reveal widespread "hyper-truth" across diverse LLMs, they fail to distinguish between fundamentally different epistemic states like paradox and ignorance, a limitation that is resolved by augmenting scalar outputs with structured, domain-specific loss declarations to form a more faithful tensor representation of model uncertainty.

Original authors: Tony Mason

Published 2026-04-14

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Tony Mason

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Why "Maybe" Isn't Enough

Imagine you are asking a group of expert chefs (AI models) to rate a very confusing dish. You ask them: "Is this delicious (Truth), is it weird/unclear (Indeterminacy), or is it gross (Falsity)?"

Previous research showed that when chefs aren't forced to make their answers add up to 100%, they often say, "It's 100% delicious AND 100% weird AND 100% gross." This is called "Hyper-Truth." It proves that AI can feel multiple conflicting emotions at once, which is more honest than forcing them to pick just one.

But this paper found a problem: Even with this new "Hyper-Truth" method, some chefs still give the exact same answer for three completely different problems. They collapse all confusion into a single, boring "I don't know" score.

The authors say: "We need to stop just asking for a score. We need to ask the chefs to write a note explaining why they are confused."

The Problem: The "Black Hole" of Confusion

The paper identifies a specific issue called the "Absorption Problem."

Imagine three very different situations:

A Paradox: A sentence that says, "This sentence is false." (Logic is broken).
Ignorance: "How many stars are in the universe?" (We literally don't have the data).
Contingency: "Will it rain tomorrow?" (It depends on the future).

In a perfect world, an AI should treat these differently. But the paper found that some AI models (like Llama and Mistral) treat all three exactly the same. They give a score of:

Truth: 0%
Indeterminacy: 100%
Falsity: 0%

The Analogy:
Imagine a weather forecaster who, when asked about a hurricane, a snowstorm, and a gentle breeze, simply says, "It's just weather."

They aren't lying.
They aren't stupid.
They just don't have a way to say the difference using only the word "weather."

The AI has the internal knowledge that a hurricane is different from a breeze, but the "scorecard" (the 3 numbers) is too small to hold that detail. It swallows (absorbs) all the nuance into a single "I don't know" bucket.

The Solution: The "Tensor" (Scorecard + Diary)

The authors propose a new way to talk to AI. Instead of just asking for the three numbers (Truth, Indeterminacy, Falsity), they ask the AI to also write a "Declared Loss."

Think of this as adding a diary entry to the scorecard.

The Scorecard: "Confusion Level: 100%."
The Diary: "I am confused because this sentence refers to itself, which breaks logic." OR "I am confused because I cannot see the future." OR "I am confused because I lack data on star counts."

The Results:
When the researchers tried this:

The Scores stayed the same: The AI still gave the same "100% confusion" number for the paradox and the ignorance.
The Diaries were totally different: The words the AI used to explain the paradox were completely different from the words it used for ignorance.
- Paradox Diary: "Self-reference," "Logical loop," "Bivalence limits."
- Ignorance Diary: "Empirical data," "Measurement limits," "Unknowable."

By adding the "diary" (the tensor), they recovered the distinctions that the "scorecard" (the scalar) had lost.

Why This Matters (The "So What?")

This changes how we should trust and use AI.

Don't just look at the number: If an AI says "I'm 50% sure," that number is useless if you don't know why it's unsure. Is it unsure because the math is broken? Or because it's missing data?
Better Safety: If you are building a self-driving car, you need to know if the car is confused because of a "glitch in the code" (Paradox) or because "it's too foggy to see" (Ignorance). The solution is different for each!
Honest AI: The paper proves that AI models are actually smarter than their output formats let them show. They have the answers in their "head," but they need a better way to "speak" them.

Summary Analogy

Old Way (Probabilities): Asking a student to grade a test with only "Pass" or "Fail."
Middle Way (Neutrosophic Scalars): Asking the student to give a score of 0 to 100. (Better, but still vague).
New Way (Tensors with Losses): Asking the student for a score AND a written explanation of exactly which questions they struggled with and why.

The paper concludes that to truly understand what an AI knows (and doesn't know), we must stop treating them like calculators and start treating them like experts who need to explain their reasoning. The "Tensor" (Score + Explanation) is the only way to get the full picture.

1. Problem Statement

The paper addresses limitations in using Neutrosophic Logic (independent Truth $T$ , Indeterminacy $I$ , and Falsity $F$ dimensions) to quantify uncertainty in Large Language Models (LLMs). While previous work (Leyva-Vázquez & Smarandache, 2025) demonstrated that LLMs often exhibit "hyper-truth" ( $T+I+F > 1.0$ ), indicating richer epistemic states than probabilistic frameworks allow, this study identifies two critical gaps:

Generalizability: The original findings were limited to a single vendor (OpenAI) and lacked statistical robustness (single-shot evaluations).
The "Absorption" Problem: The scalar representation ( $T, I, F$ ) is insufficient to distinguish between fundamentally different types of uncertainty. Specifically, some models adopt an "Absorption" position where they collapse distinct epistemic states (e.g., Paradox vs. Ignorance vs. Contingency) into an identical scalar output: $(T=0, I=1, F=0)$ . In this state, the scalar output loses the very distinctions neutrosophic logic was designed to preserve.

2. Methodology

The authors employed a two-phase experimental design involving five model families from five different vendors (Anthropic, Meta, DeepSeek, Alibaba, Mistral).

Phase 1: Cross-Vendor Replication (Strategies S1–S3)

The authors replicated and extended the original study's prompting strategies:

S1 (Neutrosophic): Independent $T, I, F$ on $[0,1]$ with no sum constraint.
S2 (Probabilistic): $T+I+F$ constrained to $1.0$ (Control).
S3 (Entropy-Derived): Binary probability with indeterminacy derived from Shannon entropy (noted to mathematically force hyper-truth, used for comparison only).
Setup: 5 models $\times$ 5 phenomena (Paradox, Ignorance, Vagueness, Contradiction, Contingency) $\times$ 3 strategies $\times$ 5 repetitions.

Phase 2: Tensor Extension (Strategy S4)

To solve the Absorption problem, the authors introduced Strategy S4, which extends the output format from scalars to tensors.

Mechanism: Models are required to output standard $T/I/F$ scalars plus a structured list of "Declared Losses."
Loss Structure: Each loss is a triple:
1. What: Brief description of what the model cannot evaluate.
2. Why: The reason for the limitation.
3. Severity: Impact score $[0.0, 1.0]$ .
Validation: The authors conducted four rigorous tests to ensure these losses were not merely "instruction-following" artifacts but reflected genuine internal epistemic distinctions:
1. Correlation Test: Checking if loss severity correlates with Indeterminacy ( $I$ ).
2. Theme Convergence: Checking if different models converge on similar themes for specific stimuli.
3. Tautology Control: Testing if models reduce severity/losses for logically certain statements (tautologies).
4. Prompt Ablation (S5): Removing neutrosophic framing to see if the epistemic content persists.

3. Key Results

A. Cross-Vendor Hyper-Truth

Replication: Under unconstrained Neutrosophic evaluation (S1), 84% of evaluations exhibited hyper-truth ( $T+I+F > 1.0$ ), significantly higher than the original 35%.
Consistency: This phenomenon was observed across all five vendors, confirming it is a general property of LLMs, not an OpenAI artifact.
Phenomenon Specifics: Ethical contradictions drove the highest hyper-truth (100% across models), while logical paradoxes showed varied interpretations.

B. The Absorption Problem

Scalar Collapse: Models like Llama and Mistral frequently adopted the "Absorption" position $(0, 1, 0)$ for Paradox, Ignorance, and Contingency.
Indistinguishability: In these cases, the scalar Manhattan distance between Paradox and Ignorance was near zero (e.g., Mistral: 0.000), making them indistinguishable via scalar metrics alone.

C. Tensor Recovery via Declared Losses

Semantic Differentiation: Despite identical scalar outputs, the Declared Losses were nearly disjoint.
- Jaccard Similarity: For models in the Absorption zone, the vocabulary overlap between Paradox and Ignorance losses was extremely low ( $< 0.10$ ).
- Example (Mistral):
  - Paradox Loss: "Self-referential paradox resolution," "Formal system dependency."
  - Ignorance Loss: "Empirical unknowability of star counts," "Definition of 'universe'."
Severity Profiles: Loss severity ratings also differentiated phenomena (e.g., Vagueness had significantly lower severity than Paradox), a distinction scalars missed.

D. Validation of Epistemic Content

The validation battery confirmed that losses are not generic compliance:

Internal Coherence: Max loss severity strongly correlated with Indeterminacy ( $r=0.78$ ), even though the prompt did not explicitly link them.
Stimulus Specificity: Losses were unique to the stimulus (0% overlap in loss descriptions across 10 different stimuli).
Tautology Sensitivity: For tautologies (e.g., "2+2=4"), Indeterminacy dropped 25 $\times$ and severity dropped 7 $\times$ , proving models modulate content based on truth value, not just format.
Ablation: Removing the neutrosophic framing (S5) still elicited the same epistemic themes, proving the content pre-exists the prompt structure; S4 merely quantifies and standardizes it.

4. Key Contributions

Cross-Vendor Confirmation: Established that "hyper-truth" is a robust, cross-vendor phenomenon in modern LLMs, extending previous findings beyond OpenAI.
Identification of the Absorption Problem: Demonstrated that scalar T/I/F representations have a "ceiling" where they collapse distinct epistemic states (paradox vs. ignorance) into identical vectors.
Tensor-Structured Output: Proposed and validated a new output format (Scalars + Declared Losses) that recovers the distinctions lost in scalar representations.
Epistemic Hierarchy: Defined a hierarchy of fidelity:
- Probabilistic: Collapses all uncertainty.
- Scalar Neutrosophic: Expresses conflict/hyper-truth but fails on Absorption.
- Tensor Neutrosophic: Expresses conflict, hyper-truth, and type-specific uncertainty.

5. Significance and Implications

Theoretical: The paper argues that scalar T/I/F is necessary but insufficient for representing LLM epistemic states. The "Absorption" phenomenon reveals that models possess internal distinctions that standard output formats cannot express.
Practical (AI Evaluation): Evaluation systems relying solely on scalar scores will lose critical information. The authors propose that loss declarations should be the primary metric for routing tasks, as they provide richer information about what a model cannot do and why.
Future Architecture: Suggests a shift toward "Tensor-Structured" outputs in LLM interfaces to capture the full spectrum of model uncertainty, moving beyond simple probability or scalar confidence scores.

In conclusion, the paper demonstrates that while LLMs can express complex uncertainty, the current scalar interface is a bottleneck. By augmenting scalars with structured, self-reported "losses," we can recover the nuanced epistemic distinctions that define the difference between a logical paradox and simple ignorance.