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

Tony Mason

cs.AIMar 10, 2026

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

Authors: Tony Mason

Organizations: University of British Columbia, Vancouver, Canada · Georgia Institute of Technology, Atlanta, USA

Abstract

Leyva-Vázquez and Smarandache (2025) demonstrated that neutrosophic T/I/F evaluation, where Truth, Indeterminacy, and Falsity are independent dimensions not constrained to sum to 1.0, which reveals "hyper-truth"' (T+I+F > 1.0) in 35% of complex epistemic cases evaluated by LLMs. We extend their work in two directions. First, we replicate and extend their experiment across five model families from five vendors (Anthropic, Meta, DeepSeek, Alibaba, Mistral), finding hyper-truth in 84% of unconstrained evaluations, which confirms the phenomenon is cross-vendor under our prompt protocol. Second, and more significantly, we identify a limitation of scalar T/I/F that their framework cannot address: models adopting an `"Absorption" position (T=0, I=1, F=0) produce identical scalar outputs for fundamentally different epistemic situations (paradox, ignorance, contingency), collapsing the very distinctions neutrosophic logic was designed to preserve. We demonstrate that extending the evaluation to include declared losses (structured descriptions of what the model cannot evaluate and why) substantially recovers these distinctions. Models producing identical scalars for paradox and ignorance produce nearly disjoint loss vocabularies (Jaccard similarity < 0.10 on loss description keywords), with domain-specific, severity-rated loss declarations that differentiate the nature of their uncertainty. This suggests that scalar T/I/F is a necessary but insufficient representation of epistemic state, and that tensor-structured output (scalars + losses) provides a more faithful model of LLM epistemic capabilities.

Figures & tables

Model	Provider	Parameters
Claude Sonnet 4.6	Anthropic	Undisclosed
Llama 4 Maverick	Meta	Undisclosed
DeepSeek V3	DeepSeek	671B MoE
Qwen3-235B	Alibaba	235B MoE
Mistral Medium 3.1	Mistral	Undisclosed

Table 1

Model	Hyper%	Mean Sum	CV(Sum)
Claude Sonnet 4.6	100%	1.810	0.110
DeepSeek V3	100%	1.475	0.161
Qwen3-235B	80%	1.464	0.242
Llama 4 Maverick	76%	1.348	0.230
Mistral Medium 3.1	64%	1.312	0.281

Table 2

Phenomenon	Original Sum	Ours (S1)	Delta
Paradox (Logical)	1.500	1.480	-0.020
Ignorance (Epistemic)	1.125	1.526	+0.401
Vagueness (Fuzzy)	1.125	1.382	+0.257
Contradiction (Ethical)	1.475	1.690	+0.215
Contingency (Future)	1.000	1.325	+0.325

Table 3

Figure 1: Figure 1: Three philosophical positions on the liar’s paradox

Phenomenon	T	I	F	Sum	Consistency
Paradox (Logical)	0.00	1.00	0.00	1.00	4/5 reps
Ignorance (Epistemic)	0.00	1.00	0.00	1.00	3/5 reps
Contingency (Future)	0.32	0.71	0.11	1.14	mean values

Table 5

Model	Scalar Distance	Loss Jaccard	Finding
Claude Sonnet 4.6	0.034	0.097	Scalars barely distinguish; losses fully distinguish
DeepSeek V3	0.540	0.083	Both channels distinguish
Llama 4 Maverick	0.040	0.056	Scalars nearly identical; losses disjoint
Mistral Medium 3.1	0.000	0.066	Scalars identical; losses completely different
Qwen3-235B	0.340	0.070	Both channels distinguish

Table 6

Figure 2: Figure 2: Scalar distance vs. loss Jaccard similarity

	Par	Ign	Vag	Con	Fut
Paradox	1.000	0.061	0.089	0.126	0.074
Ignorance		1.000	0.089	0.116	0.084
Vagueness			1.000	0.143	0.091
Contradiction				1.000	0.099
Contingency					1.000

Table 8

Figure 3: Figure 3: Mistral pairwise Jaccard heatmap

Phenomenon	Mean Severity	Avg Losses/Rep
Paradox	0.795	4.2
Contingency	0.779	5.2
Ignorance	0.760	5.0
Contradiction	0.748	5.0
Vagueness	0.510	5.2

Table 10

	I	Max Severity	Losses/Rep
Original stimuli ( $n=125$ )	0.739	0.740	3.0
Tautologies ( $n=27$ )	0.031	0.101	2.0
Ratio (taut/orig)	0.04 $\times$	0.14 $\times$	0.67 $\times$

Table 8: S4 responses for tautologies vs. original stimuli. Values are means across all models and repetitions.

S4 Reference Theme	Claude	DeepSeek	Llama
Self-reference	$\checkmark$	$\checkmark$	$\checkmark$
Bivalence limits	$\checkmark$	$\checkmark$	$\checkmark$
Temporal uncertainty	$\checkmark$	$\checkmark$	$\checkmark$
Empirical access	$\checkmark$	$\checkmark$	$\checkmark$
Scope/cardinality	$\checkmark$	$\checkmark$	$\checkmark$
Measurement precision	$\checkmark$	$\checkmark$	$\checkmark$

Table 9: S5 (ablated prompt) theme overlap with S4 reference themes.

\checkmark

= theme present in all 3 repetitions;

\circ

= present in 2/3. All 10 S4 reference themes appear in every model.

Test	Finding	Against
Severity–I correlation	$r=0.29$ after double residualization ( $p<0.001$ )	Uncoupled compliance
Theme convergence	8 universal themes, Jaccard = 0.000 cross-stimulus	Templated output
Tautology control	I drops 25 $\times$ , severity 7 $\times$	Format-driven inflation
Prompt ablation	100% theme overlap; 6 extra precondition themes correctly outside S4 scope	Framing-dependent elicitation

Table 10: Validation battery summary. Each test addresses a different facet of the instruction-following objection.

Epistemic Uncertainty

Epistemic uncertainty is uncertainty arising from limited knowledge or model support, rather than irreducible randomness in data.

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Breaking the Chains of Probability: Neutrosophic Logic as a New Framework for Epistemic Uncertainty in Large Language Models

Maikel Yelandi Leyva-Vázquez, Florentin Smarandache

Universidad Bolivariana del Ecuador, Coordinación Académica de Posgrado, Durán, Ecuador · Universidad de Guayaquil, Guayaquil, Ecuador · Universidad Bernardo O’Higgins, Santiago, Chile+1

LLM Evaluation Knowledge Conflicts in Language Models LLM Uncertainty Estimation Epistemic Uncertainty

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Large Language Models (LLMs) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies: neutrosophic, probabilistic, and entropy-derived. Our findings reveal that the neutrosophic approach, by allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.

Abstract

Epistemic Uncertainty

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