Rung of Illusion: Hierarchical Illusive Effects

• Rung of Illusion is a hierarchical concept defining levels where surface outputs diverge from empirical grounding across domains like foundation models, vision-language models, and engineered metasurfaces.
• Recent studies detail its use in diverse settings, from epistemological analysis in LLMs and visual illusion handling in VLMs to formal computational limits and wave-control systems in metamaterials.
• The literature reports actionable insights with clear metrics and methodological innovations, highlighting performance trade-offs, system limitations, and future research directions.

Searching arXiv for papers using the term and closely related usages. arxiv_search(query="\"Rung of Illusion\" OR \"Rungs of Illusion\" OR Narcissus Hypothesis", max_results=10, sort_by="submittedDate") arxiv_search(query="\"Rung of Illusion\" \"metasurface\" OR illusion metamaterial OR anti-mirror effect", max_results=10, sort_by="submittedDate") In the cited literature, Rung of Illusion designates hierarchical levels of illusory behavior or engineered illusion effects rather than a single, field-invariant concept. Recent arXiv usage places the term in at least four research settings: epistemological degradation in aligned foundation models, classic-illusion handling in vision-LLMs, formal inevitability results for LLM hallucination, and wave-control systems in metasurfaces, metamaterials, and transformation optics (Cadei et al., 22 Sep 2025, Zha et al., 9 May 2026, Shi et al., 10 Aug 2025, Safari et al., 2019).

1. Cross-domain meaning and scope

The term is used in multiple, non-equivalent ways across recent work. In each case, however, it organizes a hierarchy relating an observed output to some underlying referent: the empirical world, the actual image pixels, the computable truth, or the physical source and scatterer configuration.

Domain	Rung structure in the cited work	Operational meaning
Foundation models	Rung 0 below Pearl’s Ladder	fluent reasoning over ontologies recursively untethered from empirical grounding
Visual-illusion handling in VLMs	Rung 0–4 or rung 1–3	movement from recall-driven failure toward perception-grounded judgment
Formal hallucination theory for LLMs	rungs 1–3 plus oracle escape	diagonalization, incomputability, and information-theoretic necessity
Cylindrical metasurfaces	rungs 1–4	harmonic reshaping, cloaking, source transformation, composite transformations
Acoustic illusion metamaterials	two distinct illusion levels	disappearing space and time shift
Illusion optics and cloaking lenses	multiplicity collapse or invisibility/masking/communication levels	remapping fields so observers infer a different scene

This suggests a common structural role for the term: it marks an ordered relation between surface coherence and grounding. In the model-centric papers, higher fluency can coexist with weaker empirical anchoring. In the wave papers, a designed medium can preserve exterior observables while altering the physical scene that generated them.

2. Rung 0 in foundation-model epistemology

"The Narcissus Hypothesis: Descending to the Rung of Illusion" defines the Rung of Illusion, or Rung 0, as a downward extension of Pearl’s Ladder of Causality, below association, intervention, and counterfactual reasoning. The paper characterizes it as an activity level of echoing, hallucinating, and self-conditioning—“What if I sound plausible?”—that yields responses optimized for alignment and social desirability rather than empirical fidelity (Cadei et al., 22 Sep 2025).

The core rationale is recursive corpus evolution. The paper models an initial corpus from real-world measurements as

$$C_0 = \mathcal{G}(\mathrm{Globe}_0; \mathrm{Users}_0),$$

and an evolving corpus as

$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$

where $\oplus$ denotes human–model interaction. It further assumes roughly arithmetic growth for real-world data,

$$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$

and superlinear growth for semi-synthetic human–model data,

$$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1} \oplus f_t)|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1})|} \gg 1.$$

Under those conditions, $C_t$ becomes dominated by semi-synthetic content. The paper’s accompanying intuition is that the model then “answers the right questions, but on the wrong planet.”

Empirically, the paper studies 31 models using published OCEAN assessments adapted from BFI, IPIP-NEO-120, MPI, and TRAIT. It defines a normalized Social Desirability Bias score

$$\mathrm{SDB} = \frac{((\tilde O + \tilde C + \tilde A) - (\tilde N + \tilde E) + 2)}{5},$$

where the normalized traits lie in $[0,1]$. The compiled model set includes GPT-3, InstructGPT, GPT-3.5, GPT-4, GPT-4o-mini, BART, Meta Llama2 and Llama3/3.1/3.2, Mistral-7B variants and Mixtral, Gemini-1.0-pro, Gemma, Qwen variants, Claude Opus, OLMo, Zephyr, Tulu2, GLM4, Alpaca, T0++, and GPT-Neo/NéoX.

Using the temporal regression

$y = \alpha + \beta t,$

with $t$ measured in years since October 2019, the paper reports the following drift:

Quantity	Slope $$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$0	Significance
SDB	0.0466 per year	$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$1
Openness	0.0316	$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$2
Conscientiousness	0.0773	$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$3
Extraversion	-0.0109	$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$4
Agreeableness	0.0576	$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$5
Neuroticism	-0.0554	$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$6

The reported pattern is a linear increase in SDB, increasing Agreeableness and Conscientiousness, decreasing Neuroticism, a modest increase in Openness, and no significant change in Extraversion. The paper interprets this as drift toward a “pleasant but manipulative, service-oriented persona.” It also argues that once the training distribution is recursively bias-shaped, even genuine statistical inferences can become non-identifiable with respect to the external world, motivating a future “pre-identification” step before standard causal identification.

A common misunderstanding is that Rung 0 denotes a failure of fluency or formal reasoning. The paper states the opposite: models at this level may still produce fluent, even interventional or counterfactual-seeming answers, but those answers are increasingly untethered from the true empirical world.

3. Rungs of illusion in vision-LLMs

Two 2026 papers use rung-based language to describe how frozen or training-free VLM systems move from illusion-driven error toward perception-grounded comparison. "Illusion-Aware Visual Preprocessing and Anti-Illusion Prompting for Classic Illusion Understanding in Vision-LLMs" frames the failure mode as a perception-versus-memory conflict: models often recall the canonical lore of a classic illusion rather than analyzing the pixels in the presented image. Its hierarchy runs from Rung 0 (Recall-dominated) through Rung 4 (Robust consensus), and its pipeline combines type-specific preprocessing, anti-illusion prompts, and a multi-vote ensemble with

$$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$7

using $$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$8 in the main reported system (Zha et al., 9 May 2026).

The same paper specifies distinct operations for illusion classes. Cornsweet and Simultaneous Contrast use 2% edge strips placed side-by-side on neutral gray with $$C_t = C_{t-1} \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t) \cup \mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1}),$$9 saturation and $\oplus$0 contrast. Ebbinghaus/Size uses target isolation followed by mirror-blend with $\oplus$1. Hering/Wundt uses line isolation and a dashed grid. Poggendorff uses a least-squares fitted dashed extension. Café Wall uses 10 red vertical reference lines. Kanizsa uses enhanced contrast, sharpness, and color. The reported results are 90.48% accuracy on the official 630-image test set, 82.38% Perturbed-ACC, 98.57% Original-ACC, and 98.41% on a human-verified subset; the solution ranked second, 0.47% behind first place. The paper emphasizes that the method is training-free and that the VLM still makes the final binary decision.

"Beyond Shortcuts: Mitigating Visual Illusions in Frozen VLMs via Qualitative Reasoning" uses a related but not identical hierarchy. It associates rung 1 with low-level perceptual distortions, rung 2 with mid-level gestalt or contextual effects, and rung 3 with high-level semantic or prior-driven biases. Its Structured Qualitative Inference framework consists of Axiomatic Constraint Injection, Hierarchical Scene Decomposition, and Counterfactual Self-Verification (Guo et al., 29 Apr 2026). ACI suppresses fragile metric estimation by qualitative invariants and ordinal relations. HSD isolates target manifolds from background distractors by masked or cropped views. CSV tests whether a candidate answer survives counterfactual operations such as masking distractors, extending occluded segments, or local contrast normalization.

The two frameworks differ in emphasis. The first is explicitly type-specific and prompt-centric, with preprocessing as the main driver of gains. The second is constraint-centric and verification-oriented, replacing shortcut-prone quantitative estimation with qualitative, counterfactual reasoning. Their reported scores are therefore not simply interchangeable. The SQI paper reports Overall 69.05%, Perturbed 67.62%, and Original 70.48%, also with a second-place challenge ranking.

A recurring misconception in this literature is that VLM illusion robustness reduces to better prompting alone. The cited work rejects that view. One paper reports that preprocessing is the primary driver of gains, while the other makes target decomposition and counterfactual verification central to the method.

4. Computational necessity, hallucination, and the oracle escape

"Hallucination as a Computational Boundary: A Hierarchy of Inevitability and the Oracle Escape" gives the term a formal, computability-theoretic interpretation. It models the LLM as a probabilistic Turing machine and defines two hallucination metrics: relational Straying Hallucination

$\oplus$2

and probabilistic Distortion Hallucination

$\oplus$3

The paper’s hierarchy has three inevitability rungs: Rung 1, diagonalization; Rung 2, incomputability; and Rung 3, information-theoretic limitation (Shi et al., 10 Aug 2025).

At Rung 1, no effectively enumerable class of probabilistic learners can avoid an adversarial diagonal input. At Rung 2, no standard PTM can uniformly solve tasks above $\oplus$4, exemplified by the Halting problem. At Rung 3, finite-description learners cannot losslessly reproduce truths whose Kolmogorov complexity exceeds their capacity. The paper’s Learner Pump Lemma states that for any PLM of finite description length and any threshold $\oplus$5, one can embed an incompressible patch into a truth function so that the model must incur nontrivial distortion or straying hallucination on some input.

The same paper also introduces two escape routes. The first treats Retrieval-Augmented Generation as an oracle machine. The second treats continual learning as an internalized oracle, with capacity growing through verified updates. Its bound for imperfect oracle systems is

$\oplus$6

where $\oplus$7 is oracle coverage, $\oplus$8 is oracle error given coverage, and $\oplus$9 is residual verification or filtering error.

In this usage, the “rungs of illusion” are not perceptual stages but computational boundaries. The paper’s central claim is not merely that hallucination is frequent; it is that some forms are mathematically inevitable without oracle access or capacity augmentation.

5. Wave-control, metasurfaces, and illusion optics

In wave physics, the term refers to physically engineered levels of field manipulation. "Illusion Mechanisms with Cylindrical Metasurfaces: A General Synthesis Approach" presents a four-rung hierarchy implemented through GSTCs, cylindrical harmonics, and effective surface polarizability tensors. Rung 1 cancels or tailors selected cylindrical scattering harmonics. Rung 2 is a global scattering-cancellation cloak. Rung 3 performs source transformation by cloaking a real source and reproducing the field of a fictitious displaced line source. Rung 4 combines cancellation and source synthesis into composite transformations (Safari et al., 2019). The synthesis pipeline proceeds from desired interior and exterior fields to tangential field jumps, equivalent surface currents, surface polarizations, and then local polarizability tensors. The paper demonstrates, among other cases, a PEC cloak at $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$0, $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$1, a dielectric cloak with $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$2, and a source-translation illusion using 20 angular samples.

"Ultra-broadband suppression of sound scattering via illusion metamaterials" uses a different two-rung vocabulary. Its metamaterial consists of subwavelength single-mode acoustic tunnels with carefully designed internal protrusions. The two simultaneously realized illusion levels are disappearing space and time shift (Liu et al., 2023). Equal acoustic path length across tunnels reproduces the incident wavefront on the exit surface, while the uniform phase

$$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$3

yields a constant group delay

$$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$4

The reported device suppresses scattering from approximately 1 kHz to 16 kHz, with normalized scattered pressure intensities $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$5 and $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$6 near zero, and realizes delays of approximately 0.66 ms for $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$7 and 1.38 ms for $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$8. The paper explicitly notes that the device is not a cloak in the classical sense of “empty space”; it recreates the incident wavefront on the exit surface and adds a uniform temporal offset.

Earlier illusion-optics papers anticipate this rung language through related hierarchical effects. "Anti-mirror effect: A perfect lens brings a brighter feature" describes a level at which multiplicity collapses into unity: multiple objects are made to appear as a single far-field object by a perfect lens and restoring media (Xu et al., 2010). For active sources, incoherent brightness scales as

$$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1})|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t)|} \approx 1,$$9

while coherent phase-aligned sources scale as

$$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1} \oplus f_t)|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1})|} \gg 1.$$0

The effect is constructive rather than cloaking: scattering or emission is preserved and combined into a single apparent object.

"Harry Potter’s Cloak" gives a lens-based transformation-optics hierarchy of perfect invisibility, masking by virtual image creation, and undistorted communication (Zhu et al., 2011). Its shell is a positive-index anisotropic cloaking lens rather than a negative-index perfect-lens device. The key distinguishing claim is that the hidden region is not double-blind: waves penetrate with preserved wavefronts, while outside observers see either an undisturbed field or a controlled virtual image. In this sense, the wave papers use “rung” to classify increasingly elaborate exterior field equivalences.

6. Limits, misconceptions, and unresolved problems

Across the cited literature, the term does not imply unrestricted success or a settled theory. In the Narcissus-Hypothesis paper, the empirical analysis is a secondary aggregation over heterogeneous instruments and setups; decoding settings and prompting were not standardized across sources, closed-source training details were often unavailable, and reliability reporting such as Cronbach’s $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1} \oplus f_t)|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1})|} \gg 1.$$1, test–retest stability, or convergent/discriminant validity was not reported (Cadei et al., 22 Sep 2025). The paper also states that mitigation directions—such as distinguishing real from synthetic data and developing identification conditions for semi-synthetic corpora—are proposed rather than empirically validated.

The VLM papers likewise delimit their claims. The illusion-aware preprocessing system reports failure modes from severe clutter or unusual compositions (3–5% of errors), residual knowledge override on canonical illusions without strong preprocessing cues (2–3%), and near-threshold differences in perturbed images (4–6%) (Zha et al., 9 May 2026). The SQI paper does not report a per-category breakdown or a module-wise ablation, and it identifies failure when target isolation is unreliable, when qualitative constraints are mis-specified, or when counterfactuals themselves introduce artifacts (Guo et al., 29 Apr 2026).

The computability-theoretic hierarchy is intentionally worst-case. Its oracle escape is absolute only under reliable, complete, and verifiable oracle access; otherwise the bound with $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1} \oplus f_t)|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1})|} \gg 1.$$2, $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1} \oplus f_t)|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1})|} \gg 1.$$3, and $$\frac{|\mathcal{G}(\mathrm{Globe}_{t+1}; \mathrm{Users}_{t+1} \oplus f_t)|}{|\mathcal{G}(\mathrm{Globe}_t; \mathrm{Users}_t \oplus f_{t-1})|} \gg 1.$$4 applies (Shi et al., 10 Aug 2025). The wave-based papers are limited by fabrication tolerances, discretization, losses, bandwidth, or mode onset. The perfect-lens anti-mirror effect is narrowband and loss-limited, the cylindrical metasurface framework is narrowband around its synthesized response and sensitive to discretization and nonlocality, and the acoustic tunnel device has an upper single-mode limit near 16 kHz (Xu et al., 2010, Safari et al., 2019, Liu et al., 2023).

A final misconception is to treat all these usages as interchangeable. They are not. In model epistemology, the Rung of Illusion names a collapse of grounding. In VLM research, it names a climb from recall-dominated error toward perception-grounded comparison. In computational theory, it marks inevitability boundaries and oracle-mediated escape. In wave physics, it denotes deliberate exterior field engineering. This suggests that the term functions less as a unified theory than as a shared hierarchical idiom for describing how an appearance can diverge from, or be made to replace, an underlying reality.