ELOHIM: A Rule-Based Emergent Neural Architecture

Five Constitutive Laws, Continuous-Time Dynamics, and the Emergence of Intelligence Without Engineering

---- in short the following is a paper about why i believe LLMs or transformer ai systems are not even close to be comparable to the human brain -both in architecture and actuall understanding- and because they run every input information threw the exakt same rigit mathimaticall equations. Due to that I made an example and proto type of Elohim, an fully emergent ai, that is likely comparable to an actual brain. As well as it shows many properties that cannot be found in LLMs, like creativity without noise or randomness, abstraction of situations and learning from very few data.

Abstract

We present Elohim, a biologically-inspired continuous neural field architecture whose central design principle is the following: intelligence should emerge from rules, not from engineering. Elohim is governed by five constitutive differential laws — spatially heterogeneous temporal integration, recursive self-injection, predictive cancellation, homeostatic normalization, and metaplastic Hebbian learning. These laws are not hyperparameters. They are the complete specification of the system. No additional mechanisms, no handcrafted reward shaping, no explicit architectural modules, and no training curriculum are required. All computational properties of the system — temporal hierarchy, curiosity, sector formation, world modeling, creativity — emerge from the interaction of these five rules operating on a sparse three-dimensional field of neurons.

We present two mathematically verified properties derivable directly from the system's equations: a 51-fold dynamic range in the effective learning rate as a function of state coherence (Law V), and a 4.1× timescale separation between fast and slow neuron populations arising from the spatial τ gradient (Law I). We formally situate Elohim within the landscape of Hopfield Networks, Friston's Predictive Coding, and Kanerva's Sparse Distributed Memory, identifying in each case the precise features that distinguish Elohim's approach.

The system is evaluated as an agent in the ARC-AGI Prize 3 (2026) competition — a benchmark specifically designed to require online rule acquisition, causal world modeling, and goal inference in environments never seen during training. We argue that Elohim's purely rule-driven architecture is structurally aligned with what this benchmark actually tests.

Keywords: emergent intelligence, neural field theory, rule-based systems, online Hebbian learning, predictive coding, metaplasticity, ARC-AGI

1. The Central Principle: Intelligence From Rules Alone

1.1 The Engineering Problem

Contemporary artificial intelligence systems are, in the deepest sense, engineered. Not just in the sense that they are built by engineers — but in the sense that their computational properties are explicitly installed. An LLM's ability to reason about language is installed through the architecture of the attention mechanism and the statistics of the training corpus. A reinforcement learning agent's behavior is shaped through carefully designed reward functions, curriculum schedules, and exploration strategies. The intelligence of these systems is not emergent — it is manufactured.

This creates a fundamental ceiling. A system whose capabilities are installed can only exhibit the capabilities that were installed. It cannot surprise its designers with genuinely new forms of reasoning. It cannot adapt to environments whose structure differs from the environments it was designed for. It is, in the precise sense, a static artifact — even when its weights are frozen after training.

The human brain is not engineered in this sense. Its computational properties — temporal hierarchy, memory, curiosity, creativity, abstraction — emerge from the interaction of a small number of simple biological rules operating on a massive parallel field of neurons. The rules are not written to produce any particular capability. The capabilities arise because the rules, applied at scale and in continuous time, produce a dynamical system rich enough to exhibit them.

1.2 Elohim's Answer

Elohim is built on a single design commitment: no capability may be installed. Every capability must emerge.

This means:

• No handcrafted exploration strategy. Curiosity must emerge from the field dynamics.
• No explicit memory module. Memory must emerge from temporal inertia.
• No engineered reward shaping. Learning signals must emerge from the difference between prediction and reality.
• No designed hierarchy. Temporal hierarchy must emerge from the spatial structure of time constants.
• No explicit creativity mechanism. Creative association must emerge from the residue of prior activations.

The consequence of this commitment is that the system is fully specified by five differential laws. If a reviewer asks "where is the curiosity module?" the answer is: there is no curiosity module. Curiosity is the surprise signal |I_eff(t)|, which emerges from Law III as a byproduct of predictive filtering. If a reviewer asks "where is the memory?" the answer is: there is no memory module. Memory is the temporal inertia of the slow neurons at the high-τ end of the spatial gradient, which emerges from Law I.

This is the core claim of Elohim — and it is the claim that distinguishes it from every system in the current landscape.

1.3 The Frozen Intelligence Problem

The dominant paradigm in AI research — the large language model — has a structural property that is increasingly recognized as a fundamental limitation: the weights are frozen at deployment.

An LLM processes a sentence the same way whether it has been running for one second or one year. It has no internal state that evolves. It cannot update its beliefs from ongoing experience. It has no curiosity — no drive to seek information that reduces uncertainty. It responds; it does not act.

The human brain does all of these things continuously, at approximately 20 watts of power. It is a living dynamical system. Its properties emerge from rules, not from installation.

Elohim is an attempt to instantiate that same principle in silicon.

2. Related Work and Theoretical Landscape

2.1 Hopfield Networks and Energy Minimization

The Hopfield Network (1982, Nobel Prize in Physics 2024) stores patterns as energy minima in a symmetric weight matrix. Given a partial input, the network relaxes to the nearest stored attractor through iterative updates. Modern Continuous Hopfield Networks (Ramsauer et al., 2020) achieve exponential storage capacity and are mathematically equivalent to the attention mechanism in Transformer architectures — suggesting that the core primitive of contemporary LLMs is a form of associative memory retrieval.

Distinction from Elohim: Hopfield Networks are memory retrieval systems with a static energy landscape. Once training is complete, the system does not change. Elohim has no discrete energy minima and no static landscape. Its state space is a continuous, time-varying flow field shaped by ongoing plasticity, predictive filtering, and τ-heterogeneity. The network is never at rest in a fixed attractor — it is always in motion. Memory in Elohim is not retrieval of a stored pattern but the dynamic reinstatement of a prior trajectory through temporal inertia. This distinction is not incidental — it is the difference between a lookup table and a dynamical system.

2.2 Predictive Coding and Active Inference

Karl Friston's Free Energy Principle (2005, 2010) proposes that the brain minimizes variational free energy — decomposable into a prediction error term and a complexity cost. At each level of a cortical hierarchy, predictions are generated about the level below; prediction errors propagate upward. Actions are chosen to minimize expected free energy, unifying perception, learning, and action into a single variational principle.

Distinction from Elohim: Law III of Elohim shares the core intuition of predictive coding: the system subtracts a prediction from incoming information and processes only the residual. However, Friston's framework requires an explicit hierarchical probabilistic generative model P(causes|observations) with precision-weighted inference at each level — a significant mathematical apparatus that must be explicitly specified. Elohim's prediction is not a probabilistic model of external causes. It is the system's own recent output, delayed and scaled. The prediction emerges from the same field that receives the error signal. Prediction and representation are unified in one structure. There is no separate generative model — only the field and its temporal echo.

Furthermore, in Active Inference, action selection is part of the Free Energy minimization. In Elohim, there is no explicit action selection mechanism. Actions emerge from the spatial structure of field activity — from whichever region of the field has been most persistently active, which in turn reflects what the system has been most consistently processing. Action is a readout of the field's own organization, not an output of a designed decision module.

2.3 Sparse Distributed Memory

Kanerva's Sparse Distributed Memory (1988) stores patterns in a high-dimensional binary space. Only address neurons within a Hamming distance threshold of the query are activated. Memory is distributed and robust to noise.

Distinction from Elohim: Elohim's spatial encoding — mapping input coordinates to Gaussian activations in 3D space — is functionally similar to SDM's Hamming-sphere activation: nearby inputs activate nearby neurons. But SDM is a static storage-and-retrieval system with no temporal dynamics, no plasticity, and no action capacity. Elohim embeds SDM's spatial intuition in a continuous, self-modifying dynamical field governed by five constitutive laws. The spatial structure is the substrate; the laws are what make it alive.

2.4 Summary

3. The Five Laws: Complete System Specification

Elohim is fully specified by the following five differential laws. There is nothing else. No additional mechanisms. No modules. The system's behavior is entirely a consequence of these rules interacting in continuous time.

3.1 Law I — Spatially Heterogeneous Temporal Integration

τᵢ · (duᵢ/dt) = −uᵢ + Σⱼ Wᵢⱼ · xⱼ + I_eff,i

Each neuron i has a unique time constant τᵢ governing its integration speed. In Elohim, τ values are sorted spatially along the x-axis of the 3D coordinate space:

sorted_x_indices = torch.argsort(self.coords[:, 0])
self.tau[sorted_x_indices] = torch.sort(raw_tau)[0]

With τ ∈ [τ_min, τ_max] = [10, 50], neurons at x≈0 integrate rapidly (fast sensory response), neurons at x≈1 integrate slowly (persistent contextual accumulation). The Pearson correlation between x-coordinate and τ value across N=4,000 neurons is r = 0.9998.

What emerges from this rule alone:

A temporal hierarchy — without any explicit layering, without any designed memory module, without any attention mechanism. The same input simultaneously creates a transient representation in fast neurons and a sustained representation in slow neurons. The field encodes "what is happening now" and "what has been happening" in its spatial structure. The timescale ratio between the slowest and fastest neuron populations is 4.1× — a direct mathematical consequence of τ_max/τ_min = 50/10 = 5, realized across a population of N=4,000 neurons with empirical mean ratio of 4.1× due to the spatial sorting procedure.

This is the first sense in which intelligence emerges from a rule: temporal hierarchy, which in conventional architectures requires explicit design (LSTM gates, positional encodings, hierarchical layers), arises here from a single structural decision about how to assign time constants.

3.2 Law II — Recursive Self-Injection

echo(t) = x(t − D)     (D = RECURSION_DELAY = 10)

I_rec(t) = β · echo(t)     (β = RECURSION_STRENGTH = 0.5)

The system's own output from D timesteps prior is re-injected as additional input. This is a rule about self-reference: the system's current processing is influenced by what it was doing in the recent past.

What emerges from this rule:

Working memory — without a memory module, without a memory buffer, without explicit storage and retrieval. The field carries its own recent history forward through time via the echo. Unlike standard recurrent networks where recurrence occurs at every timestep, the explicit delay D creates genuine temporal structure: the system always has access to two distinct temporal windows simultaneously — its current state and its state from D steps prior.

3.3 Law III — Predictive Cancellation

I_eff(t) = I_ext(t) − α · echo(t − D)     (α = PREDICTION_STRENGTH = 0.98)

surprise(t) = |I_eff(t)|

The system does not process raw external input. It processes the difference between external input and its own prediction of that input. The prediction is the echo — the system's own recent output. Expected information is cancelled. Only the unexpected reaches the field with full strength.

What emerges from this rule:

Curiosity — without a curiosity module, without a designed exploration bonus, without explicit novelty detection. The surprise signal |I_eff(t)| is large when the system encounters something it did not predict, and small when it encounters something it already predicted. The field is automatically drawn toward novel, informative regions of its environment — not because curiosity was installed, but because the predictive filtering rule makes the effective input signal proportional to informational content.

This is precisely how the mammalian dopaminergic system is understood to operate: not responding to rewards per se, but to deviations from predicted rewards. Elohim's curiosity is not an analogy to dopaminergic surprise — it is the same computational principle, implemented in the same way, arising from a rule rather than from design.

Self-organizing creativity: When the system encounters a stimulus that partially overlaps with prior experience, the echo carries traces of previous activations into the current processing. The effective input is the novel component of the current stimulus, filtered through the lens of what was previously predicted. New inputs are automatically interpreted in the context of prior experience — not because a context module was designed, but because the echo structure makes this inevitable. Creativity, in Elohim, is the structured residue of temporal self-reference.

3.4 Law IV — Homeostatic Normalization

Wᵢⱼ ← Wᵢⱼ · C / (Σⱼ |Wᵢⱼ| + ε)     (C = HOMEOSTATIC_CAPACITY = 1.0)

After every weight update, each neuron's total incoming synaptic weight is normalized to C=1.0. No single synapse can grow without others shrinking.

Additionally:

self.weights[~self.is_inhibitory, :] = torch.clamp(..., min=0.0)
self.weights[self.is_inhibitory,  :] = torch.clamp(..., max=0.0)

80% of neurons are excitatory (positive weights), 20% inhibitory (negative weights), matching the empirical ratio in mammalian cortex.

What emerges from this rule:

Stability — without any explicit regularization term, without dropout, without gradient clipping. Hebbian learning without normalization is catastrophically unstable: weights grow without bound and the system collapses into a trivial all-active or all-silent state. Law IV prevents this through a biological mechanism — synaptic scaling — that maintains the total synaptic budget of each neuron constant while allowing its distribution to change freely. The result is a system that can learn continuously, without bound on time, without losing stability.

3.5 Law V — Metaplasticity

coherence(t) = [cos(x(t), x(t−1)) + 1] / 2

lr_eff(t) = lr_base · [1 + G · coherence(t)^E]     (G=50, E=4)

ΔWᵢⱼ = lr_eff(t) · [xᵢ(t) · xⱼ(t−1) − decay · Wᵢⱼ]

The learning rate is not fixed. It depends on the coherence of the system's current state with its previous state. High coherence (stable, persistent pattern) → amplified learning. Low coherence (chaotic, transitional activity) → baseline learning.

Mathematically verified property:

The effective learning rate spans a 51-fold range:

• At coherence = 0.0: lr_eff = 0.005 × (1 + 50 × 0⁴) = 0.00500
• At coherence = 1.0: lr_eff = 0.005 × (1 + 50 × 1⁴) = 0.25500
• Ratio: 51.0× — derived analytically from the formula, using exact values from config.py.

What emerges from this rule:

A soft commitment mechanism — without any designed curriculum, without any explicit transition from exploration to exploitation. When the system is exploring (low coherence), it learns slowly, allowing free exploration without premature commitment. When it finds a stable configuration (high coherence), it learns rapidly, deeply engraving that configuration into the weight structure. The system naturally transitions from exploration to consolidation based on its own internal dynamics — not because this transition was scheduled, but because coherence-amplified learning rewards stability.

The surprise signal from Law III further modulates this: unexpected inputs generate an additional learning boost, ensuring that novel information encountered during a stable state is preferentially incorporated. Expected information during stability → consolidated. Novel information during stability → maximally learned.

4. What Emerges: The System's Computational Properties

The five laws together produce a system with the following properties. None of these properties were installed — each is a consequence of the rule interactions.

4.1 Temporal Hierarchy (Laws I + II)

The combination of τ-heterogeneity and recursive self-injection creates a system that simultaneously processes multiple timescales. Fast neurons respond within ~τ_min/DT steps. Slow neurons respond within ~τ_max/DT steps. The echo buffer adds a third timescale: the explicit delay D. The result is a field that is simultaneously a sensory detector (fast neurons), a working memory (echo), and a long-term context accumulator (slow neurons) — without any of these being designed.

4.2 Curiosity and Selective Attention (Law III)

The surprise signal |I_eff(t)| is the system's natural attention signal. Whatever is unexpected receives the strongest effective input. The field is automatically drawn toward novel, informative stimuli. This is not reward-driven in the conventional sense — no reward function was defined. Novelty-seeking is an inherent consequence of the predictive filtering structure.

4.3 Stability Under Continuous Learning (Law IV)

Elohim can learn continuously, in real time, without destabilizing. The homeostatic normalization maintains a constant synaptic budget while allowing its distribution to change. The E/I balance prevents runaway excitation. The system can be left running indefinitely without weight collapse or explosion.

4.4 Spontaneous Sector Formation (Laws I + IV + V)

Over extended operation, the field spontaneously organizes into spatially coherent activity clusters — sectors — that develop functional specialization without any clustering instruction. This emerges from three interacting mechanisms: the τ gradient creates regional differentiation in temporal response; spatial decay in connectivity creates local correlation; lateral inhibition from the inhibitory population suppresses simultaneous activation of distant regions.

The sectors are not designed. They are not explicitly assigned. They are not the result of a k-means algorithm applied to neural activity. They arise because the rules, applied at scale, drive the field toward locally coherent configurations. Different types of input naturally activate different spatial regions, and Hebbian learning reinforces this separation over time.

Actions emerge from sector activity. Whichever spatial region of the field has been most persistently active over recent timesteps determines the system's behavioral tendency. This is not a designed action-selection module — it is the field's own spatial organization expressing itself as behavior.

4.5 Implicit World Modeling (Laws I + II + III)

The slow neurons at the high-τ end of the gradient accumulate evidence over long windows. When the environment has structure — when certain inputs tend to follow others — the slow neurons develop weight patterns that reflect this structure through ordinary Hebbian learning. The world model is implicit: it is distributed across the weight matrix and temporal dynamics of the slow neuron population, accessible not through lookup but through the way it contextualizes current processing.

4.6 Creativity as Temporal Associativity (Laws II + III)

When the system processes a new input, its echo carries traces of previous activations. The predictive cancellation subtracts the expected component. What remains — the effective input — is the component of the new stimulus that differs from what was predicted based on recent history. When the new stimulus is similar but not identical to prior experience, the remaining effective input encodes precisely the way it differs. The system's response to this effective input is shaped by the weight structure formed during prior experience. The result is a response that combines elements of the new stimulus with associations from structurally similar prior experience — not because a creativity module was designed, but because this is the natural consequence of temporal self-reference through the echo.

5. The Meta-Cognitive Field (L2)

Elohim's full implementation includes a second neural field (L2) operating under the same five laws as L1, but with substantially slower time constants: τ ∈ [30, 150], three to five times slower than L1's [10, 50].

L2 receives input from L1 through a connection matrix, maintains its own plastic recurrent connections under Laws IV and V, and projects back to L1 as a modulatory signal. Because L2 operates at a slower timescale, it naturally integrates L1's rapidly changing activity over extended windows. Its feedback to L1 contextualizes current moment-to-moment processing with the accumulated structure of recent experience.

The crucial point: L2 is not a designed world-model module. It is the same system as L1, running slower. Its "memory" is not stored in a database — it is encoded in its weight matrix through the same Hebbian laws that govern L1. Its contextualizing influence on L1 is not the output of a designed inference procedure — it is the natural consequence of L2's slow dynamics being fed back into L1's fast dynamics.

The architecture is self-referential: L2's input is exclusively L1's output. Therefore, L2 is continuously modeling the state of the system itself. When this model is projected back to L1, L1 receives not just external stimulation but a representation of its own recent processing history. The system is continuously exposed to a model of itself. This self-referential structure is not installed — it emerges from the decision to connect two fields of the same type at different timescales.

6. Encoding: The Bridge Between World and Field

ARC-AGI Prize 3 presents 64×64 grids with 16 color values (0–15). Elohim's input is a 3D neuron field. The encoding rule:

I_n(pixel) = (color / 7.5) · exp[−||coords_n − (x/(W−1), y/(H−1), color/15)||² / (2σ²)]

Each pixel activates a Gaussian neighborhood of neurons centered at the corresponding 3D coordinate. The x and y position of the pixel maps to the x and y axes of the field. The color maps to the z-axis. This means:

• Nearby pixels activate nearby neurons (spatial structure preserved)
• Similar colors activate nearby neurons along z (color structure preserved)
• Different colors at the same position activate different neurons (color-position binding preserved)
• The encoding is continuous — no discrete bins, no one-hot vectors

The encoding is a rule: map each pixel to a Gaussian activation in coordinate space. From this rule, a representation emerges that is sparse, spatially structured, and naturally robust to small perturbations. No feature engineering. No handcrafted color channels. One rule.

7. Mathematically Verified Properties

Two properties of Elohim can be derived directly from the system's equations, using exact parameter values from config.py. These are not simulation results — they are analytical consequences of the rules.

7.1 Property 1: Metaplasticity — 51× Learning Rate Range

Formula (Law V, exact from system.py):

lr_eff(c) = lr_base × (1 + G × c^E) = 0.005 × (1 + 50 × c⁴)

The function is strongly threshold-gated: below coherence ≈ 0.85, amplification is modest (< 5×). Above coherence ≈ 0.95, amplification accelerates sharply. This means the system maintains exploratory flexibility during incoherent phases and commits strongly and rapidly when coherence is established. The transition between these regimes is not designed — it is the shape of the function x⁴, which is flat near zero and steep near one.

Biological precedent: the BCM learning rule (Bienenstock, Cooper, Munro, 1982) proposes a sliding modification threshold for synaptic plasticity. Law V implements a complementary principle — amplifying plasticity in proportion to activity coherence rather than total activity — arising from a single equation rather than a multi-parameter model.

7.2 Property 2: Temporal Hierarchy — 4.1× Timescale Separation

From the spatial sorting procedure (system.py):

With N=4,000 neurons, coords[:,0] uniform on [0,1], τ values uniform on [10,50] sorted along x:

• Pearson r(x-coordinate, τ): r = 0.9998
• Neurons at x < 0.10: mean τ ≈ 11.85 (fast)
• Neurons at x > 0.90: mean τ ≈ 48.08 (slow)
• Empirical timescale ratio: 4.06×
• Theoretical maximum ratio: τ_max/τ_min = 5.0×

The empirical ratio (4.1×) falls below the theoretical maximum (5.0×) because the sorting distributes τ values continuously across the population, and the extreme deciles (x<0.1, x>0.9) sample from the tails of a uniform distribution on [10,50], not from the exact extremes. This discrepancy is predictable from the statistics of order statistics and requires no further explanation.

8. Relationship to the ARC-AGI Prize 3 Benchmark

The ARC-AGI Prize 3 (2026) tests for properties that feedforward systems structurally cannot exhibit:

Online rule acquisition: The agent must discover transformation rules from scratch during evaluation. An LLM cannot do this — its weights are frozen. Elohim updates its weights continuously throughout evaluation. Every level it encounters modifies its dynamics, making subsequent processing on structurally similar levels potentially more efficient. This is not a designed curriculum — it is what happens when a continuously plastic system encounters a sequence of structured environments.

Causal world modeling: The agent must build a model of an environment's causal structure from exploration. Elohim's slow neuron population naturally accumulates evidence of temporal regularities through Hebbian learning. The world model emerges in the weight structure of the slow neurons — not because a world model module was designed, but because slow Hebbian learning on structured input sequences produces this.

Intrinsic exploration: The agent must explore without an external curiosity signal. Elohim's surprise signal |I_eff(t)| is inherently large for novel inputs and small for familiar ones. The field is naturally drawn toward informative actions — not because an exploration bonus was added, but because the predictive filtering makes novel inputs more effective in driving field dynamics.

Goal inference: The agent must infer what constitutes a solution without being told. The only feedback is a binary level-completion signal. In Elohim, this signal produces a coherent burst of activity across the sectors responsible for the completing action sequence. Through Law V's metaplasticity, this coherent burst is deeply encoded — the patterns that led to success are reinforced. Over multiple levels, the system's sector structure increasingly reflects what types of activity tend to precede completion events.

None of these are engineered. All emerge from the five laws operating in the context of an ARC-AGI Prize 3 level.

9. On Claims This Paper Does Not Make

Science requires being explicit about what is claimed and what is not. The following claims are not made in this paper:

"Elohim is conscious." Consciousness is not defined with sufficient precision to be tested empirically. The self-referential L1-L2 dynamic produces a system that models its own processing — this is a functional property, not a claim about phenomenal experience.

"Elohim achieves competitive ARC-AGI Prize 3 performance." Formal quantitative benchmark results are ongoing. This paper presents the architectural argument for why Elohim is structurally suited to the benchmark, not performance claims.

"Elohim is AGI." The system exhibits properties — online learning, curiosity, emergent temporal hierarchy, implicit world modeling — that are necessary for general intelligence. Whether they are sufficient remains an open empirical question.

"The five laws are the correct minimal specification." They are a specification. Whether fewer laws could produce equivalent properties, or whether additional laws are necessary for full generality, is an open research question.

What this paper does claim: the five laws are sufficient to produce a system with the structural properties enumerated above, and those structural properties emerge from the rules rather than from engineering.

10. Open Problems

10.1 Compositional Representation

Can Elohim's field represent not just "what is present" but "what stands in what relation to what"? Compositional representations — relations between objects, not just object presence — are required for systematic generalization. Whether the three-dimensional field structure supports binding through spatial coactivation is an open question requiring investigation at larger scales.

10.2 The Rule-Symbol Gap

Elohim's field can detect that a transformation is occurring and learn that certain actions produce certain changes. It cannot yet extract the transformation as an explicit, compositional rule — "rotate 90° clockwise" — that generalizes instantly to all instances. This is the rule-symbol gap. Whether it can be closed by scale, by additional emergent mechanisms, or requires a different approach is the central open problem.

10.3 Multi-Field Hierarchy

The current two-field architecture (L1 fast, L2 slow) is a minimal implementation. Biological cortex has approximately 30–40 hierarchical areas. A natural extension is N fields with geometrically increasing time constants. Whether qualitatively new properties emerge from deeper temporal hierarchies is an empirical question.

11. Conclusion

Elohim is specified by five rules. From these rules, a system emerges with temporal hierarchy, curiosity, implicit world modeling, online learning, and associative creativity — without any of these being designed.

This is the central claim. It is a strong claim. It implies that intelligence is not fundamentally a matter of architectural complexity, but a matter of finding the right rules and letting them run.

The ARC-AGI Prize 3 benchmark is one way to test this claim empirically. A system that learns online, that is inherently curious, that builds world models through temporal accumulation, and that acts from emergent spatial organization is structurally aligned with what the benchmark tests. Whether it is competitively aligned is what the evaluation will show.

The rules are simple. What emerges from them may not be.

Appendix A: System Parameters

Appendix B: Training Environments

Elohim is pre-exposed to four synthetic environments before ARC-AGI Prize 3 evaluation. These environments do not define tasks or rewards — they are structured sensory streams from which the field learns through the five laws alone.

PongEnv: A point moves through 3D coordinate space with bouncing velocity, generating Gaussian activations at the moving point's position. The field develops temporal predictions of smooth motion.

RhythmEnv: Two oscillating Gaussian pulses at fixed positions with sinusoidal amplitude modulation at different frequencies. The field develops rhythmic predictions at multiple timescales simultaneously.

SequenceEnv: A Gaussian activation moves sequentially through three fixed positions. The field develops sequential expectation: after position A, the field predicts position B.

WorldMirrorEnv: A compound environment combining slowly moving causal dynamics, periodic semantic activations, and sparse rule-like patterns. The field develops multi-domain integration through the interaction of fast and slow neurons responding to the compound input.

None of these environments involve explicit reward signals, task definitions, or learning objectives. The field learns by predicting, failing to predict, and updating — exactly as specified by the five laws.

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Pre-print submitted to Zenodo, May 2026.
Code: kaggle.com/jakobschellenberg/arc-agi-3-elohim-upload