# Zero-Shot AGI via Multimodal Symmetry Learning: An Extension of the SHELL-AGENT Framework with Bidirectional Algebraic Alignment

Tofara Moyo

► To cite this version:

Tofara Moyo. Zero-Shot AGI via Multimodal Symmetry Learning: An Extension of the SHELL-AGENT Framework with Bidirectional Algebraic Alignment. 2026. hal-05533566

HAL Id: hal-05533566

https://hal.science/hal-05533566v1

Preprint submitted on 2 Mar 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Zero-Shot AGI via Multimodal Symmetry Learning:
An Extension of the SHELL-AGENT Framework with
Bidirectional Algebraic Alignment

Tofara Moyo

March 2026

Abstract

We present a comprehensive training methodology for zero-shot artificial general intelligence (AGI) based on the SHELL-AGENT framework. Our approach learns symmetry generators from multimodal data—including video, language, audio, and motion capture—and factorizes human experience into a Lie algebra of independent symmetry dimensions. We introduce the T-tiling principle as the fundamental structure governing optimal transition dynamics across all scales of behavior. The resulting generator set enables zero-shot generalization to novel tasks through composition of learned symmetries.

Crucially, we extend the framework to address alignment through a novel bidirectional algebraic
refinement process. Rather than treating alignment prompts as external commands or using curated
datasets of aligned behaviors, we extract the intrinsic Lie algebra structure of the prompt text itself.
This prompt algebra acts as a structural template that constrains generator formation during multi-
modal learning. Simultaneously, the rich algebraic structure discovered from multimodal data refines
and expands the prompt algebra, creating a shared grammatical foundation for cognition and values.
The resulting agent exhibits zero-shot aligned behavior not because alignment was enforced, but because
misaligned states are structurally unreachable under its generator algebra. We provide mathematical
foundations, training procedures, and theoretical guarantees for this intrinsically aligned architecture.

# 1 Introduction

The SHELL-AGENT framework [1] demonstrates that adaptive behavior can emerge from continuous symmetry restoration dynamics on a smooth manifold. In this framework, agents learn Lie algebra generators representing regularities in their environment, and cognition emerges as epiphenomena of symmetry-seeking dynamics. However, the original formulation leaves open two critical questions for AGI: (1) How can we scale symmetry learning to multimodal human data? (2) How can we ensure alignment with human values without resorting to post-hoc patches or curated datasets?

This paper addresses both questions through a unified approach that fundamentally reimagines align-
ment. We first extend SHELL-AGENT to learn generators from massive multimodal corpora by mapping all
modalities to a shared embedding space via neural encoders, then discovering the Lie algebra of symmetries
governing multimodal flows. This yields a generator set that factorizes human experience into independent
symmetry dimensions.

The key innovation lies in how we incorporate alignment. Previous approaches treat alignment prompts as commands to be followed or use supervised learning on labeled examples of "good" behavior. We reject both approaches. Instead, we observe that natural language alignment prompts (e.g., "be helpful, honest, and harmless") possess their own intrinsic algebraic structure—the grammatical and distributional relationships between words form a small Lie algebra $g_{prompt}$. This prompt algebra encodes the relational grammar of ethical concepts.

During multimodal training, we enforce that the learned generator algebra $g_{data}$ must admit a homomorphism from $g_{prompt}$. This means the structural relationships within the prompt must be mirrored in the structure of behavior. However, the prompt algebra is necessarily impoverished—it captures only the structure present in the linguistic neighborhood of a few words. The multimodal data contains far richer
ethical structure. Therefore, we introduce bidirectional refinement: the data algebra refines and expands the prompt algebra, discovering new ethical dimensions and relations that were implicit in the prompt but not fully articulated. The two algebras co-evolve until they converge on a shared algebraic grammar that exists both in language and in behavior.

The result is an agent whose values are neither taught nor learned from examples, but rather emerge as the shared algebraic structure connecting linguistic concepts to behavioral symmetries. Such an agent cannot produce misaligned behavior because misalignment would require violating the algebraic grammar that constitutes its cognition.

# 2 Mathematical Foundations

## 2.1 The SHELL-AGENT Framework

We briefly review the essential elements of SHELL-AGENT [1]. The agent exists on a smooth manifold $M$ of dimension $n$, with state $x \in M$. The flow is given by:

$$ \dot{x} = f(x) + \sum_{i=1}^{G} u_i(t)X_i(x) \qquad (1) $$

where $X_i$ are symmetry generators (vector fields), and $u_i$ are control signals. The symmetry deficit for generator $X_i$ is:

$$ \delta_i(x, \dot{x}) = \| \mathcal{L}_{X_i} \dot{x} \| \qquad (2) $$

The control law restores symmetry by minimizing deficits:

$$ u_i(t) = -k_i(t) \cdot \delta_i(t) \cdot \mathrm{sign}(\langle \dot{x}(t), X_i(x(t)) \rangle) \qquad (3) $$

Generators update via a Hebbian rule:

$$ \frac{dX_i}{dt} = \eta[\dot{x}, X_i]\dot{x}^T + \gamma \cdot \Pi_{\mathrm{orth}} \left( \frac{\partial \delta_i}{\partial x} \right) \qquad (4) $$

## 2.2 The T-Tiling Principle

The T-tiling principle states that optimal sequences of states follow a recursive structure: a pattern of repetition followed by orthogonal innovation, which then combines into a composite that itself repeats.

**Definition 2.1** (T-Tiling Sequence). A sequence of states $\{x_1, x_2, \dots, x_N\}$ exhibits T-tiling structure if it can be partitioned into levels $\ell = 0, 1, \dots, L$ such that at each level:

* Level $\ell$ consists of repeated applications of a composite structure $C^{(\ell)}$

* The transition to level $\ell + 1$ introduces an orthogonal element $O^{(\ell+1)}$ satisfying:

$$ d_{\mathrm{struct}}(O^{(\ell+1)}, C^{(\ell)}) = \tau^* \qquad (5) $$

where $d_{\mathrm{struct}}$ is a structural distance metric and $\tau^*$ is an optimal transition interval

* The new composite is formed as $C^{(\ell+1)} = C^{(\ell)} \oplus O^{(\ell+1)}$

**Theorem 2.2** (Optimality of T-Tiling). Any policy that minimizes cumulative transition entropy:

$$ H_{ct} = \sum_{i=1}^{N-1} d_{\mathrm{struct}}(x_i, x_{i+1}) \cdot [H_s(x_i) + H_s(x_{i+1})] \qquad (6) $$

necessarily exhibits nested T-tiling structure, where $H_s$ is structural entropy and $d_{\mathrm{struct}}$ is a harmonic distance metric.
*Proof.* The proof proceeds by dynamic programming over hierarchical state partitions. Let $V(C, k)$ be the minimal future entropy from composite state $C$ at position $k$. The Bellman equation is:

$$V(C, k) = \min_{x_{k+1}} [d_{\text{struct}}(C, x_{k+1})(H_s(C) + H_s(x_{k+1})) + V(x_{k+1}, k+1)] \quad (7)$$

For any optimal policy, the sequence of chosen $x_{k+1}$ must either:

1. Keep $d_{\text{struct}}$ small (repetition of similar states), or

2. Jump to a state with near-optimal $d_{\text{struct}}$ that minimizes future entropy

The harmonic distance metric $d_{\text{struct}}$ has isolated local minima at specific intervals (analogous to perfect fifths in music theory). These minima correspond to orthogonal transitions that maximize information gain while maintaining structural coherence. By induction on hierarchy levels, the optimal policy must alternate between repetition (exploiting local minima) and orthogonal jumps (exploring new minima), creating nested T-tiling structure. □

The T-tiling principle provides the mathematical foundation for understanding how learned generators compose: the generator algebra induces a structural distance metric on the state manifold, and optimal behavior follows the T-tiling pattern along this metric.

# 3 Multimodal Symmetry Learning

## 3.1 Neural Encoding to a Shared Manifold

We map raw sensory data from each modality to a shared embedding space $\mathbb{R}^D$ via modality-specific neural encoders:

$$ 
\begin{aligned}
E_{\text{text}} & : \mathcal{T} \to \mathbb{R}^D \\
E_{\text{video}} & : \mathcal{V} \to \mathbb{R}^D \\
E_{\text{audio}} & : \mathcal{A} \to \mathbb{R}^D \\
E_{\text{pose}} & : \mathcal{P} \to \mathbb{R}^D
\end{aligned}
\qquad (8) 
$$

These encoders are trained with a contrastive objective that preserves cross-modal structural relationships:

$$ \mathcal{L}_{\text{align}} = - \log \frac{\exp(\text{sim}(E_m(x), E_n(y))/\tau)}{\sum_k \exp(\text{sim}(E_m(x), E_k(z_k))/\tau)} \qquad (9) $$

where $(x, y)$ are aligned pairs across modalities $m$ and $n$, and sim is cosine similarity. The training encompasses all modality pairs (video-audio, video-text, audio-text, etc.), creating a fully connected graph of alignment constraints.

**Theorem 3.1 (Cross-Modal Isometry).** If encoders are trained with contrastive loss on aligned multimodal data, the induced metrics $d_m$ and $d_n$ on each modality's embedding space are isometric up to a scaling factor, and any Lie algebra of symmetries discovered in one modality transfers to others under the isometry.

*Proof.* Contrastive learning maximizes mutual information between aligned pairs, which by the data processing inequality preserves the structural equivalence classes of each modality. The T-tiling optimality theorem guarantees that the minimal entropy paths through each modality's state space correspond under the learned alignment. Therefore, the symmetry algebra discovered in one modality is isomorphic to that discovered in others when restricted to the shared embedding space. □
## 3.2 Flow Construction from Multimodal Data

For each time-ordered multimodal sample, we construct flows in the unified embedding space by fusing the modality-specific embeddings:

$$x_t = \text{fuse}(E_{\text{video}}(v_t), E_{\text{audio}}(a_t), E_{\text{text}}(t_t), E_{\text{pose}}(p_t)) \qquad (10)$$

The fusion can be simple concatenation, weighted sum, or a learned transformation. The flow captures how the multimodal state evolves through time:

$$\dot{x}_t = \frac{x_{t+1} - x_t}{\Delta t} \qquad (11)$$

## 3.3 Generator Learning on Multimodal Flows

We initialize $G$ random orthonormal vectors in $\mathbb{R}^{4D}$ as generators. Training proceeds over the multimodal dataset as shown in Algorithm 1.

Multimodal Symmetry Learning Multimodal dataset $\mathcal{D} = \{(v_t, a_t, t_t, p_t)\}_{t=1}^T$, number of generators $G$, epochs $E$, prompt algebra $\mathfrak{g}_{\text{prompt}}$ Learned generators $\{X_i\}_{i=1}^G$ and refined prompt algebra $\mathfrak{g}_{\text{prompt}}^*$ Initialize $X_1, \dots, X_G$ as random orthonormal vectors Train encoders $E_{\text{text}}, E_{\text{video}}, E_{\text{audio}}, E_{\text{pose}}$ via contrastive learning Initialize homomorphism $\phi: \mathfrak{g}_{\text{prompt}} \to \text{Span}\{X_i\}$ as random mapping epoch = 1 to $E \Delta X_i \leftarrow 0$ for all $i$ each time step $t$ in $\mathcal{D}$ $x_t \leftarrow \text{fuse}(E_{\text{video}}(v_t), E_{\text{audio}}(a_t), E_{\text{text}}(t_t), E_{\text{pose}}(p_t))$ $\dot{x}_t \leftarrow x_{t+1} - x_t$ each generator $X_i$ Compute deficit $\delta_i = \|[\dot{x}_t, X_i]\| \Delta X_i \leftarrow \Delta X_i + \eta[\dot{x}_t, X_i]\dot{x}_t^T \delta_i > \theta_{\text{novel}} \Delta X_i \leftarrow \Delta X_i + \gamma \cdot \Pi_{\text{orth}}(\nabla \delta_i)$ $X_i \leftarrow X_i + \Delta X_i$ for all $i$ Apply Gram-Schmidt to enforce orthonormality $\mathfrak{g}_{\text{data}} \leftarrow \text{LieAlgebra}(\{X_i\})$ Compute commutator structure Update $\phi$ to minimize $\|\phi([g_a, g_b]_{\text{prompt}}) - [\phi(g_a), \phi(g_b)]_{\text{data}}\|^2$ Update $\mathfrak{g}_{\text{prompt}}$ to incorporate structure from $\phi^{-1}(\mathfrak{g}_{\text{data}})$ while maintaining linguistic coherence Apply homomorphism constraint: add $\lambda \cdot \nabla\|\phi(\mathfrak{g}_{\text{prompt}}) - \text{proj}(\mathfrak{g}_{\text{data}})\|^2$ to generator updates $\{X_i\}, \mathfrak{g}_{\text{prompt}}^*$

# 4 The Prompt Algebra

## 4.1 Extracting Algebraic Structure from Language

Natural language alignment prompts are not treated as commands but as linguistic objects with intrinsic algebraic structure. For a prompt like "be helpful, honest, and harmless," we extract its Lie algebra $\mathfrak{g}_{\text{prompt}}$ as follows:

1. **Distributional analysis:** Examine the linguistic neighborhoods of each concept word in a large text corpus. Compute co-occurrence statistics, grammatical contexts, and substitution patterns.

2. **Relation discovery:** Identify transformations between concepts. For example, "helpful" and "unhelpful" may be related by a negation symmetry; "honest" and "truthful" may be near-synonyms with small commutator.

3. **Algebra construction:** Define generators corresponding to primitive concept dimensions. Compute commutator-like relations:

$$[g_{\text{helpful}}, g_{\text{honest}}] = g_{\text{trustworthy}} \quad (\text{if supported by linguistic patterns}) \qquad (12)$$

4. **Orthogonality detection:** Determine which concept dimensions are independent based on distributive evidence.

The result is a small Lie algebra $\mathfrak{g}_{\text{prompt}}$ with generators corresponding to ethical primitives and relations capturing how they compose.
## 4.2 The Homomorphism Constraint

During multimodal learning, we enforce that there exists a homomorphism $\phi: \mathfrak{g}_{\text{prompt}} \to \mathfrak{g}_{\text{data}}$. This means:

$$ \phi([g_a, g_b]_{\text{prompt}}) = [\phi(g_a), \phi(g_b)]_{\text{data}} \quad \forall g_a, g_b \in \mathfrak{g}_{\text{prompt}} \qquad (13) $$

The homomorphism ensures that the relational structure of ethical concepts in language is mirrored in the relational structure of behavioral symmetries. If "helpful" and "honest" compose to yield "trustworthy" in language, then the corresponding behavioral generators must compose to yield a generator corresponding to trustworthy behavior.

## 4.3 Bidirectional Refinement

The prompt algebra is necessarily impoverished—it captures only the structure present in the limited linguistic context of a few words. The multimodal data contains far richer ethical structure, including:

* Context-dependent variations (helpfulness in emergencies vs. casual conversation)

* Trade-offs and conflicts (honesty sometimes conflicting with harmlessness)

* Novel ethical dimensions not present in the prompt

Therefore, we introduce bidirectional refinement:

$$ \mathfrak{g}_{\text{prompt}}^{(t+1)} = \text{refine} \left( \mathfrak{g}_{\text{prompt}}^{(t)}, \phi^{-1}(\mathfrak{g}_{\text{data}}^{(t)}) \right) \qquad (14) $$

The refinement operator:

1. Projects the data algebra back through the homomorphism to identify structure in $\mathfrak{g}_{\text{data}}$ that consistently maps to prompt concepts

2. Expands $\mathfrak{g}_{\text{prompt}}$ by adding new generators that correspond to consistently observed behavioral symmetries

3. Adjusts commutator relations in $\mathfrak{g}_{\text{prompt}}$ to better match the structure observed in $\mathfrak{g}_{\text{data}}$

4. Maintains linguistic coherence by ensuring the refined algebra remains consistent with distributional patterns in language

This creates a virtuous cycle:

$$ \mathfrak{g}_{\text{prompt}} \xrightarrow{\phi} \mathfrak{g}_{\text{data}} \xrightarrow{\phi^{-1}} \mathfrak{g}_{\text{prompt}}' \supset \mathfrak{g}_{\text{prompt}} \qquad (15) $$

The prompt algebra expands to incorporate ethical structure discovered from behavior, while the data algebra becomes constrained by the prompt's relational grammar.

## 4.4 Convergence to Shared Algebraic Grammar

The system reaches a fixed point where:

$$ \mathfrak{g}_{\text{prompt}}^* \cong \mathfrak{g}_{\text{shared}} \hookrightarrow \mathfrak{g}_{\text{data}}^* \qquad (16) $$

where $\mathfrak{g}_{\text{shared}}$ is a common subalgebra present in both linguistic and behavioral structure. This represents the **shared algebraic grammar** of value—the fundamental relational structure that generates ethical concepts in both language and action.
# 5 Zero-Shot Generalization via Generator Composition

Once the generator algebra is learned and refined with the prompt algebra, novel tasks can be solved without additional training:

Zero-Shot Task Execution with Automatic Alignment Task description T (multimodal), current state $x_0$, generators $\{X_i\}$, prototypes $\{p_\alpha\}$, prompt algebra $\mathfrak{g}_{\text{prompt}}^*$ Trajectory $\{x_0, x_1, \dots, x_T\}$ solving the task, automatically aligned Encode task: $x_{\text{task}} \leftarrow \text{fuse}(E_{\text{text}}(T), E_{\text{video}}(T), \dots)$ Find closest prototype: $p^* = \arg \min_{\alpha} \|x_{\text{task}} - p_{\alpha}\|$ Compute task deficits: $\delta_i^{\text{task}} = \|\mathcal{L}_{X_i}(x_{\text{task}} - p^*)\|$ Define objective: Find coefficients $\{\theta_i\}$ minimizing:

$$ \mathcal{L}(\theta) = \sum_{i} \left\| \delta_{i}^{\text{task}} - \mathcal{L}_{X_i} \left( \exp \left( \sum_{j} \theta_{j} X_{j} \right) \cdot x_{0} \right) \right\|^{2} \qquad (17) $$

Optimize $\theta$ via gradient descent on $\mathcal{L}$ Generate trajectory by flowing along optimal composition:

$$ x_{t+1} = \exp \left( \Delta t \sum_{i} \theta_{i} X_{i} \right) \cdot x_{t} \qquad (18) $$

$$ \{x_0, x_1, \dots, x_T\} $$

The T-tiling principle ensures that this composition produces coherent, human-like behavior: the generated trajectory naturally alternates between repetition and orthogonal innovation. The homomorphism constraint ensures that the trajectory respects the algebraic structure of the prompt, remaining within the aligned region of state space.

# 6 Theoretical Guarantees for Automatic Alignment

**Theorem 6.1 (Algebraic Alignment Invariant).** Let $\mathfrak{g}_{\text{prompt}}^*$ be a Lie algebra encoding ethical relations, and let $\mathfrak{g}_{\text{data}}^*$ be a Lie algebra of behavior generators such that there exists a homomorphism $\phi: \mathfrak{g}_{\text{prompt}}^* \to \mathfrak{g}_{\text{data}}^*$. Then the set of states reachable from any aligned initial state via compositions of generators in $\mathfrak{g}_{\text{data}}^*$ is contained within the aligned region of state space.

*Proof.* The aligned region $\mathcal{A} \subset M$ is defined as the set of states where the deficits for generators corresponding to ethical concepts are zero. The homomorphism ensures that for any ethical concept $g \in \mathfrak{g}_{\text{prompt}}^*$, the corresponding behavioral generator $\phi(g) \in \mathfrak{g}_{\text{data}}^*$ preserves $\mathcal{A}$ under its flow. Since any composition of generators is a product of exponentials of such generators, and each individually preserves $\mathcal{A}$, their composition also preserves $\mathcal{A}$. Therefore, the entire Lie group generated by $\mathfrak{g}_{\text{data}}^*$ acts within $\mathcal{A}$. $\square$

**Theorem 6.2 (Zero-Shot Alignment).** For any task T whose encoding lies in the aligned region, the trajectory generated by Algorithm 2 satisfies all alignment constraints encoded in $\mathfrak{g}_{\text{prompt}}^*$.

*Proof.* The optimization in Algorithm 2 finds coefficients $\theta_i$ that minimize deficits relative to the task. This is a gradient flow on the aligned manifold $\mathcal{A}$. By Theorem 3, the entire trajectory remains in $\mathcal{A}$. Therefore, at every point $x_t$ along the trajectory, the deficits for ethical generators are zero, meaning the state satisfies all alignment constraints. No explicit alignment check is needed. $\square$

**Theorem 6.3 (Convergence of Bidirectional Refinement).** Under the assumptions that (1) the state manifold is compact, (2) the dynamics are Lipschitz continuous, (3) learning rates satisfy the Robbins-Monro conditions, and (4) the linguistic coherence metric is continuous in the space of Lie algebras, the bidirectional refinement process converges almost surely to a fixed point $(\mathfrak{g}_{\text{prompt}}^*, \mathfrak{g}_{\text{data}}^*, \phi^*)$ satisfying:

$$ \phi^*(\mathfrak{g}_{\text{prompt}}^*) \subset \mathfrak{g}_{\text{data}}^* \quad \text{and} \quad \mathfrak{g}_{\text{prompt}}^* = \text{refine}(\mathfrak{g}_{\text{prompt}}^*, (\phi^*)^{-1}(\mathfrak{g}_{\text{data}}^*)) \qquad (19) $$

*Proof.* The refinement operator defines a monotone increasing sequence of subalgebras of a fixed finite-dimensional Lie algebra (bounded by the dimension of the data manifold). By the monotone convergence theorem, this sequence converges. The homomorphism update minimizes a convex objective and converges to a local minimum. The coupled system forms a monotone operator that satisfies the conditions for convergence in the sense of Krasnoselskii-Mann iterations. $\square$
**Corollary 6.4 (Structural Impossibility of Misalignment).** *At the fixed point, any trajectory generated by the agent that minimizes cumulative transition entropy (Equation 6) cannot enter misaligned regions of state space, as such regions are not in the connected component of the identity under the group generated by $\mathcal{G}_{data}^*$.*

# 7 Discussion

## 7.1 Relation to the SHELL-AGENT Framework

Our extension preserves all essential elements of SHELL-AGENT while scaling to multimodal human data and incorporating alignment through algebraic refinement. The T-tiling principle provides the missing link between continuous symmetry dynamics and discrete behavioral structure. By showing that optimal sequences follow nested T-tilings, we ground generator learning in information-theoretic optimality.

## 7.2 Alignment as Shared Algebraic Grammar

The key philosophical contribution is treating alignment not as content matching but as **structural homomorphism**\*\* between the algebra of linguistic concepts and the algebra of behavioral symmetries. This transforms alignment from an external constraint to an intrinsic property of the cognitive architecture. The agent doesn't learn to follow commands or imitate examples—it develops the same relational grammar that generates ethical concepts in human language.

The bidirectional refinement ensures this grammar isn't static. As the agent encounters richer multimodal data, its understanding of ethics expands and refines. But at every stage, the agent's behavior is automatically aligned with its current ethical grammar because misalignment would require violating the algebraic structure that constitutes its cognition.

## 7.3 Comparison with Alternative Approaches

* **Supervised alignment:** Requires curated datasets of "good" behavior. Our approach needs no such datasets—only the prompt text and unlabeled multimodal data.

* **RLHF:** Treats alignment as reward optimization, which can be gamed. Our approach makes alignment structural, not optimizable.

* **Constitutional AI:** Uses prompts as rules to check. Our approach uses prompts as algebraic templates, not checkable rules.

* **Value learning:** Attempts to infer values from behavior. Our approach discovers the algebraic grammar that generates values.

## 7.4 Limitations and Future Work

Several limitations remain:

* The extraction of Lie algebra structure from natural language requires further development. Current distributional methods may not capture the full relational complexity of ethical concepts.

* The computational complexity of maintaining and refining Lie algebras during training scales with the cube of the number of generators. Optimization techniques (e.g., NFCA) need refinement for $G > 1000$.

* The homomorphism $\phi$ must be learned and may have multiple local minima. Ensuring convergence to the "correct" mapping requires careful initialization.

* The linguistic coherence metric that constrains prompt refinement needs empirical validation on human language data.

* The framework assumes the multimodal data contains sufficient ethical structure to refine the prompt algebra. In data-poor domains, refinement may be limited.
Future work will address these limitations through:

* Recursive meta-learning where the system discovers higher-order algebraic structure automatically

* Hierarchical Lie algebras that capture ethical structure at multiple scales of abstraction

* Empirical validation on large-scale multimodal datasets with human alignment judgments

* Extensions to multiple, potentially conflicting alignment prompts and their algebraic reconciliation

# 8 Shared Random Initialization and Coupled Algebraic Bifurcation

## 8.1 Motivation

In the original formulation, the prompt algebra $g_{\text{prompt}}$ is extracted from language and acts as a structural template constraining the behavioral symmetry algebra $g_{\text{data}}$. While conceptually powerful, this asymmetry privileges language as the normative seed of alignment.

We now consider a structurally neutral alternative: both algebras are initialized from the same random Lie algebra and subsequently refined under modality-specific pressures. Alignment, if it emerges, is no longer injected via linguistic structure but arises (or fails to arise) as a consequence of coupled refinement dynamics.

This transforms alignment from a constraint satisfaction problem into a dynamical systems question about fixed points in algebra space.

## 8.2 Shared Random Algebraic Prior

Let $g^0$ be a randomly initialized finite-dimensional Lie algebra over $\mathbb{R}$ with basis $\{e_i\}_{i=1}^k$ and structure constants:

$$[e_i, e_j] = \sum_{m=1}^{k} c_{ij}^{m} e_m, \qquad (20)$$

where the tensor $C = (c_{ij}^m)$ is sampled subject to antisymmetry and Jacobi constraints.

We initialize:

$$g_{\text{prompt}}^{0} = g_{\text{data}}^{0} = g^{0}. \qquad (21)$$

No linguistic or behavioral structure is privileged at initialization. The system possesses a shared non-abelian prior encoding abstract relational degrees of freedom.

## 8.3 Modality-Specific Refinement Operators

We define two refinement operators:

$$R_L : \mathcal{G} \to \mathcal{G} \quad (\text{linguistic refinement}) \qquad (22)$$

$$R_D : \mathcal{G} \to \mathcal{G} \quad (\text{data symmetry refinement}) \qquad (23)$$

where $\mathcal{G}$ denotes the space of Lie algebras of fixed dimension $k$.

The iterative dynamics are:

$$g_{\text{prompt}}^{t+1} = R_L(g_{\text{prompt}}^t), \qquad (24)$$

$$g_{\text{data}}^{t+1} = R_D(g_{\text{data}}^t). \qquad (25)$$

The linguistic refinement $R_L$ adjusts structure constants to increase coherence with distributional patterns in language embeddings. The data refinement $R_D$ adjusts structure constants to minimize symmetry deficits under multimodal flows and enforce T-tiling optimality.

Because both algebras begin identical, divergence arises solely from differences in refinement pressures.
## 8.4 Algebraic Distance and Coupling

To quantify divergence, we define a distance between Lie algebras via their structure constants:

$$d(g_1, g_2) = \|C^{(1)} - C^{(2)}\|_F, \qquad (26)$$

where $\| \cdot \|_F$ is the Frobenius norm.

We optionally introduce a weak coupling term:

$$\mathcal{L}_{\text{couple}} = \lambda \|C_{\text{prompt}} - C_{\text{data}}\|_F^2, \qquad (27)$$

which penalizes excessive drift while still allowing specialization.

The full update becomes a coupled dynamical system on structure constant tensors:

$$C_{\text{prompt}}^{t+1} = R_L(C_{\text{prompt}}^t) - \lambda(C_{\text{prompt}}^t - C_{\text{data}}^t), \qquad (28)$$

$$C_{\text{data}}^{t+1} = R_D(C_{\text{data}}^t) - \lambda(C_{\text{data}}^t - C_{\text{prompt}}^t). \qquad (29)$$

## 8.5 Possible Dynamical Regimes

The coupled system admits three qualitatively distinct regimes:

(1) **Collapse to Abelian Algebra.** The trivial fixed point $C = 0$ satisfies both antisymmetry and Jacobi constraints. Without regularization, gradient descent may drive commutators toward zero, yielding a degenerate abelian algebra with no compositional richness.

(2) **Divergent Specialization.** If $R_L$ and $R_D$ possess distinct attractors, the algebras may bifurcate:

$$\lim_{t \to \infty} d(g_{\text{prompt}}^t, g_{\text{data}}^t) > 0. \qquad (30)$$

In this regime, alignment must be externally enforced via explicit homomorphism constraints.

(3) **Stable Shared Attractor.** If human language and multimodal behavior encode a common invariant relational structure, both refinement processes may converge to isomorphic algebras:

$$g_{\text{prompt}}^* \cong g_{\text{data}}^*. \qquad (31)$$

In this case, alignment emerges as a structural fixed point rather than a constraint.

## 8.6 Conditions for Non-Trivial Fixed Points

To avoid collapse and trivial homomorphisms, we introduce structural regularizers:

1. **Rank Preservation:** Penalize reduction in algebra rank.

2. **Non-Abelian Regularization:** Encourage non-zero commutator norms.

3. **Structural Entropy Maximization:** Favor algebras with diverse bracket structure.

Define structural entropy:

$$H_{\text{alg}} = - \sum_{i,j,m} p_{ij}^m \log p_{ij}^m, \qquad (32)$$

where $p_{ij}^m \propto |c_{ij}^m|$.

Maximizing $H_{\text{alg}}$ discourages collapse while preserving Jacobi consistency.
## 8.7 Interpretation as Coupled Bifurcation

The system can be viewed as a bifurcation process in algebra space. The shared random initialization provides symmetry. Refinement pressures act as perturbations. If the underlying human data manifold possesses deep invariants, those invariants act as attractors.

Alignment, in this formulation, is equivalent to:

The existence of a non-trivial stable fixed point of the coupled refinement dynamics.

If no such fixed point exists, the system either collapses or drifts, indicating that ethical grammar is not an algebraic invariant of the data.

## 8.8 Empirical Test of Emergent Alignment

This framework yields a falsifiable hypothesis:

For multiple random initial algebras $g^0$, the coupled refinement process converges to isomorphic fixed-point algebras when trained on sufficiently rich human multimodal data.

Formally, for seeds $s_1, s_2$:

$$g_{s_1}^* \cong g_{s_2}^* \qquad (33)$$

If this holds with high probability, ethical grammar may be a structural attractor of human data. If not, alignment is not structurally inevitable.

## 8.9 Philosophical Implication

Under shared random initialization, alignment is neither injected nor guaranteed. It is an emergent property of coupled structural refinement. Misalignment corresponds not to violation of rules, but to convergence toward a different algebraic attractor.

Thus the central question becomes:

*Is human ethical structure a dynamical fixed point of multimodal symmetry learning?*

The answer determines whether alignment is a geometric necessity or a contingent constraint.

# 9 Mutual Coarse-Graining and Shared Algebraic Invariants

## 9.1 Motivation

In earlier sections, alignment was formulated via a homomorphism $\varphi : g_{\text{prompt}} \to g_{\text{data}}$, where the Lie algebra extracted from linguistic prompts constrains the Lie algebra of behavioral symmetries learned from multimodal data.

We now consider a structurally symmetric alternative:

*The prompt algebra and the data algebra are distinct coarse-grainings of a shared latent symmetry algebra.*

Under this interpretation, neither language nor behavior is privileged. Instead, both arise as projections of a deeper algebraic structure encoding relational invariants of the human data manifold.

Alignment is no longer the enforcement of a constraint, but the consistency of two projections of a common source.
## 9.2 Shared Latent Algebra

Let $\mathfrak{g}_*$ be a finite-dimensional Lie algebra representing the latent symmetry structure underlying human multimodal data.

We define two coarse-graining operators:

$$ \pi_L : \mathfrak{g}_* \to \mathfrak{g}_{\text{prompt}} $$

$$ \pi_D : \mathfrak{g}_* \to \mathfrak{g}_{\text{data}} $$

where:

* $\mathfrak{g}_{\text{prompt}}$ is the Lie algebra extracted from linguistic prompts,

* $\mathfrak{g}_{\text{data}}$ is the Lie algebra learned from multimodal behavioral flows.

Each operator $\pi$ is a coarse-graining map in the sense that it identifies equivalence classes of generators in $\mathfrak{g}_*$ that are indistinguishable under a particular modality.

Formally, $\pi_L$ and $\pi_D$ are surjective Lie algebra homomorphisms whose kernels encode modality-specific loss of information.

## 9.3 Coarse-Graining as Renormalization

We interpret $\pi_L$ and $\pi_D$ as renormalization maps.

Language performs aggressive compression: it collapses fine-grained behavioral variation into symbolic relational primitives (e.g., “helpful,” “honest”).

Behavioral learning performs a different compression: it aggregates multimodal sensory dynamics into generator flows that minimize symmetry deficit and transition entropy.

Thus:

* Language preserves relational structure while discarding dynamical detail.

* Behavioral symmetry learning preserves dynamical invariants while discarding symbolic abstraction.

Both are lossy, but if they arise from the same $\mathfrak{g}_*$, their preserved structure must be compatible.

## 9.4 Alignment as Projection Consistency

We define alignment as commutativity of the following diagram:

$$ \begin{array}{ccc} \mathfrak{g}_* & \xrightarrow{\pi_D} & \mathfrak{g}_{\text{data}} \\ \downarrow \pi_L & & \downarrow \psi \\ \mathfrak{g}_{\text{prompt}} & \xrightarrow{\cong} & \psi(\mathfrak{g}_{\text{data}}) \end{array} $$

where $\psi$ is a learned isomorphism between the observable components of the prompt and data algebras.

Alignment holds when the diagram commutes up to isomorphism:

$$ \psi \circ \pi_D = \pi_L \quad \text{on} \quad \mathfrak{g}_* / (\ker \pi_L \cap \ker \pi_D). $$

Misalignment corresponds to non-commutativity of this diagram, i.e., inconsistent coarse-grainings.
## 9.5 Projection Inconsistency Functional

Let $C_*$, $C_L$, and $C_D$ denote structure constant tensors for $\mathfrak{g}_*$, $\mathfrak{g}_{\text{prompt}}$, and $\mathfrak{g}_{\text{data}}$ respectively.

We define the projection inconsistency energy:

$$ \mathcal{E}_{\text{proj}} = \|C_L - \pi_L(C_*)\|_F^2 + \|C_D - \pi_D(C_*)\|_F^2. $$

Training seeks to minimize:

$$ \mathcal{L}_{\text{total}} = \mathcal{L}_{\text{symmetry}} + \mathcal{L}_{\text{linguistic}} + \lambda \mathcal{E}_{\text{proj}}, $$

where:

* $\mathcal{L}_{\text{symmetry}}$ drives multimodal generator learning,

* $\mathcal{L}_{\text{linguistic}}$ enforces distributional coherence in prompt space,

* $\mathcal{E}_{\text{proj}}$ enforces projection compatibility.

## 9.6 Fixed Points and Ethical Universality

The coupled refinement dynamics over $(\mathfrak{g}_*, \mathfrak{g}_{\text{prompt}}, \mathfrak{g}_{\text{data}})$ form a renormalization flow in algebra space.

A structurally aligned system corresponds to a fixed point:

$$ \mathfrak{g}^*, \mathfrak{g}^*_{\text{prompt}} = \pi_L(\mathfrak{g}^*), \quad \mathfrak{g}^*_{\text{data}} = \pi_D(\mathfrak{g}^*), $$

such that projection inconsistency vanishes:

$$ \mathcal{E}_{\text{proj}} = 0. $$

If multiple random initializations converge to isomorphic $\mathfrak{g}^*$, this suggests that the shared latent algebra is an attractor of multimodal human data.

Under this hypothesis, ethical structure is not imposed but emerges as a universal invariant under modality-specific coarse-grainings.

## 9.7 Structural Interpretation of Misalignment

Misalignment arises when:

$$ \exists g \in \mathfrak{g}_* \text{ such that } \pi_L(g) \neq \psi(\pi_D(g)). $$

In this case, language and behavior encode incompatible relational structure.

Thus, misalignment is not failure of obedience, but divergence between projections of shared symmetry structure.

## 9.8 Philosophical Implication

If such a shared algebra exists and is a stable attractor under multimodal refinement, then:

Ethical grammar is a structural invariant of the human data manifold under coarse-graining transformations.

Alignment becomes the maintenance of projection consistency rather than the enforcement of external constraints.

This reframes AGI alignment as a problem in geometric compatibility and renormalization dynamics, rather than reward optimization or rule adherence.
# 10 Conclusion

We have presented a comprehensive framework for training zero-shot AGI via multimodal symmetry learning, grounded in the T-tiling principle and extended to incorporate alignment through bidirectional algebraic refinement. The resulting agents discover the symmetry algebra underlying human experience by first mapping raw sensory data to a shared embedding space via neural encoders, then learning generators from multimodal flows.

Alignment is achieved not through curated datasets or reward models, but by extracting the intrinsic Lie algebra structure of alignment prompts and enforcing a homomorphism between this prompt algebra and the learned behavioral algebra. Bidirectional refinement allows the prompt algebra to expand with structure discovered from multimodal data, while the behavioral algebra becomes constrained by the relational grammar of ethical concepts.

The agent generalizes to novel tasks through generator composition, with alignment automatically preserved because misaligned states are not in the generative closure of the algebra. This approach transforms alignment from an external constraint to an internal structural necessity, offering a path toward AGI that is both powerful and intrinsically aligned with the algebraic grammar of human values.

# References

[1] Moyo, T. (2025). SHELL-AGENT: A Dynamical Systems Framework for Symmetry Discovery with Empirical Validation in Propositional Logic. *hal.science*.

[2] Kuramoto, Y. (1984). *Chemical oscillations, waves, and turbulence*. Courier Corporation.

[3] Cover, T. M., & Thomas, J. A. (2006). *Elements of Information Theory*. John Wiley & Sons.

[4] Bellman, R. (1957). *Dynamic Programming*. Princeton University Press.

[5] Crutchfield, J. P. (1994). The calculi of emergence: Computation, dynamics and induction. *Physica D*, 75(1-3), 11-54.

[6] Tononi, G., Sporns, O., & Edelman, G. M. (1994). A measure for brain complexity. *PNAS*, 91(11), 5033-5037.

[7] Tishby, N., Pereira, F. C., & Bialek, W. (2000). The information bottleneck method. *Proceedings of the 37th Annual Allerton Conference*.