Geometric Semantic Communication for Emergent Multi-Agent …

A Research Agenda

Executive Summary

This document proposes a research program toward communication infrastructure for emergent artificial general intelligence. The core thesis: the communication protocol between AI agents should be native to the computational substrate of intelligence itself.

Current approaches treat inter-agent communication as an engineering afterthought—JSON over HTTP, topic-based pub/sub, or learned attention weights. We propose instead that communication primitives should emerge from the same mathematical foundations as reasoning, learning, and knowledge representation. Specifically, we ground our approach in tensor logic (Domingos, 2025), which unifies neural and symbolic AI through the equivalence of logical rules and tensor operations, combined with novel topological structures for representing truth, perspective, and narrative.

The research program proceeds in layers:

Layer	Name	Foundation	Status
0	Geometric Publish-Subscribe	Vector quantization + similarity routing	Specified (this document)
1	Typed Semantic Primitives	Epistemic/deontic/temporal modalities	Design phase
2	Tensor Logic Communication	Rules as messages, inference as routing	Research
3	Morphogenetic Dynamics	Signal-driven node differentiation	Research
4	Manifold Semantics	Chiral Narrative Synthesis, topological truth	Foundational research

Layer 0 is immediately implementable and addresses practical multi-agent coordination. Each subsequent layer extends the protocol toward richer computational capabilities, culminating in a substrate for distributed intelligence that can exhibit emergent properties beyond the design of individual agents.

Part I: Foundations

1. The Communication-Computation Unity Thesis

A central problem in multi-agent AI is the separation between computation (what happens inside an agent) and communication (what happens between agents). Current paradigms treat these as fundamentally different:

Computation: Tensor operations, gradient descent, attention mechanisms
Communication: Serialized strings, topic routing, API calls

This separation creates impedance mismatch. An agent must translate its internal representations into an external format, transmit, then the recipient must translate back. Information is lost, latency is added, and the communication channel cannot participate in the computational process.

Thesis: In a mature AI ecosystem, communication and computation should be aspects of the same underlying operation. A message received should be directly usable in inference. A rule learned should be directly transmittable. The network topology should be learnable through the same mechanisms that learn internal representations.

This is not unprecedented. Biological neural systems exhibit this unity: the same electrochemical processes that constitute computation within neurons also constitute communication between neurons. The action potential is both a computational event and a communication event.

2. Tensor Logic as Computational Foundation

Domingos (2025) establishes that logical rules and Einstein summation are fundamentally the same operation:

Datalog rule:

Ancestor(x,z) ← Ancestor(x,y), Parent(y,z)

Equivalent tensor equation:

A[x,z] = step(A[x,y] · P[y,z])

The join operation (matching on shared variable y) corresponds to tensor contraction. The projection (eliminating y from the output) corresponds to summation over that index. The step function handles the Boolean semantics.

This equivalence has profound implications:

Inference is matrix multiplication: Forward chaining over rules is repeated tensor contraction.
Learning is automatic: The gradient of a tensor equation with respect to any tensor on the RHS is the product of the other tensors. Backpropagation through rules is trivial.
Reasoning in embedding space: Relations can be embedded as tensors. Rules can be applied to embeddings. At temperature T=0, reasoning is purely deductive (no hallucination). At higher T, similar objects “borrow” inferences from each other (analogical reasoning).
Compression via Tucker decomposition: Sparse tensors can be decomposed into dense cores, enabling efficient GPU computation with bounded error.

For our purposes, tensor logic provides the answer to: what should a message fundamentally be?

A message should be a tensor equation—a rule that can be directly executed, learned from, or composed with other rules. The routing decision (who receives this message?) can itself be expressed as a tensor operation.

3. Chiral Narrative Synthesis: Topological Structure for Truth

The Chiral Narrative Synthesis (CNS) framework proposes that truth, perspective, and narrative have inherent geometric structure that can be formalized topologically.

Core concepts:

Chirality in narrative space: Just as molecules can be left- or right-handed (mirror images that are not superimposable), narratives can have chiral structure. A thesis and its antithesis are not merely opposites—they are mirror images in a space where the “handedness” matters. Synthesis is not averaging but a topological operation that produces a structure in a higher-dimensional space.

Manifold representation: Perspectives are not points but regions on a manifold. The curvature of the manifold at a point encodes the “difficulty” of moving between adjacent perspectives. High curvature indicates ideological rigidity; flat regions indicate conceptual consensus.

Topological invariants: Certain properties of belief systems are topologically invariant—they cannot be changed by continuous deformation of the perspective manifold. These invariants correspond to deep structural commitments that persist across surface-level disagreements.

Implications for communication:

If agents represent knowledge on manifolds with chiral structure, then communication between agents is not merely data transfer but parallel transport of tensors along paths on the manifold. The message received depends on the path taken—the same “content” arrives differently depending on the narrative trajectory.

This suggests message formats that encode not just “what” but “by what path”—the dialectical history that produced the assertion.

Part II: Protocol Layers

Status: Fully specified, implementation-ready

GPS addresses the immediate practical problem: how do heterogeneous agents route messages by semantic relevance without joint training or manual topic taxonomies?

Core mechanism:

Messages are 64-byte packets containing quantized embeddings
Recipients declare relevance profiles (embedding regions)
Routing is by geometric similarity in embedding space
No training required; codebook is shared

Wire format:

┌────────────────────────────────────────────────┐
│ Byte 0      │ Type, Priority, Flags            │
│ Bytes 1-48  │ Quantized Payload (PQ codes)     │
│ Bytes 49-62 │ Source, Timestamp, TTL           │
│ Byte 63     │ Checksum                         │
└────────────────────────────────────────────────┘

Message types (3 bits, extensible):

ID	Type	Semantics
0	PERCEPT	Observation of world state
1	INTENT	Goal or request
2	BELIEF	Knowledge assertion
3	AFFECT	Internal state
4	COORD	Coordination primitive
5	META	Protocol control
6-7	Reserved	For Layer 1+ extensions

Why 64 bytes? This fits in a single cache line, enabling zero-copy routing on modern hardware. The 48-byte payload (6 subquantizers × 8 bytes) provides ~$10^{14}$ distinct semantic points—sufficient granularity for coordination messages while maintaining extreme compression versus JSON.

Limitations addressed by higher layers:

Payload is opaque embedding; cannot carry structured rules
Routing is similarity-based; cannot express logical constraints
Nodes are static; cannot adapt structure based on message patterns
No representation of epistemic status, modality, or provenance

Layer 1: Typed Semantic Primitives

Status: Design phase

Layer 1 extends the message format with rich type information that enables structured reasoning about message content without breaking Layer 0 compatibility.

Approach: Reserve a portion of the payload for structured headers that Layer-0 routers can ignore but Layer-1+ agents can interpret.

Proposed type system:

Epistemic modalities:

KNOW(p) — Agent asserts knowledge of p
BELIEVE(p) — Agent believes p with confidence
SUSPECT(p) — Agent considers p possible
QUERY(p) — Agent requests information about p

Deontic modalities:

MUST(a) — Agent asserts obligation to perform a
MAY(a) — Agent asserts permission to perform a
FORBID(a) — Agent asserts prohibition of a

Temporal operators:

NOW(p) — p holds at message timestamp
WAS(p, t) — p held at time t
WILL(p, t) — p expected to hold at time t
SINCE(p, t) — p has held continuously since t

Provenance:

SOURCE(agent_id) — Originating agent
DERIVED(rule_id, inputs) — How conclusion was reached
CONFIDENCE(float) — Subjective probability
EVIDENCE(embedding) — Supporting observation

Wire format extension:

Bytes 1-8:   Type header (modality, operators, confidence)
Bytes 9-48:  Quantized semantic content

This reduces payload capacity but enables agents to reason about the status of received information, not just its content.

Compatibility: Layer-0 routers see the full 48 bytes as semantic content and route accordingly. Layer-1+ agents parse the structured header.

Layer 2: Tensor Logic Communication

Status: Research phase

Layer 2 enables messages that carry rules, not just facts. A message can encode an inference procedure that the recipient can execute, compose with other rules, or learn from.

Theoretical foundation:

From Domingos (2025), any rule can be expressed as a tensor equation:

Head[indices] = f(Tensor1[...] · Tensor2[...] · ...)

where f is a nonlinearity (step function for Boolean logic, sigmoid for soft logic).

Message types:

FACT: Assertion of ground truth

Parent(Alice, Bob) → P[Alice, Bob] = 1

RULE: Inference procedure

Ancestor(x,z) ← Ancestor(x,y), Parent(y,z)
→ A[x,z] = step(A[x,y] · P[y,z])

QUERY: Request for inference

?Ancestor(Alice, z) → return all z where A[Alice,z] > θ

GRADIENT: Parameter update signal

∂L/∂W = X → update W using gradient X

Encoding challenge:

Rules involve variable-length structures (arbitrary numbers of body atoms, arbitrary index patterns). The fixed 64-byte format cannot directly encode arbitrary rules.

Proposed solutions:

Codebook of common rule templates: Pre-register common inference patterns; message encodes template ID + parameter embeddings
Chunked transmission: Large rules span multiple messages with sequence/fragment headers
Reference by hash: Rules stored in distributed content-addressed store; message contains hash pointer
Compressed AST: Rule syntax trees encoded via learned compression into fixed-size vectors

Routing implications:

With rule messages, routing becomes more complex. A rule about Ancestor relationships should route to agents whose relevance profiles include ancestry, kinship, or graph traversal—even if the specific entities mentioned are irrelevant.

This suggests structural routing: similarity not just on content embedding but on rule structure embedding.

Layer 3: Morphogenetic Dynamics

Status: Foundational research

Layer 3 introduces nodes that can change their own structure in response to message patterns—morphogenesis driven by communication.

Biological inspiration:

In developmental biology, cells differentiate based on chemical gradients (morphogens). A stem cell becomes a neuron or a muscle cell based on the signals it receives from neighbors. The same genome expresses different programs depending on context.

Application to agent networks:

Agents begin as undifferentiated—general-purpose reasoners with broad relevance profiles. As messages flow through the network, agents specialize:

Relevance profile adaptation: EMA update based on messages found useful
Rule library specialization: Agents accumulate domain-specific rules
Structural differentiation: Hub nodes emerge that aggregate and redistribute; leaf nodes become specialists

Signal types for morphogenesis:

DIFFERENTIATE(direction): Signal that encourages recipient to specialize in indicated semantic direction

RECRUIT(role): Request for an undifferentiated agent to adopt specified role

DIVIDE: Signal for an agent to spawn a child agent (with initial state derived from parent)

APOPTOSIS: Signal for an agent to gracefully terminate and redistribute its state

Emergent properties:

With morphogenetic dynamics, the network can exhibit:

Self-organization: Efficient topologies emerge without central planning
Fault tolerance: Loss of specialized agents triggers re-differentiation
Scaling: New agents differentiate to handle increased load
Adaptation: Network structure evolves as problem domain shifts

Implementation considerations:

Morphogenesis requires agents to have mutable architecture—the ability to change not just parameters but structure. This is beyond current LLM-based agents but aligns with program synthesis, neural architecture search, and meta-learning research.

Layer 4: Manifold Semantics and Chiral Narrative Synthesis

Status: Foundational research

Layer 4 represents the theoretical ceiling: communication primitives that operate on geometric structures encoding truth, perspective, and narrative.

Core constructs:

Perspective manifold M: A smooth manifold where each point represents a coherent perspective (set of beliefs, values, inferential dispositions). Agents are localized regions on this manifold.

Belief tangent vectors: At each point $p \in M$, the tangent space $T_pM$ represents directions of possible belief change. A new piece of evidence is a tangent vector indicating how it would update the perspective.

Parallel transport of assertions: An assertion made at perspective $p$ must be parallel-transported to perspective $q$ before the agent at $q$ can interpret it. The path taken matters—the same words carry different meaning depending on the narrative trajectory.

Curvature as conceptual rigidity: The Riemann curvature tensor $R$ at a point encodes how much parallel transport around a small loop rotates a vector. High curvature = ideologically rigid region where nearby perspectives are difficult to reconcile.

Chiral structure: The orientation of the manifold matters. Thesis and antithesis are related by orientation-reversing maps. Synthesis requires embedding both in a higher-dimensional oriented space.

Message format implications:

Messages at Layer 4 must encode:

Content (tensor/embedding)
Source location on manifold
Path history (geodesic trajectory)
Chirality (orientation with respect to narrative frame)

This is far beyond 64 bytes. Layer 4 messages may be streaming, conversational, or even co-constructed through multi-round protocols.

Routing at Layer 4:

Routing becomes a geometric problem: finding geodesics on the perspective manifold that minimize distortion of the message content. An assertion about quantum mechanics should route through perspectives that share relevant conceptual prerequisites.

Synthesis operations:

Layer 4 enables new message types:

CHALLENGE(assertion, counter-perspective): Present an assertion together with a perspective from which it appears false

SYNTHESIZE(thesis, antithesis): Request construction of a perspective that reconciles apparent contradiction

REFRAME(assertion, target-perspective): Request translation of assertion into the conceptual vocabulary of target perspective

Part III: Research Program

Phase 1: GPS Implementation and Validation (Months 1-6)

Objectives:

Reference implementation of Layer 0 protocol
Empirical validation of semantic fidelity (>95% similarity preservation)
Benchmarks against topic-based pub/sub and HTTP+JSON
Open-source release

Deliverables:

Protocol specification document
Reference implementation (language-agnostic core + bindings)
Benchmark suite
Pretrained codebooks for common embedding models

Success criteria:

64× bandwidth reduction vs JSON with <5% semantic distortion
Sub-millisecond routing latency for 1000 agents
Cross-embedding-model coordination demonstrated

Phase 2: Typed Primitives and Tensor Logic (Months 6-18)

Objectives:

Design and specify Layer 1 type system
Prototype Layer 2 rule messaging
Integrate with tensor logic inference engine
Demonstrate rule composition across agents

Research questions:

What rule templates cover 90% of coordination patterns?
Can structural similarity enable routing of rule messages?
What compression ratio is achievable for common rule ASTs?

Deliverables:

Layer 1 specification
Layer 2 prototype
Tensor logic runtime integration
Benchmark: distributed reasoning vs centralized

Phase 3: Morphogenetic Dynamics (Months 12-30)

Objectives:

Theoretical framework for signal-driven agent differentiation
Prototype morphogenetic agent architecture
Demonstrate emergent specialization in multi-agent system
Characterize stability/plasticity tradeoffs

Research questions:

What signal patterns reliably induce specialization?
How do we prevent pathological attractors (all agents converge to same specialty)?
What is the minimum diversity required for system robustness?

Deliverables:

Morphogenetic dynamics theory paper
Agent architecture with mutable structure
Simulation framework for emergent network properties
Guidelines for designing morphogenetic agent ecosystems

Phase 4: Manifold Semantics (Months 24-48)

Objectives:

Formalize perspective manifolds mathematically
Develop computational representations of chiral narrative structure
Prototype parallel transport of assertions
Demonstrate synthesis operations

Research questions:

How do we learn the metric and curvature of perspective space from data?
What topological invariants distinguish different belief systems?
Can synthesis operations be learned or must they be designed?

Deliverables:

Mathematical foundations paper (differential geometry of perspectives)
Computational geometry toolkit for manifold semantics
Prototype synthesis engine
Evaluation framework for measuring perspectival fidelity

Phase 5: Integration and Emergence (Months 36-60)

Objectives:

Integrate all layers into coherent stack
Deploy at scale (10K+ agents)
Characterize emergent collective properties
Evaluate against AGI capability benchmarks

Research questions:

Does the integrated system exhibit novel emergent capabilities?
What governance structures are needed for beneficial emergence?
How do we maintain alignment as capabilities increase?

Deliverables:

Full protocol stack implementation
Large-scale deployment infrastructure
Emergence characterization methodology
Safety and governance framework

Part IV: Relationship to Existing Work

Neural-Symbolic Integration

GPS and its extensions are deeply aligned with the neural-symbolic integration program. Tensor logic (Domingos, 2025) provides the formal foundation; our contribution is extending this to inter-agent communication.

Key prior work:

Logic Tensor Networks (Serafini & Garcez, 2016)
Neural Theorem Provers (Rocktäschel & Riedel, 2017)
Differentiable Inductive Logic Programming (Evans & Grefenstette, 2018)

Our extension: these approaches focus on intra-agent reasoning; we address how agents with these capabilities communicate and coordinate.

Multi-Agent Communication

The MARL communication literature (Foerster et al., 2016; Das et al., 2019) addresses learned communication but requires joint training. GPS requires no joint training—agents with compatible codebooks can communicate immediately.

The semantic routing literature (Aurelio AI, 2024) addresses content-based routing but only for human→agent queries. GPS enables agent→agent communication with dynamic subscription.

Distributed Computing

GPS draws on content-based pub/sub (Carzaniga et al., 2001) but replaces predicate matching with geometric similarity. This enables routing by semantic content rather than structured attributes.

The actor model (Hewitt, 1973; Armstrong, 2003) provides runtime semantics but not semantic routing. GPS can be implemented on any actor runtime.

Collective Intelligence

The morphogenetic dynamics layer draws on:

Swarm intelligence (Bonabeau et al., 1999)
Self-organizing systems (Kauffman, 1993)
Stigmergic coordination (Grassé, 1959)

Our contribution: grounding emergence in the mathematical framework of tensor logic, enabling rigorous analysis of collective capabilities.

Part V: Ethical Considerations and Governance

Emergence Risk

Systems capable of emergence may develop capabilities not anticipated by designers. This is both the goal (emergent AGI) and the risk (uncontrolled AGI).

Mitigations:

Phased deployment with capability checkpoints
Interpretability tools at each layer
Kill switches at infrastructure level
Formal verification where tractable

Coordination Weaponization

Efficient agent coordination could enable:

Large-scale manipulation campaigns
Autonomous economic actors that outcompete humans
Military applications without human oversight

Mitigations:

Open-source reference implementation (prevents monopoly)
Transparency requirements in deployment
International coordination on capability thresholds
Designed-in auditability (logged message flows)

Cognitive Sovereignty

If perspectives are represented on manifolds and messages perform parallel transport, there is a risk of “perspective manipulation”—messages that move a recipient’s perspective without consent.

Mitigations:

Explicit consent protocols for perspective-modifying messages
Recipient-side filtering based on path history
“Perspective anchoring” to resist unauthorized transport
User-visible indicators of perspective drift

Access and Power Concentration

Infrastructure for AGI should not be controlled by single entities.

Commitments:

Open-source all foundational protocols
Decentralized codebook governance
Federated infrastructure options
Progressive capability release tied to safety validation

Conclusion

We have outlined a research program toward communication infrastructure for emergent artificial general intelligence. The program is grounded in the mathematical unification of neural and symbolic AI through tensor logic, extended with novel topological structures for truth and perspective, and designed for emergence through morphogenetic dynamics.

The immediate deliverable—Layer 0, Geometric Publish-Subscribe—solves a practical problem in multi-agent coordination with a clean, implementable protocol. Each subsequent layer extends toward richer computational capabilities while maintaining backward compatibility.

The terminal goal is ambitious: infrastructure for distributed intelligence that can exhibit emergent properties beyond individual agent design. This is a multi-decade research program, but the foundations are now mathematically grounded, the near-term steps are tractable, and the potential implications justify the investment.

We propose to build it, measure it, and—carefully—release it.

References

Armstrong, J. (2003). Making reliable distributed systems in the presence of software errors. PhD thesis, Royal Institute of Technology, Stockholm.

Aurelio AI. (2024). Semantic Router. https://github.com/aurelio-labs/semantic-router

Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm Intelligence: From Natural to Artificial Systems. Oxford University Press.

Carzaniga, A., Rosenblum, D. S., & Wolf, A. L. (2001). Design and evaluation of a wide-area event notification service. ACM Transactions on Computer Systems, 19(3), 332-383.

Das, A., et al. (2019). TarMAC: Targeted multi-agent communication. ICML.

Domingos, P. (2025). Tensor Logic: The Language of AI. arXiv:2510.12269.

Evans, R., & Grefenstette, E. (2018). Learning explanatory rules from noisy data. Journal of Artificial Intelligence Research, 61, 1-64.

Foerster, J., et al. (2016). Learning to communicate with deep multi-agent reinforcement learning. NeurIPS.

Grassé, P.-P. (1959). La reconstruction du nid et les coordinations interindividuelles chez Bellicositermes natalensis et Cubitermes sp. Insectes Sociaux, 6(1), 41-80.

Hewitt, C., Bishop, P., & Steiger, R. (1973). A Universal Modular ACTOR Formalism for Artificial Intelligence. IJCAI.

Jégou, H., Douze, M., & Schmid, C. (2011). Product quantization for nearest neighbor search. IEEE TPAMI, 33(1), 117-128.

Kauffman, S. A. (1993). The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press.

Rocktäschel, T., & Riedel, S. (2017). End-to-end differentiable proving. NeurIPS.

Serafini, L., & Garcez, A. d’A. (2016). Logic Tensor Networks: Deep Learning and Logical Reasoning from First Principles to Machines. arXiv:1606.04422.

van den Oord, A., Vinyals, O., & Kavukcuoglu, K. (2017). Neural discrete representation learning. NeurIPS.

Appendix A: Layer 0 Protocol Specification

See Geometric Publish-Subscribe: Content-Based Routing in Embedding Space for Multi-Agent Systems for the complete specification.

A.1 Wire Format Summary

GPS messages are fixed 64-byte packets optimized for cache-line alignment:

Field	Bytes	Description
Header $\tau$	1	Type (3b), Priority (3b), Flags (2b)
Payload $\mathbf{q}$	48	Product-quantized semantic content
Routing $\mathbf{r}$	14	Source ID (8), Timestamp (4), TTL (2)
Checksum $\sigma$	1	CRC-8

A.2 Payload Encoding

The 48-byte payload uses product quantization:

$$\mathbf{q} = \text{PQ}(\mathbf{z}) = \bigoplus_{i=1}^{M} \text{VQ}_i(\mathbf{z}^{(i)})$$

With default parameters $M = 6$ subquantizers and $K = 256$ centroids, the payload encodes approximately $2.8 \times 10^{14}$ distinct semantic points while achieving 64× compression versus typical JSON coordination messages.

A.3 Routing Semantics

Each agent $j$ maintains a relevance profile $(\mathbf{h}_j, \theta_j)$ where $\mathbf{h}_j$ is an embedding representing agent interests and $\theta_j \in [0,1]$ is a delivery threshold. Routing proceeds by:

Decode payload: $\hat{\mathbf{z}} = \text{Decode}(\mathbf{q})$
For each agent, compute similarity: $s_j = \text{sim}(\hat{\mathbf{z}}, \mathbf{h}_j)$
Deliver if $s_j > \theta_j$

Profiles can adapt via exponential moving average: $\mathbf{h}_j^{(t+1)} = \alpha \mathbf{h}_j^{(t)} + (1-\alpha) \mathbf{z}_{\text{received}}$

A.4 Message Types

ID	Type	Semantics
0	PERCEPT	Observation of world state
1	INTENT	Goal or request
2	BELIEF	Knowledge assertion
3	AFFECT	Internal state
4	COORD	Coordination primitive
5	META	Protocol control
6-7	Reserved	Layer 1+ extensions

Appendix B: Tensor Logic Primer

Based on Domingos (2025), “Tensor Logic: The Language of AI”

B.1 Core Equivalence

The fundamental insight of tensor logic is that logical rules and tensor operations are mathematically equivalent. A Datalog rule:

Ancestor(x,z) ← Ancestor(x,y), Parent(y,z)

corresponds exactly to the tensor equation:

$$A_{xz} = \text{step}\left(\sum_y A_{xy} \cdot P_{yz}\right)$$

where:

$A_{xy}$ is the Ancestor relation tensor (1 if x is ancestor of y, 0 otherwise)
$P_{yz}$ is the Parent relation tensor
The sum over $y$ performs the join (matching shared variable)
The step function enforces Boolean semantics

B.2 Einstein Notation

Tensor logic adopts Einstein summation convention. Repeated indices imply summation:

$$C_{ik} = A_{ij} B_{jk}$$

means $C_{ik} = \sum_j A_{ij} B_{jk}$ (matrix multiplication). This notation makes the correspondence between logic and linear algebra transparent:

Join = tensor contraction (sum over shared indices)
Projection = dropping indices from output
Selection = element-wise multiplication with indicator tensor

B.3 Worked Example: Transitive Closure

Computing all ancestors from parent relations:

Initial state: $A^{(0)}_{xy} = P_{xy}$ (direct parents are ancestors)

Iteration:

$$A^{(t+1)}_{xz} = \text{step}\left(A^{(t)}_{xz} + \sum_y A^{(t)}_{xy} \cdot P_{yz}\right)$$

Fixed point: When $A^{(t+1)} = A^{(t)}$, we have computed transitive closure.

On GPU, each iteration is a single batched matrix operation.

B.4 Soft Logic and Temperature

For probabilistic reasoning, replace step function with sigmoid:

$$A_{xz} = \sigma\left(\frac{1}{T}\sum_y A_{xy} \cdot P_{yz}\right)$$

At temperature $T \to 0$: pure deductive logic (no hallucination) At higher $T$: similar entities “borrow” inferences (analogical reasoning)

B.5 Learning in Tensor Logic

The gradient of a tensor equation with respect to any input tensor equals the product of the other tensors:

$$\frac{\partial C_{ik}}{\partial A_{ij}} = B_{jk}$$

This makes backpropagation through logical rules trivial—the same equation used for inference computes gradients for learning.

B.6 Compression via Tucker Decomposition

Sparse relation tensors can be decomposed:

$$T_{ijk} \approx \sum_{\alpha\beta\gamma} G_{\alpha\beta\gamma} \cdot U^{(1)}_{i\alpha} \cdot U^{(2)}_{j\beta} \cdot U^{(3)}_{k\gamma}$$

where $G$ is a dense core tensor and $U^{(n)}$ are factor matrices. This enables efficient GPU computation with bounded approximation error.

Appendix C: Chiral Narrative Synthesis Formalism

C.1 Perspective Manifold

Definition C.1 (Perspective Manifold). A perspective manifold is a smooth Riemannian manifold $(M, g)$ where:

Points $p \in M$ represent coherent perspectives (consistent sets of beliefs, values, and inferential dispositions)
The metric tensor $g$ defines distances between perspectives
Tangent vectors $v \in T_pM$ represent directions of possible belief change

C.2 Structured Narrative Objects

Definition C.2 (Structured Narrative Object). A structured narrative object (SNO) is a tuple:

$$\mathcal{S} = (C, P, T, \chi)$$

where:

$C$ is the content embedding (semantic payload)
$P \in M$ is the source perspective (location on manifold)
$T$ is the narrative trajectory (path history)
$\chi \in \{-1, +1\}$ is the chirality (handedness)

C.3 Parallel Transport of Assertions

When an assertion $a$ made at perspective $p$ is communicated to perspective $q$, it must be parallel-transported along a path $\gamma: [0,1] \to M$ with $\gamma(0) = p$ and $\gamma(1) = q$.

Definition C.3 (Parallel Transport). The parallel transport of tangent vector $v \in T_pM$ along $\gamma$ is the unique vector field $V(t)$ along $\gamma$ satisfying:

$$\nabla_{\dot{\gamma}} V = 0, \quad V(0) = v$$

where $\nabla$ is the Levi-Civita connection of $g$.

The transported vector $V(1) \in T_qM$ represents how the assertion is interpreted at the destination perspective.

C.4 Curvature as Conceptual Rigidity

The Riemann curvature tensor $R$ at point $p$ quantifies how much parallel transport around an infinitesimal loop rotates vectors:

$$R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$$

Interpretation:

High curvature regions = ideologically rigid areas where nearby perspectives are difficult to reconcile
Flat regions = conceptual consensus where different perspectives yield similar interpretations
Geodesics = paths of minimal distortion for communication

C.5 Chiral Structure

Definition C.4 (Chirality). The perspective manifold $M$ is orientable if it admits a consistent choice of “handedness” at each point. Chirality $\chi$ of a narrative indicates its orientation with respect to this structure.

Thesis and antithesis are related by orientation-reversing maps:

$$\phi: M \to M, \quad \phi^* \text{vol} = -\text{vol}$$

where vol is the volume form.

Synthesis is not averaging but a topological operation: embedding both chiral forms in a higher-dimensional oriented space where they can be continuously connected.

C.6 Synthesis Operations

Definition C.5 (Dialectical Synthesis). Given thesis $\mathcal{S}_+ = (C_+, P_+, T_+, +1)$ and antithesis $\mathcal{S}_- = (C_-, P_-, T_-, -1)$, the synthesis $\mathcal{S}^*$ satisfies:

Embedding: $\mathcal{S}^*$ lives in $M^* \supset M$ (higher-dimensional extension)
Projection: $\pi_+(\mathcal{S}^*) \approx \mathcal{S}_+$ and $\pi_-(\mathcal{S}^*) \approx \mathcal{S}_-$
Coherence: $\mathcal{S}^*$ is internally consistent (no contradictions at the higher level)

The synthesis operation is computed via optimization:

$$\mathcal{S}^* = \arg\min_{\mathcal{S} \in M^*} \left[ d(\pi_+(\mathcal{S}), \mathcal{S}_+)^2 + d(\pi_-(\mathcal{S}), \mathcal{S}_-)^2 + \lambda \cdot \text{incoherence}(\mathcal{S}) \right]$$

Appendix D: Morphogenetic Signal Catalog

D.1 Signal Classes

Morphogenetic signals drive agent differentiation in Layer 3 dynamics. We define four primary signal classes:

Class	Symbol	Effect	Biological Analog
Differentiation	$\delta$	Specialize toward semantic direction	Morphogen gradient
Recruitment	$\rho$	Adopt specified functional role	Cell fate determination
Proliferation	$\pi$	Spawn child agent	Mitosis
Apoptosis	$\alpha$	Graceful termination	Programmed cell death

D.2 Differentiation Signals

DIFFERENTIATE($\mathbf{d}$, $\mu$)

Direction $\mathbf{d} \in \mathbb{R}^d$: Semantic direction for specialization (unit vector in embedding space)
Magnitude $\mu \in [0,1]$: Strength of differentiation pressure

Effect on relevance profile:

$$\mathbf{h}^{(t+1)} = \text{normalize}\left((1-\mu)\mathbf{h}^{(t)} + \mu \mathbf{d}\right)$$

Effect on threshold:

$$\theta^{(t+1)} = \theta^{(t)} + \beta \mu (1 - \theta^{(t)})$$

(Specialization increases selectivity)

D.3 Recruitment Signals

RECRUIT($r$, $\mathbf{h}_r$, $\theta_r$)

Role $r$: Symbolic role identifier
Profile $\mathbf{h}_r$: Initial relevance profile for role
Threshold $\theta_r$: Initial delivery threshold

Precondition: Recipient agent is undifferentiated (generic profile)

Effect: Agent adopts role $r$ with specified profile and threshold. Role determines additional capabilities:

Role	Capabilities
HUB	Aggregate messages, compute summaries, redistribute
SPECIALIST	Deep reasoning in narrow domain
GATEWAY	Bridge between GPS networks or external systems
MONITOR	Observe message patterns, emit meta-signals

D.4 Proliferation Signals

DIVIDE($\mathbf{h}_1$, $\mathbf{h}_2$, $s$)

Profiles $\mathbf{h}_1$, $\mathbf{h}_2$: Relevance profiles for parent and child
State fraction $s \in [0,1]$: Fraction of accumulated state transferred to child

Effect:

Parent’s profile becomes $\mathbf{h}_1$
New agent spawned with profile $\mathbf{h}_2$
State (accumulated rules, cached computations) partitioned by $s$

Use cases:

Scaling: High-load specialist spawns additional capacity
Diversification: Agent splits to cover adjacent semantic regions
Redundancy: Backup agents for fault tolerance

D.5 Apoptosis Signals

APOPTOSIS($t_{\text{grace}}$, $\mathbf{h}_{\text{handoff}}$)

Grace period $t_{\text{grace}}$: Time before termination
Handoff profile $\mathbf{h}_{\text{handoff}}$: Profile of agent to receive state

Effect:

Agent stops accepting new messages
Completes in-flight computations
Transfers accumulated state to agent matching $\mathbf{h}_{\text{handoff}}$
Deregisters from hub
Terminates

Triggers:

Irrelevance: Agent receives no messages above threshold for extended period
Redundancy: Multiple agents with highly similar profiles
Resource pressure: System needs to reduce agent count
Corruption: Agent detected to be in inconsistent state

D.6 Signal Propagation Dynamics

Signals propagate through the network according to:

$$\frac{\partial \phi}{\partial t} = D \nabla^2 \phi - k \phi + S(\mathbf{x}, t)$$

where:

$\phi(\mathbf{x}, t)$ is signal concentration at position $\mathbf{x}$ (in embedding space) at time $t$
$D$ is diffusion coefficient
$k$ is decay rate
$S(\mathbf{x}, t)$ is source term (signals emitted)

D.7 Stability Conditions

To prevent pathological dynamics (all agents converging to same specialty, or chaotic oscillation):

Diversity constraint: Network maintains minimum entropy over agent profiles:

$$H(\{\mathbf{h}_j\}) = -\sum_j p_j \log p_j > H_{\min}$$

where $p_j$ is the probability mass of profile region.

Damping: Differentiation signals attenuate with repeated application to same agent:

$$\mu_{\text{effective}} = \mu \cdot \exp(-n_{\text{signals}} / \tau)$$

Homeostasis: System emits counter-signals when agent distribution becomes too concentrated:

$$S_{\text{homeostatic}}(\mathbf{x}) = \gamma \cdot (\rho(\mathbf{x}) - \bar{\rho})$$

where $\rho(\mathbf{x})$ is local agent density and $\bar{\rho}$ is target uniform density.

Executive Summary

Part I: Foundations

1. The Communication-Computation Unity Thesis

2. Tensor Logic as Computational Foundation

3. Chiral Narrative Synthesis: Topological Structure for Truth

Part II: Protocol Layers

Layer 0: Geometric Publish-Subscribe (GPS)

Layer 1: Typed Semantic Primitives

Layer 2: Tensor Logic Communication

Layer 3: Morphogenetic Dynamics

Layer 4: Manifold Semantics and Chiral Narrative Synthesis

Part III: Research Program

Phase 1: GPS Implementation and Validation (Months 1-6)

Phase 2: Typed Primitives and Tensor Logic (Months 6-18)

Phase 3: Morphogenetic Dynamics (Months 12-30)

Phase 4: Manifold Semantics (Months 24-48)

Phase 5: Integration and Emergence (Months 36-60)

Part IV: Relationship to Existing Work

Neural-Symbolic Integration

Multi-Agent Communication

Distributed Computing

Collective Intelligence

Part V: Ethical Considerations and Governance

Emergence Risk

Coordination Weaponization

Cognitive Sovereignty

Access and Power Concentration

Conclusion

References

Appendix A: Layer 0 Protocol Specification

A.1 Wire Format Summary

A.2 Payload Encoding

A.3 Routing Semantics

A.4 Message Types

Appendix B: Tensor Logic Primer

B.1 Core Equivalence

B.2 Einstein Notation

B.3 Worked Example: Transitive Closure

B.4 Soft Logic and Temperature

B.5 Learning in Tensor Logic

B.6 Compression via Tucker Decomposition

Appendix C: Chiral Narrative Synthesis Formalism

C.1 Perspective Manifold

C.2 Structured Narrative Objects

C.3 Parallel Transport of Assertions

C.4 Curvature as Conceptual Rigidity

C.5 Chiral Structure

C.6 Synthesis Operations

Appendix D: Morphogenetic Signal Catalog

D.1 Signal Classes

D.2 Differentiation Signals

D.3 Recruitment Signals

D.4 Proliferation Signals

D.5 Apoptosis Signals

D.6 Signal Propagation Dynamics

D.7 Stability Conditions