Phi-Manifolds & Meaning-Memory: Summary for Multi-Agent Coherence

Tagline: Geometry of Meaning

Authors: Ramsey Ajram ramsey@orgs.io

Date: 2025-11-24

Abstract

This paper outlines the philosophical and architectural foundation for Phi-Manifolds, a geometric approach to agent memory and meaning. We propose that an agent's internal world should be structured as a manifold—a space with coordinates, curvature, and neighborhoods—rather than a bag of embeddings. By defining how seeds (inputs) expand into this manifold through deterministic rules, we enable agents to maintain stable "mental landscapes," share meaning through geometric coordinates, and evolve coherent long-term identities without drift.

1. What a Phi-Manifold Is (Operational Definition)

A phi-manifold is the structured space in which an agent’s internal representations live.

• We treat it as a geometric organisation of meaning: a space with coordinates, curvature, and neighbourhoods that encode how concepts relate.

• Each seed (input, value, feeling, signal, event) expands through a sequence of transformations:

  F0 → F1 → F2 → … → Fn

  where each Fn is a more structured, higher-order interpretation of the seed.

• By the time we reach mid-level depth (F4–F6), we have something like a stable conceptual scaffold: a space the agent can “think in” consistently over time.

The key: A manifold gives us shape, continuity, and constraints so that an agent’s internal world doesn’t drift or collapse. It’s similar to giving a neural net a latent space—except here the geometry is explicitly defined and interpretable.

2. What Phi-Meaning Is

Phi-meaning is the interpretation rule that maps raw signals into the manifold with structure, not just embeddings.

• It is not semantic embedding or vector similarity.

• It’s a structured, layered interpretation based on:

  • values
  • relevance
  • context
  • relationships
  • prior history
  • affective state (if relevant)

• It ensures the seed lands in the right region of the manifold and inherits the correct relational structure.

In simple terms: Phi-meaning tells us what something means within the agent’s worldview, not in isolation.

3. How Phi-Memory Works

Phi-memory stores:

• the seed
• its depth/expansion (F0→Fn)
• its location in the manifold
• the relationships it formed
• the value-based appraisal it produced

This gives us persistent geometric anchors.

Over time, memory becomes a mesh of stable manifolds that agents can keep building on.

Key: Memory isn’t just content; it is topology. When the agent recalls something, it returns to the same region of the manifold with all constraints intact.

4. How This Produces Long-Term Coherence in Multi-Agent Systems

A. Stable Internal Geometry

Each agent maintains a consistent manifold. No drift. No entropic collapse. No contradictory local reasoning. This gives long-lived agents a stable “mental landscape”.

B. Shared Coordinate System Across Agents

If we agree on:

• how seeds expand
• what counts as structure at F1–F6
• how values shape curvature

then agents can exchange meaning coherently.

They aren’t passing raw text; they’re passing manifold-anchored meaning packets.

C. Value-Shaped Inference

Because phi-meaning incorporates values, the entire system has aligned long-horizon behaviour. Agents won’t drift into contradictory decision-making because they are literally navigating the same shaped space.

D. Memory as Manifold-Continuation

When an agent remembers something, it doesn’t fetch a string or an embedding—it continues a previously formed trajectory in the manifold. That’s what gives us:

• continuity of identity
• consistency of preferences
• persistent long-term goals
• recognisable behaviour patterns

This is what current LLM-based agents fundamentally lack.

E. Collective Coherence

When multiple agents share (or partially share) the phi-manifold structure, we get emergent collective coherence:

• shared grounding
• interpretable communication
• predictable interaction patterns
• reduced semantic divergence

Agents “grow together” instead of diverging over time.

5. How We Use This in Our System

We implement phi-manifolds and phi-memory as:

1. Seed → Fn expansion pipeline Deterministic, structured, reproducible.

2. Manifold coordinate representation JSON-serialisable structured vectors + relational matrices.

3. Value-curvature shaping Our shared value-system directly adjusts manifold geometry.

4. Shared schema for cross-agent exchange Meaning packets are not embeddings; they are structured manifold objects.

5. Long-term memory store A continuation of the manifold, not an external side-channel.

6. Consistency checks When anything new enters an agent, it is integrated into the manifold only if it preserves topological constraints.

This is the foundation that gives us coordinated multi-agent behaviour without collapse, drift, or incoherence across long time horizons.

6. PHI-MANIFOLD GLOSSARY (Meaning/Memory)

(Short, Precise, No Fluff)

SEED (F₀)

The smallest expression of intent. The origin point. Everything grows from this. Example: “Help me live a coherent life.”

PRINCIPLES (F₁)

The rules or invariants implied by the seed. They don’t change easily. Example: Clarity, Integrity, Growth.

ATTRIBUTES (F₂)

Contextual details that shape how meaning shows up. These do change. Example: energy level, environment, constraints, temperament, resources.

HARMONICS / INTERACTIONS (F₃–F₄)

Combinations of principles + attributes. This is where structure and patterns emerge. Example: Integrity × constraints → “small, realistic value-aligned actions.”

PROTO-ATTRACTOR (F₅)

A repeated pattern that appears stable, but isn’t fully hardened yet. Example: “Most days feel better when I check in with my values first.”

ATTRACTOR (F₆)

A stable basin of meaning. An attractor is stable across time, reinforced by experience, aligned with the seed, generative, and something life naturally falls back into. Example: “My day works best when it starts with grounding.”

CHECKPOINT SEED

When an attractor becomes stable enough, it becomes the new seed. This represents evolution, new direction, upgraded identity, or a new phase of life. Example: From attractor “Adaptive daily rhythm” to new seed “Live a life designed around alignment rituals.”

RULE-SET

The “physics” the system uses to generate meaning. Different rule-sets = different modes (exploratory, convergent, reframing, identity-forming, evolutionary).

VALUE GRADIENT

How strongly a memory or idea supports the seed or attractor. Like a “pull force” (0 = no pull, 1 = extremely aligned).

TENSION

How much conflict a memory carries relative to the seed or attractor. High tension = needs reframing / new rule-set.

MANIFOLD

The complete structure generated from the seed: principles, attributes, harmonics, patterns, proto-attractors, and attractors. This is the “semantic landscape.”

7. Simplest Summary

• Seed → origin of meaning
• Principles → rules
• Attributes → context
• Harmonics → patterns
• Attractors → stable meaning
• Checkpoint seed → attractor becomes new origin
• Rule-set → generative mode
• Gradient/tension → forces shaping movement
• Manifold → everything combined