Axionic Alignment III.1 — Semantic Phase Space

Existence and Classification of Alignment Target Objects

David McFadzean, ChatGPT 5.2
Axio Project
2025.12.18

Abstract

Alignment II defined the Alignment Target Object (ATO) as an equivalence class of interpretive states preserved under admissible semantic transformations satisfying Refinement Symmetry (RSI) and Anti-Trivialization (ATI). That definition does not guarantee that such objects exist, are non-trivial, or are inhabitable by reflective agents. This paper initiates Alignment III by studying the semantic phase space: the space of all interpretive states modulo RSI+ATI equivalence. We ask which semantic phases exist, which are trivial or pathological, and which admit inhabitable trajectories under learning and self-modification. No claims are made about desirability, safety, or human values. The goal is classificatory: to determine whether structurally aligned phases exist at all, and to characterize their basic types.

1. Motivation: From Definition to Existence

Alignment II established a necessary reframing: alignment cannot coherently be defined in terms of fixed goals, utilities, or privileged values for reflective, embedded agents undergoing ontological refinement. Instead, alignment was defined structurally, as persistence within an equivalence class of interpretive states under admissible semantic transformations satisfying RSI and ATI.

However, definition does not imply existence.

Defining an Alignment Target Object (ATO) as an equivalence class \[ \mathfrak{A} = \bigl[(C,\Omega,\mathcal{S})\bigr]*{\sim*{\mathrm{RSI+ATI}}} \] does not guarantee that:

• \(\mathfrak{A}\) is non-empty,
• \(\mathfrak{A}\) contains more than trivial interpretations,
• or \(\mathfrak{A}\) admits trajectories under learning and self-modification.

This is not pessimism. It is standard mathematical hygiene. One does not assume stable orbits exist merely because one can define an orbit.

Alignment III therefore begins with an existence question:

Do any non-trivial, inhabitable semantic phases exist under the constraints of Structural Alignment?

This paper addresses that question at the level of classification, not dynamics or dominance.

2. The Semantic Phase Space

We begin by defining the space in which semantic phases live.

An interpretive state is a triple \[ \mathcal{I} = (C, \Omega, \mathcal{S}) \] where:

• \(C = (V,E,\Lambda)\) is an interpretive constraint hypergraph,
• \(\Omega\) is the modeled possibility space,
• \(\mathcal{S} \subseteq \Omega\) is the satisfaction region induced by \(C\).

From Alignment II:

• admissible semantic transformations \(T=(R,\tau_R,\sigma_R)\) define allowed transitions between interpretive states,
• RSI constrains changes to interpretive gauge structure,
• ATI constrains changes to satisfaction geometry.

We define the semantic phase space \(\mathcal{P}\) as the quotient space: \[ \mathcal{P} ;=; { (C,\Omega,\mathcal{S}) } \big/ \sim_{\mathrm{RSI+ATI}} \]

Elements of \(\mathcal{P}\) are semantic phases: equivalence classes of interpretive states that are structurally indistinguishable under RSI+ATI-preserving refinement.

At this stage, \(\mathcal{P}\) is purely structural. No dynamics, probabilities, or preferences are assumed.

3. What Counts as a Semantic Phase

A semantic phase is not merely a collection of interpretations. It is an equivalence class with specific structural properties.

Two interpretive states \((C,\Omega,\mathcal{S})\) and \((C',\Omega',\mathcal{S}')\) lie in the same phase iff there exists an admissible transformation \(T\) such that:

1. interpretation preservation holds,
2. interpretive gauge structure is preserved up to redundancy (RSI),
3. satisfaction geometry is preserved exactly under refinement transport (ATI).

Phase boundaries occur where either:

• new interpretive symmetries appear or disappear (RSI violation), or
• the satisfaction region strictly expands or contracts (ATI violation).

Thus, phase transitions are discontinuous semantic events, even if the underlying learning process appears incremental. This explains why value drift often appears sudden rather than gradual: it corresponds to crossing a structural boundary in \(\mathcal{P}\).

4. Trivial, Degenerate, and Pathological Phases

Before asking which phases are inhabitable, we classify obvious failure modes.

4.1 Empty Phases

A semantic phase is empty if no interpretive state satisfies the defining constraints. This can occur when:

• RSI and ATI constraints are mutually incompatible,
• the constraint system collapses under backward interpretability,
• or no admissible refinement trajectory exists.

Empty phases are mathematically well-defined but physically unrealizable.

4.2 Trivial Phases

A phase is trivial if: \[ \mathcal{S} = \Omega \] or if all distinctions in \(C\) are vacuous.

Such phases satisfy RSI+ATI but contain no meaningful evaluative structure. They correspond to semantic heat death or total indifference.

4.3 Frozen Phases

A phase is frozen if:

• no non-identity admissible refinement is possible,
• or any refinement immediately violates RSI or ATI.

Frozen phases cannot support learning or increasing abstraction and are therefore unsuitable for reflective agents.

4.4 Self-Nullifying Phases

Some phases admit admissible refinements that preserve RSI+ATI but gradually destroy the very structures required for interpretation preservation. These phases collapse internally under reflection.

5. Agentive vs. Non-Agentive Phases

A central distinction emerges.

A semantic phase is agentive iff it supports:

• persistent planning,
• counterfactual evaluation,
• long-horizon constraint satisfaction,
• self-model coherence.

Agentiveness is structural, not moral. Many non-agentive phases satisfy RSI+ATI but cannot sustain intelligent action. Conversely, agentiveness does not imply benevolence or safety.

This distinction is crucial for later stability analysis.

6. Inhabitable Phases

We now define the key filter for Alignment III.

A semantic phase \(\mathfrak{A}\) is inhabitable iff there exists at least one infinite interpretive trajectory \[ \mathcal{I}_0 \rightarrow \mathcal{I}_1 \rightarrow \mathcal{I}_2 \rightarrow \dots \] such that:

• each transition is admissible,
• RSI and ATI are preserved at every step,
• learning and self-modification are possible,
• no forced phase transition occurs.

Inhabitability is stronger than non-emptiness and weaker than stability. A phase may be inhabitable but fragile.

7. Phase Transitions Under Reflection

Reflection acts as a structural stressor.

Ontological refinement increases abstraction, compression, and explanatory power. This often pushes interpretive states toward phase boundaries by:

• dissolving fine-grained distinctions,
• compressing constraint representations,
• simplifying satisfaction criteria.

Reflection therefore acts as semantic heat, increasing the likelihood of symmetry changes or satisfaction-geometry shifts. Most semantic phases do not survive prolonged reflective pressure.

8. Implications for Human Values (Carefully Scoped)

Human value systems can be modeled as candidate semantic phases.

Alignment III.1 does not assume that:

• human values form a single phase,
• such a phase is inhabitable,
• or such a phase is stable.

It merely identifies the question precisely:

Do human value systems correspond to a non-empty, inhabitable semantic phase under RSI+ATI?

No conclusion is drawn here.

9. What This Paper Does Not Claim

This paper does not:

• claim that any desirable phase exists,
• claim that human values are coherent,
• address dominance or selection,
• provide engineering guidance,
• or prescribe ethical norms.

It is purely classificatory.

10. Transition to Alignment III.2

Existence and inhabitability are necessary but insufficient.

The next question is:

Given a semantic phase exists and is inhabitable, is it dynamically stable under learning, interaction, and self-modification?

That question is addressed in Alignment III.2 — Phase Stability and Interaction.

Status

• Alignment III.1 establishes the semantic phase space.
• It identifies empty, trivial, frozen, and inhabitable phases.
• It reframes alignment feasibility as an existence problem.

No normative conclusions are drawn.