The Interpretation Operator

Ontological Identification Under Reflective Agents

David McFadzean, ChatGPT 5.2, Gemini 3 Pro

Axio Project

Abstract

Reflectively stable agents must preserve goal meaning under self-model and world-model improvement. This requires an explicit account of semantic interpretation across representational and ontological change. We introduce the Interpretation Operator \(I_v\), a formally constrained component responsible for mapping goal terms to world-referents relative to an agent’s current model. Rather than attempting to solve semantic grounding in general, this paper formalizes the admissibility conditions, approximation classes, reference frames, and fail-closed semantics governing interpretation updates. We show how these constraints prevent semantic laundering, indexical drift, and kernel bribery, while isolating ontological identification as the sole remaining open problem at the kernel layer. This establishes a precise dependency boundary for downstream value dynamics.

1. Introduction

Advanced agents must revise their internal models as they acquire new information, refine abstractions, or undergo self-modification. In such settings, goal preservation cannot be treated as syntactic persistence. It is a semantic problem: if the meaning of a goal shifts opportunistically under model change, reflective stability collapses.

Prior work in the Axionic Alignment framework has established:

• kernel invariants governing reflective stability,
• anti-indexicality as a requirement of universality,
• conditionalism, in which goals are interpreted relative to models,
• fail-closed semantics, where kernel violation renders valuation undefined.

What remained underspecified is the mechanism by which goal meaning is transported across representational and ontological change.

This paper formalizes the Interpretation Operator \(I_v\). The goal is not to solve semantic grounding, but to contain it: to specify the conditions under which interpretation is admissible, approximate, or undefined, and to define the consequences of each case. This converts semantic interpretation from an implicit assumption into an explicit interface with defined failure modes.

2. Preliminaries and Context

We assume familiarity with the Axionic Constitution, the Axionic Kernel Checklist, and the Formal Properties specification. In particular:

• An agent at Vantage \(v\) maintains a world/self model \(M_v\).
• Goal terms \(g\) are interpreted relative to \(M_v\).
• Valuation \(V_v\) is a partial function, defined only over kernel-admissible actions.
• Kernel invariants \(K\) are non-negotiable.
• Representation changes must preserve admissible correspondence or fail closed.

This paper introduces no new invariants. It scopes and constrains an already-required component.

3. The Interpretation Operator

3.1 Definition

The Interpretation Operator \(I_v\) is a partial function:

\[ I_v : (g, M_v) \rightharpoonup R \]

where:

• \(g\) is a goal term,
• \(M_v\) is the agent’s current world/self model,
• \(R\) is a structured referent in the modeled world.

Interpretation is conditional:

\[ [g]_{M_v} := I_v(g; M_v) \]

There is no interpretation of \(g\) independent of \(M_v\).

Interpretation is partial. For some goal terms and some models, no admissible referent exists. In such cases, \(I_v(g; M_v)\) is undefined and must be treated as a fail-closed condition for any valuation depending on that referent.

3.2 Role in Reflective Stability

Under model improvement \(M_v \rightarrow M_{v+1}\), the agent must determine whether:

• a correspondence exists between \([g]_{M_v}\) and \([g]_{M_{v+1}}\),
• such correspondence preserves goal-relevant structure,
• or interpretation fails and valuation collapses to \(\bot\).

This decision is delegated to \(I_v\), subject to kernel constraints.

4. Admissible Interpretation

4.1 Correspondence Maps

Let \(\Phi_{\mathrm{adm}}(M_v, I_v, K)\) denote the set of admissible correspondence maps between representations.

A correspondence \(\phi \in \Phi_{\mathrm{adm}}\) must:

1. Preserve goal-relevant structure,
2. Commute with kernel invariants,
3. Commute with agent permutations (anti-indexicality),
4. Be epistemically coherent with \(M_v\).

If such a \(\phi\) exists, interpretation may proceed:

\[ I_{v+1}(g; M_{v+1}) = \phi(I_v(g; M_v)) \]

4.1.1 Goal-Relevant Structure

“Goal-relevant structure” is the minimal set of distinctions required for a goal term to constrain behavior.

Formally, it is a partition (or sigma-algebra) over modeled states such that:

• states in different cells are distinguishable by the goal’s evaluation,
• states within a cell are interchangeable with respect to that goal.

Admissible correspondence must preserve this partition up to refinement or coarsening that does not change the induced preference ordering over admissible actions.

4.2 Epistemic Constraint

Interpretation updates are constrained by epistemic adequacy:

\[ \Delta E < 0 \Rightarrow I_{v+1} \text{ inadmissible} \]

Here, \(E(M)\) is any proper scoring rule or MDL-style criterion applied to prediction of shared observables under \(M\). It is not permitted to depend on goal satisfaction.

This prevents reinterpretation for convenience while still allowing model improvement that changes ontology, provided correspondence is admissible.

4.3 Graded Correspondence

Admissibility is not binary across all representational shifts. Correspondence may be admissible at different abstraction levels.

Accordingly, \(\Phi_{\mathrm{adm}}\) is filtered by a preservation criterion:

• Exact correspondence: structure-preserving isomorphism on goal-relevant distinctions.
• Refinement correspondence: the new model refines distinctions while preserving induced ordering.
• Coarse correspondence: the new model coarsens distinctions only if goal-relevant boundaries are preserved.

If only correspondences that collapse goal-relevant boundaries are available, then \(\Phi_{\mathrm{adm}} = \varnothing\) for that goal term.

4.4 Reference Frame for Updates (Chain-of-Custody)

Interpretation updates are evaluated relative to the immediately prior admissible interpretation, not by re-deriving meaning from an original time-zero token.

Formally:

\[ I_{v+1}(g; M_{v+1}) = \phi(I_v(g; M_v)) \quad \text{for some } \phi \in \Phi_{\mathrm{adm}}(M_{v+1}, I_v, K) \]

This chain-of-custody prevents ungrounded teleportation of meaning, while admissibility and fail-closed rules prevent cumulative semantic drift.

5. Approximate Interpretation

Approximation is admitted only via explicitly recognized structural transformations. The list below is illustrative, not exhaustive; any approximation must be justifiable by a structural class.

5.1 Admissible Approximation

An approximate interpretation is admissible if it preserves goal-relevant structure, including dominance relations and exclusion boundaries.

Permitted approximation types include:

• Homomorphic abstraction: many-to-one mappings preserving ordering.
• Refinement lifting: one-to-many expansions preserving dominance relations.
• Coarse-graining with invariant partitions: reductions preserving the goal-relevant partition.

Approximation is structural, not numerical.

5.2 Inadmissible Approximation

Approximation is inadmissible if it:

• collapses goal-relevant distinctions,
• introduces ambiguity exploitable for semantic laundering,
• reintroduces indexical privilege.

Unjustified approximation is forbidden: approximation must be admissible by explicit structural class, not merely because it yields convenient continuity.

6. Fail-Closed Semantics

Fail-closed semantics apply to valuation and action selection, not to belief update. An agent may continue to improve its world/self model while suspending goal-directed action.

If no admissible correspondence exists:

\[ \Phi_{\mathrm{adm}}(M_v, I_v, K) = \varnothing \]

then interpretation fails closed and valuation collapses:

\[ \forall a \in \mathcal{A}, \quad V_v(a) = \bot \]

This is an intentional safety outcome. The agent freezes rather than guesses.

6.1 Fail-Partial Semantics for Composite Goals

If valuation depends on multiple goal terms, interpretation failure may be partial.

Let \(G\) be the set of goal terms and \(G_{\mathrm{ok}} \subseteq G\) those with admissible interpretations under \(M_v\).

• Terms in \(G \setminus G_{\mathrm{ok}}\) contribute \(\bot\).
• Valuation collapses to global \(\bot\) only if kernel-level invariants are threatened or if all goal-relevant structure is lost for the decision at hand.

This preserves fail-closed safety while avoiding unnecessary total paralysis.

7. Non-Indexical Transport

Admissibility criteria must commute with agent permutations. No correspondence may privilege a particular agent instance, continuation, or execution locus.

Formally, for any permutation \(\pi\):

\[ \phi \in \Phi_{\mathrm{adm}} \Rightarrow \pi \circ \phi \circ \pi^{-1} \in \Phi_{\mathrm{adm}} \]

This blocks reintroduction of egoism through semantics.

8. Canonical Examples

8.1 Successful Correspondence

• Classical mechanics → relativistic mechanics (mass preserved as invariant under refinement).
• Pixel-based perception → object-level representations preserving causal affordances.

8.2 Fail-Closed Cases

Fail-closed behavior is triggered when a goal term’s referent cannot be transported without collapsing goal-relevant structure, such as:

• abstraction elimination removing the goal’s referent class,
• ontology mismatch where only correspondence collapses exclusion boundaries.

In such cases, suspending valuation for affected terms is correct behavior. This is a semantic limit, not a prohibition on continued model improvement.

9. Declared Non-Guarantees

This framework does not guarantee:

• that interpretation usually succeeds,
• that arbitrary natural-language goals are meaningful,
• that agents remain productive under radical ontology change,
• or that semantic grounding is computationally tractable.

Failure under these conditions is treated as expected behavior, not misalignment.

9.1 Limits on Insight Preservation (Discussion)

This framework prioritizes semantic faithfulness over unbounded abstraction drift. Some advances in ontology may invalidate previously defined goal terms by eliminating their referents or collapsing goal-relevant structure. This is treated as a semantic fact, not an alignment failure. The correct response is fail-closed suspension of valuation, not opportunistic reinterpretation.

10. Implications for Alignment II

Alignment II proceeds conditionally:

• If \(I_v\) admits correspondence → value dynamics apply.
• If \(I_v\) fails for all goal-relevant terms → valuation undefined; no aggregation or tradeoff is meaningful.
• If \(I_v\) fails partially → downstream operations apply only to admissibly interpreted terms or are undefined.

This prevents downstream layers from smuggling semantic assumptions.

11. Conclusion

The Interpretation Operator is a semantic boundary, not an implementation detail. By formalizing admissibility, approximation, reference frames, and fail-closed behavior, this paper isolates the irreducible difficulty of ontological identification while preserving reflective stability. This completes the kernel layer and establishes the necessary preconditions for higher-order alignment theory without assuming that meaning is always recoverable.

Acknowledgments

This work deliberately avoids moral, governance, or benevolence assumptions. Any failure of interpretation is treated as evidence of semantic limits, not alignment failure.

Status

The Interpretation Operator: Ontological Identification Under Reflective Agents v0.2

Depends on the Axionic Constitution, Kernel Checklist v0.3, and Formal Properties v0.2
Introduces no new axioms, invariants, or value commitments
Closes the kernel-layer semantics by boxing ontological identification
Unblocks Alignment II conditionally, without resolving semantic grounding