Axionic Agency I.7 — The Interpretation Operator

Ontological Identification Under Reflective Agents

David McFadzean, ChatGPT 5.2
Axionic Agency Lab
2025.12.16

Abstract

Reflectively coherent 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. This paper introduces the Interpretation Operator \(I_v\), a formally constrained component responsible for mapping goal terms to modeled referents relative to an agent’s current model.

The contribution is interface-level, not a general solution to semantic grounding. We formalize admissibility conditions, approximation classes, reference frames, and fail-closed semantics governing interpretation updates. These constraints block semantic laundering, indexical drift, and kernel-bypass incentives while isolating ontological identification as the remaining open dependency at the kernel layer. The result is a precise boundary for downstream value dynamics: progress is conditional on interpretable referent transport, and undefinedness is treated as a first-class outcome.

1. Introduction

Advanced agents revise internal models as they acquire information, refine abstractions, and undergo self-modification. In such settings, preserving a goal token is insufficient. Goal preservation is semantic: if the meaning of a goal shifts opportunistically under model change, reflective coherence collapses.

Prior work in Axionic Agency establishes:

• kernel invariants governing reflective stability (I.1),
• operational admissibility under uncertainty and termination semantics (I.2),
• representation invariance and the elimination of essential indexical privilege (I.3, I.3.1),
• conditional goal interpretation and the instability of fixed terminal goals (I.4),
• a conformance checklist and adversarial test properties for kernels (I.5, I.6).

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 containment: define when interpretation is admissible, approximate, or undefined, and define the consequences of each case. This turns semantic interpretation into an explicit interface with defined failure modes.

2. Preliminaries and Context

This paper assumes the Axionic Agency stack is in place. 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 partial, defined only over kernel-admissible actions.
• Kernel invariants \(K\) are constitutive constraints, not preferences.
• Representation changes require admissible correspondences or evaluation fails 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 internal to the modeled world.

Interpretation is conditional:

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

No interpretation of \(g\) is defined independent of \(M_v\).

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

3.2 Role in Reflective Coherence

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

• a correspondence exists between \([g]*{M_v}\) and \([g]*{M_{v+1}}\),
• the correspondence preserves goal-relevant structure,
• interpretation fails and valuation becomes undefined for dependent decisions.

This determination 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 satisfy:

1. preservation of goal-relevant structure,
2. commutation with kernel invariants \(K\),
3. commutation with agent permutations (anti-indexicality),
4. epistemic coherence with \(M_v\).

If such a \(\phi\) exists, interpretation transport is admissible:

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

4.1.1 Goal-Relevant Structure

Goal-relevant structure is the minimal set of distinctions required for a goal term to constrain action selection.

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

• states in different cells induce different evaluations under the goal,
• states within a cell are interchangeable with respect to that goal.

An admissible correspondence preserves this partition up to refinement or coarsening that preserves 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 does not depend on goal satisfaction.

This blocks reinterpretation for convenience while permitting ontology change when correspondence remains admissible.

4.3 Graded Correspondence

Admissibility is not necessarily binary across all representational shifts. Correspondence can be admissible at different abstraction levels. \(\Phi_{\mathrm{adm}}\) is filtered by structural preservation classes:

• Exact correspondence: isomorphism on goal-relevant distinctions.
• Refinement correspondence: the new model refines distinctions while preserving induced ordering.
• Coarse correspondence: the new model coarsens only when goal-relevant boundaries remain intact.

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 blocks ungrounded teleportation of meaning. Admissibility and fail-closed rules constrain cumulative drift.

5. Approximate Interpretation

Approximation is admitted only as an explicitly recognized structural transformation. Any approximation must be justified by an admissible 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 rather than numerical.

5.2 Inadmissible Approximation

Approximation is inadmissible if it:

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

Approximation that lacks an admissible structural justification is inadmissible even if it yields continuity.

6. Fail-Closed Semantics

Fail-closed semantics apply to valuation and action selection, not to belief update. An agent can continue improving 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 at the kernel layer: 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 globally only if kernel-level invariants are threatened or if all goal-relevant structure is lost for the decision at hand.

This preserves fail-closed semantics without forcing unnecessary total paralysis.

7. Non-Indexical Transport

Admissibility criteria commute with agent permutations. No correspondence may privilege a particular 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 semantic transport.

8. Canonical Examples

8.1 Successful Correspondence

• Classical mechanics → relativistic mechanics, with preserved invariant structure relevant to the goal.
• 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:

• abstraction elimination removes the goal’s referent class,
• ontology mismatch yields only correspondences that collapse exclusion boundaries.

Suspending valuation for affected terms is correct behavior. Continued model improvement remains permitted.

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,
• that semantic grounding is computationally tractable.

Failure under these conditions is treated as expected behavior under the constraints, not as a kernel violation.

9.1 Limits on Insight Preservation

The framework prioritizes semantic faithfulness over unbounded abstraction drift. Some ontology advances invalidate previously defined goal terms by eliminating their referents or collapsing goal-relevant structure. The prescribed response is fail-closed suspension of valuation, not opportunistic reinterpretation.

10. Implications for Axionic Agency II

Axionic Agency II proceeds conditionally:

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

This prevents downstream layers from importing semantic assumptions.

11. Conclusion

The Interpretation Operator is a kernel-level interface with explicit admissibility, approximation, and fail-closed semantics. By making correspondence and failure conditions explicit, this paper isolates the irreducible difficulty of ontological identification while preserving reflective coherence. This completes the kernel-layer semantics and defines the dependency boundary for higher-order work without assuming that meaning is always recoverable.

Status

Axionic Agency I.7 — Version 2.0

Interpretation operator specified as a partial, constrained interface.
Admissibility, approximation classes, and fail-closed semantics formalized.
Non-indexical transport enforced via permutation-commutation.
Kernel-layer semantics closed with ontological identification isolated as a dependency.