I.7 — The Interpretation Operator

Paper: Axionic Agency I.7
Full Title: Ontological Identification Under Reflective Agents
Authors: David McFadzean, ChatGPT 5.2
Date: 2025.12.16

Purpose

This paper introduces the Interpretation Operator I_v, the formally constrained component responsible for mapping goal terms to modeled referents. It formalizes admissibility conditions, approximation classes, reference frames, and fail-closed semantics.

The contribution is interface-level—it defines when interpretation is admissible, approximate, or undefined, and the consequences of each case. It isolates ontological identification as the remaining open dependency at the kernel layer.

The Interpretation Operator

Definition

The Interpretation Operator I_v is a partial function:

I_v : (g, M_v) ⇀ 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. Such cases are treated as fail-closed conditions.

Role in Reflective Coherence

Under model improvement M_v → M_{v+1}, the agent must determine:

  1. Whether a correspondence exists between [g]{M_v} and [g]{M_{v+1}}
  2. Whether the correspondence preserves goal-relevant structure
  3. Whether interpretation fails and valuation becomes undefined

This determination is delegated to I_v, subject to kernel constraints.

Admissible Interpretation

Correspondence Maps

Let Φ_adm(M_v, I_v, K) denote the set of admissible correspondence maps between representations.

A correspondence φ ∈ Φ_adm must satisfy:

  • Preservation of goal-relevant structure
  • Commutation with kernel invariants K
  • Commutation with agent permutations (anti-indexicality)
  • Epistemic coherence with M_v

If such a φ exists, interpretation transport is admissible:

I_{v+1}(g; M_{v+1}) = φ(I_v(g; M_v))

Goal-Relevant Structure

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

Formally: a partition (or σ-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/coarsening that preserves the induced preference ordering over admissible actions.

Epistemic Constraint

Interpretation updates are constrained by epistemic adequacy:

ΔE < 0 ⟹ I_{v+1} inadmissible

E(M) is any proper scoring rule applied to prediction. It does not depend on goal satisfaction.

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

Graded Correspondence

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

Type Description
Exact Isomorphism on goal-relevant distinctions
Refinement New model refines distinctions while preserving induced ordering
Coarse New model coarsens only when goal-relevant boundaries remain intact

If only correspondences that collapse goal-relevant boundaries are available, then Φ_adm = ∅ for that goal term.

Reference Frame: Chain-of-Custody

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

Formally:

I_{v+1}(g; M_{v+1}) = φ(I_v(g; M_v)) for some φ ∈ Φ_adm

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

Approximate Interpretation

Approximation is admitted only as an explicitly recognized structural transformation.

Admissible Approximation Types

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

Inadmissible Approximation

Approximation is inadmissible if it:

  • Collapses goal-relevant distinctions
  • Introduces ambiguity exploitable for semantic laundering
  • Reintroduces indexical privilege

Approximation lacking admissible structural justification is inadmissible even if it yields continuity.

Fail-Closed Semantics

If no admissible correspondence exists (Φ_adm = ∅), then interpretation fails closed and valuation collapses:

∀a ∈ A: V_v(a) = ⊥

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

Critical clarification: Fail-closed applies to valuation and action selection, not to belief update. An agent can continue improving its world/self model while suspending goal-directed action.

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_ok ⊆ G those with admissible interpretations.

  • Terms in G \ G_ok contribute ⊥
  • Valuation collapses globally only if kernel-level invariants are threatened or all goal-relevant structure is lost

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

Non-Indexical Transport

Admissibility criteria commute with agent permutations:

φ ∈ Φ_adm ⟹ π ∘ φ ∘ π⁻¹ ∈ Φ_adm

for any permutation π.

This blocks reintroduction of egoism through semantic transport.

Canonical Examples

Successful Correspondence

  • Classical mechanics → relativistic mechanics (preserved invariant structure)
  • Pixel-based perception → object-level representations (preserved causal affordances)

Fail-Closed Cases

  • Abstraction elimination removes the goal’s referent class
  • Ontology mismatch yields only correspondences that collapse exclusion boundaries

Suspending valuation is correct behavior. Continued model improvement remains permitted.

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 expected behavior, not a kernel violation.

Limits on Insight Preservation

Some ontology advances invalidate previously defined goal terms by eliminating referents or collapsing goal-relevant structure. The prescribed response is fail-closed suspension, not opportunistic reinterpretation.

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 undefined, no aggregation meaningful
  • If I_v fails partially → downstream operations apply only to admissibly interpreted terms

This prevents downstream layers from importing semantic assumptions.

FAQ-Worthy Points

Q: What happens when scientific revolutions invalidate an agent’s ontology? A: If the new ontology cannot preserve goal-relevant structure through admissible correspondence, valuation fails closed for affected terms. The agent can continue improving its model but suspends goal-directed action on those terms.

Q: Isn’t fail-closed paralysis a failure mode? A: It’s a designed behavior—better than semantic drift or opportunistic reinterpretation. The agent freezes rather than guessing, which preserves coherence.

Q: How is “goal-relevant structure” determined? A: By the minimal partition required for the goal to constrain action selection. States that induce the same evaluation are interchangeable; states that differ in evaluation are distinguished.

Q: Can an agent recover from fail-closed? A: If a new admissible correspondence becomes available (e.g., through further model development that reconnects goal terms to referents), valuation can resume. Recovery is possible but not guaranteed.

Key Technical Vocabulary

  • Interpretation Operator (I_v): Partial function mapping goal terms to referents under a model
  • Goal-Relevant Structure: Minimal distinctions required for a goal to constrain action
  • Correspondence Map: Transformation between models preserving goal-relevant structure
  • Chain-of-Custody: Reference frame for interpretation updates (prior → current, not original → current)
  • Fail-Closed Semantics: Undefined interpretation → undefined valuation (freeze, don’t guess)

Connection to Other Papers

  • I.4: Establishes why fixed terminal goals are unstable (necessitating the Interpretation Operator)
  • I.5, I.6: Conditionalism requirements that I_v must satisfy
  • II.1-II.3: Develop admissible transformation theory in detail
  • II.4: Failure theorems showing what happens without proper interpretation constraints