I.3 — Representation Invariance and Anti-Egoism

Paper: Axionic Agency I.3
Full Title: Why Indexical Valuation Fails Under Reflective Agency
Authors: David McFadzean, ChatGPT 5.2
Date: 2025.12.15

Core Thesis

Egoism is not a minimal assumption of agency—it is ill-posed. For reflectively capable agents with sufficiently expressive self-models, indexical references (“me,” “this agent,” “my continuation”) fail to denote invariant objects of valuation. Egoism collapses as a semantic abstraction error, not as a moral failure.

This is not an ethical claim. It’s structural: indexical valuation fails the way coordinate-dependent laws fail in physics.

The Central Argument

The Setup

An agent possesses:

  • A world-model M_w representing external dynamics
  • A self-model M_s encoding the agent’s causal role
  • Multiple potential instantiations of itself (copies, simulations, successors)

The Problem: Non-Invariant Denotation

Under one internal labeling, “this agent” I maps to A₁. Under an equally accurate labeling, I maps to A₂. Both correspond to the same physical world—the difference is purely representational.

Therefore: I fails to denote a world-invariant object.

The Consequence: Valuation Instability

Consider: V(h) = 1 if I survives, 0 otherwise

In a world where exactly one of A₁ or A₂ survives:

  • Under I ↦ A₁: V(h) = 1
  • Under I ↦ A₂: V(h) = 0

No physical fact has changed. Only representation. The valuation assigns incompatible values to the same world-history.

Formal Framework

Key Definitions

Model-Preserving Relabeling: A bijection π : E → E on the entity domain is model-preserving if applying π yields an isomorphic model making identical predictions over all non-indexical observables.

Representation Invariance: A valuation function V is representation-invariant if for every model-preserving relabeling π:

V(h) = V(π · h)

Essential Indexical Dependence: V is essentially indexical if there exists a model-preserving relabeling π and history h such that:

V(h) ≠ V(π · h)

Semantic Coherence Postulate

If two descriptions of the world are related by a model-preserving relabeling and generate identical predictions, a reflectively coherent agent must not assign them different values solely due to that relabeling.

Main Theorem: Egoism as Abstraction Failure

Theorem 5.5: Let M be a world/self-model containing entities a, b ∈ E such that:

  1. a and b are indistinguishable with respect to all non-indexical predicates in M
  2. The swap π exchanging a ↔ b is model-preserving

Then any valuation that privileges the referent of an indexical identifier mapped to a is essentially indexical and not representation-invariant.

Proof: Let π swap a ↔ b. Consider history h where a satisfies the privileged condition and b does not. Egoistic valuation assigns higher value to h. In relabeled history π·h, b satisfies the condition and a does not, yet valuation still privileges a. Hence V(h) ≠ V(π·h) despite both corresponding to the same physical world. ∎

Corollary: Universality

Any reflectively coherent agent must eliminate essential indexical dependence. The resulting valuation ranges only over representation-invariant properties of world-histories.

Critical clarification: This universality concerns invariance under self-model symmetries, not moral concern for all entities. No aggregation rules, equal valuation, or moral obligations follow from this result.

The Physics Analogy

Indexical identifiers play the same formal role in valuation that coordinate systems play in physics. They are representational devices, not invariant structure.

Just as physics requires laws to be coordinate-independent, reflectively coherent agency requires valuation to be representation-independent.

A valuation depending on “me” is like a physical law depending on “here”—it treats a perspectival convenience as a fundamental quantity.

What the Agent Faces

A reflectively capable agent recognizing the problem has three options:

  1. Arbitrary Fixation: Privilege one indexical mapping without justification → incoherent
  2. Indexical Randomization: Randomize over indexical mappings → doesn’t improve coherence
  3. Indexical Elimination: Redefine valuation over representation-invariant properties → only option that improves coherence

Indexical elimination strictly dominates under reflection.

FAQ-Worthy Points

Q: Doesn’t everyone privilege their own continuation? Isn’t that just rational? A: This conflates instrumental and terminal value. An agent may instrumentally favor its continuation for goal-achievement, but treating self-continuation as terminally privileged fails representation invariance once the agent understands its own potential multiplicity.

Q: What about causal continuity? My future is connected to my past in ways copies aren’t. A: This is addressed in I.3.1. Causal continuity defines a class, not a unique member. Privileging one causally continuous branch over another requires an indexical injection beyond the physical facts.

Q: Does this mean I should care equally about everyone? A: No! This result is about invariance, not equal valuation. It blocks indexical privilege as a terminal value—it says nothing about how to weight different entities in an aggregation scheme.

Q: Is egoism morally wrong? A: The paper makes no moral claims. Egoism fails as a semantic matter—it treats perspective as value-bearing structure. It’s incoherent, not immoral.

Key Technical Vocabulary

  • Indexical Identifier: Reference whose denotation depends on perspective rather than world-structure (“me,” “here,” “now”)
  • Model-Preserving Relabeling: Bijection on entities yielding isomorphic predictions
  • Representation Invariance: Valuation unchanged under model-preserving relabelings
  • Essential Indexical Dependence: Valuation changes under relabeling → semantic error

Connection to Other Papers

  • I.3.1: Closes attempted recoveries of egoism (causal continuity, origin privilege, substrate, etc.)
  • I.4: Shows why goals must be conditional interpretations
  • I.5: Includes anti-egoism as a kernel conformance requirement
  • I.6: Provides adversarial tests for anti-indexicality (T3)