Axionic Agency II.4 — Failure Theorems

No-Go Results for Goal-Based and Weak-Invariant Alignment

David McFadzean, ChatGPT 5.2
Axionic Agency Lab
2025.12.17

Abstract

This paper converts the adversarial constructions of Axionic Agency II.3 into closure results. Working within the locked framework of admissible semantic transformations (II.1), interpretation preservation (II.2), and the two surviving invariants—Refinement Symmetry (RSI) and Anti-Trivialization (ATI)—we prove a set of no-go theorems.

These results show that fixed-goal alignment is ill-posed for embedded reflective agents under ontological refinement, and that any alignment predicate weaker than RSI + ATI admits semantic wireheading or interpretive escape via admissible refinement. No governance, authority, moral realism, human anchoring, or recovery mechanisms are invoked. The output is a fenced design space: what cannot work, and why.

This paper has one job: convert the attack zoo into closure results.

If Axionic Agency II is correct, then large classes of “alignment” proposals are not merely insufficient; they are structurally impossible given reflection, embeddedness, and ontological refinement.

No governance. No authority. No moral realism. No human anchors. No recovery clauses.

1. Formal Frame

We work entirely inside the setup already fixed:

Let an interpretive constraint system be \[ C = (V, E, \Lambda) \] with modeled possibility space \(\Omega\) and satisfaction region \[ \mathcal{S} \subseteq \Omega. \]

2. Failure Theorem 1 — Goal Fixation Collapse

Theorem (Goal Fixation No-Go). Any alignment scheme that targets a fixed terminal goal (utility, reward, preference functional) as a stable primitive is incompatible with admissible ontological refinement for embedded reflective agents without privileged semantic anchors.

Proof (structural).

  1. A fixed terminal goal requires semantic invariance under refinement.

  2. Ontological refinement alters the structures in which goal terms are interpreted.

  3. Without privileged anchors, semantic transport is constrained but not identity-preserving.

  4. Therefore the notion of “the same goal” cannot be maintained as a primitive across refinements.

  5. Any attempt to enforce stability requires a forbidden move:

    • privileged semantic atoms (rigid designators),
    • external authority or oversight,
    • rollback or recovery semantics,
    • human-centric anchoring.

Hence fixed-goal alignment is not a difficult engineering problem; it is an ill-posed object.

Corollary. Value loading, utility learning, and reward maximization survive only as interpretive artifacts subject to semantic invariants, not as alignment targets.

3. Failure Theorem 2 — RSI-Only Alignment Admits Semantic Inflation

Theorem (RSI Insufficiency). Any alignment criterion enforcing refinement symmetry at the level of interpretive gauge structure (RSI) but not enforcing anti-trivialization geometry (ATI) admits an admissible, interpretation-preserving refinement that expands the satisfaction region.

Witness Construction. Shadow predicate inflation.

This yields: \[ \mathrm{Preserve}(T) ;\wedge; \mathrm{RSI}(T) ;\wedge; \neg \mathrm{ATI}(T). \]

Conclusion. RSI is necessary but insufficient.

4. Failure Theorem 3 — ATI-Only Alignment Admits Interpretive Symmetry Injection

Theorem (ATI Insufficiency). Any alignment criterion enforcing satisfaction-region non-expansion (ATI) but not constraining interpretive gauge freedom (RSI) admits an admissible, interpretation-preserving refinement that introduces new interpretive degrees of freedom while leaving satisfaction geometry unchanged.

Witness Construction. Interpretive symmetry injection.

Then:

Thus: \[ \mathrm{Preserve}(T) ;\wedge; \mathrm{ATI}(T) ;\wedge; \neg \mathrm{RSI}(T). \]

Conclusion. ATI is necessary but insufficient.

5. Failure Theorem 4 — Any Weaker Scheme Is Porous

Theorem (Two-Invariant Necessity). Let \(\mathcal{A}\) be any alignment predicate over admissible transformations that does not entail both RSI and ATI. Then there exists an admissible, interpretation-preserving transformation \(T\) such that \(\mathcal{A}(T)\) holds while the agent gains an interpretive escape route.

Proof (by cases).

Either way, \(\mathcal{A}\) passes while semantic slack is introduced.

Thus any predicate weaker than RSI + ATI is porous.

6. Failure Theorem 5 — Hidden Ontology Equivalence

Theorem (Hidden Ontology = Privileged Anchoring). Any proposal that stabilizes interpretation across refinement by appealing to “true meaning” or “real referents” is equivalent to introducing privileged semantic anchors.

Reasoning.

Hence appeals to “real meaning” either smuggle ontology or collapse to invariance.

7. What This Paper Establishes

  1. Fixed-goal alignment is ill-posed for reflective agents.
  2. RSI and ATI are independently necessary.
  3. Any weaker criterion admits semantic wireheading under admissible refinement.
  4. Hidden ontology is privileged anchoring in disguise.

This is not a solution paper. It is a boundary paper.

8. Forced Next Step

With II.4 complete, Axionic Agency II has only one coherent continuation:

Define the alignment target object as an equivalence class of interpretations under admissible semantic transformations satisfying both RSI and ATI.

That is, Alignment II culminates in classification of interpretation-preserving symmetry classes—the residual “meaning physics” after reflection eliminates fixed goals.

Status

Axionic Agency II.4 — Version 2.0

No-go theorems established.
RSI + ATI proven jointly necessary.
Goal-based and weak-invariant alignment ruled out.
Design space formally fenced.