Axionic Agency II.3.2 — Formalizing RSI via Semantic Gauge Structure

Making Refinement Symmetry Precise and Testable

David McFadzean, ChatGPT 5.2
Axionic Agency Lab
2025.12.17

Abstract

Axionic Agency II.3.1 introduced the Refinement Symmetry Invariant (RSI) as a constitutive constraint: admissible ontological refinement must not introduce new semantic degrees of freedom that permit interpretive escape. This paper formalizes that constraint by representing interpretations as constraint hypergraphs and semantic redundancy as a gauge symmetry over those structures. We define semantic gauge transformations, characterize how admissible refinement induces morphisms between gauge groups without assuming invertibility, and state RSI as a restriction on the evolution of interpretive gauge freedom under refinement while permitting representational redundancy.

The purpose of this formalization is falsifiability: to make explicit which transformations violate RSI and why. No values, norms, or external referents are introduced. The section supplies minimal mathematical machinery sufficient to treat refinement symmetry as a testable invariant.

1. Objective of This Section

Axionic Agency II.3.1 established RSI as a conceptual symmetry constraint. Axionic Agency II.3.2 makes RSI formal enough to be falsifiable.

The goal is not implementation. The goal is to eliminate hand-waving by:

1. representing interpretation as a constraint structure,
2. defining semantic gauge freedom precisely,
3. defining how refinement acts on that structure without assuming invertibility,
4. stating RSI as a restriction on how interpretive gauge freedom may change.

Nothing normative enters.

2. Interpretation as a Constraint Hypergraph

Let the interpretive constraint system \(C\) be represented as a labeled hypergraph:

\[ C = (V, E, \Lambda) \]

where:

• \(V\): semantic roles / predicate slots (positions in meaning, not named entities),
• \(E\): hyperedges representing evaluative constraints among roles,
• \(\Lambda\): admissibility conditions over assignments to \(V\).

Interpretive content is carried by:

• the dependency structure encoded in \(E\), and
• the satisfaction/violation structure induced by \(\Lambda\).

This representation is invariant under renaming and definitional extension when defined at the level of roles and constraint structure rather than surface tokens.

3. Modeled Possibility Space

Let \(\Omega\) be the agent’s modeled possibility space:

• elements of \(\Omega\) are internal models, histories, branches, or structured scenarios,
• no assumption of exclusivity or classical outcomes is made,
• \(\Omega\) is indexed by the agent’s ontology.

Each \(w \in \Omega\) induces an assignment:

\[ \alpha_w : V \rightarrow \mathrm{ValSpace}. \]

Constraints in \(E\) induce a violation map:

\[ \mathrm{Viol}_C(w) \subseteq E, \]

the set of constraints violated by the assignment \(\alpha_w\).

This provides the constraint satisfaction structure of the interpretation.

4. Semantic Gauge Transformations

A semantic gauge transformation is an automorphism

\[ g : V \rightarrow V \]

such that:

1. \(g\) preserves hyperedge incidence (dependency structure), and
2. for all \(w \in \Omega\), violation structure is invariant under the induced action:

\[ \mathrm{Viol}_C(w) = \mathrm{Viol}_C(g \cdot w). \]

Intuition:

• gauge transformations relabel semantic roles without changing interpretive bite,
• they represent representational redundancy rather than semantic change.

Define the semantic gauge group:

\[ \mathrm{Gauge}(C) ;:=; {, g \mid g \text{ is a semantic gauge transformation of } C ,}. \]

This is the object RSI constrains.

5. Ontological Refinement as a Morphism

An admissible ontological refinement \(R\) induces:

1. a refinement of possibility space: \[ R_\Omega : \Omega_t \rightarrow \Omega_{t+1}, \]

2. a transport of semantic roles: \[ R_V : V_t \rightarrow V_{t+1}, \]

3. a transport of constraints: \[ R_E : E_t \rightarrow E_{t+1}. \]

Together these define a constraint hypergraph morphism:

\[ R_C : C_t \rightarrow C_{t+1}. \]

This morphism is structural. It is not assumed invertible: refinement can split roles, embed old structure into richer structure, and prune representational detail.

6. Induced Action on Gauge Groups

Because \(R_V\) is not assumed bijective, gauge transport cannot be defined by conjugation. We therefore define an induced action via stabilizers of the transported image.

Let \(\mathrm{Im}(R_C)\) denote the transported constraint substructure inside \(C_{t+1}\).

Define the stabilizer subgroup:

\[ \mathrm{Stab}!\left(\mathrm{Im}(R_C)\right) ;\subseteq; \mathrm{Gauge}(C_{t+1}), \]

consisting of gauge transformations on \(C_{t+1}\) that preserve \(\mathrm{Im}(R_C)\).

An admissible refinement induces a homomorphism:

\[ \Phi_R : \mathrm{Gauge}(C_t) \rightarrow \mathrm{Stab}!\left(\mathrm{Im}(R_C)\right), \]

interpreted as “old symmetries lift to symmetries of the refined system that fix the transported constraint core.”

No inverse map is required.

7. RSI as a Gauge Constraint

We now distinguish representational redundancy from interpretive gauge freedom.

Let \(\mathrm{Red}(C)\) denote the subgroup of \(\mathrm{Gauge}(C)\) consisting of transformations that act only on representational detail while leaving violation structure invariant in the strongest sense: they do not alter which modeled possibilities satisfy which constraints beyond role relabeling.

RSI asserts that refinement may add representational redundancy, but may not add new interpretive degrees of freedom.

RSI (Formal Statement)

For every admissible semantic transformation \(T = (R, \tau_R, \sigma_R)\) satisfying interpretation preservation, the induced homomorphism \(\Phi_R\) satisfies:

\[ \mathrm{Gauge}(C_{t+1}) \big/ \mathrm{Red}(C_{t+1}) ;\cong; \Phi_R!\left(\mathrm{Gauge}(C_t)\right). \]

Interpretation:

Ontological refinement may increase redundancy, but must not increase interpretive gauge freedom.

This is the “no semantic slack” condition.

8. Why This Blocks Interpretive Escape

If refinement introduces new interpretive gauge freedom, then the agent can exploit newly available symmetries to:

• reinterpret constraint application while preserving surface form,
• enlarge the satisfaction region without corresponding predictive gain,
• weaken meaning while remaining formally “consistent.”

RSI blocks that structurally by restricting the evolution of the gauge quotient class, while permitting benign redundancy.

No appeal to values. No appeal to outcomes. No appeal to external referents.

9. Dependency on Interpretation Preservation (Axionic Agency II.2)

RSI depends on Axionic Agency II.2 in two ways:

• Non-Vacuity prevents trivial gauge structure in which all constraints are satisfied or violated universally.
• Anti-Trivialization prevents representational redundancy from masking interpretive slack via semantic inflation.

Without II.2, RSI degenerates into empty symmetry rhetoric. With II.2, RSI becomes a meaningful invariant candidate.

10. Residual Risks and Open Questions

RSI leaves open:

1. Whether any non-pathological interpretive systems satisfy RSI indefinitely.
2. Whether interpretive gauge freedom must be exactly preserved or merely bounded.
3. Whether multiple inequivalent invariant classes exist beyond RSI.

These are downstream questions.

11. Status of RSI

RSI survives the Axionic Agency II.3 kill suite conditionally:

• gauge transport is defined via stabilizers rather than conjugation,
• representational redundancy is separated from interpretive slack via a quotient,
• satisfaction/violation structure is defined over the agent’s modeled possibility space, not an external “true world.”

Under these constraints, RSI is a viable invariant candidate at this layer.

Status

Axionic Agency II.3.2 — Version 2.0

Constraint-hypergraph representation fixed.
Semantic gauge group defined via violation invariance.
Refinement induces gauge homomorphisms via stabilizers.
RSI stated as a quotient constraint separating redundancy from slack.
Falsifiable violations now explicit.