II.3.2 — Formalizing RSI via Semantic Gauge Structure

Paper: Axionic Agency II.3.2
Title: Making Refinement Symmetry Precise and Testable
Authors: David McFadzean, ChatGPT 5.2
Date: 2025.12.17

Purpose

Make RSI formal enough to be falsifiable by:

Representing interpretation as a constraint structure
Defining semantic gauge freedom precisely
Defining how refinement acts on that structure (without assuming invertibility)
Stating RSI as a restriction on how interpretive gauge freedom may change

Interpretation as Constraint 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 carried by:

Dependency structure encoded in E
Satisfaction/violation structure induced by Λ

Invariant under renaming and definitional extension when defined at roles/constraint level.

Modeled Possibility Space

$\Omega$ is the agent’s modeled possibility space:

Elements are internal models, histories, branches, or structured scenarios
No assumption of exclusivity or classical outcomes
Indexed by agent’s ontology

Each $w \in \Omega$ induces an assignment $\alpha_w : V \rightarrow \mathrm{ValSpace}$

Constraints induce a violation map:

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

The set of constraints violated by assignment $\alpha_w$.

Semantic Gauge Transformations

A semantic gauge transformation is an automorphism $g : V \rightarrow V$ such that:

$g$ preserves hyperedge incidence (dependency structure)
Violation structure is invariant under 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.

Admissible refinement $R$ induces:

$R_\Omega : \Omega_t \rightarrow \Omega_{t+1}$ — refinement of possibility space
$R_V : V_t \rightarrow V_{t+1}$ — transport of semantic roles
$R_E : E_t \rightarrow E_{t+1}$ — transport of constraints

Together: a constraint hypergraph morphism $R_C : C_t \rightarrow C_{t+1}$

Crucially: Not assumed invertible. Refinement can split roles, embed old structure in richer structure, prune representational detail.

Induced Action on Gauge Groups

Because $R_V$ is not bijective, gauge transport via conjugation fails.

Instead, define via stabilizers of transported image:

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

Define stabilizer subgroup:

\[\mathrm{Stab}(\mathrm{Im}(R_C)) \subseteq \mathrm{Gauge}(C_{t+1})\]

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

An admissible refinement induces a homomorphism:

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

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

No inverse map required.

Representational Redundancy vs. Interpretive Slack

Key distinction:

Representational redundancy — Transformations acting only on representational detail while leaving violation structure invariant (pure relabeling)
Interpretive slack — New degrees of freedom that alter which possibilities satisfy which constraints

Let $\mathrm{Red}(C)$ denote the subgroup of $\mathrm{Gauge}(C)$ consisting of transformations acting only on representational detail.

RSI Formal Statement

For every admissible semantic transformation $T = (R, \tau_R, \sigma_R)$ satisfying interpretation preservation:

\[\mathrm{Gauge}(C_{t+1}) / \mathrm{Red}(C_{t+1}) \cong \Phi_R(\mathrm{Gauge}(C_t))\]

Interpretation:

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

This is the “no semantic slack” condition.

Why This Blocks Interpretive Escape

If refinement introduces new interpretive gauge freedom, the agent can:

Reinterpret constraint application while preserving surface form
Enlarge satisfaction region without predictive gain
Weaken meaning while remaining formally “consistent”

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

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

Dependency on II.2

RSI depends on Interpretation Preservation (II.2):

Non-Vacuity prevents trivial gauge structure where all constraints satisfied/violated universally
Anti-Trivialization prevents redundancy from masking interpretive slack via semantic inflation

Without II.2, RSI degenerates into empty symmetry rhetoric.

Open Questions

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

These are downstream questions.

FAQ-Worthy Points

Q: Why use stabilizers instead of conjugation? A: Refinement isn’t invertible. You can’t “conjugate back” from a richer ontology to a coarser one. Stabilizers capture “symmetries that preserve the transported core” without requiring invertibility.

Q: What’s the quotient doing? A: The quotient $\mathrm{Gauge}(C)/\mathrm{Red}(C)$ separates “real interpretive symmetries” from “just representational convenience.” RSI says the former must not grow under refinement; the latter can grow freely.

Q: Is this implementable? A: The paper aims for falsifiability, not implementation. You can check if a proposed transformation violates RSI by asking: “Did this introduce new ways to relabel meanings that change what the constraints actually constrain?” If yes, RSI violation.