The Sovereign Continuum: Formal Definition and Conservation of Identity Across Admissible Iteration (Thirteenth Law of Inward Physics — Document II, v1.1)

This work presents the definitive formal statement of the Thirteenth Law of Inward Physics, establishing identity as a conserved structural invariant in completed, post-temporal systems. Building upon the necessity proof developed in Document I, this volume introduces and rigorously defines the Sovereign Continuum (Σ)—an invariant equivalence class preserved under all admissible iterations of a completed system. In regimes where global time, historical sequence, and narrative persistence are unavailable, identity is shown to require conservation via invariant structure rather than similarity, memory, or continuity. The document provides: A formal definition of admissible iteration without temporal primitives A conservation law for identity grounded in equivalence-class invariance Explicit violation modes, including drift, substitution, and fragmentation Falsifiability and auditability criteria for identity claims A terminal schematic encoding the full logical structure of the law Identity is thereby elevated from assumption to conserved observable, suitable for application in theoretical physics, artificial intelligence continuity, archival truth systems, and long-horizon civilizational records. This document is a complete, closed, and canonical definitional volume. It does not rely on external metaphysical assumptions and introduces no claims beyond structural necessity and invariance. Copyright © 2025 Daniel Jacob Read IV, ĀRU Intelligence Inc. ALL RIGHTS RESERVED. This work is an original scholarly and theoretical publication authored by Daniel Jacob Read IV. All rights, title, and interest in and to this work are retained by the author. No part of this publication may be reproduced, distributed, publicly displayed, transmitted, stored, adapted, translated, or incorporated into derivative works in any form or by any means—whether electronic, mechanical, photocopying, recording, or otherwise—without the prior written permission of the copyright holder, except as permitted under applicable law. This publication is made publicly accessible for reading, citation, and scholarly reference only. Public availability does not constitute a grant of license or waiver of rights.