Paper XIV - Emergent Gauge Structure from Local Redundancy

DOI: 10.5281/zenodo.18009155 Emergent Gauge Structure from Local Redundancy Paper XIV of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We show that gauge structure arises necessarily from local redundancy in record descriptions within an information-theoretic framework where time, space, and gravity are emergent. Building on Paper XI (Operational Time Regulation), Paper XII (Emergent Spatial Locality), and Paper XIII (Gravity as Inhomogeneous Influence Propagation), we demonstrate that any attempt to represent physical states locally introduces unavoidable internal redundancy. Finite influence speed precludes global elimination of this redundancy, forcing the introduction of compensating connection variables. These variables transform as gauge fields. Gauge symmetry is thus not a postulate or mathematical convenience but a necessary consequence of local record autonomy, bounded influence propagation, and reversible dynamics. In the continuum limit, standard Abelian and non-Abelian gauge structures emerge as effective descriptions of redundancy transport on the emergent interaction graph. Contents 1 Introduction 1 2 Recap: Time, Space, and Gravity 1 3 Local Record Descriptions and Redundancy 2 3.1 Records and internal labels 2 3.2 Autonomy of local descriptions 2 4 Gauge Freedom is Forced 2 5 Gauge Fields on the Emergent Interaction Graph 3 5.1 Parallel transport of redundancy 3 5.2 Curvature and holonomy 3 6 Continuum Limit and Effective Gauge Theory 3 7 Discussion 3 8 Conclusion 4 1 Introduction Gauge symmetry occupies a central role in modern physics. Electromagnetism, the weak interaction, and the strong interaction are all described by gauge theories. Despite their empirical success, the foundational status of gauge symmetry remains ambiguous: is it a fundamental principle of nature, or a redundancy introduced by our mathematical description? In conventional formulations, gauge symmetry is imposed as a requirement of local invariance. While mathematically elegant, this approach leaves open a deeper question: why must physical theories exhibit local gauge freedom at all? In the Ordered-Dynamics Reconstruction Program, space, time, and gravity are not postulated but derived from operational constraints on finite systems. Paper XI established that operational time must be regulated relative to an ordering parameter. Paper XII showed that bounded influence propagation forces the emergence of spatial locality. Paper XIII demonstrated that gravity arises as inhomogeneous influence delay. This paper shows that once local descriptions exist within such a framework, gauge symmetry is unavoidable. It is forced by the autonomy of local records under finite influence speed. 2 Recap: Time, Space, and Gravity We briefly summarize prior results essential for the present argument. •Paper XI: Operational time differs from the underlying ordering parameter due to finite clock processing, requiring regulated local rescaling. – 1 – •Paper XII: Space emerges as a sparse interaction graph defined by bounded influence propagation. •Paper XIII: Gravity emerges from spatial variation in influence delay, encoded by a universal path-delay factor Z(x). These structures are assumed throughout. In particular, we assume: •autonomous local record-keeping, •finite influence speed, •no globally accessible labeling of internal states. 3 Local Record Descriptions and Redundancy 3.1 Records and internal labels A record is an operationally accessible effect whose outcome can be stored, compared, and reproduced. To encode records locally, an observer must choose internal labels for the subsystem state space, such as: •phase conventions, •basis choices, •orientations of internal degrees of freedom. These labels are not physical observables. Different choices can encode identical operational statistics. Definition 1 (Local redundancy).Two internal descriptions are redundantly equivalent if they yield identical statistics for all operationally accessible records. Local redundancy is therefore unavoidable. 3.2 Autonomy of local descriptions Finite influence speed forbids instantaneous coordination of internal labels across spatially separated regions. Each region must maintain its own local description autonomously. This autonomy is enforced by causal structure, not convention. 4 Gauge Freedom is Forced [Impossibility of global redundancy elimination] Global elimination of local redundancy is incompatible with finite influence propagation. Sketch. Eliminating redundancy globally requires comparing and aligning internal labels across spatially separated regions. Such comparison requires influence propagation. Finite influence speed forbids instantaneous alignment outside causal cones. [Emergence of compensating variables] Consistency of local descriptions under finite influence propagation requires the introduction of variables encoding relative redundancy between neighboring regions. These variables compensate for local relabelings and preserve invariant predictions. They are gauge connections. – 2 – 5 Gauge Fields on the Emergent Interaction Graph 5.1 Parallel transport of redundancy Let xand ybe neighboring regions in the interaction graph. Local descriptions ψ(x) and ψ(y) may differ by a redundancy transformation. Consistency requires a mapping ψ(y)7→ Uxy ψ(y), where Uxy compensates the mismatch. Under local relabeling ψ(x)→g(x)ψ(x), Uxy →g(x)Uxy g(y)−1. This is the defining transformation law of a gauge connection. 5.2 Curvature and holonomy Transporting redundancy around a closed loop γyields a holonomy H(γ) = Y (x,y)∈γ Uxy. Nontrivial holonomy signals obstruction to global redundancy elimination. Gauge curvature therefore measures redundancy mismatch, not fundamental force. 6 Continuum Limit and Effective Gauge Theory When the interaction graph admits coarse-graining, discrete connections Uxy may be approximated by continuous gauge fields Aµ(x). Local redundancy transformations become ψ(x)→g(x)ψ(x), Aµ→gAµg−1−(∂µg)g−1. Gauge dynamics arise as effective descriptions of redundancy transport subject to locality and reversibility constraints. Remark 1 (Compatibility with interaction locality).Gauge connections encode redundancy transport and do not represent direct matter–matter interactions. Accordingly, the appearance of covariant derivatives in the continuum limit does not contradict the restriction to multiplicative interaction generators derived in Paper IV. 7 Discussion Gauge symmetry is often described as a redundancy of description. Here we show that this redundancy is not optional: it is enforced by local autonomy under finite influence speed. Gauge fields mediate consistency between locally encoded records. Forces arise only when redundancy transport acquires dynamics. Gauge symmetry therefore shares a common operational origin with space and gravity. – 3 – 8 Conclusion We have shown that gauge structure emerges necessarily from local redundancy in record descriptions when influence propagation is finite. Gauge symmetry is not postulated but forced by the same constraints that give rise to space and gravity. This completes the derivation of classical gauge structure within the Ordered-Dynamics Reconstruction Program. Subsequent papers address quantum superposition, measurement, and ultraviolet consistency. References [1] P. Cooney, Operational Time Regulation from Ordered Dynamics, Paper XI of the Ordered-Dynamics Reconstruction Program (2025). [2] P. Cooney, Emergent Spatial Locality from Bounded Influence, Paper XII of the Ordered-Dynamics Reconstruction Program (2025). [3] P. Cooney, Gravity as Inhomogeneous Influence Propagation, Paper XIII of the Ordered-Dynamics Reconstruction Program (2025). [4] C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954). [5] R. D. Sorkin, Lectures on Quantum Gravity, Springer (2005). – 4 –