Surplus Temporal Linearization over an Intrinsic Relational Event Poset A Constructive Existence Result in K-star Dynamics
Year: 2026
Type: preprint
Abstract: A dynamical history is conventionally represented as a totally ordered sequence of events. Yet a physical dynamics need not itself determine all pairwise precedence relations contained in such a sequence. This paper asks when the additional ordering introduced by temporal linearization is physically redundant, and when a symmetry of the complete relational state prevents any deterministic equivariant choice of one privileged total history. The analysis does not assume a partial order in advance. Atomic structural events are defined through physical admissibility and transition rules, and intrinsic precedence is defined as a necessity relation over physically admissible executions. Numerical timestamps, solver order, labels, and implementation-level serialization are excluded as sources of precedence. A finite intrinsic event poset P is then distinguished from its set L(P) of linear extensions. Linearization is called surplus at P when the completed physical transition map is constant on L(P), equivalently when the physical law factors through the intrinsic event structure rather than the chosen totalization. Three results are established. First, exact co-admissibility or simultaneous threshold crossing is insufficient to imply incomparability: an explicit K-star counterexample contains two simultaneously eligible structural events whose alternative resolution orders produce physically inequivalent outcomes. Second, an explicit four-node K-star configuration generates a nontrivial three-element poset containing both forced precedence and genuine incomparability. Connectivity preservation requires creation of a replacement edge before deletion of a dying bridge, while an independent non-bridge deletion remains incomparable with both members of that forced chain. The poset has exactly three linear extensions, and all three produce the same complete relational state and subsequent physical law. Third, a separate symmetric four-node witness shows that the complete relational state can possess a nontrivial stabilizer that exchanges two incomparable events and hence exchanges the two linear extensions of their event antichain. Finite linear-extension rigidity then implies that no deterministic symmetry-equivariant selector can choose one unique extension from the symmetry-preserving physical data. A local-to-global extension-independence theorem is proved using context-stable adjacent swaps of incomparable events. Separately, a Physical Temporal Stabilizer Obstruction is obtained by specializing the previously established stabilizer logic to finite dynamically derived event posets. The two conclusions are logically independent: extension-independence establishes dynamical redundancy, whereas state symmetry obstructs deterministic equivariant canonicalization. The result does not imply that temporal order as such is redundant, that causal spacetime has been derived, or that time is unreal. It establishes the narrower constructive statement that a relational dynamics can preserve physically indispensable partial precedence while remaining exactly indifferent to additional ordering relations introduced solely by total linearization; and, in a symmetric realization, that no unique total representative can be privileged without breaking a physical state symmetry.
Author: Khawaldeh, Jalal
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204717893