Metric Emergence from Event-Order Geometry

Three Separation Constructions

Given a partially ordered event set, three distinct notions of separation emerge naturally from the order structure:

• Chain-length separation — for comparable (causally connected) pairs. Measures temporal distance by counting minimal chain steps.
• Observer-disagreement separation — for incomparable pairs. Measures spatial-like distance through variance of time-difference across observers.
• Transition-cost metric — for the full event set. Shortest path in the Hasse graph, satisfying all metric axioms.

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Chain-Length Separation

For comparable events — those connected by a directed path in the partial order — the chain-length separation counts the minimum number of chain steps between them. This is an observer-independent, directed measure of temporal separation.

Chain-length separation is defined only for comparable pairs — events connected by the partial order. It captures the directed, temporal aspect of separation: how many irreducible steps lie between two causally related events.

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Observer-Disagreement Separation

For incomparable events — those not connected by any directed path — different observers assign different time-differences. The variance of these assignments across the observer set defines a spatial-like separation.

The key structural distinction: for comparable events, the sign of the time-difference is observer-independent. For incomparable events, the sign is observer-dependent. This sign structure mirrors the timelike/spacelike distinction in Lorentzian geometry.

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Transition-Cost Metric

The transition-cost metric extends separation to the full event set by computing shortest paths in the Hasse graph. Unlike chain-length (comparable pairs only) or observer-disagreement (incomparable pairs only), the transition-cost metric is defined for all event pairs.

The transition-cost metric satisfies all metric axioms — identity of indiscernibles, symmetry, and triangle inequality — conditional on the Hasse graph being connected. It provides a proper distance function on the full event set.

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Honest Assessment

• d_chain is defined only for comparable pairs — it does not extend to incomparable events
• σ (observer-disagreement) fails the triangle inequality — it is a separation measure, not a metric
• d_trans is positive-definite, not Lorentzian — it does not reproduce the signed interval of general relativity

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Stated Limitations

• No Lorentzian metric is derived — d_trans is positive-definite
• No signed interval — the temporal/spatial split is structural, not metrical
• No continuous manifold — all constructions are on finite discrete event sets
• No Einstein equations — the metric is kinematic, not dynamical

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