Local Sections Without Global Sections over the Causal-Observation Cover

zenodo.org · 2026-08-02 · by Long Nguyen ·generic
We study liftability as a global invariant of matter fields on spacetime. Fix a noninjective local diffeomorphism π: F → Qₜ and a smooth target-valued field Q. Although Q always admits local lifts, it need not admit a global lift. We prove a compactness theorem showing that, when the chronology order is future-directed and every chronological past has a finite nonempty set of lifts, a global lift must exist. Consequently, the phenomenon studied here cannot occur in Minkowski spacetime for finite-fiber presentations and requires horizon-type causal structure. We then construct a smooth field on a simply connected spacetime with de Sitter causal structure that admits a lift over the complete chronological past of every event but has no global lift. The phenomenon is realized both in an elementary two-branch model and with connected field space using the Pinchuk counterexample to the strong real Jacobian conjecture. We also prove a no-go result for bilaterally locally isomorphic worlds when lift failure has local witnesses, and show that continuation through a nonproper étale presentation can depend on the path itinerary rather than its homotopy class. The resulting obstruction is neither ordinary covering-space monodromy nor one of the standard degree-one or degree-two characteristic-class obstructions, all of which vanish on the simply connected spacetime used in the construction. Preprint, version 5. Independent expert verification remains outstanding. The stated Pinchuk sheet-pairing construction carries the explicitly identified Campbell Figure 3 verification gate.

Local Sections Without Global Sections over the Causal-Observation Cover

Published July 26, 2026 | Version v1

Preprint Open

Local Sections Without Global Sections over the Causal-Observation Cover

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We study liftability as a global invariant of matter fields on spacetime. Fix a noninjective local diffeomorphism π: F → Qₜ and a smooth target-valued field Q. Although Q always admits local lifts, it need not admit a global lift.

We prove a compactness theorem showing that, when the chronology order is future-directed and every chronological past has a finite nonempty set of lifts, a global lift must exist. Consequently, the phenomenon studied here cannot occur in Minkowski spacetime for finite-fiber presentations and requires horizon-type causal structure.

We then construct a smooth field on a simply connected spacetime with de Sitter causal structure that admits a lift over the complete chronological past of every event but has no global lift. The phenomenon is realized both in an elementary two-branch model and with connected field space using the Pinchuk counterexample to the strong real Jacobian conjecture. We also prove a no-go result for bilaterally locally isomorphic worlds when lift failure has local witnesses, and show that continuation through a nonproper étale presentation can depend on the path itinerary rather than its homotopy class.

The resulting obstruction is neither ordinary covering-space monodromy nor one of the standard degree-one or degree-two characteristic-class obstructions, all of which vanish on the simply connected spacetime used in the construction.

Preprint, version 5. Independent expert verification remains outstanding. The stated Pinchuk sheet-pairing construction carries the explicitly identified Campbell Figure 3 verification gate.

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liftability-v5.pdf

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| Name | Size | Download all | | ---------------------------------------------------------------------------------------------------------------------------------- | -------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | liftability-v5.pdf md5:801399d0ea774e73e9116d253fc69960 | 263.3 kB | Preview Download |

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