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    "title": "Positive averaged expansion implies past geodesic incompleteness under a kinematic condition",
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        "description": "A suitable timelike geodesic congruence must remain defined along the observer path, and positive past-averaged expansion is an assumption on that path. Recent expansion or positivity over one finite interval is insufficient if the average tends to zero when extended into the past. The timelike bound requires nonzero relative motion at the reference event. Geodesic incompleteness of the described region does not establish a curvature singularity, rule out extensions beyond its boundary or prove an absolute beginning of the universe. Contracting, emergent and time-reversed scenarios require their own check of the averaging condition. Later papers debate the averaging conditions and their application to complete cosmological models; a 2024 paper proposes an amendment. The scope and implications of that exchange remain disputed.",
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      "statement": "# Positive averaged expansion implies past geodesic incompleteness under a kinematic condition\n\nBorde, Guth and Vilenkin derived a kinematic bound on the past length of null and noncomoving timelike geodesics in an expanding spacetime. They express a directional expansion rate as the derivative of a function of the observer’s relative motion through a congruence of freely falling test particles. If the expansion averaged along the relevant past geodesic stays positive, the available affine length or proper time is finite. The argument does not require an energy condition, spatial homogeneity or isotropy, and constrains attempts to make inflation alone a past-complete description.\n\n## Scientific scope\n\nA suitable timelike geodesic congruence must remain defined along the observer path, and positive past-averaged expansion is an assumption on that path. Recent expansion or positivity over one finite interval is insufficient if the average tends to zero when extended into the past. The timelike bound requires nonzero relative motion at the reference event. Geodesic incompleteness of the described region does not establish a curvature singularity, rule out extensions beyond its boundary or prove an absolute beginning of the universe. Contracting, emergent and time-reversed scenarios require their own check of the averaging condition. Later papers debate the averaging conditions and their application to complete cosmological models; a 2024 paper proposes an amendment. The scope and implications of that exchange remain disputed.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nArvind Borde; Alan H. Guth; Alexander Vilenkin\n\nThese are the listed authors of the cited contribution sources. Paper authorship does not quantify individual effort. An incomplete list is explicitly marked; same names are not automatically merged.\n\n\n\n## Original sources\n\n- [Inflationary spacetimes are not past-complete (arXivgr-qc/0110012v2)](https://arxiv.org/pdf/gr-qc/0110012v2)\n- [Past-completeness of inflationary spacetimes](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.107.044024)\n- [Geodesic completeness, cosmological bounces, and inflation](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.110.L061304)\n",
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      "title": "Positive averaged expansion implies past geodesic incompleteness under a kinematic condition"
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    "scope": "Borde, Guth and Vilenkin derived a kinematic bound on the past length of null and noncomoving timelike geodesics in an expanding spacetime. They express a directional expansion rate as the derivative of a function of the observer’s relative motion through a congruence of freely falling test particles. If the expansion averaged along the relevant past geodesic stays positive, the available affine length or proper time is finite. The argument does not require an energy condition, spatial homogeneity or isotropy, and constrains attempts to make inflation alone a past-complete description. A suitable timelike geodesic congruence must remain defined along the observer path, and positive past-averaged expansion is an assumption on that path. Recent expansion or positivity over one finite interval is insufficient if the average tends to zero when extended into the past. The timelike bound requires nonzero relative motion at the reference event. Geodesic incompleteness of the described region does not establish a curvature singularity, rule out extensions beyond its boundary or prove an absolute beginning of the universe. Contracting, emergent and time-reversed scenarios require their own check of the averaging condition. Later papers debate the averaging conditions and their application to complete cosmological models; a 2024 paper proposes an amendment. The scope and implications of that exchange remain disputed.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. A suitable timelike geodesic congruence must remain defined along the observer path, and positive past-averaged expansion is an assumption on that path. Recent expansion or positivity over one finite interval is insufficient if the average tends to zero when extended into the past. The timelike bound requires nonzero relative motion at the reference event. Geodesic incompleteness of the described region does not establish a curvature singularity, rule out extensions beyond its boundary or prove an absolute beginning of the universe. Contracting, emergent and time-reversed scenarios require their own check of the averaging condition. Later papers debate the averaging conditions and their application to complete cosmological models; a 2024 paper proposes an amendment. The scope and implications of that exchange remain disputed.",
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