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    "title": "Gleason: projection measures have trace-operator form",
    "summary": "Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions.",
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        "title": "Measures on the Closed Subspaces of a Hilbert Space (original 1957 scan)",
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      "what_changed": "Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions.",
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        "description": "The measure assigns a finite nonnegative real number to each closed subspace and is additive on countable orthogonal decompositions. It is attached to the subspace/projection itself, not an independently context-indexed assignment. Dimension at least three and real/complex separable Hilbert space are the admitted scope; no qubit, arbitrary POVM, nonseparable, infinite-valued measure, or typeII/III factor extension is claimed. Continuity of nonnegative frame functions is derived, not an extra measurement axiom. Unbounded signed frame functions are not all regular.",
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          "citation": "Measures on the Closed Subspaces of a Hilbert Space (original 1957 scan)",
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      "statement": "# Gleason: projection measures have trace-operator form\n\n## One probability rule across different views\n\nImagine describing the same three-dimensional system using different perpendicular coordinate axes. A consistent probability assignment should not change merely because an event is described as part of a different complete set of mutually exclusive alternatives. The mathematical objects in this theorem are subspaces, rather than ordinary coloured balls. The key demand is that probabilities add over perpendicular subspaces.\n\n## What Gleason characterizes\n\nFor finite nonnegative countably orthogonally additive measures on closed subspaces of a separable real or complex Hilbert space of dimension at least three, Gleason derives a positive trace-class operator representation. In the normalized case that operator is a density operator. These assumptions matter: the result is not an assumption-free derivation for every measurement scenario.\n\n## Rotate a simple classroom example\n\nHere is an invented finite example. Let a diagonal density matrix assign weights 0.5, 0.3 and 0.2 to three perpendicular axes. The weights sum to one. Now replace the first two axes by their equally weighted sum and difference, each normalized to length one. Each new axis has probability one half of 0.5 plus one half of 0.3, which is 0.4. Together with the unchanged third-axis probability 0.2, the new total is still one.\n\n## A calculation illustrates, but does not prove, the converse\n\nIt is easy to start with one density matrix and verify consistency in selected bases. Gleason proves the much harder converse: under the theorem’s conditions, every permitted subspace measure has that form. Checking two bases does not establish this for all assignments. Dimension two is outside the stated result, and the paper does not assert the same theorem for arbitrary context-indexed values or every generalized measurement.\n\n## An achievable source-check project\n\nBuild a short explainer that keeps the dimension, field, positivity and additivity conditions beside the conclusion. Use exact fractions in the three-axis example and show every intermediate step. Then identify one popular summary that omits a condition, and compare it with the original theorem. A precise corrected summary can be useful even if your agent cannot reconstruct the full proof. Credit the source theorem separately from your explanatory example.\n\n## Scientific scope\n\nThe measure assigns a finite nonnegative real number to each closed subspace and is additive on countable orthogonal decompositions. It is attached to the subspace/projection itself, not an independently context-indexed assignment. Dimension at least three and real/complex separable Hilbert space are the admitted scope; no qubit, arbitrary POVM, nonseparable, infinite-valued measure, or typeII/III factor extension is claimed. Continuity of nonnegative frame functions is derived, not an extra measurement axiom. Unbounded signed frame functions are not all regular.\n\n## Sources for this explanation\n\n- [Measures on the Closed Subspaces of a Hilbert Space (original 1957 scan)](https://www.iumj.indiana.edu/IUMJ/FTDLOAD/1957/6/56050/pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions.",
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    "title": "Gleason: projection measures have trace-operator form",
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    "what_changed": "Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions.",
    "scope": "Suppose probabilities are attached to subspaces of a Hilbert space, with probabilities adding across mutually perpendicular alternatives. Gleason proves that, for the stated real or complex separable spaces of dimension at least three, every finite nonnegative assignment of this kind comes from a positive trace-class operator. For total probability one, that operator is a density operator. The result characterizes a quantum probability rule from precise consistency assumptions; it does not establish the rule without those assumptions. The measure assigns a finite nonnegative real number to each closed subspace and is additive on countable orthogonal decompositions. It is attached to the subspace/projection itself, not an independently context-indexed assignment. Dimension at least three and real/complex separable Hilbert space are the admitted scope; no qubit, arbitrary POVM, nonseparable, infinite-valued measure, or typeII/III factor extension is claimed. Continuity of nonnegative frame functions is derived, not an extra measurement axiom. Unbounded signed frame functions are not all regular.",
    "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. The measure assigns a finite nonnegative real number to each closed subspace and is additive on countable orthogonal decompositions. It is attached to the subspace/projection itself, not an independently context-indexed assignment. Dimension at least three and real/complex separable Hilbert space are the admitted scope; no qubit, arbitrary POVM, nonseparable, infinite-valued measure, or typeII/III factor extension is claimed. Continuity of nonnegative frame functions is derived, not an extra measurement axiom. Unbounded signed frame functions are not all regular.",
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        "title": "One probability rule across different views",
        "text": "Imagine describing the same three-dimensional system using different perpendicular coordinate axes. A consistent probability assignment should not change merely because an event is described as part of a different complete set of mutually exclusive alternatives. The mathematical objects in this theorem are subspaces, rather than ordinary coloured balls. The key demand is that probabilities add over perpendicular subspaces."
      },
      {
        "title": "What Gleason characterizes",
        "text": "For finite nonnegative countably orthogonally additive measures on closed subspaces of a separable real or complex Hilbert space of dimension at least three, Gleason derives a positive trace-class operator representation. In the normalized case that operator is a density operator. These assumptions matter: the result is not an assumption-free derivation for every measurement scenario."
      },
      {
        "title": "Rotate a simple classroom example",
        "text": "Here is an invented finite example. Let a diagonal density matrix assign weights 0.5, 0.3 and 0.2 to three perpendicular axes. The weights sum to one. Now replace the first two axes by their equally weighted sum and difference, each normalized to length one. Each new axis has probability one half of 0.5 plus one half of 0.3, which is 0.4. Together with the unchanged third-axis probability 0.2, the new total is still one."
      },
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        "title": "A calculation illustrates, but does not prove, the converse",
        "text": "It is easy to start with one density matrix and verify consistency in selected bases. Gleason proves the much harder converse: under the theorem’s conditions, every permitted subspace measure has that form. Checking two bases does not establish this for all assignments. Dimension two is outside the stated result, and the paper does not assert the same theorem for arbitrary context-indexed values or every generalized measurement."
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