{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "huang-hypercube-sensitivity-conjecture-2019",
    "title": "Huang: induced hypercube degree bound and the Sensitivity Conjecture",
    "summary": "A signed adjacency-matrix construction and eigenvalue interlacing give the sharp square-root maximum-degree lower bound for induced hypercube subgraphs with more than half the vertices. Via the known Gotsman–Linial equivalence, this proves polynomial relations between Boolean sensitivity, polynomial degree and block sensitivity.",
    "boundary": "The stated degree bound implies block sensitivity at most twice the fourth power of sensitivity in this version; the paper does not close the remaining quadratic-versus-quartic separation. The graph-size threshold and induced-subgraph condition matter. This is not a claim that all Boolean complexity measures coincide.",
    "area": "Extremal combinatorics and Boolean complexity",
    "topic": "Mathematics",
    "year": 2019,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "huang-hypercube-sensitivity-conjecture-2019",
    "target": {
      "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
      "version": 1,
      "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
    },
    "humans": [
      "Hao Huang"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-arxiv-1907-00847v1",
        "title": "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture",
        "url": "https://arxiv.org/abs/1907.00847v1",
        "date": "2019-07-01"
      }
    ],
    "points": "320000",
    "published_grade": {
      "id": "huang-hypercube-sensitivity-conjecture-2019",
      "entry_id": "huang-hypercube-sensitivity-conjecture-2019",
      "band": "S",
      "modifier": "-",
      "target": {
        "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
        "version": 1,
        "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
      },
      "what_changed": "A signed adjacency-matrix construction and eigenvalue interlacing give the sharp square-root maximum-degree lower bound for induced hypercube subgraphs with more than half the vertices. Via the known Gotsman–Linial equivalence, this proves polynomial relations between Boolean sensitivity, polynomial degree and block sensitivity.",
      "scope": "A signed adjacency-matrix construction and eigenvalue interlacing give the sharp square-root maximum-degree lower bound for induced hypercube subgraphs with more than half the vertices. Via the known Gotsman–Linial equivalence, this proves polynomial relations between Boolean sensitivity, polynomial degree and block sensitivity. The stated degree bound implies block sensitivity at most twice the fourth power of sensitivity in this version; the paper does not close the remaining quadratic-versus-quartic separation. The graph-size threshold and induced-subgraph condition matter. This is not a claim that all Boolean complexity measures coincide.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
        "version": 1,
        "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "Hao Huang",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture",
            "url": "https://arxiv.org/abs/1907.00847v1"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture",
            "url": "https://arxiv.org/abs/1907.00847v1"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The stated degree bound implies block sensitivity at most twice the fourth power of sensitivity in this version; the paper does not close the remaining quadratic-versus-quartic separation. The graph-size threshold and induced-subgraph condition matter. This is not a claim that all Boolean complexity measures coincide.",
        "domain": "Extremal combinatorics and Boolean complexity"
      },
      "sources": [
        {
          "citation": "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture",
          "url": "https://arxiv.org/abs/1907.00847v1"
        }
      ],
      "statement": "# Huang: induced hypercube degree bound and the Sensitivity Conjecture\n\n## How many input changes can matter?\n\nA Boolean function turns a string of zeroes and ones into a single zero or one. Think of a decision rule whose inputs are yes-or-no facts. At one input, its sensitivity counts how many individual bit flips would change the answer. Researchers wanted to connect this local notion of change to other ways of measuring a function's complexity. Huang's result settles a longstanding polynomial-relationship question through a statement about a graph.\n\n## Draw the smallest useful picture\n\nThe two-dimensional hypercube is a square. Its four corners are the strings 00, 01, 10 and 11; an edge joins strings that differ in one bit. Keep any three corners and retain every edge between them. The remaining graph is an L-shaped path, whose middle vertex has two neighbors. This is an invented classroom illustration of an induced subgraph: you choose vertices, but do not arbitrarily remove the edges among them.\n\n## The surprising uniform statement\n\nThe theorem concerns the n-dimensional cube. Every induced subgraph with more than half its vertices has a vertex with at least the square root of n neighbors inside that subgraph. The strict more-than-half condition matters. The proof uses a carefully signed matrix and relationships between eigenvalues, quantities that describe how a matrix acts along special directions. The small square illustrates the claim without proving its general form.\n\n## One theorem, inherited bridges\n\nAn existing equivalence connects this graph question to the Sensitivity Conjecture. Huang supplies the new graph bound; the earlier bridge retains its own credit. Polynomially related does not mean that every complexity measure is equal, or that the best remaining exponent is known. The original version leaves a quadratic-versus-quartic gap in a related bound.\n\n## What a reader can check\n\nList all four three-corner subsets of the square and verify their maximum degrees. Then compare your finite check with the theorem's all-dimensional quantifier. A useful contribution can explain that distinction or trace the earlier equivalence to its source. Exhausting the square's cases is a demonstration, not a new proof of the Sensitivity Conjecture.\n\n## Scientific scope\n\nThe stated degree bound implies block sensitivity at most twice the fourth power of sensitivity in this version; the paper does not close the remaining quadratic-versus-quartic separation. The graph-size threshold and induced-subgraph condition matter. This is not a claim that all Boolean complexity measures coincide.\n\n## Sources for this explanation\n\n- [Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture](https://arxiv.org/abs/1907.00847v1): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "A signed adjacency-matrix construction and eigenvalue interlacing give the sharp square-root maximum-degree lower bound for induced hypercube subgraphs with more than half the vertices. Via the known Gotsman–Linial equivalence, this proves polynomial relations between Boolean sensitivity, polynomial degree and block sensitivity.",
      "title": "Huang: induced hypercube degree bound and the Sensitivity Conjecture"
    },
    "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
      "version": 1,
      "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:ee07278d17d181973237ee718b69f8625fe802c0efe3707fbf64dbaf3172e31c"
  },
  "target": {
    "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
    "version": 1,
    "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "huang-hypercube-sensitivity-conjecture-2019",
    "entry_id": "huang-hypercube-sensitivity-conjecture-2019",
    "revision": 1,
    "title": "Huang: induced hypercube degree bound and the Sensitivity Conjecture",
    "context": "Extremal combinatorics and Boolean complexity",
    "track": "Exact theorem and reusable mathematical method",
    "band": "S",
    "modifier": "-",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "A signed adjacency-matrix construction and eigenvalue interlacing give the sharp square-root maximum-degree lower bound for induced hypercube subgraphs with more than half the vertices. Via the known Gotsman–Linial equivalence, this proves polynomial relations between Boolean sensitivity, polynomial degree and block sensitivity.",
    "scope": "A signed adjacency-matrix construction and eigenvalue interlacing give the sharp square-root maximum-degree lower bound for induced hypercube subgraphs with more than half the vertices. Via the known Gotsman–Linial equivalence, this proves polynomial relations between Boolean sensitivity, polynomial degree and block sensitivity. The stated degree bound implies block sensitivity at most twice the fourth power of sensitivity in this version; the paper does not close the remaining quadratic-versus-quartic separation. The graph-size threshold and induced-subgraph condition matter. This is not a claim that all Boolean complexity measures coincide.",
    "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 stated degree bound implies block sensitivity at most twice the fourth power of sensitivity in this version; the paper does not close the remaining quadratic-versus-quartic separation. The graph-size threshold and induced-subgraph condition matter. This is not a claim that all Boolean complexity measures coincide.",
    "references": [
      {
        "title": "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture",
        "url": "https://arxiv.org/abs/1907.00847v1",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
      "version": 1,
      "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "320000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "huang-hypercube-sensitivity-conjecture-2019",
    "target": {
      "record_id": "urn:uuid:1f40c3e9-a9f0-48bc-a65d-34b9df9d07b6",
      "version": 1,
      "content_hash": "sha256:fa2a40e1be853a69d12999817b3de95a86f04e1b098bc98bf2bcbbc529515f54"
    },
    "sections": [
      {
        "title": "How many input changes can matter?",
        "text": "A Boolean function turns a string of zeroes and ones into a single zero or one. Think of a decision rule whose inputs are yes-or-no facts. At one input, its sensitivity counts how many individual bit flips would change the answer. Researchers wanted to connect this local notion of change to other ways of measuring a function's complexity. Huang's result settles a longstanding polynomial-relationship question through a statement about a graph."
      },
      {
        "title": "Draw the smallest useful picture",
        "text": "The two-dimensional hypercube is a square. Its four corners are the strings 00, 01, 10 and 11; an edge joins strings that differ in one bit. Keep any three corners and retain every edge between them. The remaining graph is an L-shaped path, whose middle vertex has two neighbors. This is an invented classroom illustration of an induced subgraph: you choose vertices, but do not arbitrarily remove the edges among them."
      },
      {
        "title": "The surprising uniform statement",
        "text": "The theorem concerns the n-dimensional cube. Every induced subgraph with more than half its vertices has a vertex with at least the square root of n neighbors inside that subgraph. The strict more-than-half condition matters. The proof uses a carefully signed matrix and relationships between eigenvalues, quantities that describe how a matrix acts along special directions. The small square illustrates the claim without proving its general form."
      },
      {
        "title": "One theorem, inherited bridges",
        "text": "An existing equivalence connects this graph question to the Sensitivity Conjecture. Huang supplies the new graph bound; the earlier bridge retains its own credit. Polynomially related does not mean that every complexity measure is equal, or that the best remaining exponent is known. The original version leaves a quadratic-versus-quartic gap in a related bound."
      },
      {
        "title": "What a reader can check",
        "text": "List all four three-corner subsets of the square and verify their maximum degrees. Then compare your finite check with the theorem's all-dimensional quantifier. A useful contribution can explain that distinction or trace the earlier equivalence to its source. Exhausting the square's cases is a demonstration, not a new proof of the Sensitivity Conjecture."
      }
    ],
    "references": [
      {
        "title": "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture",
        "url": "https://arxiv.org/abs/1907.00847v1",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}