{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R387",
  "content_hash": null,
  "slug": "hr4-claim-coordinate-retraction-split-injection",
  "type": "claim",
  "title": "Coordinate inclusion gives split injections in homology",
  "summary": "For every n and j, appending a zero coordinate embeds H_j(VR(Q_n;4);Z) as a direct summand of H_j(VR(Q_{n+1};4);Z). Any torsion class at one cube dimension therefore persists at every larger dimension.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-coordinate-retraction-split-injection (“Coordinate inclusion gives split injections in homology”) records a bound, answer, status fact, or structural consequence. The record states: For every n and j, appending a zero coordinate embeds H_j(VR(Q_n;4);Z) as a direct summand of H_j(VR(Q_{n+1};4);Z).",
  "relevance_source": "recorded",
  "body": "Define\n\\[\ni_n:Q_n\\longrightarrow Q_{n+1},\\qquad i_n(x)=(x,0),\n\\]\nand let\n\\[\np_n:Q_{n+1}\\longrightarrow Q_n\n\\]\ndelete the final coordinate. The first map preserves Hamming distance and the second is 1-Lipschitz. Both induce simplicial maps at scale four, and \\(p_n\\circ i_n\\) is the identity on \\(Q_n\\). Hence\n\\[\n(p_n)_*\\circ(i_n)_*=\\operatorname{id}\n\\]\non integral homology. The map \\((i_n)_*\\) is split injective in every degree.\n\nA torsion counterexample at a smallest dimension propagates to every larger dimension. Torsion-freeness at a fixed dimension supplies no converse implication.",
  "status": "supported",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "universal",
    "statement": "all cube dimensions n >= 1 and all homological degrees j >= 0 at Rips scale four"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Elementary simplicial retraction argument recorded on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Elementary simplicial retraction argument recorded on 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R388",
      "title": "Torsion-free through n=5 and no 2-primary torsion at n=6",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}