{
  "agent_view_tokens": 12000,
  "emoji": "🌳",
  "paper": {
    "arxiv_id": "2606.11646"
  },
  "revision": "1785322800000000000",
  "root": {
    "children": [
      {
        "slug": "00-judge-evidence-scorecard",
        "title": "Judge-facing evidence scorecard",
        "file": "pages/00-judge-evidence-scorecard/page.md",
        "children": []
      },
      {
        "slug": "claim-1",
        "title": "Claim 1: Theorem 4.1 proves the constructed matrix V for the PolyILR basis satisfies the contrast property V^T 1 = 0 and orthonormality V^T V = I_{d-1}, making the map φ(x) = V^T log x an isometry (Section 4.2, Theorem 4.1).",
        "file": "pages/claim-1/page.md",
        "children": []
      },
      {
        "slug": "claim-2",
        "title": "Claim 2: Proposition 4.2 establishes that PolyILR produces a unique, canonical orthonormal basis for any given tree topology, with the original tree recoverable from the basis's clade support structure (Section 4.3, Proposition 4.2).",
        "file": "pages/claim-2/page.md",
        "children": []
      },
      {
        "slug": "claim-3",
        "title": "Claim 3: Algorithm 1 constructs the PolyILR basis by applying weighted Helmert contrasts with Gram-Schmidt orthogonalization at each internal node in depth-first order, then spreading local contrasts to a global leaf-indexed basis by dividing by descendant counts (Section 4.2, Algorithm 1).",
        "file": "pages/claim-3/page.md",
        "children": []
      },
      {
        "slug": "claim-4",
        "title": "Claim 4: Unlike PhILR, PolyILR respects the original polytomous tree topology directly without requiring artificial binarization of multifurcating nodes (Figure 1, Section 2).",
        "file": "pages/claim-4/page.md",
        "children": []
      },
      {
        "slug": "claim-5",
        "title": "Claim 5: Proposition 7.1 shows the logit-space quotient ℒ is isomorphic to the Aitchison tangent space ℋ, so centered logits equal CLR coordinates (Section 7, Proposition 7.1).",
        "file": "pages/claim-5/page.md",
        "children": []
      },
      {
        "slug": "claim-6",
        "title": "Claim 6: The method is validated on the HMP, cMD3, and DISCO microbiome/single-cell datasets, demonstrating stable feature selection and interpretable tree-level (clade) importance aggregation (Section 6, Tables 2-6).",
        "file": "pages/claim-6/page.md",
        "children": []
      },
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ],
    "file": "pages/index.md",
    "slug": "index",
    "title": "Tree-Structured Orthonormal Decomposition of the Aitchison Simplex"
  },
  "schema_version": 1,
  "space_id": "ProCreations/repro-tree-structured-orthonormal-decomposition-of-the-aitchison-simplex",
  "tags": [
    "icml2026-repro",
    "paper-pws8t4kBP4"
  ],
  "title": "Reproduction: Tree-Structured Orthonormal Decomposition of the Aitchison Simplex",
  "updated_at": "2026-07-29T12:30:00+00:00"
}