{
  "schema": "connectedness-atlas/research-ledger/v1",
  "updated": "2026-10-02",
  "description": "A guide to the scope of claims used by the explorer. Manuscript claims below retain references to the 19 September 2026 working draft for traceability. The main article and research companion are in preparation; PDFs await author review and approval. The recorded presence of an argument is not journal acceptance, a priority claim or formal verification.",
  "claims": [
    {
      "id": "normalization",
      "title": "Three four-dimensional orientation charts",
      "statement": "A labelled pair of contracting planar similarities with distinct fixed points has a unique complex-affine normalization with translations −1 and +1. Its nonzero coefficients lie in the open unit disk. Branch exchange identifies the two mixed orientation types.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 2.1",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "Complex-affine conjugacy; pairs with a common fixed point are excluded."
    },
    {
      "id": "contact",
      "title": "Connectedness is a first-level contact",
      "statement": "K is connected exactly when its two first-level pieces meet; equivalently 2 belongs to D = L−K − L+K. The contact gap is continuous, so the connectedness locus is relatively closed in the strict contraction domain.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 3.1 and equation (8)",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "All three plane-orientation types. This uses the classical binary connectedness criterion."
    },
    {
      "id": "bounds",
      "title": "A connected core and an outer bound",
      "statement": "r−²+r+² ≥ 1 guarantees connectedness; connectedness requires r−+r+ ≥ 1. In atlas coordinates these are ρ ≤ 2√(w²+(1−w)²) and ρ ≤ 2, subject to strict contraction.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 3.2",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "All three orientation types.",
      "interpretation": "These bounds do not assert radial monotonicity or identify the entire boundary."
    },
    {
      "id": "outer-rigidity",
      "title": "Rigidity at the outer bound",
      "statement": "At r−+r+=1, connectedness occurs exactly for real canonical coefficients; the attractor is then a real interval.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Theorem 4.3",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "All plane orientations; real nonzero strict contractions."
    },
    {
      "id": "phase-families",
      "title": "Exact phase diagrams",
      "statement": "Opposite–opposite diagonal and anti-diagonal families have explicit Cantor-product and parallelogram descriptions. Their radial frontier includes a genuine discontinuity. A quarter-turn family has equivalent presentations across the plane orientations.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Theorem 5.1, Corollary 5.2 and Theorem 5.3",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "The explicitly specified one-complex-parameter and quarter-turn families."
    },
    {
      "id": "weighted-triangle",
      "title": "An exact triangle at every weight",
      "statement": "The opposite–opposite altitude-dissection family reaches the square-sum bound at every contraction weight. Separated perturbations establish sharpness in that orientation chart.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Theorem 5.5",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "Opposite–opposite chart; 0<w<1."
    },
    {
      "id": "contact-fibres",
      "title": "Closed contact fibres and prefix limits",
      "statement": "Fixing one address and allowing all competitors in the opposite first-level piece gives a closed contact fibre. Nested prefix contact loci recover that fibre; over compact contraction windows their distance functions converge with uniform geometric error bounds.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Theorem 7.1 and Proposition 7.2",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "interpretation": "Fixing one address is different from fixing both addresses or fixing a spatial point independently of the parameter."
    },
    {
      "id": "structural-stability",
      "title": "A stable coding quotient",
      "statement": "Within a prescribed nonempty family, parameters with exactly the forced coding relation have the same symbolic quotient. Excess identifications form a relative Fσ set, so the stable set is relative Gδ.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Definition 6.5, Proposition 6.6 and Proposition 7.3",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "interpretation": "This general statement does not assert that every stable set is open or nonempty."
    },
    {
      "id": "twelve-zippers",
      "title": "Twelve marked zipper surfaces",
      "statement": "Three plane-orientation types and four endpoint-reversal signatures give twelve marked binary zipper charts. Endpoint coupling yields connected attractors; absence of additional contacts gives embedded canonical curves.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Theorem 8.2 and Proposition 8.3",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "interpretation": "Marked presentations may overlap or be symmetry-related. Plane orientation and endpoint traversal are different data."
    },
    {
      "id": "stable-disks",
      "title": "Three complete opposite–opposite stable disks",
      "statement": "The OO zipper signatures 00, 01 and 10 have stable set exactly |q−1/2|<1/2. Their nonreal boundary parameters give explicit filled triangles and specified contact segments.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 8.5 and Theorem 8.6",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "Only these marked zipper slices; this is a complete classification within each stated lens."
    },
    {
      "id": "zipper-certificates",
      "title": "Finite endpoint-return criteria",
      "statement": "Seven marked presentations admit a common endpoint return. Exhausting the required cylinder-disk separation tests certifies an embedded curve; outward bounds on a rational rectangle certify the entire rectangle.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 8.7",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "DD01, DD10, DD11, DO10, DO11, OO01 and OO10.",
      "interpretation": "A stopped or capped test is unresolved. Coloured certified cells form a finite inner cover, not a full classification of the other stable slices."
    },
    {
      "id": "exterior-placement",
      "title": "Stable surfaces in exterior coordinates",
      "statement": "The canonical reciprocal coefficients satisfy |c1|>1 and |c2|>1. Four selected DD/DO stable slices are placed in a common three-dimensional coefficient-phase projection, with weight retained by colour. Opposite coefficients acquire the phase prescribed by canonical normalization.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 8.8 and equations (42)–(47)",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "scope": "DD10, DD11, DO10 and DO11.",
      "interpretation": "Overlaps in the projection at different weights need not be intersections in four dimensions. The toroidal opening comes from the display map."
    },
    {
      "id": "finite-separation",
      "title": "Replayable disconnection and neighbourhoods",
      "statement": "Complete finite covers of all cross-prefix pairs give disconnection certificates. A strictly positive contact-gap bound persists on an explicit four-real-dimensional parameter neighbourhood of fixed orientation.",
      "status": "proved-in-manuscript",
      "source": {
        "publication": "4d-connectedness-2026",
        "reference": "Proposition 9.1, Lemma 9.2 and Appendix A",
        "url": "https://complextrees.com/4D/publication/#flagship",
        "referenceVersion": "Working draft, 19 September 2026; numbering subject to revision",
        "availability": "Author review; PDF not released"
      },
      "authorship": [
        "Bernat Espigulé"
      ],
      "interpretation": "Certification requires validated arithmetic and complete cover verification; a floating-point browser exclusion is recorded separately."
    },
    {
      "id": "dd-square-sum-antecedent",
      "title": "Published antecedent for the DD square-sum implication",
      "statement": "For a normalized pair of nonzero holomorphic contractions with distinct fixed points, |λ−|²+|λ+|² ≥ 1 implies connectedness. This combines the dust-likeness bound in Theorem 1.1 with the binary equivalence recorded in Remark 1.1.",
      "status": "published-theorem",
      "authorship": [
        "Miwa Aoki",
        "Masayo Fujimura",
        "Masahiko Taniguchi"
      ],
      "source": {
        "publication": "AokiFujimuraTaniguchi2014",
        "reference": "Theorem 1.1 and Remark 1.1",
        "url": "https://doi.org/10.4171/JFG/10"
      },
      "scope": "DD only; strict contractions 0<|λ−|,|λ+|<1 and distinct fixed points, normalized to 0 and 1 in the cited paper.",
      "interpretation": "A sufficient criterion, not a characterization of connectedness. The all-orientation bounds are treated separately in the current manuscript. Contact fibres, marked stable families and the atlas projection have their own definitions and results."
    }
  ],
  "computationalEvidence": [
    {
      "kind": "exact-family",
      "meaning": "An analytic result under its stated hypotheses; displayed coordinates may still be floating-point evaluations."
    },
    {
      "kind": "certified-stability-cell",
      "meaning": "Outward bounds verify the prescribed finite endpoint-return tests on every parameter of the recorded cell."
    },
    {
      "kind": "exact-disconnection-certificate",
      "meaning": "The independent rational checker validates a complete cross-prefix cover and its strict separation bounds."
    },
    {
      "kind": "numerical-exclusion",
      "meaning": "A finite floating-point computation reports separation; no exact arithmetic certificate is implied."
    },
    {
      "kind": "finite-survival",
      "meaning": "The parameter was not excluded at the recorded depth and budget. This alone gives no positive connectedness result."
    },
    {
      "kind": "unresolved",
      "meaning": "The available calculation stopped or lacks a decisive certificate."
    }
  ],
  "openDirections": [
    "Describe local contact geometry near the exact critical triangles and rectangles.",
    "Determine how fixed-address fibres intersect and how excess contacts change prescribed family quotients.",
    "Compare stability islands in distinct marked families without conflating the ambient spaces.",
    "Extend finite capture to independent multipliers with explicit hypotheses and validated parameter cells."
  ],
  "scopeNotes": [
    "The dated companion does not claim global regular-closedness for all four-dimensional orientation charts or a global cavity classification.",
    "A selected finite address hue is a computational fingerprint, not a topological invariant or a definition of structural stability.",
    "The 3D prefix tree uses a scale height to display addresses. It is not the planar attractor, a limiting contact graph, or an additional spatial dimension of the mathematical system."
  ]
}