{
  "paper_id": "qqc-001-foundations",
  "claims": [
    {
      "claim_id": "prop-1",
      "label": "Proposition 2.1 (Normalized spinors form S^3)",
      "type": "proposition",
      "statement": "The set of normalized vectors (alpha, beta) in C^2 satisfying |alpha|^2 + |beta|^2 = 1 is the unit sphere S^3 inside R^4 under the identification C^2 ≅ R^4.",
      "status": "proved",
      "proof_sketch": "Write alpha = a + ib and beta = c + id. The normalization condition becomes a^2 + b^2 + c^2 + d^2 = 1, which is exactly the defining equation of S^3.",
      "dependencies": [],
      "evidence_location": "Section 2.1",
      "notes": "Standard mathematical fact about normalized two-component spinors."
    },
    {
      "claim_id": "prop-2",
      "label": "Proposition 2.2 (Pure states are CP^1 ≅ S^2)",
      "type": "proposition",
      "statement": "The pure-state space of a single qubit is the projective space CP^1, obtained from normalized spinors by quotienting global phase, and CP^1 is diffeomorphic to S^2 via the Bloch-sphere map.",
      "status": "proved",
      "proof_sketch": "The circle action (alpha, beta) -> (e^{i chi} alpha, e^{i chi} beta) preserves normalization and defines the projective quotient. The Bloch map is phase-invariant and has image on the unit 2-sphere.",
      "dependencies": ["prop-1"],
      "evidence_location": "Section 2.2",
      "notes": "Standard fact; the paper uses it to keep S^3 distinct from the Bloch sphere."
    },
    {
      "claim_id": "prop-3",
      "label": "Proposition 2.3 (Single-qubit gates and SU(2) representatives)",
      "type": "proposition",
      "statement": "Every single-qubit gate U in U(2) can be written as U = e^{i chi} V with V in SU(2), and the representative V is unique up to sign.",
      "status": "proved",
      "proof_sketch": "Choose chi so that e^{2 i chi} = det(U). Then V = e^{-i chi} U has determinant 1. A second such representative differs by a phase whose square is 1, hence by ±1.",
      "dependencies": [],
      "evidence_location": "Section 2.3",
      "notes": "This is the key bridge from physical gates in U(2) to quaternionic modeling through SU(2)."
    },
    {
      "claim_id": "def-4",
      "label": "Definition 3.1 (Chosen quaternion-SU(2) map)",
      "type": "definition",
      "statement": "The paper uses the explicit identification Phi(a + b i + c j + d k) = [[a - d i, -c - b i], [c - b i, a + d i]], equivalently Phi(q) = a I - i (b sigma_x + c sigma_y + d sigma_z).",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [],
      "evidence_location": "Section 3.1, Eq. (1)",
      "notes": "This is a standard variant chosen because the quaternion basis units align with the x, y, z Pauli axes."
    },
    {
      "claim_id": "prop-5",
      "label": "Proposition 3.1 (Unit quaternions are isomorphic to SU(2))",
      "type": "proposition",
      "statement": "The restriction of Phi to the unit quaternions Sp(1) = {q in H : |q| = 1} is a Lie-group isomorphism from Sp(1) onto SU(2).",
      "status": "proved",
      "proof_sketch": "Under Phi, the quaternion units satisfy the same multiplication rules as -i sigma_x, -i sigma_y, and -i sigma_z. The determinant of Phi(q) is |q|^2, so unit quaternions map into SU(2), and every SU(2) matrix of the standard form has a unique quaternion preimage.",
      "dependencies": ["def-4"],
      "evidence_location": "Section 3.1",
      "notes": "Standard Lie-group fact."
    },
    {
      "claim_id": "cor-6",
      "label": "Corollary 3.2 (Sequential gate composition is quaternion multiplication)",
      "type": "corollary",
      "statement": "If q1 and q2 in Sp(1) represent single-qubit gates through Phi, then applying the first gate and then the second corresponds to the product q2 q1.",
      "status": "proved",
      "proof_sketch": "This is the homomorphism property Phi(q2 q1) = Phi(q2) Phi(q1).",
      "dependencies": ["prop-5"],
      "evidence_location": "Section 3.1",
      "notes": "Standard consequence of the chosen identification."
    },
    {
      "claim_id": "prop-7",
      "label": "Proposition 4.1 (Double cover of SO(3))",
      "type": "proposition",
      "statement": "Conjugation of pure quaternions by unit quaternions defines a surjective homomorphism rho: Sp(1) -> SO(3) with kernel {+1, -1}; equivalently SU(2) is a double cover of SO(3).",
      "status": "proved",
      "proof_sketch": "The action q v q^{-1} preserves the pure-quaternion subspace and Euclidean norm, hence gives an element of SO(3). Every spatial rotation has an axis-angle lift to a unit quaternion, and only ±1 act trivially on all pure quaternions.",
      "dependencies": ["prop-5"],
      "evidence_location": "Section 4.3",
      "notes": "Standard rotation-theoretic fact."
    },
    {
      "claim_id": "rem-8",
      "label": "Remark 2.4 (Two distinct roles of S^3)",
      "type": "remark",
      "statement": "The manifold S^3 appears both as the space of normalized spinor representatives in C^2 and as the manifold underlying the Lie group of unit quaternions, but these are different mathematical roles and should not be conflated with the Bloch sphere S^2.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": ["prop-1", "prop-2", "prop-5"],
      "evidence_location": "Section 2.4 and Section 3.2",
      "notes": "Clarifying statement emphasized by the paper."
    },
    {
      "claim_id": "int-9",
      "label": "Interpretive Claim 5.2 (RQM framing choice)",
      "type": "interpretive_claim",
      "statement": "RQM Technologies chooses to organize standard single-qubit gate geometry by treating the quaternion model of SU(2) as a primary expository language, while explicitly preserving the standard distinctions among state vectors, projective states, gate representatives, and physical rotations.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": ["prop-2", "prop-3", "prop-5", "prop-7", "rem-8"],
      "evidence_location": "Section 5.2",
      "notes": "This is an organizational claim about presentation, not a claim of new mathematics or new physics."
    }
  ]
}