{
  "_generated": "2026-04-09T15:13:48.409255+00:00",
  "_description": "Site-wide index of all claims across all papers in the series.",
  "_source": "Generated by scripts/generate_index.py",
  "total": 42,
  "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:prop-1"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:prop-2"
    },
    {
      "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).",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:prop-3"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:def-4"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:prop-5"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:cor-6"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:prop-7"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:rem-8"
    },
    {
      "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.",
      "_paper_id": "qqc-001-foundations",
      "_paper_path": "papers/qqc-001-foundations/",
      "_global_claim_id": "qqc-001-foundations:int-9"
    },
    {
      "claim_id": "def-1",
      "label": "Definition 3.1 (Global-phase equivalence on U(2))",
      "type": "definition",
      "statement": "Two single-qubit gates U and V in U(2) are globally phase-equivalent, written U ~gp V, when there exists a real chi such that U = e^{i chi} V.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [],
      "evidence_location": "Section 3",
      "notes": "This is the gate-level equivalence relation that motivates the IR scope.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:def-1"
    },
    {
      "claim_id": "prop-2",
      "label": "Proposition 2.1 (Single-qubit gates admit SU(2) representatives)",
      "type": "proposition",
      "statement": "For every single-qubit gate U in U(2), there exist a real chi and V in SU(2) such that U = e^{i chi} V, and V is unique up to multiplication by -1.",
      "status": "proved",
      "proof_sketch": "Remove determinant phase exactly as in qqc-001-foundations:prop-3. Uniqueness up to sign follows because any second determinant-one representative differs by a phase whose square is 1.",
      "dependencies": [
        "qqc-001-foundations:prop-3"
      ],
      "evidence_location": "Section 2.1",
      "notes": "Standard mathematical fact carried forward from paper 1.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:prop-2"
    },
    {
      "claim_id": "prop-3",
      "label": "Proposition 2.2 (Unit quaternions model SU(2))",
      "type": "proposition",
      "statement": "Under the fixed convention Phi(a + b i + c j + d k) = a I - i (b sigma_x + c sigma_y + d sigma_z), the unit quaternions are in Lie-group isomorphism with SU(2).",
      "status": "proved",
      "proof_sketch": "This is the quaternion-SU(2) correspondence established in qqc-001-foundations:prop-5.",
      "dependencies": [
        "qqc-001-foundations:def-4",
        "qqc-001-foundations:prop-5"
      ],
      "evidence_location": "Section 2.1",
      "notes": "The paper treats this as inherited background, not a new theorem.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:prop-3"
    },
    {
      "claim_id": "cor-4",
      "label": "Corollary 2.3 (The only residual ambiguity is q ~ -q)",
      "type": "corollary",
      "statement": "If two unit quaternions represent the same single-qubit gate action modulo global phase under the fixed Phi convention, then they differ by sign: q' = q or q' = -q.",
      "status": "proved",
      "proof_sketch": "Combine the existence and sign-uniqueness of SU(2) representatives with the unit-quaternion model of SU(2).",
      "dependencies": [
        "prop-2",
        "prop-3"
      ],
      "evidence_location": "Section 2.1",
      "notes": "This is the exact quotient ambiguity handled by the IR.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:cor-4"
    },
    {
      "claim_id": "def-5",
      "label": "Definition 3.2 (Quaternionic single-qubit IR element)",
      "type": "definition",
      "statement": "A quaternionic single-qubit IR element u1q(w, x, y, z) is a quadruple of real numbers with w^2 + x^2 + y^2 + z^2 = 1, interpreted as the unit quaternion q = w + x i + y j + z k and therefore as an SU(2) representative of a single-qubit gate action modulo global phase.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "prop-3",
        "cor-4"
      ],
      "evidence_location": "Section 3",
      "notes": "This is the central IR definition of the paper.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:def-5"
    },
    {
      "claim_id": "def-6",
      "label": "Definition 4.1 (Normalization map)",
      "type": "definition",
      "statement": "For any nonzero quadruple q-tilde in R^4, the normalization map sends q-tilde to q-tilde / ||q-tilde||, producing a unit quaternion candidate before sign canonicalization.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [],
      "evidence_location": "Section 4.1",
      "notes": "This addresses floating-point drift rather than exact mathematical ambiguity.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:def-6"
    },
    {
      "claim_id": "def-7",
      "label": "Definition 4.2 (Sign-canonicalization map)",
      "type": "definition",
      "statement": "For a unit quaternion q = (w, x, y, z), sign-canonicalization keeps q when w > 0, negates q when w < 0, and on the equator w = 0 chooses the sign for which the first nonzero component among (x, y, z) is positive.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "cor-4"
      ],
      "evidence_location": "Section 4.2",
      "notes": "The equatorial tie-break is a deterministic convention, not a physical claim.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:def-7"
    },
    {
      "claim_id": "prop-8",
      "label": "Proposition 4.4 (Canonicalization selects exactly one representative)",
      "type": "proposition",
      "statement": "Each sign-equivalence class {q, -q} of unit quaternions contains exactly one representative in the canonical hemisphere defined by w > 0 together with the equatorial first-nonzero-positive tie-break when w = 0.",
      "status": "proved",
      "proof_sketch": "If w is nonzero, exactly one of q or -q has positive real part. If w = 0, the first nonzero component of (x, y, z) changes sign under q -> -q, so exactly one sign satisfies the tie-break rule.",
      "dependencies": [
        "def-7"
      ],
      "evidence_location": "Section 4.2",
      "notes": "This is the determinism guarantee of the IR.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:prop-8"
    },
    {
      "claim_id": "rem-9",
      "label": "Remark 4.5 (Shortest-geodesic interpretation)",
      "type": "remark",
      "statement": "For q = cos(theta/2) + sin(theta/2)(n_x i + n_y j + n_z k), the canonical condition w >= 0 restricts theta to the closed interval [0, pi]; at w = 0 both signs have the same geodesic length and the tie-break serves only to resolve representation ambiguity.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "def-7",
        "prop-8"
      ],
      "evidence_location": "Section 4.2",
      "notes": "This explains the phrase 'shortest-geodesic representative' in compiler-facing terms.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:rem-9"
    },
    {
      "claim_id": "prop-10",
      "label": "Proposition 5.1 (Named gates translate into explicit canonical tuples)",
      "type": "proposition",
      "statement": "Under the fixed Phi convention, the identity, Pauli gates, H, S, S-dagger, T, T-dagger, and the rotation families Rx(theta), Ry(theta), and Rz(theta) translate into explicit canonical unit-quaternion tuples suitable for u1q storage.",
      "status": "proved",
      "proof_sketch": "The rotation families follow from axis-angle form. Pauli and phase-family gates are coordinate-axis rotations, and H is the pi-rotation about the axis (1, 0, 1) / sqrt(2) after removing an overall phase.",
      "dependencies": [
        "prop-3",
        "prop-8"
      ],
      "evidence_location": "Section 5",
      "notes": "The translation table is one of the main operational outputs of the paper.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:prop-10"
    },
    {
      "claim_id": "prop-11",
      "label": "Proposition 6.2 (Composition of adjacent single-qubit gates)",
      "type": "proposition",
      "statement": "If g_1, ..., g_m are consecutive single-qubit gates on one wire in application order and q_1, ..., q_m are their unit-quaternion representatives, then the fused segment action is represented by Can(q_m q_{m-1} ... q_1).",
      "status": "proved",
      "proof_sketch": "Sequential application corresponds to matrix multiplication in SU(2), which corresponds under Phi to quaternion multiplication in the reverse textual order of application. Canonicalization then chooses the deterministic representative of the resulting sign class.",
      "dependencies": [
        "qqc-001-foundations:cor-6",
        "def-6",
        "def-7",
        "prop-8"
      ],
      "evidence_location": "Section 6",
      "notes": "This is the fusion rule that makes the IR compiler-friendly.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:prop-11"
    },
    {
      "claim_id": "def-12",
      "label": "Definition 3.3 (Single-qubit segment)",
      "type": "definition",
      "statement": "A single-qubit segment on wire a is a maximal consecutive run of gates that act only on wire a, bounded by circuit endpoints or by an operation outside the single-qubit unitary model for that wire, in particular by an entangling gate touching that wire.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [],
      "evidence_location": "Section 3",
      "notes": "This definition marks the intended scope boundary between the present IR and multi-qubit structure.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:def-12"
    },
    {
      "claim_id": "int-13",
      "label": "Interpretive Claim 8.1 (RQM IR framing choice)",
      "type": "interpretive_claim",
      "statement": "RQM Technologies adopts the canonical u1q quaternion tuple, the closed-hemisphere sign convention, and the segment-fusion viewpoint as a deterministic internal representation layer for single-qubit circuit structure, while explicitly preserving the standard matrix formulation and the limitation to single-qubit action modulo global phase.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "def-5",
        "prop-8",
        "prop-10",
        "prop-11",
        "def-12"
      ],
      "evidence_location": "Section 8",
      "notes": "This is an organizational design claim, not a new mathematical theorem.",
      "_paper_id": "qqc-002-canonical-ir",
      "_paper_path": "papers/qqc-002-canonical-ir/",
      "_global_claim_id": "qqc-002-canonical-ir:int-13"
    },
    {
      "claim_id": "def-1",
      "label": "Definition 3.1 (Single-qubit segment)",
      "type": "definition",
      "statement": "A single-qubit segment on wire a is a maximal consecutive run of gates acting only on wire a, bounded by circuit endpoints or by boundary operations outside the present single-qubit model for that wire, including entangling gates, measurement, reset, or classical-control boundaries.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [],
      "evidence_location": "Section 3",
      "notes": "This is the optimization unit targeted by the paper.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:def-1"
    },
    {
      "claim_id": "def-2",
      "label": "Definition 4.1 (Quaternionic fusion operator)",
      "type": "definition",
      "statement": "For a single-qubit segment seg = (g_1, ..., g_n) in circuit application order, with canonical unit-quaternion representatives q_1, ..., q_n, the quaternionic fusion operator is Fuse(seg) = Can(q_n ... q_1).",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "qqc-002-canonical-ir:def-5",
        "qqc-002-canonical-ir:def-6",
        "qqc-002-canonical-ir:def-7"
      ],
      "evidence_location": "Section 4",
      "notes": "The multiplication order matches sequential application under the fixed Phi convention.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:def-2"
    },
    {
      "claim_id": "def-3",
      "label": "Definition 5.1 (Axis-aware reconstruction)",
      "type": "definition",
      "statement": "The reconstruction operator Recon(q) emits no gate for identity, an exact named gate for recognized canonical tuples, an exact axis-aligned rotation when exactly one of the imaginary coordinates is nonzero, and otherwise a generic single-qubit primitive carrying the canonical quaternion payload.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "def-2"
      ],
      "evidence_location": "Section 5",
      "notes": "Reconstruction is a presentation layer on top of the canonical fused quaternion.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:def-3"
    },
    {
      "claim_id": "prop-4",
      "label": "Proposition 6.1 (Fusion preserves exact segment action up to global phase)",
      "type": "proposition",
      "statement": "For every single-qubit segment seg, the fused canonical quaternion Fuse(seg) represents exactly the same single-qubit action as seg modulo global phase under the fixed quaternion-SU(2) convention.",
      "status": "proved",
      "proof_sketch": "Under Phi, sequential application of the segment gates corresponds to the matrix product Phi(q_n) ... Phi(q_1), which equals Phi(q_n ... q_1). Canonicalization does not change the represented action because it only normalizes and selects one representative from the sign pair {q, -q}.",
      "dependencies": [
        "qqc-001-foundations:cor-6",
        "qqc-002-canonical-ir:prop-8",
        "def-2"
      ],
      "evidence_location": "Section 6",
      "notes": "This is the core equivalence-preservation result for the local optimization pass.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:prop-4"
    },
    {
      "claim_id": "prop-5",
      "label": "Proposition 6.2 (Determinism of the fused representative)",
      "type": "proposition",
      "statement": "For every nonempty single-qubit segment, Fuse(seg) is uniquely determined by the segment action and the fixed canonicalization convention inherited from qqc-002-canonical-ir.",
      "status": "proved",
      "proof_sketch": "Quaternion multiplication gives the segment action uniquely up to the sign ambiguity q ~ -q, and the canonicalization map Can selects exactly one representative from that sign class.",
      "dependencies": [
        "qqc-002-canonical-ir:prop-8",
        "def-2"
      ],
      "evidence_location": "Section 6",
      "notes": "This is the determinism guarantee used by the benchmark harness and by equality checks.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:prop-5"
    },
    {
      "claim_id": "cor-6",
      "label": "Corollary 6.3 (Segment replacement by one canonical payload)",
      "type": "corollary",
      "statement": "Every nonidentity single-qubit segment may be replaced by one canonical quaternionic payload without changing the represented segment action modulo global phase, while identity segments may be replaced by the empty segment.",
      "status": "proved",
      "proof_sketch": "Combine exact equivalence preservation with determinism of the fused representative; identity is the unique case in which reconstruction can emit no gate.",
      "dependencies": [
        "prop-4",
        "prop-5",
        "def-3"
      ],
      "evidence_location": "Section 6",
      "notes": "This is the immediate local optimization corollary.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:cor-6"
    },
    {
      "claim_id": "prop-7",
      "label": "Proposition 6.4 (Minimality in the quaternionic IR)",
      "type": "proposition",
      "statement": "Within the quaternionic IR itself, the fully fused representation of a nonidentity single-qubit segment has minimal gate count equal to one, while the identity class has minimal emitted gate count zero.",
      "status": "proved",
      "proof_sketch": "A canonical u1q payload is itself the storage unit of the IR, so every nonidentity segment action has a one-payload representation and no smaller nonempty representation exists inside that IR.",
      "dependencies": [
        "qqc-002-canonical-ir:def-5",
        "cor-6"
      ],
      "evidence_location": "Section 6",
      "notes": "This claim is about the quaternionic IR layer itself, not about optimality in arbitrary external gate sets.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:prop-7"
    },
    {
      "claim_id": "prop-8",
      "label": "Proposition 5.2 (Exact axis-aligned recovery)",
      "type": "proposition",
      "statement": "If the canonical fused quaternion has exactly one nonzero imaginary coordinate, then the reconstruction layer can recover an exact axis-aligned rotation Rx(theta), Ry(theta), or Rz(theta) representing the same segment action.",
      "status": "proved",
      "proof_sketch": "A quaternion with exactly one nonzero imaginary coordinate is already in axis-angle form about a coordinate axis, so its corresponding SU(2) element is one of the three standard single-axis rotation families.",
      "dependencies": [
        "def-3",
        "qqc-001-foundations:prop-5"
      ],
      "evidence_location": "Section 5",
      "notes": "This is an exact recovery statement for special cases, not a claim of globally unique named-gate reconstruction.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:prop-8"
    },
    {
      "claim_id": "emp-9",
      "label": "Empirical Result 8.1 (Reference-corpus compression)",
      "type": "empirical_result",
      "statement": "On the shipped reference benchmark corpus of 13 single-qubit segments, the quaternionic local fusion pass reduces 36 input single-qubit gates to 11 reconstructed output gates, eliminates 2 segments to identity, and passes the authoritative canonical-quaternion equivalence check on all measured segments.",
      "status": "empirical",
      "proof_sketch": "Measured by the included plain-Python harness benchmark_gate_fusion.py, with outputs recorded in benchmark_results.json.",
      "dependencies": [
        "def-2",
        "def-3",
        "prop-4",
        "prop-5"
      ],
      "evidence_location": "Section 8; artifacts/code/benchmark_results.json",
      "notes": "This is a measured result on the shipped reference corpus only, not a universal compiler-performance claim.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:emp-9"
    },
    {
      "claim_id": "int-10",
      "label": "Interpretive Claim 9.1 (Geometry-native local optimization pass)",
      "type": "interpretive_claim",
      "statement": "RQM Technologies treats quaternionic gate fusion as a geometry-native local optimization pass because the optimization rule is the native group law of the canonical single-qubit representation rather than an external symbolic heuristic layered on top of named-gate syntax.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "def-2",
        "prop-4",
        "prop-5",
        "emp-9"
      ],
      "evidence_location": "Section 9",
      "notes": "This is an organizational and methodological framing claim, not a new mathematical theorem.",
      "_paper_id": "qqc-003-gate-fusion",
      "_paper_path": "papers/qqc-003-gate-fusion/",
      "_global_claim_id": "qqc-003-gate-fusion:int-10"
    },
    {
      "claim_id": "def-1",
      "label": "Definition 4.1 (Canonical segment representative)",
      "type": "definition",
      "statement": "Let seg = (g_1, ..., g_n) be a single-qubit segment in application order, and let q_1, ..., q_n be the translated unit-quaternion representatives of those gates. The canonical segment representative is Q(seg) = Can(q_n ... q_1).",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "qqc-002-canonical-ir:def-6",
        "qqc-002-canonical-ir:def-7",
        "qqc-003-gate-fusion:def-2"
      ],
      "evidence_location": "Section 4.1",
      "notes": "This definition introduces the canonical paper-specific object Q(seg) used throughout the trust layer.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:def-1"
    },
    {
      "claim_id": "def-2",
      "label": "Definition 4.2 (Authoritative equivalence)",
      "type": "definition",
      "statement": "Two single-qubit segments seg and seg' are canonically equivalent, written Verify(seg, seg'), when Q(seg) = Q(seg').",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "def-1"
      ],
      "evidence_location": "Section 4.1",
      "notes": "This is the authoritative correctness criterion used by the paper.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:def-2"
    },
    {
      "claim_id": "prop-3",
      "label": "Proposition 4.3 (Canonical equivalence is exact segment equivalence modulo global phase)",
      "type": "proposition",
      "statement": "For single-qubit segments, Verify(seg, seg') holds if and only if seg and seg' represent the same SU(2) action up to the global-phase quotient inherited from U(2).",
      "status": "proved",
      "proof_sketch": "Single-qubit actions modulo global phase correspond exactly to sign classes {q, -q} of unit quaternions under the fixed map Phi. The canonicalization rule chooses exactly one representative from each sign class, so equality of canonical representatives is equivalent to equality of represented action modulo global phase.",
      "dependencies": [
        "qqc-001-foundations:prop-5",
        "qqc-002-canonical-ir:prop-8",
        "def-1",
        "def-2"
      ],
      "evidence_location": "Section 4.1",
      "notes": "This proposition resolves the trust question for quaternionic optimization directly in the native representation.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:prop-3"
    },
    {
      "claim_id": "def-4",
      "label": "Definition 4.4 (Quaternionic compilation invariants)",
      "type": "definition",
      "statement": "For a valid optimized replacement of a single-qubit segment, the preserved invariants are: unit norm after normalization, canonical sign choice under the fixed hemisphere rule, represented SU(2) action modulo global phase, and reproducible canonical output form.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "qqc-002-canonical-ir:def-6",
        "qqc-002-canonical-ir:def-7",
        "qqc-002-canonical-ir:prop-8"
      ],
      "evidence_location": "Section 4.2",
      "notes": "These are the core invariants used in the verifier.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:def-4"
    },
    {
      "claim_id": "prop-5",
      "label": "Proposition 4.6 (Canonical output form is deterministic)",
      "type": "proposition",
      "statement": "If two compilation runs produce valid optimized representations of the same single-qubit segment action under the fixed convention, then their canonical quaternionic payloads are identical.",
      "status": "proved",
      "proof_sketch": "Both outputs represent the same sign class {q, -q} in the unit quaternions, and the canonicalization map chooses exactly one representative from that class.",
      "dependencies": [
        "def-4",
        "qqc-002-canonical-ir:prop-8"
      ],
      "evidence_location": "Section 4.2",
      "notes": "This is the reproducibility guarantee of the trust layer.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:prop-5"
    },
    {
      "claim_id": "def-6",
      "label": "Definition 5.1 (Canonical validation procedure)",
      "type": "definition",
      "statement": "To validate a candidate optimized segment seg' against an original segment seg: translate each segment into the fixed quaternionic representation, multiply in application order to obtain raw segment products, normalize and sign-canonicalize both products, and compare the resulting canonical representatives for equality.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "def-1",
        "def-2"
      ],
      "evidence_location": "Section 5.1",
      "notes": "This is the operational trust workflow advocated by the paper.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:def-6"
    },
    {
      "claim_id": "prop-7",
      "label": "Proposition 6.4 (Canonical equality implies perfect matrix diagnostics)",
      "type": "proposition",
      "statement": "If Verify(seg, seg') holds exactly, then for the corresponding matrix representatives U and V one has MatrixEq(U, V), ProcFid(U, V) = 1, TraceDiag(U, V) = 2, and PhaseRes(U, V) = 0.",
      "status": "proved",
      "proof_sketch": "Canonical equality implies the same unit quaternion representative. Under Phi, the two segments therefore produce the same SU(2) matrix, so U dagger V = I and the stated diagnostics follow immediately.",
      "dependencies": [
        "def-1",
        "def-2",
        "def-6"
      ],
      "evidence_location": "Section 6",
      "notes": "This proposition ties the authoritative quaternionic criterion to the supporting matrix diagnostics.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:prop-7"
    },
    {
      "claim_id": "prop-8",
      "label": "Proposition 6.1 (Validation of optimized segments)",
      "type": "proposition",
      "statement": "A local single-qubit optimization is valid when the optimized segment and the original segment have equal canonical segment representatives; matrix diagnostics may be reported in addition but do not override the canonical equivalence result.",
      "status": "proved",
      "proof_sketch": "This is a direct consequence of canonical equivalence being sufficient and necessary for equality of represented single-qubit action modulo global phase.",
      "dependencies": [
        "prop-3",
        "def-6"
      ],
      "evidence_location": "Section 6",
      "notes": "This is the trust decision rule the paper recommends for toolchains.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:prop-8"
    },
    {
      "claim_id": "emp-9",
      "label": "Empirical Result 8.1 (Reference verifier behavior)",
      "type": "empirical_result",
      "statement": "On the shipped seven-case reference verification corpus, all four positive equivalence cases pass and all three negative cases are rejected under the canonical quaternion check, with matrix diagnostics consistent with those outcomes.",
      "status": "empirical",
      "proof_sketch": "Measured by the included verifier script verify_canonical_equivalence.py, with outputs recorded in verification_results.json.",
      "dependencies": [
        "def-6",
        "prop-8"
      ],
      "evidence_location": "Section 8; artifacts/code/verification_results.json",
      "notes": "This is a measured consistency result on the shipped verification corpus only.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:emp-9"
    },
    {
      "claim_id": "int-10",
      "label": "Interpretive Claim 9.1 (Trust layer for quaternionic compilation)",
      "type": "interpretive_claim",
      "statement": "RQM Technologies uses canonical quaternion equality together with secondary matrix diagnostics as the trust layer for quaternionic compilation because that combination yields deterministic native verification and interoperable external reporting without changing the underlying semantics.",
      "status": "informal",
      "proof_sketch": null,
      "dependencies": [
        "prop-3",
        "prop-7",
        "prop-8",
        "emp-9"
      ],
      "evidence_location": "Section 9",
      "notes": "This is an organizational trust claim, not a new mathematical theorem.",
      "_paper_id": "qqc-004-verification-equivalence",
      "_paper_path": "papers/qqc-004-verification-equivalence/",
      "_global_claim_id": "qqc-004-verification-equivalence:int-10"
    }
  ]
}