{
  "schema": "AN04-restored-separated-branches-proof-map/v1",
  "proofs": [
    {
      "id": "BR:R001",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Work on a time interval and a conic neighborhood",
      "dependencies": [
        "SY:H7",
        "SY:R1",
        "U001:F0-COMP"
      ],
      "scope": "Uniform gap and fixed single-eigenvalue multiplicities. BR1--BR3 specify all parameter derivatives, compact normalized margins and a single scalar eigenvalue in each repeated block; an internally splitting cluster is excluded."
    },
    {
      "id": "BR:R002",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "A global realization with bounded symbol families",
      "dependencies": [
        "BR:R001",
        "SY:R1",
        "SY:R0",
        "SY:U1"
      ],
      "scope": "Separate global energy and local reduction hypotheses. The global bounded-symbol realization is explicit; a local chart is not assumed to supply global energy or support bounds."
    },
    {
      "id": "BR:R003",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "We give the finite-dimensional argument",
      "dependencies": [
        "SY:H16",
        "U001:F0-COMP",
        "U001:P2"
      ],
      "scope": "Finite Hermitian spectral theorem. Rayleigh maximization, real/imaginary variations and invariant orthogonal-complement induction give a complete finite-dimensional proof."
    },
    {
      "id": "BR:R004",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Normalize \\(\\xi=\\rho\\omega\\)",
      "dependencies": [
        "BR:R003",
        "U001:P13.2-series",
        "U001:P2"
      ],
      "scope": "Stable resolvent contours. The Neumann inverse and exact Hermitian resolvent norm prevent eigenvalues escaping the fixed circles."
    },
    {
      "id": "BR:R005",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "\\pi_j=\\frac{1}{2\\pi i}\\int_{\\Gamma_j}",
      "dependencies": [
        "BR:R004",
        "U001:P16.1",
        "U001:P16.2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Smooth full-rank spectral projections. BR4--BR5 are evaluated pointwise; repeated resolvent derivatives and fixed rank give every parameter bound without differentiating eigenvectors."
    },
    {
      "id": "BR:R006",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Under the fixed-multiplicity hypothesis,",
      "dependencies": [
        "BR:R005",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Smooth eigenvalues with fixed multiplicity. The trace formula divides by the actual rank d_j and gives the homogeneous S1 branch with all derivatives."
    },
    {
      "id": "BR:R007",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "There is also an exact polynomial expression,",
      "dependencies": [
        "BR:R003",
        "BR:R006"
      ],
      "scope": "Polynomial projection formula. BR6 is checked on every eigenspace, with ordered matrix factors commuting because they are polynomials in H1."
    },
    {
      "id": "BR:R008",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Differentiating the denominators in (BR6),",
      "dependencies": [
        "BR:R001",
        "BR:R006",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Reciprocal gap bounds. BR7 and the differentiated reciprocal formula give S-1, including every time and base derivative."
    },
    {
      "id": "BR:R009",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "At one normalized point choose an",
      "dependencies": [
        "BR:R005",
        "BR:R006",
        "BR:S1",
        "U001:F0-COMP"
      ],
      "scope": "Local Gram frames. BR8 uses the convergent differentiated matrix binomial series on a uniform norm margin."
    },
    {
      "id": "BR:R010",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "It follows that \\(U_j^*U_j=I_{d_j}\\).",
      "dependencies": [
        "BR:R009",
        "BR:R005",
        "U001:P2"
      ],
      "scope": "Unitary block frame and domain. BR9 follows from orthogonal ranges and exact dimensions; the frame is local and homogeneous, with no bundle trivialization assumed."
    },
    {
      "id": "BR:R011",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "We use left quantization with",
      "dependencies": [
        "SY:H6",
        "SY:H7",
        "SY:H8",
        "K:K1",
        "K:K2",
        "K:K3"
      ],
      "scope": "Full ordinary finite-matrix composition. BR10 preserves ordered factors; finite matrix sums and parameter product rules receive the exact prior differentiated remainder proof."
    },
    {
      "id": "BR:R012",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Quantize \\(U\\), using proper support",
      "dependencies": [
        "BR:R010",
        "BR:R011",
        "K:K2",
        "K:K3",
        "P2:OP4",
        "P2:OP5"
      ],
      "scope": "Conjugation with the time derivative retained. BR11 includes E1*d/dt; a smoothing spatial inverse defect is not treated as an order-zero evolution error."
    },
    {
      "id": "BR:R013",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Suppose an intermediate coefficient has diagonal blocks",
      "dependencies": [
        "BR:R008",
        "BR:R012"
      ],
      "scope": "Signed gap correction. BR12 divides each off-diagonal block by i*(lambda_j-lambda_l), giving S(-m-1) and both signed cancellations."
    },
    {
      "id": "BR:R014",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Then \\(K\\in S^{-m-1}\\)",
      "dependencies": [
        "BR:R013",
        "BR:R011",
        "K:K3"
      ],
      "scope": "Conjugation remainder bookkeeping. BR13 includes time, composition, lower-order and quadratic terms; -2m-1 is at most -m-1 for m>=0."
    },
    {
      "id": "BR:R015",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Iterate this construction.",
      "dependencies": [
        "BR:R014",
        "BR:R011"
      ],
      "scope": "Ordinary block recursion. Every step keeps the complete nonclassical off-diagonal symbol; scalar principal repeated blocks retain arbitrary internal S0 matrices."
    },
    {
      "id": "BR:R016",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "For precision, asymptotic summation here",
      "dependencies": [
        "BR:R015",
        "K:K1",
        "SY:H7",
        "U001:P14.3"
      ],
      "scope": "Parameter-aware asymptotic sum. Increasing radii control the first m seminorms, all time derivatives and every fixed stronger tail order; no homogeneous expansion of C is assumed."
    },
    {
      "id": "BR:R017",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Summing the conjugators and diagonal coefficients",
      "dependencies": [
        "BR:R016",
        "BR:R011"
      ],
      "scope": "Infinite-order intertwiner. BR14 follows by comparing each prescribed residual order to a deeper finite construction; multiplication loses one spatial order and time differentiation loses none."
    },
    {
      "id": "BR:R018",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The full microlocal inverse \\(R\\) of \\(T\\)",
      "dependencies": [
        "BR:R017",
        "K:K2",
        "K:K3",
        "K:K1",
        "BR:R011"
      ],
      "scope": "Both ordered inverses. A left error is canceled by -e*U*, a right error by -U*e; both Borel sums agree modulo smoothing."
    },
    {
      "id": "BR:R019",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The second identity is microlocal,",
      "dependencies": [
        "BR:R017",
        "BR:R018",
        "P2:OP6"
      ],
      "scope": "Precise inverse-conjugation defect. BR15 explicitly retains S1*d/dt and identifies S0=RE+(RT-I)B; proper finite-order composition preserves the smoothing ideal."
    },
    {
      "id": "BR:R020",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The order-zero diagonal coefficient can be computed",
      "dependencies": [
        "BR:R011",
        "BR:R012",
        "BR:R014",
        "BR:R021"
      ],
      "scope": "Leading polarization coefficient. BR16 is an ordinary identity modulo S-1, including the left-quantization inverse correction and the additional frame-curvature term."
    },
    {
      "id": "BR:R021",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "To check the formula, the inverse of the pointwise symbol",
      "dependencies": [
        "BR:R011",
        "BR:R010",
        "BR:R018",
        "U001:P2"
      ],
      "scope": "Signed inverse correction and cancellation. r_-1=-(1/i)sum U*_xi U_x U* is obtained from r#U=I; its principal product cancels the ordered composition term."
    },
    {
      "id": "BR:R022",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Changing a local frame within this eigenspace",
      "dependencies": [
        "BR:R020",
        "BR:R021",
        "BR:R010",
        "BR:R011"
      ],
      "scope": "Ordered frame-change law. BR20 uses the full symbol inverse and retains both time and Hamilton derivatives; extra differentiated gauge terms vanish by the eigenrelations."
    },
    {
      "id": "BR:R023",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Here is the precise receiving theorem.",
      "dependencies": [
        "BR:R001",
        "BR:R002",
        "BR:R010",
        "SY:U1",
        "SY:R1",
        "CH:W1",
        "EC:G0"
      ],
      "scope": "Localized receiving theorem. BR21 declares compact input phase support, the full global energy realization, both endpoint directions and every H(-M)->H(L) differentiated remainder."
    },
    {
      "id": "BR:R024",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "We first make an auxiliary global system",
      "dependencies": [
        "BR:R003",
        "BR:S1",
        "BR:L1",
        "BR:R010"
      ],
      "scope": "Unitary matrix logarithm. Commuting Hermitian real/imaginary parts are jointly diagonalized using the proved finite theorem; the convergent logarithm is skew-Hermitian with all differentiated bounds."
    },
    {
      "id": "BR:R025",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Choose a scalar smooth cutoff \\(\\vartheta\\)",
      "dependencies": [
        "BR:R024",
        "BR:R008",
        "U001:P14.3",
        "U001:F0-COMP",
        "SY:H16"
      ],
      "scope": "Global auxiliary extension. BR22 uses one common scalar cutoff, so every extended gap is a convex combination of positive gaps; the exponential frame stays unitary and low frequencies are regularized."
    },
    {
      "id": "BR:R026",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Let \\(\\lambda_j^{\\mathrm e}",
      "dependencies": [
        "BR:R025",
        "BR:R017",
        "BR:R018",
        "SY:H7",
        "SY:H10",
        "P2:OP4"
      ],
      "scope": "Global smoothing intertwining estimates. BR23 uses global bounded symbol seminorms and sufficiently negative orders for every Sobolev pair; proper quantization adds another bounded smoothing family."
    },
    {
      "id": "BR:R027",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Each block of \\(D^{\\mathrm e}\\)",
      "dependencies": [
        "BR:R026",
        "EC:H0",
        "EC:H1",
        "EC:A1",
        "EC:D3",
        "EC:D7",
        "EC:G0",
        "EC:G3",
        "EC:G5"
      ],
      "scope": "Scalar-principal block evolutions. Each diagonal block meets the complete U047 hypotheses, including an arbitrary full ordinary finite-matrix S0 coefficient."
    },
    {
      "id": "BR:R028",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "where \\(E_j\\) denotes the constant coordinate block projection,",
      "dependencies": [
        "BR:R027",
        "BR:R026",
        "SY:U2",
        "SY:C6",
        "EC:C0",
        "EC:C1",
        "EC:C2",
        "EC:C7"
      ],
      "scope": "Auxiliary branch kernels and normalization. BR24 has residual E V R and initial value I+S0; the derivative in the second endpoint is retained."
    },
    {
      "id": "BR:R029",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "We now localize and compare with the actual system.",
      "dependencies": [
        "BR:R023",
        "BR:R028",
        "EC:A2",
        "EC:A4",
        "EC:A5",
        "EC:D4",
        "SY:H10",
        "K:K4"
      ],
      "scope": "Vanishing conic composition terms. Cutoffs and coefficient differences vanish near each whole relevant graph; every stationary term vanishes and arbitrary negative remainders are controlled with compact input/output."
    },
    {
      "id": "BR:R030",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "A full PDO may also create output outside",
      "dependencies": [
        "BR:R029",
        "P2:OP4",
        "SY:H10",
        "CE:H12",
        "EC:D5",
        "EC:D6",
        "EC:H3",
        "EC:H4"
      ],
      "scope": "Exterior full-PDO tails. Outside the compact output, positive diagonal separation and repeated frequency integration by parts give all kernel derivatives and arbitrary spatial decay, hence global Sobolev bounds."
    },
    {
      "id": "BR:R031",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "PV=R,\\qquad V(s,s)=\\Psi+B",
      "dependencies": [
        "BR:R028",
        "BR:R029",
        "BR:R030"
      ],
      "scope": "Distinct initial and PDE defects. BR25 separately identifies B and R and gives every global H(-M)->H(L) bound."
    },
    {
      "id": "BR:R032",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The two defects are distinct.",
      "dependencies": [
        "BR:R031",
        "SY:U2",
        "SY:C6",
        "CE:H4",
        "EC:C1",
        "EC:C2"
      ],
      "scope": "Actual-system energy correction. BR26 corrects both defects under the actual system evolution; finite derivative losses are supplied by arbitrary smoothing gain."
    },
    {
      "id": "BR:R033",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "All terms on the right are globally smoothing.",
      "dependencies": [
        "BR:R032",
        "BR:N1",
        "EC:C3",
        "EC:C4",
        "EC:C5",
        "EC:C6",
        "EC:C7"
      ],
      "scope": "Jointly smooth actual remainder kernel. Every endpoint derivative is handled by the two evolution identities and differentiated integral; the exact prior delta-column reconstruction applies."
    },
    {
      "id": "BR:R034",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Let a compact finite-time family of all the branch trajectories",
      "dependencies": [
        "BR:R006",
        "EC:F1",
        "EC:F2",
        "EC:F3",
        "EC:G1",
        "K:K4",
        "U001:F0-COMP"
      ],
      "scope": "Finite compact trajectory cover. A finite normalized cover and short steps are chosen for the whole compact trajectory family inside a larger uniform-gap neighborhood."
    },
    {
      "id": "BR:R035",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "In each frame patch define a full ordinary microlocal projection",
      "dependencies": [
        "BR:R017",
        "BR:R018",
        "BR:R019",
        "BR:R011"
      ],
      "scope": "Full commuting spectral projections. BR27 follows from the full two-sided inverse and differentiated intertwining; the correct inverse-family equation is R_t+BR-RA smoothing."
    },
    {
      "id": "BR:R036",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "These projections are unique to all orders",
      "dependencies": [
        "BR:R035",
        "BR:R008",
        "BR:R011"
      ],
      "scope": "Uniqueness from idempotence and the gap. BR28 first improves diagonal/non-j blocks; the commutation equation and reciprocal gap improve the remaining off-diagonal blocks. Induction works in full ordinary classes."
    },
    {
      "id": "BR:R037",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Thus projections from different frames agree",
      "dependencies": [
        "BR:R036",
        "K:K4",
        "BR:R029",
        "BR:R030"
      ],
      "scope": "Overlap compatibility and partition gluing. Unique germs agree to all orders; near each point the differentiated partition sums cancel and every remaining error is smoothing."
    },
    {
      "id": "BR:R038",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "At one short time step take a finite partition",
      "dependencies": [
        "BR:R034",
        "BR:R037",
        "K:K4",
        "BR:R028",
        "BR:R029"
      ],
      "scope": "Projected local input normalization. The local input Pi_j chi_a Pi_j sums to Pi_j at the initial endpoint and then to the identity; arbitrary extensions are used only away from the relevant graphs."
    },
    {
      "id": "BR:R039",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "after the declared compact phase localization.",
      "dependencies": [
        "BR:R038",
        "BR:R035",
        "BR:R036",
        "BR:R029",
        "BR:R030"
      ],
      "scope": "Endpoint block membership. BR29 uses the local constant block and unique full projections; applying an output projection adds only the controlled commutator residual."
    },
    {
      "id": "BR:R040",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Compose the finitely many short-step sums,",
      "dependencies": [
        "BR:R039",
        "BR:R035",
        "BR:R032",
        "BR:R033",
        "SY:U1",
        "SY:H10",
        "EC:G2",
        "EC:G4"
      ],
      "scope": "Eliminate every switched-branch word. Different consecutive branches contain Pi_j Pi_l, which is smoothing under the declared localization; finite products preserve all smoothing estimates."
    },
    {
      "id": "BR:R041",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The canonical graphs in each surviving word compose",
      "dependencies": [
        "BR:R040",
        "EC:F2",
        "RC:C3",
        "RC:C4",
        "RC:C5",
        "RC:C6",
        "RC:C7",
        "RC:C8",
        "RC:C9",
        "EC:G3"
      ],
      "scope": "Finite same-branch graph composition. BR30 retains only one branch per word; Hamilton graph diffeomorphisms compose with zero excess and complete prior ordered graph calculus gives order zero."
    },
    {
      "id": "BR:R042",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Reversing the time subdivision proves",
      "dependencies": [
        "BR:R041",
        "BR:R033",
        "BR:N1",
        "EC:G1",
        "EC:G4"
      ],
      "scope": "Two directions and all endpoint families. Common sufficiently fine affine subdivisions handle both directions and differentiated compact endpoint families; cutoffs are one on the full preceding output microsupport."
    },
    {
      "id": "BR:R043",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The transport has an invariant meaning.",
      "dependencies": [
        "BR:R027",
        "BR:R022",
        "BR:R041",
        "EC:D0",
        "EC:D1",
        "EC:D7",
        "EC:G5",
        "GS:Z6",
        "GS:Z10",
        "GS:Z11"
      ],
      "scope": "Elliptic endpoint bundle transport. BR31 has rank d_j with an inverse from ordered transport and nonzero half-density; endpoint frame changes cancel without a global eigenbasis."
    },
    {
      "id": "BR:R044",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "On an input cone where the scalar cutoff",
      "dependencies": [
        "BR:R043",
        "BR:I1",
        "BR:R039",
        "EC:G7",
        "EC:G8",
        "CH:W1"
      ],
      "scope": "Exact full-matrix kernel wavefront. Output/input full projections isolate an elliptic branch even at graph intersections; the conclusion concerns the full matrix union, not every entry."
    },
    {
      "id": "BR:R045",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "For compactly supported distributional data \\(f\\)",
      "dependencies": [
        "BR:R044",
        "BR:I1",
        "BR:R039",
        "SY:U1",
        "SY:H2",
        "EC:G9",
        "K:K3"
      ],
      "scope": "Precisely localized projected data propagation. The full Pi_j data statement requires normalized WF(f) inside K and scalar Psi=1 there; the compact smooth initial remainder remains H-infinity under energy. The elliptic bundle inverse gives the converse."
    },
    {
      "id": "BR:R046",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The theorem covers compact trajectory families",
      "dependencies": [
        "BR:R001",
        "BR:R023",
        "BR:R034",
        "BR:R045"
      ],
      "scope": "Crossing and internal-cluster scope guard. The branch theorem requires a uniform single-eigenvalue gap; energy across a crossing is not promoted to a crossing FIO theorem."
    },
    {
      "id": "BR:R047",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "**Exercise 1 — intermediate:",
      "dependencies": [
        "BR:R020",
        "BR:R013",
        "BR:R014",
        "SY:H16",
        "BR:S1",
        "U001:P16.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Rotating-frame correction and exact spectrum. Exercise1 proves both signed corrections and eigenvalues +/-i*sqrt(rho^2+omega^2), including the order-minus-one shift."
    },
    {
      "id": "BR:R048",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "**Exercise 2 — intermediate:",
      "dependencies": [
        "BR:R002",
        "BR:R008",
        "BR:R013",
        "SY:H1",
        "SY:H2",
        "SY:R1"
      ],
      "scope": "Energy with a closing gap. Exercise2 proves Fourier unitarity for every real Sobolev weight while the correction omega/(2it rho) loses uniform time bounds."
    },
    {
      "id": "BR:R049",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "**Exercise 3 — advanced:",
      "dependencies": [
        "BR:R003",
        "BR:R005",
        "BR:R010",
        "BR:S1",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P16.3",
        "U001:P15.2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Uniform gap with a nontrivial complex eigenline. Exercise3 computes both Pauli projections and local frames; the complete radial-argument winding contradiction rules out a global smooth nonzero section."
    },
    {
      "id": "BR:R050",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "**Exercise 4 — advanced:",
      "dependencies": [
        "BR:R022",
        "BR:R027",
        "BR:R043",
        "BR:R044",
        "U001:P16.1",
        "U001:P16.2",
        "EC:G8",
        "SY:C6",
        "SY:H2",
        "P3:L2"
      ],
      "scope": "Exact rotating-polarization branch kernel. Exercise4 cancels the prescribed time connection, solves both translations, proves BR33 and its initial identity, signed cone labels and rank-one endpoint maps."
    },
    {
      "id": "BR:R051",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Figure: the horizontal axis is",
      "dependencies": [
        "BR:R050",
        "U001:P16.1",
        "U001:P2"
      ],
      "scope": "Exact rays and polarization figure. The reproducible instance has s=y=0, eta=1, x=+/-t and theta=pi t/4. Separate component planes display actual unit vectors; 121 samples and all horizontal reader portions were inspected."
    },
    {
      "id": "BR:S1",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "Here is the scalar-series input",
      "dependencies": [
        "BR:R003",
        "U001:P13.2-series",
        "U001:P13.2-mertens",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full scalar binomial recurrence, positive inverse-square-root identity and every norm-valued matrix-series derivative."
    },
    {
      "id": "BR:L1",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "For completeness the logarithm identity used here",
      "dependencies": [
        "BR:S1",
        "BR:R003",
        "U001:P15.1",
        "U001:P15.2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full logarithm identity, imaginary scalar unit-circle logarithms, skew-Hermitian unitary logarithm and all matrix parameter bounds."
    },
    {
      "id": "BR:N1",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The norm estimates needed for these time differentiations",
      "dependencies": [
        "BR:R002",
        "SY:U1",
        "SY:U2",
        "SY:H10",
        "EC:H2",
        "EC:H3",
        "EC:H4",
        "CE:H4",
        "CE:H7"
      ],
      "scope": "Actual matrix-principal evolution, both oriented extra-order Taylor bounds, all endpoint norm derivatives and Bochner majorants."
    },
    {
      "id": "BR:I1",
      "source": "separated-system-branches-preparation.md",
      "source_sha256": "9cac5b6654ff09c938e2baed6831ed834b58e97ea585c004a0d71bdfa701d3a0",
      "proof_locator": "The inverse assertion for a branch can be made directly",
      "dependencies": [
        "BR:R043",
        "BR:R039",
        "K:K2",
        "K:K3",
        "K:K4",
        "GE:D9",
        "RC:C9",
        "GS:Z12",
        "EC:G7",
        "EC:G8"
      ],
      "scope": "Local rank-block graph inverse with an elliptic scalar input cutoff, projected two-sided identities, overlap compatibility and exact diagonal non-smoothness."
    }
  ],
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}