{ "schema": "AN04-restored-mixed-dirichlet-proof-map/v1", "proofs": [ { "id": "DC:T001", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Let \\(X\\) be a smooth", "dependencies": [ "HC:R1", "HC:M4", "U001:F0-COMP" ], "scope": "Complete scalar mixed-problem hypotheses. DC1 retains proper time, strict quadratic hyperbolicity, the positive time sign, negative boundary conormals and smooth complex lower terms." }, { "id": "DC:T002", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Write \\(q_x(\\xi,\\eta)\\)", "dependencies": [ "U001:P2", "SY:H16" ], "scope": "Strict quadratic discriminant and Lorentz signature. DC2 proves the negative orthogonal complement and converse, with the timelike orientation and closed convex future cone." }, { "id": "DC:T003", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "At the boundary, the metric-normal vector", "dependencies": [ "DC:T002", "U001:P2" ], "scope": "Boundary transversality and tangent timelike vector. DC3 proves the induced Lorentz signature, positive projected time square and noncritical boundary time without a Euclidean normal substitution." }, { "id": "DC:T004", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "This also constructs a future timelike vector", "dependencies": [ "DC:T003", "DC:T002", "HC:M4", "U001:P14.3", "U001:FTC-TAYLOR-COMPACT-PARAMETERS" ], "scope": "Global boundary-tangent temporal vector field. The projected vector, collar extension and convex future partition produce W with W phi=1 and positive cone margin on compact slabs." }, { "id": "DC:T005", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Here is the smooth flow argument", "dependencies": [ "DC:T004", "U001:F0-CALC", "U001:F0-COMP", "U001:FTC-TAYLOR-COMPACT-PARAMETERS" ], "scope": "Local smooth flow with all parameter derivatives. Uniform Picard contractions and their geometric Cauchy bound prove existence and uniqueness; difference quotients and the ordered variational integral equation prove every smooth parameter derivative." }, { "id": "DC:T006", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "On every finite interval of flow time", "dependencies": [ "DC:T005", "DC:T001", "U001:F0-COMP", "U001:FTC-TAYLOR-COMPACT-PARAMETERS", "U001:P15.1", "U001:P15.2" ], "scope": "Compact continuation and boundary-preserving product collars. Finite coordinate boxes give a common continuation time; W^r=r a gives the exact exponential formula and the displayed inverse of the slab product map." }, { "id": "DC:T007", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "**Mixed Dirichlet–Cauchy theorem.**", "dependencies": [ "DC:T001", "DC:T040", "HC:M4", "HC:M5", "MH:TR3" ], "scope": "Full Sobolev Dirichlet and zero-past theorem. DC4--DC5 retain every real s>=0, restricted forcing Hs, boundary H(s+1), solution restricted H(s+1), actual trace and prescribed vanishing past." }, { "id": "DC:T008", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "The algebra behind the estimate", "dependencies": [ "DC:T002", "U001:P2", "SY:H16" ], "scope": "Positive Lorentz flux and its exact converse. DC6--DC7 include the explicit strictly positive quadratic lower bound, complex Hermitian extension and the converse up to simultaneous metric sign." }, { "id": "DC:T009", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "We now work in coordinates", "dependencies": [ "DC:T002", "HC:M4", "U001:F0-COMP" ], "scope": "Differential operator sign and normalized model. DC8 uses L=-P with ordinary derivatives; g_rr=-1, bounded full coefficients, compact perturbation of a constant Lorentz metric and positive g_tt are retained." }, { "id": "DC:T010", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Choose a smooth future timelike vector", "dependencies": [ "DC:T009", "DC:T004", "DC:T008", "HC:M4", "U001:P14.3" ], "scope": "Uniform inward timelike multiplier. Tangential timelike vectors plus a small inward component, collars and convex partitions give F^r>0 with uniform positive margins, including the constant exterior." }, { "id": "DC:T011", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "For complex smooth \\(u\\) set", "dependencies": [ "DC:T009", "DC:T010", "U001:FTC-TAYLOR-COMPACT-PARAMETERS", "U001:P2" ], "scope": "Exact complex current and lower-order remainder. DC9--DC10 derive the pairwise second-derivative cancellation and mass contribution; every complex lower term enters only the bounded first-derivative quadratic remainder." }, { "id": "DC:T012", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Indeed (DC6) applies", "dependencies": [ "DC:T008", "DC:T011", "DC:T010" ], "scope": "Temporal and artificial spacelike flux coercivity. DC11 applies the Lorentz lemma to the metric-dual of F and each outward future timelike conormal, retaining the positive mass term." }, { "id": "DC:T013", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "If \\(b=u|_{r=0}\\)", "dependencies": [ "DC:T011", "DC:T010", "U001:P2" ], "scope": "Outward Dirichlet flux and free normal derivative. DC12 uses the outward conormal -dr; its exact normal square coefficient is F^r, and only tangential derivatives of the given boundary value enter the cross-term loss." }, { "id": "DC:T014", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "over \\(\\Omega_{a,T}=", "dependencies": [ "DC:T011", "DC:T012", "DC:T013", "DC:D1", "P3:M3", "U001:P14.3" ], "scope": "Weighted volume, terminal and boundary energy. DC13--DC14 apply divergence with e^(-lambda t), absorb the full coefficient remainder, and retain all three positive terms and the forcing lambda inverse." }, { "id": "DC:T015", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Reversing time uses a past timelike multiplier", "dependencies": [ "DC:T014", "DC:T010", "SY:H16", "U001:P15.1" ], "scope": "Reverse-time inward multiplier and small-time scale. DC15 uses a past timelike vector still inward, rather than -F, with e^(lambda t); lambda=1/T gives the full adjoint interior and normal boundary bound." }, { "id": "DC:T016", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "This retains the past support and the zero Dirichlet trace.", "dependencies": [ "HC:M6", "HC:M3", "MH:TR3", "P3:L1", "P3:L2", "P3:M4", "U001:P14.3" ], "scope": "Retarded smoothing and all normal orders. DC16 preserves the past support and zero Dirichlet trace; the zero-forcing HC48 application on nested collars gives every normal order needed for actual smooth energy tests." }, { "id": "DC:T017", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Here is the complete commutator argument", "dependencies": [ "P3:M3", "P3:M4", "P3:L2", "SY:H1", "U001:FTC-TAYLOR-COMPACT-PARAMETERS" ], "scope": "Full Friedrichs commutator with correct coefficient sign. DC17--DC18 derive both integral terms by integration by parts; cancellation of their smooth limits and L2 density prove uniform boundedness and strong convergence after integrating in r." }, { "id": "DC:T018", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "In \\([L,J_\\epsilon]u\\)", "dependencies": [ "DC:T016", "DC:T017", "DC:T014", "P3:L2" ], "scope": "Rough uniqueness without second-derivative assumptions. DC19 removes the pure normal commutator using constant g_rr; every remaining second-order term acts on an actual L2 first derivative. The smooth energy and residual limit prove H1 uniqueness." }, { "id": "DC:T019", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "The power uses the right-half-plane logarithm.", "dependencies": [ "HC:M1", "HC:M2", "MH:J1", "MH:J2", "MH:J3", "MH:I1", "MH:A1", "MH:A2" ], "scope": "One-sided complex powers and exact antidual. DC20--DC21 use the complete HC39--HC42 half-space proof in the temporal variable, preserving terminal support at every real order and using the restricted quotient norm." }, { "id": "DC:T020", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "The symbols in (DC20) need not be", "dependencies": [ "DC:T019", "MH:J1", "U001:FTC-TAYLOR-COMPACT-PARAMETERS", "U001:P14.2-powers" ], "scope": "Isotropic symbol distinction and exact multiplier difference. DC22 proves the first-gradient and relative-symbol bounds directly on the frequency segment; no false ordinary isotropic S^sigma class membership is asserted." }, { "id": "DC:T021", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "For a model coefficient \\(A(r,z)\\)", "dependencies": [ "DC:T020", "DC:T009", "P3:L1", "P3:L2", "P3:M3", "P3:M4" ], "scope": "Exact Fourier kernel and two Schur marginals. DC23 subtracts the constant coefficient remainder, uses rapid uniform partial Fourier decay and an integrable frequency-difference majorant, then proves the L2 bound by weighted Cauchy--Schwarz." }, { "id": "DC:T022", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Every term of \\(L^*\\) other than", "dependencies": [ "DC:T021", "DC:T009" ], "scope": "Full commutator representation on actual derivatives. DC24 absorbs one tangential derivative in each second-order term, retains at most one normal derivative and bounds every resulting coefficient uniformly in r." }, { "id": "DC:T023", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "All resulting operators preserve support", "dependencies": [ "DC:T022", "DC:T019", "MH:A2", "MH:I1" ], "scope": "Causal restriction and legitimate zero extension. All constituents preserve terminal support. Extend actual L2 first derivatives by zero to bound the restricted operator; do not differentiate that extension or use unspecified negative-time values." }, { "id": "DC:T024", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Apply (DC15).", "dependencies": [ "DC:T015", "DC:T022", "DC:T023", "DC:T019", "HC:M3", "MH:TR3" ], "scope": "All-real adjoint estimate with order-dependent time bound. DC25--DC26 commute the full adjoint, retain the coefficient error, absorb C_sigma T^2 and the normal trace, and declare the dependence of T_sigma on sigma." }, { "id": "DC:T025", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "An extension across \\(r=0\\)", "dependencies": [ "HC:M1", "HC:M2", "MH:B1", "MH:B3", "P3:M4", "P3:L2" ], "scope": "Forcing as a supported tangential Hilbert input. DC27 follows from full Fourier weight comparison and restriction infima for s>=0; no positive-order isotropic zero extension through the physical boundary is used." }, { "id": "DC:T026", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "All integrals denote the supported/restricted dual pairings", "dependencies": [ "DC:T019", "DC:T025", "DC:T013", "DC:G1" ], "scope": "Weak boundary functional and positive Green sign. DC28 retains complex conjugation, the outward -dr sign, g_rr=-1 and the boundary pairing with the actual test normal derivative." }, { "id": "DC:T027", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "In particular two tests with the same restricted", "dependencies": [ "DC:T024", "DC:T026", "CE:H1", "HC:M2", "MH:A1", "MH:A2" ], "scope": "Bound on an arbitrary image and complex extension. DC29 with sigma=-s-1 proves well-definedness on the restricted adjoint image, then uses the exact complex Hahn--Banach programme proof without a closed-range assumption." }, { "id": "DC:T028", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "The representation under the integral is", "dependencies": [ "DC:T027", "MH:I1", "MH:A1", "MH:A2", "CE:H3", "CE:H4", "CE:H5", "CT:T1", "MH:B3" ], "scope": "One global Hilbert representation and initial support. DC30 and HC42 yield one supported L2_r dotH(s+1) distribution. Tests crossing t=0 give the same equation because both tangential inputs have the declared support; no initial deltas are omitted." }, { "id": "DC:T029", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "The seed in (DC30) is", "dependencies": [ "DC:T028", "HC:M6", "MH:R1", "MH:R2" ], "scope": "Three normal-recovery inequalities and full isotropic order. DC31 checks all three HC48 inequalities with the forcing pair (s+2,0), seed (0,s+1) and target (s+1,0), including the physical boundary and initial time." }, { "id": "DC:T030", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "It remains to identify the trace from the weak identity.", "dependencies": [ "DC:T029", "DC:G1", "HC:M5", "MH:TR3" ], "scope": "Weak Green identity justified at H1 regularity. DC32 uses smooth H1 boundary approximation and H1_0 interior test approximation for Lu in L2. It does not assume a normal trace of u beyond the available regularity." }, { "id": "DC:T031", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Subtract (DC28).", "dependencies": [ "DC:T026", "DC:T030", "U001:P14.3" ], "scope": "Actual Dirichlet trace from arbitrary boundary tests. The tests v=r chi(r) h realize every compact smooth boundary normal derivative. Subtraction of the two Green identities proves gamma u=b, including across the initial time." }, { "id": "DC:T032", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "First, the energy estimate controls", "dependencies": [ "DC:T012", "DC:T014", "DC:D1", "U001:F0-COMP", "U001:FTC-TAYLOR-COMPACT-PARAMETERS" ], "scope": "Shrinking-ball causal region and positive artificial faces. DC33 has outward conormal v0 dt+d|x_sp| future timelike; intersection with a future spacelike psi face keeps nonnegative artificial flux and requires no incoming artificial boundary condition." }, { "id": "DC:T033", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "This assertion also holds for the \\(H^1\\)", "dependencies": [ "DC:T032", "DC:T016", "DC:T017", "DC:T018", "DC:T031", "P3:M3" ], "scope": "Domain of dependence for rough constructed solutions. Strict interior regions and zero forcing locally permit the normal-recovery smoothing; residual convergence, zero smoothed data and exhaustion prove the causal statement for H1 solutions." }, { "id": "DC:T034", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "Second, let \\(x_0\\in\\partial X\\)", "dependencies": [ "DC:T003", "DC:T009", "HC:M4", "U001:P3", "U001:F0-COMP", "U001:P14.3" ], "scope": "Boundary bowl coordinates and legitimate global model. DC34 preserves r=0 and the time cone after shrinking; positive normalization fixes g_rr, a small convex coefficient blend preserves the Lorentz model, and full complex lower terms are retained." }, { "id": "DC:T035", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "If \\(f,b\\) vanish for", "dependencies": [ "DC:T034", "DC:T031", "DC:T033", "HC:M4", "MH:M1" ], "scope": "Local supported existence for the original curved time. Compact data localization has t>=kappa|x_sp|^2 and stays in restriction spaces; causal localization with psi=phi proves vanishing on the original phi-past." }, { "id": "DC:T036", "source": "mixed-dirichlet-cauchy-energy-preparation.md", "source_sha256": "57fa468d42bbb8bea79c769790e808c7feb937c42bd9b046edefa4fba62c692e", "proof_locator": "For local uniqueness, take", "dependencies": [ "DC:T034", "DC:T018", "HC:L6", "HC:L7", "HC:M4", "U001:P14.3" ], "scope": "One-sided local uniqueness and exact interior provider. DC35 places all cutoff derivatives where the zero-past solution already vanishes for t