Conditional Reconstruction Closure E: Finite Element Closure of the Maxwell Sector and Local Cauchy Well-Posedness

Within the Maxwell sector fixed by the data list, this paper separates, item byitem, the connections among discrete constraint closure, the continuous Maxwellsolution, and local Cauchy uniqueness, distinguishing which connections are conditional statements, which are anchored at the literature layer, and which aremachine-check debts still to be closed. Chapter 2 first discloses the inputs, outputs, and judgments of five groups of machine-check ledgers, all of which pass,while registering the posting locations of three debts. Chapter 3 fixes the consumption interface of all subsequent statements as a six-item list: four-dimensional flatMinkowski spacetime with a fixed signature; a finite-duration Cauchy region; anexternal source satisfying dJ = 0; a linear metric constitutive law; a single commoncausal cone; and regularity and compatibility of initial values, boundary, and gauge.If any item is missing, the corresponding theorem is automatically downgraded.Chapter 4 uses Proposition 4.2 to separate the structural fixed point C3a and thedynamical solution C3b into two hierarchical objects, stipulating that the two maybe connected only through the convergence certificate E-CT2 and the identificationcondition E-CT4. Chapter 5 states Proposition 5.1 (the discrete exterior differential squares to zero, machine check MC-E-01(a) passed) on a gauge-oriented finiteelement complex, and gives two admissibility tickets for solvability in Definition 5.2,honestly registering that the failure witness for the second ticket is absent from themachine-check scope of this paper. Chapter 6 states Theorem E-CT1: when thetwo tickets, together with the boundary conditions and gauge conditions specifiedin item 6 of the list, hold, the discrete problem is solvable and unique in the quotientspace; the harmonic part and the gauge degrees of freedom are deducted separately.Chapter 7 registers the three conditions of Theorem E-CT2—consistency, stability, and regularity—separately, and declares that its positive-path machine checkis a one-dimensional scalar model proxy and does not constitute a proof of convergence for the Maxwell system on the de Rham complex. Chapter 8 splits constraintpropagation into Proposition 8.2(a), an algebraic premise, and Proposition 8.2(b), asemi-discrete propagation proposition, and the identity must not be directly namedpropagation. Chapter 9 registers the four applicable conditions of Theorem E-T3and their correspondence to the list, and closes with a forward pointer to Chapter10. Chapter 10 states and proves Theorem E-CT4: when condition (i) object convergence and condition (ii) separated observation both hold, the limit of a certifieddiscrete closure sequence is the unique Maxwell solution of the Cauchy problem under the list; identification holds only within the uniqueness class of E-T3. Chapter11 states Theorem E-CT5, its completeness direction and failure handling, with theimplementation sequence constructed from the three elements of interpolation, discrete source, and certification, and the limit identification of the certified solutioncarried out per E-CT4. Chapter 12 uses Definition 12.1 to register the trigger signals, machine-check witnesses, mandatory dispositions, and post-downgrade claimceilings of the three failure classes F1, F2, and F3, and registers item by item theobligation to close the three machine-check debts at subsequent layers. AppendicesA through F respectively register the machine-check protocol overview, the orientation ledger specification, the complete output accounts, the literature-layer reconciliation table, the failure-disposition quick-reference table, and the list-downgrademapping table.