The curved-space Dirac equation from empirical anchors and structural commitments

The curved-space Dirac equation (iγμ Dμ − μ)Ψ = 0 with Dμ = ∂μ + (1/4) ωμab σab + iqAμ/ℏ on a smooth Lorentzian four-manifold with spin structure is reconstructed step by step from the empirical content of the matter wave (five experimental anchors plus the PBR theoretical no-go) together with three structural commitments on equations for physically-real potentials (frame-independence, index matching, self-sourcing) and an explicit set of algebraic, minimality, and scope-choice conditions named at the stage each enters. The five experimental anchors are Davisson-Germer electron diffraction, Stern-Gerlach two-valued intrinsic angular momentum, the 4π-rotation sign change in neutron interferometry, Pauli exclusion through fermion antisymmetry, and the Aharonov-Bohm phase response in regions where E = B = 0. Frame-independence on the spin-1/2 anchor with the parity-symmetric mass and conserved U(1) charge conditions selects Ψ as the Dirac bispinor (1/2, 0) ⊕ (0, 1/2). Index matching with parity selection on the kinetic term and first-derivative minimality selects the kinetic operator iΨ̅ γμ ∂μ Ψ. Local U(1) invariance with the dim-4 leading-order scope on the matter sector, anchored by the Aharonov-Bohm phase whose gauge-invariant holonomy establishes Aμ as a physically operative connection rather than a calculational convenience over E and B (the Wu and Yang 1975 reading, with the local-gauge value of Aμ at a point unphysical and the holonomy exp[(iq/ℏ) ∮ Aμ dxμ] the gauge-invariant content), selects the minimal coupling Dμ = ∂μ + iqAμ/ℏ. Universal coupling under frame-independence with a torsion-free scope choice lifts the construction to curved spacetime through the tetrad and spin connection, with the matter wave acting as gravitational source through the symmetric Hilbert / Belinfante-Rosenfeld stress-energy tensor. The construction is presented as one motivated route complementary to the orthodox Tetrode-Weyl-Fock-Ivanenko geometric assembly on the spinor bundle. No new Dirac operator, no new coupling term, and no new uniqueness theorem is claimed. The contribution is the assumption-accounting reconstruction itself, in which the empirical anchors, algebraic conditions, minimality conditions, and scope choices that the orthodox geometric route treats as starting hypotheses are named explicitly at the stages they enter the chain. The three structural commitments alone do not force the equation: the algebraic, minimality, and scope-choice conditions are explicit additional inputs at the points of use, and each load-bearing input is named at the stage it enters the reconstruction. The scope is the single-particle matter-wave equation on a fixed background curved spacetime with U(1) gauge coupling. The coupled Einstein-Dirac-Maxwell back-reaction with full Ψ stress-energy source, the multi-particle Fock-space lift through spin-statistics inheritance, the internal-component structure of Ψ beyond the Dirac bispinor, the Standard Model gauge content, and the substrate-from-spinor derivation of the geometric background itself are each separate constructions out of present scope.