Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Varying a minimally gauged diagnostic action fixes the current form and normalization identities, while projection exposes missing-mode contributions.

What the written action permits

DA=∇A+iqaA,Lm=ψˉ(iγADA−m−VN)ψ−κ2(ψˉψ)2. D_A=\nabla_A+iq a_A,\qquad \mathcal L_m=\bar\psi(i\gamma^AD_A-m-V_N)\psi -\frac\kappa2(\bar\psi\psi)^2.

Here q is signed charge, A is a spacetime/sector index, and V_N is the scalar term in that diagnostic action, not an already-derived Coulomb field. Under ψ → e^{-iqχ}ψ and a → a+dχ the action is invariant. Variation gives

JA=−δSmδaA=qψˉγAψ. J^A=-\frac{\delta S_m}{\delta a_A}=q\bar\psi\gamma^A\psi.

The field equations and their adjoints give ∇_A J^A = 0. Algebraic neutral scalar interactions contribute no additional derivative with respect to a. Derivative interactions, a-dependent projectors, or added orbit-driving terms must instead be varied too. This is the usual minimal-coupling construction; see Tong's QED section. It derives a current from an assumed charged action, not electric charge or the physical U(1) identification from the master theory.

The coefficient Z in −ZF²/4 also matters: canonical normalization of the photon gives coupling q/√Z. A current formula alone does not predict the fine-structure constant.

Projection and total charge

For fixed admissible mode sections s_a with ψ = Σ_a c_a s_a, define the Gram matrix G_ab = ∫s_a†s_b dμ. The kinetic action is ∫dt [iℏ c†G ċ − c†h c]. The local charge matrix is R_ab(u) = q s_a(u)†s_b(u), and therefore

∫R(u)dμ=qG. \int R(u)d\mu=qG.

In orthonormal coordinates b = G^{1/2}c, total charge is q b†b. Multiplicity of modes does not multiply a normalized particle's electric charge. A moving orthonormal basis S(t) adds −iℏ S†Ṡ to the energy Hamiltonian; its response to a probe cannot be omitted if the basis depends on that probe.

After a specified conserved pushforward to observer space, the dipole and integrated current matrices satisfy

Di=∫xiR(x)d3x,Ii=∫ji(x)d3x=iℏ[h,Di] D_i=\int x_iR(x)d^3x,\qquad I_i=\int j_i(x)d^3x=\frac{i}{\hbar}[h,D_i]

for time-independent matrices, no boundary flux, and a closed projected subspace. This is a consistency condition, not a way to determine D_i from h alone. Choosing D_i arbitrarily and defining I_i by this equation does not derive the physical charge distribution.

Exact truncation defect

Let P be an orthogonal projector and Q = 1−P. Suppose the full local operator identity is (i/ℏ)[h,R]+div j = 0. Direct compression gives

iℏ[PhP,PRP]+P(div⁡j)P=−iℏPhQRP+iℏPRQhP. \frac{i}{\hbar}[PhP,PRP]+P(\operatorname{div}j)P =-\frac{i}{\hbar}PhQRP+\frac{i}{\hbar}PRQhP.

All operators here act on the retained space where appropriate. The right side vanishes if h preserves that space. Merely preserving charge globally is weaker than satisfying local conservation after projection.

Even if QhP = 0 without a probe, a dipole D can have QDP ≠ 0. Then the unforced orbit is exactly retained but its weak-field response need not be. Projecting D to PDP before calculating response discards intermediate Q modes. Integrating those modes out consistently produces additional, generally frequency-dependent response terms. For a linear probe with QhP = 0 at zero field, these include terms quadratic in the applied field in the effective action; their variation contributes to linear response. One cannot infer their absence from PDP = 0. The geometric example below demonstrates this explicitly without fitting a spectrum.

Charge versus degree

The degree L of O(L) is not automatically the electric charge of a particle in that section space. If every connection is the Lth tensor power of one physical fluctuating connection, its infinitesimal coupling scales with L. That is not a universal electron charge across orbit families. A model can instead specify different background fluxes and a universally normalized electromagnetic perturbation, or distinguish a geometric twist from the electromagnetic bundle. Which interpretation V2 uses must be stated in the action. Representation degree alone does not settle it.

Reproduction and boundary

projected_current_checks.py checks the projection-defect identity, its disappearance for an invariant subspace, and charge normalization with a nonorthogonal Gram metric. The selected-line and geometric-probe notes apply this framework to actual orbit evolution. The remaining physical inputs are the master-selected bundle, orbit-driving term, electromagnetic observer map, nuclear contribution, and normalization.