The Physics of Energy Flow
Part II — Corollaries and Technical Appendices
Part II — Corollaries and Technical Appendices
2026-03-18
Part I established the physical spine of the argument in thirteen chapters.
Part II gathers what follows without lengthening that main chain.
The short chapters that follow do not add new dynamics. They state conceptual consequences about cause, time, space, and force that are consistent with the reconstruction but are not needed to derive it.
The later appendices then carry the more technical derivations: transport composition, uniqueness, hydrodynamic limits, interaction laws, gravity details, and the engineering consequences of field-shaped transport.
Cause is not directly observed. What is directly observed is registered change.
We register configurations. We compare records. If the records differ, change has been noticed.
Because physical registrations do not present contradictions — a given record does not assign multiple incompatible values to the same quantity at the same location — we infer an order in the reconfiguration. That persistent order is what later receives causal language: this before that; this producing that.
Cause and consequence are therefore not primitive furniture of the world. They are compact descriptions of ordered, registered reconfiguration. Change is primary. Cause is the interpretation added afterward.
Time is the ordering of persistent flow, counted by recurrence.
The later configuration is not created from nothing. It is the same continuous energy, rotating other parts of its extent into present registration.
A stable flow registers change by continuing to reconfigure. When records of that reconfiguration recur, the recurrence can be counted. Any recurrent pattern of flow can therefore serve as a clock. Time is the count of such recurrent steps.
In this framework, the coordinate \(t\) in the equations labels ordered registrations of the changing present. Physical time is the operational order of change.
Space is the collection of distances defined by ordered signal separation.
A distance is a count of how many recurrent signal steps separate two configurations. Once such steps are ordered, they can also be described as causal steps. The total set of such separations forms the relational structure we call space.
Geometry is descriptive. It summarizes the organization of flow. When the resulting distance relations can be represented coherently in three dimensions, we use that representation and call it space.
Force is a derived description of reconfiguration and momentum transfer.
When a bounded region changes its motion, what has changed physically is the flux of momentum through its boundary. “Force” is the compact name for that boundary accounting.
This also clarifies the status of so-called force fields. If two supposedly independent substrates interact, they are not truly independent. A mediating field does not preserve the split. It exposes a deeper common structure in which the coupling is taking place.
Force is descriptive. It is bookkeeping for coupled change within one underlying physical organization.
Appendix 209 makes this exact for the electromagnetic case: the Lorentz-force form is derived there as boundary stress transfer and field-momentum bookkeeping for a localized charged closure.
These appendices supply the technical derivations that belong after the main spine of Part I and after the short corollary chapters of Part II.
They do not replace the main chain. They prove, sharpen, or extend later points without interrupting the primary derivation.
The double-curl transport closure of chapter 7 determines the local transport cone. Reapplying
\[ \nabla\times(\nabla\times\mathbf{F}) \]
acts again on field structure and raises the spatial operator. Nested transport is a different question. It belongs to kinematics: how do successive bounded transport increments compose when they occur in the same approximately uniform region and therefore share the same local transport speed \(k\)?
To keep the discussion in one lab, consider motion along one spatial direction \(x\). Let \(u\) denote the speed produced from rest by one standard transport pulse in that region. If a body is already moving at speed \(v\), let
\[ v \oplus u \]
denote the speed measured in the same lab after applying that same standard pulse again.
Because \(k\) is the local transport speed singled out by the electromagnetic closure, no transport process native to that region can push a mode outside the admissible interval \(|v|<k\). So the composition law must preserve that bound. This already rules out Galilean additivity as an exact transport law, since
\[ v+u \]
can exceed \(k\) even when both \(|v|<k\) and \(|u|<k\) separately hold. The problem is therefore not whether the bound survives composition, but what exact shape the composition law must take once that bound is respected.
The composition operation \(\oplus\) should satisfy four basic requirements:
The same law can also be derived from momentum-flux bookkeeping, and this is arguably the more physical route.
In one spatial direction, the local transport cone gives two causal channels: forward and backward. Let \(T_{++}\ge 0\) and \(T_{--}\ge 0\) denote the corresponding channel loads through a transverse cut. Then the total energy density and flux are
\[ u = T_{++}+T_{--}, \qquad S = k(T_{++}-T_{--}). \]
For a coherent moving mode, the measured lab speed is therefore
\[ v=\frac{S}{u} = k\,\frac{T_{++}-T_{--}}{T_{++}+T_{--}}. \]
So the speed is determined entirely by the ratio
\[ R(v):=\frac{T_{++}}{T_{--}} = \frac{k+v}{k-v}. \]
A standard transport pulse preserving the same two causal channels cannot alter the local bound \(k\); it can only rebalance the forward and backward channel loads. So such a pulse acts by
\[ T_{++}\mapsto a\,T_{++}, \qquad T_{--}\mapsto b\,T_{--}, \]
with \(a,b>0\). Therefore the ratio transforms multiplicatively:
\[ R \mapsto \frac{a}{b}R. \]
If one standard pulse sends rest to speed \(u\), then since rest has \(R(0)=1\), that same pulse has
\[ R(u)=\frac{a}{b}. \]
Applying it to a state already moving at speed \(v\) gives
\[ R(v\oplus u)=R(u)\,R(v). \]
Therefore
\[ \frac{k+v\oplus u}{k-v\oplus u} = \frac{k+u}{k-u}\cdot\frac{k+v}{k-v}. \]
Solving this relation gives
\[ \boxed{ v\oplus u = \frac{v+u}{1+vu/k^2} }. \]
This is the hyperbolic composition law forced by bounded momentum-flux transport.
The same result can be written geometrically from the transport cone. In the same approximately uniform region, the local transport speed \(k\) picks out the lines
\[ x = \pm kt, \]
which bound the local transport cone. Writing the corresponding null coordinates
\[ \xi = t + \frac{x}{k}, \qquad \chi = t - \frac{x}{k}, \]
any orientation-preserving linear map fixing those two directions takes the form
\[ \xi' = a\,\xi, \qquad \chi' = b\,\chi, \]
with \(a,b>0\). For speed composition only the ratio matters, so write
\[ \Lambda^2:=\frac{a}{b}. \]
Along a line of constant speed \(v\), the null-coordinate ratio is
\[ \frac{\xi}{\chi} = \frac{t+x/k}{t-x/k} = \frac{1+v/k}{1-v/k} = \frac{k+v}{k-v}. \]
So the momentum-flux ratio \(R(v)\) is exactly the null-coordinate ratio. A cone-preserving map rescales it by \(\Lambda^2\), which is the same multiplicative law obtained above from channel rebalancing.
Only after this step is it useful to introduce an additive parameter. Taking the logarithm of \(R(v)\) gives
\[ \eta(v):=\frac12\ln\!\frac{k+v}{k-v}. \]
Then
\[ \eta(v\oplus u)=\eta(v)+\eta(u). \]
Equivalently,
\[ \eta(v)=\operatorname{artanh}\!\left(\frac{v}{k}\right), \qquad v=k\tanh\eta. \]
So successive identical pulses add linearly in \(\eta\), not in \(v\). If one pulse contributes \(\eta_0\), then after \(n\) identical pulses the lab speed is
\[ v_n=k\tanh(n\eta_0). \]
So the distinction is exact:
The train-and-passenger image is therefore valid, but only at the kinematic level. One transport process may be nested inside another. The resulting composition is hyperbolic because the same local transport speed \(k\) is preserved at each step.
This appendix gives the mathematical step used structurally in chapter 7.
The result is the following.
A single real first-order self-curl evolution of a divergence-free field does not produce neutral propagating transport. The minimal local propagating closure in this class requires two coupled divergence-free fields.
The point is not to postulate Maxwell’s equations, but to show why they appear as the minimal propagating closure of source-free rotational transport.
Let
\[ \mathbf{F}(\mathbf{r},t) \]
be a vector field on three-dimensional space.
Source-free transport means
\[ \nabla \cdot \mathbf{F} = 0. \]
This expresses the absence of primitive beginnings or endings of the flow. If a primitive start or end point were enclosed by a closed surface, the net flux through that surface would not vanish. The source-free condition says that this does not happen: the enclosed net flow remains identically zero.
Assume a local first-order evolution relation
\[ \partial_t \mathbf{F} = \mathcal{D}(\mathbf{F}), \]
where \(\mathcal{D}\) is a spatial differential operator.
To preserve the source-free condition we require
\[ \nabla \cdot (\partial_t \mathbf{F}) = 0. \]
Substituting the evolution relation gives
\[ \nabla \cdot \mathcal{D}(\mathbf{F}) = 0 \]
for every divergence-free field \(\mathbf{F}\).
A natural local first-order differential operator with this property is curl, since
\[ \nabla \cdot (\nabla \times \mathbf{A}) = 0 \]
for any vector field \(\mathbf{A}\).
So a natural divergence-preserving self-update is
\[ \partial_t \mathbf{F} = k\,\nabla \times \mathbf{F}. \]
This relation is posed simultaneously for all \(\mathbf{r}\) in the extent. It is therefore a whole-field update, not a rule for tracking one individually marked point through space.
We now examine whether this relation yields propagating transport.
The simplest test is to differentiate the self-curl relation once more:
\[ \partial_t^2 \mathbf{F} = k\,\nabla\times(\partial_t\mathbf{F}). \]
Substituting
\[ \partial_t \mathbf{F} = k\,\nabla \times \mathbf{F} \]
gives
\[ \partial_t^2 \mathbf{F} = k^2\,\nabla\times(\nabla\times\mathbf{F}). \]
Using the vector identity
\[ \nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla\cdot\mathbf{F})-\nabla^2\mathbf{F}, \]
and the source-free condition
\[ \nabla\cdot\mathbf{F}=0, \]
we obtain
\[ \partial_t^2 \mathbf{F} = -k^2\nabla^2\mathbf{F}. \]
Equivalently,
\[ \partial_t^2 \mathbf{F} + k^2\nabla^2\mathbf{F}=0. \]
This is not the neutral propagating wave equation, in which the second temporal derivative term and the spatial Laplacian term appear with opposite signs. Here both second-order terms enter with the same sign. A single self-curl evolution therefore does not by itself furnish the propagating closure we seek. It preserves turning, but it does not produce the neutral propagating form.
There is also a direct obstruction to bodily transport.
Assume, for contradiction, that a nontrivial bounded closure could be carried bodily by the single self-curl relation. Then there would exist a smooth localized profile \(\mathbf{G}\) and a constant drift velocity \(\mathbf{v}\) such that
\[ \mathbf{F}(\mathbf{r},t)=\mathbf{G}(\mathbf{r}-\mathbf{v}t). \]
Differentiating gives
\[ \partial_t\mathbf{F} = -(\mathbf{v}\cdot\nabla)\mathbf{G}, \qquad \partial_t^2\mathbf{F} = (\mathbf{v}\cdot\nabla)^2\mathbf{G}. \]
Substituting this translating ansatz into
\[ \partial_t^2 \mathbf{F} + k^2\nabla^2\mathbf{F}=0 \]
gives
\[ (\mathbf{v}\cdot\nabla)^2\mathbf{G}+k^2\nabla^2\mathbf{G}=0. \]
Now take the Euclidean inner product with \(\mathbf{G}\) and integrate over all space. Because \(\mathbf{G}\) is localized, the boundary terms vanish under integration by parts. Therefore
\[ \int \mathbf{G}\cdot(\mathbf{v}\cdot\nabla)^2\mathbf{G}\,dV = -\int \left|(\mathbf{v}\cdot\nabla)\mathbf{G}\right|^2\,dV \]
and
\[ \int \mathbf{G}\cdot\nabla^2\mathbf{G}\,dV = -\int |\nabla\mathbf{G}|^2\,dV. \]
So
\[ \int \left|(\mathbf{v}\cdot\nabla)\mathbf{G}\right|^2\,dV + k^2\int |\nabla\mathbf{G}|^2\,dV = 0. \]
Both integrands are nonnegative. Hence both integrals must vanish:
\[ (\mathbf{v}\cdot\nabla)\mathbf{G}=0, \qquad \nabla\mathbf{G}=0. \]
So \(\mathbf{G}\) is constant. Since \(\mathbf{G}\) is localized, that constant must be zero. Therefore the only localized rigidly translating solution of the single self-curl relation is the trivial one.
This proves the point needed in the main text: a single self-curl update can turn a structure, but it cannot carry a nontrivial bounded closure bodily from one region to another.
The same argument rules out rigid bodily rotation of a localized closure.
Assume that a nontrivial bounded closure rotates rigidly about a fixed axis. Let \(Q(t)=e^{t\Omega}\) be the corresponding one-parameter family of rotation matrices, with \(\Omega\) a constant skew-symmetric matrix, and suppose
\[ \mathbf{F}(\mathbf{r},t)=Q(t)\,\mathbf{G}(Q(t)^{-1}\mathbf{r}). \]
Define the linear operator
\[ A_\Omega \mathbf{G} := \Omega \mathbf{G}-(\Omega\mathbf{r})\cdot\nabla \mathbf{G}. \]
Then
\[ \partial_t\mathbf{F}\big|_{t=0}=A_\Omega\mathbf{G}, \qquad \partial_t^2\mathbf{F}\big|_{t=0}=A_\Omega^2\mathbf{G}. \]
Substituting into
\[ \partial_t^2 \mathbf{F} + k^2\nabla^2\mathbf{F}=0 \]
at \(t=0\) gives
\[ A_\Omega^2\mathbf{G}+k^2\nabla^2\mathbf{G}=0. \]
Now take the \(L^2\) inner product with \(\mathbf{G}\). The operator \(A_\Omega\) is skew-adjoint on compactly supported fields:
Therefore
\[ \int \mathbf{G}\cdot A_\Omega^2\mathbf{G}\,dV = -\int |A_\Omega\mathbf{G}|^2\,dV. \]
Together with
\[ \int \mathbf{G}\cdot\nabla^2\mathbf{G}\,dV = -\int |\nabla\mathbf{G}|^2\,dV, \]
we obtain
\[ \int |A_\Omega\mathbf{G}|^2\,dV + k^2\int |\nabla\mathbf{G}|^2\,dV = 0. \]
Again both terms are nonnegative, so both must vanish. Hence
\[ A_\Omega\mathbf{G}=0, \qquad \nabla\mathbf{G}=0. \]
Thus \(\mathbf{G}\) is constant, and since it is localized, it must be zero. Therefore the only localized rigidly rotating solution of the single self-curl relation is the trivial one.
So the one-field self-curl update does not bodily move a bounded closure, either by translation or by rigid rotation. What remains possible is weaker than rigid-body motion: internal reorientation, internal deformation, or phase progression on a fixed support. Those possibilities are not classified by the present no-go result.
This is the precise content of the chapter-7 summary: single curl reorganizes locally.
Now introduce two divergence-free fields
\[ \mathbf{F}_+, \qquad \mathbf{F}_-. \]
Consider the coupled evolution
\[ \partial_t \mathbf{F}_+ = k\,\nabla \times \mathbf{F}_- \]
\[ \partial_t \mathbf{F}_- = -k\,\nabla \times \mathbf{F}_+. \]
Taking a time derivative of the first equation,
\[ \partial_t^2 \mathbf{F}_+ = k\,\nabla \times (\partial_t \mathbf{F}_-). \]
Substituting the second equation,
\[ \partial_t^2 \mathbf{F}_+ = -k^2\,\nabla \times (\nabla \times \mathbf{F}_+). \]
Using the vector identity
\[ \nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla\cdot\mathbf{F})-\nabla^2\mathbf{F}, \]
and the divergence-free condition
\[ \nabla\cdot\mathbf{F}_+ = 0, \]
we obtain
\[ \partial_t^2 \mathbf{F}_+ = k^2\nabla^2\mathbf{F}_+. \]
Thus \(\mathbf{F}_+\) satisfies the wave equation
\[ \partial_t^2\mathbf{F}_+ - k^2\nabla^2\mathbf{F}_+ = 0. \]
The same derivation holds for \(\mathbf{F}_-\).
There is also an explicit transporting branch.
Let \(\phi:\mathbb{R}\to\mathbb{R}\) be any smooth scalar profile, and define
\[ \mathbf{F}_+(\mathbf{r},t)=\phi(x-kt)\,\mathbf{e}_y, \qquad \mathbf{F}_-(\mathbf{r},t)=\phi(x-kt)\,\mathbf{e}_z. \]
Then
\[ \nabla\cdot\mathbf{F}_+=0, \qquad \nabla\cdot\mathbf{F}_-=0, \]
because each field has only one transverse component and depends only on \(x\).
Now compute the curls:
\[ \nabla\times\mathbf{F}_- = \nabla\times(0,0,\phi(x-kt)) = (0,-\partial_x\phi(x-kt),0), \]
and
\[ \nabla\times\mathbf{F}_+ = \nabla\times(0,\phi(x-kt),0) = (0,0,\partial_x\phi(x-kt)). \]
Also,
\[ \partial_t\mathbf{F}_+ = (0,-k\,\partial_x\phi(x-kt),0), \]
and
\[ \partial_t\mathbf{F}_- = (0,0,-k\,\partial_x\phi(x-kt)). \]
Therefore
\[ \partial_t\mathbf{F}_+ = k\,\nabla\times\mathbf{F}_-, \qquad \partial_t\mathbf{F}_- = -k\,\nabla\times\mathbf{F}_+. \]
So the coupled curl system admits exact translating solutions.
If the initial profile \(\phi\) is supported in an interval \([a,b]\), then at time \(t\) the transported profile is supported in the shifted interval
\[ [a+kt,b+kt]. \]
Thus the doubled structure does what the single self-curl relation cannot do: it carries a profile from one region to another. The transport is explicit. The shape is preserved, and the profile advances rigidly at speed \(k\) along the \(x\) direction.
This establishes the existence of a genuine transport branch. It is a one-direction translating profile embedded in three dimensions. Additional closure is needed later to build bounded self-sustained modes from such transport.
The analysis shows:
So the minimal propagating closure in this class is
\[ \partial_t \mathbf{F}_+ = k\,\nabla \times \mathbf{F}_- \]
\[ \partial_t \mathbf{F}_- = -k\,\nabla \times \mathbf{F}_+. \]
These equations preserve
\[ \nabla\cdot\mathbf{F}_+ = 0,\qquad \nabla\cdot\mathbf{F}_- = 0. \]
Now define
\[ \mathbf{E} \equiv \mathbf{F}_+,\qquad \mathbf{B} \equiv \mathbf{F}_-/k. \]
Then the coupled equations become
\[ \partial_t \mathbf{E} = k^2\nabla \times \mathbf{B} \]
\[ \partial_t \mathbf{B} = -\nabla \times \mathbf{E}. \]
With conventional constants absorbed into the normalization of \(k\), these correspond to the source-free Maxwell equations.
The two fields are not independent substances.
They are two complementary transverse aspects of the same organized source-free transport. Their mutual curl coupling yields the minimal propagating structure compatible with divergence-free flow.
Starting from divergence-free transport:
Maxwell dynamics therefore appears here as the minimal propagating closure of source-free rotational transport. Appendix 211 then proves that within the stated real local isotropic first-order two-field class, this closure is also unique up to real field recombination.
This appendix derives longitudinal contraction from the structure of a moving self-sustained closure. The contraction is not inserted as a coordinate rule. It is the geometric deformation required for one bounded transport mode to remain coherent while drifting uniformly through an approximately uniform region.
Throughout this appendix, let \(k\) denote the local transport speed in that region. The derivation assumes that \(k\) is effectively constant over the size of the apparatus during one run. More general backgrounds require path integrals of the same local transport law.
Fix an approximately uniform region in which the local transport speed is the constant \(k>0\).
Consider one bounded self-sustained mode. In its rest configuration there is no distinguished drift direction. Choose one closure span of length \(L_0\) along a chosen axis, and one orthogonal closure span of the same length \(L_0\).
The quantity \(L_0\) is not introduced as an external ruler length. It is the rest span of one internal closure of the mode.
One out-and-back closure across such a span has recurrence period
\[ T_0=\frac{2L_0}{k}. \]
Now let the whole mode drift uniformly with speed \(v\) along the \(x\) direction, with
\[ 0\le v<k. \]
We seek the moving longitudinal span \(L_\parallel\) for which the drifting mode remains one coherent closure.
Consider a transverse closure across a span of length \(L_0\), orthogonal to the drift.
During one half-cycle of duration \(\tau_\perp\), the receiving point of the mode shifts longitudinally by \(v\tau_\perp\). The transport still moves locally at speed \(k\), so the half-cycle geometry is
\[ (k\tau_\perp)^2=L_0^2+(v\tau_\perp)^2. \]
Rearranging,
\[ (k^2-v^2)\tau_\perp^2=L_0^2. \]
Therefore
\[ \tau_\perp=\frac{L_0}{\sqrt{k^2-v^2}}, \]
and the full transverse closure time is
\[ T_\perp=2\tau_\perp=\frac{2L_0}{\sqrt{k^2-v^2}}. \]
Let the longitudinal extent of the moving mode be \(L_\parallel\). This is the quantity to be determined.
For the forward half-cycle, the receiving boundary recedes at speed \(v\), so
\[ \tau_+=\frac{L_\parallel}{k-v}. \]
For the return half-cycle, the receiving boundary approaches at speed \(v\), so
\[ \tau_-=\frac{L_\parallel}{k+v}. \]
The full longitudinal closure time is therefore
\[ T_\parallel = \tau_+ + \tau_- = \frac{L_\parallel}{k-v}+\frac{L_\parallel}{k+v} = \frac{2kL_\parallel}{k^2-v^2}. \]
The drifting object is one self-sustained mode. Its transverse and longitudinal closures cannot recur with different periods. If they did, then after one full transverse recurrence and one full longitudinal recurrence the same moving mode would assign different local states to what is supposed to be one coherent configuration. The closure would fail to remain single-valued.
So the moving mode must satisfy
\[ T_\parallel = T_\perp. \]
Substituting the expressions above gives
\[ \frac{2kL_\parallel}{k^2-v^2} = \frac{2L_0}{\sqrt{k^2-v^2}}. \]
Solving for \(L_\parallel\),
\[ L_\parallel = L_0\,\frac{\sqrt{k^2-v^2}}{k} = L_0\sqrt{1-\frac{v^2}{k^2}}. \]
If we define
\[ \gamma=\frac{1}{\sqrt{1-v^2/k^2}}, \]
then
\[ L_\parallel=\frac{L_0}{\gamma}. \]
This is the contraction of the moving closure along the drift direction.
The previous computation proves the following statement.
Let a bounded self-sustained closure drift uniformly through an approximately uniform region with local transport speed \(k\). If the moving closure remains coherent, then its longitudinal span must be
\[ L_\parallel=L_0\sqrt{1-\frac{v^2}{k^2}}. \]
The conclusion is forced by coherence of the moving closure. It is not an additional convention.
The contraction has not been imposed as a measurement convention. It has been derived from one structural requirement only: the moving bounded mode must remain one coherent closure.
So the meaning of the result is precise:
Length contraction appears here as the geometry required for moving closure, not as an external postulate.
Now consider a Michelson-Morley interferometer built from the same bounded material closures and drifting uniformly through the same approximately uniform region.
Let each arm have rest length \(L_0\).
The arm transverse to the drift keeps that length geometrically, so its round-trip transport time is
\[ T_\perp=\frac{2L_0}{\sqrt{k^2-v^2}}. \]
The arm parallel to the drift contracts to
\[ L_\parallel=L_0\sqrt{1-\frac{v^2}{k^2}}, \]
so its round-trip transport time is
\[ T_\parallel = \frac{L_\parallel}{k-v}+\frac{L_\parallel}{k+v} = \frac{2kL_\parallel}{k^2-v^2}. \]
Substituting the contracted length,
\[ T_\parallel = \frac{2kL_0\sqrt{1-v^2/k^2}}{k^2-v^2} = \frac{2L_0}{\sqrt{k^2-v^2}} = T_\perp. \]
Therefore
\[ \Delta T = T_\parallel - T_\perp = 0. \]
So a Michelson-Morley device is blind to uniform translational drift through a region with uniform local transport speed \(k\).
The null result does not arise because nothing moved. It arises because the same transport closure that carries the signal also determines the moving geometry of the device.
This appendix addresses uniform drift in an approximately uniform region. If the surrounding transport conditions vary across the apparatus, then the local speed \(k\) must be replaced by the appropriate path-dependent transport speed. The structural point remains the same: transport and geometry must be solved together, because the bounded mode and the signal it carries are governed by the same closure.
This appendix does not introduce a new equation. It clarifies an interpretive question that naturally arises from the preceding chapters:
Why does the transport picture keep suggesting a medium?
The answer is stronger: the structures already established are those of a continuous substrate whose reorganization is governed locally across its whole extent. In that sense the present program is already a fluid-mechanics theory, with electromagnetic energy itself playing the role of the fluid.
The main chain has already established the following.
First, energy is described by a distribution
\[ u(\mathbf{r}), \]
defined across the extent of what exists.
Second, redistribution between ordered registrations is described by a flow
\[ \mathbf{S}(\mathbf{r};1,2), \]
and continuity requires that local change be accounted for by transport across neighboring regions.
Third, the source-free transport structure is expressed by a divergence-free flow
\[ \nabla\cdot\mathbf{F}=0, \]
whose admissible local reorganization is constrained by curl.
Fourth, single curl reorganizes locally, while doubled curl yields the first transporting closure.
Fifth, the local transport speed is a property of the region. In a sufficiently uniform region it is written as a constant \(k\); in general it should be understood as locally determined.
None of these statements is a particle statement. All of them are continuum statements.
The word medium is used here in a minimal sense.
A medium is a continuous substrate such that:
Under that definition, the picture developed in this book is already medium-like.
This can be seen point by point.
The primitive object is a distribution over an extent, not a list of separate particles.
The continuity statement is local and simultaneous across all \(\mathbf{r}\). It does not track one marked parcel through a background. It constrains the whole substrate at once.
The curl closures are also posed simultaneously across the whole extent. They describe local reorganization of a field, not action at a distance.
The transport speed is determined by local conditions of the region, not by an empty background independent of the substrate.
Bounded stable modes, including the apparatus used to measure transport, are made of the same substrate whose transport they register.
Taken together, these are precisely the features that make ordinary continuum mechanics intelligible. The same is true here, even though the specific closure is different.
Navier-Stokes comes to mind because the present framework is already doing continuum mechanics in the strong sense. It shares the following structural features:
So when the imagination reaches for fluid motion, it is responding to something real in the mathematics.
What it is responding to is the fact that the ontology has already shifted from point particles in empty space to organized motion in a continuous substrate. The fluid is electromagnetic energy, and the localized bodies later discussed in the book are bounded closures of that same fluid.
The point is not that the book merely resembles fluid mechanics. The point is that it already has the same continuum architecture:
The present fluid closure uses electromagnetic energy itself as the primitive fluid variable, and it organizes that fluid by a source-free doubled-curl transport relation:
\[ \partial_t\mathbf{F}_{+}=k\,\nabla\times\mathbf{F}_{-}, \qquad \partial_t\mathbf{F}_{-}=-k\,\nabla\times\mathbf{F}_{+}. \]
Pressure, viscosity, and the standard advective form appear at the coarse-grained level developed later. Appendices 207, 213, and 214 show how the familiar hydrodynamic conservation forms arise when the electromagnetic substrate is averaged into effective continuum variables.
So the right claim is:
this is fluid mechanics with electromagnetic energy as the fluid, and the later hydrodynamic variables are emergent summaries of that deeper closure.
Reading the substrate as medium-like makes several later claims easier to understand.
Particles cease to be primitive objects and become bounded organized modes of the substrate.
Charge and spin cease to be added labels and become global aspects of closed circulation in the substrate.
The Michelson-Morley null result becomes less mysterious because the signal and the moving apparatus are both closures of the same transport medium.
Geometry ceases to be something imposed from outside the substrate and instead becomes tied to the way coherent closures persist within it.
So this appendix does not add a new derivation. It identifies the ontological direction already implied by the mathematics.
The transport framework developed in the book is a continuum mechanics of one continuous substrate whose state is defined across an extent and whose changes are governed by local redistribution and closure relations.
In that sense it is already a fluid-mechanics theory. The fluid is electromagnetic energy itself, while the familiar hydrodynamic variables arise later as coarse-grained summaries of the deeper transport closure.
Appendix 219 develops one corollary of that fluid picture for passive regions: complete transport data on a closed boundary constrain the interior strongly enough to determine passive transport there, and relative unloading of a region raises its local transport speed and can therefore create a faster transport corridor.
Appendix 220 develops the complementary corollary for bounded modes: matter is the persistent closed causal loop of that same Maxwellian transport.
This appendix states the derivational route by which hydrodynamic form emerges from the principles already developed in this book. Appendix 207 carries out that derivation in the simplest uniform-region setting. Appendix 213 extends the same balance-law structure to variable background.
The point is to keep three levels distinct:
The main chain establishes the first level. The present appendix lays out the remaining derivational route.
The book already has the local continuity statement
\[ \partial_t u + \nabla\cdot\mathbf{S}=0, \]
or, in earlier notation between ordered registrations, its discrete analogue.
This is the conservation of energy under redistribution. It says that local change of stored energy is accounted for by neighboring transport.
This is the first hydrodynamic balance law of the theory. It supplies the conservation of energy under redistribution and fixes the starting point for the continuum limit.
To proceed toward a hydrodynamic limit, one needs a local momentum density
\[ \mathbf{g}(\mathbf{r},t), \]
together with a flux of momentum across neighboring regions.
At the fundamental level, the natural candidate is built from the same transport structure that already appears as energy flow. In a sufficiently uniform region, it takes the form
\[ \mathbf{g}=\mathcal{G}(u,\mathbf{S},k,\ldots), \]
where \(k\) denotes the local transport scale of the region and the ellipsis marks any additional local closure data carried by the actual substrate.
The logical requirement is:
a hydrodynamic limit emerges by building local momentum density from the same underlying transport that carries energy.
Once a momentum density is defined, the next equation is a local balance law of the form
\[ \partial_t \mathbf{g} + \nabla\cdot\mathbf{T}=0, \]
where
\[ \mathbf{T} \]
is the momentum-flux tensor, or stress tensor, of the coarse-grained transport.
This is the direct analogue, for momentum, of what chapter 3 already does for energy. The next target is therefore a stress description, not a velocity field in isolation.
The fundamental fields of this book describe transport across the whole extent. Fluid dynamics, by contrast, uses averaged fields defined over regions large compared to microscopic structure but small compared to macroscopic variation.
So an emergent hydrodynamic limit would require a coarse-graining map sending the underlying closure variables to effective continuum variables such as
\[ \rho_{\mathrm{eff}}, \qquad \mathbf{v}_{\mathrm{eff}}, \qquad \mathbf{T}_{\mathrm{eff}}. \]
These effective fields summarize unresolved local circulation, transport, and closure within each averaging region.
This is exactly where pressure, viscosity, and related effective quantities appear as observables of the coarse-grained substrate.
At the coarse-grained level, a Navier-Stokes-like system would require a stress tensor of the approximate form
\[ \mathbf{T}_{\mathrm{eff}} = \rho_{\mathrm{eff}}\,\mathbf{v}_{\mathrm{eff}}\otimes\mathbf{v}_{\mathrm{eff}} + p_{\mathrm{eff}}\,\mathbf{I} - \boldsymbol{\tau}_{\mathrm{eff}}, \]
where:
If the last term is further approximated by a local rate-of-strain closure, then one obtains the familiar Navier-Stokes form.
So the real derivation target is:
These steps produce the hydrodynamic limit of the present program.
In the present framework, pressure and viscosity emerge from unresolved internal organization:
pressure and viscosity, if they appear, are effective summaries of unresolved local closure and transport.
This is the point at which the medium intuition becomes operational rather than merely suggestive.
The derivation program can therefore be stated in a strict order.
Keep the fundamental level: \[ u,\quad \mathbf{S},\quad \mathbf{F},\quad \mathbf{F}_{+},\mathbf{F}_{-}. \]
Derive a local momentum density from the same closure.
Derive a local momentum-flux tensor.
Prove the corresponding local momentum balance law.
Coarse-grain these fields over regions containing many local closures.
Identify the effective density, velocity, and stress variables of that coarse-grained description.
Determine when the effective stress reduces to Euler-like or Navier-Stokes-like form.
This derivation path begins from continuity and source-free transport closure, then asks what macroscopic medium theory emerges.
The hydrodynamic objective is therefore clear:
derive momentum density, momentum flux, and stress from the same underlying transport, then coarse-grain them to obtain the emergent continuum limit.
Appendix 207 shows that this program can already be completed in the simplest uniform-region case. There, Navier-Stokes-like dynamics appears as an effective large-scale theory of organized energy transport.
This appendix carries out, in the simplest uniform-region setting, the derivation sketched in Appendix 206.
The result is exact up to the constitutive choice for the coarse-grained stress. That last step is where Euler-like or Navier-Stokes-like behavior enters.
Work in a region where the local transport scale \(k\) may be treated as constant over the coarse-graining cell and over the time interval of interest.
In that region the already-derived electromagnetic transport quantities are
\[ u, \qquad \mathbf{S}, \qquad \mathbf{g}=\frac{\mathbf{S}}{k^2}, \qquad \partial_t \mathbf{g}-\nabla\cdot\mathbf{T}=0. \]
The last equation is the exact local momentum continuity law already obtained in chapter 12.
We now coarse-grain these quantities.
Let \(\langle \cdot \rangle\) denote averaging over a cell large compared to local closure structure and small compared to macroscopic variation.
Assume, in the usual continuum approximation, that averaging commutes with space and time differentiation at the resolved scale:
\[ \langle \partial_t f\rangle = \partial_t \langle f\rangle, \qquad \langle \partial_i f\rangle = \partial_i \langle f\rangle. \]
Then the exact local conservation laws average to
\[ \partial_t \langle u\rangle + \nabla\cdot\langle \mathbf{S}\rangle = 0, \]
and
\[ \partial_t \langle \mathbf{g}\rangle - \nabla\cdot\langle \mathbf{T}\rangle = 0. \]
These are still exact at the coarse-grained level. No constitutive assumption has entered yet.
Define the effective density by
\[ \rho := \frac{\langle u\rangle}{k^2}. \]
Define the effective velocity by
\[ \rho \mathbf{v} := \langle \mathbf{g}\rangle = \frac{\langle \mathbf{S}\rangle}{k^2}. \]
Equivalently,
\[ \mathbf{v}=\frac{\langle \mathbf{S}\rangle}{\langle u\rangle}. \]
This is the coarse-grained transport velocity of the energy content of the cell.
Now divide the averaged energy continuity equation by \(k^2\):
\[ \partial_t \rho + \nabla\cdot(\rho \mathbf{v}) = 0. \]
So the usual continuity equation of continuum mechanics appears immediately.
At this point nothing has been postulated beyond:
The averaged momentum equation is
\[ \partial_t(\rho \mathbf{v}) - \nabla\cdot\langle \mathbf{T}\rangle = 0. \]
This still contains the full coarse-grained momentum-flux tensor
\[ \langle \mathbf{T}\rangle. \]
To isolate the transported momentum of the mean motion, define the residual stress tensor
\[ \boldsymbol{\Sigma} := \rho\,\mathbf{v}\otimes\mathbf{v}-\langle \mathbf{T}\rangle. \]
This is an exact definition, not an approximation. It says simply that the total coarse-grained momentum flux is decomposed into:
Substituting gives the exact equation
\[ \partial_t(\rho \mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) -\nabla\cdot\boldsymbol{\Sigma} = 0. \]
This is already the hydrodynamic form. What remains is to parameterize \(\boldsymbol{\Sigma}\).
Decompose the residual stress into isotropic and traceless parts:
\[ \boldsymbol{\Sigma} = p\,\mathbf{I} - \boldsymbol{\tau}, \]
where
\[ p := \frac{1}{3}\,\mathrm{tr}(\boldsymbol{\Sigma}), \]
and
\[ \mathrm{tr}(\boldsymbol{\tau})=0. \]
This gives
\[ \partial_t(\rho \mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) + \nabla p - \nabla\cdot\boldsymbol{\tau} = 0. \]
Equivalently,
\[ \partial_t(\rho \mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) = -\nabla p + \nabla\cdot\boldsymbol{\tau}. \]
This is the exact coarse-grained momentum balance once the residual stress is split into isotropic and traceless parts.
Use the continuity equation
\[ \partial_t \rho + \nabla\cdot(\rho \mathbf{v})=0 \]
to rewrite the momentum equation in convective form.
Expand
\[ \partial_t(\rho \mathbf{v})+\nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) = \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) + \mathbf{v}\left(\partial_t\rho+\nabla\cdot(\rho\mathbf{v})\right). \]
The second term vanishes by continuity, so
\[ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p + \nabla\cdot\boldsymbol{\tau}. \]
This is the standard continuum momentum equation, still exact up to the form of \(\boldsymbol{\tau}\).
At this stage the derivation branches according to constitutive closure.
If the deviatoric stress is neglected,
\[ \boldsymbol{\tau}=0, \]
then
\[ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p. \]
This is the Euler form.
If the unresolved stress is approximated by the Newtonian constitutive form
\[ \boldsymbol{\tau} = \eta\left(\nabla\mathbf{v}+(\nabla\mathbf{v})^{\mathsf T} -\frac{2}{3}(\nabla\cdot\mathbf{v})\mathbf{I}\right) + \zeta(\nabla\cdot\mathbf{v})\mathbf{I}, \]
then
\[ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p + \nabla\cdot\boldsymbol{\tau}. \]
This is the compressible Navier-Stokes form.
In the incompressible limit
\[ \nabla\cdot\mathbf{v}=0, \qquad \rho=\text{const.}, \]
it reduces to
\[ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p + \eta \nabla^2\mathbf{v}. \]
So the familiar hydrodynamic equations arise as constitutive limits of the coarse-grained momentum balance.
Up to the introduction of \(\boldsymbol{\Sigma}\), everything in this appendix is an exact rewriting of:
The unresolved stress \(\boldsymbol{\Sigma}\) is therefore the place where specific hydrodynamic material behavior enters.
So the exact conclusion is:
the present program already yields the exact hydrodynamic conservation form.
Choosing an Euler closure sets
\[ \boldsymbol{\tau}=0. \]
Choosing a Newtonian viscous closure gives the Navier-Stokes form written in section 207.7.
This derivation matters because it says where fluid behavior would come from in the present ontology.
The effective density is coarse-grained stored energy divided by the local transport scale squared.
The effective velocity is coarse-grained energy transport divided by coarse-grained stored energy.
Pressure and viscosity are not primitive substances or forces. They are summaries of unresolved local closure and transport encoded in \(\boldsymbol{\Sigma}\).
So if Navier-Stokes-like dynamics is correct at macroscopic scales, it appears here as an emergent continuum limit of the same energy substrate, not as a separate ontology.
In a region with approximately uniform local transport scale \(k\), the already derived conservation laws imply:
\[ \partial_t \rho + \nabla\cdot(\rho \mathbf{v})=0, \]
and
\[ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p + \nabla\cdot\boldsymbol{\tau}, \]
with \(\rho\), \(\mathbf{v}\), \(p\), and \(\boldsymbol{\tau}\) defined by exact coarse-graining identities together with the chosen hydrodynamic closure.
Thus the book does not merely point toward hydrodynamics in the abstract. It already contains the detailed route by which Euler-like and Navier-Stokes-like equations emerge from continuity, momentum flux, and coarse-graining of the same underlying transport substrate. Appendix 213 then extends the same derivation to variable background.
The emergent hydrodynamic form derived above inherits the underlying transport bound of the electromagnetic substrate. In a uniform region,
\[ |\mathbf{S}| \le k\,u. \]
So for any bounded region \(\Omega\) with outward normal \(\mathbf n\),
\[ \frac{d}{dt}\int_\Omega u\,dV = -\int_{\partial\Omega}\mathbf S\cdot\mathbf n\,dA \le \int_{\partial\Omega}|\mathbf S|\,dA \le k\int_{\partial\Omega}u\,dA. \]
This says that the rate at which energy can be driven into a region is bounded by what is already being transported across its boundary.
That is the decisive physical point. The emergent fluid here is not an unrestricted Newtonian continuum in which arbitrarily large amounts of energy or momentum may be delivered into an arbitrarily small region with no transport ceiling. It is a causally bounded transport medium. Any concentration process must be fed through the boundary at finite speed.
So the physical blow-up picture is dissolved at the level of ontology:
This does not attempt to rescue every nonphysical branch of an unrestricted Newtonian PDE idealization. It makes the narrower and stronger claim relevant to the present book:
in the electromagnetic-fluid ontology derived here, the physical phenomenon of finite-time blow-up is excluded because transport itself is bounded.
This appendix derives the Lorentz-force form directly from the source-free stress transfer of a compact toroidal charged mode.
No effective charge density or current density is introduced. The bounded mode is a self-closing toroidal organization of one continuous field. The familiar Lorentz formula appears as the exact point-mode limit of its interaction with a smooth external Maxwell field.
For definiteness, take the compact mode to be an axisymmetric torus with equal winding class \((m,m)\) and symmetry axis \(\hat{\mathbf a}\). Nothing essential depends on that simplifying choice. It only makes the geometric bookkeeping cleaner. The leading interaction depends solely on the signed through-hole flux class carried along the torus axis. All finer toroidal structure enters only through higher multipoles.
Let
\[ K_\varepsilon \]
be a coherent toroidal charged mode of size \(\varepsilon\), centered at the worldline
\[ X(\tau), \]
with proper time \(\tau\).
Chapter 10 associated charge with the signed through-hole flux class across the torus aperture. Write that class as the scalar
\[ q. \]
In the instantaneous rest frame of the mode, choose Cartesian coordinates with origin at the center of energy and let
\[ \mathbf r = R\,\mathbf n, \qquad |\mathbf n|=1. \]
The compact toroidal mode is assumed to have the far-field behavior established in chapter 10:
\[ \mathbf E_{\mathrm s}(\mathbf r) = \frac{q}{4\pi\varepsilon_0}\frac{\mathbf n}{R^2} + \mathbf e_{\mathrm{rem}}(\mathbf r), \]
\[ \mathbf B_{\mathrm s}(\mathbf r) = \mathbf b_{\mathrm{rem}}(\mathbf r), \]
with bounds
\[ |\mathbf e_{\mathrm{rem}}(\mathbf r)| \le C_E\frac{\varepsilon}{R^3}, \qquad |\mathbf b_{\mathrm{rem}}(\mathbf r)| \le C_B\frac{\varepsilon}{R^3}, \]
for every
\[ R\ge 2\varepsilon. \]
So the leading exterior field of the compact torus is the inverse-square monopole term determined by the through-hole flux class, while all finer toroidal structure decays at least one power faster.
Let the external Maxwell field be smooth near the mode center. On a sphere
\[ S_R:=\{\,\mathbf x : |\mathbf x-X(\tau)|=R\,\}, \]
write
\[ \mathbf E_{\mathrm e}(X(\tau)+R\mathbf n,\tau) = \mathbf E_0(\tau)+\mathbf E_1(R,\mathbf n,\tau), \]
\[ \mathbf B_{\mathrm e}(X(\tau)+R\mathbf n,\tau) = \mathbf B_0(\tau)+\mathbf B_1(R,\mathbf n,\tau), \]
with
\[ |\mathbf E_1(R,\mathbf n,\tau)|\le C'_E R, \qquad |\mathbf B_1(R,\mathbf n,\tau)|\le C'_B R. \]
This is just the first-order smoothness expansion of the external field near the mode center.
Let the total field be
\[ \mathbf E=\mathbf E_{\mathrm s}+\mathbf E_{\mathrm e}, \qquad \mathbf B=\mathbf B_{\mathrm s}+\mathbf B_{\mathrm e}. \]
The total field is source-free everywhere. We evaluate the balance on spheres lying outside the compact toroidal core only so the exterior asymptotic form of the compact closure can be used cleanly. The exact local momentum balance is
\[ \partial_t g_i - \partial_j T_{ij}=0, \]
where
\[ \mathbf g=\varepsilon_0\,\mathbf E\times\mathbf B, \]
and
\[ T_{ij} = \varepsilon_0\left(E_iE_j-\frac{1}{2}\delta_{ij}\mathbf E^2\right) + \frac{1}{\mu_0}\left(B_iB_j-\frac{1}{2}\delta_{ij}\mathbf B^2\right). \]
Decompose
\[ \mathbf g = \mathbf g_{\mathrm s} + \mathbf g_{\mathrm e} + \mathbf g_{\times}, \]
\[ \mathbf T = \mathbf T_{\mathrm s} + \mathbf T_{\mathrm e} + \mathbf T_{\times}, \]
where the cross terms are
\[ \mathbf g_{\times} := \varepsilon_0\bigl( \mathbf E_{\mathrm s}\times\mathbf B_{\mathrm e} + \mathbf E_{\mathrm e}\times\mathbf B_{\mathrm s} \bigr), \]
and
\[ (T_{\times})_{ij} := \varepsilon_0\left( E_{{\mathrm s}i}E_{{\mathrm e}j} + E_{{\mathrm e}i}E_{{\mathrm s}j} - \delta_{ij}\,\mathbf E_{\mathrm s}\cdot\mathbf E_{\mathrm e} \right) + \frac{1}{\mu_0}\left( B_{{\mathrm s}i}B_{{\mathrm e}j} + B_{{\mathrm e}i}B_{{\mathrm s}j} - \delta_{ij}\,\mathbf B_{\mathrm s}\cdot\mathbf B_{\mathrm e} \right). \]
Subtracting the self and external balances from the total balance gives the exact cross-balance
\[ \partial_t(\mathbf g_{\times})_i - \partial_j(T_{\times})_{ij} = 0. \]
Integrating over the ball
\[ B_R:=\{\,\mathbf x : |\mathbf x-X(\tau)|\le R\,\} \]
gives
\[ \frac{d}{dt}\int_{B_R}\mathbf g_{\times}\,dV = \int_{S_R}\mathbf T_{\times}\cdot\mathbf n\,dA. \]
For the monist reading used in this book, the right-hand side is the exact rate at which external stress transfers momentum into the compact closure across the surrounding sphere.
Define therefore
\[ \mathbf F_R(\tau) := \int_{S_R}\mathbf T_{\times}\cdot\mathbf n\,dA. \]
The Lorentz force will be the exact limit of \(\mathbf F_R\) as the mode is shrunk to a point while the surrounding sphere shrinks with it but remains outside the toroidal core.
Choose an event on the worldline and work in the instantaneous rest frame of the toroidal mode at that event.
Let
\[ \alpha(R):=\frac{q}{4\pi\varepsilon_0 R^2}. \]
On \(S_R\) write
\[ \mathbf E_{\mathrm s} = \alpha(R)\,\mathbf n + \mathbf e_{\mathrm{rem}}, \]
with
\[ |\mathbf e_{\mathrm{rem}}|\le C_E\frac{\varepsilon}{R^3}, \qquad |\mathbf B_{\mathrm s}|\le C_B\frac{\varepsilon}{R^3}. \]
The electric cross term on the sphere is
\[ \mathbf T_{\times}^{(E)}\cdot\mathbf n = \varepsilon_0\left[ (\mathbf E_{\mathrm s}\cdot\mathbf n)\mathbf E_{\mathrm e} + (\mathbf E_{\mathrm e}\cdot\mathbf n)\mathbf E_{\mathrm s} - (\mathbf E_{\mathrm s}\cdot\mathbf E_{\mathrm e})\mathbf n \right]. \]
Substitute
\[ \mathbf E_{\mathrm s}=\alpha\mathbf n+\mathbf e_{\mathrm{rem}}, \qquad \mathbf E_{\mathrm e}=\mathbf E_0+\mathbf E_1. \]
The leading terms simplify exactly:
\[ \varepsilon_0\left[ (\alpha\mathbf n\cdot\mathbf n)\mathbf E_0 + (\mathbf E_0\cdot\mathbf n)\alpha\mathbf n - (\alpha\mathbf n\cdot\mathbf E_0)\mathbf n \right] = \varepsilon_0\,\alpha\,\mathbf E_0. \]
So
\[ \mathbf T_{\times}^{(E)}\cdot\mathbf n = \frac{q}{4\pi R^2}\,\mathbf E_0 + \mathbf R_E(R,\mathbf n), \]
where the remainder satisfies
\[ |\mathbf R_E(R,\mathbf n)| \le C_1\frac{|q|}{R^2}\,R + C_2\frac{\varepsilon}{R^3}. \]
Therefore
\[ \int_{S_R}\mathbf T_{\times}^{(E)}\cdot\mathbf n\,dA = q\,\mathbf E_0 + O(R) + O\!\left(\frac{\varepsilon}{R}\right). \]
For the magnetic cross term,
\[ \mathbf T_{\times}^{(B)}\cdot\mathbf n = \frac{1}{\mu_0}\left[ (\mathbf B_{\mathrm s}\cdot\mathbf n)\mathbf B_{\mathrm e} + (\mathbf B_{\mathrm e}\cdot\mathbf n)\mathbf B_{\mathrm s} - (\mathbf B_{\mathrm s}\cdot\mathbf B_{\mathrm e})\mathbf n \right], \]
so
\[ \left| \int_{S_R}\mathbf T_{\times}^{(B)}\cdot\mathbf n\,dA \right| \le C_3\,R^2\sup_{S_R}|\mathbf B_{\mathrm s}|\,\sup_{S_R}|\mathbf B_{\mathrm e}| = O\!\left(\frac{\varepsilon}{R}\right). \]
Hence
\[ \mathbf F_R = q\,\mathbf E_0 + O(R) + O\!\left(\frac{\varepsilon}{R}\right). \]
Take a two-scale limit in which
\[ \varepsilon\to 0, \qquad R\to 0, \qquad \frac{\varepsilon}{R}\to 0. \]
Then
\[ \boxed{ \lim_{\varepsilon\to 0}\mathbf F_R = q\,\mathbf E_0 }. \]
So a compact toroidal charged mode at rest experiences exactly the electric force
\[ \mathbf F_{\mathrm{rest}}=q\,\mathbf E_{\mathrm e}(X). \]
No magnetic term appears in the rest frame, because the static toroidal mode has no magnetic monopole part. The toroidal details affect only the discarded higher multipoles.
The rest-frame theorem gives the monopole coupling of the toroidal charge class to a smooth electric load. To get the moving magnetic term, one should not jump immediately to a covariant ansatz. The torus itself already tells us what has to be sampled: a moving aperture of one common field.
Let
\[ \Sigma_\varepsilon(t) \]
be a spanning surface across the torus aperture, transported with the compact mode, and let
\[ \mathbf u(\mathbf y,t) \]
be the local velocity of the material point
\[ \mathbf y\in \Sigma_\varepsilon(t). \]
For any moving surface, Maxwell-Faraday transport gives
\[ \frac{d}{dt}\int_{\Sigma_\varepsilon(t)}\mathbf B_{\mathrm e}\cdot d\mathbf A = -\oint_{\partial\Sigma_\varepsilon(t)} \bigl( \mathbf E_{\mathrm e} + \mathbf u\times \mathbf B_{\mathrm e} \bigr)\cdot d\boldsymbol\ell. \]
So the local field sampled by a moving aperture is not
\[ \mathbf E_{\mathrm e} \]
alone, but
\[ \mathbf E_{\mathrm e}+\mathbf u\times\mathbf B_{\mathrm e}. \]
This is the transport meaning of the magnetic term: the moving toroidal aperture samples the surrounding momentum-flux geometry through its transport across the transverse external field.
For a rigidly drifting compact torus, decompose
\[ \mathbf u(\mathbf y,t)=\mathbf v(t)+\mathbf u_{\mathrm{int}}(\mathbf y,t), \]
where
\[ \mathbf v(t)=\dot{\mathbf X}(t) \]
is the drift of the torus center and
\[ \mathbf u_{\mathrm{int}} \]
is the internal helical traversal of the closure.
Define the aperture average of a field over \(\Sigma_\varepsilon(t)\) by
\[ \langle \mathbf W\rangle_{\Sigma_\varepsilon(t)} := \frac{1}{A(\Sigma_\varepsilon(t))} \int_{\Sigma_\varepsilon(t)}\mathbf W\,dA. \]
Because the external field is smooth on the compact scale,
\[ \langle \mathbf E_{\mathrm e}\rangle_{\Sigma_\varepsilon(t)} = \mathbf E_{\mathrm e}(\mathbf X(t),t)+O(\varepsilon), \]
\[ \langle \mathbf B_{\mathrm e}\rangle_{\Sigma_\varepsilon(t)} = \mathbf B_{\mathrm e}(\mathbf X(t),t)+O(\varepsilon). \]
For the equal-winding axisymmetric torus, the internal traversal has no monopole average across the aperture:
\[ \langle \mathbf u_{\mathrm{int}}\rangle_{\Sigma_\varepsilon(t)}=0. \]
Geometrically, opposite points of the aperture carry opposite tangential traversal velocities, so the internal helical motion cancels at monopole order. It affects only higher multipoles.
Therefore
\[ \left\langle \mathbf E_{\mathrm e} + \mathbf u\times\mathbf B_{\mathrm e} \right\rangle_{\Sigma_\varepsilon(t)} = \mathbf E_{\mathrm e}(\mathbf X(t),t) + \mathbf v(t)\times\mathbf B_{\mathrm e}(\mathbf X(t),t) + O(\varepsilon). \]
The moving compact torus therefore samples the smooth external field through the effective transport load
\[ \boxed{ \mathbf L_{\mathrm{mov}} = \mathbf E_{\mathrm e}(\mathbf X(t),t) + \mathbf v(t)\times\mathbf B_{\mathrm e}(\mathbf X(t),t) }. \]
Section 209.3 proved that the compact toroidal charge class couples at monopole order by
\[ q\times(\text{smooth load sampled across the aperture}). \]
For a static torus, that sampled load is
\[ \mathbf E_{\mathrm e}(X). \]
For a translating torus, section 209.4 shows that the sampled load is
\[ \mathbf L_{\mathrm{mov}} = \mathbf E_{\mathrm e}(\mathbf X(t),t) + \mathbf v(t)\times\mathbf B_{\mathrm e}(\mathbf X(t),t). \]
Therefore, in the same compact two-scale limit,
\[ \boxed{ \mathbf F = q\,\mathbf L_{\mathrm{mov}} = q\bigl( \mathbf E_{\mathrm e} + \mathbf v\times\mathbf B_{\mathrm e} \bigr) }. \]
This is the Lorentz-force form at compact toroidal monopole order.
The associated power law follows immediately:
\[ \frac{dE}{dt} = \mathbf F\cdot\mathbf v = q\,\mathbf E_{\mathrm e}\cdot\mathbf v, \]
because
\[ (\mathbf v\times\mathbf B_{\mathrm e})\cdot\mathbf v=0. \]
So the magnetic part redirects transport but does no work.
Force here is therefore the rate at which cross stress transfers momentum into the compact closure. Power is the corresponding energy-transfer rate obtained by contracting that momentum transfer with the drift velocity.
The argument used only the following ingredients:
No effective source density was needed.
The role of the equal-winding \((m,m)\) torus was only to give a clean axis and a clean aperture-flux class. The Lorentz law depends only on the resulting scalar \(q\) at monopole order. All finer toroidal geometry survives only in higher multipole corrections beyond the point-mode limit.
Within this framework, the Lorentz force is not a primitive rule about a particle being pushed by an external field.
It is the compact expression of one exact statement:
Geometrically, the toroidal charge class is carried by an axial energy-moment orthogonal to the local transverse pair \((\mathbf F_+,\mathbf F_-)\). When that axis drifts through the external transverse organization, the sampled momentum-flux load resolves into the lateral transport term $ vB_{e}. $
So the Lorentz law is not imported. It is the point-mode limit of toroidal boundary stress transfer together with moving-aperture transport.
For a compact toroidal charged mode, the exact source-free cross-stress transfer across a sphere \(S_R\) is
\[ \mathbf F_R = \int_{S_R}\mathbf T_{\times}\cdot\mathbf n\,dA. \]
In the instantaneous rest frame, the compact-mode limit gives
\[ \lim_{\varepsilon\to 0}\mathbf F_R = q\,\mathbf E_{\mathrm e}(X). \]
For a moving compact torus, the transported aperture samples the smooth external field through
\[ \mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}. \]
Therefore the compact toroidal monopole force is
\[ \frac{d\mathbf p}{dt} = q\bigl(\mathbf E_{\mathrm e}+\mathbf v\times\mathbf B_{\mathrm e}\bigr). \]
Thus the Lorentz-force form is derived here directly from first principles at compact toroidal monopole order: compact toroidal charge, exact source-free stress transfer, and moving-aperture transport of one common electromagnetic substrate.
This appendix derives the interaction of two charged modes directly from the same compact toroidal closure used in appendix 209.
The static interaction is first obtained as the exact compact-limit cross energy of two bounded toroidal closures. The Coulomb law then follows by taking the gradient of that interaction potential. The moving interaction is obtained by applying appendix 209’s compact transport theorem to each mode in the field generated by the other.
No effective charge density or current density is introduced anywhere in the derivation.
Let
\[ K_{1,\varepsilon_1}, \qquad K_{2,\varepsilon_2} \]
be two disjoint coherent toroidal charged modes of sizes
\[ \varepsilon_1,\qquad \varepsilon_2, \]
centered at
\[ \mathbf X_1,\qquad \mathbf X_2. \]
Write their signed through-hole flux classes as
\[ q_1,\qquad q_2. \]
Assume the two compact modes are separated by a distance
\[ d:=|\mathbf X_1-\mathbf X_2| \]
with
\[ d\gg \varepsilon_1+\varepsilon_2. \]
In the compact toroidal limit of chapter 10 and appendix 209, the exterior field of each mode has the asymptotic form
\[ \mathbf E_a(\mathbf r) = \frac{q_a}{4\pi\varepsilon_0} \frac{\mathbf r-\mathbf X_a}{|\mathbf r-\mathbf X_a|^3} + \mathbf e_{a,\mathrm{rem}}(\mathbf r), \]
\[ \mathbf B_a(\mathbf r) = \mathbf b_{a,\mathrm{rem}}(\mathbf r), \]
with bounds
\[ |\mathbf e_{a,\mathrm{rem}}(\mathbf r)| \le C_a\frac{\varepsilon_a}{|\mathbf r-\mathbf X_a|^3}, \]
\[ |\mathbf b_{a,\mathrm{rem}}(\mathbf r)| \le C'_a\frac{\varepsilon_a}{|\mathbf r-\mathbf X_a|^3}, \]
for every point outside the toroidal core.
At static leading order, the magnetic terms are higher multipoles and the interaction is governed by the electric cross energy.
Work first in a frame in which the two toroidal centers are instantaneously at rest and retain only the static leading interaction.
The fields are source-free everywhere. On the exterior domain outside small balls enclosing the two toroidal cores they are static, so
\[ \nabla\times\mathbf E_a=0, \qquad \nabla\cdot\mathbf E_a=0. \]
Therefore there exist harmonic exterior potentials
\[ \phi_1,\qquad \phi_2 \]
such that
\[ \mathbf E_a=-\nabla\phi_a \]
outside the cores, with asymptotics
\[ \phi_a(\mathbf r) = \frac{q_a}{4\pi\varepsilon_0|\mathbf r-\mathbf X_a|} + \phi_{a,\mathrm{rem}}(\mathbf r), \]
\[ |\phi_{a,\mathrm{rem}}(\mathbf r)| \le C''_a\frac{\varepsilon_a}{|\mathbf r-\mathbf X_a|^2}. \]
Fix radii
\[ \rho_1,\rho_2,R \]
such that
\[ 2\varepsilon_1<\rho_1\ll d, \qquad 2\varepsilon_2<\rho_2\ll d, \qquad R\gg d. \]
Let
\[ \Omega_{R,\rho_1,\rho_2} := B_R(0)\setminus \bigl(B_{\rho_1}(\mathbf X_1)\cup B_{\rho_2}(\mathbf X_2)\bigr), \]
where
\[ B_\rho(\mathbf X):=\{\,\mathbf r:|\mathbf r-\mathbf X|<\rho\,\}. \]
Define the static cross energy by
\[ U_\times(R,\rho_1,\rho_2) := \varepsilon_0 \int_{\Omega_{R,\rho_1,\rho_2}} \mathbf E_1\cdot\mathbf E_2\,dV. \]
Since
\[ \mathbf E_a=-\nabla\phi_a, \]
this becomes
\[ U_\times(R,\rho_1,\rho_2) = \varepsilon_0 \int_{\Omega_{R,\rho_1,\rho_2}} \nabla\phi_1\cdot\nabla\phi_2\,dV. \]
Because
\[ \Delta\phi_2=0 \]
throughout \(\Omega_{R,\rho_1,\rho_2}\), Green’s identity gives
\[ U_\times(R,\rho_1,\rho_2) = \varepsilon_0 \int_{\partial\Omega_{R,\rho_1,\rho_2}} \phi_1\,\partial_n\phi_2\,dA. \]
We now evaluate the three boundary pieces.
On the outer sphere \(|\mathbf r|=R\),
\[ \phi_1=O(R^{-1}), \qquad \partial_n\phi_2=O(R^{-2}), \]
so
\[ \int_{|\mathbf r|=R}\phi_1\,\partial_n\phi_2\,dA = O(R^{-1}). \]
Near \(\mathbf X_1\), the potential \(\phi_2\) is smooth because the second mode is disjoint from the first. Hence
\[ \partial_n\phi_2=O(1) \]
on \(|\mathbf r-\mathbf X_1|=\rho_1\), while
\[ \phi_1=O(\rho_1^{-1}). \]
Therefore
\[ \int_{|\mathbf r-\mathbf X_1|=\rho_1}\phi_1\,\partial_n\phi_2\,dA = O(\rho_1). \]
On the sphere
\[ S_{2,\rho_2}:=\{\,\mathbf r:|\mathbf r-\mathbf X_2|=\rho_2\,\}, \]
write
\[ \mathbf r-\mathbf X_2=\rho_2\,\mathbf n_2, \qquad |\mathbf n_2|=1. \]
The outward normal of the punctured domain \(\Omega_{R,\rho_1,\rho_2}\) on this inner boundary is
\[ \mathbf n=-\mathbf n_2. \]
Hence
\[ \partial_n\phi_2 = \frac{q_2}{4\pi\varepsilon_0\rho_2^2} + O\!\left(\frac{\varepsilon_2}{\rho_2^3}\right). \]
Since \(\phi_1\) is smooth near \(\mathbf X_2\),
\[ \phi_1(\mathbf X_2+\rho_2\mathbf n_2) = \phi_1(\mathbf X_2)+O(\rho_2). \]
Therefore
\[ \varepsilon_0 \int_{S_{2,\rho_2}} \phi_1\,\partial_n\phi_2\,dA = q_2\,\phi_1(\mathbf X_2) + O(\rho_2) + O\!\left(\frac{\varepsilon_2}{\rho_2}\right). \]
Collecting the three boundary pieces, we obtain
\[ U_\times(R,\rho_1,\rho_2) = q_2\,\phi_1(\mathbf X_2) + O(R^{-1}) + O(\rho_1) + O(\rho_2) + O\!\left(\frac{\varepsilon_2}{\rho_2}\right). \]
Now let
\[ R\to\infty, \qquad \rho_1\to 0, \qquad \rho_2\to 0, \qquad \frac{\varepsilon_2}{\rho_2}\to 0. \]
Then
\[ \boxed{ U_\times = q_2\,\phi_1(\mathbf X_2) }. \]
By symmetry the same argument also gives
\[ \boxed{ U_\times = q_1\,\phi_2(\mathbf X_1) }. \]
Using the compact monopole asymptotic of \(\phi_1\) at \(\mathbf X_2\),
\[ \phi_1(\mathbf X_2) = \frac{q_1}{4\pi\varepsilon_0 d} + O\!\left(\frac{\varepsilon_1}{d^2}\right), \]
the compact-limit interaction potential is
\[ \boxed{ U_\times = \frac{q_1q_2}{4\pi\varepsilon_0 d} }. \]
So the Coulomb potential is the exact compact-limit cross energy of two toroidal charged closures.
The force on the first toroidal mode is the negative gradient of the interaction energy with respect to its center:
\[ \mathbf F_{1\leftarrow 2} := -\nabla_{\mathbf X_1}U_\times. \]
Using
\[ U_\times=q_1\,\phi_2(\mathbf X_1), \]
we obtain
\[ \mathbf F_{1\leftarrow 2} = -q_1\nabla\phi_2(\mathbf X_1) = q_1\,\mathbf E_2(\mathbf X_1). \]
Similarly,
\[ \mathbf F_{2\leftarrow 1} = q_2\,\mathbf E_1(\mathbf X_2). \]
In the compact monopole limit,
\[ \boxed{ \mathbf F_{1\leftarrow 2} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_1-\mathbf X_2}{|\mathbf X_1-\mathbf X_2|^3} }, \]
\[ \boxed{ \mathbf F_{2\leftarrow 1} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_2-\mathbf X_1}{|\mathbf X_2-\mathbf X_1|^3} = -\mathbf F_{1\leftarrow 2} }. \]
So the static Coulomb law is recovered exactly as the gradient of the compact toroidal cross-energy potential.
Now allow the two compact toroidal modes to move.
Appendix 209 already proved that a compact toroidal charged mode samples a smooth external field through the compact transport load
\[ \mathbf E+\mathbf v\times\mathbf B \]
at monopole order in the compact limit. Therefore the same toroidal mode obeys
\[ \frac{d\mathbf p}{dt} = q\bigl(\mathbf E+\mathbf v\times\mathbf B\bigr). \]
Apply that theorem to each mode in the field generated by the other. Then
\[ \boxed{ \mathbf F_{1\leftarrow 2} = q_1\bigl( \mathbf E_2(\mathbf X_1,t) + \mathbf v_1\times\mathbf B_2(\mathbf X_1,t) \bigr) }, \]
\[ \boxed{ \mathbf F_{2\leftarrow 1} = q_2\bigl( \mathbf E_1(\mathbf X_2,t) + \mathbf v_2\times\mathbf B_1(\mathbf X_2,t) \bigr) }. \]
So the moving two-torus interaction is not a new law layered on top of the static Coulomb potential. It is the compact-limit same-field interaction of each toroidal closure with the propagated field of the other.
The result can now be read in the same monist way as appendix 209.
So neither Coulomb nor Lorentz interaction is primitive action-at-a-distance. Both are compact-limit expressions of energy transfer and momentum transfer in one common electromagnetic substrate.
For two compact toroidal charged modes, the static cross energy is
\[ U_\times = \varepsilon_0\int \mathbf E_1\cdot\mathbf E_2\,dV. \]
In the compact limit this is exactly
\[ U_\times = \frac{q_1q_2}{4\pi\varepsilon_0|\mathbf X_1-\mathbf X_2|}. \]
Taking the gradient gives the exact static interaction
\[ \mathbf F_{1\leftarrow 2} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf X_1-\mathbf X_2}{|\mathbf X_1-\mathbf X_2|^3}. \]
For moving compact modes, appendix 209 therefore gives
\[ \mathbf F_{1\leftarrow 2} = q_1\bigl( \mathbf E_2(\mathbf X_1,t) + \mathbf v_1\times\mathbf B_2(\mathbf X_1,t) \bigr), \]
and similarly for mode 2.
Thus the two-charge interaction is derived here directly from compact toroidal closure, cross energy, and transported same-field stress.
This appendix proves the uniqueness claim that chapter 7 can only suggest in words.
The claim is not that Maxwell closure is unique among all imaginable local relations whatsoever. The claim is narrower and precise:
Within the class of real, linear, local, homogeneous, isotropic, first-order, purely differential, divergence-preserving two-field closures, every neutral isotropic transporting closure is equivalent, by a real linear recombination of the two fields, to the Maxwell pair \[ \partial_t\mathbf{F}_{+}=k\,\nabla\times\mathbf{F}_{-}, \qquad \partial_t\mathbf{F}_{-}=-k\,\nabla\times\mathbf{F}_{+}. \]
This is the exact sense in which the closure of chapter 7 is unique.
Let
\[ \mathbf{F}_1(\mathbf{r},t), \qquad \mathbf{F}_2(\mathbf{r},t) \]
be two real vector fields on \(\mathbb{R}^3\), each constrained by
\[ \nabla\cdot\mathbf{F}_1=0, \qquad \nabla\cdot\mathbf{F}_2=0. \]
We consider closures of the form
\[ \partial_t\mathbf{F}_a=\mathcal{L}_{ab}\mathbf{F}_b, \qquad a,b\in\{1,2\}, \]
with the following assumptions.
Real-linear: \(\mathcal{L}\) is linear over the real numbers.
Local and homogeneous: \(\mathcal{L}\) has constant coefficients and depends only on the value of the fields and their derivatives at the same point.
First-order in space: \(\mathcal{L}\) contains at most one spatial derivative.
Purely differential: there is no algebraic zero-order mixing term. This restriction is intentional. Chapter 6 separated algebraic updates from spatial reorganization; the present theorem concerns the transporting closure class.
Isotropic: the closure is equivariant under proper spatial rotations.
Divergence-preserving: if \(\nabla\cdot\mathbf{F}_a=0\) initially, then \(\nabla\cdot(\partial_t\mathbf{F}_a)=0\).
The task is to classify all such closures and determine which of them yield neutral isotropic transport.
Because the closure is linear, local, homogeneous, first-order, and purely differential, there exist constants \(B_{abijk}\) such that
\[ (\partial_t\mathbf{F}_a)_i = B_{abijk}\,\partial_j(\mathbf{F}_b)_k. \]
Isotropy under proper rotations means the tensor \(B_{abijk}\) must satisfy
\[ R_{i\ell}\,B_{ab\ell mn}\,R_{jm}\,R_{kn} = B_{abijk} \]
for every rotation matrix \(R\in SO(3)\).
For fixed field indices \(a,b\), this is the classification problem for an isotropic rank-three tensor on Euclidean space. Up to a scalar multiple, the only such tensor is the Levi-Civita symbol \(\varepsilon_{ijk}\). Therefore
\[ B_{abijk}=M_{ab}\,\varepsilon_{ijk} \]
for some real \(2\times 2\) matrix \(M=(M_{ab})\).
Hence every closure in the stated class has the form
\[ \partial_t\mathbf{F}_a = M_{ab}\,\nabla\times\mathbf{F}_b. \]
Equivalently, if we write
\[ \mathbb{F} := \begin{pmatrix} \mathbf{F}_1\\ \mathbf{F}_2 \end{pmatrix}, \]
then
\[ \partial_t\mathbb{F} = M\,(\nabla\times)\mathbb{F}, \]
where \((\nabla\times)\) acts componentwise on the two fields.
This classification is exact.
For any real matrix \(M\),
\[ \nabla\cdot\bigl(M_{ab}\,\nabla\times\mathbf{F}_b\bigr) = M_{ab}\,\nabla\cdot(\nabla\times\mathbf{F}_b)=0. \]
So every closure in the classified form preserves the divergence-free condition identically.
Thus the classification problem is reduced to the field-space matrix \(M\).
Differentiate once more in time:
\[ \partial_t^2\mathbf{F}_a = M_{ab}\,\nabla\times(\partial_t\mathbf{F}_b). \]
Substitute the closure again:
\[ \partial_t^2\mathbf{F}_a = M_{ab}\,\nabla\times\bigl(M_{bc}\,\nabla\times\mathbf{F}_c\bigr) = (M^2)_{ac}\,\nabla\times(\nabla\times\mathbf{F}_c). \]
Because each field is source-free,
\[ \nabla\times(\nabla\times\mathbf{F}_c) = \nabla(\nabla\cdot\mathbf{F}_c)-\nabla^2\mathbf{F}_c = -\nabla^2\mathbf{F}_c. \]
Hence
\[ \partial_t^2\mathbf{F}_a = -(M^2)_{ac}\,\nabla^2\mathbf{F}_c. \]
In block form,
\[ \partial_t^2\mathbb{F} = -M^2\,\nabla^2\mathbb{F}. \]
So the entire second-order transport content of the closure is encoded by the matrix \(-M^2\).
In the present appendix, neutral isotropic transport means the following:
there exists a real field-space change of variables
\[ \mathbb{G}=P^{-1}\mathbb{F}, \qquad P\in GL(2,\mathbb{R}), \]
and a positive constant \(k\) such that each transformed field satisfies the same wave equation
\[ \partial_t^2\mathbf{G}_1-k^2\nabla^2\mathbf{G}_1=0, \qquad \partial_t^2\mathbf{G}_2-k^2\nabla^2\mathbf{G}_2=0. \]
Because \(P\) is constant, it commutes with \(\partial_t\) and \(\nabla^2\). So the second-order equation becomes
\[ \partial_t^2\mathbb{G} = -P^{-1}M^2P\,\nabla^2\mathbb{G}. \]
Therefore neutral isotropic transport holds if and only if
\[ -P^{-1}M^2P=k^2I. \]
Since the identity is invariant under similarity, this is equivalent to
\[ M^2=-k^2I. \]
So the transport criterion is exact:
A closure in the stated class yields neutral isotropic transport with one common speed \(k\) if and only if its field-space matrix satisfies \(M^2=-k^2I\).
We now classify all real \(2\times 2\) matrices satisfying
\[ M^2=-k^2I, \qquad k>0. \]
Let \(\mathbf{e}\in\mathbb{R}^2\) be any nonzero vector.
The vectors \(\mathbf{e}\) and \(M\mathbf{e}\) are linearly independent. For if they were dependent, we would have
\[ M\mathbf{e}=\lambda\mathbf{e} \]
for some real \(\lambda\), and then
\[ M^2\mathbf{e}=\lambda^2\mathbf{e}, \]
which would imply
\[ \lambda^2=-k^2, \]
impossible over the real numbers.
So the vectors
\[ \mathbf{e}_1:=\mathbf{e}, \qquad \mathbf{e}_2:=-\frac{1}{k}M\mathbf{e} \]
form a basis of \(\mathbb{R}^2\).
In this basis,
\[ M\mathbf{e}_1 = k\,\mathbf{e}_2, \]
and
\[ M\mathbf{e}_2 = -\frac{1}{k}M^2\mathbf{e} = -\frac{1}{k}(-k^2)\mathbf{e} = -k\,\mathbf{e}_1. \]
Therefore the matrix of \(M\) in the basis \((\mathbf{e}_1,\mathbf{e}_2)\) is
\[ \begin{pmatrix} 0 & -k\\ k & 0 \end{pmatrix}. \]
After exchanging the two basis vectors if desired, this becomes the canonical matrix
\[ J_k := \begin{pmatrix} 0 & k\\ -k & 0 \end{pmatrix}. \]
So there exists a real invertible matrix \(P\) such that
\[ P^{-1}MP=J_k. \]
Now set
\[ \begin{pmatrix} \mathbf{F}_{+}\\ \mathbf{F}_{-} \end{pmatrix} := P^{-1} \begin{pmatrix} \mathbf{F}_1\\ \mathbf{F}_2 \end{pmatrix}. \]
Then the closure becomes
\[ \partial_t \begin{pmatrix} \mathbf{F}_{+}\\ \mathbf{F}_{-} \end{pmatrix} = \begin{pmatrix} 0 & k\\ -k & 0 \end{pmatrix} (\nabla\times) \begin{pmatrix} \mathbf{F}_{+}\\ \mathbf{F}_{-} \end{pmatrix}, \]
that is,
\[ \partial_t\mathbf{F}_{+}=k\,\nabla\times\mathbf{F}_{-}, \qquad \partial_t\mathbf{F}_{-}=-k\,\nabla\times\mathbf{F}_{+}. \]
This is exactly the canonical Maxwell closure of chapter 7.
We have therefore proved:
Every neutral isotropic transporting closure in the stated class is equivalent, by a real linear recombination of the two fields, to the Maxwell pair.
That is the promised uniqueness theorem.
For a one-field closure in the same class, the matrix \(M\) is just a real scalar \(m\), so the condition for neutral isotropic transport would be
\[ m^2=-k^2, \]
which has no real solution.
So within the same class:
This corollary places appendix 203 and the present appendix in one chain:
The theorem is exact, but its scope is the class stated at the beginning.
It does not address:
So the theorem should be read correctly:
That is already a strong result.
Within the real, linear, local, homogeneous, isotropic, first-order, purely differential, divergence-preserving two-field class:
Thus Maxwell closure is unique in the stated class.
This appendix develops the weak-field truncation of the gravity interaction used in chapter 13 as far as the static benchmark set.
The framework is not meant to stop at weak field. Exact interaction is the actual target. This appendix keeps only the leading weak-field term because the benchmark observables treated here are measured in that regime.
The point is not to assume spacetime curvature first and then reinterpret it. The point is to begin from the exterior mass-potential of a bounded trapped closure, write its weak-field transport summary, derive the corresponding transport geometry, and then recover from that one geometry the standard leading weak-field observables:
Appendix 215 explains why the light-bending factor of two should be traced more deeply to the two-aspect stress of a null Maxwell probe, and appendix 216 derives the same weak exterior factor directly from the sign-symmetric axial loading of a static toroidal closure. The present appendix does not replace that deeper derivation. It keeps the symmetric constitutive closure as the weak-field macroscopic writing of the same exterior mass-potential interaction and derives its static benchmark consequences.
The time-dependent radiative sector is not treated here.
Chapter 13 used the exterior mass-potential
\[ \eta(r):=\frac{GM}{rc^2}, \qquad \eta\ll 1. \]
This is the spherical unresolved exterior of a bounded mass closure, in direct analogy with the charge chapter’s far-field monopole summary. The weak-field constitutive writing used here is then
\[ \varepsilon_{\mathrm{eff}}(r)=\varepsilon_0\bigl(1+2\eta(r)\bigr), \qquad \mu_{\mathrm{eff}}(r)=\mu_0\bigl(1+2\eta(r)\bigr), \]
The corresponding local propagation speed is
\[ k(r) := \frac{1}{\sqrt{\varepsilon_{\mathrm{eff}}(r)\mu_{\mathrm{eff}}(r)}} = \frac{c}{1+2\eta(r)} = c\bigl(1-2\eta(r)\bigr)+O(\eta^2). \]
So this constitutive summary determines the local transport speed of weak electromagnetic disturbances in the background generated by the central mass.
At leading weak-field order in \(\eta\), the most general static spherically symmetric isotropic metric can be written as
\[ ds^2 = c^2\bigl(1+2A\eta(r)\bigr)\,dt^2 - \bigl(1+2B\eta(r)\bigr)\,\bigl(dr^2+r^2d\Omega^2\bigr) + O(\eta^2), \]
where \(A\) and \(B\) are constants still to be determined.
We now fix them by two conditions.
For a slowly moving bounded configuration, write the action
\[ S=-mc\int ds. \]
Using \(v^2=\dot r^2+r^2\dot\Omega^2\) and dividing by \(dt\), the Lagrangian is
\[ L = -mc^2 \sqrt{ \bigl(1+2A\eta\bigr) - \bigl(1+2B\eta\bigr)\frac{v^2}{c^2} } +O(\eta^2). \]
Expand to leading order in \(\eta\) and \(v^2/c^2\):
\[ L = -mc^2 \left( 1+A\eta-\frac{v^2}{2c^2} \right) +O\!\left(\eta\frac{v^2}{c^2},\,\eta^2,\,\frac{v^4}{c^4}\right). \]
Up to the irrelevant additive constant \(-mc^2\), this becomes
\[ L = \frac{1}{2}mv^2 -mAc^2\eta +O(c^{-2}). \]
Since
\[ c^2\eta=\frac{GM}{r}, \]
the effective potential is
\[ V(r)=mAc^2\eta=\frac{AGMm}{r}. \]
To recover the Newtonian attractive potential
\[ V_{\mathrm{N}}(r)=-\frac{GMm}{r}, \]
we must have
\[ A=-1. \]
For a radial null path, \(ds^2=0\) and \(d\Omega=0\), so
\[ 0 = c^2\bigl(1+2A\eta\bigr)\,dt^2 - \bigl(1+2B\eta\bigr)\,dr^2. \]
Hence the radial coordinate speed is
\[ \frac{dr}{dt} = c\sqrt{\frac{1+2A\eta}{1+2B\eta}} = c\bigl(1+(A-B)\eta\bigr)+O(\eta^2). \]
But the constitutive closure already fixed the transport speed to be
\[ k(r)=c\bigl(1-2\eta\bigr)+O(\eta^2). \]
Therefore
\[ A-B=-2. \]
Since \(A=-1\), it follows that
\[ B=1. \]
The weak-field transport geometry selected by the adopted constitutive closure is therefore
\[ ds^2 = c^2\bigl(1-2\eta(r)\bigr)\,dt^2 - \bigl(1+2\eta(r)\bigr)\,\bigl(dr^2+r^2d\Omega^2\bigr) + O(\eta^2). \]
This is exactly the weak-field Schwarzschild metric in isotropic coordinates.
So the constitutive summary used in chapter 13 does not merely reproduce light bending. It fixes the full static weak-field metric at leading order.
For a stationary clock at radius \(r\), we have \(dr=d\Omega=0\), so
\[ d\tau = \sqrt{1-2\eta(r)}\,dt = \bigl(1-\eta(r)\bigr)\,dt+O(\eta^2). \]
Thus the proper time of a static localized mode runs more slowly deeper in the potential well.
In a static metric, the covector component \(p_t\) of a photon is conserved. The frequency measured by a static observer with four-velocity proportional to \(\partial_t\) is therefore proportional to \(1/\sqrt{g_{tt}}\). Hence
\[ \frac{\nu_{\mathrm{obs}}}{\nu_{\mathrm{em}}} = \sqrt{\frac{g_{tt}(r_{\mathrm{em}})}{g_{tt}(r_{\mathrm{obs}})}} = \sqrt{\frac{1-2\eta(r_{\mathrm{em}})}{1-2\eta(r_{\mathrm{obs}})}}. \]
At leading weak-field order,
\[ \frac{\Delta\nu}{\nu} := \frac{\nu_{\mathrm{obs}}-\nu_{\mathrm{em}}}{\nu_{\mathrm{em}}} = \eta(r_{\mathrm{obs}})-\eta(r_{\mathrm{em}}) +O(\eta^2). \]
If the observer is far away, \(\eta(r_{\mathrm{obs}})\approx 0\), then
\[ \frac{\Delta\nu}{\nu} = -\frac{GM}{r_{\mathrm{em}}c^2}. \]
So the signal is redshifted when it climbs out of the central field.
For null propagation in a static isotropic metric, the optical index is
\[ n(r) := \sqrt{\frac{1+2\eta(r)}{1-2\eta(r)}} = 1+2\eta(r)+O(\eta^2) = 1+\frac{2GM}{rc^2}+O(\eta^2). \]
This is exactly the index used heuristically in chapter 13, but it is now derived from the same weak-field metric that also yields redshift and orbital precession.
Take a ray passing the mass with impact parameter \(b\). At leading weak-field order, use the straight-line approximation
\[ r(z)=\sqrt{b^2+z^2}. \]
The transverse gradient of the refractive index is
\[ \partial_b n(r(z)) = \frac{d}{db} \left( 1+\frac{2GM}{c^2\sqrt{b^2+z^2}} \right) = -\frac{2GM\,b}{c^2(b^2+z^2)^{3/2}}. \]
The total bending magnitude is therefore
\[ \theta = \int_{-\infty}^{\infty}|\partial_b n|\,dz = \frac{2GM\,b}{c^2} \int_{-\infty}^{\infty}\frac{dz}{(b^2+z^2)^{3/2}}. \]
Since
\[ \int_{-\infty}^{\infty}\frac{dz}{(b^2+z^2)^{3/2}} = \frac{2}{b^2}, \]
we obtain
\[ \theta = \frac{4GM}{bc^2}. \]
This is the standard weak-field light-bending value.
For a null path,
\[ dt=\frac{n(r)}{c}\,ds. \]
So the coordinate travel time between two points \(A\) and \(B\) is
\[ T = \frac{1}{c}\int_A^B n(r)\,ds = \frac{1}{c}\int_A^B ds + \frac{2GM}{c^3}\int_A^B \frac{ds}{r} +O(\eta^2). \]
The first term is the flat-space travel time. The second is the gravitational delay.
Approximate the path by a straight line with impact parameter \(b\), coordinate \(z\), and endpoints at \(z=z_A\) and \(z=z_B\). Then
\[ r(z)=\sqrt{b^2+z^2}, \qquad ds=dz, \]
so
\[ \Delta T = \frac{2GM}{c^3} \int_{z_A}^{z_B}\frac{dz}{\sqrt{b^2+z^2}}. \]
Using
\[ \int \frac{dz}{\sqrt{b^2+z^2}} = \ln\!\left(z+\sqrt{b^2+z^2}\right), \]
we obtain
\[ \Delta T = \frac{2GM}{c^3} \ln\!\left( \frac{z_B+\sqrt{b^2+z_B^2}} {z_A+\sqrt{b^2+z_A^2}} \right). \]
Writing the endpoint radii as
\[ r_A=\sqrt{b^2+z_A^2}, \qquad r_B=\sqrt{b^2+z_B^2}, \]
and the coordinate separation as
\[ R=z_B-z_A, \]
this is the standard one-way Shapiro form
\[ \Delta T = \frac{2GM}{c^3} \ln\!\left( \frac{r_A+r_B+R}{r_A+r_B-R} \right). \]
The perihelion calculation is easiest in the equivalent areal-radius form of the same weak-field metric.
Define
\[ R:=r\bigl(1+\eta(r)\bigr)=r+\frac{GM}{c^2}. \]
At leading order in \(GM/(Rc^2)\), the metric becomes
\[ ds^2 = c^2\left(1-\frac{2GM}{Rc^2}\right)dt^2 - \left(1-\frac{2GM}{Rc^2}\right)^{-1}dR^2 - R^2d\Omega^2 + O(c^{-4}). \]
This is the standard weak-field Schwarzschild form, equivalent to the isotropic metric above at the order being kept.
Restrict to equatorial motion, \(\theta=\pi/2\), and parameterize the worldline by proper time \(\tau\). Write
\[ \alpha:=\frac{GM}{c^2}. \]
Then the metric is
\[ ds^2 = c^2\left(1-\frac{2\alpha}{R}\right)dt^2 - \left(1-\frac{2\alpha}{R}\right)^{-1}dR^2 - R^2d\phi^2. \]
The two conserved quantities are
\[ E := c^2\left(1-\frac{2\alpha}{R}\right)\dot t, \]
\[ h := R^2\dot\phi, \]
where a dot denotes \(d/d\tau\).
The timelike normalization condition is
\[ c^2 = c^2\left(1-\frac{2\alpha}{R}\right)\dot t^2 - \left(1-\frac{2\alpha}{R}\right)^{-1}\dot R^2 - R^2\dot\phi^2. \]
Substitute the conserved quantities:
\[ \dot t=\frac{E}{c^2\left(1-\frac{2\alpha}{R}\right)}, \qquad \dot\phi=\frac{h}{R^2}. \]
Then
\[ \dot R^2 = \frac{E^2}{c^2} - \left(1-\frac{2\alpha}{R}\right) \left(c^2+\frac{h^2}{R^2}\right). \]
Now set
\[ u(\phi):=\frac{1}{R}. \]
Since
\[ \dot R=\frac{dR}{d\phi}\dot\phi = \left(-\frac{u'}{u^2}\right)(hu^2) = -hu', \]
where a prime denotes \(d/d\phi\), the radial equation becomes
\[ h^2u'^2 = \frac{E^2}{c^2} - \bigl(1-2\alpha u\bigr)\bigl(c^2+h^2u^2\bigr). \]
Differentiate with respect to \(\phi\):
\[ 2h^2u'u'' = 2\alpha u'\bigl(c^2+h^2u^2\bigr) - \bigl(1-2\alpha u\bigr)(2h^2uu'). \]
Divide by \(2u'\):
\[ h^2u'' = \alpha\bigl(c^2+h^2u^2\bigr) - h^2u\bigl(1-2\alpha u\bigr). \]
Therefore
\[ h^2u'' = \alpha c^2 -h^2u +3\alpha h^2u^2, \]
or
\[ u''+u = \frac{\alpha c^2}{h^2} +3\alpha u^2. \]
Since \(\alpha c^2=GM\), this is
\[ u''+u = \frac{GM}{h^2} +\frac{3GM}{c^2}u^2. \]
The Newtonian orbit equation is the same without the final term. Its bound solution is
\[ u_0(\phi) = \frac{GM}{h^2}\bigl(1+e\cos\phi\bigr). \]
Now write
\[ u=u_0+u_1, \]
where \(u_1\) is first order in \(GM/c^2\). Substituting into
\[ u''+u = \frac{GM}{h^2} + \frac{3GM}{c^2}u^2 \]
and retaining only first order in \(GM/c^2\) gives
\[ u_1''+u_1 = \frac{3GM}{c^2}u_0^2. \]
Since
\[ u_0^2 = \left(\frac{GM}{h^2}\right)^2 \bigl(1+2e\cos\phi+e^2\cos^2\phi\bigr) \]
and
\[ \cos^2\phi=\frac{1+\cos 2\phi}{2}, \]
the forcing becomes
\[ u_1''+u_1 = \frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2 \left( 1+\frac{e^2}{2}+2e\cos\phi+\frac{e^2}{2}\cos 2\phi \right). \]
The constant term and the \(\cos 2\phi\) term change the detailed shape of the orbit but do not accumulate a secular phase shift. The resonant term
\[ 2e\,\frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2\cos\phi \]
does.
To isolate that secular part, solve
\[ y''+y=A\cos\phi. \]
A direct check shows that
\[ y_p(\phi)=\frac{A}{2}\phi\sin\phi \]
is a particular solution, because
\[ \left(\frac{A}{2}\phi\sin\phi\right)'' + \frac{A}{2}\phi\sin\phi = A\cos\phi. \]
Therefore the resonant contribution to \(u_1\) is
\[ u_{1,\mathrm{res}}(\phi) = \frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2 e\,\phi\sin\phi. \]
Now compare with a precessing ellipse
\[ \frac{GM}{h^2} \Bigl(1+e\cos((1-\delta)\phi)\Bigr), \qquad \delta\ll 1. \]
Using
\[ \cos((1-\delta)\phi) = \cos\phi+\delta\,\phi\sin\phi+O(\delta^2), \]
the secular correction produced by a precession \(\delta\) is
\[ \frac{GM}{h^2}e\,\delta\,\phi\sin\phi. \]
Matching this with \(u_{1,\mathrm{res}}\) gives
\[ \frac{GM}{h^2}e\,\delta = \frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2 e, \]
so
\[ \delta = \frac{3G^2M^2}{h^2c^2}. \]
For a Newtonian ellipse,
\[ h^2=GMa(1-e^2), \]
therefore
\[ \delta = \frac{3GM}{a(1-e^2)c^2}. \]
After one orbit, the perihelion advance is
\[ \Delta\omega = 2\pi\delta = \frac{6\pi GM}{a(1-e^2)c^2}. \]
This is the standard weak-field perihelion precession.
Starting from the symmetric constitutive summary
\[ \varepsilon_{\mathrm{eff}}=\varepsilon_0(1+2\eta), \qquad \mu_{\mathrm{eff}}=\mu_0(1+2\eta), \qquad \eta=\frac{GM}{rc^2}, \]
the local transport speed is
\[ k(r)=c\bigl(1-2\eta(r)\bigr)+O(\eta^2). \]
Combining that transport speed with the Newtonian slow-mode limit determines the unique leading weak-field static isotropic metric
\[ ds^2 = c^2(1-2\eta)\,dt^2 - (1+2\eta)\,(dr^2+r^2d\Omega^2) + O(\eta^2). \]
From that one weak-field closure follow:
So within the symmetric constitutive summary used here, chapter 13 no longer rests on light bending alone. The full static weak-field benchmark set is recovered from one and the same transport geometry. Appendices 215 and 216 then isolate the deeper factor-of-two point: a null electromagnetic probe carries two equal stress channels, and a static toroidal closure samples both through its axial line.
Appendix 207 derived the emergent hydrodynamic form in an approximately uniform region, where the local transport scale \(k\) could be treated as constant. This appendix extends that derivation to the case in which
\[ k=k(\mathbf{r},t) \]
varies across the resolved continuum.
The balance-law structure derived here is exact once two variable-background ingredients are adopted:
Appendix 214 derives the first of these exactly inside the symmetric constitutive closure used in the gravity chapters. What is not yet derived here is the full substrate-specific closure that fixes both objects from the underlying transport. The novelty is that once \(k\) varies, one must distinguish carefully between:
Let
\[ \beta(\mathbf{r},t):=\frac{1}{k(\mathbf{r},t)^2}. \]
At the resolved scale, keep the local energy density and energy flux
\[ u, \qquad \mathbf{S}, \]
with exact local energy continuity
\[ \partial_t u+\nabla\cdot\mathbf{S}=0. \]
Adopt, as the variable-background extension of the uniform-region relation, the local momentum density
\[ \mathbf{g}:=\beta\,\mathbf{S}=\frac{\mathbf{S}}{k^2}. \]
When the background varies, the resolved subsystem need not be momentum-closed by itself. Define the exact background-exchange density
\[ \mathbf{f}_{\mathrm{bg}}(\mathbf{r},t) \]
by the local balance law
\[ \partial_t\mathbf{g}-\nabla\cdot\mathbf{T} = \mathbf{f}_{\mathrm{bg}}. \]
This is not a new force law. It is the exact residual bookkeeping term once \(\mathbf{g}\) and \(\mathbf{T}\) are specified: whatever momentum is not balanced by the resolved stress transport is, by definition, momentum exchanged with the unresolved background constitutive organization.
In a uniform region,
\[ \nabla k=0, \qquad \partial_t k=0, \qquad \mathbf{f}_{\mathrm{bg}}=0, \]
and the derivation of appendix 207 is recovered.
Let \(\langle\cdot\rangle\) again denote averaging over a cell large compared to local closure structure and small compared to resolved macroscopic variation.
Assume, at the resolved scale, that averaging commutes with differentiation:
\[ \langle\partial_t f\rangle=\partial_t\langle f\rangle, \qquad \langle\partial_i f\rangle=\partial_i\langle f\rangle. \]
Then the exact coarse-grained energy balance is still
\[ \partial_t\langle u\rangle+\nabla\cdot\langle\mathbf{S}\rangle=0. \]
This remains source-free. Energy itself is still only redistributed.
The effective momentum balance is
\[ \partial_t\langle\mathbf{g}\rangle \;-\;\nabla\cdot\langle\mathbf{T}\rangle = \langle\mathbf{f}_{\mathrm{bg}}\rangle. \]
So the background enters through momentum exchange, not through failure of energy continuity.
When \(k\) varies, it is no longer enough to write
\[ \rho=\frac{\langle u\rangle}{k^2} \]
with one constant \(k\) pulled out of the cell.
The exact effective inertial density is instead
\[ \rho:=\langle \beta u\rangle = \left\langle \frac{u}{k^2}\right\rangle. \]
This should be distinguished from the coarse-grained stored-energy density
\[ \bar u:=\langle u\rangle. \]
Only \(\bar u\) obeys a source-free continuity equation. The quantity \(\rho\) is an effective inertial density, and when \(k\) varies it does not satisfy a source-free continuity law by itself.
Define the coarse-grained momentum density by
\[ \rho\mathbf{v}:=\langle\mathbf{g}\rangle = \left\langle\frac{\mathbf{S}}{k^2}\right\rangle. \]
In a slowly varying background, where \(k\) changes little across the cell, this reduces to
\[ \rho \approx \frac{\bar u}{k(\mathbf{X},t)^2}, \qquad \rho\mathbf{v} \approx \frac{\langle\mathbf{S}\rangle}{k(\mathbf{X},t)^2}, \]
so that
\[ \mathbf{v}\approx\frac{\langle\mathbf{S}\rangle}{\langle u\rangle}. \]
Thus the transport velocity remains the ratio of coarse-grained energy flux to coarse-grained energy density, but the effective inertial density now carries the background weighting.
Differentiate \(\rho=\langle\beta u\rangle\):
\[ \partial_t\rho = \left\langle \beta\,\partial_t u+u\,\partial_t\beta\right\rangle. \]
Also,
\[ \nabla\cdot(\rho\mathbf{v}) = \nabla\cdot\langle\beta\mathbf{S}\rangle = \left\langle \beta\,\nabla\cdot\mathbf{S}+\mathbf{S}\cdot\nabla\beta\right\rangle. \]
Add the two expressions and use \(\partial_t u+\nabla\cdot\mathbf{S}=0\):
\[ \partial_t\rho+\nabla\cdot(\rho\mathbf{v}) = \left\langle u\,\partial_t\beta+\mathbf{S}\cdot\nabla\beta \right\rangle. \]
Define the exact variable-background source term
\[ \sigma_k := \left\langle u\,\partial_t\beta+\mathbf{S}\cdot\nabla\beta \right\rangle. \]
Then
\[ \partial_t\rho+\nabla\cdot(\rho\mathbf{v})=\sigma_k. \]
This is the exact balance law for the effective inertial density once \(\rho=\langle u/k^2\rangle\) has been adopted.
Now write it directly in terms of \(k\). Since
\[ \beta=k^{-2}, \]
we have
\[ \partial_t\beta=-2\beta\,\partial_t\ln k, \qquad \nabla\beta=-2\beta\,\nabla\ln k. \]
So
\[ \sigma_k = -2\left\langle \beta\left(u\,\partial_t\ln k+\mathbf{S}\cdot\nabla\ln k\right) \right\rangle. \]
In a slowly varying resolved cell, this becomes
\[ \sigma_k \approx -2\rho\left(\partial_t+ \mathbf{v}\cdot\nabla\right)\ln k. \]
Therefore the continuity equation becomes
\[ \partial_t\rho+\nabla\cdot(\rho\mathbf{v}) \approx -2\rho\,D_t\ln k, \]
where
\[ D_t:=\partial_t+\mathbf{v}\cdot\nabla. \]
This equation says something precise:
when a transported configuration moves into a region where the local transport scale decreases, the same coarse-grained energy corresponds to a larger effective inertial density.
So the variable-\(k\) medium modifies inertia even before any constitutive stress assumption is made.
The coarse-grained momentum balance is
\[ \partial_t(\rho\mathbf{v}) \;-\;\nabla\cdot\langle\mathbf{T}\rangle = \mathbf{f}, \]
where
\[ \mathbf{f}:=\langle\mathbf{f}_{\mathrm{bg}}\rangle. \]
Define, exactly as before, the residual stress tensor
\[ \boldsymbol{\Sigma} := \rho\,\mathbf{v}\otimes\mathbf{v}-\langle\mathbf{T}\rangle. \]
Then
\[ \partial_t(\rho\mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) -\nabla\cdot\boldsymbol{\Sigma} = \mathbf{f}. \]
Decompose
\[ \boldsymbol{\Sigma}=p\,\mathbf{I}-\boldsymbol{\tau}, \]
with
\[ p:=\frac{1}{3}\operatorname{tr}(\boldsymbol{\Sigma}), \qquad \operatorname{tr}(\boldsymbol{\tau})=0. \]
Then the exact variable-background momentum equation is
\[ \partial_t(\rho\mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) = -\nabla p+\nabla\cdot\boldsymbol{\tau}+\mathbf{f}. \]
Expand the left-hand side:
\[ \partial_t(\rho\mathbf{v}) + \nabla\cdot(\rho\,\mathbf{v}\otimes\mathbf{v}) = \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) + \mathbf{v}\left(\partial_t\rho+\nabla\cdot(\rho\mathbf{v})\right). \]
Using the exact variable-background continuity equation,
\[ \partial_t\rho+\nabla\cdot(\rho\mathbf{v})=\sigma_k, \]
we obtain
\[ \rho\left(\partial_t\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p+\nabla\cdot\boldsymbol{\tau}+\mathbf{f}-\mathbf{v}\,\sigma_k. \]
This is the exact convective form in a variable background.
In the slowly varying resolved approximation,
\[ \rho D_t\mathbf{v} \approx -\nabla p+\nabla\cdot\boldsymbol{\tau} + \mathbf{f} + 2\rho\,\mathbf{v}\,D_t\ln k. \]
So the variable background enters in two distinct ways:
These two effects should not be conflated.
If the background is static, then
\[ \partial_t k=0, \]
and the continuity source reduces to
\[ \sigma_k = \langle \mathbf{S}\cdot\nabla\beta\rangle \approx -2\rho\,\mathbf{v}\cdot\nabla\ln k. \]
So
\[ \partial_t\rho+\nabla\cdot(\rho\mathbf{v}) \approx -2\rho\,\mathbf{v}\cdot\nabla\ln k. \]
If, in addition, the background momentum exchange is potential-like, write
\[ \mathbf{f}=-\rho\,\nabla\Phi_k. \]
Then
\[ \rho D_t\mathbf{v} \approx -\nabla p+\nabla\cdot\boldsymbol{\tau} -\rho\,\nabla\Phi_k + 2\rho\,\mathbf{v}\,\mathbf{v}\cdot\nabla\ln k. \]
For the gravity closure of appendix 212, the weak-field metric gave
\[ g_{tt}=\frac{k}{c}+O(\eta^2), \]
so the corresponding slow-mode potential is
\[ \Phi_k := \frac{c^2}{2}\ln\frac{k}{c}. \]
Since
\[ k=c(1-2\eta)+O(\eta^2), \]
we obtain
\[ \Phi_k = -c^2\eta+O(\eta^2) = -\frac{GM}{r}+O(\eta^2), \]
and therefore
\[ -\nabla\Phi_k = -\frac{GM}{r^2}\,\hat{\mathbf r}+O(\eta^2), \]
which is the Newtonian gravitational acceleration.
So the variable-background hydrodynamic form contains the gravity closure of appendix 212 as one special constitutive case.
The exact equations above reduce to the familiar forms once the unresolved stress is closed.
If
\[ \boldsymbol{\tau}=0, \]
then
\[ \rho D_t\mathbf{v} = -\nabla p+\mathbf{f}-\mathbf{v}\sigma_k. \]
This is the variable-background Euler form.
If the deviatoric stress is approximated by the Newtonian constitutive form
\[ \boldsymbol{\tau} = \eta_v\left(\nabla\mathbf{v}+(\nabla\mathbf{v})^{\mathsf T} -\frac{2}{3}(\nabla\cdot\mathbf{v})\mathbf{I}\right) + \zeta_v(\nabla\cdot\mathbf{v})\mathbf{I}, \]
then
\[ \rho D_t\mathbf{v} = -\nabla p+\nabla\cdot\boldsymbol{\tau} + \mathbf{f} -\mathbf{v}\sigma_k. \]
This is the variable-background Navier-Stokes-like form.
Compared with the uniform-region case, the new terms are exactly those tied to
Up to the introduction of the constitutive form of \(\boldsymbol{\tau}\), the derivation is exact once
\[ \mathbf{f}_{\mathrm{bg}} \]
is defined as the exact residual momentum exchange with the unresolved background.
So the exact conclusions are:
What remains constitutive is:
When the local transport scale varies,
\[ k=k(\mathbf{r},t), \]
the exact coarse-grained effective inertial density is
\[ \rho=\left\langle\frac{u}{k^2}\right\rangle, \]
and the exact momentum density is
\[ \rho\mathbf{v} = \left\langle\frac{\mathbf{S}}{k^2}\right\rangle. \]
They satisfy
\[ \partial_t\rho+\nabla\cdot(\rho\mathbf{v})=\sigma_k, \]
with
\[ \sigma_k = \left\langle u\,\partial_t(k^{-2})+\mathbf{S}\cdot\nabla(k^{-2}) \right\rangle, \]
and
\[ \rho D_t\mathbf{v} = -\nabla p+\nabla\cdot\boldsymbol{\tau} + \mathbf{f} -\mathbf{v}\sigma_k. \]
Thus the variable-background hydrodynamic limit is no longer just the uniform-region Euler/Navier-Stokes form with \(k(\mathbf r,t)\) inserted by hand.
It has a definite new structure:
This is a consistent variable-background extension of appendix 207 within the matched constitutive relation \(\mathbf{g}=\mathbf{S}/k^2\), derived in appendix 214 for the symmetric closure and otherwise still to be fixed by the chosen background structure.
Appendix 213 used the variable-background relation
\[ \mathbf{g}=\frac{\mathbf{S}}{k^2} \]
as an adopted extension of the uniform-region momentum density.
This appendix derives that relation exactly for the specific constitutive closure already used in the gravity chapters, once the resolved transport momentum is taken to be the constitutive momentum density
\[ \mathbf{g}:=\mathbf{D}\times\mathbf{B}. \]
With that choice, the relation used in appendix 213 follows identically inside this constitutive class:
\[ \varepsilon(\mathbf{r},t)=\varepsilon_0\,\alpha(\mathbf{r},t), \qquad \mu(\mathbf{r},t)=\mu_0\,\alpha(\mathbf{r},t), \]
with
\[ \alpha=\frac{c}{k}. \]
It also derives the exact energy-exchange term for time-dependent background and the leading background force for radiative transport in the geometric-optics limit.
What it does not derive is the full exact background force for arbitrary resolved field configurations. That remains open.
Take a source-free linear isotropic medium with constitutive relations
\[ \mathbf{D}=\varepsilon\,\mathbf{E}, \qquad \mathbf{B}=\mu\,\mathbf{H}, \]
and assume the symmetric constitutive scaling
\[ \varepsilon=\varepsilon_0\,\alpha, \qquad \mu=\mu_0\,\alpha. \]
Then the local transport speed is
\[ k=\frac{1}{\sqrt{\varepsilon\mu}} = \frac{1}{\sqrt{\varepsilon_0\mu_0}}\frac{1}{\alpha} = \frac{c}{\alpha}. \]
The local impedance is
\[ Z=\sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{\mu_0}{\varepsilon_0}} = Z_0. \]
So this closure changes the propagation speed while keeping the impedance fixed.
The Poynting vector is
\[ \mathbf{S}=\mathbf{E}\times\mathbf{H}. \]
Take the resolved transport momentum density to be
\[ \mathbf{g}:=\mathbf{D}\times\mathbf{B}. \]
Under the symmetric closure,
\[ \mathbf{g} = (\varepsilon_0\alpha\mathbf{E})\times\mathbf{B}. \]
Since
\[ \mathbf{H}=\frac{\mathbf{B}}{\mu} = \frac{\mathbf{B}}{\mu_0\alpha}, \]
we have
\[ \mathbf{S} = \mathbf{E}\times\mathbf{H} = \frac{1}{\mu_0\alpha}\,\mathbf{E}\times\mathbf{B}. \]
Therefore
\[ \mathbf{E}\times\mathbf{B} = \mu_0\alpha\,\mathbf{S}, \]
and hence
\[ \mathbf{g} = \varepsilon_0\mu_0\alpha^2\,\mathbf{S} = \frac{\alpha^2}{c^2}\,\mathbf{S}. \]
But
\[ \alpha=\frac{c}{k}, \]
so
\[ \frac{\alpha^2}{c^2}=\frac{1}{k^2}. \]
Therefore
\[ \boxed{ \mathbf{g}=\frac{\mathbf{S}}{k^2} }. \]
This is exact for the adopted symmetric constitutive closure.
So appendix 213’s background-weighted momentum density is not arbitrary inside this constitutive class. It is exactly the constitutive momentum density \(\mathbf D\times\mathbf B\).
The electromagnetic energy density is
\[ u := \frac{1}{2}\bigl(\mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B}\bigr) = \frac{1}{2}\bigl(\varepsilon E^2+\mu H^2\bigr). \]
Maxwell’s equations in the source-free medium are
\[ \nabla\cdot\mathbf{D}=0, \qquad \nabla\cdot\mathbf{B}=0, \]
\[ \nabla\times\mathbf{E}=-\partial_t\mathbf{B}, \qquad \nabla\times\mathbf{H}=\partial_t\mathbf{D}. \]
Take the scalar product of the second curl equation with \(\mathbf E\) and of the first with \(\mathbf H\), then subtract:
\[ \mathbf E\cdot(\nabla\times\mathbf H)-\mathbf H\cdot(\nabla\times\mathbf E) = \mathbf E\cdot\partial_t\mathbf D+\mathbf H\cdot\partial_t\mathbf B. \]
Using
\[ \nabla\cdot(\mathbf E\times\mathbf H) = \mathbf H\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf H), \]
this becomes
\[ \partial_t u+\nabla\cdot\mathbf S = -\frac{1}{2}\bigl(E^2\,\partial_t\varepsilon+H^2\,\partial_t\mu\bigr). \]
For the symmetric constitutive closure,
\[ \partial_t\varepsilon=\varepsilon_0\,\partial_t\alpha, \qquad \partial_t\mu=\mu_0\,\partial_t\alpha. \]
Also,
\[ u = \frac{\alpha}{2}\bigl(\varepsilon_0E^2+\mu_0H^2\bigr). \]
So the right-hand side becomes
\[ -\frac{1}{2}\bigl(\varepsilon_0E^2+\mu_0H^2\bigr)\partial_t\alpha = -\frac{u}{\alpha}\,\partial_t\alpha. \]
Therefore
\[ \partial_t u+\nabla\cdot\mathbf S = -u\,\partial_t\ln\alpha. \]
Since \(\alpha=c/k\),
\[ \partial_t\ln\alpha=-\partial_t\ln k, \]
and hence
\[ \boxed{ \partial_t u+\nabla\cdot\mathbf S = u\,\partial_t\ln k }. \]
This is the exact energy balance for the time-dependent symmetric constitutive closure.
Consequences:
So a time-dependent background already breaks the stronger claim that the resolved subsystem always has source-free energy continuity.
To derive the leading background force for transport itself, restrict to the geometric-optics regime of a narrow radiative packet.
The local dispersion relation is
\[ \omega(\mathbf{r},\mathbf{q},t)=k(\mathbf{r},t)\,|\mathbf{q}|. \]
Treat this as the packet Hamiltonian
\[ H(\mathbf{r},\mathbf{p},t)=k(\mathbf{r},t)\,|\mathbf{p}|. \]
Hamilton’s equations are
\[ \dot{\mathbf{r}}=\nabla_{\mathbf p}H = k\,\frac{\mathbf p}{|\mathbf p|}, \]
\[ \dot{\mathbf p}=-\nabla_{\mathbf r}H = -|\mathbf p|\,\nabla k. \]
The packet energy is
\[ U=H=k|\mathbf p|, \]
so
\[ |\mathbf p|=\frac{U}{k}. \]
Therefore
\[ \boxed{ \dot{\mathbf p} = -\frac{U}{k}\,\nabla k = -U\,\nabla\ln k }. \]
This is the leading force on a radiative packet due to spatial variation of the local transport speed.
Likewise the energy changes as
\[ \dot U=\partial_t H=|\mathbf p|\,\partial_t k=\frac{U}{k}\partial_t k, \]
so
\[ \boxed{ \dot U=U\,\partial_t\ln k }. \]
This is exactly the packet version of the local energy-exchange law derived above.
For a narrow radiative packet with local energy density \(u\) and negligible internal stress compared to the background gradient scale, the geometric-optics force density is
\[ \boxed{ \mathbf{f}_{\mathrm{rad}} = -u\,\nabla\ln k }. \]
Equivalently, using \(\mathbf g=\mathbf S/k^2\) and \(|\mathbf S|=uk\) for pure radiation,
\[ \mathbf{f}_{\mathrm{rad}} = -k\,|\mathbf g|\,\nabla\ln k. \]
This formula is not the full exact background force for arbitrary resolved field configurations. It is the leading transport-force density for radiative packets in the geometric-optics limit.
So the situation is now clear:
Appendix 212 used the same symmetric constitutive closure to derive the static weak-field metric and the corresponding benchmark observables.
Appendix 213 used
\[ \mathbf g=\frac{\mathbf S}{k^2} \]
as the adopted variable-background momentum relation inside the hydrodynamic balance-law extension.
The present appendix now clarifies the scope:
For the symmetric constitutive closure
\[ \varepsilon=\varepsilon_0\,\frac{c}{k}, \qquad \mu=\mu_0\,\frac{c}{k}, \]
the exact field momentum density is
\[ \mathbf g=\mathbf D\times\mathbf B=\frac{\mathbf S}{k^2}. \]
The exact energy balance is
\[ \partial_t u+\nabla\cdot\mathbf S=u\,\partial_t\ln k. \]
So:
For radiative packets in geometric optics, the background force is
\[ \mathbf f_{\mathrm{rad}}=-u\,\nabla\ln k. \]
This provides the constitutive origin of the background-weighted momentum used in appendix 213 and identifies the next real open problem:
derive the full exact background force and stress exchange for general variable-background field configurations, not only for radiative transport.
Chapter 13 and appendix 212 used a symmetric constitutive summary to recover the weak-field light-bending value
\[ \theta=\frac{4GM}{bc^2}. \]
That summary is useful, but it should not be mistaken for the deepest explanation of the factor of two. The deeper point is that the probe is a null Maxwell mode. It carries two equal aspects, electric and magnetic, and a static toroidal closure must sample both when it loads the probe along its axial transport line.
This appendix isolates exactly what is already forced by that fact and what still remains open.
Take a narrow electromagnetic probe in a region that is approximately uniform over one wavelength. Write its fields locally as
\[ \mathbf{E}, \qquad \mathbf{B}, \]
with propagation direction \(\mathbf{n}\) and local transport speed \(k\).
For a null electromagnetic mode,
\[ \mathbf{n}\cdot\mathbf{E}=0, \qquad \mathbf{n}\cdot\mathbf{B}=0, \qquad \mathbf{B}=\frac{1}{k}\,\mathbf{n}\times\mathbf{E}. \]
This means the probe has no trapped rest component. Its transport is fully carried along \(\mathbf n\), so locally
\[ |\mathbf S|=ku. \]
Its energy density splits into electric and magnetic pieces:
\[ u=u_E+u_B, \]
\[ u_E=\frac{1}{2}\varepsilon E^2, \qquad u_B=\frac{1}{2\mu}B^2. \]
If the local impedance is
\[ Z=\sqrt{\frac{\mu}{\varepsilon}}, \]
then
\[ |\mathbf{H}|=\frac{|\mathbf{E}|}{Z}, \qquad \mathbf{B}=\mu\mathbf{H}, \]
and therefore
\[ u_B = \frac{1}{2}\mu H^2 = \frac{1}{2}\mu\frac{E^2}{Z^2} = \frac{1}{2}\mu E^2\frac{\varepsilon}{\mu} = \frac{1}{2}\varepsilon E^2 = u_E. \]
So for any null Maxwell mode,
\[ \boxed{ u_E=u_B=\frac{u}{2} }. \]
Now split the Maxwell stress tensor into electric and magnetic pieces:
\[ T_{ij}=T^{(E)}_{ij}+T^{(B)}_{ij}, \]
with
\[ T^{(E)}_{ij} = \varepsilon\left(E_iE_j-\frac{1}{2}\delta_{ij}E^2\right), \]
\[ T^{(B)}_{ij} = \frac{1}{\mu}\left(B_iB_j-\frac{1}{2}\delta_{ij}B^2\right). \]
Choose local coordinates so the probe moves in the \(+z\) direction. Then one may take
\[ \mathbf{E}=(E,0,0), \qquad \mathbf{B}=\left(0,\frac{E}{k},0\right). \]
The longitudinal momentum-flux component is \(-T_{zz}\). For the electric part,
\[ -T^{(E)}_{zz} = \frac{1}{2}\varepsilon E^2 = u_E. \]
For the magnetic part,
\[ -T^{(B)}_{zz} = \frac{1}{2\mu}B^2 = u_B. \]
Hence
\[ \boxed{ -T_{zz}=u_E+u_B=u }, \]
and the electric and magnetic sectors contribute equally to the longitudinal transport stress of the probe.
That equality is exact. It does not depend on weak gravity. It is a structural fact about null Maxwell transport.
Now consider a weak static bounded mass mode. The result above means that any correct interaction cannot treat the probe as one channel only.
The remaining question is not whether there are two equal sectors. That is already forced. The remaining question is how a static bounded closure samples them.
Appendix 216 answers that directly. A static toroidal closure interacts through its axial line, and because it is static it samples the two opposite axial directions symmetrically. When that symmetric axial load is written in terms of the probe transport data, it becomes
\[ u+\Pi_n. \]
For a null Maxwell probe, \(\Pi_n=u\), so the static closure sees
\[ u+\Pi_n=2u. \]
That is the structural origin of the factor of two.
Write the total fields as linear superposition of background and probe:
\[ \mathbf{E}=\mathbf{E}_1+\mathbf{E}_2, \qquad \mathbf{B}=\mathbf{B}_1+\mathbf{B}_2. \]
Then the cross part of the Maxwell stress tensor is
\[ (T_\times)_{ij} = \varepsilon_0\left(E_{1i}E_{2j}+E_{2i}E_{1j} -\delta_{ij}\mathbf{E}_1\cdot\mathbf{E}_2\right) + \frac{1}{\mu_0}\left(B_{1i}B_{2j}+B_{2i}B_{1j} -\delta_{ij}\mathbf{B}_1\cdot\mathbf{B}_2\right). \]
This identity is exact, but by itself it does not complete the weak-field gravity derivation.
There are two reasons.
First, if the background is modeled as purely static and electric at leading order, then
\[ \mathbf{B}_1\approx 0, \]
so the explicit magnetic part of the raw cross stress vanishes at that order. The second half of the factor of two is therefore not sitting in the formula as an extra \(B_1B_2\) term waiting to be read off.
Second, linear Maxwell superposition in vacuum does not make one source-free solution bend another. The missing object is the actual interaction law by which the bounded mass closure and the passing null probe reorganize one common field.
So the exact role of the argument above is not to claim that raw vacuum superposition already gives the full interaction law. Its role is sharper:
Chapter 13 and appendix 212 summarized the weak interaction by the symmetric constitutive modification
\[ \varepsilon_{\mathrm{eff}}=\varepsilon_0(1+2\eta), \qquad \mu_{\mathrm{eff}}=\mu_0(1+2\eta). \]
That summary should now be read more carefully.
It is not the deepest explanation of the factor of two. Rather, it is the macroscopic summary of an interaction law that must, if it is correct, act equally on the two stress sectors of a null Maxwell probe.
So the logical order is:
Appendix 216 now completes the weak exterior factor-of-two derivation by doing exactly that axial sampling step. What still remains open is not the factor of two itself, but the full exact interaction beyond the weak exterior regime:
For a null Maxwell probe:
\[ u_E=u_B=\frac{u}{2}, \]
and the electric and magnetic sectors contribute equally to the longitudinal transport stress.
Therefore any admissible leading weak-field interaction term that treats the two Maxwell aspects symmetrically must produce
\[ \text{full null interaction} = 2\times\text{one-channel effect}. \]
So the weak-field factor of two should not be explained by arbitrary constitutive symmetry. It belongs more deeply to the two-aspect stress structure of the probe itself and to the sign-symmetric axial loading by a static toroidal closure.
Appendix 216 completes that weak exterior derivation directly from axial transport.
Appendix 215 established that a null Maxwell probe carries two equal stress sectors. That alone explains why a one-channel account gives only half the full bending, but it does not yet say how a static bounded mass closure samples those two sectors.
This appendix derives that missing step directly from axial transport.
The key geometric point is simple. A compact toroidal closure interacts through its axial line. For a static closure there is no preferred sign along that line, so the leading weak exterior interaction must sample the probe through both axial directions equally. When that symmetric axial load is written in terms of the probe’s energy-momentum tensor, it is exactly
\[ u+\Pi_n, \]
where \(u\) is the probe energy density and \(\Pi_n\) is its longitudinal momentum flux. For a slow bounded mode this reduces to \(u\). For a null Maxwell probe it becomes \(2u\). The weak-field factor of two therefore follows from axial transport itself, not from a later adjustment.
Work in the rest frame of the static bounded mass closure.
For a narrow probe packet, write its local energy density, momentum density, and spatial stress as
\[ u, \qquad \mathbf g, \qquad T_{ij}, \]
with
\[ \mathbf g=\frac{\mathbf S}{c^2}. \]
Introduce the corresponding energy-momentum tensor
\[ \Theta^{00}=u, \qquad \Theta^{0i}=c\,g_i, \qquad \Theta^{ij}=-T_{ij}. \]
Let
\[ \mathbf n \]
be the local transport direction of the narrow probe, with
\[ |\mathbf n|=1. \]
Define the longitudinal momentum-flux density by
\[ \Pi_n:=-n_in_jT_{ij}. \]
For a null Maxwell probe, appendix 215 gives
\[ \Pi_n=u. \]
Equivalently,
\[ |\mathbf S|=cu, \]
so the probe carries no trapped rest component. It is transport all the way through.
At a point outside a static toroidal closure, the leading interaction is carried along the local axial line. Because the closure is static, that line has no preferred sign. It therefore presents two opposite transport channels, forward and backward along \(\mathbf n\).
Introduce the two null axial directions
\[ k_+^\mu:=(1,\mathbf n), \qquad k_-^\mu:=(1,-\mathbf n). \]
Their loads against the probe energy-momentum tensor are
\[ \Theta_{\mu\nu}k_+^\mu k_+^\nu, \qquad \Theta_{\mu\nu}k_-^\mu k_-^\nu. \]
The sign-symmetric axial load seen by the static closure is therefore
\[ \Lambda_n := \frac{1}{2}\Theta_{\mu\nu}k_+^\mu k_+^\nu + \frac{1}{2}\Theta_{\mu\nu}k_-^\mu k_-^\nu. \]
Expanding in the closure rest frame gives
\[ \Theta_{\mu\nu}k_+^\mu k_+^\nu = u+2c\,\mathbf g\cdot\mathbf n+\Pi_n, \]
\[ \Theta_{\mu\nu}k_-^\mu k_-^\nu = u-2c\,\mathbf g\cdot\mathbf n+\Pi_n. \]
Adding and dividing by two, the mixed momentum terms cancel exactly:
\[ \boxed{ \Lambda_n=u+\Pi_n }. \]
This identity is exact. No weak-field approximation has been used yet.
It says that a static closure samples two things at once:
That is the flow-theoretic origin of the doubling.
For a slowly moving bounded probe, the longitudinal momentum flux is smaller by order \(v^2/c^2\), so in the strict slow-mode limit
\[ \Pi_n=o(u), \]
and therefore
\[ \Lambda_n=u+o(u). \]
So a slow bounded mode loads the static closure essentially by its stored energy density alone.
For a null Maxwell probe, appendix 215 gives
\[ \Pi_n=u, \]
hence
\[ \boxed{ \Lambda_n=2u }. \]
So the same static closure sees twice the axial load from a null probe that it would see from a one-channel or slow-mode treatment of the same energy.
This is the exact weak-field factor-of-two point.
Let the static mass closure generate the exterior strength
\[ \eta(\mathbf r):=\frac{GM}{rc^2}, \qquad r=|\mathbf r|. \]
In the weak exterior regime, the leading local interaction must be:
Under these conditions, the unique axial scalar carried by the probe is
\[ \Lambda_n=u+\Pi_n. \]
Therefore the leading local interaction energy density is
\[ \boxed{ w_{\mathrm{int}}=-\eta\,\Lambda_n = -\eta\,(u+\Pi_n) }. \]
This is not an arbitrary fit. The overall unit is already fixed by the definition of \(\eta\) through the Newtonian slow-mode limit. The only question was what scalar of the probe a static axial closure must sample. The answer is the symmetric two-channel load \(\Lambda_n\).
For a narrow packet centered at \(\mathbf X(t)\), with support small compared to the background scale, we may treat \(\eta\) as constant across the packet to leading order:
\[ U_{\mathrm{int}}(t) = \int w_{\mathrm{int}}\,dV = -\eta(\mathbf X(t)) \int \Lambda_n\,dV. \]
Define the total axial load of the packet by
\[ L_n:=\int \Lambda_n\,dV. \]
Then
\[ U_{\mathrm{int}}(t)=-\eta(\mathbf X(t))\,L_n. \]
For a null Maxwell probe,
\[ \Lambda_n=2u, \]
so
\[ L_n=2U, \qquad U:=\int u\,dV. \]
Hence
\[ U_{\mathrm{int}}(t) = -2U\,\eta(\mathbf X(t)). \]
The transverse force on the packet is the negative gradient of the interaction energy:
\[ \mathbf F_\perp := -\nabla_\perp U_{\mathrm{int}} = 2U\,\nabla_\perp\eta. \]
Because
\[ \eta(r)=\frac{GM}{rc^2}, \]
the vector \(\nabla\eta\) points inward, so the force is attractive.
The momentum magnitude of the null packet is
\[ P=\frac{U}{c}. \]
Therefore the infinitesimal change of direction is
\[ d\theta = \frac{|\mathbf F_\perp|}{P}\,dt = 2c\,|\nabla_\perp\eta|\,dt. \]
Along the unperturbed straight path,
\[ dt=\frac{dz}{c}, \]
so
\[ d\theta = 2|\nabla_\perp\eta|\,dz. \]
Take the ray to pass the mass with impact parameter \(b\). Then
\[ r(z)=\sqrt{b^2+z^2}, \]
and
\[ |\nabla_\perp\eta| = \frac{GM\,b}{c^2(b^2+z^2)^{3/2}}. \]
Hence
\[ \theta = 2\int_{-\infty}^{\infty} \frac{GM\,b}{c^2(b^2+z^2)^{3/2}}\,dz = \frac{2GM\,b}{c^2} \int_{-\infty}^{\infty}\frac{2\,dz}{(b^2+z^2)^{3/2}}. \]
Using
\[ \int_{-\infty}^{\infty}\frac{dz}{(b^2+z^2)^{3/2}} = \frac{2}{b^2}, \]
we obtain
\[ \boxed{ \theta=\frac{4GM}{bc^2} }. \]
So the full weak-field light-bending value is derived here directly from the axial two-channel loading of a null Maxwell probe.
If one keeps only the slow-mode channel, one uses
\[ \Lambda_n\approx u \]
instead of
\[ \Lambda_n=u+\Pi_n. \]
Then the interaction energy would be
\[ U_{\mathrm{int}}^{\mathrm{one\ channel}} = -U\,\eta, \]
and the same calculation would give
\[ \theta_{\mathrm{one\ channel}} = \frac{2GM}{bc^2}. \]
So the Newtonian half-value is not mysterious. It is simply the result of counting only one axial loading channel of the probe.
The full null value appears when the static closure is allowed to sample the probe through both axial directions, as a toroidal same-substrate interaction must.
The factor of two is no longer an arbitrary constitutive choice, and it is no longer hidden in an undetermined coefficient. It has been derived from:
What remains open is not the factor of two itself. What remains open is the full exact interaction beyond the weak exterior regime:
For a narrow probe with transport direction \(\mathbf n\), the sign-symmetric axial load seen by a static closure is
\[ \Lambda_n = \frac{1}{2}\Theta_{\mu\nu}k_+^\mu k_+^\nu + \frac{1}{2}\Theta_{\mu\nu}k_-^\mu k_-^\nu = u+\Pi_n. \]
For a slow bounded mode, this reduces to
\[ \Lambda_n\approx u. \]
For a null Maxwell probe,
\[ \Lambda_n=2u. \]
Therefore the weak exterior interaction energy is
\[ U_{\mathrm{int}}=-\eta L_n, \]
and the resulting null deflection is exactly
\[ \theta=\frac{4GM}{bc^2}. \]
So the weak-field factor of two is derived here from first principles of flow: a static toroidal closure samples both axial transport channels of the passing null probe.
The earlier chapters already derived the transport core. Once one further step is admitted, namely that Maxwellian transport can organize into a bounded toroidal mode, the structures usually postulated in string descriptions follow directly.
This appendix proves the exact part of that statement. It does not assume a fundamental string ontology. It shows that a bounded thin-tube Maxwellian mode already carries:
In that precise sense, string-theoretic structure is an effective description of bounded electromagnetic topology.
Let
\[ X : \mathbb{R}/L\mathbb{Z} \to \mathbb{R}^3 \]
be a smooth closed curve parameterized by arclength \(s\), so
\[ |X'(s)| = 1, \qquad \tau(s) := X'(s). \]
For sufficiently small \(\varepsilon > 0\), let \(N_\varepsilon(X)\) be the thin tube about \(X\), and let \(\Sigma_\varepsilon(s)\) denote the transverse cross-section orthogonal to \(\tau(s)\) at \(X(s)\).
Assume a bounded Maxwellian mode is concentrated in that tube. Write its energy density and flux as
\[ u_\varepsilon(x,t), \qquad S_\varepsilon(x,t). \]
Assume further that the transporting part of the mode is tangent to the tube, so in the thin-tube limit
\[ S_\varepsilon(x,t) = k\,u_\varepsilon(x,t)\,\tau(s) + O(\varepsilon), \]
where \(k\) is the local transport speed of the background region. In vacuum, \(k=c\).
Suppose the thin tube lies on an invariant torus. A torus has two independent non-contractible cycles. Any closed tangent transport line on that torus is therefore labeled by an integer pair
\[ (m,n) \in \mathbb{Z}^2, \]
counting its windings about those two cycles.
This is the same topological statement established in chapter 8. The present point is only that, once the bounded mode is reduced to a thin closed tube, those integers are already the winding numbers of an effective string.
Define the line energy density of the tube by integrating the energy density over each transverse section:
\[ \mathcal{T}_\varepsilon(s,t) := \int_{\Sigma_\varepsilon(s)} u_\varepsilon(x,t)\,dA. \]
The total energy contained in the tube is then
\[ E_\varepsilon(t) = \int_0^L \mathcal{T}_\varepsilon(s,t)\,ds. \]
In the effective one-dimensional description, energy per unit length is exactly the string tension. So define
\[ \mathcal{T}(s,t) := \lim_{\varepsilon\to 0}\mathcal{T}_\varepsilon(s,t). \]
If the closed mode is approximately uniform along the tube, then
\[ \mathcal{T}(s,t) = \mathcal{T}, \]
and therefore
\[ E = \mathcal{T} L. \]
So the effective tension is not postulated. It is the line reduction of the Maxwellian energy density.
Define the momentum density by
\[ g_\varepsilon := \frac{S_\varepsilon}{k^2}. \]
Its tangential line density is
\[ \mathcal{P}_\varepsilon(s,t) := \int_{\Sigma_\varepsilon(s)} g_\varepsilon(x,t)\cdot\tau(s)\,dA. \]
Using the tangential transport relation from section 217.1,
\[ g_\varepsilon(x,t)\cdot\tau(s) = \frac{u_\varepsilon(x,t)}{k} + O(\varepsilon), \]
so
\[ \mathcal{P}_\varepsilon(s,t) = \frac{1}{k}\mathcal{T}_\varepsilon(s,t) + O(\varepsilon). \]
Passing to the thin-tube limit gives
\[ \mathcal{P}(s,t)=\frac{\mathcal{T}(s,t)}{k}. \]
The corresponding inertial line density is therefore
\[ \mu(s,t) := \frac{\mathcal{P}(s,t)}{k} = \frac{\mathcal{T}(s,t)}{k^2}. \]
In vacuum,
\[ \mu = \frac{\mathcal{T}}{c^2}. \]
So the effective inertial density is also not postulated. It is fixed by the Maxwellian momentum density of the bounded mode.
Now consider a small transverse displacement
\[ \xi(s,t) \]
of the effective line. Assume the unperturbed closed mode is uniform enough that \(\mathcal{T}\) and \(\mu\) may be treated as constants to leading order.
Take a short segment \([s,s+ds]\). The tension vectors at its endpoints are
\[ \mathcal{T}\,\partial_s(X+\xi)(s,t), \qquad \mathcal{T}\,\partial_s(X+\xi)(s+ds,t). \]
Subtracting and keeping the leading transverse term yields the net transverse force
\[ dF_\perp = \mathcal{T}\,\partial_s^2 \xi(s,t)\,ds. \]
The inertial mass of that segment is
\[ dm = \mu\,ds, \]
so momentum balance gives
\[ \mu\,\partial_t^2 \xi = \mathcal{T}\,\partial_s^2 \xi. \]
Using
\[ \mu=\frac{\mathcal{T}}{k^2}, \]
we obtain
\[ \partial_t^2 \xi - k^2 \partial_s^2 \xi = 0. \]
So the effective one-dimensional disturbance speed is exactly the underlying Maxwellian transport speed.
Appendix 214 already derived the geometric-optics transport law in a static background speed field \(k(x)\). For a narrow radiative packet with Hamiltonian
\[ H(x,p)=k(x)\,|p|, \]
Hamilton’s equations are
\[ \dot{x}=k\,\frac{p}{|p|}, \qquad \dot{p}=-|p|\,\nabla k. \]
Let
\[ \hat{\mathbf t}:=\frac{p}{|p|} \]
be the unit transport tangent. Then
\[ \dot{x}=k\,\hat{\mathbf t}. \]
Write
\[ p=|p|\,\hat{\mathbf t}. \]
Differentiating gives
\[ \dot{p}=\dot{|p|}\,\hat{\mathbf t}+|p|\,\dot{\hat{\mathbf t}}. \]
Project this orthogonally to \(\hat{\mathbf t}\). Since the tangential part drops out,
\[ |p|\,\dot{\hat{\mathbf t}} = -|p|\,\nabla_\perp k, \]
where
\[ \nabla_\perp k := \nabla k-(\hat{\mathbf t}\cdot\nabla k)\hat{\mathbf t} \]
is the transverse gradient. Therefore
\[ \dot{\hat{\mathbf t}}=-\nabla_\perp k. \]
Now parametrize the ray by arclength \(s\). Since
\[ \dot{x}=k\,\hat{\mathbf t}, \]
we have
\[ \frac{ds}{dt}=k. \]
Hence
\[ \frac{d\hat{\mathbf t}}{ds} = \frac{\dot{\hat{\mathbf t}}}{k} = -\nabla_\perp \ln k. \]
But for a curve with curvature \(\kappa\) and principal normal \(N\),
\[ \frac{d\hat{\mathbf t}}{ds}=\kappa N. \]
So the exact trapping condition is
\[ \boxed{ \kappa N=-\nabla_\perp \ln k }. \]
This equation is the clean transport statement of self-refraction. A narrow transport branch remains trapped on a curved support exactly when the inward transverse gradient of the local transport speed supplies the required curvature of the ray.
For a circular orbit of radius \(R\) in a radially symmetric static profile \(k(r)\), the curvature is \(\kappa=1/R\) and the condition reduces to
\[ \frac{1}{R} = \frac{1}{k(R)}\frac{dk}{dr}(R). \]
So a closed circular transport line is self-trapped precisely when the outward increase of \(k\) balances the inward bending required by the orbit.
In a self-trapped bounded Maxwellian mode, that profile is not imposed from outside. It is generated by the same total trapped load of the closure itself. The field is both the transported thing and the thing that shapes the transport path.
Because the support is closed,
\[ \xi(s+L,t)=\xi(s,t). \]
Expand in Fourier modes,
\[ \xi(s,t)=\sum_{n\in\mathbb{Z}} a_n(t)\,e^{i2\pi ns/L}. \]
Substituting into the line equation gives
\[ \ddot{a}_n + \omega_n^2 a_n = 0, \]
with
\[ \omega_n = \frac{2\pi k}{L}|n|. \]
So the closed Maxwellian tube carries a discrete oscillator spectrum.
The reason is exactly the same as in chapter 8: closure forces periodic matching. The effective string equation only makes that spectral consequence explicit.
From bounded Maxwellian transport on a thin closed toroidal support, the following effective string data are forced:
These are precisely the structures usually introduced axiomatically in a string description.
String structure is not primitive here. It is the effective one-dimensional description of a bounded Maxwellian mode.
The deeper object is the electromagnetic closure itself. The string appears when that closure is reduced to its thin-tube, closed-line, periodic support.
The main text derives the transport core first and identifies the two complementary aspects \(\mathbf E\) and \(\mathbf B\) with the two transporting aspects \(\mathbf F_+\) and \(\mathbf F_-\). This appendix does not replace that derivation. It proves only the local converse: once a local energy density and energy flux are given, and once they obey the transport bound, one can reconstruct at least one Maxwell pair that reproduces them.
This is a representation theorem, not a dynamical one. It shows what the local observables \((u,\mathbf S)\) are sufficient to encode. It does not by itself determine how the fields evolve. That still belongs to Maxwellian transport.
Suppose a pointwise energy density and flux are given:
\[ u(\mathbf r,t) > 0, \qquad \mathbf S(\mathbf r,t). \]
Assume the transport bound
\[ |\mathbf S| \le c\,u. \]
We ask whether there exist vectors \(\mathbf E\) and \(\mathbf B\) such that
\[ u = \frac{\varepsilon_0}{2}|\mathbf E|^2 + \frac{1}{2\mu_0}|\mathbf B|^2, \]
and
\[ \mathbf S = \frac{1}{\mu_0}\,\mathbf E\times\mathbf B. \]
The answer is yes.
For every pair \((u,\mathbf S)\) with \(u>0\) and \(|\mathbf S|\le cu\), there exists at least one pair \((\mathbf E,\mathbf B)\) satisfying
\[ u = \frac{\varepsilon_0}{2}|\mathbf E|^2 + \frac{1}{2\mu_0}|\mathbf B|^2, \]
\[ \mathbf S = \frac{1}{\mu_0}\,\mathbf E\times\mathbf B. \]
Moreover, the reconstruction is not unique.
If \(\mathbf S\neq 0\), let
\[ \hat{\mathbf s}:=\frac{\mathbf S}{|\mathbf S|}. \]
If \(\mathbf S=0\), choose any unit vector \(\hat{\mathbf s}\).
Choose any unit vector \(\hat{\mathbf e}\) orthogonal to \(\hat{\mathbf s}\), and define
\[ \hat{\mathbf b}:=\hat{\mathbf s}\times\hat{\mathbf e}. \]
Then \((\hat{\mathbf e},\hat{\mathbf b},\hat{\mathbf s})\) is a right-handed orthonormal frame.
Now define
\[ \mathbf E := E\,\hat{\mathbf e}, \qquad \mathbf B := B\,\hat{\mathbf b}, \]
with scalars \(E,B\ge 0\) to be chosen.
Because
\[ \hat{\mathbf e}\times\hat{\mathbf b}=\hat{\mathbf s}, \]
we have
\[ \mathbf E\times\mathbf B = EB\,\hat{\mathbf s}. \]
So the flux condition becomes
\[ \frac{1}{\mu_0}EB = |\mathbf S|. \]
The energy condition becomes
\[ u = \frac{\varepsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2. \]
It is convenient to rescale:
\[ X:=\sqrt{\varepsilon_0}\,E, \qquad Y:=\frac{B}{\sqrt{\mu_0}}. \]
Then the two conditions become
\[ u=\frac{X^2+Y^2}{2}, \]
and, since
\[ c^2=\frac{1}{\mu_0\varepsilon_0}, \]
also
\[ |\mathbf S| = c\,X\,Y. \]
So it is enough to find nonnegative \(X,Y\) such that
\[ X^2+Y^2=2u, \qquad XY=\frac{|\mathbf S|}{c}. \]
Set
\[ \rho := \frac{|\mathbf S|}{c\,u}. \]
By assumption,
\[ 0\le \rho \le 1. \]
Choose an angle \(\theta\in[0,\pi/4]\) such that
\[ \sin(2\theta)=\rho. \]
Now define
\[ X:=\sqrt{2u}\,\cos\theta, \qquad Y:=\sqrt{2u}\,\sin\theta. \]
Then
\[ \frac{X^2+Y^2}{2} = \frac{2u(\cos^2\theta+\sin^2\theta)}{2} = u, \]
and
\[ cXY = c(2u\cos\theta\sin\theta) = cu\sin(2\theta) = |\mathbf S|. \]
So the reconstructed pair
\[ \mathbf E = \frac{X}{\sqrt{\varepsilon_0}}\,\hat{\mathbf e}, \qquad \mathbf B = \sqrt{\mu_0}\,Y\,\hat{\mathbf b} \]
satisfies the required relations exactly.
Thus the reconstruction exists for every \((u,\mathbf S)\) satisfying \(|\mathbf S|\le cu\).
The same inequality is also necessary.
For any vectors \(\mathbf E,\mathbf B\),
\[ |\mathbf S| = \frac{1}{\mu_0}|\mathbf E\times\mathbf B| \le \frac{1}{\mu_0}|\mathbf E|\,|\mathbf B|. \]
Using the arithmetic-geometric mean inequality,
\[ \frac{1}{\mu_0}|\mathbf E|\,|\mathbf B| \le \frac{c}{2}\left( \varepsilon_0|\mathbf E|^2+\frac{1}{\mu_0}|\mathbf B|^2 \right) = cu. \]
So any Maxwell pair must satisfy
\[ |\mathbf S|\le cu. \]
The bound is therefore not an extra decoration. It is exactly the condition under which the local observables admit Maxwell representation.
The reconstruction is not unique.
First, the choice of transverse unit vector \(\hat{\mathbf e}\) is arbitrary up to rotation in the plane orthogonal to \(\hat{\mathbf s}\). That already gives a continuous family of solutions.
Second, even after one pair \((\mathbf E,\mathbf B)\) is fixed, the duality rotation
\[ \mathbf E'=\mathbf E\cos\phi + c\,\mathbf B\sin\phi, \]
\[ c\,\mathbf B'=-\mathbf E\sin\phi + c\,\mathbf B\cos\phi \]
leaves both \(u\) and \(\mathbf S\) unchanged.
So \((u,\mathbf S)\) do not determine \((\mathbf E,\mathbf B)\) uniquely. The underdetermination is exactly the local polarization or duality freedom of the two-aspect transport.
The pair \((u,\mathbf S)\) determines:
It does not determine:
Those missing pieces are not flaws in the reconstruction. They are precisely what requires the full Maxwellian transport closure of chapter 7.
The Maxwell pair is not an arbitrary superstructure placed on top of energy density and flux. Its primary origin in this book is still the double-curl transport closure. The present appendix says only that whenever the local observables satisfy
\[ |\mathbf S|\le cu, \]
they already admit at least one exact Maxwell representation.
the already-derived two-aspect transport admits at least one local Maxwell representation whose scalar and vector observables are \(u\) and \(\mathbf S\). What the representation alone cannot do is fix the evolution. That is the role of Maxwellian transport.
The earlier appendices establish two ingredients that can now be combined.
First, passive Maxwellian transport in a region is governed by local source-free closure. Second, the local transport speed is a property of the region itself and may vary from place to place.
This appendix draws two consequences:
The second point is not a claim of super-causal propagation. It is only a relative statement: a more lightly loaded region transports faster than a more heavily loaded one.
Consider a bounded spatial region \(\Omega\) with smooth closed boundary \(\partial\Omega\). For passive source-free Maxwellian transport, the local state in \(\Omega\) is governed by the transport closure together with whatever data are fed across the boundary.
So for a passive region, complete transport data on the enclosing boundary over the relevant causal interval determine the interior evolution. The interior is not an independent second ontology. It is the continuation of the same transport constrained by the boundary history.
This is the right sense in which the picture is boundary-determined. One does not need a primitive substance hidden behind the surface. One needs the full transport data on that surface and the closure law of the same substrate.
This boundary-determined picture applies cleanly only to passive regions.
If a region contains a persistent causal loop carrying internally retained organization, then one boundary slice is not enough to exhaust what that region can do next. The loop can reorganize incoming transport using structure it already carries.
That distinction matters later for living or imprint-sensitive organization. But for inert transport without such internal steering, the passive boundary-determined picture is the correct one.
Appendix 214 already fixed the constitutive relation
\[ k(\mathbf r)=\frac{1}{\sqrt{\varepsilon(\mathbf r)\mu(\mathbf r)}}. \]
In the symmetric constitutive class used there,
\[ \varepsilon=\varepsilon_0\alpha, \qquad \mu=\mu_0\alpha, \qquad k=\frac{c}{\alpha}. \]
So a more heavily loaded region has larger \(\alpha\) and therefore smaller local transport speed \(k\). A more lightly loaded region has smaller \(\alpha\) and therefore larger \(k\).
This is the precise sense in which unloading a region speeds transport there.
Consider a tubular region \(\Gamma\) joining two endpoints \(A\) and \(B\). Let \(s\in[0,L]\) denote arclength along its axis, and suppose its local transport speed is
\[ k_\Gamma(s). \]
For a narrow radiative packet constrained to follow that tube, the travel time is
\[ T_\Gamma = \int_0^L \frac{ds}{k_\Gamma(s)}. \]
Now compare this with another route through a more heavily loaded region, with speed
\[ k_{\mathrm{ext}}(s). \]
If
\[ k_\Gamma(s) > k_{\mathrm{ext}}(s) \qquad \text{for all } s, \]
then
\[ T_\Gamma < \int_0^L \frac{ds}{k_{\mathrm{ext}}(s)}. \]
So the less-loaded tube is a faster transport corridor.
This is the exact sense in which one may speak of a high-speed communication tube. It is not faster than the tube’s own local causal speed. It is faster than transport through neighboring more heavily loaded regions.
The speed advantage does not by itself fix how the corridor turns or guides a packet. That is a separate design problem.
For a narrow radiative packet in a static background, appendix 214 gives
\[ \dot{\mathbf p}=-U\,\nabla\ln k. \]
So passive static gradients bend rays toward lower \(k\), not toward higher \(k\).
Therefore, if a tubular region has larger \(k\) than its surroundings, then:
Making such a corridor turn, branch, or remain tightly guided is an engineering problem. It requires additional structure, for example:
So the exact derivational statement is:
Passive source-free regions are boundary-determined in the sense that complete transport data on the enclosing boundary determine the passive interior evolution.
Within that same framework, lowering the relative electromagnetic loading of a region raises its local transport speed. A suitably maintained tubular region of lower loading is therefore a faster transport corridor relative to more heavily loaded surroundings. Appendix 221 derives the corresponding lensing and guidance laws for such field-shaped transport profiles, and appendix 222 derives boundary superposition as one fundamental unloading mechanism.
Chapter 9 already states that matter is Maxwellian transport under closure. This appendix sharpens that statement:
matter is a persistent closed causal loop of Maxwellian transport.
The point is not metaphorical. It is already forced by the transport spine once a bounded self-trapped mode exists.
The transport core of the book is Maxwellian transport: the double-curl closure of chapter 7. In a region with local transport speed \(k\), the propagating part of the mode moves locally at that causal speed.
For pure transport,
\[ |\mathbf S| = k\,u. \]
So the basic moving thing is not a particle. It is organized transport at local causal speed.
Now suppose the transport does not remain open. Suppose instead that it closes on a bounded support.
Let
\[ X : \mathbb{R}/L\mathbb{Z}\to\mathbb{R}^3 \]
be the closed support curve of a thin bounded mode, parameterized by arclength \(s\). If the transporting branch is tangent to that support, then in the thin tube limit
\[ S_\varepsilon = k\,u_\varepsilon\,\tau(s)+O(\varepsilon), \qquad \tau(s)=X'(s). \]
One complete traversal of the closed support takes the recurrence time
\[ T_{\mathrm{loop}} = \oint \frac{ds}{k}. \]
For constant \(k\) this is simply
\[ T_{\mathrm{loop}}=\frac{L}{k}. \]
So the closure is literally a causal loop: later transport around the support is generated from earlier transport around the same support after a finite causal recurrence time.
Not every closed path gives matter. The loop must also persist.
Appendix 217 derived the exact self-trapping condition
\[ \kappa N=-\nabla_\perp\ln k. \]
So a bounded closed loop persists only when the transport it carries also generates the transverse profile required to keep later transport returning into the same closure.
That is why matter is not just any loop. It is a persistent closed causal loop.
Chapter 9 already derived the mass statement:
\[ m=\frac{E_0}{c^2} \]
in the rest frame of the bounded closure.
Appendix 217 sharpened the same structure in thin-tube form:
\[ \mathcal T = \text{line energy density}, \qquad \mu = \frac{\mathcal T}{k^2}. \]
So the matter-like object is not a thing carrying transport as an attribute. It is the transport closure itself, and its mass is the trapped load of that closure.
The tighter the closure, the more trapped load can be stored per extent. In that sense denser matter corresponds to tighter persistent causal closure.
Because the transport remains local-causal everywhere along the loop, matter is not slow because its underlying transport slows down. It is slow because not all of that transport is available for net translation.
Part of it is locked into circulation.
That is why the bounded mode as a whole can drift at
\[ |\mathbf v_{\mathrm{drift}}|<k \]
even though its constitutive transport remains local at speed \(k\) throughout the loop.
Light is the untrapped case. Matter is the closed case.
An inert bounded mode already counts as matter if it is a persistent closed causal loop.
A richer loop may carry internally retained organization that alters how it reorganizes future transport. That is a further development, not part of the minimal definition of matter here.
So the distinction is:
The second belongs to the later theory of self. The first already belongs to TPOEF.
Matter is not something different from Maxwellian transport. Matter is Maxwellian transport under persistent self-trapped closure.
In that exact sense, matter is a closed causal loop of energy transport. What we later measure as mass, inertia, density, persistence, and other physical effects are the externally visible consequences of that looped transport, because transport itself is what makes things happen and therefore what makes their effects measurable.
Appendix 214 derived the radiative packet law in a static variable-speed background:
\[ \dot{\mathbf x}=k(\mathbf x)\,\hat{\mathbf t}, \qquad \frac{d\hat{\mathbf t}}{ds}=-\nabla_\perp\ln k. \]
Appendix 219 used that law to show that lower loading can create a faster transport corridor. The present appendix derives the corresponding lensing and guidance consequences. Appendix 222 derives one fundamental way such unloading can be produced: boundary subtraction of a selected passive mode.
The point is simple. Once a static field configuration creates a spatial loading profile and therefore a spatial transport-speed profile
\[ k(\mathbf x), \]
later Maxwellian transport is bent by that profile. Field-shaped transport therefore acts as lens, guide, or corridor according to the geometry of \(k\).
Within the symmetric constitutive class of appendix 214,
\[ \varepsilon=\varepsilon_0\alpha(\mathbf x), \qquad \mu=\mu_0\alpha(\mathbf x), \qquad k(\mathbf x)=\frac{c}{\alpha(\mathbf x)}. \]
So a static organized electromagnetic loading profile is equivalently a static transport-speed profile.
All the results below follow from that profile alone. No extra force law is introduced.
For a narrow radiative packet, appendix 214 gives
\[ \dot{\mathbf p}=-U\,\nabla\ln k. \]
Equivalently, with \(\hat{\mathbf t}= \mathbf p/|\mathbf p|\) and arclength \(s\),
\[ \frac{d\hat{\mathbf t}}{ds}=-\nabla_\perp\ln k. \]
This is the exact transport-curvature law.
Its content is immediate:
So lensing and guidance are not extra phenomena added afterward. They are the direct geometric consequences of the field-shaped transport profile.
Take an axisymmetric static profile about the \(z\) axis, and let
\[ r_\perp = \sqrt{x^2+y^2} \]
denote the transverse radius.
For a ray nearly parallel to the axis, the paraxial approximation gives
\[ \frac{d^2\mathbf r_\perp}{dz^2} = -\nabla_\perp \ln k(\mathbf r_\perp,z). \]
This is the lens equation of the present framework.
So:
The first is focusing. The second is defocusing.
Expand near the axis.
If the transport speed has a local minimum on the axis, write
\[ k(r_\perp)=k_0\!\left(1+\frac{\beta}{2}r_\perp^2\right)+O(r_\perp^4), \qquad \beta>0. \]
Then
\[ \ln k(r_\perp)=\ln k_0+\frac{\beta}{2}r_\perp^2+O(r_\perp^4), \]
so
\[ \nabla_\perp\ln k = \beta\,\mathbf r_\perp + O(r_\perp^3). \]
Therefore the paraxial ray equation becomes
\[ \frac{d^2\mathbf r_\perp}{dz^2}+\beta\,\mathbf r_\perp=0. \]
This is harmonic confinement. Rays oscillate about the axis. A low-speed core is therefore a passive focusing guide.
If the transport speed has a local maximum on the axis, write
\[ k(r_\perp)=k_0\!\left(1-\frac{\beta}{2}r_\perp^2\right)+O(r_\perp^4), \qquad \beta>0. \]
Then
\[ \ln k(r_\perp)=\ln k_0-\frac{\beta}{2}r_\perp^2+O(r_\perp^4), \]
so
\[ \nabla_\perp\ln k = -\beta\,\mathbf r_\perp + O(r_\perp^3). \]
Hence
\[ \frac{d^2\mathbf r_\perp}{dz^2}-\beta\,\mathbf r_\perp=0. \]
This is exponential defocusing. A high-speed core does not passively trap transport.
The previous section separates two different design objectives.
If the goal is passive focusing or confinement, the axis must be a local minimum of \(k\). Then transport is slower on axis but naturally guided.
If the goal is faster transit than through neighboring regions, the corridor must have larger \(k\) than those neighbors. Then transport through the core is faster, but the core is not passively confining.
So a fast corridor and a passive guide are not the same object.
A high-speed corridor can still be made useful. The point is simply that its guidance is an engineering problem rather than an automatic consequence of the speed contrast.
The necessary extra structure may come from:
So the exact derived consequence is this:
Organized electromagnetic loading shapes the local transport-speed profile \(k(\mathbf x)\). Once that happens, later Maxwellian transport is bent by the exact ray law
\[ \frac{d\hat{\mathbf t}}{ds}=-\nabla_\perp\ln k. \]
This yields:
So lensing, guidance, and corridor transport are not external technologies added to the framework. They are direct engineering consequences of taking the transport ontology seriously.
Appendices 219 and 221 showed that a lower-loading region can act as a faster transport corridor. The missing step is the unloading mechanism itself.
For passive source-free Maxwellian transport, that mechanism is already available. It is the combination of:
So the fundamental point is not yet engineering. It is simply this: given boundary control of a passive region, one can subtract as well as reinforce the interior field.
Let \(\Omega\) be a passive bounded region with smooth closed boundary \(\partial\Omega\). Let \(B\) denote the complete boundary transport data fed through \(\partial\Omega\) over the relevant causal interval.
For source-free linear Maxwellian transport, the interior resolved field depends linearly on that boundary data. Writing
\[ \mathcal F[B] = (\mathbf E[B],\mathbf H[B]), \]
linearity gives
\[ \mathcal F[B_1+B_2]=\mathcal F[B_1]+\mathcal F[B_2]. \]
Therefore boundary control can add or subtract admissible interior field components.
This is the first fundamental fact.
Take any boundary-driven interior field component
\[ (\mathbf E_0,\mathbf H_0) \]
in the target region.
Now choose a second boundary control that excites the same component with opposite phase and relative amplitude \(\lambda\), where
\[ 0\le \lambda \le 1. \]
That is,
\[ (\mathbf E_1,\mathbf H_1) = -\lambda(\mathbf E_0,\mathbf H_0). \]
By superposition, the total field becomes
\[ (\mathbf E_\lambda,\mathbf H_\lambda) = (1-\lambda)(\mathbf E_0,\mathbf H_0). \]
The electromagnetic energy density is
\[ u = \frac12\bigl(\varepsilon E^2+\mu H^2\bigr), \]
and the Poynting flux is
\[ \mathbf S = \mathbf E\times\mathbf H. \]
So for this same-mode subtraction,
\[ u_\lambda=(1-\lambda)^2u_0, \qquad \mathbf S_\lambda=(1-\lambda)^2\mathbf S_0. \]
This reduction is exact. It is not heuristic. It follows from the quadratic form of energy density and the bilinear form of flux.
At \(\lambda=1\), the selected component is canceled completely.
In a tube or corridor geometry, let the dominant passive mode have the form
\[ (\mathbf E_0,\mathbf H_0) = A\,(\mathbf e,\mathbf h)(x_\perp)\,e^{i(\beta z-\omega t)}. \]
If the boundary actuation injects the same mode with amplitude
\[ -\lambda A, \]
then the resulting mode amplitude inside the target region is
\[ (1-\lambda)A. \]
So the interior loading of that mode is reduced by the exact factor
\[ (1-\lambda)^2. \]
This is the fundamental boundary-unloading mechanism for a transport corridor: mode-wise subtraction by phase-opposed superposition.
The exact boundary pattern needed to realize the desired subtraction is the engineering problem. The unloading mechanism itself is already forced by the linearity of the passive interior.
Appendices 214 and 219 work in the symmetric constitutive class
\[ \varepsilon=\varepsilon_0\alpha, \qquad \mu=\mu_0\alpha, \qquad k=\frac{c}{\alpha}. \]
Within that class, lower local loading means larger local transport speed. Therefore any boundary program that lowers the resolved background load in a region raises the local transport speed there relative to the more heavily loaded case.
That is the fundamental basis of a high-speed transport corridor.
The important limit is also clear. Exact cancellation of the selected carrier component gives zero load in that component; it does not by itself provide a working guide or an infinite-speed theorem. Corridor design therefore uses controlled unloading, or unloading of a background component while preserving a separate signal-bearing structure.
Given boundary control of a passive region, one can subtract admissible Maxwellian modes as well as reinforce them. Because the field equations are linear and the electromagnetic energy density is quadratic, phase-opposed boundary control lowers interior energy exactly.
So the corridor idea does have a clean first-principles derivation:
Appendix 221 then gives the corresponding lensing and guidance consequences once such a loading profile has been engineered.