The Physics of Energy Flow
Part III - Open Regimes and Explorations
Part III - Open Regimes and Explorations
2026-03-29
Chapter 12 of The Physics of Energy Flow recovered the weak-field Newtonian limit by summing the positive scalar energies of many closures and taking the far field of a compact aggregate. That argument is correct for a roughly compact, mixed, and orientation-averaged body.
A spiral galaxy is not such a body.
It is:
The scalar monopole therefore cannot be the whole story. In a rotating disk, the first moment of the organized flow may cancel while the second moment survives. That surviving second moment is stress.
The dark-matter question is therefore recast as follows:
can the flat outer rotation curves of galaxies arise from a surviving azimuthal stress of organized energy flow, rather than from additional unseen matter?
This text sits in Part III rather than in the main book or its technical appendices because the galactic regime is not yet closed. What is developed here is a serious candidate mechanism for the flat-curve regime, together with the matching logarithmic weak-lensing law inside the same weak constitutive closure already used in the gravity chapters. What is not yet recovered is the full galactic microphysics that would make those outer-disk assumptions follow automatically.
Let \(\hat{\mathbf e}_\phi(\phi)\) denote the local azimuthal direction in the galactic plane.
Around a full annulus,
\[ \int_0^{2\pi}\hat{\mathbf e}_\phi(\phi)\,d\phi = 0. \]
So any vector sum of the azimuthal transport can vanish.
But the second moment does not vanish:
\[ \hat{\mathbf e}_\phi\otimes\hat{\mathbf e}_\phi \neq 0. \]
This is the basic structural point. A rotating galaxy can have no net vector flux around the annulus and still carry a nonzero azimuthal momentum-flux tensor.
That is exactly what a monopole reduction throws away. The monopole keeps the scalar energy and discards the directional part. A rotating disk keeps a directional second moment, and that second moment contributes to radial balance.
This point can be written directly. Let
\[ \mathbf A(R,z)=A_\phi(R,z)\,\hat{\mathbf e}_\phi \]
be a purely azimuthal axisymmetric transport field. Then
\[ \nabla\cdot\mathbf A = \frac{1}{R}\partial_\phi A_\phi = 0. \]
So the first moment is divergence-free.
But now form its second moment:
\[ \mathbf Q:=\mathbf A\otimes\mathbf A. \]
Its only nonzero component is
\[ Q_{\phi\phi}=A_\phi^2. \]
Using the cylindrical divergence formula,
\[ \bigl(\nabla\cdot\mathbf Q\bigr)_R = \partial_R Q_{RR} + \frac{Q_{RR}-Q_{\phi\phi}}{R} + \partial_z Q_{Rz} = -\frac{A_\phi^2}{R}. \]
So a purely azimuthal divergence-free flow still carries an inward radial load through the divergence of its second moment. That is the exact mathematical form of hoop stress.
This is the sense in which a vortex-like (m,n)
organization can add pull without adding source or sink: not through
\(\nabla\cdot\mathbf A\), but through
\(\nabla\cdot(\mathbf A\otimes\mathbf
A)\) and its coarse-grained stress descendants.
It is important not to confuse two different notions of “second moment.”
A scalar multipole expansion of a compact source is one thing. The surviving directional second moment of organized transport is another.
For a compact scalar exterior field, the far expansion has the form
\[ \eta(r,\Omega) = \frac{M_0}{r} + \frac{D_1(\Omega)}{r^2} + \frac{Q_2(\Omega)}{r^3} + \cdots \]
where the angular factors encode dipole, quadrupole, and higher scalar multipoles.
The corresponding radial loading scales as
\[ g_r \sim \partial_r \eta \sim \frac{1}{r^2}, \frac{1}{r^3}, \frac{1}{r^4}, \ldots \]
and the circular-balance contribution of each finite compact multipole scales as
\[ v_\phi^2 = r\,g_r \sim \frac{1}{r}, \frac{1}{r^2}, \frac{1}{r^3}, \ldots \]
So a neglected compact multipole tail can modify the angular structure of the field, but it cannot sustain a flat outer curve. Every such term decays too quickly.
A flat regime requires instead
\[ v_\phi(R)\approx v_f=\text{const.}, \]
so the required inward loading is
\[ g_R(R)\approx \frac{v_f^2}{R}, \]
which corresponds to a logarithmic effective scalar, not to any finite compact scalar multipole tail.
That is why the galactic dark-matter effect should not be read here as the sum of neglected higher terms in the scalar mass expansion of chapter 12. The missing object is tensorial rather than scalar: the surviving azimuthal second moment of organized transport. It enters through the stress tensor and its cylindrical divergence, not as a scalar quadrupole correction to the monopole field.
Appendix 207 already recovered the exact coarse-grained momentum equation
\[ \partial_t(\rho\mathbf v) + \nabla\cdot(\rho\,\mathbf v\otimes\mathbf v) - \nabla\cdot\boldsymbol{\Sigma} = 0, \]
where
\[ \rho=\frac{\langle u\rangle}{k^2} \]
is the effective inertial density,
\[ \rho\mathbf v=\left\langle \frac{\mathbf S}{k^2}\right\rangle \]
is the mean transport momentum density, and \(\boldsymbol{\Sigma}\) is the exact residual stress tensor of the unresolved transport.
Take a steady axisymmetric disk in cylindrical coordinates \((R,\phi,z)\) with
\[ \partial_t=0, \qquad \partial_\phi=0, \qquad \mathbf v = v_\phi(R,z)\,\hat{\mathbf e}_\phi, \]
and negligible mean radial or vertical drift:
\[ v_R=v_z=0. \]
Then the radial component of the convective term is the usual centripetal term,
\[ \bigl[\nabla\cdot(\rho\,\mathbf v\otimes\mathbf v)\bigr]_R = -\rho\,\frac{v_\phi^2}{R}. \]
So the exact radial balance is
\[ \rho\,\frac{v_\phi^2}{R} = -\bigl(\nabla\cdot\boldsymbol{\Sigma}\bigr)_R. \]
For an axisymmetric stress tensor, the radial divergence is
\[ \bigl(\nabla\cdot\boldsymbol{\Sigma}\bigr)_R = \partial_R \Sigma_{RR} + \frac{\Sigma_{RR}-\Sigma_{\phi\phi}}{R} + \partial_z \Sigma_{Rz}. \]
Therefore
\[ \rho\,\frac{v_\phi^2}{R} = -\partial_R \Sigma_{RR} - \frac{\Sigma_{RR}-\Sigma_{\phi\phi}}{R} - \partial_z \Sigma_{Rz}. \]
This equation is exact. No dark matter has been inserted. No modified force law has been inserted. Everything now depends on the structure of the unresolved stress.
In the outer disk, suppose the unresolved transport is dominated by aligned azimuthal circulation. Then the residual stress is anisotropic, with
\[ \Sigma_{\phi\phi} \gg \Sigma_{RR}, \qquad \Sigma_{\phi\phi} \gg R\,|\partial_R\Sigma_{RR}|, \qquad \Sigma_{\phi\phi} \gg R\,|\partial_z\Sigma_{Rz}|. \]
Under these explicit assumptions, the radial balance reduces to
\[ \rho\,\frac{v_\phi^2}{R} \approx \frac{\Sigma_{\phi\phi}}{R}, \]
so
\[ \boxed{ v_\phi^2 \approx \frac{\Sigma_{\phi\phi}}{\rho}. } \]
This is the key equation.
Flat rotation curves therefore do not require an additional scalar mass distribution if the outer galaxy carries a residual azimuthal stress whose ratio to the effective inertial density is approximately constant.
Appendix 216 already gives the local transport-stress magnitude of a narrow null Maxwell packet. If \(\mathbf n\) is the packet direction, then the longitudinal momentum-flux density is
\[ \Pi_n = -n_i n_j T_{ij} = u. \]
So a narrow transport element moving in the azimuthal direction carries a positive azimuthal transport-stress magnitude equal to its energy density.
For a coarse-grained ensemble of co-rotating closures, let
\[ u_\phi(R,z) \]
be the part of the local energy density stored in unresolved azimuthal transport. Then the corresponding leading residual stress is
\[ \Sigma_{\phi\phi}\approx u_\phi. \]
This is the coarse-grained form of the packet statement above: each unresolved azimuthal transport element contributes its local energy density to the azimuthal second moment, and those contributions add.
Inside the same constitutive class already used in the gravity appendices,
\[ \rho = \frac{u}{k^2}, \]
so if a fraction
\[ f(R,z):=\frac{u_\phi(R,z)}{u(R,z)} \]
of the local coarse-grained energy sits in aligned unresolved azimuthal transport, then
\[ \Sigma_{\phi\phi}\approx f\,u = f\,\rho\,k^2. \]
Substituting into the outer-disk balance gives
\[ \boxed{ v_\phi^2 \approx f\,k^2. } \]
This is the strongest compact form of the result.
The same mechanism already supplies a lower and upper bound on what this channel can explain. The derivation is step by step.
First, return to the exact radial balance from section 3:
\[ \rho\,\frac{v_\phi^2}{R} = -\partial_R \Sigma_{RR} - \frac{\Sigma_{RR}-\Sigma_{\phi\phi}}{R} - \partial_z \Sigma_{Rz}. \]
Multiply by \(R/\rho\) and regroup the terms:
\[ v_\phi^2 = \left( -\frac{R}{\rho}\partial_R \Sigma_{RR} -\frac{\Sigma_{RR}}{\rho} -\frac{R}{\rho}\partial_z \Sigma_{Rz} \right) + \frac{\Sigma_{\phi\phi}}{\rho}. \]
Define the non-azimuthal baseline by
\[ v_{\mathrm{base}}^2 := -\frac{R}{\rho}\partial_R \Sigma_{RR} -\frac{\Sigma_{RR}}{\rho} -\frac{R}{\rho}\partial_z \Sigma_{Rz}. \]
Then the circular speed splits exactly as
\[ \boxed{ v_\phi^2 = v_{\mathrm{base}}^2 + \frac{\Sigma_{\phi\phi}}{\rho}. } \]
This is the key structural point. The full observed radial load
\[ \frac{v_\phi^2}{R} \]
is not partly outside the stress description. It is exactly what the stress tensor accounts for once all contributing terms are retained. The azimuthal term is an additional surviving part of that same tensorial load, not a second force laid on top from elsewhere.
So the excess above the non-azimuthal baseline is
\[ \boxed{ \Delta v_\phi^2 := v_\phi^2-v_{\mathrm{base}}^2 = \frac{\Sigma_{\phi\phi}}{\rho}. } \]
Second, bound \(\Sigma_{\phi\phi}\) from the unresolved transport itself. Inside the same narrow-packet coarse-graining already used in section 5, resolve the local unresolved transport into elements labeled by \(a\), with local energy densities \(u_a\) and unit directions \(\hat{\mathbf n}_a\). For each element, the packet result gives the local momentum-flux tensor
\[ \mathbf Q^{(a)} \approx u_a\,\hat{\mathbf n}_a\otimes\hat{\mathbf n}_a. \]
Its azimuthal component is therefore
\[ Q^{(a)}_{\phi\phi} \approx u_a\,(\hat{\mathbf n}_a\cdot\hat{\mathbf e}_\phi)^2. \]
Coarse-graining over all unresolved elements gives
\[ \Sigma_{\phi\phi} \approx \sum_a u_a\,(\hat{\mathbf n}_a\cdot\hat{\mathbf e}_\phi)^2. \]
Now
\[ 0\le (\hat{\mathbf n}_a\cdot\hat{\mathbf e}_\phi)^2 \le 1 \]
for every element, so summing yields
\[ 0 \le \Sigma_{\phi\phi} \le \sum_a u_a. \]
But the local coarse-grained energy density is just
\[ u=\sum_a u_a, \]
so
\[ \boxed{ 0\le \Sigma_{\phi\phi}\le u. } \]
Equivalently, the azimuthal fraction obeys
\[ \boxed{ 0\le f:=\frac{\Sigma_{\phi\phi}}{u}\le 1. } \]
Third, convert that stress bound into a velocity-squared bound. Inside the same constitutive class,
\[ \rho=\frac{u}{k^2}, \qquad k=\frac{1}{\sqrt{\varepsilon\mu}}. \]
Therefore
\[ 0\le \frac{\Sigma_{\phi\phi}}{\rho}\le \frac{u}{\rho}=k^2, \]
that is,
\[ \boxed{ 0\le \Delta v_\phi^2 = \frac{\Sigma_{\phi\phi}}{\rho}\le k^2. } \]
This is the envelope expression.
If one uses an observational baryonic baseline \(v_{\mathrm{bar}}^2\) to represent the same non-azimuthal contribution, then the observable excess
\[ \Delta v_\phi^2:=v_{\mathrm{obs}}^2-v_{\mathrm{bar}}^2 \]
should satisfy the same envelope:
\[ \boxed{ 0\le \Delta v_\phi^2 \le k^2. } \]
So the structural test is immediate. If the galactic dark-matter cases really come from this azimuthal-stress channel, they should fall inside that envelope. A case requiring
\[ \Delta v_\phi^2 > k^2 \]
would lie outside the capacity of this mechanism.
The envelope above only bounds what the azimuthal-stress channel can supply. To connect that bound directly to the coherent-overlap logic of the self-refraction chapters, now write the interaction term as an averaged product term.
Let the resolved co-rotating baryonic families be labeled by \(I\), with local positive azimuthal amplitudes
\[ A_I(R,z)\ge 0. \]
Choose local oscillatory representatives
\[ f_I(t;R,z) := \sqrt{2}\,A_I(R,z)\cos\bigl(\omega t+\phi_I(R,z)\bigr). \]
Then each family contributes the diagonal average
\[ \langle f_I^2\rangle = A_I^2. \]
Therefore the time-averaged square of the summed local family field is
\[ \left\langle \left(\sum_I f_I\right)^2 \right\rangle = \sum_I A_I^2 + 2\sum_{I<J}A_IA_J\,C_{IJ}, \]
where the pair correlators are
\[ C_{IJ}(R,z) := \left\langle \cos\bigl(\phi_I-\phi_J\bigr)\right\rangle. \]
This is the exact averaged-product form of the coherent interaction term. The ordinary baryonic addition keeps only the diagonal piece and drops the positive product term.
The local 4u result is the two-family maximal case of
this same formula. If
\[ A_1=A_2=\sqrt{u}, \qquad C_{12}=1, \]
then
\[ \left\langle (f_1+f_2)^2\right\rangle = u+u+2u = 4u. \]
So the galactic interaction term is the coarse-grained many-family descendant of the same coherent-overlap rule.
Now define the diagonal family energy content by
\[ U_I:=A_I^2. \]
The ordinary baryonic baseline then keeps only the diagonal contribution:
\[ V_N^2=\sum_I U_I=\sum_I A_I^2. \]
If the resolved families contribute coherently to the aligned azimuthal second moment, the corresponding coarse-grained energy-flow curve is
\[ V_{\mathrm{EF}}^2 = V_N^2 + 2\sum_{I<J}A_IA_J\,C_{IJ}. \]
In the constructive aligned sector,
\[ 0\le C_{IJ}\le 1. \]
So the coherent excess above the diagonal baryonic baseline is
\[ \Delta v_{\mathrm{coh}}^2 \approx 2\sum_{I<J}A_IA_J\,C_{IJ}. \]
Because each correlator lies between zero and one, the resolved-family coherent term obeys the exact coarse bounds
\[ \boxed{ 0 \le \Delta v_{\mathrm{coh}}^2 \le 2\sum_{I<J}A_IA_J. } \]
Equivalently,
\[ \boxed{ V_N^2 \le V_{\mathrm{EF}}^2 \le \left(\sum_I A_I\right)^2. } \]
So the diagonal Newtonian curve is the lower bound, while the fully
coherent resolved-family sum is the upper bound. No detailed knot shape
is needed for this step. The local 4u principle enters in
exactly the right place: it is the maximal two-family value of the same
averaged product term.
For the resolved SPARC families gas, disk, and bulge, take
\[ A_g:=\sqrt{\max(V_g|V_g|,0)}, \qquad A_d:=\sqrt{\Upsilon_d}\,V_d, \qquad A_b:=\sqrt{\Upsilon_b}\,V_b, \]
with the SPARC fiducial values
\[ \Upsilon_d=0.5, \qquad \Upsilon_b=0.7. \]
Then the resolved-family correlator band becomes
\[ \boxed{ V_N^2(R) \le V_{\mathrm{EF}}^2(R) \le \bigl(A_g(R)+A_d(R)+A_b(R)\bigr)^2. } \]
and the maximum resolved coherent excess is
\[ \boxed{ \Delta v_{\mathrm{coh,max}}^2(R) = 2\bigl(A_gA_d+A_gA_b+A_dA_b\bigr). } \]
If one compresses the three resolved pair correlators into a single effective correlator
\[ C_{\mathrm{eff}}(R), \]
then
\[ \boxed{ V_{\mathrm{EF}}^2(R) = V_N^2(R) + 2\,C_{\mathrm{eff}}(R)\bigl(A_gA_d+A_gA_b+A_dA_b\bigr). } \]
and the observed galaxy defines the required effective correlator
\[ \boxed{ C_{\mathrm{req}}(R) := \frac{v_{\mathrm{obs}}^2(R)-V_N^2(R)} {2\bigl(A_gA_d+A_gA_b+A_dA_b\bigr)}. } \]
Whenever
\[ 0\le C_{\mathrm{req}}(R)\le 1, \]
the observed excess can be recovered inside the resolved gas-disk-bulge correlator band alone. If instead
\[ C_{\mathrm{req}}(R)>1, \]
then the resolved families undercount the full positive cross term, and finer baryonic decomposition or additional unresolved baryonic families must still contribute.
The previous band still treats each resolved visible family as though it were a single coherent channel. That is too restrictive.
A visible family can itself contain several coherent winding sectors. Write
\[ A_I^2(R)=\sum_k A_{Ik}^2(R), \]
where the index k labels the active winding channels
carried by resolved family I.
Then the full constructive energy-flow curve is
\[ V_{\mathrm{EF}}^2(R) = \sum_{I,k}A_{Ik}^2(R) + 2\sum_{(I,k)<(J,\ell)}A_{Ik}(R)A_{J\ell}(R)\,C_{Ik,J\ell}(R). \]
This matters for two reasons.
First, same-family coherent products now appear automatically through
the terms with fixed I and different winding labels
k\neq \ell. So the coarse resolved-family denominator no
longer collapses merely because only one visible family dominates at
some radius.
Second, the visible coarse amplitudes still control the constructive envelope. By Cauchy,
\[ \sum_k A_{Ik}(R) \le \sqrt{N_I(R)}\,A_I(R), \]
where N_I(R) is the number of active winding channels
carried by resolved family I at that radius.
Therefore
\[ V_{\mathrm{EF}}^2(R) \le \left(\sum_I\sum_k A_{Ik}(R)\right)^2 \le \left(\sum_I \sqrt{N_I(R)}\,A_I(R)\right)^2. \]
If one common ceiling N_\ast(R) bounds the active
winding multiplicity of the visible families,
\[ N_I(R)\le N_\ast(R) \qquad \text{for all }I, \]
then
\[ \boxed{ V_{\mathrm{EF}}^2(R) \le N_\ast(R)\bigl(A_g(R)+A_d(R)+A_b(R)\bigr)^2. } \]
This is the rigorous winding-channel lift of the coarse baryonic ceiling.
The coarse resolved-family band corresponds to
\[ N_\ast=1. \]
If each visible family carries at most one dominant conjugate pair,
for example (m,n) together with (n,m),
then
\[ N_\ast=2, \]
and the constructive ceiling becomes
\[ \boxed{ V_{\mathrm{EF}}^2(R) \le 2\bigl(A_g(R)+A_d(R)+A_b(R)\bigr)^2. } \]
So a preferred conjugate winding pair does not merely change interpretation. It strictly raises the admissible baryonic ceiling and restores same-family coherent products that the coarse resolved-family formula omits.
The observed galaxy then defines the required common winding ceiling
\[ \boxed{ N_{\ast,\mathrm{req}}(R) := \frac{v_{\mathrm{obs}}^2(R)} \bigl(A_g(R)+A_d(R)+A_b(R)\bigr)^2. } \]
Whenever
\[ N_{\ast,\mathrm{req}}(R)\le 1, \]
the observed point lies inside the coarse one-channel ceiling. Whenever
\[ 1 < N_{\ast,\mathrm{req}}(R)\le 2, \]
a single dominant conjugate pair per visible family is enough. If instead
\[ N_{\ast,\mathrm{req}}(R)>2, \]
then even that is not enough, and more active winding sectors, finer baryonic decomposition, or additional unresolved baryonic families must still contribute.
Nothing in this argument rejects the astronomical observations themselves.
The observations give resolved baryonic organization:
What standard practice then does is reduce that observed organization to a scalar baryonic mass profile and assume that this scalar profile is the whole dynamically relevant object.
In symbols, the usual ontological reduction is
\[ \text{observed baryonic light/tracers} \;\longrightarrow\; \rho_{\mathrm{bar}}(R) \;\longrightarrow\; \text{gravity from scalar mass alone}. \]
This text keeps the observational first step but rejects the last reduction as ontologically incomplete for an extended rotating galaxy.
The observed baryonic profile still gives the diagonal content:
\[ V_N^2(R)=\sum_I A_I^2(R). \]
But the same observed baryonic organization can also carry a surviving directional second moment, and that second moment contributes through the correlator term
\[ \Delta v_{\mathrm{coh}}^2(R) = 2\sum_{I<J}A_I(R)A_J(R)\,C_{IJ}(R). \]
So the collective dynamical object recovered from the observations is not just a scalar mass density. It is
\[ \text{diagonal baryonic content} + \text{surviving correlator/stress structure}. \]
That is the ontological correction. The data are not discarded. What changes is the dynamical reading of those data.
This is also why the galactic excess need not imply either additional unseen matter or an error in the observed baryonic profile. It can arise because a structured rotating baryonic system is being forced into a scalar-only gravitational reading. In that misreading, the correlator/stress contribution has nowhere to appear except as fictitious missing mass.
Equation
\[ v_\phi^2 \approx f\,k^2 \]
shows immediately how a plateau arises.
If, over the outer galactic regime,
\[ f(R,z)\approx f_0, \qquad k(R,z)\approx k_0, \]
with both varying only slowly, then
\[ v_\phi(R)\approx \sqrt{f_0}\,k_0 = \text{const.} \]
The rotation curve is flat.
The ontology is then clear:
Observers often translate the measured circular speed into an inferred enclosed mass by the spherical Newtonian relation
\[ M_{\mathrm{inf}}(R)=\frac{R\,v_\phi^2(R)}{G}. \]
If \(v_\phi(R)\) is flat, this gives
\[ M_{\mathrm{inf}}(R)\propto R, \]
which is then read as evidence for a massive unseen halo.
In the present ontology, that linear growth is not necessarily the profile of an unseen scalar mass. It is the scalar mass one would back-fit to motion that is actually supported by anisotropic azimuthal stress:
\[ M_{\mathrm{inf}}(R) \approx \frac{R}{G}\,\frac{\Sigma_{\phi\phi}(R)}{\rho(R)}. \]
So the “dark matter” can be, at the level of flat rotation curves, a stress misread as mass.
Adopt the same weak constitutive summary already recovered in the gravity chapters for null probes:
\[ n=1+2\eta, \qquad k=\frac{c}{1+2\eta}. \]
The slow radial load in the flat outer regime is
\[ a_R(R)=-\frac{v_f^2}{R}. \]
Appendix 213 gives the slow-mode potential
\[ \Phi_k=-c^2\eta. \]
Therefore the radial acceleration is
\[ a_R=-\partial_R\Phi_k=c^2\,\partial_R\eta_{\mathrm{gal}}. \]
Substituting the flat-curve load gives
\[ \partial_R\eta_{\mathrm{gal}}=-\frac{v_f^2}{c^2R}, \]
so
\[ \boxed{ \eta_{\mathrm{gal}}(R) = \eta_0-\frac{v_f^2}{c^2}\ln\!\frac{R}{R_0}, } \]
up to an irrelevant additive constant.
Therefore the corresponding refractive index is
\[ n(R) = 1+2\eta_0-\frac{2v_f^2}{c^2}\ln\!\frac{R}{R_0}. \]
Take a null probe with impact parameter \(b\) in the scale-free outer regime, and write
\[ R^2=b^2+z^2 \]
along the unperturbed path. Then
\[ \partial_\perp n = -\frac{2v_f^2}{c^2}\,\frac{b}{b^2+z^2}. \]
The sign is inward. Using the same weak-ray law as chapter 12, the deflection magnitude is
\[ \Delta\alpha = \int_{-\infty}^{\infty}\bigl|\partial_\perp n\bigr|\,dz = \frac{2v_f^2}{c^2} \int_{-\infty}^{\infty}\frac{b\,dz}{b^2+z^2} = \frac{2v_f^2}{c^2} \left[\arctan\!\frac{z}{b}\right]_{-\infty}^{\infty}. \]
Therefore
\[ \boxed{ \Delta\alpha = \frac{2\pi v_f^2}{c^2}. } \]
This is the characteristic logarithmic-lens result of the flat regime:
This derivation explains the original flat-curve trigger of the dark-matter problem inside the energy-flow ontology, together with the corresponding logarithmic lensing law of the same regime:
But it does not yet explain everything commonly grouped under the dark matter label.
Still open are:
So the present result should be read narrowly and exactly:
flat galactic rotation curves, together with the corresponding logarithmic lensing law of the same outer regime, can be recovered in this ontology from the surviving azimuthal stress of organized co-rotating transport in an extended axisymmetric disk.
The correct collective object for a rotating galaxy is not the scalar monopole of a compact random aggregate. It is the stress tensor of an extended organized disk.
The vector part of the azimuthal transport can cancel around the galaxy. The second moment does not. That surviving second moment is an azimuthal stress, and its cylindrical hoop-stress term supplies the inward radial loading needed for circular motion.
So the completed galactic attraction must coincide with what is observed, and in this framework that completed load is read as stress-tensor structure all the way through: the non-azimuthal baseline plus the surviving azimuthal term.
Under the explicit outer-disk assumptions above,
\[ v_\phi^2 \approx \frac{\Sigma_{\phi\phi}}{\rho}\approx f\,k^2, \qquad \Delta\alpha \approx \frac{2\pi v_f^2}{c^2}, \]
so a slowly varying azimuthal transport fraction produces a flat rotation curve, and the same weak constitutive summary yields the matching logarithmic-regime lensing strength.
At that level, the missing mass is not missing matter. It is missing stress.