UMFT — Geometric and proper-time comparisons

These calculations test particular metric-gradient, charge-constraint, and telegraph equations. They do not define UMFT, supersede its evolved field-closure model, or establish that every geometric formulation has the same limitations. The main paper develops the current model and compares its numerical outputs with measurements.

XVIII. LINDGREN’S GEOMETRIC MAXWELL PROGRAM

What could hold a field without adding a container?

Linear Maxwell theory carries energy, momentum, stress, and pressure. It does not make one vacuum wavepacket attract itself into a persistent charged rest state. Lindgren, Kovacs, and Liukkonen approach that missing operation by varying spacetime geometry itself.¹³

Their construction proposes a metric-gradient action over a spacetime region \(\Omega\), a harmonic-metric equation, and a rank-one electromagnetic contribution to Minkowski space. The connection and allowed variations determine whether these are consequences of one variational problem:

\[ \mathcal I[g,\Gamma]=\int_\Omega \nabla_\sigma g^{\mu\nu}\nabla^\sigma g_{\mu\nu} \sqrt{|g|}\,d^4x, \qquad \delta\mathcal I=0, \qquad \delta g_{\mu\nu}|_{\partial\Omega}=0, \tag{45} \] \[ \Delta_g g_{\mu\nu}=\nabla^\sigma\nabla_\sigma g_{\mu\nu}=0, \qquad g_{\mu\nu}=\eta_{\mu\nu}+A_\mu A_\nu, \tag{46} \] \[ \nabla_\sigma A_\nu\nabla^\sigma A_\mu +A_\nu\Delta_gA_\mu +\nabla_\sigma A_\mu\nabla^\sigma A_\nu +A_\mu\Delta_gA_\nu=-\Delta_g\eta_{\mu\nu}. \tag{47} \]

The paper’s Generalized Maxwell Equation sets the right-hand side of Equation (47) to zero. That step requires \(\Delta_g\eta_{\mu\nu}=0\), not merely coordinate-constant components of \(\eta\). Covariant derivatives also act on its tensor indices. The displayed derivative order follows the source’s initial definition; raising the derivative before applying the outer derivative is a different operator in a nonmetric geometry.

What does the metric-gradient action actually vary?

With the Levi-Civita connection of \(g\), metric compatibility makes \(\nabla g=0\) identically, so this particular action vanishes for every metric. With the ordinary Weyl connection in Equation (48), the inverse-metric identity instead gives

\[ \nabla_\sigma g_{\mu\nu}=2\phi_\sigma g_{\mu\nu},\qquad \nabla_\sigma g^{\mu\nu}=-2\phi_\sigma g^{\mu\nu},\qquad (\nabla^\sigma g^{\mu\nu})(\nabla_\sigma g_{\mu\nu})=-16\phi_\sigma\phi^\sigma. \tag{47a} \] \[ \nabla_\mu V^\mu= \frac1{\sqrt{|g|}}\partial_\mu\!\left(\sqrt{|g|}V^\mu\right) -4\phi_\mu V^\mu. \tag{47b} \]

For independent fields \((g,\phi)\), the resulting action is algebraic in \(\phi\); unrestricted variation of that one-form gives \(\phi=0\), not a propagating Maxwell field. A different constraint or coupling can change the problem, but must be supplied before the binding force can be calculated. Equation (47b) also leaves a bulk nonmetricity term in integration by parts even when the boundary term vanishes.²⁰ Thus the harmonic-metric equation is retained as a candidate, not an established consequence of the bare action.

The following figure tests the simpler flat-background product identity. For a transverse potential \(A_y(x,t)\), in units \(c=1\), set \(\Box_0=\partial_x^2-\partial_t^2\):

\[ A_y=a_R\cos[k(x-t)]+a_L\cos[k(x+t)+\delta],\qquad \Box_0A_y=0,\qquad \Box_0(A_y^2)=2\left[(\partial_xA_y)^2-(\partial_tA_y)^2\right]. \tag{47c} \]

This establishes that a harmonic potential need not have a harmonic square. It does not compute the full Weyl operator, a self-consistent metric, or an attractive stress.

FIGURE 7

When does a wave’s square remain harmonic?

Both traveling components satisfy the flat-background wave equation. Their squared sum generally does not. All three traces use Equation (47c), with \(a_R=k=1\); they do not evolve the Weyl connection.

Counter-wave amplitude0.00 Relative phase0.00π Physical time0.00

linear residual max: 0.000000
flat dyad residual max: 0.000000
linear equation: both waves pass
flat harmonic dyad: satisfied

XIX. CHARGE AS GEOMETRIC FLOW

Where does charge enter the metric?

Charge must determine how a state responds to an external field, not merely name a scalar in the equations. The paper proposes to obtain that response from Weyl semimetricity and a constraint on the connection. The constraint can be checked before assigning its terms a charge interpretation:

\[ \nabla_\sigma g_{\mu\nu}=2\phi_\sigma g_{\mu\nu}, \qquad \Gamma^\mu{}_{\nu\lambda}= \left\{\!\begin{matrix}\mu\\\nu\lambda\end{matrix}\!\right\} -\delta^\mu_\nu\phi_\lambda-\delta^\mu_\lambda\phi_\nu +g_{\nu\lambda}\phi^\mu, \tag{48} \]

Let \(C^\mu{}_{\nu\lambda}\) denote the three Weyl correction terms after the Christoffel symbol in Equation (48). The source’s full cancellation condition is

\[ B_{\nu\lambda}=\tfrac12(\partial_\nu A_\lambda+\partial_\lambda A_\nu), \qquad A^\mu B_{\nu\lambda}+C^\mu{}_{\nu\lambda}=0, \tag{48a} \] \[ b=g^{\nu\lambda}B_{\nu\lambda},\qquad bA^\mu+2\phi^\mu=0. \tag{48b} \]

For nonzero \(A\), the trace in Equation (48b) makes \(\phi\) parallel to \(A\). Contracting the remaining upper index in (48a) with a nonzero covector orthogonal to \(A^\mu\) then forces that proportionality to vanish. Thus the full constraint requires \(\phi=0\) and \(B=0\), including when \(A\) is null. At \(A=0\), it still requires \(\phi=0\). It cannot coexist with the source’s later assignment \(\phi^\mu=u^\mu\) for a nonzero four-velocity.

This calculation requires a nondegenerate metric. The bare dyad \(A_\mu A_\nu\) has rank one and no four-dimensional inverse; using it alone does not evade the constraint while retaining the ordinary Weyl connection. Requiring cancellation only after contraction with one chosen velocity is weaker and can hold, but is a different problem from Equation (48a).

A directional tensor can still be meaningful without being the entire metric. But a rank-one pseudoinverse cannot provide the electromagnetic kinetic term. Any symmetric rank-one raising tensor \(G^{\mu\nu}=s v^\mu v^\nu\) annihilates every antisymmetric two-form:

\[ G^{\mu\alpha}G^{\nu\beta}F_{\alpha\beta} =s^2v^\mu v^\nu(v^\alpha v^\beta F_{\alpha\beta})=0, \qquad F_{\mu\nu}G^{\mu\alpha}G^{\nu\beta}F_{\alpha\beta}=0. \tag{48c} \]

This is stronger than the vanishing scalar invariant of an ordinary null electromagnetic field: the entire raised field vanishes, for every input. The dyad must therefore remain a directional tensor alongside a nondegenerate causal metric or a separately specified electromagnetic constitutive operator.

With a supplied metric \(g\), the rank-one update \(\widetilde g_{\mu\nu}=g_{\mu\nu}+\lambda A_\mu A_\nu\) has a well-defined inverse on the branch \(D=1+\lambda A_\mu A^\mu>0\). Here \(\lambda\) supplies the required units and unmarked contractions use \(g\):

\[ \widetilde g^{\mu\nu}=g^{\mu\nu}-\frac{\lambda}{D}A^\mu A^\nu, \qquad \sqrt{|\widetilde g|}=\sqrt{|g|}\sqrt D, \] \[ W_\nu=A^\mu F_{\mu\nu},\qquad F_{\mu\nu}\widetilde g^{\mu\alpha}\widetilde g^{\nu\beta}F_{\alpha\beta} =F_{\mu\nu}F^{\mu\nu}-\frac{2\lambda}{D}W_\nu W^\nu. \tag{48d} \]

This retains electromagnetic kinetics even for a null potential, where \(D=1\). It is a candidate constitutive relation, not a derived binding mechanism. For fixed \(g\), it generally changes under \(A\mapsto A+d\chi\) while \(F\) stays unchanged. Its physical gauge symmetry and the variation of its potential-dependent metric must be specified before it supplies field equations.

The force term itself has a valid factorization. If an admissible prescription removes the extra connection term, the geodesic equation takes the conditional form

\[ \tfrac12g^{\mu\alpha}(A_\nu F_{\alpha\lambda}+A_\lambda F_{\alpha\nu})u^\nu u^\lambda =(A_\nu u^\nu)g^{\mu\alpha}F_{\alpha\lambda}u^\lambda, \] \[ u^\mu=\frac{dx^\mu}{d\tau},\qquad \kappa=-A_\nu u^\nu, \qquad \frac{d^2x^\mu}{d\tau^2}=\kappa g^{\mu\alpha}F_{\alpha\nu}u^\nu. \tag{49} \]

The coefficient \(\kappa\) is not yet a conserved intrinsic charge-to-mass ratio. Under an ordinary gauge shift \(A\mapsto A+d\chi\), it changes by \(-d\chi/d\tau\) even though \(F\) does not. The potential-dependent metric changes too. A model can assign physical meaning to additional potential structure, but must then define its actual gauge symmetry and measurement coupling.

A conditional identity survives if the rank-one dyad is treated as a tensor \(h\), with a separate invertible Weyl metric used to raise indices. At nonzero \(A\), recurrence of \(h\) implies recurrence of \(A\), but the two possible divergence contractions have opposite signs:

\[ h_{\mu\nu}=A_\mu A_\nu,\quad \nabla_\sigma h_{\mu\nu}=2\phi_\sigma h_{\mu\nu} \Rightarrow\nabla_\sigma A_\mu=\phi_\sigma A_\mu, \] \[ g^{\mu\nu}\nabla_\mu A_\nu=\phi^\nu A_\nu, \qquad \nabla_\mu A^\mu=-\phi^\nu A_\nu. \tag{49a} \]

Here recurrence means a covariant transport law, not a repeated orbit in physical time. Equation (49a) is an identity under that supplied constraint, not a proof that the full metric’s dynamics produce a charged particle. It neither repairs Equation (48a) nor identifies a measured current.

The separate scalar-gradient current proposal can be written with explicit flat-space signs. For \(x^0=ct\), signature \((+---)\), and \(\Box_+=c^{-2}\partial_t^2-\nabla^2\), a conserved candidate is

\[ S=\partial_\mu A^\mu,\qquad J^\mu=\frac1{\mu_0}\partial^\mu S, \qquad \rho=\frac1{\mu_0c^2}\frac{\partial S}{\partial t}, \qquad \mathbf J=-\frac1{\mu_0}\nabla S, \qquad \partial_t\rho+\nabla\!\cdot\mathbf J=\frac1{\mu_0}\Box_+ S=0. \tag{50} \]

The spatial minus sign is required by continuity with this convention. Conservation of the candidate in Equation (50) does not identify it with the worldline coefficient in (49) or either contraction in (49a). A common action and coupling must establish those relations.

The physical comparison is already demanding: proton and antiproton charge-to-mass ratios have been compared with a fractional uncertainty of 16 parts per trillion, giving \(-(q/m)_p/(q/m)_{\bar p}=1.000000000003(16)\).²¹ A universal prescribed connection assigns one autoparallel acceleration to the same event and velocity; opposite charge responses require an internal state or additional coupling. Deriving that response from finite field closures remains possible in principle, but is not supplied by the bare geodesic rule.

Lindgren et al. then locate the proposed transition between linear and generalized behavior in circulation rather than field intensity:

\[ (E\gamma_t,\mathbf p)=eA, \qquad k(2\pi r)=2\pi n, \qquad \Phi_{\rm orb}=\oint A_\perp\!\cdot d\boldsymbol\lambda =n\frac{h}{e}, \qquad n\in\mathbb Z. \tag{51} \]

Equation (51) recovers flux quantization once \(e\) is supplied. The paper does not solve Equation (47) for a localized stable charge that selects the observed \(e\), electron mass, proton mass, or fine-structure constant. Those remain the nonlinear eigenvalue problem inherited by UMFT.

XX. PROPER TIME AS AN EVOLUTION COORDINATE

Can the spacetime part carry its own clock?

Lindgren’s 2026 Stueckelberg paper supplies a second operator.¹⁴ A transition amplitude \(\varphi(\tau,t,\mathbf x)\) evolves in a proper-time parameter \(\tau\), independently of its spacetime arguments. With a self-adjoint generator and suitable boundary conditions, its norm integrated over spacetime is conserved under \(\tau\)-evolution:

\[ i\hbar\frac{\partial\varphi}{\partial\tau} =-\frac{\hbar^2}{2m}\Box\varphi+V(\mathbf x)\varphi, \qquad \Box=-\frac1{c^2}\frac{\partial^2}{\partial t^2}+\nabla^2, \qquad \varphi\in L^2, \tag{52} \] \[ \rho=\varphi^*\varphi, \qquad J^\mu=\frac{i\hbar}{2m} \left(\varphi\nabla^\mu\varphi^*-\varphi^*\nabla^\mu\varphi\right), \qquad \frac{\partial\rho}{\partial\tau}+\nabla_\mu J^\mu=0, \qquad \int|\varphi|^2d^4x=1. \tag{53} \]

The current sign in Equation (53) follows the forward-proper-time convention of Equation (52). Reversing the sign of the proper-time generator reverses that current. The paper next proposes a fixed-speed coordinate-time telegraph equation, written here for a distinct reduced field \(\psi\):

\[ d\tau=\sqrt{1-\frac{v^2}{c^2}}\,dt, \qquad \frac{i\hbar}{\sqrt{1-v^2/c^2}}\frac{\partial\psi}{\partial t} -\frac{\hbar^2}{2mc^2}\frac{\partial^2\psi}{\partial t^2} =\left[-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf x)\right]\psi. \tag{54} \]

This is a candidate reduction, not a general change of variables in Equation (52). For the explicit restriction \(\psi(t,\mathbf x)=\varphi(t/\gamma,t,\mathbf x)\), the chain rule gives \(\partial_t\psi=\gamma^{-1}\partial_\tau\varphi+\partial_t\varphi\). The second derivative also contains mixed derivatives. A worldline relation between clock readings cannot by itself discard these terms while retaining the original coordinate-time second derivative. Equation (54) can be tested as an equation in its own right; its equivalence to Equation (52) requires an additional reduction prescription.

For a self-adjoint Hamiltonian \(\mathcal Hf=\lambda f\), \(T(t)=e^{rt}\) reduces Equation (54) to

\[ \alpha r^2+\beta r-\lambda=0, \qquad \alpha=-\frac{\hbar^2}{2mc^2}, \qquad \beta=\frac{i\hbar}{\sqrt{1-v^2/c^2}}, \qquad r_\pm=\frac{-\beta\pm\sqrt{\beta^2+4\alpha\lambda}}{2\alpha}, \tag{55} \] \[ a=\frac{\hbar^2}{2mc^2},\quad b=\hbar\gamma,\quad q_0=\frac{b}{2a},\quad \Omega_j^2=q_0^2+\frac{\lambda_j}{a}, \qquad \lambda_j>-\frac{mc^2\gamma^2}{2} \Rightarrow r_{j\pm}=i(q_0\mp\Omega_j), \tag{56} \] \[ \psi(t,\mathbf x)=\sum_j f_j(\mathbf x) \left(A_j e^{r_{j+}t}+B_j e^{r_{j-}t}\right), \qquad \|\psi(t)\|^2=\sum_j\left|A_j e^{r_{j+}t}+B_j e^{r_{j-}t}\right|^2. \tag{56a} \]

Each exponential has unit modulus in this spectral range. Their sum generally does not. The two roots multiply the same spatial eigenfunction, so spatial orthogonality leaves their interference term intact. The source’s two-root expansion therefore does not establish conservation of the ordinary scalar norm. Even at the repeated imaginary root, the general solution contains \(e^{iq_0t}(A+Bt)\), which can grow.

The initial-value test below uses one normalized spatial mode and real branch amplitudes \(1-\zeta\) and \(\zeta\), with \(0\le\zeta\le1\). The fraction is an amplitude choice, not a probability:

\[ -a\ddot u+ib\dot u=\lambda u,\qquad u(0)=1,\qquad\dot u(0)=i[q_0+(2\zeta-1)\Omega], \tag{56b} \] \[ u(t)=(1-\zeta)e^{i(q_0-\Omega)t}+\zeta e^{i(q_0+\Omega)t}, \qquad |u(t)|^2=1-4\zeta(1-\zeta)\sin^2(\Omega t). \tag{56c} \] \[ \chi=e^{-iq_0t}u,\qquad\ddot\chi+\Omega^2\chi=0, \qquad \mathcal E_\chi=|\dot\chi|^2+\Omega^2|\chi|^2=\mathrm{constant}. \tag{56d} \] \[ \Xi=\begin{pmatrix}\chi\\i\dot\chi/\Omega\end{pmatrix}, \qquad i\dot\Xi=\Omega\begin{pmatrix}0&1\\1&0\end{pmatrix}\Xi, \qquad\|\Xi\|^2=\mathcal E_\chi/\Omega^2. \tag{56e} \]

For \(\Omega^2>0\), Equation (56e) is a precise two-component unitary construction: a decrease in one component is accompanied by an increase in the other. Its norm is not the scalar Born norm \(|u|^2\). Alternatively, selecting only one branch, with its compatible initial derivative, gives unitary scalar evolution. A physical model must specify which state space and observable it means; these two scalar components do not become a spinor or two neutrino species by being drawn separately.

The source’s d’Alembertian in Equation (52) uses the opposite, \((-+++)\), metric convention to the earlier Maxwell and invariant-mass equations. The coordinate-time telegraph form fixes \(v\) and therefore partially gives up the Lorentz covariance of the proper-time equation, as the paper states. The source also supplies a confining potential to obtain its discrete spatial modes; it does not derive autonomous confinement.

FIGURE 8

Two steady phases do not make a steady norm.

The phase pointers and frequency curves retain the two roots. The lower plot evolves both initial data in Equation (56b), compares the scalar norm with the conserved doubled-state norm, and exposes their difference. Units are \(\hbar=m=c=1\).

Hamiltonian eigenvalue λ1.00 Frame speed v/c0.00 High-branch amplitude fraction ζ0.50 Coordinate time0.00

r+ branch frequency: −0.7321
r− branch frequency: 2.7321
branch modulus: 1.0000 / 1.0000
polynomial residual: 0.00e+0
scalar norm |u|²: 1.000000
doubled norm / initial: 1.000000

At equal amplitudes the scalar norm can vanish while its derivative remains nonzero. No energy or probability loss has been established by that zero: the scalar norm has not been shown to be the conserved probability of this two-branch system. At either amplitude endpoint, only one branch remains and the scalar norm stays one.

XXI. WHAT THIS BORROWING EARNS

Has geometry produced the atom’s parts?

Not yet. The borrowed program supplies candidate equations for potential-dependent geometry, propagating divergence, and independent proper-time evolution. The first simulation tests a flat-background product identity; the nonmetric variational problem remains to be completed. The second separates oscillatory roots from a unitary state evolution: a branch prescription or a justified enlarged state space is still required. Neither calculation produces a localized charged particle.

The decisive calculation remains open: a finite-energy solution of Equations (45)-(50) must localize, select charge magnitude, acquire half-spin, and yield the electron or proton invariants without those measurements as coefficients. Until that solution exists, “waveknot” names a required behavior, not a solved particle.

Can one geometric field action produce stable charged closures, multi-closure nuclear pressure, and the measured transition matrix elements under a single set of coefficients?

References

J. Lindgren, A. Kovacs, and J. Liukkonen, “Electromagnetism as a purely geometric theory”, Journal of Physics: Conference Series 2987, 012001 (2025).
J. Lindgren, “On the Unitarity of the Stueckelberg Wave Equation and Measurement as Bayesian Update from Maximum Entropy Prior Distribution”, Quantum Reports 8, 18 (2026).
A. Delhom, “Minimal coupling in presence of non-metricity and torsion”, The European Physical Journal C 80, 728 (2020). Metric-volume and affine divergences differ in nonmetric geometry. Equations (47a)-(47b) are evaluated here in the convention of Equation (48).
M. J. Borchert et al., “A 16-parts-per-trillion measurement of the antiproton-to-proton charge–mass ratio”, Nature 601, 53–57 (2022).