Maxwell, Einstein and Dirac as projections of a single Hessian

[Roberto Pavani]

TL;DR

Assuming Maxwell, Einstein or Dirac valid for any observer, is it possible to describe those equations as three different projections of a single mathematical object, with what contraint? The proposed structure link the three equations using the Hessian of an Helmholtz field in Euclidean R4 space. The constraints are non-trivial and produce falsifications but also zero-parameters predictions.

Roberto Pavani — Independent Researcher, Sestri Levante (GE), Italy

ORCID: 0009-0002-9098-1203



Full papers on Zenodo:

Lean version (proofs): DOI: 10.5281/zenodo.19599262
Extended version (phenomenology): DOI: 10.5281/zenodo.19598026


1. What this is (and what it is not)



This is not a theory of everything. It does not derive Maxwell, Einstein or Dirac — it takes all three as established empirical facts.



The question is: if all three equations are true, can they be described as three different projections of a single mathematical object? And if so, what constraints follow from this simultaneous containment?



The answer is yes. The object is the Hessian $$\Theta_{ab} = \nabla_a \nabla_b \theta$$ of the phase of a Helmholtz field in flat Euclidean $$\mathbb{R}^4$$. The constraints are non-trivial and produce zero-parameter predictions.





2. The starting point



Consider a flat, static, four-dimensional Euclidean space $$(\mathbb{R}^4, \delta_{ab})$$ — no time, no causality, no dynamics. In this space, define a complex field H obtained via the Clarke transform (positive-sequence projection) of a balanced three-phase Helmholtz triplet:



$$H = \frac{2}{3}(h_1 + h_2\,e^{i2\pi/3} + h_3\,e^{i4\pi/3}), \qquad (\Delta + k^2)h_\tau = 0$$



The three components are not summed algebraically but projected onto a complex field via the Clarke transform (positive-sequence projection). Three is the minimum number producing a rotating phase with constant amplitude (a singlet has no rotating phase; a doublet has no balancing; $$N \geq 4$$ reduces to triplets).



The phase $$\theta = \arg(H)$$ and its gradient define a unit vector $$u^a = \nabla^a\theta / \lVert\nabla\theta\rVert$$ and a tetrad [imtexath]\{u^a, E_i^a\}[/tex]. The tensor



$$g_{ab} := \delta_{ab} - 2u_a u_b$$



has Lorentzian signature $$(-,+,+,+)$$ by construction (trivial check: $$g_{ab}u^a u^b = 1 - 2 = -1$$, $$g_{ab}E_i^a E_j^b = \delta_{ij}$$).



Time is not postulated — it arises as the direction of maximum phase variation.





3. Three sectors from one Hessian



The Hessian $$\Theta_{ab} = \nabla_a \nabla_b \theta$$ decomposes with respect to the tube axis $$w^a$$ into three orthogonal blocks (tensorial identity, valid for any symmetric 2-tensor):



$$\Theta_{ab} = \underbrace{w_a w_b (w^c w^d \Theta_{cd})}_{\text{LL (time scale)}} + \underbrace{w_a \mathcal{V}_b + \mathcal{V}_a w_b}_{\text{Mixed (EM-like)}} + \underbrace{\Pi^w_a{}^c \Pi^w_b{}^d \Theta_{cd}}_{\text{TT (GR-like)}}$$



Mixed block → antisymmetric extractor $$\mathcal{F}_{ab} = w_a \mathcal{V}_b - \mathcal{V}_a w_b$$ satisfies $$d\mathcal{F} = 0$$ exactly for constant $$w^a$$ (proved: the third derivative $$T_{cab} = \nabla_c \nabla_a \nabla_b \theta$$ is totally symmetric in flat space). This is Maxwell's first equation.
TT block → extrinsic curvature $$K_{ab} = (\mathbb{G}_{\text{op}})_{ab}/\lVert\nabla\theta\rVert$$ of the equal-phase hypersurfaces. Via Gauss–Codazzi–Ricci, this gives the full Riemann tensor of $$g_{ab}$$. By Lovelock's theorem, the unique field equation is Einstein's equation $$G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}$$.
Helmholtz factorisation → $$({\not\!\partial} - ik)\Psi = 0$$ via $${\not\!\partial}^2 = \Delta$$. The complexified Clifford isomorphism $$\text{Cl}(4,0) \otimes \mathbb{C} \cong \text{Cl}(1,3) \otimes \mathbb{C}$$ maps this to the Dirac equation.




4. The fundamental constraint: Pythagorean decomposition



Since the three blocks are orthogonal projections of the same tensor:



$$|\Theta|^2 = (\text{LL})^2 + |\mathcal{F}|^2 + |\mathbb{G}_{\text{op}}|^2$$



This is a pointwise algebraic identity. The physical consequence: the three sectors share a fixed total norm. A change in one must be compensated by the others. Inter-sector energy convertibility (EM ↔ gravitational ↔ mechanical) is a theorem, not an empirical input. No action, Lagrangian or Noether symmetry is required.





5. Quantitative predictions (zero free parameters)



Three lepton families



The radial profile $$j_1(\kappa r)$$ (spherical Bessel, $$\ell = 1$$) has zeros at $$a_1 \approx 4.493$$, $$a_3 \approx 10.904$$, $$a_5 \approx 17.221$$. The three-phase twist-closure condition selects exactly the odd zeros (the even zeros miss by exactly $$120° = 2\pi/3$$, one triphase offset). This gives three families:



Electron: reference

Muon: $$m_\mu/m_e$$ predicted = 212.0, experimental = 206.8 (deviation +2.5%)

Tau: $$m_\tau/m_e$$ predicted = 3308, experimental = 3477 (deviation −4.9%)



Mass formula: $$m_i/m_e = (a_i/a_1)^6 \cdot (\phi(a_1)/\phi(a_i))^2$$, where the exponent $$6 = 2 \times \dim(\ker u)$$ (volume scaling × energy quadraticity) and $$\phi(a_n)$$ is the accumulated twist of the transverse frame.



Anomalous magnetic moment



The Dirac factorisation gives $$g = 2$$ pointwise. The finite radial profile of $$j_1$$ creates a mismatch between the internal energy distribution (TT block, $$\int j_1^2\,dr$$) and the far-field reconstruction (TL block, $$\int j_1'^2\,dr$$). An exact Bessel identity gives:



$$\mathcal{E}_{\text{TL}} = \mathcal{E}_{\text{TT}} - \int_0^{a_1} \frac{j_1^2}{r^2}\,dr$$



Combined with the Malus projection (holonomy of the transverse frame, $$\sin^2\phi(a_1)$$) and the angular normalisation $$3/(4\pi)$$:



$$(g-2)/2 = 0.001\,160 \quad \text{(elliptic closure, deviation } +0.016\%\text{)}$$



Experimental: $$0.001\,159\,652\dots$$ — zero free parameters.



The structure $$\alpha/(2\pi)$$ is identical to Schwinger's formula, but here $$\alpha$$ is the geometric closure asymmetry forced by $$d\mathcal{F} = 0$$ (no monopoles), not the fine-structure constant inserted by hand.





6. Falsification conditions



The framework makes explicit predictions that can be violated:



Magnetic monopole discovered → framework falsified ($$d\mathcal{F} = 0$$ is a theorem)
Point charge observed (exactly zero size) → falsified ($$C^\infty$$ regularity)
Fourth lepton family → falsified (three-phase closure admits exactly 3)
Lepton with extra quantum number beyond spin and charge → falsified ($$\ell = 1$$ exhausts the degrees of freedom)
Curvature everywhere accompanied by EM signal → falsified (nodal regions necessarily exist)
White hole observed → falsified ($$d\theta/ds > 0$$ everywhere)
Gravitational dynamics beyond Einstein (massive spin-2, non-metric torsion) → falsified


All conditions are consistent with current observations.





7. What the framework does NOT do



Does not derive Maxwell, Einstein or Dirac — takes them as facts
Does not explain why Helmholtz, why three phases, why $$\ell = 1$$
Does not replace QED, QCD or the Standard Model
$$\kappa$$ (Newton's constant) is empirical, not derived


The framework provides an algebraic bridge between the three theories whose consequences — energy conservation across sectors and retroprojection constraints — are theorems, not assumptions.





8. Summary



Starting from three postulates (flat $$\mathbb{R}^4$$, three-phase Helmholtz field, operational observer), the Hessian of the phase decomposes into Maxwell, Einstein and Dirac sectors. The sectors share a fixed total norm (Pythagorean constraint), producing:



Structural constraints: no monopoles, no point sources, zero total charge, EM-silent curvature regions, exactly 3 lepton families
Zero-parameter predictions: mass ratios (2–5% accuracy), $$(g-2)/2 = 0.001160$$ (+0.016% deviation)
Falsification conditions: 7 explicit conditions, all consistent with current data


The image $$\text{Im}(\pi) \subsetneq \mathcal{S}_{\text{EM}} \times \mathcal{S}_{\text{GR}} \times \mathcal{S}_{\text{Dirac}}$$ is a proper subset: the restrictions are the content of the framework.





References:

R. Pavani, Lorentzian Structure, Maxwell and Dirac Equations Linked by the Algebraic Decomposition of the Hessian of a Harmonic Phase, Zenodo (2026), DOI: 10.5281/zenodo.19599262
R. Pavani, Physical Structures Contained in the Hessian of a Harmonic Phase: a Spectral Functional Framework, Zenodo (2026), DOI: 10.5281/zenodo.19598026


Critical feedback welcome, particularly on the TT/TL identification with the gyromagnetic ratio and the twist-closure family selection.[/tex]