Papers
The complete MFT corpus is published on Zenodo as 16 self-contained papers (P0–P15) under one always-current DOI. Each paper is downloadable as a PDF with full LaTeX source. Citing the corpus DOI ensures readers always get the most recent version of every paper.
Always-current corpus DOI
This DOI always resolves to the latest version of the entire corpus. Individual paper DOIs (which change with each version) are not used; cite the corpus DOI to reference any paper.
How to read the corpus
The 16 papers form a single derivation chain: each step depends on the previous. For first-time readers we recommend starting with the Understanding Guide or the Synthesis page, then choosing one of three reading orders:
- Foundations-first (most rigorous): P0 → P1 → P2 → P9 (the flagship), then branch into the three sectors via the topic pages.
- Results-first (most accessible): P9 (flagship) → P4 (mass spectrum) → P3 (F3 theorem) → P5 (galactic) → P8 (microphysics). Theorems and quantum completions can come after.
- Topic-by-topic (most curated): use the Foundations, Particles, Gravity, Quantum pages. Each weaves the relevant papers together.
The 16-paper corpus
Gravitational Field Equations
Derives the modified Einstein equations from the MFT action with non-minimal coupling. Establishes the scalar–tensor structure, the 3D spatial formulation, and the role of the ordering parameter $\tau$. The action's structural form before any specification of $F(\varphi)$, $V_6(\varphi)$, or matter content.
Key result: The MFT action $S[h, \varphi] = \int d^3x \sqrt{h}\, [F(\varphi) R^{(3)} - \frac{\kappa}{2} D_i\varphi D^i\varphi - V_6(\varphi) + \mathcal{L}_{\text{matter}}]$ and its field equations.
Why $K_4 < 0$ — Sign Constraints from First Principles
Proves all four sign constraints on the sextic potential coefficients from elementary physical requirements: $\lambda_6 > 0$ (energy bounded below), $m_2 > 0$ (localisation), $\lambda_4 > 0$ from Derrick's theorem in 3D, and $\lambda_4^2 > 4 m_2 \lambda_6$ for barrier existence. There are no choices.
Key result: The answer to "why $K_4 < 0$?" is "because particles exist in three dimensions" — Derrick's theorem applied to Q-ball solitons forces an attractive quartic.
The Symmetric Back-Reaction Theorem
Derives the potential shape condition $\lambda_4^2 = 8 m_2 \lambda_6$ from the requirement that the contraction field's gravitational back-reaction is self-consistent at both vacua simultaneously ($\Sigma(\varphi_b) = \Sigma(\varphi_v)$). This is the origin of the silver ratio $\delta = 1 + \sqrt{2}$. The extended theorem then derives $F(\varphi) = F_0 e^{\beta\varphi}$ uniquely.
Key result: The silver ratio is the unique fixed point of the back-reaction map $r \mapsto 2 + 1/r$. Everything else follows.
The Family-of-Three Stability Theorem
Proves that the constrained Morse index of the Q-ball energy functional in the silver-ratio sextic potential admits exactly three physically admissible radial modes: two stable ($n = 0, 1$) and one metastable ($n = 2$). All higher modes are multiply unstable. This is MFT's structural answer to "why three generations?"
Key result: $m_{\text{phys}}(u_0) = 0,\ m_{\text{phys}}(u_n) = n - 1$ for $n \ge 1$. Stable: electron, muon. Metastable: tau. $n \ge 3$: structurally excluded.
Lepton, Quark, and Gauge Boson Mass Hierarchies
Solves the Q-ball soliton equation with sector-specific Coulomb coupling for charged leptons ($Z = 1$), quarks ($Z = 1$ up-type, $Z = 2$ down-type), and electroweak bosons ($Z = 9/5$). With one calibration ($m_e$), reproduces all mass ratios and the Weinberg angle.
Key results: $m_\mu/m_e$ to 1.25%, $m_\tau/m_\mu$ to 0.81%, $m_Z/m_W$ to 0.11%, $m_H/m_W$ to 0.41%, $\sin^2\theta_W$ to 0.4%.
Galactic Rotation Curves from the Silver Ratio
Solves the nonlinear contraction-field BVP on galactic scales with baryon source. The contraction field forms halos whose density profiles reproduce observed rotation curves without dark-matter particles, using the same potential and same coupling $\beta$ as the microphysics sectors. The halo profile shape is derived from the SBR theorem, not fitted.
Key result: $\Sigma\chi^2/\text{dof} = 1.17$ across six spirals (MW, M31, NGC 3198, NGC 2403, NGC 7793, UGC 2259) using two parameters per galaxy plus one global $\beta$.
Spin-½ Emergence from the Q-ball Internal Frequency
Derives fermion vs boson classification from the Q-ball internal frequency $\omega^2$ via emergent Lorentz invariance and the Finkelstein–Rubinstein constraint. Includes five no-go theorems showing that no purely 3D-spatial-slice mechanism can derive spin-½ — emergent Lorentz invariance is essential.
Key result: Fermions $\omega^2/m_2 \in [0.926, 0.958]$, bosons $\omega^2/m_2 \in [0.050, 0.066]$. Structural 14× gap with zero overlap. $g = 2$ follows by Weinberg's theorem.
The Fractional-Charge Confinement Theorem
Proves that the MFT–Skyrme reduction admits only integer-baryon-number solitons as finite-energy asymptotic states, via $\pi_3(SU(2)) \cong \mathbb{Z}$. Fractional-$B$ configurations require bridge winding with linear energy growth — an emergent string tension fixed by the same potential parameters that control the lepton spectrum. Confinement without a separate gauge sector.
Key result: No isolated fractional-charge soliton can satisfy both finite energy and the Derrick virial; bridge tension $T(\lambda_4, \lambda_6) > 0$.
Microphysics: Hadrons, Neutrinos, and the Electroweak Boson Sector
The most comprehensive companion paper. Extends MFT to all of microphysics from the same sextic potential. Includes the MFT–Skyrme reduction, the converged $B = 1$ hedgehog soliton, the chiral-stiffness closure, the neutral pion at $Z = 0$, the neutrino mass-hierarchy ratio $\delta^4 - 1$, the absolute neutrino masses from one-loop gravitational self-energy with universal screening, and the boson coupling $Z_{\text{boson}} = 9/5$ from SO(3) mode counting.
Key results: $f_\pi^2 = \delta$ (0.03%); $m_{\pi^0}$ (0.2%); $m_\sigma = 2 f_\pi$; decuplet 1–6%; $\Delta m^2_{32}/\Delta m^2_{21} = \delta^4 - 1$ (1.2%); absolute neutrino masses $\sim$6%.
Monistic Field Theory: The Complete Foundations (flagship)
The flagship paper. Presents the monistic principle, the benchmark action with every Lagrangian term, the parameter ledger (which parameters are free, which are derived), canonical normalisation connecting MFT to measured coupling constants, field equations in four physical regimes, and the step-by-step derivation of emergent Lorentz invariance and $E = mc^2$ from the contraction dynamics.
Key result: $\sim 20$ free parameters of SM + $\Lambda$CDM replaced by $1$ calibration ($m_e$) + $1$ coupling ($\beta$). Lorentz invariance derived in five steps.
Quantum Completion: Hamiltonian, Constraints, and Fock Space
Constructs the Hamiltonian formulation of MFT: canonical momenta, primary and secondary (Gauss-law) constraints, Coulomb gauge fixing, linearisation around the vacuum, free dispersion relations, and the Fock-space construction for scalar and photon-like quanta. Provides the framework for quantum field theory on the MFT medium.
Key result: Standard QFT structure with constraint-reduced transverse Hilbert space; emergent speed of light $c_{\text{eff}} = \sqrt{Z_{B0}/Z_{E0}}$ shared by all low-energy excitations.
Linearised Propagators and Feynman Rules
Derives the linearised propagators for the scalar contraction mode, the massless photon, and the massive gauge bosons ($W^\pm, Z$) in $(\omega, \mathbf{k})$ space. Provides the perturbative toolkit for loop corrections and scattering amplitudes. Shows that scalar–gauge–gauge couplings vanish under $\varepsilon = 1$.
Key result: Three boxed propagators — scalar $\Delta_\varphi$, photon $D_{ij}^{(\gamma)}$, massive gauge $D_{ij}^{(a)}$. The framework that P14 uses to verify $F_{\text{MFT}} = F_{\text{QED}}$.
Cosmology from Spatial Contraction
Derives cosmological expansion from the relaxation of the contraction field in cosmic voids, without a cosmological constant. Late-time acceleration emerges naturally with $w_\varphi \approx -1$. Under $\varepsilon(\varphi) = 1$, photons accumulate redshift only through the geometric expansion of the medium volume; $\alpha_{\text{EM}}$ is automatically constant. The Hubble tension is flagged as an open problem requiring quantitative numerical work.
Key result: Effective Friedmann equation $H_{\text{eff}}^2 = (8\pi G_{\text{eff}}/3)[\rho_{\text{matter}} + \rho_\varphi]$ with no $\Lambda$ added by hand.
Compact Objects, Screening, and Singularity Resolution
Derives the Yukawa screening mechanism for the contraction field in dense regions, the resulting PPN compatibility with Solar-System tests, and the saturation behaviour inside neutron stars and black holes. The elastic ceiling at $\varphi_v$ bounds the medium's density and prevents curvature singularities.
Key results: $\omega_{\text{BD}} > 40{,}000$ at $\beta \sim 10^{-4}$ (Cassini bound passed); black holes saturate at the elastic ceiling — no singularity at the centre.
The Electromagnetic Form Factor and $g = 2$
Computes the electromagnetic form factor of the electron soliton. The classical soliton charge radius is shown to be a renormalisation artifact; after all-orders quantum dressing, S-matrix equivalence gives $F_{\text{MFT}}(q^2) = F_{\text{QED}}(q^2) + \mathcal{O}(m_e^2/M_{\text{Pl}}^2)$. Includes one-loop verification confirming the scalar-QED log coefficient. Derives $g = 2$ from the Weinberg theorem.
Key results: $F_{\text{MFT}} = F_{\text{QED}} + \mathcal{O}(10^{-45})$; $g = 2$ from emergent Lorentz + FR-spin-½ + minimal coupling; $g - 2 = \alpha/(2\pi)$ at one loop (Schwinger).
3D Finiteness and Renormalisability
Resolves the renormalisability question. Proves that the Q-ball ansatz $\Phi = \varphi(\mathbf{x}) e^{-i\omega\tau}$ exactly separates the ordering-parameter dependence from the spatial profile, reducing quantum corrections to a 3D spectral problem. In 3D, the sextic coupling $[\lambda_6] = 0$ is marginal; the one-loop effective potential is finite; no counterterms are needed. The silver-ratio condition is preserved under quantum corrections.
Key result: $V_{\text{1-loop}}^{(3D)}(\varphi) = -[V''(\varphi)]^{3/2}/(12\pi)$ — closed-form, finite, no counterterms. Silver ratio preserved under RG flow.
How to cite
Cite the always-current corpus DOI rather than individual paper DOIs. This ensures readers always retrieve the most recent version of each paper:
Wahl, D. (2026). Monistic Field Theory: The Complete Foundations and Companion Papers (P0–P15). Zenodo. https://doi.org/10.5281/zenodo.19343255
When citing a specific result, reference the relevant paper number (e.g., "the Family-of-Three theorem (P3, in Wahl 2026)" or "the silver-ratio identity $f_\pi^2 = \delta$ (P8 §3.2, in Wahl 2026)"). Within the always-current DOI, P0–P15 are stable identifiers.
Source code and reproducibility
Every numerical result in the corpus is reproducible from a Python script bundled with the LaTeX source on Zenodo. Curated copies of the most relevant scripts are also available directly from this site, organised by topic page:
- Foundations scripts — gravitational verification, $\lambda$-ratio derivation, $F(\varphi)$ derivation, Lorentz verification, flagship cross-checks (P0, P2, P9).
- Particles scripts — F3 theorem, Q-ball mass spectrum, vector bosons, cross-sector, spin-½, confinement, microphysics, hadrons, neutrinos, Skyrme reduction, hedgehog BVP (P3, P4, P6, P7, P8).
- Gravity scripts — galactic nonlinear BVP, cosmology, compact objects (P5, P12, P13).
- Quantum scripts — quantum completion, propagators, scattering form factor and $g$-factor, 3D finiteness (P10, P11, P14, P15).
All scripts are pure Python (NumPy, SciPy, matplotlib). They run on a standard scientific Python install with no special dependencies. Total runtime for the full corpus is on the order of minutes.