Foundations

Established Physics That PM Builds Upon

This page showcases the well-established equations, constants, and experimental measurements from the scientific literature that Principia Metaphysica builds upon. These are NOT predictions or new formulas—they are the proven foundation of modern physics.

... Foundational Equations
... Physics Domains

Trace any formula back to b3

Every prediction in Principia Metaphysica is reducible to one topological seed: the third Betti number of the G2 manifold, b3 = 24. Click any of the examples below to open an interactive dependency graph showing the symbolic (Arithma) and operator-token (EML) walks side by side. Each graph terminates at the same root: b3 = 24.

Dark energy w0 equation of state Derives the DESI-measured w0 directly from -23/24, where 24 = b3.
Three generations ngen from topology Three fermion generations arise as χeff/(4·b3) = 144/48 = 3.
Fine-structure constant α-1 geometric α-1 resolves to a closed-form expression rooted in b3 via the G2 volume.
Foundational Equations — Standard Notation Foundational Equations — EML Mirror Phase Operator Form
Einstein Field Equations
Albert Einstein (1915)
General Relativity
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$
The Einstein field equations relate the curvature of spacetime (encoded in the Einstein tensor Gμν) to the energy-momentum content (Tμν). The cosmological constant Λ drives the accelerated expansion measured by DESI 2025, consistent with the PM prediction w₀ = −23/24 ≈ −0.9583.
G_uv = ops.sub(ops.ricci_tensor(g_uv), ops.mul(ops.div(1,2), ops.mul(ops.ricci_scalar(g_uv), g_uv)))
EFE = ops.add(G_uv, ops.mul(Lambda, g_uv)) == ops.mul(ops.div(ops.mul(8, pi_val), ops.pow(c, 4)), T_uv)
EML Mirror Phase form: Curvature ops (ops.ricci_tensor, ops.ricci_scalar) act on the metric tensor g_uv. The cosmological constant term ops.mul(Lambda, g_uv) enters as an additive operator. In PM: Λ is not a free parameter — it is constrained by the b₃=24 topology via Lambda = ops.inv(ops.mul(b3, Vol_V7)) (vacuum residue after G₂ compactification). Mirror Phase Mathematics form — same numerical result as standard notation.
Dirac Equation
Paul Dirac (1928)
Quantum Field Theory
$$(i \gamma^\mu \partial_\mu - m)\psi = 0$$
The Dirac equation describes relativistic spin-½ fermions. In the PM framework, the 4096-component Primordial Spinor Field ΨP arises from the Clifford algebra Cl(26,1) of the M27(24,1,2) bulk. The three fermion generations (ngen = χeff/(4·b₃) = 144/48 = 3) are spectral residues of the Dirac operator on the G₂ holonomy manifold V₇.
dirac_op = ops.sub(ops.mul(ops.mul(i_unit, gamma_mu), partial_mu), m_fermion)
dirac_eq = ops.apply(dirac_op, psi) == eml_scalar(0)
EML Mirror Phase form: The Dirac operator is built from ops.mul of the imaginary unit, gamma matrices, and the partial derivative operator, minus the mass term. In PM: n_gen = ops.div(chi_eff, ops.mul(eml_scalar(4), b3)) = ops.div(eml_scalar(144), eml_scalar(48)) = eml_scalar(3) — three generations from G₂ topology, not a free parameter. Mirror Phase Mathematics form — same numerical result as standard notation.
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