Quantengravitation im Regge-Grenzraum

The Ricci–Planck Programme: A Unified Geometric Framework for Fundamental Physics M. Nolden — Preprint Series, March 2026 We present a unified geometric framework in which gravity, electromagnetism, quantum mechanics, charge quantisation, spin, parity violation, the arrow of time, and the hydrogen spectrum all emerge from two primitives: a new geometric object called the Regge limit space G, and a single stochastic equation on it. No new physical postulates are introduced at any step. Every physical law in the programme is a theorem. The starting point is a question about local coordinates in four-dimensional physics. If one wishes to compute physics at each space node independently — without destroying the global topological invariants of the space (holes, winding numbers, linking numbers) — what is the minimal local coordinate unit? The Gauss–Bonnet theorem answers this question unambiguously: in four dimensions, one needs a 4-simplex (pentachoron). Simplices of lower dimension cannot enclose three-dimensional voids, and therefore cannot represent topological defects in local computations. This is the four-dimensional analogue of the well-known fact in computer graphics that triangles — not edges or points — are the minimal unit for representing two-manifolds with holes. The 10 edge lengths of a pentachoron encode exactly the 10 independent components of the Ricci tensor; its boundary is homeomorphic to S³ and can enclose a three-dimensional surgery site. The pentachoron is the natural simulation unit of four-dimensional physics. Taking the Gromov–Hausdorff closure of all pentachoral approximations to a smooth Riemannian 4-manifold M defines the Regge limit space G, a subspace of the classical DeWitt superspace of diffeomorphism-invariant metrics. G is dense in the DeWitt superspace, complete, separable, and decomposed into topological sectors by winding numbers and homology classes. This is the central innovation of the programme: not a new physical postulate, but the identification of the correct mathematical arena. Every subsequent result is a consequence of working in G rather than in flat space or in the full superspace, where the relevant measures and equations are ill-defined. On G, a single stochastic Langevin equation — the Ricci–Planck equation — governs the evolution of metric configurations. Its drift is the negative gradient of the Regge action (the discrete Einstein–Hilbert action), and its diffusion is Planck-scale white noise with amplitude equal to the squared Planck length. The Ricci flow component smooths all geometric fluctuations, driving the metric toward saddle points of the action; the Planck noise continuously injects new curvature. Their competition is the engine of all physics. The Regge action splits orthogonally under the Hodge decomposition into self-dual and anti-self-dual sectors, forced by the Lie algebra decomposition SO(4) = SU(2)+ × SU(2)−. The self-dual sector yields the Einstein field equations; the anti-self-dual sector yields the Maxwell equations. Gravity and electromagnetism are not two theories unified by an external principle — they are the two eigenspaces of parity acting on the same geometric object, the deficit angle at each hinge of the lattice. Charged particles arise as surgery sites: topological defects where a four-ball has been excised from the manifold and reglued with a map of integer winding number q, an element of pi_3(S³) ≅ Z. Charge is winding number. The Hopf fibration of the surgery boundary identifies the anti-self-dual deficit angle with a U(1) holonomy: after a full loop around a surgery site, the local coordinate frame has rotated by 2πq in the anti-self-dual sector. This is the geometrical content of electric charge — a place where the way one measures distances runs once in a circle. Charge quantisation follows unconditionally from the topology of pi_3(S³); no magnetic monopole is required. The Kolmogorov forward equation of the Ricci–Planck process is the Fokker–Ricci–Planck equation on G. This is the physically fundamental equation — not the Schrödinger equation. The Schrödinger equation is a derived object, obtained by writing the probability density as the squared modulus of a complex amplitude (which forces U(1) gauge invariance and thereby the electromagnetic potential as a mathematical necessity) and rotating to imaginary time. Its eigenmodes are the stable vibrational modes — the solitons — of topological objects in G. Every permanently stable solution of the Fokker–Planck equation must carry topological holes; all matter without holes disappears in finite time under the Ricci–Planck flow. Matter is not an input to the theory. It is the residue that the Ricci flow cannot eliminate. Two structural theorems reveal the global architecture of G. The Double Projection Theorem states that every smooth curve in four-dimensional space arises as a test-particle geodesic under some metric in G: what appears curved in spacetime is straight in G, the superspace analogue of Einstein's equivalence principle lifted one level. The Representation Theorem provides a long exact sequence decomposing the homology of G into observable physics and topological invariants; the Gauss–Bonnet–Chern theorem emerges as the rank formula for the kernel, a consequence of the representation structure rather than an independent postulate. The framework resolves several long-standing conceptual problems. The measurement problem is resolved: wavefunction collapse is a surgery event, a discontinuous topological jump between sectors of G induced by Perelman neck-pinch singularities of the Ricci flow; the Born rule follows as a theorem from the normalisation of the Fokker–Planck distribution. Hawking radiation is topological pair production at the event horizon, where strong curvature makes spontaneous surgery probable, creating particle-antiparticle pairs with opposite winding numbers. The arrow of time is the Perelman W-functional monotonicity of the Ricci–Planck flow. There is no Big Bang as a singular initial moment: the topologically trivial empty sector of G is ergodically unstable under Planck noise, and will produce matter with probability 1. There is no end of the universe: after all matter annihilates, the empty sector produces new matter again. Time is unceasing because the Planck noise amplitude is strictly positive, which is equivalent to the condition for quantum mechanics to exist. A universe without quantum mechanics would be topologically frozen. The programme is presented in five self-contained papers. The single open problem — Mosco convergence of the discrete Dirichlet forms, which would make the continuum limit fully rigorous — is a well-posed problem in the Kuwae–Shioya framework, with two of three required ingredients already established as theorems.