Motivation: Holonomy plays a central role in math, quantum mechanics, quantum computing, and elsewhere. So what kind of "intelligent" process is connected with geometric phase shifts? First of all, holonomy from parallel transport can be quantified as a scalar geometric phase shift or extra vector rotation (precession) after a closed loop, which carries the information about the curvature enclosed by the loop. Reversely, if there is a spin-precession connection leading to a kind of advanced rolling or vortex dynamics (driven by a relativistic connection, mechanical gear-type rolling connection, Berry connection and magnetic monopole in Hilbert space) one should experience feedback effects at some distinct magic precession angles or phases. It was proposed in 2002 that this should happen with electrons in atomic orbits. Serendipitously, there already exists since 1973 a patented mechanical toy (Powerball) exactly showing this kind of chaotic recurrence relation with a magic precession angle on a macroscopic scale. Although it clearly shows chaotic behavior there was no publication mentioning that a strange attractor based on holonomy could be at work in this device. Due to this funny discovery, recurrent holonomy was in 2007/2008 named "Magic Angle Precession" (MAP) equipped with additional online simulations and presented at STAIF 2008.



Recurrent holonomy and connection of this extreme kind has not been found in literature so far. Since both, holonomy and curvature are directly related via connection, we have a situation where curvature is almost directly reacting back onto curvature, which evolves more ore less "noisy" but deterministically towards strange attractors in the basin of attraction showing fixed points, limit cycles, bifurcations ... of course depending on some initial conditions and the manifold. According to the Poincaré-Bendixson theorem this can only happen in a continuous dynamical system if it has three or more dimensions. In addition to idealized Lagrangian systems and methods approximating the low energy response, where degrees of freedom are formally treated as being independent, recurrent holonomy could provide for exact solutions in the high-energy range where degrees of freedom and boundaries become dynamic. The need for an advanced recurrent strategy can be shown with an example related to holonomy: General Relativity works very well but has problems to describe situations analytically if curvature is coupling back non-linearly to the source of curvature. Think about a spinning mass precessing in curved space-time next to a second spinning mass, which also precesses due to the curvature generated by the other mass (de Sitter effect in the linearized case). Since curvature becomes modulated by precession both, precessions and spins will synchronize recursively in a strictly nonlinear process. Taking the equivalence principle such a situation can arise if different gyroscopes (rotating rotors) are recursively accelerating each other, where spin currents are exchanged or transformed into precession. In a recurrent network situation enormous exchange currents can arise. The extension from the MAP unit to a spin-driven recurrent holonomy network was recently initiated. It can already describe the pairing condition and synchronization properties revealing the nature of a net charge. At the present stage the discussion feedback encourages to look at further scenarios - especially "anomalies" - and there are some real candidates, here is a table. Could it be that the problems regarding the Gravity Probe B data analysis can be traced to a model ignoring such effects?
2008 MAP presentation in Grete and New Mexico.



Trying to get a solution to these puzzles MAP could enter as probably beeing the most elemental strange attractor of recurrent holonomy in SO(3), where precession is related to spin not only as an effect but also as a cause. It appears that MAP can be locked into a U(1) qubit (magnetic monopole), a robust recurrent holonomy memory quantum carrying charge via precession and vice versa. Normally, holonomy given by a transcendental Hannay-Ishlinskii-Berry geometric phase is the cause of precession in only one way. But as already mentioned, you can buy a patented mechanical gyroscopic device for about €10, where high frequency spin can be controlled by a low frequency external precession the way back (you may know it, it is called Powerball, Gyrotwister, ...). The device shows clearly how a linear precession-spin coupling can be arranged mechanically in addition to the spin-precession connection based on curvature and acceleration. What we get is holonomy "caught" in a closed nonlinear loop, where the effect couples back to the cause (we can allow for a small "adiabatic" time delay given by the rotor frequency and get memory stored in a precession strange attractor).
Here is the state of art describing how an advanced description of a recurrent holonomy on a simple manifold could look like:
MAP (CHAOS 2008 presentation notes, paper, STAIF 2008 in New Mexico) is a chaotic attractor that can be extended as a neural unit to a recurrent holonomy network. The corresponding magic precession and nutation angles are singularities (in SO(3) intersections between linear and nonlinear magneto-resistance terms, see Chua's circuit model), which increase in number after passing the bifurcation coupling strength limit of a synchronized spin/precession holonomy network. The MAP dynamics shows characteristic singularities given by the cosine map (intersections between the linear and the cosine term), well known as a chaotic map belonging to the family of exponential maps. The chaotic attractor recursion relation can directly be obtained from Euler's dynamical equations. Since MAP is based on a recurrent holonomy controlled by geometric phases a neural gauge theory emerges, where the precession angle as a neural property becomes in a physical sense a local gauge potential driving a spin vector current iteratively. The dynamical structure naturally carries magnetic monopoles with charge strength and number depending on the symmetry and topology, providing for related winding numbers and other topological invariants. MAP can be easily extended to a higher-dimensional recurrent neural net exchanging spin current via precession gradients. So MAP could be a promising approach to neural quantum computing.



There are already some new ideas how to proceed on a deeper math level.

2. Unit-Scale Recurrent Flux Normalization and Inversion

Geometric fluxes like geometric spin currents in MAP obey Gauss law, where the flux density depends on the topology and dimensionality. Applying Gauss law in the isotropic situation the dimensionality is manifest in the exponent of the power-law radial flux weighting. If we want to compare different-dimensional fluxes there is a trick: at the unit scale (radius = 1) the radial exponents containing the dimensionality are not relevant since all powers of radius = 1 are 1. So we must take care about the normalization constants given by the surfaces of n-spheres with unit radius, which provide for the flux densities necessary to calculate the correct coupling strengths and mass-energy densities. The new strategy here proposed does not take the Planck scale as the Unit Scale. Comparing cosmic currents and quantum currents we choose the scale where the corresponding forces are measured and normalized. The intersection scale from normalization and inversion can be obtained as follows: Calculate the gravitational bulk mass where the orbital dynamics shows the human artificial geometric units of length (1 meter) and time (1 second). The unit scale generating bulk mass is given by |m_G|= |4 p²/G|. Doing so, the common unit scale connects scaling laws of different fields with different topology. Assume there is a global exchange of angular momentum currents between a local particle and a cosmic entity, where in the small quantum range local degrees of freedom become "frozen" governed by a considerable change in flux topology and dimensionality (in MAP at least one degree of freedom).

Assigning to the cosmic image (bulk) a 4d, to the quantum memory (baryon) a 2d, and to the unit inversion scale a 3d spherical topology
(a) requires to apply a Unit scale intersection/normalization in the 3d mid scale comparing the surfaces of unit spheres in different dimensions and scales, which can
(b) reduce the number of independent fundamental constants from three to two, and
(c) shows that a quantum particle scale limit is near to the Compton/Fermi scale and not at the unmeasurable small Planck scale.
(d) Finally, it can explain the cosmic expansion, see Friedmann Propulsion in a Flat Holographic Universe, where the quantum critical density exchanging a minimum cosmic current corresponds to 1 memory unit (baryon) in the flat FRW cosmic test model, which could be the MAP U(1) qubit, a cosmic holonomy memory quantum.
(e) On the cosmic scale it shows how a wrong flux normalization (not taking into account topological and dimensional differences between the microscale and macroscale) can be related to the dark energy/matter discrepancy. In the standard approach staying in 3d on all scales (3d-3d-3d) the quantum critical density is instead of 1 baryon about 25 baryons per unit scale volume, which is not representing the real flux density near to the singularities. Therefore, the 3d-3d-3d cosmic flux density with expansion dynamics related to the Hubble constant can force the wrong conclusion that 24/25 or 96% mass-energy is missing and that there could be "dark energy" and "dark mass". In other words, the quantum pressure of one 2d baryon with respect to a 4d cosmic background (responsible for expansion) is as strong as a the pressure generated by 25 particles with a 3d volume and 2d surface interacting with a 3d cosmic background. So "dark energy" and "dark mass" can be assigned to a special low-dimensional baryon topology different from what we expect as 3d observers.

All points above fit very well if the cosmic critical density is obtained from the correct baryon dimensionality with a 2d-3d-4d cosmic flux relation hierarchy. To relate global cosmological and local quantum currents having different topologies and dimensionalities (like in a holographic system) it is helpful to introduce a pure baryon number scaling, where the baryon is the natural mass and interaction flux quantum, see Friedmann Propulsion in a Flat Holographic Universe, Scaling and Extra-Dimensions. This helps to compare an interaction on the quantum level (i.e. the  electromagnetic flux quantum) with a higher-dimensional interaction on the macroscopic level (i.e. the gravitational flux of a bulk mass). Since the unit scale intersection constraint reduces the number of independent fundamental constants from three to two, there is an additional relation connecting the proton mass, the Newton constant, Hubble parameter, and/or the Planck length. These can now be calculated from the unit intersection of quantum gravitational and electromagnetic fluxes knowing the action quantum and light velocity. The Eddington-Dirac Number N_G can be found by dividing m_G into baryons or quantum mass units µ, where the Planck scale limit is obtained by dividing the unit length by N_Gµ c⁴. With this exact number scaling based on a pure geometric input the measured mass and coupling numbers can be tested. What we find is a macroscopic number/mass defect of about 0.9% which can be assigned to nuclear interaction as a typical value for lead and iron, see the Talk and presentation at STAIF 2007. STAIF 2007, STAIF 2008.