The name is absent

A.P. Kirilyuk

and {ψ0n (ξ),П0ⁿ} is the eigen-solution set for the second-level truncated system:

The same mechanism of dynamic multivaluedness due to the essentially nonlinear EP
dependence on the eigen-solutions to be found in eqs. (16)-(17) leads to the second level
of splitting, this time of auxiliary system solutions entering the first-level expressions
(8)-(12), into many mutually incompatible realisations (numbered by index r′):

{ψn⁰i (^ξ)^, ηn⁰i }→{ψn r '(^ξ)^{, η0 r}_i'} .                             ⁽¹⁹⁾

We can continue to trace this hierarchy of dynamical splitting by applying the
same EP method to ever more truncated systems of equations, such as eqs. (18), and
obtaining corresponding levels of dynamically multivalued structures with the attached
intrinsic space and time, until we obtain a directly integrable equation for one unknown
function. The maximum number of levels in this dynamically multivalued hierarchy is
equal to the number of component states (excitations) NQ , although in practice each of
them need not be resolved. We can now specify the detailed, probabilistically fractal
structure of the complete general solution to the interaction problem, eq. (13):

V^ +

P(ξ, Q)= ∑ Pr>_rr'r,..(ξ, Q) ,                      (20)

r,r′,r′′...

with indexes r, r′, r′′,... enumerating permanently, chaotically changing realisations of
consecutive levels of dynamic (probabilistic) fractality, naturally emerging thus as the
unreduced, truly exact solution to any real many-body problem, eqs. (1), (2), (5). The
time-averaged expectation value for the dynamically fractal density is given by

v^

ρ(^ξ,Q)= Σ ^αrr'r∙...Prr'r"... (^ξ,Q) ^{,                        (21)}

r,r′,r′′...

where the dynamically determined probabilities of the respective fractality levels are
obtained in a form analogous to eq. (14)

Multivalued fractal solution of eqs. (20)-(22) can be obtained in a number of
versions, but with the same essential result of probabilistically adapting hierarchy of
realisations. Consecutive level emergence of unreduced dynamic fractality should be
distinguished from perturbative series expansion: the latter provides a qualitatively
incorrect, generically “diverging” (because of dynamic single-valuedness [9])
approximation for a single level of structure, while the series of levels of dynamic
fractality corresponds to really emerging structures, where each level is obtained in its
unreduced, dynamically multivalued and entangled version. In fact, the ultimately
complete, dynamically fractal version of the general solution demonstrates the genuine,
physically transparent origin of a generic problem “nonintegrability” (absence of a
“closed”, unitary solution) and related “nonseparability” (now being clearly due to the
physical, fractally structured and chaotically changing component entanglement).

The dynamically probabilistic fractal thus obtained is a natural extension of the
ordinary, dynamically single-valued (basically regular) fractality, which is especially

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