imaginary angle $i\eta$, $L_\eta = I - \eta/2 \vec{n} \cdot \vec{\sigma}$. The operator $L_\eta$ is purposed to generate the multifractal measure eq.(33) if $\eta$ is rough, that is if

$$ \int_0^t \eta(s)/2 \, \vec{\sigma} \cdot \vec{n} \, ds = \int_0^t dB^0(s), \qquad (56) $$

with various rescaled “contracting/dilating” directions. The branch cut for closed loops forcing phase singularities eq.(52) must be recovered for the full evolution composition operator :

$$ L = L_{\delta\theta}L_{\eta}. \qquad (57) $$

The global structure of real rotations $SO(3) \sim SU(2)$ is exactly preserved at any given time because the non-commuting generators of real and imaginary rotations are part of the wider group $SO(3, \mathbb{C}) \sim SO(3, 1)$, the Lie algebra of which is isomorphic to the direct sum : $so(3, 1) \sim su(2) \oplus su(2)$. Just as in the Abelian complex case, if the evolution is unpredictable, at least, the algebra $su(2)$ averaged over the noise is therefore preserved. In other words, an averaged Hamiltonian evolution acts only on the real angle, as rough random real and imaginary angle fluctuations are hidden.

This development reveals incidentally a possible connexion between Chern numbers and the multifractal cumulants in the expansion eq.(53). The gauge connexion 2-form curvature $\Omega = d\omega + \frac{1}{2}\omega \wedge \omega$ is indeed non trivial because of singularities densely dispersed, yet Chern characters $Ch(n) = \frac{1}{n!}tr\{(\frac{i}{2\pi}\Omega)^n\}$ are the nth terms in the expansion of the generating function

$$ tr\{\exp\left(\frac{i}{2\pi}\Omega\right)\} \qquad (58) $$

and their integration over a compact region of space, which are the Chern numbers, can be understood as “moments”, whereas Chern classes would correspond to cumulants (please compare the definitions in footnote 39 and eq.(58), both can be found e.g in [60] and in [19]). In that respect, the multifractal parameter $c_2$ should be related in such theories to the second Chern class.

## 8.3 Dealing with internal symmetries

Given the joint probabilistic machinery described in section 2, the measurement process does live in a given algebra, e.g. here $su(2)$, as opposed to the over simplistic and ill-founded approach from Bell’s theorems, in which only randomly picked a priori projection states along any axis are supposed to be independently predetermined by the set of hidden variables and hence contextuality is completely lost.

The Lie algebra has been found to be bond to the noisy evolution of excitations of $\rho$ eq.(57), in subsection 8.2. At the same time, it allows
“polarizer” or “magnetic field” and gets out deterministically, albeit unpredictably, with a projection statistically compatible with their shared property, on the one hand, and compatible with the probability law describing the interaction dynamics with the apparatus vector potentials. The sharing holds at any given time of a stationary environment until the measurement is done.

Again, this preservation necessitates that chaos arises only along individual particle paths but that nonlocal coherence is preserved in the background, here by topological constraints and hyperbolic (or ballistic) transport. Otherwise, it would be arguable that a Lyapunov time scale would erase memory, lost in microscopic degrees of freedom with elliptic (or diffusive) transport. That’s one main difference with the setting of Brownian entanglement [21].

The subsequent evolution of the phase through the singular medium is chaotic but remains deterministic. As we argued, weak solutions preserve Lie groups symmetries, because their generators are part of the wider indefinite orthogonal group when including multifractal scaling generators in L. Thus, as quantization is a topological invariant, the locally preserved memory of initial conditions in the real part of the phase is not affected by the noise, because the noise does not affect averages. Consequently, initial conditions are well preserved throughout the two paths followed by the pair, until an event such as a measurement resets the phase in at least one of them. Hence, along the paths to the experimental apparatus we have:

$$f(\phi_i(x, t)) = f(\phi_i(t_0, x_0)), \qquad (69)$$

for some function f.

In a yet more explicit form, consider a pair excitation, at time $t_0$ and position $x_0$, both corpuscles have opposite projections along a direction $\vec{n}$ of their angular momenta, for example define f as:

$$f(\phi_1(t_0, x_0)|n) = \theta_0/2 \vec{\sigma} \cdot \vec{n} = -f(\phi_2(t_0, x_0)|n), \qquad (70)$$

where the projection along $\vec{n}$ is made explicit as in subsection (8.3). Therefrom, one can directly recover formally any case as, for instance, eq.(15) in the case of photons with the three dimensional representation of SU(2). This is possible if one can make sure that for such a medium, the excitations are really conveying those oscillations internally.

Wrapping up this mathematically, wavelet transforms express the locally deterministic and globally statistical state of the pair excitations as a tensor product of eq.(46):

$$\Psi_{1,2} = \Psi_1 \otimes \Psi_2. \qquad (71)$$

By construction, each local excitation experiences the self background noise along their characteristics, e.g. $L_{\delta\theta}L_{\eta}$. So, statistical summation into non separable states is clearly preparable, just as it is the case for the specific SU(2) group, exemplified using Jones vectors in section (2).
Noether’s theorem to give rise on average to conservation laws, in the weak sense via the optimal Lagrangian. So that, an initially shared property between two bodies can be conserved throughout time and space, before a substantial perturbation occurs.

This is directly related to the autonomous dynamics of an excitation within its reference frame, conserving the action phase integral, as long as the piece of medium crossed is adiabatically homogeneous and no external perturbation occurs.

If we go back to our reasoning, because photon measurements involve the electromagnetic field (or electrons in Stern-Gerlach type of experiments), we are entitled to postulate that initial conditions are hidden in the polarization vector’s phase, because this phase is the single aspect affecting those experiments involving the vector or scalar potentials ($\vec{A}, \Phi$).

Let us draw the parallel for concreteness with two isolated excitations generated on the medium, prepared simultaneously at the same location, such that the total angular momentum is null. Imagine a nonlinear scenario, in which a soliton and an anti-soliton emerge simultaneously counter-propagating, see annex 9.1. As we saw previously, projections of angular momentum on any axis, for any of the two excitations, vary maybe randomly, but there is quantization of the action phase integral.

Determinism is here important for measurement outcomes, since measurements are physical processes between a macroscopic field (the apparatus) and the excitations. Typical experimental examples are :

* A polarizer, which acts as a layer of atoms dephasing the excitation phase plane as it crosses the layer.

* A spin (Stern-Gerlach) analyzer, basically a magnetic field, with the phase and the gauge connexion intertwined from Aharanov-Bohm effect.

Since the global action phase integral is conserved on average, there is a quantity embarked within each excitation with conserved null sum. So, we will assume that the phase, if known, sets reproducibly the outcome of a measurement, although sampling this quantity at various realizations with time $t_i$ generates a random series, because it is not known$^{40}$.

Thus, when one excitation goes through its “polarizer” or “magnetic field” and gets out with a given result of projection on some axis, the other one goes through its own “polarizer” or “magnetic field” and gets out deterministically, albeit unpredictably, with a result (often intuitively thought

<footnote>$^{40}$It’s already true in a Poissonian description of symplectic dynamics that one cannot simply project on fixed directions a specific outcome, if not knowing the initial phase, because of the Poisson brackets commutation relations, as is exemplified with the Larmor precession of a magnet around an applied magnetic field, ref Stone Goldbart chap 11.</footnote>
## 8.5 Entangled field excitations

By virtue of eq.(46), we can now calculate the transition due to the unpredictable measurement at time $t_a$ of one eigenstate, with spin aligned along an axis denoted $\vec{a}$, of one excitation passing through the analyser $\sigma_a$. It is expressed as:

$$ \Psi_1(\vec{a} \cdot \vec{\sigma}_a, t_a) = \langle \vec{a} \cdot \vec{\sigma}_a | \rho(t_a) \rangle = \left\langle \vec{a} \cdot \vec{\sigma}_a \left| e^{-i \int_{t_0}^{t_a} \left(\frac{\delta\theta_1(s)+i\eta(s))}{2} \vec{\sigma} \cdot \vec{n} \right)} \right| \rho(t_0) \right\rangle, \quad (72) $$

and similarly along axis b for the other excitation, where $\vec{a} \cdot \vec{\sigma}_a |\rho(t_a)\rangle$ is employed to take eigenstates that are parallel or antiparallel to $\vec{a}$, as it was used in eq.(6). A tensorial product $|a, b\rangle$ denotes eigenstates along a and b.

The square modulus of eq.(72) converges to the probability density distribution of Born rule as it is sampled empirically. This probability density takes as argument the angle, as in eq.(9). Now, the density matrix, derived from projections over those eigenstates, can also account for a mixture, such as a pair of outcomes along axis a:

$$ \frac{1}{2}(a + a'). \qquad (73) $$

Product states $a \otimes b$, with a tensorial product of solutions written above in eq.(72), do account for both eigenvectors being distributed at random among the two excitations. But, as previously, a singlet superposition of the pair states:

$$ \frac{1}{\sqrt{2}}(ab' + ba') \qquad (74) $$

emerges when it is statistically impossible to determine the result of an operation $\vec{a} \cdot \vec{\sigma}_a$ with certainty, but only with respect to some specific law incorporating latent information shared with the other member of the pair acted on by $\sigma_b$, as in eq.(9). Therefore, we have again that the joint probabilities is not separable:

$$ p_{a,b} \neq p_a p_b. \qquad (75) $$

This results from the interactions with both analysers $\sigma_a$ and $\sigma_b$, exactly in the same way as in eq.(11), but now easily generalizable to any symmetry algebra and apparatus law. It forces the density matrix to be non diagonalizable, see eq.(55), which is the signature of a statistically entangled pair state like in eq.(13), leading to cross correlations as in eq.(15):

$$ \sum_{a,b} (p_{a,b} - p_a p_b) |a, b\rangle\langle a, b| \neq 0. \qquad (76) $$

## 9 Short discussion and conclusion

Two main results have been derived, the first one discards any nonlocal non-realistic interpretation of quantum mechanics, the second one investigates possible origins of
as “opposite”) statistically compatible with their shared property and inter-action dynamics with the apparatus vector potentials, such as Malus’s law. The sharing holds at any given time of a stationary environment until the measurement is done.

Again, this preservation necessitate that chaos arises only along indi-vidual particle paths but that nonlocal coherence is preserved in the back-ground, here by topological constraints and hyperbolic (or ballistic) trans-port. Otherwise, it would be arguable that a Lyapunov time scale would erase memory, lost in microscopic degrees of freedom with elliptic (or diffu-sive) transport.

The subsequent evolution of the phase through the singular medium is chaotic but remains deterministic. As we argued, weak solutions preserve Lie groups symmetries, because their generators are part of the wider indefinite orthogonal group when including multifractal scaling generators in L. Thus, as quantification is a topological invariant, the locally preserved memory of initial conditions in the real part of the phase is not affected by the noise, because the noise does not affect averages. Consequently, initial conditions being preserved throughout the two paths followed by the pair, until an event such as a measurement resets the phase in at least one of them, so that along the paths to the experimental apparatus we have:

$$f(\Phi_i(x,t)) = f(\Phi_i(t_0, x_0)), \qquad (59)$$

for some function $f$. In a more explicit form, consider a pair excitation, at time $t_0$ and position $x_0$ of the event, both corpuscles have opposite projections along a direction $\vec{n}$ of their angular momenta, for example define $f$ as:

$$f(\Phi_1(t_0, x_0)|n) = \theta_0/2 \vec{\sigma} \cdot \vec{n} = -f(\Phi_2(t_0, x_0)|n), \qquad (60)$$

where the projection along $\vec{n}$ is made explicit. We can retrieve formally any case directly, as recovering eq.(13) in the photon case with the three dimen-sional representation of SU(2). This is possible if one can make sure that for such a medium, the excitations are really conveying those oscillations internally.

Wrapping up this mathematically, wavelet transforms express the locally deterministic and globally statistical state of the pair excitations as a tensor product $\Psi_{1,2} = \Psi_1 \otimes \Psi_2$. By construction, each local excitation experiences the self background noise along their characteristics, e.g. $L_{\delta\theta}L_{\eta}$. So statisti-cal summation into non separable states is clearly prepareable as with Jones vectors in sec.(2).

By virtue of eq.(42), the transition due to the unpredictable measure-ment at time $t_a$ of one eigenstate, with spin aligned along an axis denoted $\vec{a}$, of one excitation passing through the analyser $\sigma_a$ is expressed as:

$$\Psi_1(\vec{a} \cdot \vec{\sigma}_a, t_a) = \langle \vec{a} \cdot \vec{\sigma}_a | \rho(t_a) \rangle = \langle \vec{a} \cdot \vec{\sigma}_a | e^{-i \int_{t_0}^{t_a} (\delta\theta_1(s) + i\eta(s))/2 \vec{\sigma} \cdot \vec{n}} | \rho(t_0) \rangle, \qquad (61)$$
the unique property of quantum systems that is duality, close in spirit to de Broglie's first insights. Because quantum systems are here shown to abide by the tenets of classical physics, the realm and overarching scope of the latter are broaden. The EPR paradox has lived as such, it is now upgraded to the founding remark which engulfs quantum physics within statistical physics.

Regarding the demonstration of realism, it was reminded that the only important assumption in Bell's theorem is the independence of measurements, as Jaynes was first to criticise in the late 80s and Khrennikov more recently. This constitutes altogether its major loophole, because obviously quantum mechanics breaks that assumption, but no one had ever proved that classical mechanics could evade such independence until Spreeuw's thought experiment in the late 90s. Consequently, in this essay, entanglement is presented as a natural statistical outcome when a non separable joint probability of some property is evaluated by joint measurements. The latent information is encoded in the phase of localized "dual objects." Just as in the Bohmian interpretation but without the need of a macroscopic nonlocal pilot-wave, because the phase interferes with the measuring apparatus, outputs are not independent. It constitutes an initial condition for the deterministic interaction dynamics at play within the apparatus, between the internal field state and the analyser's macroscopic field. For instance, conservation of angular momentum allows unaltered spin state information to be carried along any inertial particle track. In the case of a pair of elementary excitations in a singlet state, the latent statistical variable is encoded in the phase of the oscillating internal electromagnetic vector fields. That is the insight that connects the findings derived in this short essay to walking droplets, for example.

In that respect, the major added information, compared to a statistical description involving a partition function, of a quantum description, that builds on what was called naively at first the “square root” of a partition function, in fact simply a generating function for any observable moments, so resides in the phase, not surprisingly. Put again differently, phases of entangled entities interfere if we bring them back together. They do not confer to quantum phenomenology a more “ontic” attribute than to the “epistemic” phenomenology of any other system in the class of exact analogues about which knowledge is lacking. Quantum field theory was constructively introduced as an effective theory starting from the generalized kinetic susceptibility and averaging over high frequencies of chaotic oscillations, in an excitable medium. Showing that it is possible to probe beyond the quantum of action, it was defined as a Poincaré invariant, in a way measuring the amount of thermal energy within a period of spontaneous oscillations. The class of quantum analogues comprises those generating interacting self-focusing objects: Particles are localized topological waves emerging from the medium relaxation oscillations. Such entities are typically complex vector solitons keeping within themselves all properties based upon the symmetries of the background, all concealed within their phases. For instance, special relativity, that would govern the dynamics of their envelop, can be forced in via the appropriate dispersion relation. This has radical consequences for the interpretation of relativity: Because dispersion relations emerge at the thermodynamic level of coarse graining, for example via Gibbs relation and equipartition, the inferred internal structure of fundamental particles can be superluminal without any trouble, as Ohanian discussed in
and similarly on axis *b* for the other excitation, where $\vec{a}|\rho(t_a)\rangle$ is a notation trick to make eigenstates parallel or antiparallel to $\vec{a}$, as in eq.(4). The square moduli of eq.(61) has to converge to the probability density distribution by sampling empirically, as in eq.(7). This probability density takes as argument the angle.

The density matrix, derived from projections over those eigenstates, now
can account for a mixture such as $\frac{1}{2}(a+a')$. Product states $a \otimes b$ do account
for both eigenvectors being distributed at random among the two excitations
1 and 2. But, as previously, a singlet superposition of states for the two
excitations together $\frac{1}{\sqrt{2}}(ab'+ba')$ emerges when it is impossible to determine
the result of an operation $\vec{a} \cdot \vec{\sigma}_a$ with certainty, but only with respect to some
specific law whatever the set of latent variables, as in eq.(7). Therefore, the
joint probabilities $p_{a,b}$ of the results of the interactions with both analysers
$\sigma_a$ and $\sigma_b$, as in eq.(10), forces the density matrix to be non diagonalizable
to reproduce the results $\sum_{a,b} p_{a,b}|a,b\rangle\langle a,b|$.

9 Conclusion

Two main results have been derived, the first one discards any non-local non-
realistic interpretation of quantum mechanics, the second one investigates
possible origins of the unique property of quantum systems that is duality.
Because quantum systems are here shown to abide by the tenets of classical
physics, the realm and overarching scope of the latter are broaden.

Regarding the demonstration of realisms, it was reminded that the only
important assumption in Bell's theorem is the independence of measure-
ments, which was Jaynes' remark and criticism in the late 80s. This consti-
tutes altogether its main and major loophole, because obviously quantum
mechanics breaks that assumption, but no one had ever proved classical me-
chanics to fail such dependence. Spreeuw actually showed the contrary for
the first time in the late 90s.

Entanglement was hence exposed as the natural statistical property of
joint probabilities for the unpredictable process of joint measurements of
"dual objects"carrying conserved information within latent variables, for
some definite probabilistic law of outputs at an analyser. For instance,
conservation of angular momentum allows unaltered spin state information
to be carried along any inertial particle track. In the case of a pair of
elementary excitations in a singlet state, the latent statistical variable is
encoded in the phase of the oscillating internal vector fields. It constitutes
an initial condition for the interacting dynamics of the internal field state
and the analyser's macroscopic field. No need for a pilot-wave but instead
"dual corpuscles" are required who have memory in the form of conservation
laws.

In that respect, a mathematical foundation of quantum mechanics and
the 80s. The peculiar reconciliation between relativity and quantum fields can gain a description: Since internal symmetries are now understood, less abstractly than in the standard model, but as being imprinted topologically, it becomes calculable quite easily to show that a particle may be morphed into its antiparticle in a boosted reference frame, as emission and absorption are swapped.

The demonstration was given in an operational way, explicitly defining Dirac's bra and ket bilinear space acting on wave functions. Entanglement being the non diagonalizability property of density matrices, they must be derived also explicitly from first principles, which is given too. The operation goes quite straightforwardly by means of projections with wavelet transforms and annealing over the effective randomness of the background. Onsager relations and scale invariance let us write the effect of randomness as multiplicative chaos. The background is thus dominated by a singular measure which allows non-differential paths to induce non-abelian algebra of operators. The rationale for that assumption is heuristically that critical dynamics self-generate parametric fluctuations of the “1/f” type. Note that criticality is compatible for instance with the two important phenomena of long range static forces and confinement. It is also an important parameter for walking droplets. Quantization of momentum and energy emerges from the logarithmic branch cut as one integrates over the “1/f” fluctuations. Such topological invariants ensure also the necessary memory compatibility of the background. It is straightforward to infer that energy levels for the annealed dynamics are described by the spectrum of the annealed evolution operator, identified with a Hamiltonian in standard ways. Conservation laws are guaranteed, in the mathematically weak sense, by Noether’s theorem. The multifractal background is essential at high enough energies, when second quantization is required and ladder operators become necessary descriptors. In any event, the multifractal background is also well consistent with the requirement of renormalization.

Madelung’s “reduced” picture of phase and quantum potential is thus reversed and broaden. Dirac, Klein-Gordon, Schrödinger’s equations, or any other relevant one, are seen as the normal forms, hiding nonlinearities, of more complicated coupled partial differential equations describing some underlying reaction-diffusion medium and lying maybe in one vaster universality class. As it seems, the approach could be coined a “stochastic zitterbewebung dynamics” but with more constraints, as we just listed. The compelling arguments were funnelled to achieve the appropriate algebraic structure of quantum phenomenology. For example, droplet walkers are localized Fara-day waves with unpredictable paths in important instances, and quantization there arguably follows also by the topology of wave packets described in a Hilbert space.

A deep connection between topological invariants like Chern classes and the non-smooth geometry generated by the multifractal background is worth exploring in more detail. It seems to indicate, for instance, the emergence of spatial curvature via covariant derivatives changing with defect (matter) content, in large volumes and at annealing scales. A clue can be intuited thanks to the powerful theorems of algebraic and geometric topology, such as those relating characteristic classes of Chern and Euler, thus a step away from general relativity coming out as another effective field theory. Hopefully, newly gained insights about the microscopic underlying “dual”
field theory is constructively introduced as an effective theory from aver-
aging over the high frequencies of possibly chaotic sustained oscillations of
driven dissipative excitable media. There, the class of quantum media is one
generating interacting self-focusing entities over the dynamical and memory-
ful background. Such entities are typically complex solitons keeping within
themselves all properties based upon symmetries of the background envi-
ronment, concealed within their phase. Special relativity governing their
dynamics is for example an instance of the various dispersion relations. An
operational description is shown by means of projections with wavelet trans-
forms and annealing. The joint probabilities are calculated in terms of wave
functions, defined as some kind of generating functions.

The background is dominated by a singular measure which allows non-
differential paths to induce non-abelian algebra of operators, and scale in-
variance when the measure becomes effectively multifractal. The rationale
for that assumption is heuristically that critical dynamics self-generate para-
metric fluctuations of the “1/f” type. Note that, interestingly, criticality is
compatible for instance with the two important phenomena of long range
static forces and confinement. It is also an important parameter for walking
droplets. Quantization of momentum and energy emerges from topological
invariants. It is straightforward to infer that energy levels for the annealed
dynamics are described by the spectrum of the annealed evolution opera-
tor, identified with a Hamiltonian in standard ways. Conservation laws are
guaranteed, in the mathematically weak sense, by Noether’s theorem.

More general in scope than Madelung’s “reduced” picture of phase and
quantum potential, Dirac, Klein-Gordon, or Schrödinger’s equations, and
any other relevant, are seen as mere normal form equations for more com-
plicated coupled partial differential equations describing some underlying
reaction-diffusion medium, hiding the nonlinearities. Nonlinearities nonethe-
less possibly fall in different universality classes.

There might be different mechanisms, though similar, by which to achieve
the appropriate algebraic structure, as it seems the approach could be com-
pared to a combined “stochastic zitterbewebung dynamics”. For example,
droplet walkers have unpredictable paths in important instances, and quan-
tization there arguably follows by applying wave mechanics, since solutions
live in a Hilbert space because of the underlying Faraday waves.

It may be so that the multifractal background would only be manifest at
high enough energies, when second quantization is required and ladder oper-
ators become necessary descriptors. This class should plausibly encompass
some regimes of droplet walkers. In any event, the multifractal background
is well consistent with the requirement of renormalization and is here essen-
tial in order to enable the quantization of energy-momentum, thanks to the
complex logarithmic branch cut.

This opportune situation reveals a deep connection between topological
invariants like Chern classes and the non-smooth geometry generated by
physics that can handle the dimensionality of Newton’s constant could help solving
the quantization of gravity conundrum.

Another related connection between the multifractal spectrum, understood as the
entropy of local scaling exponents within a range of scales, and entanglement entropy,
or mutual information, also lays the ground for further work, since both derive from
transition probabilities eq.(46). Note that some quantum computing concepts may
need a change of perspective since state superposition is only statistical, as now proven,
however with an identical computing power, dimensionally speaking.

In short, the description of quantum physics was done in terms of emergence at coarse grained levels of nonlinear localized oscillatory excitations of finite energy living over some dynamical scale invariant “foam.” The preservation of information is encoded in the phase as symmetry operators and guaranteed topologically.

# Appendix

## 9.1 Sine-Gordon pair excitation

Sine-Gordon equation can be derived as the normal form of some coupled PDEs. This equation preserves Lorentz invariance. Typically, pairs of kinks or soliton anti-soliton are excited, see e.g. Fig.(2) from [39], please refer to it and the references contained therein for details.

## 9.2 Ladder operators

Creation and annihilation operators are legitimate operators for a Fock space descrip-
tion of classical excitable systems. This works when differentiability of trajectories is
lost, or when the involved chemicals (at the discrete level) react into products that
diffuse [35, 36].

Textbook second quantization shows that eq.(26) is obeyed by the ladder operators
$a = \frac{1}{\sqrt{(2\omega)}}(\omega x + ip)$ for a harmonic oscillator $\frac{d}{dt}a = -i\omega a$, which means that the
field named “$\rho$” throughout the text is in fact, in the chosen terminology, the “pre-
ladder” operator, as intuition would well tell us. More formally, ladder operators are a
consequence of the non commutativity implied by non-differentiability of wave packets
trajectories [52].

## Acknowledgment

Dedicated to you my late dear friend*, you see, I won this bet. I know you wouldn't
mind! *Mmanu Fleurence.

## References

[1] David BOHM, *A suggested interpretation of the quantum theory in terms of “hidden” variables*. I., Phys. Rev. 85, 166-179 (1952). https://doi.org/10.1103/PhysR ev.85.1662.
the chaotic background, which is worth exploring in more detail. It seems to indicate, for instance, the emergence of spatial curvature via covariant derivatives changing with matter content, in large volumes and at annealing scales. A clue can be intuited thanks to the powerful theorems of algebraic and geometric topology, such as those relating characteristic classes of Chern and Euler, thus a step away from general relativity coming out as another effective field theory. Hopefully, newly gained insights about the microscopic underlying “dual” physics that can handle the dimensionality of Newton’s constant could help solving the quantization of gravity conundrum.

A relationship between the multifractal spectrum, understood as the entropy of local scaling exponents within a range of scales, and entanglement entropy may also lay the ground for further work, since both derive from transition probabilities eq.(42). Note that some quantum computing concepts may need a change of perspective since state superposition is only statistical, however with an identical computing power dimensionally speaking.

To summarize, the description of quantum physics in terms of linear operators is emergent through statistical averaging. Nonlinear excitations live over some dynamical background “foam”, with an effectively stochastic linear kinetic response allowing the preservation of information hidden in their phase and internally oscillating fields.

# Annex

NB : To be fully completed.

## 9.1 Sine-Gordon pair excitation

Sine-Gordon equation derived as the normal form of coupled PDEs example. Pair of kink or soliton, anti-soliton is pedagogical, e.g. Fig.(2). Preserved Lorentz invariance.

## 9.2 Ladder operators

Creation and annihilation operators are legitimate operators for a Fock space description of classical excitable systems. This works when differentiability of their trajectories is lost, or when the involved chemicals react into products that diffuse [30, 29].

Quantum field theory shows that eq.(23) is obeyed by the ladder operators$^{41}$ for a harmonic oscillator, which means that the field named “$\rho$” throughout the text is in fact the “pre-ladder” operator, as intuition would

$$^{41}a = \frac{1}{\sqrt{(2\omega)}}(\omega x + ip)$$
<figure><img src="image_4.png" /><figcaption>**Fig. 2** Snapshot example of a nonlinear relaxation pair mode akin to a pair of kin-antikink in 1D simulation of an infinite dimensional (like Navier Stokes turbulence) instability. Both reactant $|A|$ and $|B|$ amplitudes are displayed spatially (X) at a given time in normalized units. The mode was generated spontaneously far from the driving source in a system like eq.(24). Taken from [39].</figcaption></figure>

[2] David BOHM, *A suggested interpretation of the quantum theory in terms of “hidden” variables*. II. Phys. Rev. 85, 180–193 (1952). https://doi.org/10.1103/PhysR ev.85.180

[3] David BOHM, Jean Pierre VIGIER, *Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations*. Phys. Rev. 96, 208–216 (1954). https://doi.org/10.1103/PhysR ev.96.208

[4] Louis DE BROGLIE, *La thermodynamique “cachée” des particules*, Ann. Inst. Henri Poincaré, Vol. I, n° 1, p1-19, (1964), Section A : Physique théorique.

[5] Louis DE BROGLIE, *Interpretation of quantum mechanics by the double solution theory* Annales de la Fondation Louis de Broglie, Volume 12, no.4, (1987).
<figure><img src="image_3.png" /><figcaption>Figure 2: Example of a nonlinear relaxation mode akin to a pair of kin-antikink. Both reactant amplitudes are displayed spatially at a given time in normalized units. The mode was generated spontaneously far from the driving source in a system like eq.(21). From [36], please refer to it for more details.</figcaption></figure>

well tell us. More formally, ladder operators are a consequence of the non
commutativity implied by non-differentiability [59].

9.3 Characteristic classes

Chern characters and classes, generating function. Derived from Ω or ω and
relation to multifractal generating function. Projection.

9.4 Entanglement entropy

Partial trace over tensor product.

***

Acknowledgment

To my late dear friend*, have you seen? I won this bet! \*Mmanu Fleurence
[6] Yves COUDER, Emmanuel FORT, C.-H. GAUTIER, Arezki BOUDAOUD, *From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath*. Phys. Rev. Lett. 94, 177801, 1 (2005).

[7] Antonin EDDI, Emmanuel FORT, Frédéric MOISY, and Yves COUDER, *Unpredictable Tunneling of a Classical Wave-Particle Association*. Phys. Rev. Lett. 102, 240401, (2009).

[8] Yves COUDER, Arezki BOUDAOUD, Suzie PROTIÈRE and Emmanuel FORT, *Walking droplets, a form of wave-particle duality at macroscopic scale?*, Europhysics News Vol. 41, No. 1, (2010), pp. 14-18.

[9] Sthéphane PERRARD, et al. *Self-organization into quantized eigen-states of a classical wave-driven particle*. Nat. Commun. 5:3219 doi: 10.1038/ncomms4219 (2014).

[10] Daniel M. HARRIS, Julien MOUKHTA, Emmanuel FORT, Yves COUDER, and John W. M. Bush, *Wavelike statistics from pilot-wave dynamics in a circular corral*. Phys. Rev. E 88, 011001(R), (2013).

[11] John W. M. BUSH, Anand U. OZA and Jan MOLÁČEK, *The wave-induced added mass of walking droplets*. Fluid Mech., vol. 755, R7, (2014). doi:10.1017/jfm.2014.459

[12] Yuval DAGAN, and John W. M. BUSH, *Hydrodynamic quantum field theory: the freeparticle*. Comptes Rendus Mécanique, 348, no 6-7, p.555-571, (2020). https://doi.org/10.5802/crmeca.34

[13] John W. M. BUSH, Anand U. OZA, *Hydrodynamic quantum analogs.*, Rep. Prog. Phys. 84 (2020) 017001 (41pp). https://doi.org/10.1088/1361-6633/abc22c

[14] Konstantinos PAPATRYFONOS, M. RUELLE, C. BOURDIOL, A. NACHBIN, John W. M. BUSH, and Matthieu LABOUSSE, *Hydrodynamic superradiance in wave-mediated cooperative tunneling*, Commun. Phys. 5, 142 (2022).

[15] V. FRUMKIN, John W. M. BUSH, and Konstantinos PAPATRYFONOS, *Superradiant droplet emission from parametrically excited cavities*, Phys. Rev. Lett. 130, 064002 (2023).

[16] Konstantinos PAPATRYFONOS, Jemma W. SCHRODER and Valeri FRUMKIN, *Superradiant droplet emission from a single hydrodynamic cavity near a reflective boundary*, arXiv: 2408.02620 (2024).

[17] Konstantinos PAPATRYFONOS, L. VERVOORT, A. NACHBIN, Matthieu LABOUSSE, and John W. M. BUSH, *Bell test in a classical pilot-wave system*, arXiv: 2208.08940 (2022).
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[67] Paul C. MARTIN, Eric D. SIGGIA and Harald A. ROSE, *Statistical Dynamics of Classical Systems* Phys. Rev. A 8, 423, 1973

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[71] Konstantinos PAPATRYFONOS, L. VERVOORT, A. NACHBIN, Matthieu LABOUSSE, and John W. M. BUSH, *Bell test in a classical pilot-wave system*, arXiv: 2208.08940 (2022).

[72] Abel J. ABRAHAM, Stepan MALKOV, Frane A. LJUBETIC, Matthew DUREY, Pedro J. SÁENZ, *Anderson localization of walking droplets*, arXiv:2310.16000 (2023)

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