Rickey W. Austin
St Claire Scientific Research, Development, and Publishing
Draft outline —
The preceding Q-series developed a deterministic exchange-sector transport law for closure density and transport-derived phase, and showed that these variables admit a complex representation of Schrödinger type. However, full equivalence with the Madelung form of nonrelativistic quantum mechanics requires an additional amplitude-curvature contribution to the phase evolution equation. The purpose of the present manuscript is to identify this missing contribution within the NUVO framework.
We introduce a closure-density curvature functional measuring the structural cost of spatially inhomogeneous closure density. Under locality, positivity, rotational invariance, normalization compatibility, and lowest-order derivative assumptions, the minimal admissible curvature cost is proportional to
When this cost is included in the transport action for closure density and phase, variation yields the continuity equation together with the Hamilton–Jacobi–Madelung phase equation containing the density-curvature term
The resulting pair of real transport equations is then shown to be equivalent, under the representation
to the Schrödinger equation with scalar-geometric potential . In this interpretation, the quantum potential is not introduced as an external quantum postulate; it is the representation-level expression of closure-density curvature required for coherent transport across spatially inhomogeneous closure distributions.
The preceding papers of the Q-series developed the exchange-sector
transport structure of scalar–conformal NUVO systems. In that
development, admissible bound configurations arise from closure
conditions on exchange cycles rather than from externally imposed
quantization postulates. The early Q-series established holonomic
return structure, closure-compatible excitation states, and the
hydrogenic spectral ladder. The later Q-series then introduced
transport-derived phase, closure density, and a deterministic transport
law governing the evolution of these quantities.
In Q11, the exchange-sector transport law was expressed in terms of a
closure density , a transport-derived phase , and
a velocity field determined by the admissible transport structure. In
schematic form, the transport system contains a continuity relation
together with a phase-evolution relation driven by exchange interaction
and scalar geometry. This formulation is deterministic and geometric:
is not introduced as a probability density, is not
introduced as the phase of a primitive wavefunction, and no operator
formalism is assumed at the foundational level.
In Q12, the pair was combined into the complex
representation
where denotes the representation scale. This construction
showed that the transport system admits a compact complex encoding and
that, under appropriate normalization and structural conditions, the
resulting representation takes a Schrödinger-type form. The purpose
of that result was not to introduce a wave ontology, but to show that a
familiar complex evolution equation can arise as a representation of
underlying closure transport.
However, the passage from a Schrödinger-type representation to full
nonrelativistic quantum correspondence requires a further step. In the
standard Madelung decomposition of the Schrödinger equation, writing
yields two coupled real equations: a continuity equation and a
Hamilton–Jacobi-type phase equation modified by the amplitude-curvature
term
This term is often called the quantum potential in the Madelung–Bohm
formulation. Its presence is not optional: it is the term that makes
the real phase equation equivalent to the full Schrödinger equation
rather than to a purely classical Hamilton–Jacobi transport system.
The existing Q11–Q12 bridge establishes the closure-density and phase
representation needed for such a structure, but it does not yet isolate
the intrinsic origin of the amplitude-curvature contribution. If the
phase evolution is driven only by an external scalar or exchange
potential depending on the background scalar field , then
the resulting dynamics remain incomplete from the standpoint of full
Madelung equivalence. The missing ingredient is a structural term that
depends not on the external scalar geometry alone, but on the spatial
inhomogeneity of the closure density itself.
The purpose of the present paper is to supply that missing ingredient.
We show that an inhomogeneous closure-density distribution carries a
minimal local deformation cost. This cost is determined by the spatial
curvature of the amplitude and is represented, at lowest
derivative order, by the functional
When the representation scale is identified with the empirical exchange
action scale, , this becomes
The central claim of this paper is that this term should not be
understood as an imported quantum postulate. Within the NUVO exchange
sector, it is interpreted as the lowest-order structural cost required
to maintain coherent transport across a spatially nonuniform closure
density. Uniform closure density carries no such cost. Spatially
varying closure density, by contrast, requires additional structural
compatibility between neighboring closure elements, and this
compatibility is measured by the curvature of the represented amplitude.
We therefore introduce a closure-density transport action of the form
where denotes the scalar/exchange potential associated
with the ambient NUVO geometry. Variation with respect to
yields the continuity equation, while variation with respect to
yields the modified phase equation
Thus the amplitude-curvature term appears as a consequence of the
closure-density deformation functional.
Together with the continuity equation
this phase equation is precisely the Madelung system associated with
the Schrödinger equation
Accordingly, the present paper upgrades the Q12 result from a
Schrödinger-type representational correspondence to a full
Madelung-equivalent transport representation, subject to the structural
assumptions made explicit below.
The scope of the result is deliberately limited. We do not claim here
to derive all of quantum mechanics, nor do we introduce measurement,
probability, spin, relativistic dynamics, or many-body entanglement.
Those topics belong to later representational and correspondence
developments. The aim of the present paper is narrower and more
foundational: to identify the missing closure-density curvature term
required for exact equivalence between NUVO closure transport and the
Madelung form of nonrelativistic Schrödinger dynamics.
The paper proceeds as follows. Section~2 reviews the transport-density
and phase structure inherited from Q11 and Q12. Section~3 motivates the
need for an intrinsic closure-density curvature cost. Section~4 derives
the minimal local functional associated with closure-density
inhomogeneity. Section~5 introduces the closure-density transport
action. Sections~6 and~7 derive the continuity and modified phase
equations by variation. Section~8 states the resulting Madelung
transport theorem. Section~9 proves equivalence with the complex
Schrödinger representation. Sections~10 and~11 interpret the
amplitude-curvature term within the NUVO framework and clarify the
relationship between the present result and Q12. The final sections
record correspondence checks, limitations, and directions for subsequent
development.
The present paper builds on two structural inputs established in the
preceding exchange-sector development. First, Q11 formulated a local
transport law for closure density and transport-derived phase. Second,
Q12 showed that this pair admits a lossless complex representation
which, under suitable reduction conditions, takes a Schrödinger-type
form. We recall only the elements needed for the present derivation.
Let
denote the local closure density of an admissible exchange-sector
configuration. This quantity measures the local density of
closure-compatible exchange structure. It is not introduced as a
probability density at the foundational level, although probabilistic
interpretations may arise later at the representational or measurement
level.
Let
denote the transport-derived phase associated with cumulative exchange
transport. In the Q-series, this phase is not postulated as the phase
of a primitive wavefunction. It is a geometric bookkeeping quantity
measuring accumulated transport compatibility along admissible exchange
paths.
In the local integrable regime, the pair
provides the minimal real state description of closure transport. The
density records how closure-compatible exchange structure is
distributed, while the phase records the local transport
orientation and accumulated coherence state. The present paper assumes
this local integrable regime throughout, so that and
may be treated as smooth fields on a spatial domain
over a time interval of interest.
The local conservation of closure transport gives the continuity
equation
where is the admissible transport velocity field. This
equation expresses conservation of closure density under exchange-sector
transport. It does not require a probabilistic interpretation of
; it is a transport-conservation law for closure-compatible
structure.
For the nonrelativistic Madelung correspondence, the admissible
velocity field is specialized to the phase-gradient form
This relation is the local nonrelativistic reduction of the
exchange-sector phase-gradient law: phase gradients determine the local
direction and rate of admissible closure transport, and the parameter
supplies the inertial scale of the represented closure structure.
Substituting Eq. eq:q13-phase-gradient-velocity into
Eq. eq:q13-continuity-general gives
This is the continuity equation that appears in the Madelung
decomposition of the nonrelativistic Schrödinger equation. In the
present framework, however, it is interpreted as closure-density
transport rather than as probability-current conservation at the
foundational level.
The transport-derived phase evolves under exchange interaction and
scalar geometry. Prior to the refinement introduced in the present
paper, the phase evolution may be represented schematically as
where denotes the exchange-sector driving contribution and
is the corresponding conversion factor between exchange
transport and phase accumulation.
For comparison with nonrelativistic Hamilton–Jacobi structure, the
same phase evolution may be written in reduced form as
where denotes the scalar-geometric or exchange-sector
potential contribution associated with the ambient NUVO geometry. The
notation is used to emphasize that this term is determined by
the scalar/exchange background and not by the internal shape of the
closure-density distribution itself.
Equation Eq. eq:q13-hj-unrefined has the form of a classical
Hamilton–Jacobi phase equation. By itself it is not yet the full
Madelung phase equation. The missing contribution is an internal
closure-density curvature term depending on the spatial inhomogeneity of
. The purpose of the present paper is to derive this term and
to show that the refined phase equation becomes
The final term in Eq. eq:q13-hj-refined-preview is the
closure-density curvature contribution. In conventional Madelung
language it corresponds to the quantum potential. In the present
framework it is interpreted instead as the representation-level image of
the structural cost required to maintain coherent transport across a
spatially inhomogeneous closure-density distribution.
Q12 introduced the complex representation
where denotes the representation action scale. In the
empirically calibrated nonrelativistic correspondence this scale is
identified with the usual Planck-reduced action scale , but the
notation is retained to emphasize its role as the NUVO
representation scale.
The representation Eq. eq:q13-complex-representation is lossless
wherever . Indeed,
up to the usual branch structure of phase. Thus does not add
new ontology to the transport system. It repackages the two real
fields into a single complex object.
This distinction is important for the present paper. The complex state
is not assumed as fundamental, and the Schrödinger equation
is not taken as a primitive dynamical law. Instead, the goal is to show
that once the closure-density curvature contribution is included, the
real transport system for is exactly equivalent to
the standard complex Schrödinger representation
The remaining sections derive the missing curvature term from a
closure-density deformation functional and then prove the equivalence
between the resulting real transport system and
Eq. eq:q13-schrodinger-preview.
The preceding section reviewed the closure-density and phase variables
inherited from Q11 and Q12. Those variables are sufficient to describe
local transport in the integrable regime, but the unrefined phase
equation remains incomplete if it contains only a scalar-geometric
potential contribution. Full Madelung equivalence requires an additional
term depending on the spatial structure of the closure density itself.
The purpose of the present section is to identify the corresponding
density-dependent contribution. We do not introduce it as a quantum
postulate. Instead, we characterize it as the minimal local cost
associated with maintaining coherent exchange transport across a
spatially inhomogeneous closure-density distribution.
A spatially uniform closure density carries no internal spatial
deformation. If
on a spatial region, then neighboring points in that region carry the
same closure-density weight. In such a region, coherent transport
requires compatibility of phase accumulation, but there is no additional
density-gradient structure to reconcile.
An inhomogeneous closure density is different. If
then the local amount of closure-compatible exchange structure varies
from point to point. Maintaining coherent transport across such a
distribution requires compatibility not only of phase gradients, but also
of density gradients. Adjacent regions of different closure density
must remain mutually compatible under transport, otherwise the
represented exchange structure would fail to define a coherent local
state.
Thus, in addition to the scalar-geometric or exchange-sector potential
, there must be an internal contribution measuring the
structural cost of spatial variation in . This cost should
vanish when is uniform, should be nonnegative, and should
depend locally on the spatial variation of closure density in the
lowest-order nonrelativistic theory.
It is useful to emphasize the limited role of this term. It is not a
new force field, not a separate scalar source, and not an externally
imposed quantum potential. It is a deformation cost internal to the
closure-density representation. Its role is to account for the extra
compatibility requirement introduced when closure support is spatially
curved.
We now record the structural requirements imposed on such a cost
functional. Let denote a scalar functional measuring
the internal deformation cost of a closure-density distribution on a
spatial domain .
Principle (Admissible closure-density curvature cost).
A local closure-density curvature cost functional in
the nonrelativistic integrable regime should satisfy the following
conditions:
These requirements do not uniquely determine all possible higher-order
corrections. They determine the leading local curvature cost in the
same sense that a lowest-order field theory is determined by symmetry,
positivity, and derivative order. Additional terms involving higher
derivatives, nonlinear density powers, or couplings to scalar geometry
may be considered in later refinements, but they are not part of the
minimal Madelung correspondence.
Because the cost must vanish for uniform density, the leading term must
depend on spatial derivatives of . Rotational invariance
requires that the first-derivative contribution enter through the scalar
combination
for some local function . Compatibility with the complex
representation singles out the amplitude
Indeed, in the representation
the modulus is the object whose spatial curvature appears in the
Laplacian of the complex state. Therefore the leading local
density-curvature cost should be quadratic in .
The minimal admissible functional is therefore proportional to
Equivalently, wherever ,
and hence
This expression is the Fisher-information-type density-gradient
functional, but no probabilistic interpretation is being imposed here.
In the present setting it is interpreted as the leading local measure of
closure-density deformation.
The coefficient is fixed by dimensional correspondence with the
transport mass and the representation action scale .
The combination
has the required dimensions to convert the amplitude-gradient measure
into an energy-density contribution after integration over space. This
is also the coefficient required for equivalence with the standard
nonrelativistic Schrödinger representation.
Definition (Closure-density curvature functional).
The closure-density curvature functional is
Equivalently, on regions where ,
The functional has the required structural properties.
It is local, since its density depends on and its first
spatial derivatives. It is rotationally invariant, since it depends on
the Euclidean scalar $ |\nabla\sqrt{\rho}|^2 $. It is nonnegative,
because
pointwise. It vanishes for spatially uniform closure density, since
The functional is also compatible with fixed total closure content. If
the admissible state space is restricted by
where denotes the total closure content, then
measures only the spatial deformation of the density
profile and not the total amount of closure content itself. Uniform
redistributions carry zero curvature cost, while spatially structured
distributions carry positive cost according to their amplitude
gradients.
Finally, the use of , rather than itself, is
not arbitrary. It is required by compatibility with the complex
representation. Since is the amplitude of ,
the functional is precisely the amplitude-gradient
contribution that appears when the Schrödinger kinetic term is written
in polar variables. The next sections show that its variational
derivative produces the closure-density curvature potential required
for full Madelung form.
The functional measures the structural cost of
maintaining coherent closure transport across a spatially curved density
profile. It is not a probability postulate and does not presuppose wave
ontology. It is a local deformation cost for closure density.
In NUVO language, a nonuniform closure-density profile represents a
spatially varying distribution of exchange-sector closure support. The
phase field governs the accumulated transport orientation of
this support, while the amplitude records how strongly
closure support is represented at each point. When the amplitude varies
spatially, neighboring regions of the exchange-sector representation
must remain compatible under coherent transport. The curvature
functional measures the leading cost of maintaining
that compatibility.
This interpretation distinguishes the present construction from both a
classical Hamilton–Jacobi theory and a postulated quantum theory. In a
purely classical Hamilton–Jacobi system, the phase evolves under an
external potential and no intrinsic amplitude-curvature contribution is
present. In the present framework, by contrast, the closure-density
distribution itself contributes to the phase evolution through its
spatial curvature. At the same time, this contribution is not inserted
as an external quantum rule. It follows from the admissible local
curvature cost associated with inhomogeneous closure density.
Thus the conventional quantum potential is reinterpreted here as the
Madelung representation of closure-density curvature. In later
sections we will show explicitly that variation of
contributes the term
to the phase equation, thereby completing the real transport system
needed for equivalence with the nonrelativistic Schrödinger equation.
The previous section identified the leading local curvature cost
associated with spatially inhomogeneous closure density. We now combine
that cost with the phase-gradient transport structure inherited from
Q11 and Q12. The result is a variational principle for the pair
. Its Euler–Lagrange equations will yield the
continuity equation and the modified phase equation containing the
closure-density curvature term.
Let be a spatial domain and let
be a time interval. We assume that
and are sufficiently smooth on
, with boundary behavior specified below.
Define the closure-density transport action by
The first term records the coupling between closure density and phase
evolution. The second term,
is the nonrelativistic phase-gradient transport contribution. It is
the local kinetic transport term obtained when the admissible velocity
field is reduced to
The third term,
records the scalar-geometric or exchange-sector potential contribution.
The final term,
is the closure-density curvature cost introduced in the previous
section.
The overall sign convention in
Eq. eq:q13-transport-action is not essential. One may multiply the
entire action by without changing the variational content,
provided the sign convention is handled consistently. The convention
chosen here is arranged so that variation with respect to
yields the Hamilton–Jacobi–Madelung phase equation in the form
It is sometimes useful to rewrite the curvature term directly in
. On regions where ,
so that the action may equivalently be written as
The amplitude form Eq. eq:q13-transport-action will be used in the
main derivation because it makes the origin of the curvature term more
transparent.
The term represents the scalar-geometric or exchange-sector
contribution inherited from the NUVO scalar modulation. In applications
to bound systems, encodes the effective exchange or
scalar-geometric potential experienced by the closure structure. Its
precise form depends on the sectoral reduction under consideration.
For the purposes of the present paper, is treated as a given
external potential on the spatial domain. This does not mean that it is
external to the NUVO framework. Rather, it means that is
external to the local variation of and performed in
this paper. It is inherited from the ambient scalar/exchange
configuration and is not produced by varying the closure-density
curvature functional.
This distinction is essential. The scalar-geometric potential and the
closure-density curvature contribution have different origins:
The first depends on the scalar or exchange background in which the
closure structure is embedded. The second depends on the spatial shape
of the closure density itself.
With the notation used here, the internal density-curvature contribution
is
Thus the refined phase equation takes the form
The point of the present construction is precisely that
is not absorbed into . If the only
potential contribution were , the phase equation would have
the structure of a classical Hamilton–Jacobi equation. Full Madelung
equivalence requires the additional internal term
, which arises from the curvature of the closure-density
amplitude.
The variational calculation assumes boundary behavior sufficient to
remove surface terms generated by integration by parts. We impose one
of the following standard conditions:
For the time boundary, the variations are taken to vanish at the
endpoints of the interval :
Only the vanishing of at the time endpoints is needed
for the phase variation, but it is convenient to impose endpoint
conditions on both variables.
There is one further technical restriction. The expression
is singular at nodes where . The first theorem of the
present paper is therefore stated on nodal-free regions where
This restriction is standard in polar decompositions of complex fields.
The treatment of nodes, branch changes of phase, and distributional
contributions at zero-density sets is a separate technical problem and
is deferred to later work. For the purpose of establishing the local
Madelung correspondence, it is sufficient to work on connected regions
where the closure density is strictly positive and the phase can be
chosen smoothly.
We now derive the first Euler–Lagrange equation associated with the
transport action. The phase field enters the action only
through its time derivative and spatial gradient. Therefore variation
with respect to yields a conservation law. In the present
framework, this conservation law is the continuity equation for
closure-density transport.
Lemma (Phase variation yields continuity).
Let and be smooth on
, and assume boundary conditions such that all
surface terms vanish. Variation of the action
with respect to yields
Proof.
Let , where
is a smooth test variation satisfying the boundary conditions
specified in the previous section. The closure density is
held fixed during this variation. The terms in the action depending on
are
Therefore the first variation with respect to is
which equals
We integrate the first term by parts in time:
The endpoint term vanishes because the variation is assumed to
vanish at and .
Similarly, integrating the spatial-gradient term by parts gives
The boundary term vanishes under the assumed spatial boundary
conditions.
Combining the two integrations by parts, we obtain
Since the variation is arbitrary in the interior
of , stationarity of the action implies
This proves the lemma.
The resulting equation has exactly the continuity form required for
Madelung correspondence. In NUVO terms, however, it is interpreted as
the conservation of closure density under admissible exchange-sector
transport. The associated transport current is
Thus the continuity law may also be written as
The current is the local flux of closure-compatible exchange
structure in the nonrelativistic integrable regime.
We next derive the second Euler–Lagrange equation associated with the
transport action. This variation is the crucial step in the present
paper. Variation with respect to the phase yielded closure-density
continuity. Variation with respect to the closure-density amplitude
yields the modified phase equation containing the density-curvature
term.
It is convenient to work with the amplitude variable
On nodal-free regions, , and the transformation between
and is smooth and invertible.
In terms of , the transport action becomes
Define the unrefined Hamilton–Jacobi expression
Then the action may be written compactly as
Lemma (Amplitude variation yields the density-curvature phase equation).
Let and be smooth on , and assume
boundary conditions such that all surface terms vanish. Variation of
with respect to yields
Equivalently, since ,
where
Proof.
Let , where is a smooth test
variation satisfying the boundary conditions specified above. The
phase is held fixed during this variation. Using
Eq. eq:q13-action-amplitude-compact, the first variation is
equals
The second term is integrated by parts in space:
The boundary term vanishes under the assumed spatial boundary
conditions. Therefore
Since is arbitrary in the interior of
, stationarity of the action implies
On nodal-free regions where , division by gives
Substituting the definition of from
Eq. eq:q13-hphase-definition yields
Finally, using gives
and hence the equivalent form
This proves the lemma.
The density-curvature term has therefore appeared as the variational
derivative of the closure-density curvature functional. It has not
been introduced as an independent force, an external potential, or a
probabilistic postulate. It is the phase-level contribution required by
stationarity of the closure-density transport action when spatial
inhomogeneity of the closure density is included.
In terms of the closure-density amplitude, the internal contribution is
This is formally identical to the quantum potential of the standard
Madelung decomposition. Within the present framework, however, it is
interpreted as the closure-density curvature potential: the
representation-level expression of the cost required to maintain
coherent exchange transport across a nonuniform closure-density profile.
We now combine the two variational results. Variation with respect to
produced the closure-density continuity equation, while
variation with respect to the amplitude produced the
phase equation corrected by the density-curvature term. Together, these
two equations form the Madelung transport system associated with
closure-density dynamics in the nonrelativistic integrable regime.
Theorem (NUVO Madelung transport system).
Let be a spatial domain and let
be a time interval. Suppose
are smooth on , and assume boundary conditions such
that all surface terms vanish in the variational calculation. In the
local integrable nonrelativistic closure-transport regime, stationarity
of the action
with respect to independent variations of and
yields the coupled system
and
Proof.
The first equation follows from variation of the action with respect to
, as shown in the phase-variation lemma. The second equation
follows from variation with respect to the amplitude
, as shown in the amplitude-variation lemma. Since
the variations are independent and arbitrary in the interior of
, stationarity requires both Euler–Lagrange equations
to hold simultaneously. Substituting into the
amplitude equation gives exactly
Eq. eq:q13-madelung-phase.
The first equation may be written in conservative form
where the closure-density current is
This current is the nonrelativistic transport flux of
closure-compatible exchange structure. In standard quantum notation,
the analogous expression is interpreted as probability current. In the
present framework, the more primitive interpretation is transport of
closure density. Any probabilistic interpretation is therefore
secondary to the closure-transport structure.
The second equation can be written more compactly by defining
Then Eq. eq:q13-madelung-phase becomes
This is a Hamilton–Jacobi transport equation corrected by the
closure-density curvature potential. The correction is not an external
force and is not part of the ambient scalar-geometric potential
. It is the phase-level expression of the structural cost of
maintaining coherent transport across a spatially inhomogeneous
closure-density profile.
The distinction between and is central:
whereas
Thus the refined phase equation contains both an external geometric
contribution and an internal density-curvature contribution.
The system
is the NUVO Madelung transport system in the local nonrelativistic
integrable regime.
Equation Eq. eq:q13-madelung-system-a expresses conservation of
closure density under phase-gradient transport. Equation
Eq. eq:q13-madelung-system-b expresses phase evolution under three
contributions:
The third contribution is the term missing from the unrefined
Hamilton–Jacobi transport equation. Its inclusion is precisely what
distinguishes the full Madelung transport system from a classical
Hamilton–Jacobi system with a transported density.
In conventional terminology, is the quantum potential. In the
NUVO interpretation developed here, it is not treated as mysterious or
as separately postulated. It is the representation-level image of the
minimal local deformation cost associated with nonuniform
closure-density support. When the closure density is spatially uniform,
vanishes. When the closure density is spatially curved,
contributes to phase evolution because coherent transport must
remain compatible across the inhomogeneous density profile.
Thus the Madelung correction is reinterpreted as closure-density
curvature. This closes the gap between the deterministic
closure-transport law and the full real form of nonrelativistic
Schrödinger dynamics.
The preceding section derived the real transport system for closure
density and transport-derived phase. We now show that this system is
equivalent to a complex Schrödinger representation. This establishes
the precise sense in which the Schrödinger equation arises in the
present framework: it is the compact complex encoding of the
closure-density continuity equation together with the
density-curvature-corrected phase equation.
Let
On any connected nodal-free region where , this encoding is
locally invertible up to the usual branch structure of phase:
Thus contains no additional local information beyond the pair
. It is a representation of the real closure-transport
state, not an additional primitive field.
Theorem (Equivalence with Schrödinger form).
Assume , and suppose , , and are
sufficiently smooth on . Then the Madelung transport
system
is equivalent, under the encoding
to the complex Schrödinger-type equation
Proof.
Let
We compute the time derivative and Laplacian of :
and
Substituting Eq. eq:q13-time-derivative-psi and
Eq. eq:q13-laplacian-psi into
Eq. eq:q13-schrodinger-equation, and dividing by the common factor
, gives
equals
Equating imaginary parts yields
Using , this becomes
Multiplying by , and using , gives
equals
Therefore
which is Eq. eq:q13-equivalence-continuity.
Equating real parts of
Eq. eq:q13-separated-before-real-imag gives
Using again , we obtain
Dividing by and rearranging gives
which is Eq. eq:q13-equivalence-phase.
Thus the complex Schrödinger equation implies the two real Madelung
transport equations.
Conversely, suppose that
Eq. eq:q13-equivalence-continuity and
Eq. eq:q13-equivalence-phase hold. Reversing the preceding
calculation shows that the imaginary part and the real part of
Eq. eq:q13-schrodinger-equation are both satisfied under the
encoding . Therefore the two real equations
recombine into the single complex equation
This proves the equivalence.
The theorem establishes an exact local equivalence between two
descriptions:
The equivalence is mathematical, but its interpretation within NUVO is
specific. The complex equation is not taken as the primitive starting
point. Rather, it is the compact representation of the real
closure-density transport system derived from the variational principle.
This is the point at which the gap in the earlier Q11–Q12 bridge is
closed. Without the closure-density curvature term, the real phase
equation would reduce to a classical Hamilton–Jacobi form and would
not be equivalent to the full Schrödinger equation. With the
curvature term included, the real transport system has exactly the
Madelung structure required for recombination into the complex equation.
The theorem has been stated using the NUVO representation scale
. In the empirical nonrelativistic correspondence, this scale
is identified with the reduced Planck action . Under that
identification,
which is the standard Schrödinger equation with potential
.
Within the NUVO program, the role of is not merely to serve
as a formal constant in the complex representation. It is the action
scale inherited from exchange-cycle closure and calibrated in the
hydrogenic sector. Thus the same scale that governs closure action
also controls the curvature cost of the closure-density amplitude. This
identification is what connects the local Madelung bridge developed
here with the earlier Q-series derivation of the action scale.
The equivalence theorem has been stated on regions where . This
restriction is not a physical claim that nodes cannot occur. It is a
technical restriction required by the polar representation and by the
term
At nodes, the phase may become singular or multi-valued, and additional
topological or distributional structure may be required. Such behavior
is familiar from the standard Madelung decomposition and is not unique
to the present framework.
For the purpose of establishing the local equivalence between
closure-density transport and Schrödinger representation, it is
sufficient to work on connected nodal-free regions. A complete
treatment of nodes, phase defects, vortices, and global branch structure
belongs to a later technical development.
The present paper is best understood as a refinement and completion of
the bridge begun in Q12. Q12 introduced the complex representation of
closure density and transport-derived phase and showed that the
transport system can be expressed in a Schrödinger-type form under
appropriate structural conditions. The present work identifies one of
those structural conditions explicitly and derives its consequence: the
closure-density curvature term required for full Madelung equivalence.
Q12 established that the real exchange-sector variables
admit the complex encoding
This representation is lossless on regions where . The
closure density is recovered from the modulus,
and the transport-derived phase is recovered from the argument,
up to the ordinary branch structure of phase.
The significance of Q12 was not merely notational. It showed that the
transport variables of the Q-series can be combined into a single
complex representational object without introducing a primitive wave
ontology, a probability postulate, or an operator formalism. The
complex state was therefore interpreted as an encoding of
closure transport rather than as a fundamental physical substance.
Q12 also showed that the complex representation naturally produces
second-order spatial structure. In particular, the Laplacian of
contains contributions from both the phase and the amplitude.
When
the expression contains terms involving
Thus the mathematical structure needed for a Schrödinger-type
representation was already present in the complex encoding.
However, Q12 left open the precise origin of the amplitude-curvature
term in the real phase equation. It showed that the transport system
can be represented in a Schrödinger-type form under structural
conditions, but it did not fully isolate the closure-density
admissibility condition responsible for the term
As a result, Q12 established a strong representational bridge, but not
yet the complete Madelung transport derivation.
The present paper supplies the missing ingredient by identifying
closure-density curvature admissibility as a structural condition of
local coherent transport.
The key observation is that an inhomogeneous closure-density profile
requires more than phase-gradient compatibility. If the closure density
varies across space, then coherent transport must also preserve
compatibility across density gradients. This introduces a local
deformation cost depending on the spatial curvature of the amplitude
Under locality, rotational invariance, nonnegativity, vanishing for
uniform density, lowest-order derivative dependence, and compatibility
with the complex representation, the minimal such cost is
Including this term in the closure-density transport action produces,
by variation, the density-curvature potential
This is the precise term that appears in the Madelung decomposition of
the Schrödinger equation. In the present framework, however, it is
not introduced as an external quantum potential. It is derived as the
phase-level expression of the structural cost of maintaining coherent
transport across a spatially inhomogeneous closure-density profile.
Thus Q13 upgrades the Q12 bridge in two ways:
The result is that the real transport system no longer has the structure
of a classical Hamilton–Jacobi equation with a transported density.
Instead, it becomes the full Madelung system:
and
The combined Q11–Q13 result should therefore be stated in a more
precise form than the preliminary Q12 language. Rather than saying only
that a Schrödinger-type equation emerges, the revised claim is:
The exchange-sector transport system of scalar–conformal NUVO theory
admits a local integrable regime in which closure density and
transport-derived phase satisfy the Madelung equations. Under the
complex representation
this system is equivalent to the nonrelativistic Schrödinger equation
with scalar-geometric potential .
This formulation is stronger than the original Q12 claim because it
specifies the real transport equations and includes the
density-curvature term required for exact Schrödinger equivalence. It
is also more precise because it states the assumptions under which the
claim holds:
These qualifications are important. They prevent the result from being
overstated while preserving its significance. Q13 does not claim to
derive every aspect of quantum theory. It establishes a specific and
load-bearing bridge: the deterministic closure-transport variables of
the Q-series can satisfy the full Madelung form, and therefore can be
represented exactly by the nonrelativistic Schrödinger equation in the
regime described above.
With this refinement, the logical sequence of the Q-series becomes more
stable. The early papers establish closure, holonomic return, action
scale, dynamic-loop coupling, and hydrogenic spectral structure. Q10
and Q11 then introduce phase and closure-density transport. Q12
introduces the complex representation. Q13 supplies the missing
closure-density curvature condition that converts the representation
from Schrödinger-type to Madelung-equivalent.
The resulting chain may be summarized as
This sequence preserves the NUVO interpretive order. The complex
Schrödinger equation is not the starting point. It is the terminal
representation of a real closure-density and phase transport system
once the minimal curvature cost of density inhomogeneity is included.
The preceding sections established the variational origin of the
density-curvature term and proved equivalence between the resulting
Madelung transport system and the complex Schrödinger representation.
We now clarify the physical and structural meaning of this term within
the NUVO framework.
The interpretive point is important. The term
has the same mathematical form as the quantum potential in the standard
Madelung decomposition. In the present framework, however, it is not
introduced through Bohmian ontology, probabilistic assumptions, or an
external quantum rule. It is derived as the local curvature contribution
associated with spatially inhomogeneous closure density.
In the standard Madelung rewriting of the Schrödinger equation, the
term
is usually called the quantum potential. It appears when a complex
wavefunction is decomposed into amplitude and phase,
The real part of the Schrödinger equation then becomes a
Hamilton–Jacobi-type equation corrected by .
In the present paper the same mathematical structure appears, but its
interpretation is different. The corresponding NUVO term is
and is interpreted as the closure-density curvature potential.
It measures how the spatial curvature of the closure-density amplitude
contributes to the phase evolution of an admissible exchange-sector
transport state.
The terminology reflects the origin of the term. In the NUVO
derivation, is not introduced as a probability density. It is
the local density of closure-compatible exchange structure. Its square
root,
is the amplitude in the complex representation, and spatial variation
of this amplitude carries a deformation cost. The variational
derivative of that cost produces
.
Thus the conventional phrase "quantum potential" is mathematically
accurate in the sense of Madelung correspondence, but it is not the most
structurally transparent term within NUVO. The term is better
understood as the potential-like expression generated by
closure-density curvature.
The closure-density curvature potential should not be interpreted as a
primitive force field. It is not an independently existing field on
space, and it is not sourced in the same manner as the scalar-geometric
potential . It is a functional of the closure density itself:
This distinguishes it from the scalar-geometric contribution
. The term represents the ambient
scalar/exchange background experienced by the closure structure. Its
origin lies in the scalar-conformal and exchange-sector environment. By
contrast, depends on the internal spatial shape of the
represented closure-density distribution.
The distinction may be summarized as
while
For this reason, the closure-density curvature potential does not
represent a force acting on the closure density from outside. Rather,
it is the phase-level consequence of requiring coherent transport to be
maintained across an inhomogeneous density profile. If the density is
spatially uniform, there is no amplitude curvature and the contribution
vanishes. If the density varies spatially, the transport phase must
adjust to preserve coherence across the curved density profile.
In this sense, is closer to an internal compatibility pressure
or deformation response than to a primitive force. It records the fact
that nonuniform closure support cannot be transported coherently as if
each point were independent of its neighbors. Spatially adjacent
regions of different closure density are linked by the requirement of
coherent exchange transport, and the curvature potential is the
representation-level expression of that linkage.
The curvature functional introduced above may be written as
on regions where . This expression resembles the Fisher
information functional when is interpreted as a probability
density. However, no such interpretation is assumed in the present
derivation.
In this paper, remains closure density. The functional
measures the spatial deformation of closure support,
not the information content of a probability distribution. The
similarity to Fisher information arises because both constructions seek
a positive, local, lowest-order measure of density variation. The
mathematical form is shared, but the interpretation is different.
This distinction is necessary for the logical order of the NUVO program.
The present paper operates before the introduction of measurement
weights, event frequencies, or probabilistic interpretation. Its only
inputs are closure density, transport-derived phase, the
scalar-geometric potential, and the local curvature cost of
inhomogeneous closure density.
Probabilistic interpretation, if introduced later, belongs to the
QB/QM measurement correspondence. In that later context, one may
interpret as a weight or probability density under
additional structural assumptions involving event projectors,
coherence-gated interactions, and frequency realization. None of those
assumptions are required for the derivation of the Madelung transport
system itself.
Thus the derivation proceeds in the following order:
Only after this chain has been established may one ask whether the
resulting representation also supports a probability interpretation.
The use of the amplitude is not merely a formal
device. It reflects the fact that the complex representation stores
closure density through the modulus:
The amplitude is therefore the field whose spatial curvature enters the
second-order representation.
A nonzero value of
indicates that the closure-density amplitude is spatially curved. The
ratio
then measures this curvature relative to the local amplitude itself.
The corresponding contribution to phase evolution is
This form has a natural structural interpretation. Regions where the
amplitude bends sharply impose a larger compatibility burden on coherent
transport than regions where the amplitude is nearly flat. The
curvature potential measures this burden locally. Its dependence on the
ratio expresses the fact that the relevant quantity is not
absolute curvature alone, but curvature relative to the local amount of
closure support.
If the density-curvature contribution is omitted, the phase equation
reduces to
which is the classical Hamilton–Jacobi form for a phase or action
function evolving under the potential . Coupled to a continuity
equation, this gives a classical ensemble-like transport system.
The Madelung system differs precisely by the addition of
:
This term couples the phase evolution to the shape of the density
profile. The phase is no longer determined only by the external
potential and its own gradient. It also responds to the curvature of
the closure-density amplitude.
This is the structural feature that makes the system equivalent to the
Schrödinger equation. In NUVO terms, it says that coherent
exchange-sector transport is not merely classical transport of a density
along phase-gradient trajectories. It is transport constrained by the
curvature of the closure-density support itself.
The term conventionally called the quantum potential is reinterpreted in
this paper as the closure-density curvature potential:
Its role is threefold:
It is not a primitive force, not an external scalar source, and not a
probability postulate. It is the internal structural contribution
required when coherent transport is maintained across a spatially
inhomogeneous closure-density distribution.
The preceding section established the mathematical equivalence between
the NUVO Madelung transport system and the Schrödinger representation.
We now record three immediate correspondence consequences: free-particle
spreading, interference, and hydrogenic stationary states. These
examples are not introduced as new assumptions. They follow from the
Schrödinger representation once the closure-density curvature term has
been included.
The purpose of this section is therefore limited. We do not rederive
the full phenomenology of nonrelativistic quantum mechanics. Rather,
we identify how familiar quantum behavior is interpreted in the NUVO
closure-transport language.
Consider the case
The Schrödinger representation then becomes
This is the free nonrelativistic Schrödinger equation with
representation scale . It supports the usual dispersive
evolution of localized wave packets.
In the Madelung variables, the corresponding real system is
and
The second equation shows that even in the absence of an external
scalar-geometric potential, a nonuniform closure-density profile
contributes to phase evolution through its curvature. This is the
source of free-particle spreading in the real transport description.
In NUVO language, free-particle dispersion is therefore not interpreted
as the spreading of a primitive wave substance. It is the evolution of
a represented closure-density distribution under phase-gradient
transport, corrected by the internal curvature of the closure-density
amplitude. A sharply localized profile has significant amplitude
curvature and therefore a nontrivial curvature contribution to phase
evolution. A broad, slowly varying profile has a smaller curvature
contribution and correspondingly weaker dispersive behavior.
Thus the free-particle case makes clear why the density-curvature term
is essential. Without it, the system would reduce to classical
Hamilton–Jacobi transport with no intrinsic mechanism for the standard
Schrödinger dispersion of localized states.
Interference is also naturally represented once the complex encoding is
available. Suppose two locally admissible represented states are written
as
Their representational superposition is
The represented closure-density profile associated with is
equals
The cross term depends on the relative transport-derived phase. This
is the usual interference structure of the Schrödinger representation.
In NUVO language, interference is not introduced as the collision or
overlap of physical wave substances. It is the phase-sensitive
recombination of represented closure-density amplitudes. The underlying
real quantities remain closure density and transport-derived phase. The
complex representation encodes how different admissible transport
branches recombine when expressed in a common representational space.
The density-curvature term is again essential. Interference patterns
are spatially structured density profiles. Once such profiles form,
their subsequent evolution depends on the curvature of
. Thus the same structural term responsible for
Madelung equivalence also governs the evolution of the spatial density
structure produced by phase-sensitive recombination.
This interpretation is compatible with the earlier Q-series account of
matter-wave coherence. The observable interference pattern is not
taken as evidence for a primitive matter wave. Rather, it is the
representation-level density structure generated by coherent phase
relations among admissible exchange-transport histories.
Now consider a scalar-geometric or exchange-sector potential
corresponding to the hydrogenic proton–electron exchange sector. In
the Schrödinger representation, stationary states have the form
where satisfies the time-independent equation
For the Coulombic hydrogenic correspondence, this equation admits the
standard discrete bound-state spectrum and stationary mode structure.
Within the NUVO program, this result should be compared with, but not
used to replace, the independent closure-hierarchy derivation of the
hydrogen spectral ladder developed earlier in the Q-series. In that
earlier derivation, the spectral structure arises from exchange-cycle
closure, holonomic admissibility, and the discrete hierarchy of
closure-compatible configurations. The present Schrödinger
representation supplies a complementary local transport representation
of the same nonrelativistic sector.
The relationship should therefore be understood as follows. The Q7
closure-hierarchy derivation identifies the discrete hydrogenic ladder
from global exchange-cycle closure. The present Madelung bridge shows
that, once closure-density curvature is included, the local
closure-transport representation is equivalent to the usual
Schrödinger equation whose stationary solutions also exhibit the
hydrogenic spectrum.
Thus the two descriptions support one another but play different roles:
This distinction prevents circularity. The hydrogen spectrum is not
first assumed through the Schrödinger equation and then claimed as a
closure result. Rather, the closure hierarchy and the Schrödinger
representation are two corresponding descriptions of the same
hydrogenic exchange sector, one global and holonomic, the other local
and differential.
The inclusion of closure-density curvature provides the missing
mechanism needed for the usual nonrelativistic quantum behaviors to
appear in the local representation:
These correspondences support the central claim of the present paper:
the closure-density curvature term is the structural ingredient needed
to convert the Q11–Q12 transport representation into a full
Madelung-equivalent description of the nonrelativistic quantum regime.
The present paper closes a specific gap in the Q-series: it identifies
the closure-density curvature term required for full Madelung
equivalence and shows that the resulting real transport system is
equivalent to the nonrelativistic Schrödinger representation. The
scope of this result is intentionally limited. Several technical and
conceptual issues remain outside the present derivation and should be
treated in later work.
The derivation above assumes that the closure density is strictly
positive on the region of interest:
Equivalently, the amplitude
is assumed to satisfy . This assumption allows the phase
to be chosen smoothly and permits division by in the
density-curvature term
This restriction is standard in local Madelung-type decompositions, but
it is not sufficient for a complete global theory. In ordinary
Schrödinger systems, nodes occur naturally in excited states,
interference patterns, angular momentum eigenstates, and scattering
configurations. At such points the amplitude vanishes, the phase may
become undefined or multi-valued, and the ratio may become
singular.
A full NUVO treatment of nodes must therefore address at least four
issues:
The present paper avoids these complications by proving the local result
on nodal-free regions. This is sufficient for establishing the
Madelung bridge, but not for a complete global account of all
Schrödinger states. A later technical development should treat nodes,
phase defects, and singular closure-density profiles explicitly.
The present derivation is nonrelativistic. This restriction enters in
several places. First, the admissible transport velocity is reduced to
the phase-gradient form
Second, the kinetic transport contribution is taken to be
Third, the resulting complex representation is the nonrelativistic
Schrödinger equation
These are appropriate assumptions for the local nonrelativistic
closure-transport regime, but they are not expected to remain exact in
relativistic settings. A relativistic extension must account for:
Accordingly, the present result should not be interpreted as a
derivation of relativistic quantum mechanics. It establishes the
nonrelativistic Madelung bridge. The relativistic extension belongs to
the RQM development, where the density-curvature idea may require a
covariant reformulation.
The present derivation is written over physical three-space for a
single closure-density field:
This is sufficient for the single-body or effective one-body
nonrelativistic correspondence. However, standard many-body quantum
mechanics is formulated on configuration space, with wavefunctions of
the form
The associated density lives on a -dimensional configuration
space rather than ordinary physical space.
A full many-body NUVO extension must therefore clarify whether the
appropriate object is:
This issue is not merely technical. Entanglement, nonseparability, and
multi-particle interference depend on the structure of the state space.
If NUVO is to reproduce the many-body quantum formalism, it must explain
why and when a configuration-space representation is required, and how
that representation relates to physical exchange-sector closure among
bundled structures.
The present paper does not solve that problem. It establishes the
single-field Madelung bridge. The many-body extension should be treated
as a separate development, likely connecting the Q-series transport
framework to the QB/QM treatment of representational spaces,
projectors, and entanglement.
The coefficient of the closure-density curvature functional was chosen
as
This coefficient is required for dimensional compatibility and for exact
correspondence with the Schrödinger kinetic operator. With this
choice, the variational derivative produces
and the resulting Madelung system recombines into
This is sufficient for the present correspondence theorem, but it is
not yet the deepest possible NUVO derivation of the coefficient. A
stronger future derivation should connect both factors in the
coefficient to prior NUVO structures.
First, the action scale should be tied explicitly to the
closure-action scale developed in the Q-series. Earlier Q-series work
identified an invariant exchange-cycle action scale through closure
transport and hydrogenic calibration. The present paper uses
as the representation scale controlling both phase and
closure-density curvature. A deeper derivation should show directly
why the same closure-action scale controls the amplitude-curvature cost.
Second, the mass parameter should be connected to the
support-sector invariant intake structure. In the support sector,
persistent anchored structures are characterized by invariant structural
intake. The nonrelativistic transport mass appearing in the Madelung
system should therefore be interpreted not as an imported inertial
parameter, but as the reduction of the support-sector persistence scale
into the exchange-sector transport representation.
The deeper coefficient problem can therefore be stated as:
The present paper assumes this coefficient at the level required for
Madelung correspondence. A later foundational refinement should derive
it from the already developed NUVO action and support-sector structures.
The closure-density curvature functional
was motivated as the minimal local functional satisfying the
admissibility requirements listed above. This is a strong structural
argument, but it is not the same as deriving the functional uniquely
from a more primitive substrate dynamics.
The present paper therefore establishes a conditional result:
If the local nonrelativistic closure-density deformation cost is given
by the minimal admissible curvature functional, then the resulting
transport system is exactly Madelung-equivalent and hence
Schrödinger-equivalent.
A stronger future result would derive this functional directly from a
deeper NUVO substrate model. Possible routes include:
For the purposes of closing the Q11–Q12 Madelung gap, the minimal
functional is adequate. But for a fully foundational derivation, the
status of should remain marked as a structural
principle awaiting deeper substrate-level justification.
The result of the present paper should be stated carefully. It does
not claim to derive the entirety of quantum mechanics. It does not
derive measurement, Born weights, spin, relativistic wave equations,
field quantization, or many-body entanglement. Those topics require
additional structures developed elsewhere in the NUVO program.
What the paper does establish is narrower and more precise:
This closes a specific structural gap in the Q-series. It does not
complete the entire quantum reconstruction program. Rather, it secures
one of its central bridges: the passage from deterministic
closure-density transport to the full nonrelativistic Madelung form.
The present manuscript addressed a specific structural gap in the
exchange-sector development of scalar–conformal NUVO theory. Previous
work established closure density, transport-derived phase, and a
lossless complex representation of the pair
through
That construction provided a Schrödinger-type representation of
closure transport, but it did not yet isolate the internal
density-dependent contribution required for full Madelung equivalence.
The missing contribution is the amplitude-curvature term
In standard Madelung language this term is called the quantum potential.
In the present framework it has been reinterpreted as the
closure-density curvature potential. It is not introduced as an
external force, an independent scalar source, a probability postulate,
or a primitive quantum assumption. It arises as the phase-level
expression of the minimal local deformation cost associated with a
spatially inhomogeneous closure-density profile.
The central structural input was the closure-density curvature
functional
This functional is local, nonnegative, rotationally invariant, vanishes
for uniform closure density, and is compatible with the complex
representation. Including it in the closure transport action gives
Variation with respect to the phase yields the continuity equation
while variation with respect to the amplitude
yields the corrected phase equation
Together these equations form the NUVO Madelung transport system in the
local nonrelativistic integrable regime.
Under the complex representation
the two real transport equations are equivalent to
Thus the Schrödinger equation appears as the compact complex
representation of the real closure-density transport system once the
density-curvature cost is included.
This result sharpens the claim of the preceding Q-series papers. The
appropriate statement is not merely that a Schrödinger-type equation
can be written for closure transport. Rather, in the local integrable
nonrelativistic regime, closure density and transport-derived phase
satisfy the Madelung equations, and their complex encoding is equivalent
to the Schrödinger equation with scalar-geometric potential
.
The conclusion should be read with its limitations intact. The present
paper does not derive measurement theory, Born weights, spin,
relativistic quantum dynamics, field quantization, or many-body
entanglement. It also assumes a nodal-free local region and takes the
coefficient at the correspondence level, with deeper
derivation from closure action and support-sector persistence deferred
to future work. What it does establish is narrower but essential: the
specific amplitude-curvature term needed to close the gap between
deterministic closure transport and full Madelung form arises naturally
as closure-density curvature.
Accordingly, the conventional quantum potential is reinterpreted within
NUVO as the representation-level expression of the structural cost of
maintaining coherent transport across a spatially inhomogeneous
closure-density distribution. This closes the transport-to-Madelung
bridge and provides a more precise foundation for the subsequent
Schrödinger, Hilbert-space, and quantum-measurement correspondences in
the broader NUVO program.
This appendix records the detailed variation of the closure-density
curvature functional used in the main text. Let
and consider the spatial functional
We assume that is smooth on the spatial domain
, and that variations satisfy boundary
conditions sufficient to remove surface terms. For example, one may
assume compact support, sufficiently rapid decay at infinity, periodic
boundary conditions, or
Let
where is a smooth test variation. Then
Expanding to first order in , we obtain
Therefore,
Using integration by parts,
where denotes the outward unit normal on .
Under the assumed boundary conditions, the boundary term vanishes:
Hence
Since , this gives
Multiplying by the coefficient appearing in the closure-density
curvature functional,
we obtain
This is the term used in the amplitude variation in Section~6.
When combined with the variation of
where
one obtains
Stationarity for arbitrary gives
On nodal-free regions where , division by yields
which is the density-curvature-corrected phase equation.
This appendix records the standard polar decomposition of the
Schrödinger representation used in the equivalence theorem. Consider
and write
For compactness, define
Then
First compute the time derivative:
Therefore,
Since ,
so
Next compute the spatial derivatives. The gradient is
Taking another divergence gives
Using , we get
Since
this may also be written as
Substituting into the Schrödinger equation and dividing by
, we obtain
Separating real and imaginary parts gives two equations.
The imaginary part is
Canceling ,
Multiplying by , we find
Since ,
and
Thus
Using
we obtain
This is the Madelung continuity equation.
The real part is
Dividing by gives
Rearranging,
Since , this becomes
This is the Madelung phase equation.
Thus the polar decomposition of the Schrödinger representation yields
the two real equations
and
Conversely, if these two real equations hold, reversing the calculation
recombines them into the complex Schrödinger representation.
This appendix gives a controlled structural argument for the use of
as the minimal local curvature cost for closure density. The result is
not presented as a final global uniqueness theorem. Rather, it is a
minimality lemma: under the stated local assumptions and to lowest
nontrivial derivative order, this is the natural scalar functional
compatible with the NUVO closure-density representation and with
Madelung correspondence.
Lemma (Minimal local closure-density curvature cost).
Let be a smooth closure-density field on a spatial domain
. Suppose a local curvature cost
functional satisfies the following leading-order
requirements:
Then, to lowest derivative order and up to an overall positive
coefficient, the natural admissible curvature cost is
Proof (Structural argument).
Because the cost must vanish for spatially uniform closure density, its
leading nontrivial contribution must depend on spatial derivatives of
. A zeroth-order term of the form
would generally not vanish for uniform density unless were
trivial or specially constrained. Such a term would represent a local
density potential rather than a curvature or deformation cost. It is
therefore excluded from the leading curvature functional.
The lowest nontrivial derivative order is first order in spatial
gradients. Rotational invariance requires that first derivatives enter
through scalar contractions. Thus the leading local density-gradient
cost must have the schematic form
or equivalently,
for some monotone local amplitude variable .
Nonnegativity requires the coefficient of the quadratic gradient term
to be nonnegative. The cost then vanishes for uniform , since
The remaining question is which local amplitude variable
is selected. Compatibility with the complex representation selects
This is because the complex encoding is
with
Thus , rather than itself, is the amplitude field whose
spatial curvature appears in the Laplacian of the complex
representation.
The Schrödinger kinetic term is
When , the real part of this second-order
term contains
Therefore a closure-density curvature functional compatible with the
Schrödinger/Madelung representation must vary to produce a term
proportional to
in the phase equation.
The functional
has exactly this property. Its variation is
and, when combined with the variation of the term , it
produces
Thus the amplitude-gradient functional is the minimal first-derivative
functional whose Euler–Lagrange contribution has the required
second-order curvature form.
Since ,
Equivalently,
so
This is the leading Fisher-information-type density-gradient functional,
although no probabilistic interpretation is assumed in the present
context.
Other local functionals are possible if one allows higher derivatives,
higher powers, nonminimal density weights, or additional coupling to the
scalar background. For example, one could consider terms of the form
or
Such terms may be relevant as higher-order corrections or sectoral
refinements, but they are not part of the minimal lowest-order
curvature cost.
Therefore, under the stated leading-order assumptions, the minimal
local scalar cost compatible with the closure-density representation is
with . Madelung–Schrödinger correspondence fixes
This yields
The preceding lemma should be read as a controlled structural
minimality argument. It does not claim that no other density-dependent
terms can ever appear in a more complete NUVO theory. Rather, it shows
that if one seeks the lowest-order local, positive, rotationally
invariant curvature cost compatible with the amplitude representation
then the amplitude-gradient functional
is the natural minimal choice.
This is sufficient for the purpose of the present manuscript: to supply
the missing density-curvature contribution required for local Madelung
equivalence. A deeper future derivation may attempt to obtain the same
functional from an underlying discrete closure network, a finite-capacity
substrate model, or a more primitive exchange-sector coherence metric.
| Symbol | Meaning |
|---|---|
| closure density | |
| transport-derived phase | |
| R=\sqrt | closure amplitude |
| representation/action scale | |
| transported closure mass parameter | |
| scalar-geometric or exchange-sector potential | |
| closure-density curvature potential | |
| complex representation of |