Within the scalar--conformal NUVO framework, the support sector describes physical systems as localized anchors embedded in a uniformly delivered capacity field. The physical state of an anchor is given by its boundary flux distribution, subject to the invariant total intake condition
[
\int_{\partial S} \Phi_n , dA = mc^2.
]
Previous work has established that inertial and accelerated motion correspond, respectively, to stationary and evolving boundary flux distributions, and that scalar modulation of the delivery geometry induces admissible changes in these distributions. However, a closed evolution law governing this response has not yet been formulated.
In this work we derive the first dynamical law of the NUVO support sector. We show that the evolution of the boundary flux distribution is a local, constraint-preserving redistribution process driven by mismatch between the current boundary state and a locally preferred configuration determined by the scalar--conformal delivery geometry.
The scalar field encodes the geometric structure of delivery,
while the normalized scalar response
[
\Afield(x) = \frac{\Lambda(x)}{\Lambda_0}
]
provides a dimensionless diagnostic of this structure. Analogous to temperature in thermodynamics: it encodes the local conditions under which delivery is presented, without constituting a force or transported quantity.
The resulting evolution law is expressed as a projected relaxation on the admissible boundary state space. In the weak-field limit, this law reduces to a dipole response proportional to , yielding the inverse-square gravitational form for a static spherical source. Dynamics therefore arise as admissible evolution of boundary intake, and acceleration appears as an emergent kinematic consequence of boundary-state adjustment rather than as a primitive force.
The scalar--conformal NUVO framework reformulates physical dynamics in terms of capacity delivery and boundary response, rather than force and transport. In this setting, space is equipped with a scalar field that encodes the local structure of capacity delivery, while physical systems appear as localized anchors that consume this delivery at a fixed invariant rate.
A central result of the support-sector development is that capacity is not a transported substance. Instead, it is uniformly delivered across space, and anchors act as localized consumers of this delivery. The physical state of an anchor is therefore not determined by internal variables or external forces, but by the distribution of boundary intake across its surface, represented by the boundary flux density .
This perspective leads to a natural redefinition of motion. An inertial state corresponds to a stationary boundary flux distribution,
[
\partial_\tau \Phi_n = 0,
]
while acceleration corresponds to evolution of this distribution,
[
\partial_\tau \Phi_n \neq 0.
]
Gravitational response arises not from a force acting on the anchor, but from scalar modulation of the delivery geometry, which alters the set of admissible boundary flux configurations.
The preceding paper (M7) established the first response structure linking scalar geometry to boundary evolution. In particular, it identified the existence of a locally preferred boundary configuration determined by the scalar field and its spatial variation, and showed how weak-field gravitational response emerges from the resulting boundary asymmetry.
However, that formulation did not yet provide a closed dynamical law. While it characterized the qualitative structure of the response, it left implicit the admissible state space of boundary configurations, the constraint-preserving nature of the evolution, and the precise form of the evolution operator.
The purpose of the present paper is to close this gap. We derive a complete evolution law for the boundary flux distribution within the support sector, expressed entirely in terms of scalar--conformal geometry and boundary admissibility.
A key conceptual element is the interpretation of the scalar field. We distinguish between the scalar field and its local evaluation \Afield = \frac{\Lambda(x)}{\Lambda_0}. The field encodes the geometric structure of delivery, while \Afield serves as a local diagnostic of that structure. In this sense, \Afield plays a role analogous to temperature in thermodynamics: it characterizes the local conditions under which delivery is presented, without representing a force or a transported quantity.
The evolution law derived here takes the form of a local, causal redistribution of boundary flux within the admissible state space defined by fixed total intake. The dynamics of an anchor are thus governed by the requirement that its boundary intake remain compatible with the locally presented delivery geometry.
In the weak-field limit, the resulting law reduces to a dipole response proportional to the gradient of the scalar field, recovering the inverse-square gravitational form for a static spherical source. More generally, the framework shows that acceleration is not a primitive quantity, but an emergent kinematic image of boundary-state evolution.
This establishes the first closed dynamical formulation of the NUVO support sector, completing the M-series as a self-contained geometric and dynamical framework independent of exchange-sector assumptions.
The support sector of the NUVO framework is defined on a scalar--conformal geometric structure in which the spacetime metric takes the form
[
g_{\mu\nu} = \Lambda^2(x),\eta_{\mu\nu},
]
where is the Minkowski metric and is a positive scalar field defined over spacetime.
The scalar field encodes the local structure of capacity delivery. It does not represent a potential or a force-generating field, but rather a geometric descriptor that determines how capacity is presented locally to embedded systems.
For local discussions it is convenient to denote the scalar value evaluated at the
anchor location by
[
\Afield = \frac{\Lambda(x)}{\Lambda_0}.
]
Thus denotes the scalar field globally, while \Afield denotes its local
value when emphasis on pointwise delivery conditions is helpful.
Gradients of the scalar field, , represent spatial variation in the delivery geometry. They do not correspond to forces or transport processes, but instead determine how the locally preferred boundary configuration varies across space.
A central principle of the support sector is that capacity is not a transported substance. There is no flow of capacity between regions of space. Instead, capacity is uniformly delivered throughout spacetime as a background process.
The scalar field modulates the structure of this delivery, but does not alter its fundamental nature as a uniform presentation. Physical effects arise only through how this delivery is locally received and processed by embedded systems.
Accordingly, no conservation law for transported capacity is assumed or required. All physically meaningful quantities arise from local interaction with the delivered capacity at the boundary of an anchor.
Physical systems in the support sector appear as localized anchored structures. Each anchor is characterized by its interaction with the uniformly delivered capacity at its boundary.
The state of an anchor is given by its boundary flux distribution,
[
\Phi_n : \partial S \to \mathbb{R}_{\ge 0},
]
which represents the rate of capacity intake per unit area across the boundary .
The total intake is invariant and defines the rest energy of the anchor:
[
\int_{\partial S} \Phi_n , dA = mc^2.
]
This condition is not dynamical, but structural: it defines the admissible state space of the anchor.
Thus, the physical state of an anchor is not given by position or velocity, but by the angular distribution of boundary intake encoded in .
Within this framework, motion is defined in terms of the evolution of the boundary flux distribution along the worldline of the anchor.
An inertial state corresponds to a stationary boundary configuration,
[
\partial_\tau \Phi_n = 0,
]
while an accelerated state corresponds to a changing boundary configuration,
[
\partial_\tau \Phi_n \neq 0.
]
This definition makes no reference to force. Acceleration is not imposed externally, but arises as a property of the evolving boundary intake structure.
Importantly, anisotropy of alone does not imply acceleration. A boundary distribution may be anisotropic yet stationary. Only the temporal evolution of the distribution corresponds to acceleration.
The scalar field influences physical behavior by modifying the set of admissible boundary flux distributions.
At each point, the locally evaluated scalar value \Afield = \frac{\Lambda(x)}{\Lambda_0}, together with its spatial variation , determines the boundary configurations that are compatible with the presented delivery geometry.
In particular, for a given local scalar environment, there exists a class of admissible boundary flux distributions satisfying:
\begin{itemize}
\item positivity: ,
\item fixed total intake: ,
\item local compatibility with the scalar-modulated delivery structure.
\end
Physical response arises when the current boundary configuration of an anchor is not compatible with the locally preferred configuration determined by the scalar geometry. The resulting evolution is a redistribution of boundary intake within this admissible class.
All constructions in this work are restricted to the support sector. No exchange-sector quantities, interactions, or coherence conditions are invoked.
In particular:
\begin{itemize}
\item No force laws are introduced.
\item No transport or flow of capacity is assumed.
\item No quantum or electromagnetic structure is used.
\end
The goal is to derive a closed dynamical law entirely from scalar--conformal geometry, uniform capacity delivery, and boundary admissibility.
Let denote a localized anchor with boundary .
The physical state of the anchor is given by its boundary flux distribution
[
\Phi_n : \partial S \to \mathbb{R}_{\ge 0},
]
which assigns to each boundary point the local rate of capacity intake per unit area.
The total intake is fixed by the structural condition
[
\int_{\partial S} \Phi_n , dA = mc^2.
]
This constraint defines the admissible class of boundary configurations and is not a consequence of dynamical evolution.
\left{
\Phi_n ;\middle|;
\Phi_n \ge 0,;
\int_{\partial S} \Phi_n , dA = mc^2,;
\Phi_n \ \text{compatible with the local scalar environment}
\right}.
]
Here, compatibility refers to the requirement that the boundary configuration be consistent with the locally presented delivery structure determined by the scalar field and its derivatives.
Thus, is a constrained function space of non-negative boundary distributions with fixed total intake, parameterized by the local scalar geometry.
Physical evolution of the boundary state corresponds to continuous deformations of within the admissible state space.
An admissible variation must preserve the total intake constraint:
[
\int_{\partial S} \delta \Phi_n , dA = 0.
]
\left{
\delta \Phi_n ;\middle|;
\int_{\partial S} \delta \Phi_n , dA = 0
\right}.
]
These variations represent pure redistributions of boundary intake, with no net change in total capacity consumption.
In addition to preserving total intake, admissible boundary configurations must satisfy the positivity condition
[
\Phi_n \ge 0.
]
Thus, the admissible state space is not a linear space, but a constrained subset of a function space with a boundary defined by .
Physical evolution must therefore remain within the interior of this space or evolve along its boundary in a manner consistent with the positivity constraint. In particular, no admissible evolution may produce negative boundary intake.
The admissible state space depends on the local scalar environment through the scalar field and its spatial derivatives.
In general, the admissibility of a given boundary configuration is determined by local scalar data of the form
[
\Lambda(x), \quad \nabla \Lambda(x), \quad \nabla\nabla \Lambda(x), \quad u^\mu,
]
where is the four-velocity of the anchor.
Thus, the admissible state space is not fixed globally, but varies from point to point in spacetime as the scalar field varies.
A boundary configuration is said to be locally compatible with the scalar environment if it belongs to and remains stationary under admissible evolution.
Such configurations correspond to steady states of the support-sector dynamics:
[
\partial_\tau \Phi_n = 0.
]
In general, for each local scalar environment there exists a class of compatible boundary configurations, among which a preferred configuration will be identified in the following section.
The admissible state space encodes the physically realizable boundary configurations of an anchor at a given location.
Dynamics in the support sector consist of continuous evolution within this space, subject to:
\begin{itemize}
\item preservation of total intake,
\item non-negativity of boundary flux,
\item compatibility with the locally presented delivery geometry.
\end
No notion of force or transport is required. The evolution of an anchor is entirely determined by how its boundary configuration adjusts within the admissible state space as the scalar environment varies.
At each spacetime point , the scalar--conformal delivery geometry
determines not only the admissible boundary state space ,
but also a locally preferred boundary configuration.
We denote this preferred configuration by
[
\Phi_n^*(x),
]
and interpret it as the boundary flux distribution that is most
compatible with the locally presented delivery structure.
The existence of such a configuration follows from the requirement that
the support sector admit steady states in which
[
\partial_\tau \Phi_n = 0.
]
Thus, represents a locally compatible steady-state
configuration within .
The preferred configuration depends on the local scalar environment,
including both the value of the scalar field and its derivatives.
In general, we write
[
\Phi_n^(x) = \Phi_n^!\left[\Lambda(x), \nabla \Lambda(x), \nabla\nabla \Lambda(x), u^\mu\right],
]
where is the four-velocity of the anchor.
Thus, the preferred boundary configuration is a local functional of the
scalar geometry and the state of motion of the anchor.
In the weak-field regime, where
[
\Lambda(x) = \Lambda_0 \bigl(1 + \epsilon(x)\bigr), \quad |\epsilon(x)| \ll 1,
]
the preferred configuration may be expanded perturbatively.
To leading order, the deviation from isotropy is governed by the scalar
gradient,
[
\nabla \Lambda(x).
]
\Phi_0
+
\delta \Phi_n(\theta,\phi),
]
where is the uniform baseline flux and is a
direction-dependent correction proportional to .
The preferred configuration satisfies the total intake constraint,
[
\int_{\partial S} \Phi_n^* , dA = mc^2,
]
and therefore lies within the admissible state space .
The preferred boundary configuration represents the
locally stable equilibrium state of the anchor in the given scalar
environment.
When the actual boundary configuration differs from
, the anchor is out of equilibrium with the local delivery
geometry. The resulting mismatch drives the evolution of the boundary
state.
We now formulate the evolution law governing the boundary flux
distribution.
The central principle is that evolution is driven by mismatch between
the current boundary configuration and the locally preferred
configuration .
Thus, the evolution must depend on the difference
[
\Phi_n - \Phi_n^*.
]
The evolution must preserve the total intake condition
[
\int_{\partial S} \Phi_n , dA = mc^2.
]
Therefore, the time derivative must satisfy
[
\int_{\partial S} \partial_\tau \Phi_n , dA = 0.
]
This implies that evolution occurs within the tangent space
defined in Section 3.
To enforce the constraint, we introduce a projection operator
onto the tangent space of admissible variations:
[
\mathcal{P} : L^2(\partial S) \to T_{\Phi_n}\mathcal{S}(x),
]
such that for any function on ,
[
\int_{\partial S} \mathcal{P}[f] , dA = 0.
]
This operator removes any component of that would change the total
intake.
-,\kappa,
\mathcal{P}!\left[\Phi_n - \Phi_n^*\right],
]
where is a scalar response coefficient.
This equation defines a projected relaxation process within the
admissible state space.
The evolution law has the following properties:
Locality: The evolution depends only on local scalar data and the
current boundary configuration.
Constraint preservation: The total intake is conserved.
Positivity preservation: For sufficiently small time steps, the
evolution preserves the non-negativity of .
Relaxation to equilibrium: The preferred configuration
is a fixed point of the evolution.
The evolution law describes a relaxation process toward local
compatibility with the scalar delivery geometry.
Acceleration arises as the macroscopic manifestation of this boundary
state evolution along the worldline of the anchor.
Thus, motion is not driven by forces, but by the requirement that the
boundary configuration remain compatible with the locally presented
delivery conditions.
We have derived a closed evolution law for the boundary flux
distribution of anchored structures in scalar--conformal NUVO space.
The law takes the form of a projected relaxation toward a locally
preferred configuration determined by the scalar field.
Dynamics in the support sector are therefore governed by:
Acceleration emerges as the kinematic consequence of boundary flux
redistribution rather than as a primitive dynamical input.
This completes the derivation of the support-sector dynamics in the
NUVO framework.