Internal Frame

The Internal Frame is the discrete, count-based configuration domain in which fundamental measures are defined relative to the fundamental units $l_f$, $m_f$, and $t_f$. Relational structure is encoded as integer count configurations. The Internal Frame does not itself possess continuous geometric structure; geometry arises only through realization into the System Frame.

System Frame

The System Frame is the non-discrete relational encoding induced by the Frames mapping. It has no independent fundamental measure. Instead, observable quantities appear as continuous relational separations derived from discrete count structure. Physically meaningful observables are defined in this realized frame.

Frames mapping

The Frames mapping is the kinematic transformation that relates discrete count configurations in the Internal Frame to continuous relational structure in the System Frame. It preserves realization consistency and imposes closure conditions on relational configurations. Observable geometry and interaction structure emerge only after this mapping is applied.

Frames Principle

The Frames Principle states that all physically meaningful observables arise from the Frames mapping between the Internal Frame and the System Frame. No observable quantity is intrinsic to the Internal Frame alone. Measurement is therefore relational and mapping-dependent, replacing the assumption that measurable geometry must begin with a pre-existing continuous spacetime manifold.

MQ Action

The MQ Action is the fundamental kinematic functional defined over Internal Frame count structure. It governs the evolution and consistency of relational configurations under the Frames mapping. Its realizations depend on count separation: fixed interaction-scale realizations correspond to electromagnetic behavior at the electromagnetic demarcation, while count-resolved realizations correspond to gravitational behavior through the mapping-induced metric structure.

In both regimes, observable dynamics arise from normalization and realization of conserved count under the mapping. Electromagnetic interaction and gravitational curvature are therefore treated as scale-dependent realizations of the same underlying action rather than as independently introduced dynamical components.

MQ variational principle

The MQ variational principle states that physically realized configurations in the System Frame correspond to stationary points of the MQ Action over admissible realizations of encoded availability derived from the Internal Frame. Let $M:\omega \to ℛ$ denote the Frames mapping, let ${\omega }^{\prime }\subset \omega$ be a relational region of the Internal Frame, and let $\Omega =M\left({\omega }^{\prime }\right)\subset ℛ$ be its realized configuration. With ${\rho }_{ℛ}$ denoting the corresponding coarse-grained encoded availability field, the action is

Here, the Lagrangian density is constructed from count-based relations induced by the mapping, and $d\mu$ is the induced measure on the realized domain. Admissible realizations satisfy $\delta S=0$ under variations consistent with the mapping constraints and the applicable conservation relations.

Informativity differential

The Informativity differential is the non-Lorentz length contraction associated with discrete Internal Frame count structure. It arises from normalization of conserved count under the Frames mapping and quantifies the difference between count-defined relational separation and its realized representation in the System Frame.

Its magnitude is count-dependent: the effect increases at small separation and diminishes toward the upper-count limit. It is distinct from curvature. The two are separate geometric manifestations of realization: one appears as a finite-count contraction of realized length, the other as emergent curvature associated with count-dependent realization.

Metric differential

The metric differential characterizes the discrete-to-continuous mismatch that arises when conserved Internal Frame count structure is expressed as a continuous observable in the System Frame. It is not itself a physical deformation. Rather, it identifies the structural requirement that integer-resolved count relations be represented by normalized, non-discrete quantities.

For a fundamental measure defined by count $n$, realization requires a mapping normalization that preserves count conservation while enforcing continuity. The resulting representation-level mismatch is the metric differential.

Electromagnetic demarcation

The electromagnetic demarcation is the fixed interaction count distance $n_L=\frac{276}{{\theta }_{\mathrm{si}}}\approx 84.60055$ fundamental-length counts, at which known electromagnetic interactions are realized in MQ. At this finite scale, the Informativity differential remains significant. Its influence increases toward smaller interaction distances and rapidly diminishes with increasing count separation.

Count separation

Count separation is the number of fundamental length units $l_f$ between relational configurations in the Internal Frame. It sets the scale at which mapping-induced deformation is evaluated and determines whether finite-count effects remain physically significant.

Realization manifold

A realization manifold is the set of admissible System Frame realizations compatible with a specified Internal Frame configuration. It may be written inline as $ℛ\left({\Omega }_I\right)$ . The manifold is not a primitive geometric object of the Internal Frame; it is the image of conserved count structure after application of the Frames mapping and any applicable realization constraints. Probabilities are assigned only after a normalized measure is defined on the admissible realization manifold.

Mapping-induced deformation

Mapping-induced deformation is the modification of realized separation that follows from the discrete-to-continuous Frames mapping. It is determined by the mapping structure and is not introduced as an independent physical field.

At a fixed count separation, the deformation appears as a local scale-specific modification governed by the Informativity differential. Across extended count separation, the same underlying mapping effect is realized through geometric curvature. These are scale-dependent manifestations of a common mapping structure rather than unrelated mechanisms.

Fixed interaction scale

A fixed interaction scale is a count-defined distance at which a physical interaction is realized under the Frames mapping. For electromagnetic interactions, this is the electromagnetic demarcation, $n_L=\frac{276}{{\theta }_{\mathrm{si}}}$ . Both the finite-count length effect and the corresponding curvature are therefore evaluated at one specific point in the discrete count structure rather than in an asymptotic limit.

Curvature-resolved regime

A curvature-resolved regime describes realized geometric structure as a function of count separation. Different separations correspond to different realized curvature values in the System Frame. This is the regime associated with gravitational phenomena, for which measurements probe a range of separations rather than a single fixed interaction scale.

Bounded realization

Bounded realization is the MQ condition in which the gravitational relation available from a physical source is realized in the System Frame only to the fraction permitted by the count-constrained Frames mapping. The realized fraction is bounded between zero and unity and approaches complete realization when baryonic acceleration is large compared with the MQ acceleration scale.

At low baryonic acceleration, incomplete realization produces a realized gravitational response larger than the value inferred by applying the Newtonian baryonic source relation alone. No additional gravitating matter is introduced, and the baryonic source law itself is not modified; the distinction lies in how much of the available gravitational relation is realized.

Realization fraction

The realization fraction $𝒫$ is the dimensionless fraction of gravitational availability from a baryonic source that is realized through the Frames mapping in the System Frame. For the galactic realization,

Here, $g_{\mathrm{bar}}\left(R\right)$ is the Newtonian baryonic acceleration and $a_{\mathrm{MQ}}$ is the MQ acceleration scale. The realization fraction approaches unity in the high-acceleration limit and decreases continuously below unity as the available baryonic acceleration decreases. The corresponding realized acceleration is $g_{\mathrm{real}}=\frac{g_{\mathrm{bar}}}{𝒫}$ .

Galactic realization law

The galactic realization law is the fixed MQ relation that maps an independently determined Newtonian baryonic source to its realized System Frame acceleration or orbital velocity through the realization fraction and the first-principles MQ acceleration scale $a_{\mathrm{MQ}}$. It contains no galaxy-specific MQ fitting parameter and leaves the reconstructed baryonic source relation unchanged.

The law approaches Newtonian baryonic dynamics continuously at high acceleration and yields an enhanced realized response in the low-acceleration regime.

Realized gravitational response

The realized gravitational response is the gravitational acceleration, potential, metric response, or associated motion produced in the System Frame after gravitational availability from a physical source has been realized through the Frames mapping.

It is distinct from the Newtonian baryonic source quantity, which specifies the gravitational availability supplied by the reconstructed baryonic mass distribution. In the stationary galactic limit, the realized response is related to that source through the realization fraction; in a covariant description it is represented through the System Frame metric and its action-derived stress-energy source.

Gravitational availability

Gravitational availability is the gravitational relation supplied by the physical source before realization through the Frames mapping. In the weak-field galactic description it is represented by the independently reconstructed Newtonian baryonic quantities $M_{\mathrm{bar}}$, ${\rho }_{\mathrm{bar}}$, ${\Phi }_{\mathrm{bar}}$, and $g_{\mathrm{bar}}$. It is therefore a source property rather than an additional material component.

Dual weak-field representation

The dual weak-field representation is the first-order variational representation of MQ weak-field gravitational realization obtained by replacing the primal nonlinear realization functional with an auxiliary spatial field and its exact Legendre-dual function.

The dual form is algebraically equivalent to the primal weak-field action and preserves the same stationary MQ realization law. Its purpose is representational: it expresses the zero-field limit in variables for which the spatial metric Hessian is regular on a homogeneous FLRW background. It introduces neither a new gravitational interaction nor an independent physical degree of freedom.