Notes

Primitive definitions

• ${\mathrm{\mathbb{N}}}_0$

  the set of nonnegative integers {0,1,2, …}, used as the codomain for discrete count variables (e.g., n_I, n_S prior to density normalization).

• $\mathrm{\mathbb{Z}}$

  is the set of integers, used as the codomain for signed Internal Frame count-transfer quantities such as adjacency flux J_ij. Positive and negative signs denote orientation of transfer along an admissible adjacency link and do not imply geometric direction in the System Frame.

• ${\mathrm{ℝ}}^3$

  three–dimensional System Frame configuration space for single–subsystem spatial coordinates $\mathit{r}$.

• ${\mathrm{ℝ}}^{3\mathit{N}}$

  System Frame configuration space for an N–subsystem (or N–particle) encoded state $\mathrm{\Psi }({\mathit{r}}_1,\mathit{\dots },{\mathit{r}}_{\mathit{N}},\mathit{t})$.

Discrete Internal Frame primitives

• $\mathrm{𝐴}$

  The Internal Frame update index is a discrete ordering parameter labeling successive configurations of the Internal Frame count medium. It indexes deterministic updates of relational count assignments ${\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$ across the configuration domain $\mathit{\omega }$, with regional evolution defined on subsets ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$ through conservation and adjacency relations. The update index $\mathit{A}$ is not a physical time coordinate and carries no metric, geometric, or causal structure. It does not correspond to System Frame time $\mathit{t}$, nor to any observable temporal parameter. Instead, it provides an intrinsic ordering of count redistribution steps within the Internal Frame. Evolution with respect to $\mathit{A}$ defines the deterministic conservation and redistribution of count via adjacency relations, as expressed in Internal Frame conservation laws of the form $\frac{\partial {\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})}{\partial \mathit{A}}+{\mathrm{div}\mathit{}}_{\mathit{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$ = 0. The mapping between the Internal Frame update index $\mathit{A}$ and System Frame time $\mathit{t}$ is established only through the Frames mapping $\mathit{M}$, and is generally nontrivial. Consequently, $\mathit{A}$ should be interpreted as a kinematic sequencing parameter intrinsic to the discrete count substrate, rather than as a temporal observable.

• ${\mathrm{𝑛}}_{\mathrm{𝐼}}\left(\mathrm{𝜉}\mathrm{,}\mathrm{𝐴}\right)$

  is the Internal Frame discrete measurement availability count at relational cell $\mathit{\xi }$ and internal update index $\mathit{A}$; integer–valued with ${\mathit{n}}_{\mathit{I}}\in {\mathrm{\mathbb{N}}}_0$. (No System Frame coordinates allowed.)

• ${\mathrm{𝜌}}_{\mathrm{𝐼}}\mathrm{(}\mathrm{𝜉}\mathrm{,}\mathrm{𝐴}\mathrm{)}$

  is the Internal Frame count density at relational cell $\mathit{\xi }$ and update index $\mathit{A}$. The quantity ${\mathit{\rho }}_{\mathit{I}}$ represents the local distribution of measurement availability within the discrete relational structure of the Internal Frame. It is defined exclusively on the relational domain $\mathit{\omega }$ and carries no geometric, metric, coordinate, or continuum interpretation. The local count density ${\mathit{\rho }}_{\mathit{I}}$ is constructed from the underlying count variable ${\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$ and serves as the local conserved quantity appearing in Internal Frame conservation relations and adjacency–flux evolution equations. The evolution of ${\mathit{\rho }}_{\mathit{I}}$ is governed by the discrete continuity relation $\frac{\partial {\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})}{\partial \mathit{A}}+{\mathrm{div}\mathit{}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A})=0$, or equivalently, $\frac{\partial {\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})}{\partial \mathit{A}}+\sum _{{\mathit{\xi }}^{\mathrm{\text{'}}}\in \partial \mathrm{\Omega }\left(\mathit{\xi }\right)}{\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A})=0$. The quantity ${\mathit{\rho }}_{\mathit{I}}$ remains an Internal Frame object and must not be interpreted as a continuum mass density, probability density, or geometric field. Under the Frames mapping $\mathit{M}$, conservation of ${\mathit{\rho }}_{\mathit{I}}$ provides the count structure from which realized System Frame observables and encoded availability measures are constructed.

• ${\mathrm{𝑁}}_{\mathrm{𝐼}}\mathrm{(}{\mathrm{𝜔}}^{\mathrm{\text{'}}}\mathrm{,}\mathrm{𝐴}\mathrm{)}$

  total Internal Frame count in a relational region $\mathrm{\Omega }$ at internal update index $\mathit{A}$, typically ${\mathit{N}}_{\mathit{I}}({\mathit{\omega }}^{\mathrm{\text{'}}},\mathit{A})={\sum }_{\mathit{\xi }\in {\mathit{\omega }}^{\mathrm{\text{'}}}}{\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$ (or the appropriate internal measure analogue).

• $\mathrm{𝜉}$

  is a discrete relational cell label indexing elements of the Internal Frame configuration domain $\mathit{\omega }$. Each $\mathit{\xi }$ identifies a primitive micro-configuration at which Internal Frame count variables ${\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$, density ${\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})$, and adjacency flux ${\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A})$ are defined. It may be described as $\mathit{\xi }\in \mathit{\omega }$, ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$, and ${\sum }_{\mathit{\xi }\in {\mathit{\omega }}^{\mathrm{\text{'}}}}{\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$. The symbol $\mathit{\xi }$ is a pure relational index and not a coordinate. It carries no metric, geometric, spatial, temporal, tangent-space, or continuum meaning. It is defined only through membership in $\mathit{\omega }$, adjacency relations $\mathit{\xi }\sim {\mathit{\xi }}^{\mathrm{\text{'}}}$, and serves as a summation index and domain element for count density and flux. Under the Frames mapping, realizations derived from collections of relational cells contribute to emergent System Frame structures, but $\mathit{\xi }$ itself never becomes a coordinate or vector in the realized continuum description.

• $\mathrm{𝜔}$

  The Internal Frame configuration domain $\mathit{\omega }$ is an abstract relational domain containing all admissible Internal Frame count assignments and constraint systems on which candidate Frames mappings may be defined. It carries no metric, geometric, or temporal structure and does not correspond to any System Frame coordinate domain. A referenceable universe corresponds to a maximal subdomain $\mathit{U}\subset \mathit{\omega }$ on which a Frames mapping exists satisfying the global referenceability constraints (unique realization, normalization, and holonomy–free closure). Regional subsets ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$ are used to define finite count measures, conservation laws, and boundary–flux relations within the Internal Frame. The corresponding realized System Frame configurations are given by $\mathit{\Omega }=\mathit{M}({\mathit{\omega }}^{\mathrm{\text{'}}})\subset \mathcal{R}$.

• ${\mathrm{𝜔}}^{\mathrm{\text{'}}}$

  Is a relational region of the Internal Frame, with ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$. This term denotes a finite subset of the Internal Frame relational adjacency domain used to define regional count measures, conservation laws, and boundary–flux relations. It carries no geometric or metric structure and does not correspond to a System Frame coordinate domain. Regional Internal Frame counts are written ${\mathit{N}}_{\mathit{I}}({\mathit{\omega }}^{\mathrm{\text{'}}},\mathit{A})$.

• $\mathrm{\partial }\mathrm{𝜔}$

  Adjacency boundary of an Internal Frame consisting of oriented adjacency links connecting cells in ${\mathit{\omega }}^{\mathrm{\text{'}}}$ to cells outside ${\mathit{\omega }}^{\mathrm{\text{'}}}$. It appears in Internal Frame conservation relations through boundary–flux terms such as ${\mathit{\varphi }}_{\mathit{I}}(\partial {\mathit{\omega }}^{\mathrm{\text{'}}},\mathit{A})$.

• ${\mathrm{\Omega }}_{\mathrm{1}}\mathrm{,}{\mathrm{\Omega }}_{\mathrm{2}}$

  Internal Frame relational subsets used in finite additivity relations. They satisfy ${\mathrm{\Omega }}_1\cap {\mathrm{\Omega }}_2=\varnothing$ and ${\mathit{N}}_{\mathit{I}}({\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2,\mathit{A})={\mathit{N}}_{\mathit{I}}({\mathrm{\Omega }}_1,\mathit{A})+{\mathit{N}}_{\mathit{I}}({\mathrm{\Omega }}_2,\mathit{A})$.

• ${\mathrm{div}}_{\mathrm{adj}}$

  is the adjacency divergence operator acting on the Internal Frame count–flux ${\mathit{J}}_{\mathit{I}}$. It is defined by ${\mathrm{div}\mathit{}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A}):=\sum _{{\mathit{\xi }}^{\mathrm{\text{'}}}\in \partial \mathrm{\Omega }\left(\mathit{\xi }\right)}{\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A})$, where $\partial \mathit{\Omega }\left(\mathit{\xi }\right)$ denotes the set of admissible neighboring relational cells of $\mathit{\xi }$. The operator returns the net count imbalance associated with relational cell $\mathit{\xi }$ during update step $\mathit{A}$ by summing count–flux contributions over all admissible adjacency links incident on $\mathit{\xi }$. It is a purely combinatorial operator defined on the adjacency structure of the Internal Frame and carries no metric, geometric, directional, vector, or differential structure. The operator measures local count imbalance within the conserved count network and appears in the Internal Frame continuity relation $\frac{\partial {\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})}{\partial \mathit{A}}+{\mathrm{div}\mathit{}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A})=0$. It is a purely combinatorial operator defined on the adjacency structure of the Internal Frame and carries no metric, geometric, directional, vector, or differential structure. The operator measures local count imbalance within the conserved count network and appears in the Internal Frame continuity relation $\frac{\partial {\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})}{\partial \mathit{A}}+{\mathrm{div}\mathit{}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A})=0$. Equivalently, $\frac{\partial {\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A})}{\partial \mathit{A}}+\sum _{{\mathit{\xi }}^{\mathrm{\text{'}}}\in \partial \mathit{\Omega }\left(\mathit{\xi }\right)}{\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A})=0$. Unlike a continuum divergence, ${\mathrm{div}\mathit{}}_{\mathrm{adj}}$ does not act on a vector field and does not involve derivatives. It operates exclusively on count transfers between admissible neighboring relational cells. Under the Frames mapping, normalized realizations of the count–flux ${\mathit{J}}_{\mathit{I}}$ and adjacency divergence ${\mathrm{div}\mathit{}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}$ correspond to the conserved current and its divergence in the System Frame, respectively.

• $\mathrm{Adj(\xi )}$

  is the set of all relational cells ξ′ connected to ξ by an admissible adjacency link within the Internal Frame configuration domain ω. The adjacency neighborhood is a purely combinatorial object defined on the relational structure of the Internal Frame. It carries no metric, geometric, coordinate, vector, or continuum interpretation. Adj(ξ) specifies the relational cells that participate in local count redistribution through the adjacency count-flux ${\mathit{J}}_{\mathit{I}}$(ξ → ξ′, A). It therefore provides the neighborhood structure used in the definitions of the adjacency divergence operator div_adj, the local conservation law, and boundary-flux relations. Under the Frames mapping, realizations derived from Adj(ξ) may contribute to emergent System Frame structures, but Adj(ξ) itself remains an Internal Frame object and is not a geometric neighborhood in the System Frame.

• ${\mathrm{𝐽}}_{\mathrm{𝐼}}$

  is the Internal Frame adjacency count–flux object defined on oriented adjacency links $\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}}$ within the configuration domain $\mathit{\omega }$. For each admissible link, ${\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A})$ specifies the signed count transfer between relational cells during update step $\mathit{A}$. The object ${\mathit{J}}_{\mathit{I}}$ is the discrete flux structure associated with the conserved count field ${\mathit{n}}_{\mathit{I}}$ and provides the local transfer mechanism appearing in the Internal Frame continuity relation ${\mathrm{\Delta }}_{\mathit{A}}{\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})+{\mathrm{div}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A})=0$. ${\mathit{J}}_{\mathit{I}}$ is a purely combinatorial object defined exclusively on the adjacency structure of $\mathit{\omega }$. It carries no metric, geometric, directional, vector, or differential structure. The relational cell labels $\mathit{\xi }$ and ${\mathit{\xi }}^{\mathrm{\text{'}}}$ denote adjacency relationships only and do not represent positions in a geometric space. Consequently, ${\mathit{J}}_{\mathit{I}}$ is not a vector field and no geometric differential operators act upon it. The associated adjacency divergence operator is defined by ${\mathrm{div}\mathit{}}_{\mathrm{adj}}{\mathit{J}}_{\mathit{I}}(\mathit{\xi },\mathit{A}):=\sum _{{\mathit{\xi }}^{\mathrm{\text{'}}}\in \partial \mathrm{\Omega }\left(\mathit{\xi }\right)}{\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A})$, where $\mathit{N}\left(\mathit{\xi }\right)$ denotes the set of admissible neighboring relational cells of $\mathit{\xi }$. This operator returns the net count imbalance associated with relational cell $\mathit{\xi }$ during update step $\mathit{A}$ and measures the local departure from count balance produced by adjacency transfers. Under the Frames mapping $\mathit{M}$, ${\mathit{J}}_{\mathit{I}}$ provides the discrete count-flux structure whose mapping-normalized realization defines the corresponding conserved current in the System Frame. The mapping does not reinterpret ${\mathit{J}}_{\mathit{I}}$ as a geometric vector. Rather, continuum current representations emerge through normalization of conserved count flow under the Frames mapping.

• ${\mathrm{𝜙}}_{\mathrm{𝐼}}\mathrm{(}\mathrm{\partial }{\mathrm{𝜔}}^{\mathrm{\text{'}}}\mathrm{,}\mathrm{𝐴}\mathrm{)}$

  is the Internal Frame boundary count–flux associated with a relational region ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$ at update index $\mathit{A}$. It is defined as the net outward signed sum of adjacency flux contributions across all oriented boundary links connecting cells in ${\mathit{\omega }}^{\mathrm{\text{'}}}$ to cells in $\mathit{\omega }\setminus {\mathit{\omega }}^{\mathrm{\text{'}}}$, ${\mathit{\varphi }}_{\mathit{I}}\left(\partial {\mathit{\omega }}^{\mathit{\text{'}}},\mathit{A}\right)={\sum }_{\left(\mathit{\xi },{\mathit{\xi }}^{\mathit{\text{'}}}\right)\in \partial {\mathit{\omega }}^{\mathit{\text{'}}}}\mathit{J}\left(\mathit{\xi }\to {\mathit{\xi }}^{\mathit{\text{'}}},\mathit{A}\right)$, where $\partial {\mathit{\omega }}^{\mathrm{\text{'}}}$ consists of all admissible oriented adjacency links with $\mathit{\xi }\in {\mathit{\omega }}^{\mathrm{\text{'}}}$ and ${\mathit{\xi }}^{\mathit{\text{'}}}\notin {\mathit{\omega }}^{\mathrm{\text{'}}}$. This quantity represents the exact discrete boundary transport of measurement availability and has units of counts per update step. It is not a geometric surface integral and involves no notion of area, normal vectors, or metric structure. The boundary flux is related to the adjacency divergence of the count–flux through the discrete divergence relation, expressing regional conservation as the sum of local count imbalances over ${\mathit{\omega }}^{\mathrm{\text{'}}}$. Under the Frames mapping, preservation of ${\mathit{\varphi }}_{\mathit{I}}$ constrains the mapping normalization functional, thereby fixing how conserved Internal Frame flux is realized as a System Frame conserved current.

• $\mathcal{l}$

  an oriented adjacency link (edge) in the Internal Frame relational adjacency domain; elements of a closed loop $\mathit{\gamma }$ are sequences of such links.

Frames Mapping and Realization Structures

• $\mathrm{𝑈}$

  is a referenceable universe domain when a maximal subdomain $\mathit{U}\subset \mathit{\omega }$ of the Internal Frame configuration domain on which a valid Frames mapping $\mathit{M}$ exists satisfying normalization, unique realization, and holonomy–free closure on all contractible reference cycles. The corresponding System Frame reference structure is given by the relational image $\mathit{M}(\mathit{U})\subset \mathcal{R}$. The domain $\mathit{U}$ is maximal in the sense that no proper extension ${\mathit{U}}^{\mathrm{\text{'}}}\supset \mathit{U}$ admits a Frames mapping satisfying the same global referenceability constraints. Distinct universes correspond to maximal subdomains whose associated Frames mappings cannot be simultaneously extended over their union without violating at least one global constraint.

• $\mathrm{\gamma }$

  A reference cycle is a closed admissible relational path within a referenceable domain $\mathit{U}\subset \mathit{\omega }\setminus \mathit{D}$ used to evaluate loop functionals and consistency conditions associated with the Frames mapping. A reference cycle consists of a finite ordered sequence of admissible adjacency links returning to its origin. Contractible cycles must satisfy holonomy–free closure, while non–contractible cycles may carry defect-induced holonomy. Reference cycles are used to probe closure structure, measure consistency, and global referenceability conditions within admissible universe domains.

• $\mathcal{C}$

  The count functional is a scalar functional acting on conserved Internal Frame count structure, mapping ${\mathit{N}}_{\mathit{I}}({\mathit{\omega }}^{\mathrm{\text{'}}},\mathit{A})$ (and, where required, its associated density and flux structure) to a System Frame–compatible scalar prior to normalization. It encodes how relational count configurations contribute to realized observables independent of mapping normalization. More explicitly, it may be expressed as $\mathcal{C}:\{{\mathit{\rho }}_{\mathit{I}},{\mathit{J}}_{\mathit{I}},\partial {\mathit{\omega }}^{\mathrm{\text{'}}}\}\to {\mathrm{ℝ}}^{+}$, with the following constraints: i) consistency with conservation laws, ii) invariance under admissible Internal Frame relabelings, and iii) independence from System Frame coordinates.

• $\mathit{f}(\mathit{r})$

  A System Frame observable function defined on the realized spatial coordinate domain $\mathit{r}\in {\mathrm{ℝ}}^3$. The function $\mathit{f}(\mathit{r})$ represents an arbitrary measurable quantity evaluated on a realized configuration and is used to construct expectation values over the normalized encoded availability density ${\mathit{\rho }}_{\mathit{P}}(\mathit{r},\mathit{t})$. It carries no primitive ontological status and does not correspond to an Internal Frame count variable, update index, or conservation quantity. Observable averages are obtained through expressions of the form $\mathit{⟨}\mathit{f}\mathit{⟩}={\int }_{{\mathrm{ℝ}}^3}\mathit{f}(\mathit{r}){\mathit{\rho }}_{\mathit{P}}(\mathit{r},\mathit{t}){\mathit{d}}^3\mathit{r}$, or equivalently through the wave function encoding representation. The symbol $\mathit{f}$ serves as a generic test function for realized System Frame observables and is not restricted to any particular physical quantity.

• $\mathcal{N}[{\mathit{\omega }}^{\mathrm{\text{'}}}]$

  The mapping normalization functional is a normalization operator induced by the Frames mapping that converts conserved Internal Frame count measures into System Frame observables by enforcing consistency with boundary–flux conservation and holonomy constraints. It acts on conserved count structure over a relational region ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$ and determines the normalization applied in the realization of $\mathit{M}({\mathit{\omega }}^{\mathrm{\text{'}}})$. Formally, it is given as $\mathcal{N}[{\mathit{\omega }}^{\mathrm{\text{'}}}\left]=1/\mathit{Z}({\mathit{\omega }}^{\mathrm{\text{'}}}\right)\mathrm{with}\mathit{Z}({\mathit{\omega }}^{\mathrm{\text{'}}})=\sum _{\mathit{\xi }\in {\mathit{\omega }}^{\mathrm{\text{'}}}}{\mathit{n}}_{\mathit{I}}(\mathit{\xi },\mathit{A})$. Modified under discrete effects this is $\mathcal{N}[{\mathit{\omega }}^{\mathrm{\text{'}}}]\to \mathcal{N}[{\mathit{\omega }}^{\mathrm{\text{'}}}]\cdot (2{\mathit{Q}}_{\mathit{L}}{\mathit{n}}_{\mathit{L}})$.

• ${\mathcal{L}}_{\mathit{R}}\left[\mathrm{\Omega }\right]$

  Realized separation functional is a System Frame observable defined as a functional of conserved Internal Frame count structure over a relational region ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$, obtained via the Frames mapping $\mathit{M}$. It assigns a measurable separation to a configuration of count density and adjacency relations without presupposing metric structure in the Internal Frame. The realized configuration is given by $\mathrm{\Omega }=\mathit{M}({\mathit{\omega }}^{\mathrm{\text{'}}})\subset \mathcal{R}$. Formally, it is described as ${\mathcal{L}}_{\mathit{R}}\left[\mathrm{\Omega }\right]=$ $\mathcal{F}\left({\mathit{\rho }}_{\mathit{I}}(\mathit{\xi },\mathit{A}),{\mathit{J}}_{\mathit{I}}(\mathit{\xi }\to {\mathit{\xi }}^{\mathrm{\text{'}}},\mathit{A}),\partial {\mathit{\omega }}^{\mathrm{\text{'}}}\right)$, with realization ${\mathcal{L}}_{\mathit{R}}\left[\mathrm{\Omega }\right]\equiv \mathcal{N}[{\mathit{\omega }}^{\mathrm{\text{'}}}]\cdot \mathcal{C}\left({\mathit{N}}_{\mathit{I}}({\mathit{\omega }}^{\mathrm{\text{'}}},\mathit{A})\right)$ where $\mathcal{N}$ is the normalization induced by mapping and $\mathcal{C}$ is the count functional consistent with conservation laws.

• $\mathcal{R}$

  The System Frame realization structure $\mathcal{R}$ is the relational image induced by the Frames mapping $\mathit{M}:\mathit{\omega }\to \mathcal{R}$, where $\mathit{\omega }$ is the Internal Frame relational domain. The structure $\mathcal{R}$ comprises all admissible realized relational configurations consistent with the Frames mapping and the Frames Principle. It is not a pre-existing geometric manifold and does not constitute primitive ontology. Rather, it is the realized relational encoding through which observable System Frame quantities are represented. A realized physical system corresponds to a realized configuration $\mathit{\Omega }=\mathit{M}({\mathit{\omega }}^{\mathrm{\text{'}}})\subseteq \mathcal{R},$ where ${\mathit{\omega }}^{\mathrm{\text{'}}}\subseteq \mathit{\omega }$ is an admissible Internal Frame relational region. Realized coordinates, observables, realization manifolds, and encoded probability measures arise only within $\mathcal{R}$.

• $\mathfrak{R}$

  is a realization manifold. $\mathfrak{R}\subseteq \mathcal{R}$ is the set of admissible System Frame realizations compatible with a specified Internal Frame configuration and any applicable realization constraints. Realization manifolds are induced by the Frames mapping and need not be unique. They may arise from constraints imposed by count separation, conservation laws, boundary conditions, contextual restrictions, or other admissible realization conditions. Probability measures are assigned only after normalization over the admissible realization manifold.

• $\mathrm{𝑀}$

  is the Frames mapping is a relational mapping $\mathit{M}:\mathit{\omega }\to \mathcal{R}$ that assigns Internal Frame count configurations in the relational domain $\mathit{\omega }$ to elements of the uniquely realizable relational structure $\mathcal{R}$ defining the emergent System Frame. A valid realization exists only on referenceable subdomains ${\mathit{\omega }}^{\mathrm{\text{'}}}\subseteq \mathit{\omega }$ that satisfy normalization, repeatable realization, and holonomy-free closure on all contractible reference cycles. For any admissible subdomain ${\mathit{\omega }}^{\mathrm{\text{'}}}$, the corresponding realized configuration is $\mathit{\Omega }=\mathit{M}({\mathit{\omega }}^{\mathrm{\text{'}}})\subseteq \mathcal{R}$. The mapping establishes consistent reference relations from which System Frame coordinates, observables, and realization structures are derived. Observable quantities arise only after application of the Frames mapping under the Frames Principle.

• ${\mathcal{R}}_{\mathit{\xi }}\in \mathcal{R}$

  is a System Frame realization element ${\mathcal{R}}_{\mathit{\xi }}$ is an individual realized relational configuration associated with an admissible Internal Frame configuration $\mathit{\xi }$ under the Frames mapping $\mathit{M}$. ${\mathcal{R}}_{\mathit{\xi }}\in \mathcal{R}$. The collection $\left\{{\mathcal{R}}_{\mathit{\xi }}\right\}$ forms the realization structure $\mathcal{R}$. Realization elements encode relational information only and do not represent independent physical entities. Observable quantities arise from relations among realization elements within admissible realized configurations.

• ${\mathrm{𝑅}}_{\mathrm{𝑖}}$

  is the observer-specific macroscopic record state space associated with observer $\mathit{i}$. It consists of stable, observer-accessible realized states encoding measurement records and contextual outcomes. $\mathit{r}\in {\mathit{R}}_{\mathit{i}}$. Elements of ${\mathit{R}}_{\mathit{i}}$ correspond to persistent realized record states that remain invariant under repeated observation and admissible evolution. Record states arise only after contextual restriction and normalization within the System Frame and therefore belong to the empirical layer of description rather than the primitive Internal Frame count structure.

• $\mathrm{𝐸}$

  is the admissible interaction set associated with a referenceable universe domain $\mathit{U}$. Elements $\mathit{x}\in \mathit{E}$ denote admissible interaction configurations supporting realization under the Frames mapping. The set $\mathit{E}$ is defined relationally within the Internal Frame and carries no continuum coordinate, metric, or geometric structure. Realization mappings assign outcome labels through $\mathit{R}:\mathit{E}\to \mathit{O}\mathrm{.}$ The set $\mathit{E}$ specifies the admissible interaction structure on which realization and record encodings are defined.

• $\mathrm{𝑂}$

  is the outcome label space associated with a referenceable Frames mapping. Elements $\mathit{o}\in \mathit{O}$ are abstract realization labels assigned by the realization mapping $\mathit{R}:\mathit{E}\to \mathit{O}$ to admissible interactions $\mathit{x}\in \mathit{E}$. The set $\mathit{O}$ is not a geometric or physical state space and carries no metric, coordinate, or dynamical structure. It indexes realizable outcomes within the System Frame encoding and serves as the domain of record encoding mappings $\mathit{\rho }:\mathit{O}\to \mathcal{R}\mathrm{.}$ The realized outcome subset is ${\mathit{O}}_{\mathit{R}}:=\mathit{R}\left(\mathit{E}\right)\subseteq \mathit{O},$ consisting of outcomes actually realized on a given admissible interaction structure.

• $\mathrm{𝜏}$

  is a record relabeling bijection induced by an admissible outcome relabeling under a Frames automorphism. The map $\mathit{\tau }:{\mathcal{R}}_{\mathit{a}}\to {\mathcal{R}}_{\mathit{b}}$ acts on realized macroscopic record states and preserves observable equivalence under relabeling of realized outcomes. For an outcome relabeling $\mathit{\sigma }:{\mathit{O}}_{\mathit{R}}\to {\mathit{O}}_{\mathit{R}}$, the induced record relabeling satisfies $\mathit{\tau }({\mathit{\rho }}_{\mathit{a}}\left(\mathit{o}\right))={\mathit{\rho }}_{\mathit{b}}\left(\mathit{\sigma }\right(\mathit{o}\left)\right),\mathit{o}\in {\mathit{O}}_{\mathit{R}}\mathrm{.}$ The map $\mathit{\tau }$ preserves stable record distinguishability and introduces no change in geometric, closure, or measurable structure.

• ${\mathrm{𝜌}}_{\mathrm{𝑖}}\mathrm{:}\mathrm{𝑂}\mathrm{\to }{\mathrm{𝑅}}_{\mathrm{𝑖}}$

  Record encoding mapping from outcome space $\mathit{O}$ to macroscopic record states ${\mathit{R}}_{\mathit{i}}$. The mapping is injective on the realized outcome set ${\mathit{O}}_{\mathit{R}}$, ensuring that distinct realized outcomes correspond to distinct observable records.

• $\mathcal{H}\left(\mathit{\gamma }\right)$

  A holonomy functional is the closure mismatch obtained by transporting reference data around a closed reference cycle $\mathit{\gamma }$ under the Frames mapping. For holonomy–free mappings, $\mathcal{H}\left(\mathit{\gamma }\right)=0$ on all contractible cycles.

• $\mathrm{𝐷}$

  A mapping defect class is a global obstruction in the Frames mapping preventing extension of a holonomy–free mapping across the full configuration domain. Persistent defects anchor universe initiation and determine the holonomy class of the resulting reference structure.

• $\mathrm{𝛷}$

  A frames automorphism is an invertible transformation $\mathit{\Phi }:\mathit{U}\to \mathit{U}$ acting on a referenceable universe domain $\mathit{U}\subset \mathit{\omega }$ that preserves the structural constraints of the Frames mapping $\mathit{M}$, including normalization, repeatable realization, and holonomy–free closure on contractible cycles. Two universe realizations related by Φ are physically equivalent descriptions of the same System Frame structure.

• ${\mathcal{C}}_{\mathit{s}}$

  is the set of admissible Internal Frame configurations that correspond to a single System Frame state $\mathit{s}$ under a valid Frames mapping. It represents the pre-image of $\mathit{s}$ and defines the domain over which encoded availability weights are assigned and normalized.

• $\mathrm{𝒫}\mathrm{(R)}$

  is the dimensionless gravitational realization fraction at galactocentric radius R. It specifies the fraction of baryonic gravitational availability realized through the Frames mapping in the System Frame and is defined by $\mathcal{P}=1-\exp \mathit{}\left[-\sqrt{{\mathit{g}}_{\mathrm{bar}}/{\mathit{a}}_{\mathrm{MQ}}}\right]$. For finite positive baryonic acceleration, $0<\mathcal{P}<1$, with $\mathcal{P}\to 1$ in the high-acceleration limit. The realized acceleration satisfies ${\mathit{g}}_{\mathrm{real}}=\frac{{\mathit{g}}_{\mathrm{bar}}}{\mathcal{P}}$.

• ${\mathrm{𝜈}}_{\mathrm{MQ}}\left(\mathrm{𝑦}\right)$

  is the MQ realization multiplier, defined by ${\mathit{\nu }}_{\mathrm{MQ}}\left(\mathit{y}\right)={\left[1-\exp \left(-\sqrt{\mathit{y}}\right)\right]}^{-1}$. It is the inverse of the realization fraction when $\mathit{y}=\frac{{\mathit{g}}_{\mathrm{bar}}}{{\mathit{a}}_{\mathrm{MQ}}}$ and maps the baryonic gravitational field to the realized gravitational response. The function is fixed by the MQ realization law rather than introduced as an empirical interpolation function.

• ${\mathrm{𝑄}}_{\mathrm{MQ}}\left(\mathrm{𝑧}\right)$

  is the scalar primitive of the MQ realization multiplier used in the weak-field variational formulation. It is defined by $\frac{\mathit{d}{\mathit{Q}}_{\mathrm{MQ}}}{\mathit{dz}}={\mathit{\nu }}_{\mathrm{MQ}}\left(\sqrt{\mathit{z}}\right)$, with normalization fixed by the lower integration bound. The function is derived from the established MQ realization law and is not an independent phenomenological function.

• ${\mathrm{𝑌}}_{\mathrm{MQ}}$

  is the dimensionless covariant spatial realization invariant formed from the norm of the baryonic potential gradient projected orthogonally to the timelike flow and normalized by ${\mathit{a}}_{\mathrm{MQ}}$. It provides the covariant scalar argument of ${\mathit{\nu }}_{\mathrm{MQ}}$ in the spatial realization action.

• ${\mathcal{Y}}_{\mathrm{MQ}}^{\mathit{\mu }}$

  is the auxiliary MQ dual realization vector introduced in the exact first-order representation of the spatial weak-field action. It is determined algebraically from ${\mathit{Y}}_{\mathrm{MQ}}$, ${\mathit{\nu }}_{\mathrm{MQ}}$, the spatial projector, and the gradient of ${\mathit{\Phi }}_{\mathit{b}}$. It introduces no independent physical degree of freedom and may be eliminated to recover the original realization action exactly.

• ${\mathrm{𝑊}}_{\mathrm{MQ}}$

  is the dimensionless invariant magnitude of the MQ dual realization vector ${\mathcal{Y}}_{\mathrm{MQ}}^{\mathit{\mu }}$. It satisfies ${\mathit{W}}_{\mathrm{MQ}}={\mathit{Y}}_{\mathrm{MQ}}{\mathit{\nu }}_{\mathrm{MQ}}\left({\mathit{Y}}_{\mathrm{MQ}}\right)$ and provides the natural scalar variable of the dual weak-field representation.

• ${\mathcal{R}}_{\mathrm{MQ}}\left(\mathit{W}\right)$

  is the exact dual realization function associated with ${\mathit{Q}}_{\mathrm{MQ}}\left({\mathit{Y}}^2\right)$. It is defined through the Legendre transformation relating W and Y and is constructed such that eliminating the auxiliary dual field returns the original spatial realization action without changing the MQ gravitational realization law.

• ${\mathrm{𝑋}}_{\mathrm{MQ}}$

  is the dimensionless timelike mass-shell invariant formed from the inverse System Frame metric and the gradient of the action phase S, normalized by m²c². On the physical timelike mass-shell branch, ${\mathit{X}}_{\mathrm{MQ}}=1$.

• ${\mathrm{𝐾}}_{\mathrm{MQ}}\left({\mathrm{𝑋}}_{\mathrm{MQ}}\right)$

  is the timelike constraint contribution to the covariant MQ realization Lagrangian. It enforces the mass-shell condition ${\mathit{X}}_{\mathrm{MQ}}=1$ through the associated source-density multiplier and generates the corresponding timelike stress-energy contribution.

• ${\mathrm{𝐽}}_{\mathrm{MQ}}\left({\mathrm{𝑌}}_{\mathrm{MQ}}\right)$

  is the spatial realization contribution to the covariant MQ realization Lagrangian. It contains the bounded-realization dependence on the covariant spatial invariant ${\mathit{Y}}_{\mathrm{MQ}}$ and complements ${\mathit{K}}_{\mathrm{MQ}}$ in the decomposition of the full realization Lagrangian.

• ${\mathrm{𝐿}}_{\mathrm{MQ}}$

  is the covariant MQ realization Lagrangian constructed from the timelike and spatial contributions according to ${\mathit{L}}_{\mathrm{MQ}}={\mathit{K}}_{\mathrm{MQ}}\left({\mathit{X}}_{\mathrm{MQ}}\right)+{\mathit{J}}_{\mathrm{MQ}}\left({\mathit{Y}}_{\mathrm{MQ}}\right)$. It embeds the stationary bounded-realization law within the covariant System Frame action.

• ${\mathrm{𝑀}}_{\mathrm{𝑐}}$

  is the completed contextual Frames mapping for a fixed measurement context c. It maps each complete admissible interaction configuration in E to one realized point in the System Frame outcome domain ${\mathrm{\Omega }}_{\mathrm{out}}$. Referenceability requires ${\mathit{M}}_{\mathit{c}}$ to be single-valued for a completed interaction; the realized outcome label is assigned only after composition with the outcome-partition label map κ.

• $\mathrm{𝜅}$

  is the System Frame outcome-label map defined on ${\mathrm{\Omega }}_{\mathrm{out}}$. It assigns $\mathit{\kappa }\left(\mathit{y}\right)={\mathit{o}}_{\mathit{k}}$ exactly when the realized configuration y lies in outcome region ${\mathit{A}}_{\mathit{k}}$. For a completed interaction, the realized-outcome map is $\mathit{R}=\mathit{\kappa }\mathrm{\circ }{\mathit{M}}_{\mathit{c}}$.

• ${\mathrm{𝜇}}_{\mathrm{𝐸}}$

  is the finite pre-probabilistic complete-interaction count measure on the admissible interaction set E. It is inherited from conserved Internal Frame integer availability through the deterministic completion construction and is not an independently selectable probability weighting.

• ${\mathrm{𝑃}}_{\mathrm{𝐸}}$

  is the normalized complete-interaction probability measure obtained from the count measure ${\mathit{\mu }}_{\mathit{E}}$ by ${\mathit{P}}_{\mathit{E}}\left(\mathit{D}\right)=\frac{{\mathit{\mu }}_{\mathit{E}}\left(\mathit{D}\right)}{{\mathit{\mu }}_{\mathit{E}}\left(\mathit{E}\right)}$ for measurable $\mathit{D}\subseteq \mathit{E}$. Its pushforward through the realized-outcome map gives the empirical distribution of completed record labels.

• ${\mathrm{𝜇}}_{\mathrm{𝑆}}$

  is the realized System Frame encoded-availability measure induced as the pushforward of ${\mathit{\mu }}_{\mathit{E}}$ through the completed contextual realization ${\mathit{M}}_{\mathit{c}}$. For measurable $\mathit{B}\subseteq {\mathrm{\Omega }}_{\mathrm{out}}$, ${\mathit{\mu }}_{\mathit{S}}\left(\mathit{B}\right)={\mathit{\mu }}_{\mathit{E}}\mathit{}\left({\mathit{M}}_{\mathit{c}}^{-1}\left(\mathit{B}\right)\right)$.

• ${\mathrm{𝑂}}_{\mathrm{𝑅}}$

  is the realized outcome subset of the outcome-label space O. It contains the labels actually realized by complete admissible interactions and is the domain on which the record-encoding map ${\mathit{\rho }}_{\mathit{i}}$ is required to be injective.

• ${\mathrm{𝑜}}_{\mathrm{𝑘}}$

  is the realized outcome label associated with outcome region ${\mathit{A}}_{\mathit{k}}$. For a completed interaction $\mathit{x}\in \mathit{E}$, ${\mathit{o}}_{\mathit{k}}$ is fixed by the unique partition membership of ${\mathit{M}}_{\mathit{c}}$(x) and is subsequently encoded as a stable macroscopic record by ${\mathit{\rho }}_{\mathit{i}}$.

• ${\mathrm{𝑃}}_{\mathrm{MQ}}\left(\mathrm{𝛼}\mathrm{,}\mathrm{𝛽}\mathrm{\mid }\mathrm{𝑎}\mathrm{,}\mathrm{𝑏}\right)$

  is the MQ joint probability for dichotomic outcomes α and β conditioned on analyzer settings a and b. It is obtained from normalized composite System Frame availability and is used to evaluate the Bell correlator and CHSH combination; it is not assumed to factorize into Bell-local response probabilities.

Emergent System Frame Continuum Structures

• $\mathrm{\Omega }$

  It denotes the System Frame image of a relational region ${\mathit{\omega }}^{\mathrm{\text{'}}}\subset \mathit{\omega }$ under the Frames mapping $\mathit{M}$, such that $\mathit{\Omega }=\mathit{M}({\mathit{\omega }}^{\mathrm{\text{'}}})$. The set $\mathrm{\Omega }$ consists of encoded relational configurations corresponding to physically realized systems and carries the continuous relational structure in which observables are defined. It is not an Internal Frame domain and does not support count–based conservation laws directly.

• $\mathcal{A}$

  System Frame log–gradient availability potential. Vector field defined on the System Frame continuum encoding by $\mathcal{A}\equiv \nabla \ln \mathit{}{\mathit{\rho }}_{\mathit{R}},$ where ${\mathit{\rho }}_{\mathit{R}}$ is the System Frame encoded availability density. This field measures the fractional spatial variation of encoded availability and represents the natural connection structure associated with multiplicative rescaling of ${\mathit{\rho }}_{\mathit{R}}$. It emerges from the Frames mapping of the Internal Frame count measure and has no counterpart in the Internal Frame. Under a multiplicative rescaling of encoded availability, ${\mathit{\rho }}_{\mathit{R}}\to {\mathit{\rho }}_{\mathit{R}}^{\mathrm{\text{'}}}={\mathit{e}}^{\mathit{\chi }}{\mathit{\rho }}_{\mathit{R}}$, the field transforms as $\mathcal{A}\to {\mathcal{A}}^{\mathrm{\text{'}}}=\mathcal{A}+\nabla \mathit{\chi }$, establishing its role as an emergent Abelian gauge potential in the System Frame representation.

• $\widehat{\mathcal{A}}\left(\mathit{y}\right)$

  is the normalized realized availability field after application of the mapping normalization functional.

• ${\mathrm{𝑛}}_{\mathrm{𝑘}}\mathrm{(}\mathrm{𝑡}\mathrm{)}$

  encoded availability in partition ${\mathit{A}}_{\mathit{k}}$ ${\mathit{n}}_{\mathit{k}}\left(\mathit{t})={\int }_{{\mathit{A}}_{\mathit{k}}}{\mathit{\rho }}_{\mathit{S}}\right(\mathit{r},\mathit{t}){\mathit{d}}^3\mathit{r}$. Represents the pre–normalized encoded availability in region ${\mathit{A}}_{\mathit{k}}$. Used to construct conditional probabilities via normalization: $\mathit{P}\left(\mathit{k})={\mathit{n}}_{\mathit{k}}\right(\mathit{t})/{\mathit{N}}_1\left(\mathit{t}\right)$.

• ${\mathit{\rho }}_{\mathit{S}}(\mathit{r},\mathit{t})$

  System Frame encoded availability density ${\mathit{\rho }}_{\mathit{S}}(\mathit{r},\mathit{t})=\frac{\mathit{d}{\mathit{N}}_{\mathit{S}}}{{\mathit{d}}^3\mathit{r}}=\underset{\mathrm{\Delta }\mathit{V}\to 0}{\lim \mathit{}}\frac{{\mathit{n}}_{\mathit{S}}(\mathrm{\Delta }\mathit{V},\mathit{t})}{\mathrm{\Delta }\mathit{V}}$. Continuum density obtained from the deterministic Frames mapping applied to the Internal Frame count measure. It is the Radon–Nikodym derivative of the induced System Frame count measure with respect to Lebesgue volume. Non–negative and integrable over finite regions. The total encoded availability is ${\mathit{N}}_1\left(\mathit{t})=\int {\mathit{\rho }}_{\mathit{S}}\right(\mathit{r},\mathit{t}){\mathit{d}}^3\mathit{r}$. Not normalized in general. Represents encoded measurement availability prior to probabilistic interpretation. Serves as the fundamental System Frame scalar from which ${\mathit{\rho }}_{\mathit{p}}$ and all observable statistics are derived.

• ${\mathit{n}}_{\mathit{S}}(\mathit{r},\mathit{t})$

  discretized System Frame availability count field. Discrete encoded availability count associated with a System Frame cell (index $\mathit{j}$) or region $\mathit{A}$ at time $\mathit{t}$, obtained via the Frames mapping from Internal Frame availability. Integer–valued when discretized. Used for counts and partitions (e.g., ${\mathit{n}}_{\mathit{S},\mathit{j}}\left(\mathit{t}\right)$ or ${\mathit{n}}_{\mathit{S}}(\mathit{A},\mathit{t})$), not as the argument of continuum differential operators.

• $\mathit{g}(\mathit{r},\mathit{t})$

  System Frame log–gradient availability field (dimension $1/\mathrm{length}$), defined by $\mathit{g}\left(\mathit{r},\mathit{t})=\nabla \ln \mathit{}{\mathit{\rho }}_{\mathit{S}}\right(\mathit{r},\mathit{t})=\nabla {\mathit{\rho }}_{\mathit{S}}\left(\mathit{r},\mathit{t})/{\mathit{\rho }}_{\mathit{S}}\right(\mathit{r},\mathit{t})$; measures fractional spatial variation of encoded availability density.

• $\mathit{r}$

  is the System Frame spatial coordinate vector $\mathit{r}\in {\mathrm{ℝ}}^3$ of the realized System Frame continuum. Used to parameterize encoded availability densities, probability densities, currents, velocity fields, wavefunctions, and other nonrelativistic observables arising from the Frames mapping. Appears in expressions such as ${\mathit{\rho }}_{\mathit{S}}(\mathit{r},\mathit{t})$, $\mathit{P}(\mathit{r},\mathit{t})$, $\mathit{J}(\mathit{r},\mathit{t})$, $\mathit{v}(\mathit{r},\mathit{t})$, $\mathit{Q}(\mathit{r},\mathit{t})$, and $\mathit{\psi }(\mathit{r},\mathit{t})$. Represents position within the realized three-dimensional spatial domain $\mathrm{\Omega }$. Distinct from the spacetime coordinate ${\mathit{x}}^{\mathit{\mu }}$ used in relativistic and geometric formulations.

• ${\mathrm{𝑉}}^{\mathrm{𝜇}}\mathrm{(}\mathrm{𝑥}\mathrm{)}$

  is a smooth contravariant vector field defined on the emergent System Frame continuum. It is used to represent tangent vectors, transport directions, currents, variational displacements, and other geometric quantities arising after application of the Frames mapping. The field carries standard continuum geometric meaning with respect to the emergent metric ${\mathit{g}}_{\mathit{\mu \nu }}^{\mathrm{sys}}$ and is acted upon by covariant derivatives, connections, and curvature operators. Unlike the Internal Frame relational cell label $\mathit{\xi }$, the quantity ${\mathit{V}}^{\mathit{\mu }}$ is not a count index and possesses no direct interpretation within the discrete count substrate. It exists only in the realized continuum description induced by the Frames mapping. Generic continuum identities such as ${\nabla }_{\mathit{\mu }}{\mathit{V}}^{\mathit{\mu }}=\frac{1}{\sqrt{-{\mathit{g}}_{\mathrm{sys}}}}{\partial }_{\mathit{\mu }}\left(\sqrt{-{\mathit{g}}_{\mathrm{sys}}}{\mathit{V}}^{\mathit{\mu }}\right)$ are therefore statements about emergent System Frame geometry rather than primitive Internal Frame count structure.

• ${\mathrm{𝑥}}^{\mathrm{𝜇}}$

  is the System Frame spacetime coordinate ${\mathit{x}}^{\mathit{\mu }}=({\mathit{x}}^0,{\mathit{x}}^1,{\mathit{x}}^2,{\mathit{x}}^3)$. Four-dimensional coordinate on the emergent System Frame spacetime manifold. Used in continuum geometric constructions including the metric tensor ${\mathit{g}}_{\mathit{\mu \nu }}^{\mathrm{sys}}$, connection coefficients ${\mathrm{\Gamma }}_{\mathit{\mu \nu }}^{\mathit{\rho }}$, curvature tensors, geodesics, action functionals, and covariant field equations. Coordinates are defined on the realized manifold induced by the Frames mapping. The spatial components of ${\mathit{x}}^{\mathit{\mu }}$ reduce locally to the realized spatial coordinates represented by $\mathit{r}$, but ${\mathit{x}}^{\mathit{\mu }}$ is reserved for relativistic and differential geometric formulations.

• $\mathrm{𝑡}$

  System Frame time coordinate used in the continuum encoding and Schrödinger evolution.

• ${\mathrm{𝑟}}_{\mathrm{𝐴}}\mathrm{,}{\mathrm{𝑟}}_{\mathrm{𝐵}}$

  System Frame spatial coordinate vectors for subsystems $\mathit{A}$ and $\mathit{B}$ in a bipartite (entanglement/Bell) configuration.

• ${\mathrm{𝛷}}_{\mathrm{sys}}$

  is the gravitational potential realized in the System Frame in the stationary weak-field limit. Its spatial gradient determines the corresponding realized gravitational response. It is the weak-field potential obtained from the realized System Frame metric rather than an independently introduced gravitational field.

• ${\mathrm{𝑀}}_{\mathrm{sys}}\left(\mathrm{<}\mathrm{𝑅}\right)$

  is the enclosed gravitational source associated with the realized System Frame weak-field response within galactocentric radius R. It is derived from the mapped baryonic gravitational availability and does not represent an independently postulated dark-matter mass component.

• ${\mathrm{𝜌}}_{\mathrm{src}}$

  is the scalar mass-density multiplier associated with the timelike source branch of the covariant realization action. It enforces the mass-shell condition through ${\mathit{K}}_{\mathrm{MQ}}$, enters the conserved current ${\mathit{\rho }}_{\mathrm{src}}{\mathit{u}}^{\mathit{\mu }}$, and generates the corresponding on-shell stress-energy tensor. It is distinct from the canonical MQ geometric density ${\mathit{\rho }}_{\mathrm{sys}}$.

• ${\mathrm{ℎ}}^{\mathrm{𝜇}\mathrm{𝜈}}$

  is the inverse spatial projection tensor associated with the timelike System Frame four-velocity ${\mathit{u}}^{\mathit{\mu }}$, defined by ${\mathit{h}}^{\mathit{\mu \nu }}={\mathit{g}}_{\mathrm{sys}}^{\mathit{\mu \nu }}+\frac{{\mathit{u}}^{\mathit{\mu }}{\mathit{u}}^{\mathit{\nu }}}{{\mathit{c}}^2}$. It projects covectors onto the spatial hypersurface orthogonal to ${\mathit{u}}^{\mathit{\mu }}$ and satisfies ${\mathit{h}}^{\mathit{\mu \nu }}{\mathit{u}}_{\mathit{\nu }}=0$.

• ${\mathrm{𝑢}}^{\mathrm{𝜇}}$

  is the normalized timelike four-velocity associated with the System Frame source branch. It satisfies ${\mathit{u}}^{\mathit{\mu }}{\mathit{u}}_{\mathit{\mu }}=-{\mathit{c}}^2$ and, in the MQ mass-shell construction, is generated from the action phase according to ${\mathit{u}}_{\mathit{\mu }}=\frac{{\nabla }_{\mathit{\mu }}\mathit{S}}{\mathit{m}}$.

Realized separation and metric structure

• ${\mathrm{𝑁}}_{\mathrm{1}}\mathrm{(}\mathrm{𝑡}\mathrm{)}$

  total encoded availability count in the single–subsystem System Frame normalization, defined by the inline equation ${\mathit{N}}_1\left(\mathit{t})={\int }_{{\mathrm{ℝ}}^3}{\mathit{\rho }}_{\mathit{S}}\right(\mathit{r},\mathit{t}){\mathit{d}}^3\mathit{r}$.

• ${\mathrm{𝑁}}_{\mathrm{2}}\mathrm{(}\mathrm{𝑡}\mathrm{)}$

  total encoded availability count in the bipartite (two–subsystem) System Frame normalization, defined by the inline equation ${\mathit{N}}_2(\mathit{t})={\int }_{{\mathrm{ℝ}}^3}{\int }_{{\mathrm{ℝ}}^3}{\mathit{\rho }}_{\mathit{S}}({\mathit{r}}_{\mathit{A}},{\mathit{r}}_{\mathit{B}},\mathit{t}){\mathit{d}}^3{\mathit{r}}_{\mathit{A}}{\mathit{d}}^3{\mathit{r}}_{\mathit{B}}$.

• ${\mathit{\rho }}_{\mathit{P}}\left(\mathit{r},\mathit{t}\right)$

  normalized System Frame probability density ${\mathit{\rho }}_{\mathit{P}}(\mathit{r},\mathit{t})=\frac{{\mathit{\rho }}_{\mathit{S}}(\mathit{r},\mathit{t})}{{\mathit{N}}_{\mathit{S}}\left(\mathit{t}\right)},{\int }_{\mathrm{\Omega }}{\mathit{\rho }}_{\mathit{P}}(\mathit{r},\mathit{t}){\mathit{d}}^3\mathit{r}=1$. Dimensionless normalized probability density derived from the realized encoded availability density ${\mathit{\rho }}_{\mathit{S}}(\mathit{r},\mathit{t})$. Defines realized outcome probabilities and detection statistics within the System Frame. The MQ wave function satisfies ${\mathit{\psi }}_{\mathit{MQ}}=\sqrt{{\mathit{\rho }}_{\mathit{P}}}{\mathit{e}}^{\mathit{i\varphi }},\mathrm{\mid }{\mathit{\Psi }}_{\mathit{MQ}}{\mathrm{\mid }}^2={\mathit{\rho }}_{\mathit{P}}$. Probability in MQ arises exclusively from normalization of realized encoded availability ${\mathit{\rho }}_{\mathit{S}}$, not from primitive Internal Frame randomness. All probabilistic and field-theoretic constructions in the System Frame are defined with respect to ${\mathit{\rho }}_{\mathit{P}}$.

• ${\mathit{\rho }}_{\mathit{P}1}(\mathit{r},\mathit{t}),{\mathit{\rho }}_{\mathit{P}2}(\mathit{r},\mathit{t})$

  marginal probability densities for subsystems 1 and 2 obtained by integrating the joint density over the other subsystem’s coordinates.

• ${\mathcal{O}}_{\mathrm{out}}$

  The System Frame output domain ${\mathit{O}}_{\mathrm{out}}\subset \mathcal{R}$ is the realized relational region associated with measurement readout, whose measurable partition $\left\{{\mathit{A}}_{\mathit{k}}\right\}$ defines distinct macroscopic record classes. These subsets satisfy ${\bigcup }_{\mathit{k}}{\mathit{A}}_{\mathit{k}}={\mathit{O}}_{\mathrm{out}}$ and ${\mathit{A}}_{\mathit{i}}\cap {\mathit{A}}_{\mathit{j}}=\varnothing$ for $\mathit{i}\ne \mathit{j}$. The domain ${\mathit{O}}_{\mathrm{out}}$ exists only in the System Frame as a partitioned subset of the realized relational structure and has no counterpart in the Internal Frame. Probabilities associated with outcomes are obtained by normalization of encoded availability over the partition elements ${\mathit{A}}_{\mathit{k}}$, and distinct subsets correspond, via the record encoding mapping, to distinct macroscopic record states.

• ${\mathrm{𝐴}}_{\mathrm{𝑘}}$

  is subset ${\mathit{A}}_{\mathit{k}}\subset {\mathit{O}}_{\mathrm{out}}\subset \mathcal{R}$ is a measurable region of the System Frame output domain corresponding to a distinct macroscopic record class or measurement outcome channel. The collection $\left\{{\mathit{A}}_{\mathit{k}}\right\}$ forms a mutually exclusive and exhaustive partition satisfying ${\mathit{A}}_{\mathit{i}}\cap {\mathit{A}}_{\mathit{j}}=\varnothing$ for $\mathit{i}\ne \mathit{j}$ and ${\bigcup }_{\mathit{k}}{\mathit{A}}_{\mathit{k}}={\mathit{O}}_{\mathrm{out}}$. Probabilities are obtained by normalization of encoded availability over these subsets, with outcome weights given by ${\mathit{n}}_{\mathit{k}}(\mathit{t})={\int }_{{\mathit{A}}_{\mathit{k}}}\mathit{\rho }(\mathbf{r},\mathit{t}){\mathit{d}}^3\mathit{r}$, where $\mathit{\rho }$ is the System Frame encoded availability density. The subsets ${\mathit{A}}_{\mathit{k}}$ exist only within the realized System Frame encoding and have no counterpart in the Internal Frame. Distinct subsets correspond, via the record encoding mapping, to distinct macroscopic record states. Indicator functions associated with ${\mathit{A}}_{\mathit{k}}$ define restriction and conditioning operations within the encoded Hilbert space representation.

• ${\mathit{\chi }}_{{\mathit{A}}_{\mathit{k}}}\left(\mathit{r}\right)$

  is associated with each subset ${\mathit{A}}_{\mathit{k}}$ is the indicator function ${\mathit{\chi }}_{{\mathit{A}}_{\mathit{k}}}\left(\mathit{r}\right)$, defined by ${\mathit{\chi }}_{{\mathit{A}}_{\mathit{k}}}(\mathit{r})=1$ for $\mathit{r}\in {\mathit{A}}_{\mathit{k}}$ and $0$ otherwise, which implements restriction and conditioning of encoded availability and wave function support to the corresponding measurement region.

• $\mathit{\psi }(\mathit{r},\mathit{t})$

  is the encoded System Frame wave function in the position representation; $\mathit{\psi }\in {\mathit{L}}^2\left({\mathrm{ℝ}}^3\right)$. The wave function provides the position space representation of the encoded System Frame state $\mathrm{\mid }\mathit{\Psi }\mathit{⟩}$ and may be expressed in Madelung form as $\mathit{\psi }(\mathit{r},\mathit{t})=\sqrt{{\mathit{\rho }}_{\mathit{P}}(\mathit{r},\mathit{t})}{\mathit{e}}^{\mathit{i\varphi }(\mathit{r},\mathit{t})}=\sqrt{{\mathit{\rho }}_{\mathit{P}}(\mathit{r},\mathit{t})}{\mathit{e}}^{\mathit{iS}(\mathit{r},\mathit{t})/\mathrm{\hslash }}\mathrm{.}$

• $\mathrm{\mid }\mathrm{𝛹}\mathrm{⟩}$

  is the encoded System Frame state vector in Hilbert space; used for basis expansions, projection updates, and expectation values.

• ${\mathrm{𝑃}}_{\mathrm{𝑘}}\left(\mathrm{𝑡}\right)$

  is the normalized System Frame outcome weight, and after the realization-frequency identification the empirical probability, associated with outcome region ${\mathit{A}}_{\mathit{k}}$ at time t. It is obtained by dividing the encoded availability in ${\mathit{A}}_{\mathit{k}}$ by the total encoded availability over ${\mathrm{\Omega }}_{\mathrm{out}}$.

• ${\mathit{\Pi }}_{\mathit{k}}$

  is the Hilbert-space projection operator associated with realized outcome subspace k. It satisfies ${\mathit{\Pi }}_{\mathit{k}}^2={\mathit{\Pi }}_{\mathit{k}}={\mathit{\Pi }}_{\mathit{k}}^{\mathrm{†}}$ and represents post-outcome conditioning of the already derived MQ state; it is not a primitive dynamical collapse operator.

• ${\mathit{\psi }}_{\mathrm{MQ}}\left(\mathbf{r},\mathit{t}\right)$

  is the MQ System Frame wave-function representation constructed from normalized encoded availability and the realized phase field. It is a derived amplitude-phase encoding and is not a primitive Internal Frame field.

• $\mathrm{|}{\mathrm{𝛹}}_{\mathrm{MQ}}\mathrm{⟩}$

  is the normalized MQ state vector in the derived Hilbert-space representation. It encodes the same normalized System Frame availability represented by ψ_MQ and is used for spectral, projection, composite-state, and operator constructions after Hilbert completion.

• ${\mathit{\rho }}_{\mathit{P}}\left({\mathit{r}}_{\mathit{A}},{\mathit{r}}_{\mathit{B}},\mathit{t}\right)$

  is the normalized joint System Frame probability density for a bipartite realization with subsystem coordinates ${\mathit{r}}_{\mathit{A}}$ and ${\mathit{r}}_{\mathit{B}}$. In entangled or otherwise nonseparable realizations it need not factorize into the product of its marginal densities ${\mathit{\rho }}_{\mathit{P}1}\left({\mathit{r}}_{\mathit{A}},\mathit{t}\right)$ and ${\mathit{\rho }}_{\mathit{P}2}\left({\mathit{r}}_{\mathit{B}},\mathit{t}\right)$.

Differential and operator structure

• $\mathrm{\nabla }$

  System Frame gradient operator acting on continuum–encoded fields.

• $\mathrm{\nabla }\mathrm{\cdot }$

  System Frame divergence operator acting on geometric vector fields.

• $\mathrm{𝐻}$

  System Frame Hamiltonian operator governing unitary Schrödinger evolution (in the encoded continuum representation), typically $\mathit{H}=-{\mathrm{\hslash }}^2/\left(2\mathit{m}\right){\nabla }^2+\mathit{V}(\mathit{r},\mathit{t})$.

• $\widehat{\mathrm{𝐻}}$

  System Frame Hamiltonian operator governing unitary Schrödinger evolution in the encoded continuum representation. For a non-relativistic encoded state, $\widehat{\mathit{H}}=-({\mathrm{\hslash }}^2/2\mathit{m}){\nabla }^2+\mathit{V}(\mathit{r},\mathit{t})$. The operator $\widehat{\mathit{H}}$ is the continuum realization of the admissible discrete redistribution generator ${\mathit{H}}_{\mathit{d}}$ obtained under the Frames mapping. It acts on encoded System Frame wave functions $\mathit{\Psi }(\mathit{r},\mathit{t})$ and generates Schrödinger-form evolution through $\mathit{i}\mathrm{\hslash }\frac{\partial \mathit{\Psi }}{\partial \mathit{t}}=\widehat{\mathit{H}}\mathit{\Psi }$. The operator represents redistribution dynamics of normalized encoded availability under closed evolution. Under the admissibility and boundary conditions, $\widehat{\mathit{H}}$ is Hermitian and generates unitary evolution of the encoded state.

• $\mathrm{𝑈}\mathrm{(}\mathrm{𝑡}\mathrm{)}$

  unitary time–evolution operator in the System Frame Hilbert representation: $\mathit{U}\left(\mathit{t}\right)={\mathit{e}}^{-\mathit{iHt}/\mathrm{\hslash }}$.

• $\mathit{V}(\mathit{r},\mathit{t})$

  potential energy function in the System Frame Schrödinger equation; may include external fields and interactions as encoded in the continuum representation.

• $\mathit{\varphi }(\mathit{r},\mathit{t})$

  is the dimensionless System Frame phase field of the encoded wave function $\mathit{\psi }(\mathit{r},\mathit{t})$. The phase determines interference structure and generates momentum and velocity fields through its spatial and temporal gradients.

• $\mathit{S}(\mathit{r},\mathit{t})$

  is the phase action field of the encoded wave function. $\mathit{S}(\mathit{r},\mathit{t})$ carries units of action and is related to the dimensionless phase field by $\mathit{S}(\mathit{r},\mathit{t})=\mathrm{\hslash }\mathit{\varphi }(\mathit{r},\mathit{t})$.

• $\mathit{v}(\mathit{r},\mathit{t})$

  is the System Frame velocity field in Madelung form, $\mathit{v}(\mathit{r},\mathit{t})=(\mathit{\hslash }/\mathit{m})\nabla \mathit{\varphi }(\mathit{r},\mathit{t})=(1/\mathit{m})\nabla \mathit{S}(\mathit{r},\mathit{t})$.

• $\mathit{p}(\mathit{r},\mathit{t})$

  momentum field in the System Frame phase gradient representation, $\mathit{p}(\mathit{r},\mathit{t})=\mathrm{\hslash }\nabla \mathit{\varphi }(\mathit{r},\mathit{t})=\nabla \mathit{S}(\mathit{r},\mathit{t})$, with $\mathit{p}=\mathit{m}\mathit{v}$ when a velocity field is defined.

• $\widehat{\mathit{p}}$

  momentum operator in position representation: $\widehat{\mathit{p}}=-\mathit{i}\mathrm{\hslash }\nabla$.

• ${\mathit{J}}_{\mathit{\rho }}(\mathit{r},\mathit{t})$

  System Frame probability current density defined with respect to the normalized density ${\mathit{\rho }}_{\mathit{p}}$. It satisfies the continuity equation ${\partial }_{\mathit{t}}{\mathit{\rho }}_{\mathit{p}}+\nabla \cdot {\mathit{J}}_{\mathit{\rho }}=0$. In Madelung form, ${\mathit{J}}_{\mathit{\rho }}={\mathit{\rho }}_{\mathit{p}}\mathit{v},\mathit{v}=\frac{\mathrm{\hslash }}{\mathit{m}}\nabla \mathit{\varphi }\mathrm{.}$ Represents conserved probability flow in the System Frame. This current is not defined for ${\mathit{\rho }}_{\mathit{S}}$, ensuring consistency between conservation laws and normalized observables.

• ${\mathrm{\Delta }}_{\mathrm{𝑑}}$

  Discrete Laplacian (difference–operator Laplacian) defined on the lattice/adjacency structure of the Internal Frame. It is the discrete analog of the continuum operator ${\nabla }^2$, constructed from adjacency relations $\mathit{\xi }\sim {\mathit{\xi }}^{\mathrm{\text{'}}}$ and acting on count–defined fields. In the continuum limit, it approximates the standard Laplacian via ${\nabla }^2\approx \frac{{\mathrm{\Delta }}_{\mathit{d}}}{(\mathrm{\Delta }\mathit{x}{)}^2}$. This operator is used in the lattice/adjacency discretization of dynamical equations, including Schrödinger–type evolution, and encodes second–order relational variation across adjacent Internal Frame configurations.

• ${\mathrm{𝐻}}_{\mathrm{𝑑}}$

  discrete Hamiltonian operator used in lattice/discretized dynamics, typically ${\mathit{H}}_{\mathit{d}}=-{\mathrm{\hslash }}^2/(2\mathit{m}){\mathrm{\Delta }}_{\mathit{d}}/(\mathrm{\Delta }\mathit{x}{)}^2+\mathit{V}$, with ${\mathrm{\Delta }}_{\mathit{d}}$ the discrete Laplacian defined on the chosen adjacency/lattice and $\mathrm{\Delta }\mathit{x}$ the spatial step.

• $\mathit{Q}(\mathit{r},\mathit{t})$

  quantum potential (Madelung form) $\mathit{Q}(\mathit{r},\mathit{t})=-\frac{{\mathrm{\hslash }}^2}{2\mathit{m}}\frac{{\nabla }^2\sqrt{{\mathit{\rho }}_{\mathit{p}}(\mathit{r},\mathit{t})}}{\sqrt{{\mathit{\rho }}_{\mathit{p}}(\mathit{r},\mathit{t})}}$ Defined strictly in terms of the normalized density ${\mathit{\rho }}_{\mathit{p}}$. Encodes quantum corrections to the Hamilton–Jacobi–like equation arising from the encoded availability structure after normalization.

• ${\mathrm{𝑇}}_{\mathrm{𝑗}}\left(\mathrm{𝑎}\right)$

  is the strongly continuous unitary translation operator that realizes a displacement a along System Frame coordinate direction j. It acts on the coordinate representation by translation and its self-adjoint generator is the momentum operator ${\mathit{P}}_{\mathit{j}}$.

• ${\mathrm{𝑃}}_{\mathrm{𝑗}}$

  is the self-adjoint generator of the System Frame translation group ${\mathit{T}}_{\mathit{j}}\left(\mathit{a}\right)$. The MQ phase-action scale fixes its normalization, yielding the position-representation action ${\mathit{P}}_{\mathit{j}}=-\mathit{i}\mathrm{\hslash }{\partial }_{\mathit{j}}$ on its regular domain.

• ${\mathrm{𝑋}}_{\mathrm{𝑗}}$

  is the System Frame position operator for Cartesian coordinate ${\mathit{x}}_{\mathit{j}}$, acting by multiplication in the realized coordinate representation. Translation covariance gives ${\mathit{T}}_{\mathit{j}}{\left(\mathit{a}\right)}^{\mathrm{†}}{\mathit{X}}_{\mathit{k}}{\mathit{T}}_{\mathit{j}}\left(\mathit{a}\right)={\mathit{X}}_{\mathit{k}}+\mathit{a}{\mathit{\delta }}_{\mathit{jk}}\mathit{I}$.

• ${\mathrm{𝑉}}_{\mathrm{𝑘}}\left(\mathrm{𝑏}\right)$

  is the bounded unitary phase operator generated by the System Frame position operator for displacement parameter b. Together with ${\mathit{T}}_{\mathit{j}}\left(\mathit{a}\right)$ it satisfies the Weyl relation from which the canonical position-momentum commutator follows on a common dense invariant core.

• ${\mathrm{𝜎}}_{\mathrm{𝑖}}$

  are the Pauli matrices acting on the derived two-component spin representation. They generate the spin operators ${\mathit{S}}_{\mathit{i}}=\frac{\mathrm{\hslash }}{2}{\mathit{\sigma }}_{\mathit{i}}$ and satisfy the standard Pauli algebra after the MQ rotational and Clifford construction has selected the spin-1/2 representation.

• ${\mathrm{𝛾}}^{\mathrm{𝜇}}$

  are the Dirac gamma matrices of the realized Lorentz-covariant System Frame description. They satisfy the Clifford algebra associated with the emergent metric and enter the MQ Dirac equation and conserved Dirac current.

• ${\mathrm{𝐷}}_{\mathrm{𝜇}}$

  is the gauge-covariant derivative acting on charged System Frame matter fields. In the electromagnetic completion it contains the realized U(1) connection and appears in the MQ Dirac and QED expressions.

• ${\mathrm{𝐴}}_{\mathrm{𝜇}}$

  is the realized System Frame electromagnetic four-potential used in the U(1) gauge-covariant derivative. It is a continuum field of the System Frame and is not a primitive Internal Frame variable.

• ${\mathrm{𝐹}}_{\mathrm{𝜇}\mathrm{𝜈}}$

  is the electromagnetic field-strength tensor constructed from the System Frame four-potential ${\mathit{A}}_{\mathit{\mu }}$. It enters the Maxwell and QED realization of the MQ continuum action.

Fundamental constant and measures

• ${\mathrm{2}\mathrm{𝑄}}_{\mathrm{𝐿}}{\mathrm{𝑛}}_{\mathrm{𝐿}}$

  also known as the Informativity differential describes the length contraction associated with discrete measure. It is a dimensionless normalization modifier arising from the discrete count structure of the Internal Frame, representing the deviation between integer–resolved count measures and their realized System Frame encoding under the Frames mapping $\mathit{M}:\mathit{\omega }\to \mathcal{R}$. It quantifies how conserved Internal Frame count ${\mathit{N}}_{\mathit{I}}\left(\mathit{\omega }\mathit{\text{'}},\mathit{A}\right)$ is redistributed and normalized when expressed as a System Frame observable.

• ${\mathrm{𝑙}}_{\mathrm{𝑓}}\mathrm{,}{\mathrm{𝑚}}_{\mathrm{𝑓}}\mathrm{,}{\mathrm{𝑡}}_{\mathrm{𝑓}}$

  are the fundamental measures, more precise expressions for Planck’s units–length, mass, and time–that consider the effects of length contraction associated with discrete measure.

• $\mathrm{G}$

  is the gravitational constant, G=6.67407794(17) $\times$ 10^–11 m³ kg^–1 s^–2 [24] such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure (e.g., G=6.67384483(17) $\times$ 10^–11 m³ kg^–1 s^–2 [25] at the blackbody demarcation ([9], Appx. AB)).

• ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$

  The invariant dimensionless MQ coefficient. Its labelled angular realization is expressed in radians and its labelled momentum realization in kilogram metres per second. The adopted numerical correspondence preserves those different dimensions. The cited X-ray calculation provides context; it does not establish an experimental determination of the MQ coefficient.

• $\mathrm{𝑐}$

  the speed of light which may also be written as c=n_Ll_f/n_Tt_f =299,792,458 m/s [27] such that n_L=n_T=1 is physically significant.

• ${\mathrm{𝐻}}_{\mathrm{sys}}$

  denotes the System Frame realization of the dimensionful expansion rate, expressed in km s^-1 Mpc^-1, corresponding observationally to the CMB-resolved expansion rate formerly denoted ${\mathit{H}}_{\mathrm{CMB}}$. It is the non-discrete System Frame realization of the scalar expansion under mapping normalization and is given by ${\mathit{H}}_{\mathrm{sys}}=(\mathit{\pi }/({\mathit{\theta }}_{\mathit{si}}{\mathit{A}}_{\mathit{U}}\left)\right)(\mathrm{km}\mathrm{Mp}{\mathrm{c}}^{-1}/(\mathrm{s}/{10}^9\left)\right)$.

• ${\mathrm{𝐻}}_{\mathrm{int}}$

  denotes the Internal Frame realization of the dimensionful expansion rate, expressed in km s^-1 Mpc^-1, corresponding observationally to the Cepheid/distance ladder expansion rate formerly denoted ${\mathit{H}}_{\mathrm{cep}}$. It is the discrete frame realization of the scalar expansion under the complementary mapping normalization and is given by ${\mathit{H}}_{\mathrm{int}}=\left(\right(2{\mathit{\theta }}_{\mathit{si}}-\mathit{\pi })/({\mathit{\theta }}_{\mathit{si}}{\mathit{A}}_{\mathit{U}}\left)\right)$ $(\mathrm{km}\mathrm{Mp}{\mathrm{c}}^{-1}/(\mathrm{s}/{10}^9\left)\right)$.

• ${\mathrm{𝐻}}_{\mathrm{𝑏}}$

  denotes the baseline dimensionful expansion rate lying at the midpoint between the System Frame and Internal Frame realizations. It provides the frame neutral comparison value ${\mathit{H}}_{\mathit{b}}=({\mathit{H}}_{\mathrm{sys}}+{\mathit{H}}_{\mathrm{int}})/2$, equivalently ${\mathit{H}}_{\mathit{b}}=(1/{\mathit{A}}_{\mathit{U}}\left)\right(\mathrm{km}\mathrm{Mp}{\mathrm{c}}^{-1}/(\mathrm{s}/{10}^9\left)\right)$.

• $\mathrm{𝐻}$

  denotes the dimensionful local expansion rate expressed in observational units, typically km s^-1 Mpc^-1. It is the unit-resolved expansion rate used for observational comparison before specifying a frame realization such as ${\mathit{H}}_{\mathrm{sys}}$, ${\mathit{H}}_{\mathrm{int}}$, or ${\mathit{H}}_{\mathit{b}}$.

• $\mathrm{H}$

  denotes the dimensionless scalar expansion magnitude associated with the increasing radius of the universe. The non-italic form $\mathrm{H}$ is used intentionally to distinguish this scalar from the italicized dimensionful expansion rates. For the realized universe, $\mathrm{H}={\mathit{\theta }}_{\mathit{si}}$.

• ${\mathrm{𝐻}}_{\mathrm{0}}$

  denotes an observed dimensionful Hubble rate value inferred from a specified observational channel. In MQ, ${\mathit{H}}_0$ is not a single primitive expansion parameter. It is a measurement channel realization of the dimensionful rate $\mathit{H}$. Thus CMB-like determinations correspond to ${\mathit{H}}_{\mathrm{sys}}$, distance ladder/Cepheid-like determinations correspond to ${\mathit{H}}_{\mathrm{int}}$, and comparisons between observational determinations should specify the relevant frame realization.

• ${\mathrm{𝑙}}_{\mathrm{𝑝}}\mathrm{,}{\mathrm{𝑚}}_{\mathrm{𝑝}}\text{and}{\mathrm{𝑡}}_{\mathrm{𝑝}}$

  are Planck’s units for length, mass, and time.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}}\mathrm{,}{\mathrm{𝑛}}_{\mathrm{𝑀}}\text{and}{\mathrm{𝑛}}_{\mathrm{𝑇}}$

  are physically significant discrete counts of l_f, m_f and t_f respectively.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}}$

  describes the count of l_f representative of a change in position of an observable measured with respect to the observer’s frame of reference.

• ${\mathrm{𝑛}}_{\mathrm{𝜃}}$

  is the System Frame count of θ_si resolved as a function of the Internal Frame demarcation adjusted for the metric differential (i.e., ⌊84.60055/2⌉ = 42).

• $\mathrm{ℎ}$

  is Planck’s constant.

• $\mathrm{ħ}$

  is the reduced Planck constant, 1.054571817 10^–34 m² kg s^–1. When accounting for the Informativity differential at the upper count bound, this term is not italicized (i.e., ħ=1.054534984(26) 10^–34 m² kg s^–1).

• ${\mathrm{𝐴}}_{\mathrm{𝑒}\mathrm{𝑒}}$

  is the dilated age of the universe as measured from the current expansionary epoch. Its value is a function of expansion with respect to the total elapsed time representing the quantum epoch.

• ${\mathrm{𝐴}}_{\mathrm{𝑞}\mathrm{𝑒}}$

  is the non–dilated age of the universe corresponding to the quantum epoch.

• ${\mathrm{𝑉}}_{\mathrm{𝑞}\mathrm{𝑒}}$

  describes the volume of the universe at the end of the quantum epoch.

• ${\mathrm{\Omega }}_{\mathit{vis}}$

  is that domain of the universe that is presently visible.

• ${\mathrm{\Omega }}_{\mathit{obs}}$

  is that domain of the universe that is presently visible or will be visible given infinite elapsed time.

• ${\mathrm{\Omega }}_{\mathit{dk}}$

  is that domain of the universe that will never be visible.

• ${\mathrm{\Omega }}_{\mathit{uobs}}$

  is that domain of the universe that will eventually be visible but is not presently visible.

• ${\mathrm{\Omega }}_{\mathit{tot}}$

  is the entire domain that makes up the universe (i.e., 1 or 100%).

• ${\mathrm{\Omega }}_{\mathit{f}}$

  is that domain of the universe that is fundamental. Its physical significance describes a pivot point with respect to the remaining domains.

• ${\mathrm{𝑀}}_{\mathrm{𝑎}\mathrm{𝑐}\mathrm{𝑟}}$

  is the rate of mass accretion in the universe.

• ${\mathrm{𝑚}}_{\mathrm{𝑒}}$

  is the mass of an electron.

• ${\mathrm{𝑎}}_{\mathrm{0}}$

  is the ground state orbital of an atom, also known as the Bohr radius.

• $\mathrm{𝛼}$

  is the fine structure constant, a dimensionless fundamental physical constant that characterizes the strength of the electromagnetic interaction between electrically charged particles. The recommended numerical value is published by CODATA based on the least-squares adjustment of the best available experimental and theoretical data.

• ${{\mathrm{𝛼}}_{\mathrm{𝑓}}}^{\mathrm{–1}}\mathrm{,}{{\mathrm{𝛼}}_{\mathrm{𝑝}}}^{\mathrm{–1}}\text{, and}{{\mathrm{𝛼}}_{\mathrm{𝑐}}}^{\mathrm{–1}}$

  are the fundamental, Planck–like, and classical forms of the inverse fine structure constant. α_f^–1=42θ_si is defined with respect to the Internal Frame; the remaining two against the System Frame. α_c^–1 differs from α_p^–1, accounting for the length contraction associated with discrete measure.

• $\mathrm{\gamma }$

  is the conventional electromagnetic transformation parameter used in the gamma approach to resolve the electromagnetic constants. It has a value of 1.1996724235 m⁶ C^-1 s^-4 and is determined from an independent source measurement of one of the electromagnetic constants (typically ${\mathit{k}}_{\mathit{e}}$, ${\mathit{\epsilon }}_0$, or ${\mathit{\mu }}_0$) and subsequently used to derive the remaining electromagnetic quantities while preserving the independence of the selected source measurement.

• ${\mathrm{\gamma }}_{\mathrm{𝐹}}$

  is the fundamental gamma, a first principles realization parameter that replaces the conventional gamma transform within the MQ framework. Unlike γ, which is resolved from experimentally measured electromagnetic constants, γ_F is derived entirely from fundamental quantities and serves as the canonical electromagnetic realization parameter from which the electromagnetic constants are independently determined.

• $\mathrm{\Lambda }$

  describes Einstein’s cosmological constant.

• ${\mathrm{\varepsilon }}_{\mathrm{0}}$

  is the electric constant (vacuum permittivity), which characterizes the ability of free space to permit electric field formation. Conventionally, ɛ₀ = 1/u₀c² = 8.854 187 8188(14) $\times$ 10-12 F m^-1.

• ${\mathrm{𝑢}}_{\mathrm{0}}$

  is the magnetic constant (vacuum permeability), which characterizes the magnetic properties of free space. Conventionally, u₀ = 2αh/e²c = 1.256 637 06127(20) $\times$ 10^-6 N A^-2.

• ${\mathrm{𝑘}}_{\mathrm{𝑒}}$

  is Coulomb's constant, the proportionality constant governing the electrostatic force between two point-charges. Conventionally, k_e = 1/4π ɛ₀ = 8.987 551 7923(14) x 109 N m² C^-2.

• $\mathrm{𝑒}$

  is the elementary charge, the magnitude of the electric charge carried by a proton (or equivalently the magnitude of the negative charge carried by an electron). In the SI, e = 1.602 176 634 x 10^-19 C exactly.

• ${\mathrm{𝐸}}_{\mathrm{𝐶}\mathrm{𝑀}\mathrm{𝐵}}$

  is the total energy associated with the CMB.

• ${\mathrm{𝐸}}_{\mathrm{𝜌}\mathrm{:}\mathrm{𝑐}\mathrm{𝑚}\mathrm{𝑏}}$

  is the energy density associated with the CMB.

• $\mathrm{𝑘}$

  is the Boltzmann constant.

• $\mathrm{𝜌}$

  is the energy density accumulated at a given age of the universe.

• ${\mathrm{𝑇}}_{\mathrm{𝐶}\mathrm{𝑀}\mathrm{𝐵}}$

  is the temperature of the Cosmic Microwave Background.

• ${\mathrm{𝑀}}_{\mathrm{vis}}$

  is the total mass corresponding to the visible domain ${\mathrm{\Omega }}_{\mathrm{vis}}$, representing mass contained within the portion of the universe presently visible from a given reference point.

• ${\mathrm{𝑀}}_{\mathrm{obs}}$

  is the total mass corresponding to the observable domain ${\mathrm{\Omega }}_{\mathrm{obs}}$, representing mass contained within the portion of the universe that is presently visible or will become visible given infinite elapsed time.

• ${\mathrm{𝑀}}_{\mathrm{dk}}$

  is the total mass corresponding to the dark domain ${\mathrm{\Omega }}_{\mathrm{dk}}$, representing mass contained within the portion of the universe that will never become observable from a given reference point.

• ${\mathrm{𝑀}}_{\mathrm{uobs}}$

  is the total mass corresponding to the unobserved domain ${\mathrm{\Omega }}_{\mathrm{uobs}}$, representing mass contained within the portion of the universe that is not presently visible but will become observable given sufficient elapsed time.

• ${\mathrm{𝑀}}_{\mathrm{tot}}$

  is the total mass of the universe, including all mass-equivalent forms of energy.

• ${\mathrm{𝑀}}_{\mathrm{𝑓}}$

  is the mass corresponding to the fundamental domain ${\mathrm{\Omega }}_{\mathit{f}}$. It defines the mass threshold associated with the fundamental partition of the universe, corresponding to the limiting rate at which distinguishable mass events can be resolved at a reference point.

• ${\mathrm{𝑀}}_{\mathrm{e}}\mathrm{(}\mathrm{𝑅}\mathrm{)}$

  is the MQ effective radial mass enclosed within orbital radius $\mathit{R}$, representing the realized gravitational mass that reproduces the observed orbital dynamics under the Frames mapping.

• ${\mathrm{𝑀}}_{\mathrm{b}}\mathrm{(}\mathrm{𝑅}\mathrm{)}$

  is the local mass-frequency bound associated with orbital radius $\mathit{R}$, representing the maximum mass that may be realized during a single increment of elapsed time before the mass-to-length frequency bound is reached. It is a bound on instantaneous mass realization rather than on the total mass that may be observed over finite elapsed time.

• ${\mathrm{𝑀}}_{\mathrm{\odot }}$

  denotes one solar mass.

• ${\mathrm{𝑛}}_{{\mathrm{𝑀}}_{\mathrm{𝑢}}}$

  is a count of ${\mathit{m}}_{\mathit{f}}$equal to the total mass-equivalent content of the universe.

• ${\mathrm{𝑛}}_{{\mathrm{𝑇}}_{\mathrm{𝑢}}}$

  is a count of ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ equal to the age of the universe.

• ${\mathrm{𝑛}}_{{\mathrm{𝐿}}_{\mathrm{𝑢}}}$

  is a count of ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ equal to the diameter of the universe.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑜}}$

  describes the count of ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ corresponding to an observed realized separation.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}}$

  describes the count of ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ corresponding to the realized separation between a reference point and a center of gravity.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑙}}$

  describes the count of ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ corresponding to a separation resolved in the local frame of reference.

• ${\mathrm{𝑛}}_{\mathrm{𝑇}\mathrm{𝑙}}$

  describes the count of ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ corresponding to elapsed duration in the local frame of reference.

• ${\mathrm{𝑛}}_{\mathrm{𝑇}\mathrm{𝑜}}$

  describes the count of ${\mathrm{𝑡}}_{\mathrm{𝑓}}$ corresponding to observed elapsed duration.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑚}}$

  describes the count of ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ corresponding to the change in realized position of an observed mass.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑐}}$

  describes the count of ${\mathrm{𝑙}}_{\mathrm{𝑓}}$ corresponding to the change in realized position of light.

• ${\mathrm{𝑣}}_{\mathrm{e}}$

  is the effective orbital velocity of a mass relative to a galaxy’s center of mass, obtained from the mapping-normalized effective radial mass.

• ${\mathrm{𝑣}}_{\mathrm{New}}$

  is the Newtonian orbital velocity predicted from the adopted baryonic mass model for a mass relative to a galaxy's center of mass.

• ${\mathrm{𝑣}}_{\mathrm{o}}$

  is the published observed orbital velocity of a mass relative to a galaxy's center of mass.

• ${\mathrm{𝑣}}_{\mathrm{real}}$

  is the MQ realized orbital velocity of a mass relative to a galaxy's center of mass, obtained from the bounded realization of the Newtonian baryonic gravitational response under the Frames mapping.

• ${\mathrm{𝑣}}_{\mathrm{b}}$

  is the orbital velocity corresponding to the local mass-frequency bound, representing the maximum orbital velocity associated with the upper realizable mass-event frequency before the mass-to-length frequency bound is exceeded.

• ${\mathrm{𝑎}}_{\mathrm{MQ}}$

  is the MQ acceleration scale governing bounded gravitational realization. It determines the transition between nearly complete gravitational realization at high baryonic acceleration and incomplete realization at low baryonic acceleration. In the galactic realization law it enters through the dimensionless ratio ${\mathit{g}}_{\mathrm{bar}}$/${\mathit{a}}_{\mathrm{MQ}}$. Its present-epoch value is derived from MQ first principles rather than calibrated to galaxy rotation data.

• ${\mathrm{𝛩}}_{\mathrm{𝑠}\mathrm{𝑖}}$

  The angular realization of the invariant MQ coefficient, expressed in radians. It is dimensionally distinct from the dimensionless coefficient and its momentum realization. The adopted numerical realization convention does not establish an experimental determination of the MQ coefficient.

• ${\mathrm{𝑝}}_{\mathrm{𝑠}\mathrm{𝑖}}$

  The momentum realization of the invariant MQ coefficient, expressed in kilogram metres per second. It enters the dimensional fundamental relation. Its numerical correspondence with the dimensionless coefficient depends on the adopted realization convention; their physical dimensions remain distinct.

• $\mathrm{𝑣}$

  A scalar speed or relative motion parameter, expressed in metres per second. Its source context specifies the moving object and reference interval. It is distinct from the vector velocity field of the continuum construction.

• ${\mathrm{𝑛}}_{\max }$

  The dimensionless upper-bound count parameter of the specified admissibility domain, normalized using the adopted reference interval. The reported numerical parameter does not determine a unique endpoint integer.

• $\mathrm{𝑛}$

  A manuscript-local positive whole-unit count or dimensionless whole-unit multiplier. Its admissibility domain and role are specified locally. It is a placeholder and does not introduce an additional MQ primitive.

• ${\mathrm{𝑚}}_{\mathrm{𝑙}}$

  The local mass, expressed in kilograms. The local label identifies a measure in the framework of the observable; the observed label identifies a measure in the framework of the observer. These labels are not interchangeable.

• ${\mathrm{𝑚}}_{\mathrm{𝑜}}$

  The observed mass, expressed in kilograms. The local label identifies a measure in the framework of the observable; the observed label identifies a measure in the framework of the observer. These labels are not interchangeable.

• ${\mathrm{𝑙}}_{\mathrm{𝑙}}$

  The local length, expressed in metres. The local label identifies a measure in the framework of the observable; the observed label identifies a measure in the framework of the observer. These labels are not interchangeable.

• ${\mathrm{𝑙}}_{\mathrm{𝑜}}$

  The observed length, expressed in metres. The local label identifies a measure in the framework of the observable; the observed label identifies a measure in the framework of the observer. These labels are not interchangeable.

• ${\mathrm{𝑡}}_{\mathrm{𝑙}}$

  The local duration, expressed in seconds. The local label identifies a measure in the framework of the observable; the observed label identifies a measure in the framework of the observer. These labels are not interchangeable.

• ${\mathrm{𝑡}}_{\mathrm{𝑜}}$

  The observed duration, expressed in seconds. The local label identifies a measure in the framework of the observable; the observed label identifies a measure in the framework of the observer. These labels are not interchangeable.

• ${\mathrm{𝛼}}_{\mathrm{𝑐}}$

  The dimensionless classical fine-structure coupling. Its inverse is defined in the existing fine-structure entry. The fundamental and System Frame realizations retain their respective count normalization and source context.

• ${\mathrm{𝛼}}_{\mathrm{𝑝}}$

  The dimensionless Planck-like fine-structure coupling. Its inverse is defined in the existing fine-structure entry. The fundamental and System Frame realizations retain their respective count normalization and source context.

• ${\mathrm{𝛼}}_{\mathrm{𝑓}}$

  The dimensionless fundamental fine-structure coupling. Its inverse is defined in the existing fine-structure entry. The fundamental and System Frame realizations retain their respective count normalization and source context.

• ${\mathrm{𝑛}}_{\mathrm{𝑀}\mathrm{𝑙}}$

  The dimensionless local mass count relative to the fundamental mass measure. Its local or observed assignment follows the observer–observable convention and is distinct from a physical mass in kilograms.

• ${\mathrm{𝑛}}_{\mathrm{𝑀}\mathrm{𝑜}}$

  The dimensionless observed mass count relative to the fundamental mass measure. Its local or observed assignment follows the observer–observable convention and is distinct from a physical mass in kilograms.

• ${\mathrm{𝛾}}_{\mathrm{𝐹}\mathrm{,}\mathrm{sys}}$

  The mapped electromagnetic realization of fundamental gamma, expressed in squared coulombs in the supplied construction. It identifies a physical representation of the configuration parameter rather than an independently fitted second parameter.

• ${\mathrm{𝑒}}_{\mathrm{𝑓}}$

  The fundamental charge representation in the supplied MQ charge construction. Configuration expressions and their dimensional realizations must be distinguished. After the stated Frames mapping, the charge realization is expressed in coulombs. It is distinct from the dimensionless exponential base.

• ${\mathrm{𝑒}}_{\mathrm{𝑝}}$

  The Planck-like charge representation in the supplied MQ charge construction. Configuration expressions and their dimensional realizations must be distinguished. After the stated Frames mapping, the charge realization is expressed in coulombs. It is distinct from the dimensionless exponential base.

• ${\mathrm{𝑒}}_{\mathrm{𝑐}}$

  The classical charge representation in the supplied MQ charge construction. Configuration expressions and their dimensional realizations must be distinguished. After the stated Frames mapping, the charge realization is expressed in coulombs. It is distinct from the dimensionless exponential base.

• ${\mathrm{𝑛}}_{\mathrm{𝑒}}$

  The charge-construction intermediate or residual. Some source constructions describe a dimensionless configuration residual; later realized constructions assign charge units. The source-specific representation and units must therefore be stated explicitly. These meanings are retained rather than treated as interchangeable.

• $\mathrm{𝑏}$

  A dimensionless right-triangle count-coordinate leg in the local count construction. The side labels and their normalization are specified with the equation. This use is distinct from a Bell analyzer setting.

• ${\mathrm{𝑎}}_{\max }$

  The dimensionless upper trial endpoint of the first count-coordinate leg. It is local to the trial-bound construction and does not define a universal physical limit.

• ${\mathrm{𝑏}}_{\min }$

  The dimensionless lower trial endpoint of the second count-coordinate leg. It is local to the trial-bound construction and does not define a universal physical limit.

• ${\mathrm{𝑐}}_{\mathrm{bound}}$

  The dimensionless bound hypotenuse count coordinate. It is local to the trial-bound construction and does not define a universal physical limit.

• ${\mathrm{𝑛}}_{\min }$

  The dimensionless positive lower reference count. It is local to the trial-bound construction and does not define a universal physical limit.

• ${\mathrm{𝑛}}_{\mathrm{bound}}$

  The dimensionless bound count returned by the specified construction. It is local to the trial-bound construction and does not define a universal physical limit.

• ${\mathrm{𝐸}}_{\mathrm{𝑙}}$

  The realized electromagnetic energy expression in the selected MQ normalization, expressed in joules. It is distinct from the baryonic energy normalization; their ratio does not establish different measured energies for a single physical process.

• ${\mathrm{𝐸}}_{\mathrm{𝑚}}$

  The realized baryonic energy expression in the selected MQ normalization, expressed in joules. Its comparison with the electromagnetic expression must retain the stated count and upper-count assumptions.

• ${\mathrm{𝑅}}_{\mathrm{\infty }}$

  The Rydberg constant, expressed in inverse metres. It is distinct from a radial distance and from the script realization structure or fraktur realization manifold. Its shared calibration dependence must be retained.

• ${\mathrm{𝑘}}_{\mathrm{B}}$

  The Boltzmann constant, expressed in joules per kelvin. This explicit label is an alternative to the existing symbol k when k denotes Boltzmann’s constant. It must be distinguished from the gravitational coupling coefficient k.

• ${\mathrm{𝐺}}_{\mathrm{EM}}$

  The gravitational realization evaluated at the electromagnetic demarcation, expressed in cubic metres per kilogram per squared second. The base G is italic here. The electromagnetic label is upright and the quantity is distinct from the upright upper-count realization.

• ${\mathrm{ħ}}_{\mathrm{up}}$

  The MQ reduced-action realization at the upper count limit, expressed in joule seconds. The base and upper-count label are upright. It is distinct from the conventional SI reduced Planck constant and the italic electromagnetic-demarcation realization.

• ${\mathrm{𝑄}}_{{\mathrm{𝑛}}_{\mathrm{𝜃}}}$

  The dimensionless residual of the supplied fundamental charge construction. It is distinct from the integer interaction count and from a dimensional charge-valued intermediate.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑎}}$

  The dimensionless fundamental-length count of the first perpendicular side of the local right triangle. All participating lengths must use the same fundamental reference. It is distinct from a Bell analyzer label.

• ${\mathrm{𝑛}}_{\mathrm{𝐿}\mathrm{𝑏}}$

  The dimensionless fundamental-length count of the second perpendicular side of the local right triangle. All participating lengths must use the same fundamental reference. It is distinct from a Bell analyzer label.

• $\mathrm{𝑝}$

  A scalar momentum observable, expressed in kilogram metres per second. Its source definition and state context are retained. It is distinct from the vector momentum field, the momentum operator, and the specially labelled MQ momentum realization.

• $\mathrm{𝜃}$

  The common signal and idler direction angle, expressed in radians, in the specified equal-angle X-ray calculation. This unsuffixed angle is distinct from the invariant dimensionless MQ coefficient and from its labelled angular realization.

• $\mathrm{𝐿}$

  The arc length at the fundamental radius in the supplied angular illustration, expressed in metres. This ordinary italic capital letter is distinct from the script Lagrangian density and the subscript of a length count.

• $\mathrm{𝑎}$

  The blackbody radiation constant, expressed in joules per cubic metre per fourth power of kelvin. It is evaluated from the stated SI constants. It is distinct from the cosmological scale function, a triangle coordinate, and a Bell analyzer setting.

• $\mathrm{e}$

  The dimensionless base of natural logarithms and the natural exponential. It is upright. It is distinct from elementary charge, the electron label in electron mass, and effective-mass or effective-velocity labels.

• $\mathrm{𝑟}$

  A scalar radial separation or radial threshold, expressed in metres, in the specified source construction. It is distinct from the bold spatial coordinate vector and from the Pearson correlation coefficient.

• $\mathrm{𝑅}$

  A positive scalar radial separation or galactocentric radius, expressed in metres. Its local reference point and source geometry are specified with the equation. It is distinct from the script realization structure, fraktur manifold and Ricci scalar.

• ${\mathrm{𝑣}}_{\mathrm{m}}$

  The historical speed-valued mass proxy at fixed radius in the supplied escape-speed construction, expressed in metres per second. It is distinct from an independently measured orbital velocity or the Newtonian baryonic circular velocity.

• ${\mathrm{𝑣}}_{\mathrm{m}}$

  The supplied or modelled orbital velocity profile in the bound-mass and effective-mass construction, expressed in metres per second. It is an input to the response prescription; agreement of transformed velocities near the bound is not independent observational evidence.

• $\mathrm{𝑀}$

  The source mass in the scalar point-source escape relation, expressed in kilograms. It may be represented by the mass count times the fundamental mass. This italic scalar is distinct from the Frames mapping.

• ${\mathrm{𝑣}}_{\mathrm{e}}$

  The point-source escape speed in the explicitly identified escape relation, expressed in metres per second. It is distinct from the effective orbital velocity using the same displayed notation in other passages; the equation context must identify the intended meaning.

• ${\mathrm{𝑛}}_{\mathrm{𝑎}}$

  The dimensionless first orthogonal leg count in the local right-triangle construction. These are local geometric count labels, distinct from Bell analyzer settings.

• ${\mathrm{𝑛}}_{\mathrm{𝑏}}$

  The dimensionless second orthogonal leg count in the local right-triangle construction. These are local geometric count labels, distinct from Bell analyzer settings.

• ${\mathrm{𝑙}}_{\mathrm{𝑐}}$

  The light-distance measure over the same represented interval as the displacement comparison, expressed in metres. Its ratio to that interval gives the speed of light.

• ${\mathrm{𝑣}}_{\mathrm{escape}}$

  The classical escape speed for an isolated point mass or the stated spherical exterior, expressed in metres per second. The reference potential vanishes at infinity. This expression does not determine escape speed in an arbitrary extended galaxy.

• ${\mathrm{𝑣}}_{\mathrm{bc}}$

  The unscaled bound speed preceding the coefficient realization that gives the bound velocity in the specified construction, expressed in metres per second.

• ${\mathrm{𝑛}}_{\mathrm{𝑏}\mathrm{𝑛}}$

  The derived dimensionless hypotenuse count formed from the two leg counts in the local right-triangle construction. Its value need not be an integer even when the primitive leg counts are whole units.

• $\mathrm{𝜎}$

  The Stefan–Boltzmann constant, expressed in watts per square metre per fourth power of kelvin. It enters the equilibrium blackbody radiation relation and is distinct from a statistical standard deviation.

• ${\mathrm{𝑛}}_{{\mathrm{𝑀}}_{\mathrm{b}}}$

  The dimensionless reference mass count associated with the fixed one-kilogram normalization in the stated bound construction. It is distinct from the radius-dependent count of the bound mass.

• $\mathrm{\pi }$

  The dimensionless circular constant. It is always upright in this notation, including in equations and subscripts.

• $\mathrm{𝑥}$

  The scalar position observable canonically conjugate to the momentum in the stated uncertainty relation, expressed in metres. It is distinct from the dimensionless rounding argument and spacetime coordinate components.

• $\mathrm{𝑚}$

  The mass argument of the speed-valued function in the historical velocity-proxy construction, expressed in kilograms and identified there with the source mass. This scalar is distinct from the upright metre unit and from the fundamental mass reference.

• $\mathrm{𝑓}\mathrm{(}\mathrm{𝑚}\mathrm{)}$

  The speed-valued dependence on the mass argument at fixed radius in the historical velocity-proxy construction, expressed in metres per second. It is distinct from the fundamental-measure label and an arbitrary continuum observable function.

• $\mathrm{𝑙}$

  The scalar displacement over the represented interval in the velocity comparison, expressed in metres. It is distinct from the invariant internal reference scale, an oriented adjacency link and the fundamental length measure.

• $\mathrm{𝑎}$

  The first dimensionless count-coordinate leg in the local right-triangle bound construction. It is distinct from the blackbody radiation constant, the scale function and a Bell analyzer setting.

• $\mathrm{𝑐}$

  The dimensionless hypotenuse count coordinate in the explicitly stated local right-triangle construction. It is distinct from the speed of light; the equation context must distinguish these uses.

Expansion and cosmological scalars

• $\mathrm{𝑡}$

  denotes the global ordering parameter labeling the one-parameter family of realized configurations in the System Frame. It provides the monotonic parameterization of expansion as defined by the evolution of the mapping-realized separation functional and does not represent a primitive temporal coordinate in the Internal Frame.

• $\mathrm{𝑎}\mathrm{(}\mathrm{𝑡}\mathrm{)}$

  denotes a derived System Frame scale function used to parameterize the evolution of realized separations under mapping normalization. It is strictly increasing and piecewise differentiable but does not constitute a fundamental degree of freedom; rather, it provides a convenient parametrization of expansion extracted from the underlying separation functional.

• $\mathrm{𝑙}$

  denotes an invariant internal reference scale associated with the mapping-normalized separation structure. It remains constant under expansion and serves to fix the reference normalization relating discrete count structure to realized System Frame separations, without introducing an independent dynamical scale.

• ${\mathrm{𝑄}}_{\mathrm{𝐿}}$

  denotes the mapping-induced residual count contribution entering the realized separation functional. In the right-triangle count construction, n_L gives the Internal Frame count separation, while n_L + Q_L gives the realized hypotenuse whose continuum encoding defines the System Frame separation. Q_L therefore replaces the older Internal-to-System scale-ratio notation by expressing expansion and curvature through count-resolved residual structure rather than through a global ratio.

• ${\mathrm{𝐴}}_{\mathrm{𝑈}}$

  is the System Frame age of the universe. The quantity ${\mathit{A}}_{\mathit{U}}\left(\mathit{z}\right)$ denotes the elapsed universe age at redshift z, as determined from the System Frame cosmological expansion history. Its present-epoch value is denoted ${\mathit{A}}_{\mathit{U},0}$.

• ${\mathrm{𝐴}}_{\mathrm{𝑈}\mathrm{,}\mathrm{0}}$

  is the present-epoch System Frame age of the universe, defined by ${\mathit{A}}_{\mathit{U},0}={\mathit{A}}_{\mathit{U}}\left(0\right)$. It provides the reference age used in the redshift dependence of ${\mathit{a}}_{\mathrm{MQ}}$ and related cosmological realization quantities.

• ${\mathrm{𝐻}}_{\mathrm{𝑓}}$

  The Hubble frequency assigned to the specified mapped representation, expressed in inverse seconds. Its relation to a mapped temporal scale requires the separate assumption stated in the source construction.

• ${\mathrm{𝐴}}_{\mathrm{CMB}}$

  The apparent CMB age in the fixed-physical-diameter comparison, expressed in the stated time unit. It is distinct from the physical System Frame age of the universe.

• ${\mathrm{𝐴}}_{\mathrm{cep}}$

  The apparent Cepheid age in the fixed-physical-diameter comparison, expressed in the stated time unit. It is distinct from the physical System Frame age of the universe.

• $\mathrm{𝛥}{\mathrm{𝐴}}_{\mathrm{CMB}}$

  The CMB apparent-age offset relative to the physical universe age at fixed physical diameter, expressed in the same time unit as those ages.

• $\mathrm{𝛥}{\mathrm{𝐴}}_{\mathrm{cep}}$

  The Cepheid apparent-age offset relative to the physical universe age at fixed physical diameter, expressed in the same time unit as those ages.

• ${\mathrm{𝑀}}_{\mathrm{CMB}}$

  The CMB radiation mass equivalent or allocated CMB mass-equivalent content in the specified construction, expressed in kilograms. Any identification of primordial allocation with present radiation energy retains the source’s temporal normalization and evolution assumptions.

• ${\mathrm{𝑉}}_{\mathrm{𝑈}}$

  The spherical mapped or realized universe volume, expressed in cubic metres. Its radius, age and representation must be stated locally. A conditional pre-referenceable mapped volume is not automatically the present realized volume.

• ${\mathrm{𝑀}}_{\mathrm{𝑈}}$

  The total universe mass equivalent, expressed in kilograms, in the specified cosmological allocation. It is an alternative source notation for the total mass and must retain the adopted normalization and allocation assumptions.

• ${\mathrm{𝑣}}_{\mathrm{𝑖}}$

  The mapped concluding expansion rate in the specified epoch construction, expressed in metres per second after realization. Its numerical value remains conditional on the stated mapping normalization.

• ${\mathrm{𝑅}}_{\mathrm{𝑈}}$

  The realized radial distance of the universe, expressed in metres. It is the radius used in the spherical volume construction and is distinct from the universe diameter.

• ${\mathrm{𝑅}}_{\mathrm{qe}}$

  The integrated pre-referenceable radius expressed in metres within the specified mapped representation. Its integration reference pair remains free until a stated conditional convention is imposed.

• ${\mathrm{𝐴}}_{\mathrm{qe,0}}$

  A positive reference value of the mapped SI-equivalent quantum-epoch temporal variable, expressed in seconds. Together with its associated radius it fixes the integration reference pair when a particular convention is imposed.

• ${\mathrm{𝐷}}_{\mathrm{𝑈}}$

  The physical diameter of the realized universe, expressed in metres. Both the capital letter and its universe label are italic. It is distinct from the radius used in a spherical volume.

• ${\mathrm{𝐻}}_{\mathrm{CMB}}$

  The physical CMB age-offset comparison rate in the fixed-diameter construction, expressed in kilometres per second per megaparsec. The base is italic in this dimensional context. It is distinct from the dimensionless scalar and the frame-specific expansion rates.

• ${\mathrm{𝐻}}_{\mathrm{cep}}$

  The physical Cepheid age-offset comparison rate in the fixed-diameter construction, expressed in kilometres per second per megaparsec. The base is italic in this dimensional context. It is distinct from the dimensionless scalar and the frame-specific expansion rates.

Mapping and normalization symbols

• ${\mathrm{𝜌}}_{\mathrm{𝑓}}$

  denotes an invariant, dimensionless internal geometric normalization factor used in the mapping of homogeneous expansion. It is not the FRW scale factor and does not define the general emergent metric. Its role is limited to fixing the reference normalization through which discrete count evolution is related to continuum expansion in the homogeneous limit.

• ${\mathrm{\Delta }}_{\mathrm{𝐻}}$

  denotes the mapping-induced normalization offset between System Frame and Internal Frame realizations of the scalar expansion parameter. It is defined by ${\mathrm{\Delta }}_{\mathit{H}}={\mathit{\theta }}_{\mathit{si}}-\mathrm{\pi }$ and represents a frame-resolution offset under the Frames mapping, not an independent dynamical parameter or a global count-expansion mismatch.

• $\mathrm{H}$

  denotes the dimensionless scalar rate of radial expansion defined at the level of the System Frame realization of the Frames mapping. It represents the intrinsic expansion magnitude governing the growth of realized separations and is independent of observational units. Within MQ, H = θ_si, so that H is not a measured local rate but the mapping-defined scalar controlling the expansion geometry and associated mapping-consistent scalar relations.

• ${\mathrm{𝐻}}_{\mathrm{0}}$

  denotes the dimensionful expansion rate expressed in observational units (e.g., km s^-1Mpc^-1). It is obtained through mapping-consistent normalization of the scalar H, converting the dimensionless expansion into a measurable quantity. It does not constitute an independent parameter, but rather the observational representation of H under a specified measurement convention at the present epoch.

• ${\mathrm{𝐻}}_{\mathrm{sys}}$

  is the System Frame realization of the expansion rate, obtained by expressing the scalar H under mapping normalization consistent with the non-discrete relational encoding. It corresponds to expansion values inferred from CMB-based measurements and represents the continuum realization of the underlying scalar expansion.

• ${\mathrm{𝐻}}_{\mathrm{int}}$

  is the Internal Frame realization of the expansion rate, obtained by expressing the scalar H under mapping normalization consistent with discrete count structure. It corresponds to expansion values inferred from local distance-ladder measurements and differs from H_sys by a mapping-induced normalization offset rather than by an independent physical parameter.

• $\mathrm{𝐻}$

  denotes the dimensionful local expansion rate expressed in observational units (e.g., km s^-1 Mpc^-1), prior to specification of a particular frame realization. It represents the normalized, unit-resolved form of the scalar expansion parameter H and serves as the general observable expansion rate from which specific realizations such as H₀, H_sys, and H_int are defined.

• $\mathrm{𝑆}$

  denotes the single MQ action functional defined over the count structure and its coarse-grained realization under the Frames mapping. It supplies the common variational origin of the metric, curvature, source, coupling, and scalar contributions, so no independent gravitational or matter action is fundamental at the primitive level.

• ${\mathrm{𝑆}}_{\mathrm{𝑔}}$

  denotes the geometric contribution to the realized action when the single MQ action is expressed in continuum form. It is a derived representation of the curvature constructed from the emergent metric, not an independently postulated Einstein-Hilbert action.

• ${\mathrm{𝑆}}_{\mathrm{𝑚}}$

  denotes the matter contribution to the realized action when the single MQ action is expressed in continuum form. It defines the stress-energy tensor through variation with respect to the emergent System Frame metric and is not an independently specified matter description. The paper states that the stress-energy tensor is obtained by varying the same action with respect to the emergent metric.

• $\mathcal{L}$

  denotes the realized Lagrangian density appearing in the MQ action. It is constructed from mapping-induced quantities and contains no independent gravitational coupling or separate primitive decomposition.

• ${\mathcal{L}}_{\mathit{m}}$

  denotes the matter Lagrangian density in the realized continuum representation. It is constructed from encoded availability, its derivatives, and the emergent metric, with admissible terms fixed by the MQ variational principle rather than by an independently postulated matter description.

• ${\mathrm{𝐶}}_{\mathrm{𝑝}\mathrm{𝜃}}$

  The dimensional conversion factor from the angular realization to the momentum realization of the MQ coefficient. Its dimensions are momentum per radian. The angular and momentum quantities must retain their distinct units.

• $\mathrm{𝐾}$

  The dimensionless product of the right-triangle remainder and radial length count used in the charge inversion. On the specified finite positive-count branch it lies strictly between zero and one half. It is distinct from a dimensional conversion factor, a gravitational coupling, and the kelvin unit.

• ${\mathrm{𝑅}}_{\mathrm{𝑝}}$

  The MQ momentum realization operator acting on the dimensionless invariant coefficient to produce the momentum realization. Its output has momentum dimensions; it does not identify a count or an angle with momentum.

• ${\mathrm{𝑟}}_{\mathrm{𝑞}}$

  The dimensionless quantization ratio evaluated at the specified electromagnetic demarcation and count constraint. Its value follows the stated whole-unit construction rather than an additional measured input.

• ${\mathrm{𝛥}}_{\mathrm{mdif}}$

  The dimensionless factor specifying the direction of the exploratory coordinate rescaling in the observable-domain construction. Its reciprocal reverses that rescaling. It is not established as a Lorentz time-dilation factor between specified inertial clocks.

• $\mathrm{𝜒}$

  The dimensionless gravitational realization depth in the stationary galactic construction for positive baryonic acceleration and positive MQ acceleration scale. It is distinct from an indicator function.

• $\mathrm{𝑃}$

  The dimensionless gravitational realization fraction in the local stationary galactic construction. It describes the realized fraction of baryonic gravitational availability and is distinct from a probability measure.

• ${\mathrm{𝛥}}_{\mathrm{𝑓}\mathrm{(}{\mathrm{𝑣}}_{\mathrm{m}}\mathrm{)}}$

  The dimensionless fractional difference between modelled and bound velocities in the supplied bound construction. It supplies the velocity-ratio input to the effective response prescription; it is not an observational error estimate.

• ${\mathrm{𝑛}}_{{{\mathrm{𝐿}}_{\mathrm{𝑢}}}^{\mathrm{\prime }}}$

  The primed dimensionless ratio used in the specified quantum-epoch Frames mapping. It is a derived allocation ratio, rather than a primitive integer count or the integrated radius.

• ${\mathrm{𝑡}}_{\mathrm{ref}}$

  The adopted reference duration, equal to one second in the specified SI numerical realization. It fixes a normalization rather than an independently derived physical clock within the configuration domain.

• ${\mathrm{H}}_{\mathrm{CMB}}$

  The dimensionless CMB realization of the radial expansion coefficient in the fixed-diameter comparison. The base is upright in this scalar context. It is distinct from the dimensional comparison rate with the same label.

• ${\mathrm{H}}_{\mathrm{cep}}$

  The dimensionless Cepheid realization of the radial expansion coefficient in the fixed-diameter comparison. The base is upright in this scalar context. It is distinct from the dimensional comparison rate with the same label.

Field equation components

• ${\mathrm{𝜌}}_{\mathrm{𝑡}\mathrm{𝑜}\mathrm{𝑡}}$

  is the total density ${\mathit{\rho }}_{\mathit{tot}}=3{\mathit{H}}_{\mathrm{sys}}^2/8\mathit{\pi G}$ inferred from the MQ System Frame expansion rate corresponds to the observationally inferred cosmic energy density when the MQ kinematic expansion law is interpreted within the standard gravitational framework.

• ${\mathrm{𝜌}}_{\mathrm{𝑐}}$

  is the critical density ${\mathit{\rho }}_{\mathit{c}}=3{\mathit{H}}^2/8\mathit{\pi G}$ is the total energy density required for spatial flatness in the standard Friedmann–Lemaître cosmological model. It provides the conventional normalization scale against which fractional density components are defined$\mathrm{.}$

• ${\mathrm{𝜌}}_{\mathrm{\Lambda }}$

  is the vacuum density ${\mathit{\rho }}_{\mathrm{\Lambda }}=\mathrm{\Lambda }{\mathit{c}}^2/8\mathit{\pi G}$ is the effective energy density associated with the cosmological constant term appearing in Einstein’s field equations. In the MQ framework, this quantity corresponds to the dynamical realization, within general relativity, of the geometric scalar derived from invariant measure relations.

• ${\mathrm{𝜌}}_{\mathrm{𝑚}}$

  is the matter density ${\mathit{\rho }}_{\mathit{m}}=(1-{\mathit{\Omega }}_{\mathrm{\Lambda }}){\mathit{\rho }}_{\mathit{c}}$ represents the non–scalar contribution to the total cosmic energy density. Under the assumption of spatial flatness, it corresponds to the complement of the scalar fraction and includes both baryonic and dark matter components.

• ${\mathrm{𝜌}}_{\mathrm{sys}}$

  is the MQ geometric density ${\mathit{\rho }}_{\mathrm{sys}}=\sqrt{\mathrm{\Lambda }}{\mathit{c}}^2/2{\mathit{l}}_{\mathit{f}}$ is the density scale arising directly from invariant fundamental measure normalization in the Internal Frame. It depends only on the geometric scalar $\mathrm{\Lambda }$ and the fundamental length measure ${\mathrm{𝑙}}_{\mathrm{𝑓}}$, and does not involve the gravitational coupling constant. This quantity represents the kinematic density associated with the induced scalar prior to any dynamical interpretation.

• ${\mathit{\Omega }}_{\mathrm{\Lambda }}$

  is the fractional scalar density parameter ${\mathit{\Omega }}_{\mathrm{\Lambda }}={\mathit{\rho }}_{\mathrm{\Lambda }}/{\mathit{\rho }}_{\mathit{c}}=\mathrm{\Lambda }{\mathit{c}}^2/3{\mathit{H}}^2$ specifies the proportion of the total cosmic energy density attributed to the scalar contribution. Within MQ it is determined entirely by invariant internal relations.

• ${\mathit{\Omega }}_{\mathit{b}}$

  is the baryonic matter fraction (ΛCDM comparison bookkeeping).

• ${\mathit{\Omega }}_{\mathit{c}}$

  is the cold dark matter fraction (ΛCDM comparison bookkeeping).

• $\mathrm{𝜑}$

  is the MQ dynamical field defined over the coarse-grained realized continuum domain $\mathrm{\Omega }$ in the System Frame. It is the effective field variable appearing in the realized MQ action functional ${\mathit{S}}^{\mathrm{sys}}[\mathit{\phi },{\mathit{g}}^{\mathrm{sys}}]$ and represents the continuum realization of underlying count-based dynamics under the Frames mapping. The field is not fundamental at the Internal Frame level and does not constitute primitive ontology. Rather, it is an emergent coarse-grained degree of freedom derived from conserved count structure. Variation of the realized action with respect to $\mathit{\phi }$ yields the admissible continuum field dynamics through the MQ variational principle.

• ${\mathrm{𝑔}}_{\mathrm{𝜇}\mathrm{𝜈}}^{\mathrm{sys}}$

  denotes the emergent metric tensor in the realized System Frame. It defines the local continuum metric structure associated with mapped measurable separations under the Frames mapping. Unlike standard general relativity, where ${\mathit{g}}_{\mathit{\mu \nu }}$ is treated as a fundamental spacetime field, MQ treats ${\mathit{g}}_{\mathit{\mu \nu }}^{\mathrm{sys}}$ as a derived object obtained through coarse–graining of the realized separation functional. Its normalization is fixed by mapping constraints and is not independently specified.

• ${\mathrm{𝐺}}_{\mathrm{𝜇}\mathrm{𝜈}}^{\mathrm{sys}}$

  denotes the Einstein tensor in the realized System Frame. It is constructed from the emergent metric tensor ${\mathit{g}}_{\mathit{\mu \nu }}^{\mathrm{sys}}$ through the associated Levi–Civita connection and resulting curvature contractions. In MQ, ${\mathit{G}}_{\mathit{\mu \nu }}^{\mathrm{sys}}$ represents the continuum realization of higher-order discrete variation induced by the Frames mapping. Its divergence-free property follows from the metric-compatible connection and associated Bianchi identity, rather than from an independently postulated geometric structure.

• ${\mathrm{𝑇}}_{\mathrm{𝜇}\mathrm{𝜈}}^{\mathrm{sys}}$

  denotes the stress–energy tensor in the realized System Frame. In MQ, it is defined variationally from the MQ Action through metric variation, ${\mathit{T}}_{\mathit{\mu \nu }}^{\mathrm{sys}}=-\frac{2}{\sqrt{-{\mathit{g}}^{\mathrm{sys}}}}\frac{\mathit{\delta }{\mathit{S}}_{\mathrm{matter}}}{\mathit{\delta }{\mathit{g}}_{\mathrm{sys}}^{\mathit{\mu \nu }}}$, and therefore encodes the response of realized normalized count density to variations in the emergent metric structure. It is not independently specified and inherits its structure from encoded availability and mapping normalization.

• ${\mathcal{K}}_{\mathrm{sys}}$

  denotes the MQ gravitational coupling factor appearing in the realized field equation of the System Frame. It defines the proportional coupling between the realized curvature structure and the mapping-induced stress–energy structure. Unlike standard general relativity, ${\mathcal{K}}_{\mathrm{sys}}$ is not introduced as an independent phenomenological constant, but is derived from the MQ primitive measures and the Frames mapping. In the realized continuum limit of the System Frame, ${\mathcal{K}}_{\mathrm{sys}}=8\mathit{\pi G}/{\mathit{c}}^4=4\mathrm{\pi }{\mathit{t}}_{\mathit{f}}/{\mathit{\theta }}_{\mathit{si}}$. The quantity therefore encodes the mapping-normalized gravitational response implied by the conserved Internal Frame count structure under coarse-grained realization.

• ${\mathrm{\Lambda }}_{\mathrm{sys}}$

  denotes the realized cosmological term appearing in the System Frame field equation. In MQ, it is not a fundamental constant but a derived scalar arising from the same realized separation structure that defines the emergent metric tensor ${\mathit{g}}_{\mathit{\mu \nu }}^{\mathrm{sys}}$. It reflects the large-scale normalization of the System Frame geometry induced by the Frames mapping and is therefore fixed by the underlying count geometry rather than introduced as an independent phenomenological parameter.

• ${\mathrm{\Lambda }}_{\mathrm{int}}$

  denotes the Internal Frame cosmological scalar prior to continuum realization. In MQ, it is a count-derived scalar contribution determined by admissible count evolution, mapping normalization, and the realized separation structure encoded within the Internal Frame count geometry. It does not represent a geometric cosmological constant in the continuum sense. Rather, it is the pre-realization scalar quantity whose Frames mapping yields the realized cosmological term ${\mathrm{\Lambda }}_{\mathrm{sys}}$ appearing in the System Frame field equations.

• ${\mathrm{𝑔}}_{\mathrm{bar}}\left(\mathrm{𝑅}\right)$

  is the Newtonian baryonic gravitational acceleration at galactocentric radius R, calculated from the independently specified baryonic mass distribution before bounded realization through the Frames mapping. It represents the gravitational availability supplied by the baryonic source and is the source acceleration entering the MQ realization fraction.

• ${\mathrm{𝑔}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$

  is the gravitational acceleration realized in the System Frame at galactocentric radius R after bounded realization of the baryonic gravitational availability. In the stationary galactic limit it is determined from ${\mathit{g}}_{\mathrm{bar}}$, ${\mathit{a}}_{\mathrm{MQ}}$, and the realization fraction without a galaxy-specific MQ fitting parameter.

• $\mathrm{\Phi b}$

  is the baryonic gravitational potential generated by the independently specified baryonic source distribution before bounded realization. Its spatial gradient determines the Newtonian baryonic acceleration ${\mathit{g}}_{\mathrm{bar}}$. The quantity is distinguished from the gravitational potential realized in the System Frame.

• ${\mathrm{𝑀}}_{\mathrm{bar}}\left(\mathrm{<}\mathrm{𝑅}\right)$

  is the total baryonic mass enclosed within galactocentric radius R. It is determined from the independently specified baryonic source distribution and supplies the Newtonian gravitational availability entering the MQ realization law.

• ${\mathrm{𝜌}}_{\mathrm{𝑏}}$

  is the baryonic mass density of the independently specified physical source distribution. In the weak-field construction it determines the baryonic potential through the Newtonian Poisson relation and remains distinct from quantities derived after bounded realization.

• ${\mathrm{𝑆}}_{\mathrm{WF}}^{\mathrm{MQ}}$

  is the MQ weak-field realization action whose Euler-Lagrange equations produce both the Newtonian baryonic source equation and the MQ bounded gravitational realization equation. Its realization dependence is fixed by ${\mathit{Q}}_{\mathrm{MQ}}$ and therefore by the pre-existing MQ galactic realization law.

• ${\mathrm{𝑆}}_{\mathrm{MQ}}^{\mathrm{dual}}$

  is the exact first-order dual form of the MQ spatial weak-field action. It replaces the primal nonlinear realization functional by the auxiliary field ${\mathcal{Y}}_{\mathrm{MQ}}^{\mathit{\mu }}$ and the dual function ${\mathcal{R}}_{\mathrm{MQ}}$. Eliminating the auxiliary field recovers the primal MQ action exactly.

• ${\mathrm{𝑆}}_{\mathrm{𝐾}}$

  is the timelike constraint action generated by ${\mathit{K}}_{\mathrm{MQ}}\left({\mathit{X}}_{\mathrm{MQ}}\right)$. Variation of ${\mathit{S}}_{\mathit{K}}$ with respect to its source-density multiplier imposes the physical mass-shell condition, while variation with respect to the action phase produces the conserved timelike source current.

• ${\mathrm{𝑇}}_{\mathrm{𝜇}\mathrm{𝜈}}^{\mathrm{𝐾}}$

  is the stress-energy tensor generated by metric variation of the timelike constraint action ${\mathit{S}}_{\mathit{K}}$. On the physical mass-shell branch it assumes the pressureless timelike form determined by the source-density multiplier and the four-velocity ${\mathit{u}}^{\mathit{\mu }}$.

• ${\mathrm{𝑔}}_{\mathrm{𝜇}\mathrm{𝜈}}$

  The dimensionless System Frame metric tensor in length coordinates. The Greek indices identify spacetime components. This is the unsuffixed notation used in the local geometric reconstruction; it does not introduce an independent primitive metric.

• ${\mathrm{𝑅}}_{\mathrm{𝜇}\mathrm{𝜈}}$

  The Ricci curvature tensor of the realized metric. Its components have inverse-square-length dimensions in length coordinates. It is distinct from a scalar radial distance and from the script realization structures.

• ${\mathrm{𝑇}}_{\mathrm{𝜇}\mathrm{𝜈}}$

  The realized stress–energy tensor. In the normalized local frame of the supplied construction its components are expressed in joules per cubic metre. Its source and metric conventions must be retained.

• ${\mathrm{𝜆}}_{\mathrm{mq}}$

  The dimensionless baseline cosmological coefficient obtained by normalizing the baseline cosmological term with the squared fundamental length. The label identifies the baseline comparison using the specified expansion input; it is distinct from the frame-specific cosmological terms.

• ${\mathrm{𝑀}}_{\mathrm{bar}}$

  The total baryonic source mass in the finite-source exterior circular-motion limit, expressed in kilograms. It is the asymptotic total of the enclosed baryonic mass and is distinct from the bound or effective mass.

• $\mathrm{𝑅}$

  The scalar contraction of the Ricci tensor of the realized metric, with inverse-square-length dimensions in length coordinates. It is distinct from scalar radius and from the script and fraktur realization structures.

• $\mathrm{𝑘}$

  The gravitational coupling coefficient in the original Einstein-equation notation used for the comparison. It has units of squared seconds per kilogram per metre. It is distinct from Boltzmann’s constant and from the dimensionless count product K.

• ${\mathrm{\Lambda }}_{\mathrm{mq}}$

  The baseline cosmological term in the comparison evaluated using the specified baseline expansion input. Its dimensions are inverse square length; multiplication by the squared fundamental length produces the dimensionless baseline coefficient. The capital Lambda is never script.

• ${\mathrm{𝑅}}^{\mathrm{N}}$

  The dimensionless normalized Ricci scalar obtained by multiplying the curvature scalar by the squared fundamental length. The upright N denotes normalization and is not a tensor index.

• ${\mathrm{𝑅}}_{\mathrm{𝜇}\mathrm{𝜈}}^{\mathrm{N}}$

  The dimensionless normalized Ricci tensor obtained by multiplying the Ricci tensor by the squared fundamental length. The upright N denotes normalization and the Greek indices remain component indices.

• ${\mathrm{𝑘}}^{\mathrm{N}}$

  The normalized gravitational coupling coefficient obtained by multiplying the original coefficient k by the squared fundamental length. It has units of metres times seconds squared per kilogram. The upright N denotes normalization.