Addendum 60 — Friedmann Partition Occurs After Recursive Throughput
Matter-Energy, Curvature, the Cosmological Constant, and the Nonprimitive Decomposition of H 2
Abstract
Addendum 59 established a conditional bridge from primitive recursive counting to the physical cosmological rate.
For a dominant cosmological transition class χ, realized through a representative subordinate peer closure a,
\[ H=f_a\delta_{A,\chi}\rho_{\zeta\mid\zeta-1,a}.\]
where
\[ f_a\]
is the physical clock calibration,
\[\delta_{A,\chi}=\frac12\ln\frac{A_{\zeta,k}}{A_{\zeta,i}}\]
is the logarithmic geometric step associated with one cosmological occurrence, and
\[\rho_{\zeta\mid\zeta-1,a}=\frac{N_\chi^\zeta}{N_C^{(a)}}\]
is the recursive throughput of enclosing cosmological occurrences per representative subordinate closure.
The R²D law gives
\[\rho_{\zeta\mid\zeta-1,a}=\frac{\Delta A_{\zeta-1,a,C}}{C_\zeta\lambda_\zeta}\]
Thus
\[ H^2=f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
This physical rate magnitude exists conceptually before the conventional Friedmann decomposition is assigned.
Define
\[ T_\zeta\equiv f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
The symbol
\[ T_\zeta\]
is introduced only as shorthand for the physically calibrated recursive throughput magnitude.
It is not a new primitive R²D quantity.
Under the conditional bridge of Addendum 59,
\[ T_\zeta=H^2\]
Conventional FLRW cosmology then represents this one rate magnitude as
\[ H^2=\frac{8\pi G}{3c^2}\epsilon-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\]
Define
\[ F_\epsilon=\frac{8\pi G}{3c^2}\epsilon\]
\[ F_k=-\frac{kc^2}{a^2}\]
and
\[ F_\Lambda=\frac{\Lambda c^2}{3}\]
Then
\[ T_\zeta=H^2=F_\epsilon+F_k+F_\Lambda\]
The three terms do not have equivalent physical or logical ownership.
The matter-energy contribution
\[ F\epsilon\]
is a source-property read.
The curvature contribution
\[ Fk\]
is a geometric read.
The cosmological-constant contribution
\[ F\Lambda\]
may be retained as a geometric constant or algebraically reassigned to an effective source density.
Indeed, define
\[\epsilon_\Lambda=\frac{\Lambda c^4}{8\pi G}\]
Then
\[\frac{8\pi G}{3c^2}\epsilon_\Lambda=\frac{\Lambda c^2}{3}\]
and the Friedmann equation becomes
\[ H^2=\frac{8\pi G}{3c^2}(\epsilon+\epsilon_\Lambda)-\frac{kc^2}{a^2}\]
Likewise define an effective curvature density
\[\epsilon_k=-\frac{3kc^4}{8\pi G a^2}\]
Then
\[\frac{8\pi G}{3c^2}\epsilon_k=-\frac{kc^2}{a^2}\]
and therefore
\[ H^2=\frac{8\pi G}{3c^2}(\epsilon+\epsilon_\Lambda+\epsilon_k)\]
Nothing about the physical value of H 2 has changed.
Only the bookkeeping partition has changed.
The same cosmological rate magnitude can therefore be represented as
source + geometry + constant geometry,
or as
sum of effective source densities.
The decomposition itself consequently cannot determine primitive ontology.
This has an immediate R²D implication.
One must not infer identities such as
\[ P_\zeta=\epsilon_\Lambda\]
\[ R_\zeta=\epsilon_m\]
or
\[ A_\zeta=\epsilon_\Lambda-\epsilon_m\]
merely because conventional cosmological terms have suggestive signs.
The R²D quantities
\[ P_\zeta,\quad R_\zeta,\quad A_\zeta\]
belong to the pre-gravitational causal architecture.
The Friedmann terms belong to a later gravitational physical projection.
Their decomposition is representation dependent.
Thus the correct ordering is
↓
gravitational physical projection
\[ H^2\]
↓
conventional Friedmann partition
\[ F_\epsilon+F_k+F_\Lambda\]
The central statement is therefore:
Central statement. The Friedmann equation does not expose three primitive causes of cosmological evolution. It partitions one already-defined physical rate magnitude into matter-energy, curvature, and cosmological-constant coordinates within the gravitational projection. Because that partition can be rearranged without changing \(H^2\), its individual terms cannot be identified with primitive R²D quantities without independent projection bridges.
And hence:
Friedmann partition occurs after recursive throughput.
1. Addendum 59 Produced One Physical Rate Magnitude
The R²D throughput bridge gives
\[ H=f_a\delta_{A,\chi}\rho_{\zeta\mid\zeta-1,a}.\]
Therefore,
\[ H^2=f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
This is one physical quantity.
No Friedmann partition has yet been assigned.
2. Define a Shorthand for Projected Recursive Throughput
Define
\[ T_\zeta\equiv f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
Then
\[ T_\zeta=H^2\]
The symbol is only shorthand.
It does not add a new primitive to R²D.
3. The R²D Law Gives Its Recursive Component
The throughput factor is
\[\rho_{\zeta\mid\zeta-1,a}=\frac{\Delta A_{\zeta-1,a,C}}{C_\zeta\lambda_\zeta}\]
Therefore,
\[ T_\zeta=f_a^2\delta_{A,\chi}^2\left(\frac{\Delta A_{\zeta-1,a,C}}{C_\zeta\lambda_\zeta}\right)^2\]
4. This Expression Is Already Physically Calibrated
The dimensionless recursive relation has acquired:
\[ f_a\]
a temporal calibration,
and
\[\delta_{A,\chi}\]
a geometric calibration.
Thus
\[ T_\zeta\]
belongs to the physical projection layer.
5. G Has Not Yet Been Required to Define the Throughput Decomposition
The recursive throughput itself is defined without G.
The geometric area step is defined without G.
The clock frequency is separately calibrated.
Only when the same physical rate is expressed through gravitational source variables does G enter explicitly.
6. Now Introduce the Friedmann Representation
For an FLRW projection,
\[ H^2=\frac{8\pi G}{3c^2}\epsilon-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\]
Therefore,
\[ T_\zeta=\frac{8\pi G}{3c^2}\epsilon-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\]
7. Friedmann Is a Partition of Tζ
Define
\[\begin{aligned}F_\epsilon&=\frac{8\pi G}{3c^2}\epsilon,\\ F_k&=-\frac{kc^2}{a^2},\end{aligned}\]
and
\[ F_\Lambda=\frac{\Lambda c^2}{3}\]
Then
\[ T_\zeta=H^2=F_\epsilon+F_k+F_\Lambda\]
The Friedmann equation partitions the rate magnitude among physical coordinates.
8. The Three Terms Are Not Ontologically Homogeneous
The notation makes them look like three additive terms of one type.
They are not.
They enter the physical theory from different descriptive categories.
9. Fε Is a Source-Property Read
The term
\[ F_\epsilon=\frac{8\pi G}{3c^2}\epsilon\]
contains the physical energy-density read of cosmological peer organization.
It also contains the gravitational conversion G.
Thus it is already downstream of both count realization and physical dimensionalization.
10. Energy Density Is Not Primitive Multiplicity
The physical quantity
\[\epsilon\]
must not be identified with
\[ W_\zeta\]
\[ S_\zeta\]
\[ P_\zeta\]
or
\[ R_\zeta\]
without an explicit physical bridge.
It is a mature physical source coordinate.
11. Fk Is a Geometric Read
The curvature term is
\[ F_k=-\frac{kc^2}{a^2}\]
It belongs to the spatial geometry of the FLRW projection.
It is not conventionally a material source term.
12. Distinguish k From k/a2
The quantity
\[ k\]
labels the spatial-curvature class of the FLRW projection.
The quantity
\[\frac{k}{a^2}\]
is the corresponding physical curvature scale.
Thus:
\[ k\ne\frac{k}{a^2}\]
Their boundary ownership differs.
13. k Is Primarily a Projection-Class Label
Within a fixed FLRW model,
\[ k\]
does not evolve as the cosmological state evolves.
It specifies which geometric projection class has been selected.
Thus it behaves more like a classification parameter than a dynamical macrostate coordinate.
14. The Curvature Scale Is State Dependent
By contrast,
\[\frac{k}{a^2}\]
changes when
\[ a\]
changes.
Therefore the physical curvature read is state dependent even when the curvature label k is fixed.
15. FΛ Has Different Ownership Again
The cosmological-constant term is
\[ F_\Lambda=\frac{\Lambda c^2}{3}\]
For constant Λ,
\[\Lambda\]
does not change with the cosmological state.
It acts as a fixed rate-squared contribution within the chosen gravitational projection.
16. A Constant Λ Does Not Distinguish Macrostates
If
\[\Lambda_i=\Lambda_k\]
then Λ itself cannot distinguish the transition
\[ i\to k\]
Its macrostate factorization is therefore trivial.
17. This Suggests Projection-Level Ownership
A truly constant
\[\Lambda\]
may be more naturally associated with the gravitational projection class than with the changing cosmological macrostate itself.
This does not establish its R²D origin.
It identifies only its logical status within the physical projection.
18. The Friedmann Sum Therefore Mixes Logical Types
The right-hand side combines:
state-dependent source or geometry versus state-independent parameter
state-dependent source or geometry versus state-independent parameter
and
state-dependent source or geometry versus state-independent parameter
This should immediately caution against primitive term-by-term R²D identification.
19. The Cosmological Constant Can Be Reassigned Algebraically
Define the effective vacuum energy density
\[\epsilon_\Lambda=\frac{\Lambda c^4}{8\pi G}\]
Then
\[\frac{8\pi G}{3c^2}\epsilon_\Lambda=\frac{\Lambda c^2}{3}\]
20. Friedmann Can Then Be Written With Λ as Source Density
The same equation becomes
\[ H^2=\frac{8\pi G}{3c^2}(\epsilon+\epsilon_\Lambda)-\frac{kc^2}{a^2}\]
The physical rate has not changed.
Only the placement of Λ has.
21. Geometry Has Been Rewritten as Source Bookkeeping
In this representation,
\[\Lambda\]
has moved from an explicit geometric term into an effective energy-density term.
Therefore its algebraic location cannot determine its primitive causal status.
22. Curvature Can Be Rewritten the Same Way
Define
\[\epsilon_k=-\frac{3kc^4}{8\pi G a^2}\]
Then
\[\frac{8\pi G}{3c^2}\epsilon_k=-\frac{kc^2}{a^2}\]
23. The Entire Friedmann Equation Can Be Written as Effective Density
Therefore:
\[ H^2=\frac{8\pi G}{3c^2}(\epsilon+\epsilon_\Lambda+\epsilon_k)\]
All three contributions now appear algebraically as source densities.
24. This Does Not Turn Curvature Into Matter
The effective quantity
\[\epsilon_k\]
is a bookkeeping coordinate.
It does not mean that spatial curvature has become a material substance.
The transformation only demonstrates nonuniqueness of the partition.
25. The Same Is True of Vacuum-Energy Representation
Writing
\[\epsilon_\Lambda\]
does not by itself establish that a primitive material vacuum fluid is the ontological source of Λ.
It is a physically useful representation.
Primitive causality remains a separate question.
26. Principle — Friedmann Partition Nonuniqueness
The same physical quantity
\[ H^2\]
can be represented through multiple algebraically equivalent partitions.
Therefore:
\[\text{Friedmann placement}\ne\text{primitive origin}\]
27. R²D Requires Independent Projection Bridges
To identify any Friedmann contribution with a primitive R²D relation, one must separately derive the mapping.
No such identity follows merely from the form of the Friedmann equation.
28. Do Not Identify Λ With Possibility
The sign of
\[ F_\Lambda\]
may make
\[ F_\Lambda\]
look attractive.
But no count bridge has established that identity.
Therefore:
\[ P_\zeta\ne\Lambda\]
at the present stage.
29. Do Not Identify Matter Density With Retention
Likewise,
\[\epsilon_m\]
does not become
\[ R_\zeta\]
merely because matter often opposes accelerated expansion in a conventional dynamical interpretation.
Thus:
\[ R_\zeta\ne\epsilon_m\]
without derivation.
30. Do Not Build Aζ From Friedmann Signs
A decomposition such as
\[ A_\zeta\overset{?}{=}\epsilon_\Lambda-\epsilon_m\]
would import the gravitational projection back into the primitive thermodynamic layer.
That reverses the correct explanatory order.
31. The Primitive Asymmetry Already Exists Earlier
For a nonterminal cosmological boundary,
\(A_\zeta=P_\zeta-R_\zeta\)
is defined before energy density, curvature, Λ, or G have been introduced.
The Friedmann partition therefore cannot define it retroactively.
32. The Throughput Magnitude Is Prior to the Partition
The appropriate ordering is:
\[ A_\zeta\]
↓
structural branch
\[\chi\zeta\]
↓
recursive throughput
\[\rho_{\zeta\mid\zeta-1,a}\]
↓
geometric and clock calibration
\[ H^2=T_\zeta\]
↓
Friedmann partition
\[ F_\epsilon+F_k+F_\Lambda\]
33. The Friedmann Terms Are Coordinates of the Physical Projection
They describe the rate magnitude in the mature gravitational language.
They are not the primitive count architecture from which the rate originated.
34. Apply the Physical-Read Factorization Criterion
Addendum 56 established that a physical observable belongs to the cosmological macrostate only when it factors through
\[\pi_\zeta\]
The same requirement applies term by term.
35. Energy Density Requires Macrostate Factorization
For energy density to be a cosmological macrostate coordinate, require
\[\epsilon\circ\Pi_G=\bar\epsilon_\zeta\circ\pi_\zeta\]
Then
\[\epsilon_{\zeta,i}\]
is well defined.
36. Curvature Requires Macrostate Factorization
Define the physical curvature read
\[\frac{k}{a^2}\]
For it to represent the R²D cosmological macrostate,
\[ K\circ\Pi_G=\bar K_\zeta\circ\pi_\zeta\]
37. k Itself Need Not Be a State Coordinate
Because k remains fixed within one projection class, it can label the physical geometry without distinguishing states within that class.
This is another reason to separate projection parameters from macrostate variables.
38. Constant Λ Factorizes Trivially
If
\[\Lambda\]
is fixed throughout the projection,
\[\Lambda[\Pi_G(\mu)]=\Lambda\]
for all
\[\mu\in\Omega_\zeta\]
Its factorization through πζ therefore carries no state information.
39. G Has Different Status Again
The gravitational constant
\[ G\]
does not label one cosmological macrostate rather than another.
It converts between physical source and gravitational geometric place-value dictionaries.
Thus:
\[ G\]
not
G = cosmological state variable.
40. The Friedmann Equation Mixes State Reads and Projection Constants
Its right-hand side therefore contains both:
state-dependent coordinates
and
\[\text{projection-defining constants}\]
This mixed ownership is another indication that the sum is downstream of primitive R²D structure.
41. Normalize the Partition
For
\[ H^2>0\]
define
\[\Omega_\epsilon=\frac{F_\epsilon}{H^2}\]
\[\Omega_k=\frac{F_k}{H^2}\]
and
\[\Omega_\Lambda=\frac{F_\Lambda}{H^2}\]
42. The Normalized Friedmann Budget Sums to One
Then
\[1=\Omega_\epsilon+\Omega_k+\Omega_\Lambda\]
This is the conventional normalized partition of the physical rate constraint.
43. These Ω Values Are Not R²D Multiplicity Fractions
The notation must not invite confusion.
They do not represent:
\[\frac{W_i}{W_{\mathrm{tot}}}\]
nor fractions of primitive possibility.
They are normalized contributions to the projected Friedmann rate equation.
44. The Cosmological Budget Is Therefore a Projected Budget
The conventional cosmic budget describes how
\[ H^2\]
is partitioned in one gravitational coordinate system.
R²D accepts its observational usefulness while assigning it downstream status.
45. A Different Algebraic Partition Gives a Different Budget
If Λ and curvature are rewritten as effective densities, the apparent category structure changes.
The same H 2 remains.
This reinforces that normalized budget categories are representation dependent.
46. The Whole Physical Rate Is More Invariant Than Its Partition
The quantity
\[ H^2\]
is common to the equivalent representations.
The decomposition
\[ F_\epsilon+F_k+F_\Lambda\]
is less primitive.
Thus the whole precedes its physical bookkeeping pieces in the R²D interpretation.
47. This Mirrors the General R²D Causal Inversion
R²D repeatedly finds that a stable whole-system relation is identified before mechanistic attribution among subordinate terms.
Friedmann cosmology displays the same architecture.
The physically constrained whole is
\[ H^2\]
The source partition is a later representation.
48. Matter-Energy Requires Its Own Derivation
The first downstream problem is therefore:
How do recursively organized peer boundaries acquire the physical source-density coordinate / εζ ?
This question cannot be answered by simply inserting mass density into P or R.
49. The Source Read Must Begin With Peer Coupling
Addendum 55 already supplied the bridge
\[\mu_\alpha\to m_\alpha\]
for a peer boundary.
A cosmological energy density must therefore emerge from the aggregate physical coupling read of the peer organization together with its projected support.
50. Curvature Requires a Different Derivation
The curvature term asks:
How does the enclosing geometric organization of the peer boundary acquire the physical / read k/a2 ?
This is a boundary-geometry problem, not a source-density problem.
51. Λ Requires a Third Derivation
The cosmological constant asks:
Why does the gravitational projection admit a state-independent inverse-length-squared / offset?
That question is logically separate from both matter-energy density and evolving spatial curvature.
52. These Problems Should Not Be Solved by One Renaming
The three physical terms have different boundary ownership.
Therefore one should not expect one primitive identity such as
\[ P-R\]
to decompose automatically into
\[\epsilon,\quad k,\quad\Lambda\]
Separate physical bridges are required.
53. Principle — Recursive Throughput Precedes Friedmann Partition
Define
\[ T_\zeta\equiv f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
When the R²D and FLRW projections describe the same physical rate,
\[ T_\zeta=H^2\]
The Friedmann equation then partitions this rate magnitude.
54. Principle — Friedmann Partition
The conventional physical decomposition is
\[ T_\zeta=H^2=F_\epsilon+F_k+F_\Lambda\]
This decomposition belongs to the gravitational projection.
55. Principle — Partition Categories Are Not Primitive Count Types
The terms
\[ F_\epsilon,\quad F_k,\quad F_\Lambda\]
do not correspond by definition to
\[ P,\quad R,\quad A\]
or any other primitive R²D count type.
Their identities must be independently mapped.
56. Principle — Algebraic Relocation Does Not Change the Physical Rate
The transformations
\[\frac{8\pi G}{3c^2}\epsilon_\Lambda=\frac{\Lambda c^2}{3}\]
and
\[\frac{8\pi G}{3c^2}\epsilon_k=-\frac{kc^2}{a^2}\]
leave
\[ H^2\]
unchanged.
Therefore the algebraic side on which a term is written does not establish primitive causal ownership.
57. Principle — Projection-Class Parameters and State Variables Must Be Distinguished
Quantities such as
\[ k,\quad\Lambda,\quad G\]
may remain fixed while the cosmological macrostate changes.
Quantities such as
\[\frac{k}{a^2}\]
may vary with the state.
These different ownership classes should not be collapsed.
58. Logical Status
\[\epsilon,\quad a,\quad k/a^2\]
Recursive throughput, geometric step, and physical clock calibration produce
\[ H^2=f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
New Shorthand
\[ T_\zeta\equiv H^2\]
within the conditional throughput projection.
This is not a new primitive quantity.
Imported Friedmann Relation
\[ H^2=\frac{8\pi G}{3c^2}\epsilon-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\]
New Interpretation
The Friedmann equation partitions the already-defined physical rate magnitude into conventional gravitational coordinates.
Algebraic Nonuniqueness
\[\epsilon_\Lambda=\frac{\Lambda c^4}{8\pi G}\]
and
\[\epsilon_k=-\frac{3kc^4}{8\pi G a^2}\]
allow the same physical rate to be written
\[ H^2=\frac{8\pi G}{3c^2}(\epsilon+\epsilon_\Lambda+\epsilon_k)\]
Consequence
The individual Friedmann terms cannot be identified with primitive R²D variables solely from their algebraic placement.
Not Established Here
This addendum does not establish:
- the R²D derivation of \(\epsilon\);
- the R²D derivation of \(k/a^2\);
- the R²D derivation of \(\Lambda\);
- \(P_\zeta=\epsilon_\Lambda\);
- \(R_\zeta=\epsilon_m\);
- \(A_\zeta=\epsilon_\Lambda-\epsilon_m\);
- \(C_\zeta=k\);
- curvature literally being an energy density;
- vacuum energy as the primitive ontology of \(\Lambda\);
- a complete derivation of Friedmann dynamics from primitive R²D counting.
59. The Complete Ordering
Primitive cosmological direction is defined first:
\[ A_\zeta=P_\zeta-R_\zeta.\]
A structural branch is selected:
\[ i_\zeta^\ast\to k_\zeta^\ast\]
The selected enclosing occurrence is realized through subordinate closure recurrence:
\[\rho_{\zeta\mid\zeta-1,a}\]
A geometric step is assigned:
\[\delta_{A,\chi}\]
A physical clock is calibrated:
\[ f_a\]
These produce one physical rate magnitude:
\[ T_\zeta=f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2=H^2\]
Only afterward does the gravitational projection write
\[ T_\zeta=\frac{8\pi G}{3c^2}\epsilon-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\]
Thus:
recursive throughput
↓
physical rate magnitude
↓
Friedmann partition.
60. Conclusion
The Friedmann equation is commonly read as a competition among physical causes.
Matter-energy contributes one term.
Spatial curvature contributes another.
The cosmological constant contributes a third.
Their sum determines the squared Hubble rate:
\[ T_\zeta=H^2=F_\epsilon+F_k+F_\Lambda\]
R²D begins earlier.
Addendum 59 identifies a single physically calibrated recursive throughput magnitude:
\[ H^2=f_a^2\delta_{A,\chi}^2\rho_{\zeta\mid\zeta-1,a}^2\]
When both descriptions refer to the same cosmological realization,
\[ T_\zeta=H^2\]
The Friedmann equation therefore partitions a rate magnitude that has already acquired physical meaning through recursive throughput, geometric projection, and clock calibration.
The partition does not possess one unique ontological form.
A cosmological constant term can be rewritten as an effective vacuum energy density:
\[\epsilon_\Lambda=\frac{\Lambda c^4}{8\pi G}\]
A curvature term can likewise be rewritten as an effective density:
\[\epsilon_k=-\frac{3kc^4}{8\pi G a^2}\]
Thus the same physical equation may be written as
\[ H^2=\frac{8\pi G}{3c^2}\epsilon-\frac{kc^2}{a^2}+\frac{\Lambda c^2}{3}\]
or
\[ H^2=\frac{8\pi G}{3c^2}(\epsilon+\epsilon_\Lambda+\epsilon_k)\]
The physical rate has not changed.
Only its coordinate decomposition has.
Therefore the location of a term on the “source” or “geometry” side cannot by itself establish primitive causal status.
This is decisive for R²D cosmology.
It prevents the primitive turbine
\(A_\zeta=P_\zeta-R_\zeta\) from being reconstructed after the fact by renaming conventional Friedmann terms.
Possibility is not dark energy by definition.
Retention is not matter density by definition.
Asymmetry is not their conventional difference.
Curvature is not primitive constraint merely because it enters with the opposite sign.
These mappings, if physically real, must be derived independently.
The correct explanatory order is therefore:
primitive count asymmetry
↓
structural realization
↓
recursive throughput
↓
physical rate magnitude
↓
Friedmann partition.
The whole physical rate relation occurs before its conventional partition into source and geometry.
The central statement is therefore:
Central statement. The Friedmann equation does not expose three primitive causes of cosmological evolution. It partitions one already-defined physical rate magnitude into matter-energy, curvature, and cosmological-constant coordinates within the gravitational projection. Because that partition can be rearranged without changing \(H^2\), its individual terms cannot be identified with primitive R²D quantities without independent projection bridges.
And hence:
Friedmann partition occurs after recursive throughput.