Addendum 56 — Cosmological Boundary Area Occurs Before the Scale Factor
Macrostate Factorization, Areal Place Value, and the Geometric Read of Cosmological Peer Reorganization
Abstract
Addendum 54 defined the primitive cosmological state space.
A cosmological boundary
\[B_\zeta\]
contains subordinate peer boundaries whose compatible joint realizations form the formal microstate domain
\[\Omega_\zeta\]
The cosmological boundary classifies those realizations through
\[\pi_\zeta:\Omega_\zeta\to\mathcal M_\zeta.\]
A cosmological macrostate
\[i \in \mathcal M_\zeta\]
is therefore a boundary-readable classification of compatible peer realization.
Its structural state is determined by occupancy,
\[i_\zeta^\ast=\arg\max_i c_{\zeta,i}.\]
\[i\]
A structural cosmological transition is
\[i_\zeta^\ast \to k_\zeta^\ast\]
Addendum 55 then introduced the gravitational projection
\[\Pi_G:\Omega_\zeta\to\Gamma_G.\]
through which an already-defined peer-boundary relation may acquire mass-energy and geometric coordinates.
The next question is now precise:
Which gravitational observable faithfully reads the cosmological macrostate transition?
R²D cannot answer this merely by identifying a conventional cosmological variable that changes monotonically with time.
A physical quantity can represent a cosmological macrostate only if it is insensitive to the unresolved formal microstate distinctions contained within that macrostate.
Let
\[Q:\Gamma_G\to\mathcal Q\]
be a candidate physical observable.
For Q to be a physical read of the R²D macrostate classification, there must exist a macrostate-level map
\[\bar Q_\zeta:\mathcal M_\zeta\to\mathcal Q\]
such that
\[Q\circ\Pi_G=\bar Q_\zeta\circ\pi_\zeta\]
Equivalently,
\[\pi_\zeta(\mu_\alpha)=\pi_\zeta(\mu_\beta)\Longrightarrow Q[\Pi_G(\mu_\alpha)]=Q[\Pi_G(\mu_\beta)]\]
The physical observable must therefore be constant over each cosmological macrostate fiber.
This is the macrostate physical-read factorization condition.
The gravitational development preceding this addendum provides a natural first candidate.
R²D maps a gravitational boundary onto physical area:
\[B_G \to A_G\]
Apply this to the cosmological boundary.
If physical area satisfies the factorization condition,
\[A\circ\Pi_G=\bar A_\zeta\circ\pi_\zeta\]
then every cosmological macrostate i possesses a well-defined areal read
\[A_{\zeta,i}=\bar A_\zeta(i)\]
A structural cosmological transition
\[i_\zeta^\ast \to k_\zeta^\ast\]
then acquires the geometric projection
\[A_{\zeta,i} \to A_{\zeta,k}\]
Area therefore becomes the first candidate gravitational read of cosmological peer reorganization.
Radius enters only afterward.
For an isotropic spherical area read,
\[r_{\zeta,i}=\sqrt{\frac{A_{\zeta,i}}{4\pi}}\]
Thus:
\[B_\zeta \to A_\zeta \to r\zeta\]
This preserves the R²D result that area occurs before radius.
The FLRW scale factor enters one step later.
For a fixed comoving spherical label χ, the conventional FLRW metric may be written
\[ds^2=-c^2dt^2+a^2(t)[d\chi^2+S_k^2(\chi)d\Omega^2]\]
The physical area of that fixed comoving two-sphere is
\[A_\chi(t)=4\pi a^2(t)S_k^2(\chi)\]
For the same comoving boundary at two physical epochs,
\[\frac{A_\chi(t_2)}{A_\chi(t_1)}=\frac{a^2(t_2)}{a^2(t_1)}\]
Therefore,
\[\frac{a(t_2)}{a(t_1)}=\sqrt{\frac{A_\chi(t_2)}{A_\chi(t_1)}}\]
If the cosmological R²D boundary area is identified with this isotropic geometric read, then
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
The scale factor is therefore not the primitive cosmological state.
It is the normalized linear geometric coordinate of a change in the projected area of that state.
No G is required for this geometric bridge.
The relation
\[A \propto a^2\]
is geometric.
The gravitational constant enters later, when the theory asks why the cosmological boundary-area read evolves as it does under the physical source projection of the peer organization.
The causal sequence is therefore
\[i_\zeta^\ast \to k_\zeta^\ast\]
↓
\[A_{\zeta,i} \to A_{\zeta,k}\]
↓
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
and only after a physical time coordinate is introduced,
\[H=\frac{\dot a}{a}=\frac12\frac{d\ln A_\zeta}{dt}\]
Thus
\[H\]
is a physical-time rate of projected boundary-area change.
It is not primitive cosmological asymmetry:
\[H\ne A_\zeta=P_\zeta-R_\zeta\]
Nor is it the terminal orientation:
\[H\ne \Xi_\star\]
The central statement is therefore:
the cosmological scale factor does not define the cosmological state. Under an isotropic gravitational
And hence:
cosmological boundary area occurs before the scale factor.
1. The Projection Problem Is Now Well Defined
Addendum 54 established the cosmological structural transition
\[i_\zeta^\ast \to k_\zeta^\ast\]
Addendum 55 established that this transition may acquire gravitational physical coordinates.
The unresolved problem is therefore not whether a physical projection exists.
It is which observable faithfully represents the R²D macrostate.
2. A Physical Observable Cannot Be Chosen by Association Alone
Suppose some gravitational quantity
\[Q\]
changes during cosmological evolution.
That correlation does not establish that Q represents the cosmological macrostate.
The quantity may instead depend on subordinate physical distinctions that remain unresolved by
\[\pi_\zeta\]
A stronger criterion is required.
3. Define the Physical Projection
Let
\(\Pi_G:\Omega_\zeta\to\Gamma_G\) maps compatible joint peer realizations into their gravitational physical representation.
An element
\[\mu \in \Omega_\zeta\]
may therefore acquire physical coordinates
\[\Pi_G (\mu) \in \Gamma_G\]
4. Define a Candidate Physical Observable
Let
\[Q:\Gamma_G\to\mathcal Q\]
The composite
\[Q ∘ \Pi_G\]
assigns a physical observable to every formal cosmological microstate.
5. The Observable Must Respect the Macrostate Classification
Suppose
\[\mu_\alpha , \mu_\beta \in \Omega_\zeta\]
are different compatible joint peer realizations but satisfy
\[\pi_\zeta (\mu_\alpha ) = \pi_\zeta (\mu_\beta ) = i\]
If Q is to represent macrostate i, then it must satisfy
\[Q[\Pi_G (\mu_\alpha )] = Q[\Pi_G (\mu_\beta )]\]
Otherwise Q distinguishes physical information that Bζ does not classify as a macrostate distinction.
6. Principle — Macrostate Physical-Read Factorization
A physical observable Q is a valid physical coordinate of the R²D macrostate classification only if there exists
\[\bar Q_\zeta:\mathcal M_\zeta\to\mathcal Q\]
such that
\[Q\circ\Pi_G=\bar Q_\zeta\circ\pi_\zeta\]
This is the factorization condition.
7. The Factorization Condition Can Be Written as a Commuting Diagram
The required relation is
\[\begin{array}{ccc}\Omega_\zeta&\xrightarrow{\Pi_G}&\Gamma_G\\\pi_\zeta\downarrow&&\downarrow Q\\\mathcal M_\zeta&\xrightarrow{\bar Q_\zeta}&\mathcal Q\end{array}\]
Both routes must give the same physical read.
8. This Is Stronger Than Correlation
A variable may correlate statistically with macrostate classification without satisfying
\[Q\circ\Pi_G=\bar Q_\zeta\circ\pi_\zeta\]
Such a variable is not yet a faithful macrostate coordinate.
Factorization requires invariance across the complete macrostate fiber.
9. This Provides a General Test for R²D Projection
The criterion is not unique to cosmology.
Any proposed physical reading of an R²D macrostate should satisfy the same condition.
Thus the addendum introduces a general projection test, specialized here to cosmology.
10. Gravitational Area Is the Natural First Candidate
Earlier gravitational development established the mapping
\[B_G \to A_G\]
The boundary itself therefore has an established candidate spatial read.
This makes area a natural observable to test against the cosmological macrostate classification.
11. Define the Cosmological Area Observable
Let
\[A:\Gamma_G\to\mathbb R_{>0}\]
assign the relevant gravitational boundary area to a projected peer realization.
For area to represent the cosmological macrostate,
\[A\circ\Pi_G=\bar A_\zeta\circ\pi_\zeta\]
must hold.
12. Area Must Be Constant Across a Macrostate Fiber
If
\[\pi_\zeta(\mu_\alpha)=\pi_\zeta(\mu_\beta)=i\]
then the area criterion requires
\[A[\Pi_G(\mu_\alpha)]=A[\Pi_G(\mu_\beta)]\]
Only then does macrostate i possess one well-defined area read.
13. Define the Macrostate Area
When factorization holds, define
\[A_{\zeta,i}=\bar A_\zeta(i)\]
The physical area is now indexed to the R²D cosmological macrostate rather than to an unresolved subordinate realization.
14. The Structural State Acquires a Boundary Area
For the realized structural macrostate
\[i_\zeta^\ast\]
define
\[A_\zeta^\ast=A_{\zeta,i_\zeta^\ast}\]
This is the area read of the currently realized cosmological organization.
15. Structural Change Can Now Acquire a Geometric Read
If
\[i_\zeta^\ast \to k_\zeta^\ast\]
then
\[A_{\zeta,i} \to A_{\zeta,k}\]
The R²D structural transition has now been physically projected.
16. Area Change Is Not the Primitive Transition
The primitive transition remains
\[i_\zeta^\ast \to k_\zeta^\ast\]
The area change is its geometric read.
Thus:
structural change → area change,
not the reverse.
17. Area Does Not Cause the Cosmological Reorganization
The physical quantity
\[A_\zeta\]
does not generate
\[A_\zeta=P_\zeta-R_\zeta.\]
The primitive direction remains upstream in the count architecture.
18. The Boundary Owns the Area Read
The area belongs to the projected enclosing boundary.
It should not be interpreted as the sum of independently primitive subordinate peer areas unless such a mapping is separately established.
Thus:
Aζ = boundary-level physical read.
19. Area Comes Before Radius
For an isotropic spherical projection,
\[r_{\zeta,i}=\sqrt{\frac{A_{\zeta,i}}{4\pi}}\]
Therefore:
\[A_{\zeta,i} \to r_{\zeta,i}\]
The radius is derived from the boundary area.
20. The Radius Is Not a Primitive Container
R²D does not begin with a pre-existing sphere of radius r containing the peer boundaries.
It begins with the count boundary and its peer realization domain.
The radius appears only after gravitational projection.
21. Cosmological Distinguishability Can Then Acquire a Radial Read
A primitive count distinction
\[\lambda_{\zeta,i\to k}\]
may now acquire the physical read
\[\lambda_{\zeta,i\to k} \longrightarrow r_{\zeta,k} - r_{\zeta,i}\]
This is a candidate projection relation, not an identity of primitives.
22. This Preserves the Existing Gravity Dictionary
The resulting hierarchy remains
boundary → area,
and
distinguishability → radial distinction.
Cosmology does not require a different gravitational ontology.
23. The Scale Factor Must Come After the Area Read
The cosmological scale factor is already a geometric coordinate.
Therefore it cannot define the primitive R²D macrostate if the aim is to derive cosmological geometry from R²D.
It must enter after
\[A_{\zeta,i}\]
24. Import the FLRW Geometric Relation
For a homogeneous and isotropic FLRW projection,
\[ds^2=-c^2dt^2+a^2(t)[d\chi^2+S_k^2(\chi)d\Omega^2]\]
This is an imported mature cosmological relation.
It is not derived here from primitive R²D counting.
25. A Fixed Comoving Sphere Has Physical Area
For fixed comoving radial label
\[\chi\]
the associated physical two-sphere has area
\[A_\chi(t)=4\pi a^2(t)S_k^2(\chi)\]
The factor
\[S_k(\chi)\]
is constant when the same comoving boundary is compared at different epochs.
26. Area Ratios Therefore Determine Scale-Factor Ratios
For the same comoving boundary,
\[\frac{A_\chi(t_2)}{A_\chi(t_1)}=\frac{a^2(t_2)}{a^2(t_1)}\]
Therefore:
\[\frac{a(t_2)}{a(t_1)}=\sqrt{\frac{A_\chi(t_2)}{A_\chi(t_1)}}\]
27. This Is the Candidate R²D Scale-Factor Bridge
If the R²D cosmological area read
\[A_{\zeta,i}\]
maps onto the corresponding FLRW boundary area, then
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
This is a physical mapping.
28. The Absolute Scale-Factor Normalization Remains Conventional
One may choose
\[a(t_0)=1\]
Likewise, only area ratios are required for the scale-factor bridge.
No absolute primitive value of a is needed.
29. The Scale Factor Is a Linearized Area Coordinate
Because
\[A \propto a^2\]
the scale factor is proportional to the square root of the projected boundary area:
\[a\propto\sqrt{A_\zeta}\]
Thus the scalar spatial place value introduced previously can be written
\[L_\zeta\propto\sqrt{A_\zeta}\]
30. The Earlier Place-Value Result Is Therefore Recovered
The earlier provisional relation
\[a \propto L_\zeta\]
becomes more specific:
\[L_\zeta\propto r_\zeta\propto\sqrt{A_\zeta}\]
Spatial place value is therefore the linear coordinate derived from the boundary-area projection.
31. Cosmological Expansion Now Has a Precise R²D Ordering
Primitive:
\[i_\zeta^\ast \to k_\zeta^\ast\]
Geometric projection:
\[A_{\zeta,i} \to A_{\zeta,k}\]
Normalized linear read:
\[a_i \to a_k\]
Thus:
peer reorganization → boundary-area change → scale-factor change.
32. Expansion Is Not the Primitive Cause
The increase
\[a_k > a_i\]
does not cause
\[i \to k\]
It is the physical geometric representation of the structural change under the stated projection.
33. Expansion Is Not Recursive Carry
Suppose the count boundary remains
\[B_\zeta\]
while its structural state changes:
\[i_\zeta^\ast \to k_\zeta^\ast\]
That is within-boundary structural evolution.
It may project as cosmological expansion.
34. Boundary Replacement Is Different
Recursive carry changes the count boundary:
\[B_\zeta\to B_{\zeta+1}\]
This defines a new state domain and a new classification.
It is not equivalent to
\[a_i \to a_k\]
35. Therefore Scale-Factor Growth Does Not Count Recursive Levels
Even very large growth in
\[a\]
need not imply a change in recursive boundary index.
Thus:
\[\ln\frac{a_k}{a_i}\ne\text{recursive depth}\]
by definition.
36. Inflationary E-Folds Remain Geometric
The number of e-folds is
\[N=\ln\frac{a_k}{a_i}\]
Under the area projection,
\[N=\frac12\ln\frac{A_{\zeta,k}}{A_{\zeta,i}}\]
This is a logarithmic geometric ratio.
It is not automatically a count of recursive closures.
37. No G Is Required for the Area-to-Scale-Factor Bridge
The relation
\[a\propto\sqrt{A}\]
is purely geometric.
Therefore:
macrostate → area → a
does not itself require the gravitational constant.
38. This Is Not a Contradiction With Addendum 55
Addendum 55 identified G as part of the physical conversion between mass-energy and gravitational spatial geometry.
The present bridge asks a different question:
Given an already-defined geometric boundary area, how is its isotropic linear scale represented?
That step is geometric and does not require another G.
40. Physical Time Must Also Be Introduced Separately
The discrete structural relation
\[i\to k\]
does not require a physical time coordinate.
Nor does the area ratio
\[A_k/A_i\]
A rate requires an additional physical clock calibration.
41. The Hubble Parameter Is Therefore Later Than the Scale-Factor Ratio
Once physical time
\[t\]
has been independently defined,
\[H=\frac{\dot a}{a}\]
Using
\[a\propto\sqrt{A_\zeta}\]
we obtain
\[H=\frac12\frac{d\ln A_\zeta}{dt}\]
42. Hubble Expansion Is an Area-Rate Read
The Hubble parameter therefore measures the logarithmic physical-time rate of projected cosmological boundary-area change.
Schematically,
H = physical rate read of Aζ .
43. H Is Not the Primitive Asymmetry
The cosmological turbine remains
\[A_\zeta=P_\zeta-R_\zeta.\]
No relation derived here establishes
\[A_\zeta = H\]
Therefore:
\[H\ne A_\zeta\]
44. H Is Not the Terminal Orientation
Likewise,
\[H\ne \Xi_\star\]
The terminal orientation belongs to the unresolved nonclosure of the largest realized path.
The Hubble parameter is a downstream physical rate.
45. The Occurrence Coordinate Is Not Cosmic Time
The R²D occurrence coordinate
\[\tau = \ln \nu\]
is an occurrence statistic.
It is not automatically the physical time coordinate appearing in
\[H\]
Thus:
\[\tau_\zeta \ne t\]
A physical clock bridge remains separately required.
46. Redshift Comes After the Geometric Bridge
Standard cosmology gives
\[1+z=\frac{a_0}{a_e}\]
Using the area relation,
\[1+z=\sqrt{\frac{A_{\zeta,0}}{A_{\zeta,e}}}\]
This is a downstream physical propagation relation.
47. Radiation Does Not Define the Cosmological Area
The redshift relation does not imply that radiation frequency creates
\[A_\zeta\]
or a.
Instead:
boundary-area evolution → scale-factor evolution → radiation redshift.
This reverses the earlier provisional CMB-first ordering.
48. The CMB Becomes a Historical Read of Area Evolution
For adiabatic radiation,
\[T_{\mathrm{CMB}}\propto a^{-1}\]
Therefore,
\[T_{\mathrm{CMB}}\propto A_\zeta^{-1/2}\]
The CMB records the cosmological geometric evolution.
It does not define the primitive cosmological state.
49. The Area Need Not Be a Cosmological Horizon Area
The physical boundary used in this mapping need not be:
• a Hubble sphere; • a particle horizon; • an event horizon; • a de Sitter horizon; • or the edge of the universe.
The scale-factor relation requires only a consistent boundary read participating in the isotropic cosmological projection.
50. A Fixed Comoving Boundary Is Sufficient for the Geometric Ratio
For fixed
\[\chi\]
the conventional FLRW area satisfies
\[A_\chi \propto a^2\]
No claim that the comoving sphere is an ontological terminal boundary is required.
51. Observable Horizon and Count Boundary Remain Distinct
An observational horizon may determine which regions are currently readable.
That need not coincide with the R²D count boundary owning the macrostate classification.
Thus:
observational horizon ≠ Bζ
by definition.
52. Isotropy Is a Physical Projection Condition
A single scalar
\[a(t)\]
describes isotropic spatial scaling.
Therefore the mapping
\[a\propto\sqrt{A_\zeta}\]
requires an isotropic physical projection.
R²D does not derive isotropy merely by defining a cosmological macrostate.
53. Without Isotropy, One Scalar Area May Be Insufficient
If the gravitational projection is anisotropic, then the structural state may require multiple directional scale reads or a tensorial spatial description.
In that case,
\[a(t)\]
would not fully represent the projected macrostate transition.
54. The Scale-Factor Bridge Is Therefore Conditional
The statement
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
is a conditional projection theorem requiring:
- a valid gravitational projection; 2. area factorization through the cosmological macrostate classification; 3. comparison of the corresponding boundary read;
- isotropic FLRW-type spatial scaling.
55. Principle — Physical Macrostate Read
A physical variable can represent an R²D macrostate only when it factors through the macrostate classification:
\[Q\circ\Pi=\bar Q\circ\pi\]
This prevents subordinate physical detail from being mistaken for the boundary-level state.
56. Principle — Cosmological Area Factorization
Cosmological boundary area is a valid physical macrostate read only if
\[A\circ\Pi_G=\bar A_\zeta\circ\pi_\zeta\]
Then
\[A_{\zeta,i}=\bar A_\zeta(i)\]
is well defined.
57. Principle — Area Read of Structural Transition
When area factorization holds,
\[i_\zeta^\ast \to k_\zeta^\ast\]
has the geometric representation
\[A_{\zeta,i} \to A_{\zeta,k}\]
The area change is the physical read of the structural transition, not its primitive cause.
58. Principle — Area Before Cosmological Radius
Under an isotropic spherical projection,
\[r_{\zeta,i}=\sqrt{\frac{A_{\zeta,i}}{4\pi}}\]
Thus cosmological radius remains downstream of boundary area.
59. Principle — Area Before the Scale Factor
Under a fixed-boundary FLRW projection,
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
The scale factor is therefore a normalized linear read of cosmological boundary-area change.
60. Principle — Hubble Rate Is a Later Physical-Time Read
Once a physical clock coordinate has been supplied,
\[H=\frac12\frac{d\ln A_\zeta}{dt}\]
This rate does not define the primitive R²D asymmetry.
61. Logical Status
Canonical R²D
The cosmological boundary classifies compatible joint peer realizations into macrostates:
\[\pi_\zeta:\Omega_\zeta\to\mathcal M_\zeta.\]
Addendum 54
The structural cosmological state is an occupancy-defined organization of those peer boundaries.
Addendum 55
A gravitational projection
\(\Pi_G:\Omega_\zeta\to\Gamma_G\) assigns physical source and geometric reads to the already-defined peer relation.
New General Projection Criterion
A physical observable Q represents an R²D macrostate only if
\[Q\circ\Pi_G=\bar Q_\zeta\circ\pi_\zeta\]
New Cosmological Candidate
Boundary area is proposed as the first gravitational observable to satisfy this factorization:
\[A\circ\Pi_G=\bar A_\zeta\circ\pi_\zeta\]
Conditional Area-to-Scale-Factor Mapping
Under isotropic FLRW projection,
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
Imported Physical Relations
The FLRW metric,
\[A_\chi=4\pi a^2S_k^2(\chi).\]
the Hubble definition,
\[H=\frac{\dot a}{a}.\]
and cosmological redshift,
\[1+z=\frac{a_0}{a_e}\]
are conventional physical relations.
They are not derived here from primitive R²D counting.
Not Established Here
This addendum does not establish:
- that cosmological area actually satisfies the factorization condition;
- the unique physical boundary whose area represents Bζ ;
- that this area is a cosmological horizon area;
- isotropy from primitive R²D counting;
- a physical clock from the occurrence coordinate;
- Aζ = H ;
- \(\Xi_\star=H\);
- the rate at which Aζ evolves;
- the source relation responsible for that rate;
- the Friedmann equations from primitive R²D counting;
- or the identities of ρ, p, k , and Λ within the R²D turbine.
These are downstream problems.
62. The Full Cosmological Projection Sequence
The cosmological peer boundaries are defined first:
\[{B_{\zeta-1,\alpha} }\]
Their compatible joint realization defines
\[\Omega_\zeta\]
The cosmological boundary classifies those realizations:
\[\Omega_\zeta\xrightarrow{\pi_\zeta}\mathcal M_\zeta\]
Occupancy determines the structural macrostate:
\[i_\zeta^\ast\]
Enclosing possibility and retention bias a structural transition:
\[A_\zeta=P_\zeta-R_\zeta.\]
The gravitational projection maps the state into a physical domain:
\[\Omega_\zeta\xrightarrow{\Pi_G}\Gamma_G\]
If area factors through the macrostate classification,
\[A\circ\Pi_G=\bar A_\zeta\circ\pi_\zeta\]
then the structural transition receives an area read:
\[i_\zeta^\ast \to k_\zeta^\ast\]
↓
\[A_{\zeta,i} \to A_{\zeta,k}\]
Under isotropic spatial projection,
\[r_\zeta=\sqrt{\frac{A_\zeta}{4\pi}}\]
and
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
Only after a physical time coordinate is defined does the rate become
\[H=\frac12\frac{d\ln A_\zeta}{dt}\]
Thus the complete ordering is
peer organization
↓
structural macrostate
↓
boundary area
↓
areal radius
↓
scale factor
↓
Hubble rate.
63. Conclusion
R²D cosmology now possesses a defined pre-gravitational state architecture.
The cosmological boundary counts compatible joint realizations of subordinate peer boundaries.
It classifies those realizations as cosmological macrostates.
A structural transition is therefore defined before gravitational geometry:
\[i_\zeta^\ast \to k_\zeta^\ast\]
The remaining problem is to identify a physical observable that genuinely belongs to that macrostate classification.
Correlation is insufficient.
The observable must be insensitive to all formal microstate differences that the cosmological boundary classifies as the same state.
This requirement is expressed by the physical-read factorization condition:
\[Q\circ\Pi_G=\bar Q_\zeta\circ\pi_\zeta\]
For cosmology, boundary area is the natural first candidate.
If
\[A\circ\Pi_G=\bar A_\zeta\circ\pi_\zeta\]
then macrostate i possesses a well-defined physical area
\[A_{\zeta,i}\]
The structural transition
\[i_\zeta^\ast \to k_\zeta^\ast\]
then projects as
\[A_{\zeta,i} \to A_{\zeta,k}\]
Radius is defined only afterward:
\[r_{\zeta,i}=\sqrt{\frac{A_{\zeta,i}}{4\pi}}\]
Under an isotropic FLRW projection, a fixed comoving boundary satisfies
\[A_\chi \propto a^2\]
so that
\[\frac{a_k}{a_i}=\sqrt{\frac{A_{\zeta,k}}{A_{\zeta,i}}}\]
The scale factor is therefore not the cosmological state.
It is not the primitive driver of that state.
It is not recursive depth.
It is the normalized linear geometric coordinate of a boundary-area change that has already been produced by a transition in cosmological peer organization.
No G is required for this geometric conversion.
G becomes necessary when the source read of the peer organization is related to the physical evolution of its geometry.
That problem comes next.
The central statement is therefore:
a physical cosmological coordinate can represent an R²D macrostate only if it factors through the bo
And hence:
cosmological boundary area occurs before the scale factor.