Addendum 47 — Horizon Count–Clock Calibration Persists Across Boundary Replacement
Abstract
Addendum 46 established that the Nariai limit is a boundary-clock replacement problem.
As the Schwarzschild–de Sitter black-hole and cosmological horizon branches coalesce, their multiplicity gap vanishes,
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}\longrightarrow0,\]
while the ordinary static-patch clock also degenerates:
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
The normalization defining that clock vanishes at the same boundary,
\[N_{\mathrm{stat}}\longrightarrow0.\]
The replacement Nariai landscape nevertheless possesses the finite boundary-appropriate phase-frequency read
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Thus neither the old clock nor the old two-horizon multiplicity classification survives unchanged.
The present addendum asks the next R²D question:
If the physical count and physical clock both change under boundary replacement, does a deeper relation remain coherent?
Pure de Sitter provides an independent horizon calibration within the same Λ-defined gravitational family.
Its horizon radius is
\[r_{\mathrm{dS}}=\sqrt{\frac3\Lambda},\]
its horizon multiplicity coordinate is
\[S_{\mathrm{dS}}=\frac{3\pi}{\Lambda\ell_P^2},\]
and its Gibbons–Hawking phase-frequency read is
\[\omega_{\mathrm{dS}}=\frac{c}{2\pi}\sqrt{\frac\Lambda3}.\]
The replacement Nariai horizon has
\[r_N=\frac1{\sqrt\Lambda},\]
\[S_N=\frac\pi{\Lambda\ell_P^2},\]
and
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Therefore,
\[S_N=\frac13S_{\mathrm{dS}},\]
while
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}}.\]
Neither physical coordinate is invariant.
But their combined relation is:
\[S_N\omega_N^2=S_{\mathrm{dS}}\omega_{\mathrm{dS}}^2.\]
Explicitly,
\[S\omega^2=\frac{c^2}{4\pi\ell_P^2}=\frac{c^5}{4\pi G\hbar}.\]
Define the dimensionless gravitational count–clock coupling
\[\alpha_G^\omega\equiv\frac{G\hbar\omega^2}{c^5}.\]
Then both horizon calibrations satisfy
\[4\pi\alpha_G^\omega S=1.\]
This relation is not introduced here as a new primitive R²D law.
It follows algebraically whenever two physical calibrations hold:
\[S=\frac{\pi r^2}{\ell_P^2},\]
and
\[\omega=\frac{c}{2\pi r}.\]
Its significance is instead relational.
Pure de Sitter and Nariai possess different radii, different horizon multiplicities, different physical clocks, and different boundary classifications. Yet both realize the same dimensionless count–clock calibration.
Thus boundary replacement need not preserve either physical coordinate separately.
It can preserve the relation connecting them.
This is the R²D meaning of replacement coherence:
state identity changes; relational organization persists.
The result is not a conservation law along the full Schwarzschild–de Sitter family.
It does not imply
\[\frac{d}{dM}(S\omega^2)=0.\]
Nor does it imply that pure de Sitter is the state immediately preceding Nariai.
Instead, the result identifies a common gravitational calibration class realized by two different horizon landscapes.
Therefore:
horizon count–clock calibration persists across boundary replacement.
More precisely:
boundary replacement can change both the physical horizon count and the physical clock while pres… [PDF line clipped; author review required.]
1. Addendum 46 Left an Invariance Question
Addendum 46 established that the ordinary Schwarzschild–de Sitter clock does not survive the Nariai boundary unchanged.
The ordinary static-patch frequencies satisfy
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
The normalization defining the old clock also satisfies
\[N_{\mathrm{stat}}\longrightarrow0.\]
A replacement Nariai clock is instead read as
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Thus physical frequency itself is not what remains unchanged.
The next question is therefore not
Which clock is the true clock?
It is:
What relation, if any, remains coherent when the clock changes?
2. R²D Does Not Require Physical Coordinates to Survive Replacement
R²D assigns physical coordinates relative to count boundaries.
A change of boundary classification can therefore change:
\[r,\]
\[A,\]
\[S,\]
\[\omega,\]
\[T,\]
and other dimensional reads.
No contradiction follows merely because these coordinates do not remain numerically constant.
A deeper invariant, if one exists, should be sought in the relation among the reads rather than in their individual values.
3. Recursive Replacement Preserves Relation More Naturally Than Identity
The R²D architecture does not require subordinate realization identities to survive recursive reclassification.
What can survive is relational organization.
Schematically,
\[\mathcal R_{\mathrm{old}}\longrightarrow\mathcal R_{\mathrm{new}},\]
while
relational form
may remain coherent even when the individual physical realizations differ.
This is the appropriate standard for the de Sitter–Nariai comparison.
4. Pure de Sitter Provides a Reference Horizon Calibration
For pure de Sitter space,
\[M=0.\]
The cosmological horizon radius is
\[r_{\mathrm{dS}}=\sqrt{\frac3\Lambda}.\]
Its spherical boundary area is
\[A_{\mathrm{dS}}=4\pi r_{\mathrm{dS}}^2.\]
Therefore,
\[A_{\mathrm{dS}}=\frac{12\pi}{\Lambda}.\]
5. The de Sitter Horizon Multiplicity Coordinate
Using the horizon multiplicity calibration,
\[S=\frac A{4\ell_P^2},\]
we obtain
\[S_{\mathrm{dS}}=\frac{12\pi/\Lambda}{4\ell_P^2}.\]
Thus,
\[S_{\mathrm{dS}}=\frac{3\pi}{\Lambda\ell_P^2}.\]
And therefore,
\[W_{\mathrm{dS}}=\exp\!\left(\frac{3\pi}{\Lambda\ell_P^2}\right).\]
6. The de Sitter Horizon Has a Thermal Phase-Clock Read
Define the de Sitter Hubble scale
\[H_{\mathrm{dS}}=c\sqrt{\frac\Lambda3}.\]
The Gibbons–Hawking thermal phase-frequency read is
\[\omega_{\mathrm{dS}}=\frac{H_{\mathrm{dS}}}{2\pi}.\]
Therefore,
\[\omega_{\mathrm{dS}}=\frac{c}{2\pi}\sqrt{\frac\Lambda3}.\]
This is a physical frequency calibration.
It is not identified with primitive R²D recurrence.
7. Radius and Clock Are Reciprocal in the de Sitter Calibration
Using
\[r_{\mathrm{dS}}=\sqrt{\frac3\Lambda},\]
we obtain
\[\omega_{\mathrm{dS}}=\frac{c}{2\pi r_{\mathrm{dS}}}.\]
Equivalently,
\[2\pi r_{\mathrm{dS}}\omega_{\mathrm{dS}}=c.\]
This is the physical cycle–phase calibration of the de Sitter horizon.
8. The Replacement Nariai Horizon Has a Different Radius
At Nariai,
\[r_N=\frac1{\sqrt\Lambda}.\]
Therefore,
\[\frac{r_N}{r_{\mathrm{dS}}}=\frac{1/\sqrt\Lambda}{\sqrt{3/\Lambda}}=\frac1{\sqrt3}.\]
Hence,
\[r_N=\frac{r_{\mathrm{dS}}}{\sqrt3}.\]
The replacement horizon is smaller by a factor of 3.
9. Its Boundary Area Is Smaller by a Factor of Three
Since
\[A=4\pi r^2,\]
we obtain
\[A_N=4\pi r_N^2=\frac{4\pi}{\Lambda}.\]
Thus,
\[A_N=\frac13A_{\mathrm{dS}}.\]
This is a change in physical boundary area.
10. Its Multiplicity Coordinate Changes by the Same Factor
Using
\[S=\frac A{4\ell_P^2},\]
we obtain
\[S_N=\frac\pi{\Lambda\ell_P^2}.\]
Therefore,
\[S_N=\frac13S_{\mathrm{dS}}.\]
The replacement boundary does not preserve the old horizon count.
11. Its Physical Clock Changes in the Opposite Direction
Addendum 46 established the boundary-appropriate Nariai phase-frequency read
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Compare this with
\[\omega_{\mathrm{dS}}=\frac{c}{2\pi}\sqrt{\frac\Lambda3}.\]
Therefore,
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}}.\]
The smaller replacement boundary carries the faster physical clock calibration.
12. Neither Coordinate Is Preserved
We have
\[S_N\ne S_{\mathrm{dS}},\]
and
\[\omega_N\ne\omega_{\mathrm{dS}}.\]
More specifically,
\[S_N=\frac13S_{\mathrm{dS}},\]
while
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}}.\]
Thus neither horizon multiplicity nor horizon phase frequency is individually invariant.
13. Their Changes Are Reciprocal
Square the frequency relation:
\[\omega_N^2=3\omega_{\mathrm{dS}}^2.\]
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}}.\]
Multiply by
\[S_N=\frac13S_{\mathrm{dS}}.\]
Then
\[S_N\omega_N^2=\frac13S_{\mathrm{dS}}(3\omega_{\mathrm{dS}}^2).\]
Therefore,
\[S_N\omega_N^2=S_{\mathrm{dS}}\omega_{\mathrm{dS}}^2.\]
A relational quantity survives even though its individual physical coordinates do not.
15. The Nariai Horizon Gives the Same Result
For Nariai,
\[S_N=\frac\pi{\Lambda\ell_P^2},\]
and
\[\omega_N^2=\frac{c^2\Lambda}{4\pi^2},\]
Hence,
\[S_N\omega_N^2=\frac\pi{\Lambda\ell_P^2}\frac{c^2\Lambda}{4\pi^2}.\]
Therefore,
\[S_N\omega_N^2=\frac{c^2}{4\pi\ell_P^2}.\]
The two boundary realizations share exactly the same count–clock relation.
16. Remove the Planck-Length Calibration
Using
\[\ell_P^2=\frac{G\hbar}{c^3},\]
we obtain
\[\frac{c^2}{4\pi\ell_P^2}=\frac{c^5}{4\pi G\hbar}.\]
Therefore,
\[S\omega^2=\frac{c^5}{4\pi G\hbar}.\]
This is the common dimensional form of the calibration relation.
17. Define a Dimensionless Gravitational Clock Coupling
Define
\[\alpha_G^\omega\equiv\frac{G\hbar\omega^2}{c^5}.\]
The quantity is dimensionless.
Then the shared count–clock relation becomes
\[4\pi\alpha_G^\omega S=1.\]
Thus both pure de Sitter and Nariai satisfy
\[4\pi\alpha_{G,\mathrm{dS}}^\omega S_{\mathrm{dS}}=1,\]
and
\[4\pi\alpha_{G,N}^\omega S_N=1.\]
18. The Coupling Changes While the Product Does Not
Because
\[\omega_N^2=3\omega_{\mathrm{dS}}^2,\]
we have
\[\alpha_{G,N}^\omega=3\alpha_{G,\mathrm{dS}}^\omega.\]
At the same time,
\[S_N=\frac13S_{\mathrm{dS}}.\]
Therefore,
\[\alpha_G^\omega\propto S^{-1}.\]
across these two horizon calibrations.
The dimensionless product remains fixed.
19. This Is Not Conservation of Entropy
The relation
\[S_N=\frac13S_{\mathrm{dS}}\]
already proves that horizon entropy is not conserved under this comparison.
Nor is the summed entropy of the two equal Nariai horizon components equal to the original pure de Sitter entropy.
The invariant is therefore not total entropy conservation.
It is a relation among boundary-calibrated quantities.
20. This Is Not Conservation of Frequency
Likewise,
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}}.\]
Frequency changes.
Thus no physical clock value is being transported unchanged from one landscape to the other.
21. Nor Is It Conservation of Radius or Area
We also have
\[r_N=\frac{r_{\mathrm{dS}}}{\sqrt3},\]
and
\[A_N=\frac13A_{\mathrm{dS}}.\]
The physical geometry changes.
Therefore the shared relation is not hidden conservation of a dimensional spatial coordinate.
22. The Invariant Comes From the Joint Scaling Structure
For the two calibrated spherical horizons,
\[S\propto r^2,\]
while
\[\omega\propto r^{-1}.\]
Therefore,
\[\omega^2\propto r^{-2}.\]
Hence,
\[S\omega^2\propto r^2r^{-2}=1.\]
The cancellation is a direct consequence of the opposed scaling of boundary count and physical clock.
23. Boundary Projects Onto Area
Addenda 41–43 established the relevant gravitational hierarchy:
\[\text{boundary}\longrightarrow A.\]
Under spherical projection,
\[A=4\pi r^2.\]
The horizon multiplicity calibration gives
\[S=\frac A{4\ell_P^2}.\]
Therefore,
\[S\propto r^2.\]
The quadratic count scaling comes from the area projection of the boundary.
24. The Physical Clock Is Reciprocal to the Boundary Scale
For both boundary calibrations considered here,
\[\omega=\frac c{2\pi r}.\]
Therefore,
\[\omega\propto r^{-1}.\]
The physical clock scales inversely with the boundary radius.
25. The Product Removes the Physical Radius
Combining
\[S=\frac{\pi r^2}{\ell_P^2},\]
with
\[\omega=\frac c{2\pi r},\]
gives
\[S\omega^2=\frac{\pi r^2}{\ell_P^2}\frac{c^2}{4\pi^2r^2}.\]
The physical radius cancels:
\[S\omega^2=\frac{c^2}{4\pi\ell_P^2}.\]
Thus the relation removes the particular gravitational scale at which the boundary is physically realized.
26. This Is Not Entropy as Circumference
Nothing in the relation requires
\[S\propto r.\]
Nothing requires entropy to be circumferential.
The horizon multiplicity coordinate satisfies
\[S\propto r^2\]
because the boundary is physically projected through area.
The phase clock satisfies
\[\omega\propto r^{-1}.\]
The invariant product therefore arises from
\[r^2\times r^{-2}.\]
No circumferential entropy coordinate is introduced.
27. Nor Is This a Primitive \(\lambda\)–\(\tau\) Relation
The equation
\[S\omega^2=\frac{c^5}{4\pi G\hbar}.\]
is written entirely in physically calibrated quantities.
It does not imply a primitive R²D relation such as
\[\tau=f(\lambda).\]
It does not establish
\[\frac{d\tau}{d\lambda}.\]
And it does not identify physical phase frequency with the primitive occurrence coordinate.
The relation remains a physical calibration relation.
28. The Algebra Alone Does Not Establish a New Fundamental Law
Whenever both
\[S=\frac{\pi r^2}{\ell_P^2}\]
and
\[\omega=\frac c{2\pi r}\]
hold, the relation
\[4\pi\alpha_G^\omega S=1\]
follows algebraically.
Therefore this addendum does not claim that R²D independently derives the relation from primitive counting.
The novelty is not the cancellation itself.
The novelty is that distinct gravitational boundary realizations reproduce the same relation.
29. The Result Is Therefore a Calibration-Class Invariant
Define
\[I_G\equiv4\pi\alpha_G^\omega S.\]
For pure de Sitter,
\[I_{G,\mathrm{dS}}=1.\]
For the replacement Nariai horizon,
\[I_{G,N}=1.\]
Thus the two physical landscapes belong to the same count–clock calibration class.
30. This Is Replacement Coherence, Not Transport
The correct R²D statement is not
\(I_G\) was physically carried from de Sitter into Nariai.
No such transport has been demonstrated.
The correct statement is
two different boundary realizations instantiate the same relation.
This is relational coherence.
31. Pure de Sitter Is Not the State Immediately Before Nariai
The Schwarzschild–de Sitter family contains a continuous range of source masses between
\[M=0\]
and
\[M=M_N.\]
Pure de Sitter is the zero-mass endpoint.
Nariai is the maximum-mass endpoint.
Therefore,
\[\text{de Sitter}\ne\text{immediate pre-Nariai state}.\]
The comparison in this addendum is between two distinguished horizon calibrations within the same Λ- defined gravitational family.
32. The Relation Is Not Claimed to Be Constant Along the Full Family
The present result does not assert
\[\frac{d}{dM}(S_h\omega_h^2)=0.\]
For generic intermediate Schwarzschild–de Sitter states, the horizon clock normalizations are more complicated.
The simple relation
\[\omega=\frac c{2\pi r}\]
does not serve as the same horizon-clock formula for every member of the two-horizon family.
Thus the invariant identified here is not a continuous first integral in M .
33. The Replacement Statement Is Local to Nariai
As the ordinary two-horizon landscape approaches Nariai,
\[S_b,S_c\longrightarrow S_N.\]
Under the boundary-appropriate normalization,
\[\omega_b^{\mathrm{BH}},\omega_c^{\mathrm{BH}}\longrightarrow\omega_N.\]
The limiting landscape therefore possesses the definite pair
\[(S_N,\omega_N).\]
The replacement boundary is physically well calibrated even though the old static clock has disappeared.
35. This Is the R²D Pattern of Relation Without Identity
The two landscapes differ in:
• horizon radius; • horizon area; • horizon multiplicity; • physical phase frequency; • temperature; • and boundary classification.
Yet they share
\[4\pi\alpha_G^\omega S=1.\]
Therefore,
identity is replaced; relation remains coherent.
This is precisely the kind of structure recursive replacement is expected to preserve.
36. The Result Mirrors the General R²D Replacement Architecture
The generic R²D pattern is
subordinate realization \(\longrightarrow\) recursive reclassification \(\longrightarrow\) new count meaning.
The constituent representation need not survive.
What can survive is a relation among the newly classified counts.
The de Sitter–Nariai comparison supplies a physical example in which:
\[S\]
changes,
\[\omega\]
changes,
but
\[\alpha_G^\omega S\]
does not.
37. This Does Not Mean the Two Boundaries Are the Same Boundary
The common calibration does not imply
\[B_{\mathrm{dS}}=B_N.\]
The physical geometries differ.
Their multiplicities differ.
Their clocks differ.
Therefore the correct statement is
\[B_{\mathrm{dS}}\ne B_N,\]
while
\[I_G(B_{\mathrm{dS}})=I_G(B_N).\]
Boundary identity is unnecessary for relational coherence.
38. The Invariant Is Also Not an Energy Conservation Law
The relation
\[4\pi\alpha_G^\omega S=1\]
does not represent transport of energy.
No conserved energetic substance is being transferred from the de Sitter horizon into the Nariai horizon.
The relation is a compatibility of physical calibrations.
Thus:
replacement coherence \(\ne\) energy conservation. [PDF line clipped; author review required.]
39. Nor Is It a Statement of Total Horizon Entropy Conservation
At Nariai there are two equal horizon components.
If one defines a summed horizon entropy,
\[S_{\mathrm{tot},N}=2S_N,\]
then
\[S_{\mathrm{tot},N}=\frac23S_{\mathrm{dS}}.\]
Therefore,
\[S_{\mathrm{tot},N}\ne S_{\mathrm{dS}}.\]
The count–clock invariant applies to the calibrated individual horizon relation considered here.
It is not a statement of conserved total entropy.
40. The Factor of Two Between Nariai Horizons Does Not Alter the Calibration Result
At Nariai the two limiting horizon branches possess equal area and equal multiplicity coordinate:
\[S_b=S_c=S_N.\]
They also approach the same boundary-normalized physical frequency:
\[\omega_b^{\mathrm{BH}}=\omega_c^{\mathrm{BH}}=\omega_N.\]
Therefore each horizon component separately realizes
\[4\pi\alpha_G^\omega S_N=1.\]
The calibration is componentwise.
41. Principle — Horizon Count–Clock Calibration
For a spherical gravitational horizon possessing the physical relations
\[S=\frac A{4\ell_P^2}=\frac{\pi r^2}{\ell_P^2},\]
and
\[\omega=\frac c{2\pi r},\]
define
\[\alpha_G^\omega=\frac{G\hbar\omega^2}{c^5},\]
Then
\[4\pi\alpha_G^\omega S=1.\]
This is a physical count–clock calibration identity.
42. Principle — Replacement Calibration Coherence
Suppose two different gravitational boundary realizations both satisfy the same count–clock calibration relation
\[I_G=1.\]
Then replacement of one realization by another need not preserve
\[S,\]
or
\[\omega,\]
individually.
It may instead preserve
\[I_G.\]
This defines calibration coherence across boundary replacement.
43. Corollary — de Sitter–Nariai Reciprocal Rescaling
For pure de Sitter and Nariai,
\[\frac{S_N}{S_{\mathrm{dS}}}=\frac13,\]
while
\[\frac{\omega_N}{\omega_{\mathrm{dS}}}=\sqrt3.\]
Therefore,
\[\frac{S_N\omega_N^2}{S_{\mathrm{dS}}\omega_{\mathrm{dS}}^2}=1.\]
The physical count and physical clock rescale reciprocally.
44. Corollary — Boundary Scale Is Eliminated From the Relation
Because
\[S\propto r^2\]
and
\[\omega^2\propto r^{-2},\]
the physical boundary scale cancels.
Therefore,
\[I_G\]
does not depend on the particular radius at which the calibrated boundary is realized.
This is why different horizon scales can occupy the same count–clock calibration class.
45. The Result Suggests Where a Deeper Gravitational Invariant May Live
The physical coordinates
\[r,\quad A,\quad S,\quad\omega,\quad T\]
all depend on boundary calibration.
The dimensionless quantity
\[I_G=4\pi\frac{G\hbar\omega^2}{c^5}S\]
does not contain an explicit horizon radius.
This makes it a more natural candidate for a deeper gravitational relation than either area or clock alone.
But the present addendum does not promote it to a universal R²D law.
46. A Universal Claim Would Require More Tests
To elevate
\[I_G=1\]
from a calibration-class identity to a general gravitational principle would require showing that the same dimensionless relation emerges independently in additional gravitational boundary classes.
Possible future tests include:
• charged horizons; • rotating horizons; • Kerr–Newman horizons; • dynamically evolving horizons; • and other replacement geometries.
Those tests have not yet been performed here.
47. Logical Status
Canonical R²D
R²D permits recursive replacement to change realization identity while preserving relational organization.
Physical coordinates are boundary relative.
Multiplicity is
\[S=\ln W.\]
Physical recurrence rates are later calibrations and are not automatically primitive recurrence counts.
Established Physical Calibration
For a spherical gravitational horizon,
\[S=\frac A{4\ell_P^2}.\]
For the de Sitter and boundary-normalized Nariai horizon reads considered here,
\[\omega=\frac c{2\pi r}.\]
Exact de Sitter Relations
\[r_{\mathrm{dS}}=\sqrt{\frac3\Lambda},\]
\[S_{\mathrm{dS}}=\frac{3\pi}{\Lambda\ell_P^2},\]
and
\[\omega_{\mathrm{dS}}=\frac c{2\pi}\sqrt{\frac\Lambda3}.\]
Exact Nariai Relations
\[r_N=\frac1{\sqrt\Lambda},\]
\[S_N=\frac\pi{\Lambda\ell_P^2},\]
and
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Derived Exact Comparison
\[S_N=\frac13S_{\mathrm{dS}},\]
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}},\]
and therefore
\[S_N\omega_N^2=S_{\mathrm{dS}}\omega_{\mathrm{dS}}^2.\]
Derived Calibration Identity
\[S\omega^2=\frac{c^5}{4\pi G\hbar},\]
or equivalently,
\[4\pi\alpha_G^\omega S=1.\]
New R²D Interpretation
The common dimensionless relation is interpreted as calibration coherence between distinct gravitational boundary realizations.
It is not interpreted as conservation of horizon entropy, clock frequency, radius, area, or energy.
Not Established Here
This addendum does not establish:
• that \(4\pi\alpha_G^\omega S=1\) is a universal R²D gravitational law; • that the invariant is continuously conserved through all intermediate Schwarzschild–de Sitter states; • that pure de Sitter is the immediate predecessor of Nariai; • that physical phase frequency is primitive recurrence; • that the de Sitter and Nariai horizons are the same boundary; • or that the relation is independent of the physical calibrations from which it follows.
48. The Central R²D Inversion
A conventional comparison begins with two geometries and notes their different horizon areas and temperatures.
R²D asks instead what relational structure survives those changing physical reads.
For pure de Sitter,
\[B_{\mathrm{dS}}\longrightarrow(S_{\mathrm{dS}},\omega_{\mathrm{dS}}).\]
For the replacement Nariai landscape,
\[B_N\longrightarrow(S_N,\omega_N).\]
The pairs differ:
\[(S_{\mathrm{dS}},\omega_{\mathrm{dS}})\ne(S_N,\omega_N).\quad\text{[Source glyph after relation clipped; author review required.]}\]
Yet both satisfy
\[4\pi\alpha_G^\omega S=1.\]
Thus the deeper hierarchy is
boundary realization
\(\Downarrow\)
area read \(\longrightarrow S\)
and independently
boundary-scale clock read \(\longrightarrow\omega\)
such that
\(S\omega^2\) = same gravitational calibration.
When the boundary changes, the physical reads change.
The relational calibration persists.
49. Conclusion
Addendum 46 showed that the Nariai limit replaces the clock belonging to the ordinary two-horizon Schwarzschild–de Sitter landscape.
The old static frequencies vanish:
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
The replacement Nariai landscape instead possesses the finite physical phase-frequency read
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Thus physical clock identity is not preserved under the boundary replacement.
The horizon multiplicity also changes.
Pure de Sitter possesses
\[S_{\mathrm{dS}}=\frac{3\pi}{\Lambda\ell_P^2},\]
while the Nariai horizon possesses
\[S_N=\frac\pi{\Lambda\ell_P^2}.\]
Therefore,
\[S_N=\frac13S_{\mathrm{dS}}.\]
The Nariai physical clock changes oppositely:
\[\omega_N=\sqrt3\,\omega_{\mathrm{dS}}.\]
Consequently,
\[S_N\omega_N^2=S_{\mathrm{dS}}\omega_{\mathrm{dS}}^2.\]
Both evaluate to
\[S\omega^2=\frac{c^5}{4\pi G\hbar}.\]
Defining
\[\alpha_G^\omega=\frac{G\hbar\omega^2}{c^5},\]
the relation becomes
\[4\pi\alpha_G^\omega S=1.\]
This is not a claim that the horizon entropy is conserved.
It is not a claim that the clock is conserved.
It is not a claim that pure de Sitter evolves directly into Nariai while carrying an invariant substance.
Nor is it a new primitive equation inserted into R²D.
The relation follows from the joint physical calibrations
\[S=\frac{\pi r^2}{\ell_P^2},\]
and
\[\omega=\frac c{2\pi r}.\]
Its R²D significance lies elsewhere.
Pure de Sitter and Nariai possess different:
• radii; • areas; • multiplicities; • clock rates; • temperatures; • and boundary classifications.
Yet the relation connecting horizon count to horizon clock is identical.
Thus recursive boundary replacement need not preserve the physical realization.
It can preserve the relational architecture by which that realization is calibrated.
The count changes.
The clock changes.
The boundary changes.
The relation remains coherent.
Therefore:
horizon count–clock calibration persists across boundary replacement.
And more specifically:
R²D replacement coherence is expressed not by persistence of physical coordinates, but by persisten… [PDF line clipped; author review required.]