Addendum 45 — Multiplicity-Gap Closure Occurs Before the Nariai Limit
Abstract
Addendum 44 proposed a generic R²D condition for gravitational saturation.
Let a subordinate gravitational boundary have multiplicity
\[W_b,\]
and let its enclosing gravitational boundary have multiplicity
\[W_c.\]
Define their logarithmic multiplicity gap
\[\Delta S_{c\mid b}=S_c-S_b=\ln\frac{W_c}{W_b}.\]
The nested classification remains multiplicatively distinct while
\[\Delta S_{c\mid b}>0.\]
Its marginal condition is
\[\Delta S_{c\mid b}=0.\]
Then
\[W_b=W_c.\]
If both boundaries share the same horizon-area calibration,
\[S=\frac A{4\ell_P^2},\]
the same condition is physically read as
\[A_b=A_c.\]
Addendum 44 did not identify a physical system that satisfies this relation. It deliberately stopped before assigning a maximum mass.
Schwarzschild–de Sitter geometry provides an exact independent test.
For positive cosmological constant Λ, the static projection is
\[ds^2=-f(r)c^2dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2,\]
where
\[f(r)=1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3}.\]
Within its nonextremal two-horizon regime, the equation
\[f(r)=0\]
has two positive roots,
\[r_b<r_c,\]
representing the black-hole and cosmological horizon reads.
They are not independent surfaces inserted into a fixed container. They are two roots of one shared gravitational equation.
Their exact coupling implies
\[r_b^2+r_br_c+r_c^2=\frac3\Lambda.\]
Define
\[q\equiv\frac{r_b}{r_c},\qquad0\le q\le1.\]
Then
\[r_c^2=\frac3{\Lambda(1+q+q^2)},\]
and
\[r_b^2=\frac{3q^2}{\Lambda(1+q+q^2)}.\]
The corresponding areas are
\[A_c=\frac{12\pi}{\Lambda(1+q+q^2)},\]
and
\[A_b=\frac{12\pi q^2}{\Lambda(1+q+q^2)}.\]
Therefore,
\[\frac{A_b}{A_c}=q^2.\]
Because both horizons share the same Bekenstein–Hawking calibration,
\[S_b=\ln W_b=\frac{A_b}{4\ell_P^2},\]
and
\[S_c=\ln W_c=\frac{A_c}{4\ell_P^2}.\]
Their multiplicity gap is therefore
\[\Delta S_{c\mid b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
For
\[0\le q<1,\]
this quantity is positive:
\[\Delta S_{c\mid b}>0.\]
Thus throughout the ordinary two-horizon Schwarzschild–de Sitter regime,
\[W_c>W_b.\]
The physical solution independently reaches its limiting mass when the two positive roots become a double root:
\[f(r_N)=0,\qquad f'(r_N)=0.\]
These conditions give
\[r_N=\frac1{\sqrt\Lambda},\]
and
\[M_N=\frac{c^2}{3G\sqrt\Lambda}.\]
This is the Nariai limit.
At that independently derived maximum mass,
\[r_b=r_c=r_N.\]
Therefore,
\[q=1,\]
and hence
\[A_b=A_c=\frac{4\pi}{\Lambda}.\]
Consequently,
\[S_b=S_c,\]
and
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}=0.\]
The exact known maximum-mass condition therefore occurs precisely where the independently proposed R²D subordinate–enclosing multiplicity gap closes.
The result is stronger than area equality alone.
At Nariai:
\[W_b=W_c,\]
\[S_b=S_c,\]
\[A_b=A_c,\]
\[r_b=r_c,\]
and the two previously distinct positive roots of the same gravitational equation coalesce into one double root.
Thus the old two-horizon classification becomes mathematically marginal at exactly the same point at which the R²D count distinction vanishes.
The enclosing cosmological boundary is not fixed during this process.
At zero black-hole mass,
\[r_c=\sqrt{\frac3\Lambda},\]
so
\[A_{\mathrm{dS}}=\frac{12\pi}{\Lambda}.\]
At Nariai,
\[A_c=\frac{4\pi}{\Lambda}=\frac13A_{\mathrm{dS}}.\]
The subordinate area grows while the enclosing area contracts.
The Nariai limit is therefore not a local black-hole well growing until it fills a pre-existing cosmological box.
It is a coupled reorganization of two boundary reads belonging to one gravitational relation.
The logical order is
shared gravitational relation
\(\Downarrow\)
\[W_b(M),\quad W_c(M)\]
\(\Downarrow\)
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}.\]
\(\Downarrow\)
\[\Delta S_{c\mid b}\longrightarrow0\]
\(\Downarrow\)
\[A_b\longrightarrow A_c\]
\(\Downarrow\)
\[r_b\longrightarrow r_c\]
\(\Downarrow\)
double-root boundary
\(\Downarrow\)
\[M\longrightarrow M_N.\]
The maximum mass comes last.
Therefore:
multiplicity-gap closure occurs before the Nariai limit.
More precisely:
the Nariai maximum mass is the physical source coordinate of the point at which the Schwarzschild–
1. Addendum 44 Made a Prediction Before Choosing a Physical Solution
Addendum 44 considered a generic subordinate gravitational multiplicity well.
Its multiplicity was
\[W_H,\]
and the multiplicity of its enclosing gravitational count boundary was
\[W_E.\]
The remaining count distinction was defined as
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
The proposed marginal condition was
\[\Delta S_{E\mid H}=0.\]
Nothing in that construction assumed Schwarzschild–de Sitter geometry.
Nothing assumed a cosmological constant.
Nothing assumed the Nariai mass.
Therefore the physical solution can now be used as an independent test.
2. The Test Requires Two Coupled Gravitational Boundaries
A useful test system must contain:
- a subordinate gravitational horizon;
- an enclosing gravitational horizon;
- a continuously varying gravitational source coordinate;
- a regime in which both boundaries remain separately readable;
- and a known limiting mass at which the two-horizon classification becomes marginal.
Schwarzschild–de Sitter possesses exactly this architecture.
3. Begin With Schwarzschild–de Sitter Geometry
For positive cosmological constant,
\[\Lambda>0,\]
the static Schwarzschild–de Sitter metric is
\[ds^2=-f(r)c^2dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2,\]
where
\[f(r)=1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3}.\]
Horizons occur where
\[f(r)=0.\]
4. The Ordinary Static Regime Contains Two Positive Horizon Roots
For an allowed nonextremal source mass,
\[f(r)=0\]
contains two positive roots:
\[r_b<r_c.\]
The smaller root,
\[r_b,\]
is the black-hole horizon.
The larger root,
\[r_c,\]
is the cosmological horizon.
There is also one negative algebraic root that does not represent a positive horizon radius.
5. The Two Horizons Belong to One Equation
Both
\[r_b\]
and
\[r_c\]
satisfy
\[f(r)=0.\]
Therefore changing M changes both roots.
The black-hole and cosmological horizon reads are not independently variable physical surfaces.
They are coupled solutions of one gravitational relation.
This already matches the structural requirement from Addendum 44:
\[W_H=W_H(M),\]
while
\[W_E=W_E(M,\ldots).\]
The enclosing boundary need not remain fixed while the subordinate well changes.
6. Multiply the Horizon Equation by r
From
\[1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3}=0,\]
multiply by r :
\[r-\frac{2GM}{c^2}-\frac{\Lambda r^3}{3}=0.\]
Rearrange:
\[r^3-\frac3\Lambda r+\frac{6GM}{\Lambda c^2}=0.\]
This cubic makes the coupling among the roots explicit.
7. Introduce the Third Root
Let the third root be
\[r_n<0.\]
The cubic contains no r 2 term.
Therefore the sum of its roots vanishes:
\[r_b+r_c+r_n=0.\]
Hence,
\[r_n=-(r_b+r_c).\]
8. The Pairwise Root Relation Gives the Coupled Horizon Identity
For the cubic,
\[r_br_c+r_br_n+r_cr_n=-\frac3\Lambda.\]
Substitute
\[r_n=-(r_b+r_c).\]
Then
\[r_br_c-r_b(r_b+r_c)-r_c(r_b+r_c)=-\frac3\Lambda.\]
Simplifying gives
\[-r_b^2-r_br_c-r_c^2=-\frac3\Lambda.\]
Therefore,
\[r_b^2+r_br_c+r_c^2=\frac3\Lambda.\]
This is the exact coupled horizon relation.
9. The Enclosing Horizon Is Not a Fixed Box
The equation
\[r_b^2+r_br_c+r_c^2=\frac3\Lambda\]
shows immediately that increasing one positive root changes the allowed value of the other.
Thus the cosmological horizon cannot remain fixed while the black-hole horizon changes.
The exact physical problem is a coupled boundary realization.
10. Define a Convenient Relative Radius Coordinate
Define
\[q\equiv\frac{r_b}{r_c}.\]
Throughout the ordinary two-horizon regime,
\[0\le q<1.\]
At horizon coalescence,
\[q\longrightarrow1.\]
This ratio is a convenient physical coordinate.
It is not introduced as a primitive R²D quantity.
11. Solve the Coupled Horizon Identity
Using q
Write
\[r_b=qr_c.\]
Then
\[q^2r_c^2+qr_c^2+r_c^2=\frac3\Lambda.\]
Therefore,
\[r_c^2(1+q+q^2)=\frac3\Lambda.\]
Hence,
\[r_c^2=\frac3{\Lambda(1+q+q^2)}.\]
And
\[r_b^2=\frac{3q^2}{\Lambda(1+q+q^2)}.\]
12. Project the Two Boundaries Onto Area
For spherical horizon projection,
\[A_b=4\pi r_b^2\]
and
\[A_c=4\pi r_c^2.\]
Therefore,
\[A_b(q)=\frac{12\pi q^2}{\Lambda(1+q+q^2)},\]
and
\[A_c(q)=\frac{12\pi}{\Lambda(1+q+q^2)}.\]
The two areas are therefore explicitly coupled.
13. Their Area Ratio Is Simple
Take the ratio:
\[\frac{A_b}{A_c}=q^2.\]
Define
\[\sigma_{b\mid c}\equiv\frac{A_b}{A_c}.\]
Then
\[\sigma_{b\mid c}=q^2.\]
For
\[0\le q<1,\]
we have
\[0\le\sigma_{b\mid c}<1.\]
At equality,
\[\sigma_{b\mid c}=1.\]
14. The Area Ratio Is Not Yet the Primitive R²D Gap
The ratio
\[\sigma_{b\mid c}\]
is physically convenient.
But Addendum 44 identified a more primitive count comparison:
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}.\]
The present addendum should therefore test that quantity directly.
15. Apply the Same Horizon Multiplicity Calibration to Both Boundaries
For the black-hole horizon,
\[S_b=\ln W_b=\frac{A_b}{4\ell_P^2}.\]
For the cosmological horizon,
\[S_c=\ln W_c=\frac{A_c}{4\ell_P^2}.\]
The common calibration allows their logarithmic multiplicities to be compared directly.
16. The Logarithmic Capacity Ratio Equals the Area Ratio
Because both use the same coefficient,
\[\frac{S_b}{S_c}=\frac{A_b}{A_c}=q^2.\]
Thus
\[\frac{S_b}{S_c}=\sigma_{b\mid c}.\]
This is a ratio of logarithmic multiplicity coordinates.
It is not the same as the ratio
\[W_b/W_c.\]
17. Define the Exact Multiplicity Gap
The R²D saturation coordinate from Addendum 44 is
\[\Delta S_{c\mid b}\equiv S_c-S_b.\]
Since
\[S=\ln W,\]
we have
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}.\]
This is the actual multiplicative distinction between the two horizon classes.
18. Substitute the Exact Coupled Areas
Using
\[A_c=\frac{12\pi}{\Lambda(1+q+q^2)},\]
and
\[A_b=\frac{12\pi q^2}{\Lambda(1+q+q^2)},\]
we obtain
\[\Delta S_{c\mid b}=\frac{A_c-A_b}{4\ell_P^2}.\]
Therefore,
\[\Delta S_{c\mid b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
Equivalently,
\[\ln\frac{W_c}{W_b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
19. The Ordinary Two-Horizon Regime Has Positive Multiplicity Gap
For
\[0\le q<1,\]
\[1-q^2>0.\]
The denominator
\[1+q+q^2\]
is positive.
Therefore,
\[\Delta S_{c\mid b}>0.\]
Hence,
\[W_c>W_b.\]
The enclosing cosmological horizon remains multiplicatively distinguishable from the subordinate black- hole horizon throughout the ordinary two-horizon regime.
20. Schwarzschild–de Sitter Therefore Realizes the Generic Addendum 44 Ordering
Addendum 44 required
\[\Delta S_{E\mid H}>0\]
for the distinguishable nested regime.
Schwarzschild–de Sitter gives exactly
\[\Delta S_{c\mid b}>0\]
for every ordinary two-horizon state.
Thus the physical solution occupies the expected side of the proposed R²D marginal boundary.
21. Pure de Sitter Is the Zero-Black-Hole Limit
Take
\[M\longrightarrow0.\]
Then the black-hole positive root disappears:
\[r_b\longrightarrow0.\]
Thus
\[q\longrightarrow0.\]
The remaining cosmological horizon satisfies
\[r_{\mathrm{dS}}=\sqrt{\frac3\Lambda}.\]
Its area is
\[A_{\mathrm{dS}}=\frac{12\pi}{\Lambda}.\]
This is the enclosing-area read at the zero-black-hole end of the family.
22. Increasing Mass Reorganizes Both Horizon Reads
As the physical source mass increases within the two-horizon regime,
\[r_b\uparrow\]
while
\[r_c\downarrow.\]
Consequently,
\[A_b\uparrow,\]
and
\[A_c\downarrow.\]
The subordinate multiplicity increases while the enclosing multiplicity decreases under their common calibration.
Thus the multiplicity gap closes from both sides.
23. This Is Not Filling a Fixed Cosmological Capacity
A simpler provisional model would hold
\[A_c\]
fixed while
\[A_b\]
increases.
The exact solution does not behave this way.
Instead,
\[A_b=A_b(M),\qquad A_c=A_c(M).\]
The black-hole and cosmological boundaries jointly reorganize.
The enclosing count domain participates in the physical realization.
24. Derive the Physical Limiting Condition Independently
The maximum-mass two-horizon solution occurs when the two positive roots coalesce.
Mathematically this is a double root.
Therefore require
\[f(r_N)=0\]
and
\[f'(r_N)=0.\]
No R²D saturation condition has yet been imposed.
This makes the result an independent physical test.
25. Differentiate the Schwarzschild–de Sitter Function
From
\[f(r)=1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3},\]
we obtain
\[f'(r)=\frac{2GM}{c^2r^2}-\frac{2\Lambda r}{3}.\]
At the double root,
\[f'(r_N)=0.\]
Therefore,
\[\frac{GM_N}{c^2}=\frac{\Lambda r_N^3}{3}.\]
c2 3
26. Insert This Into the Horizon Equation
At the limiting root,
\[f(r_N)=0.\]
Thus
\[1-\frac{2GM_N}{c^2r_N}-\frac{\Lambda r_N^2}{3}=0.\]
Using
\[\frac{GM_N}{c^2}=\frac{\Lambda r_N^3}{3},\]
gives
\[1-\frac{2\Lambda r_N^2}{3}-\frac{\Lambda r_N^2}{3}=0.\]
Therefore,
\[1-\Lambda r_N^2=0.\]
Hence,
\[r_N=\frac1{\sqrt\Lambda}.\]
27. The Physical Nariai Mass Follows
Using
\[\frac{GM_N}{c^2}=\frac{\Lambda r_N^3}{3},\]
and
\[r_N=\frac1{\sqrt\Lambda},\]
we obtain
\[M_N=\frac{c^2}{3G\sqrt\Lambda}.\]
Equivalently,
\[\frac{3GM_N\sqrt\Lambda}{c^2}=1.\]
This maximum has been obtained solely from the exact gravitational solution.
28. Now Test the R²D Prediction
At the Nariai limit,
\[r_b=r_c=r_N.\]
Therefore,
\[q=1.\]
Insert this into the exact R²D multiplicity gap:
\[\Delta S_{c\mid b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
At
\[q=1,\]
the numerator vanishes.
Therefore,
\[\Delta S_{c\mid b}=0.\]
29. The Horizon Multiplicities Become Equal
Since
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b},\]
the Nariai condition gives
\[\ln\frac{W_c}{W_b}=0.\]
Therefore,
\[W_c=W_b.\]
This is exactly the independently proposed Addendum 44 saturation condition.
30. The Horizon Areas Become Equal
At
\[r_b=r_c=\frac1{\sqrt\Lambda},\]
the spherical horizon areas are equal:
\[A_b=A_c.\]
Their common value is
\[A_N=\frac{4\pi}{\Lambda}.\]
Thus the physical area projection reproduces the multiplicity-gap marginality.
31. Their Entropy Coordinates Become Equal
Using
\[S=\frac A{4\ell_P^2},\]
we obtain
\[S_b=S_c.\]
Their common value is
\[S_N=\frac\pi{\Lambda\ell_P^2}.\]
Therefore,
\[S_c-S_b=0.\]
The logarithmic multiplicity distinction has vanished.
32. The Ratio Coordinate Simultaneously Reaches Unity
Because
\[\sigma_{b\mid c}=\frac{A_b}{A_c}=q^2,\]
at Nariai,
\[\sigma_{b\mid c}=1.\]
Thus the ratio and gap forms of the saturation condition agree:
\[\sigma_{b\mid c}=1\]
if and only if
\[\Delta S_{c\mid b}=0.\]
within the two-horizon Schwarzschild–de Sitter family.
33. Multiplicity Equality Alone Would Not Normally Prove Boundary Identity
In general,
\[W_1=W_2\]
does not require two macrostates to be identical.
Degenerate states may share the same multiplicity.
The Nariai result is stronger.
At the same limiting point,
\[W_b=W_c,\]
\[S_b=S_c,\]
\[A_b=A_c,\]
\[r_b=r_c,\]
and the two positive roots of one common horizon equation merge into a double root.
The entire physical classification becomes marginal simultaneously.
34. Root Coalescence Is the Geometric Read of Classification Loss
Before Nariai,
\[r_b<r_c.\]
The horizon equation therefore supplies two distinguishable positive branches.
At Nariai,
\[r_b=r_c.\]
The two branches no longer remain separately labeled by distinct positive root values.
Thus the conventional mathematical description itself loses the distinction on which the old two-horizon classification depended.
This is precisely the type of boundary event proposed by Addendum 44.
35. The Enclosing Area Has Changed Substantially
At zero black-hole mass,
\[A_{\mathrm{dS}}=\frac{12\pi}{\Lambda}.\]
At Nariai,
\[A_c^N=\frac{4\pi}{\Lambda}.\]
Therefore,
\[\frac{A_c^N}{A_{\mathrm{dS}}}=\frac13.\]
The enclosing cosmological area has contracted by a factor of three.
36. The Enclosing Multiplicity Changes by the Same Factor in Logarithmic Coordinate
Because
\[S_c=\frac{A_c}{4\ell_P^2},\]
the pure de Sitter entropy coordinate is
\[S_{\mathrm{dS}}=\frac{3\pi}{\Lambda\ell_P^2}.\]
At Nariai,
\[S_c^N=\frac\pi{\Lambda\ell_P^2}.\]
Thus,
\[S_c^N=\frac13S_{\mathrm{dS}}.\]
The enclosing logarithmic multiplicity capacity is not fixed either.
37. Meanwhile the Subordinate Well Grows
At the zero-mass end,
\[A_b\longrightarrow0.\]
At Nariai,
\[A_b=\frac{4\pi}{\Lambda}.\]
Thus the black-hole multiplicity coordinate grows from its zero-horizon limit to
\[S_b^N=\frac\pi{\Lambda\ell_P^2}.\]
The subordinate and enclosing count coordinates approach one another through coupled change.
38. The Gap Closes From Both Sides
The physical evolution of the family is therefore not
\[S_b\uparrow\quad\text{with}\quad S_c=\text{constant}.\]
It is
\[S_b\uparrow,\qquad S_c\downarrow.\]
Hence,
\[\Delta S_{c\mid b}=S_c-S_b\downarrow0.\]
The saturation boundary is a coupled realization.
39. Mass Can Be Written as a Function of the Boundary Ratio
The cubic root product gives
\[r_br_c(r_b+r_c)=\frac{6GM}{\Lambda c^2}.\]
Using
\[r_b=qr_c\]
and
\[r_c^2=\frac3{\Lambda(1+q+q^2)},\]
one obtains
\[\frac{GM}{c^2}=\frac{\sqrt3}{2\sqrt\Lambda}\frac{q(1+q)}{(1+q+q^2)^{3/2}}.\]
This puts the physical mass after the coupled horizon coordinate.
40. Normalize by the Nariai Mass
Since
\[M_N=\frac{c^2}{3G\sqrt\Lambda},\]
we obtain
\[\frac{M}{M_N}=\frac{3\sqrt3}{2}\frac{q(1+q)}{(1+q+q^2)^{3/2}}.\]
At
\[q=1,\]
the right-hand side is unity.
Therefore,
\[q\longrightarrow1\quad\Longrightarrow\quad M\longrightarrow M_N.\]
41. Write Mass in Terms of the Areal Saturation Coordinate
Because
\[\sigma_{b\mid c}=q^2,\]
we have
\[q=\sqrt{\sigma_{b\mid c}}.\]
Therefore,
\[\frac{M}{M_N}=\frac{3\sqrt3}{2}\frac{\sqrt{\sigma_{b\mid c}}(1+\sqrt{\sigma_{b\mid c}})}{(1+\sqrt{\sigma_{b\mid c}}+\sigma_{b\mid c})^{3/2}}.\]
Thus mass is explicitly a downstream function of the coupled boundary relation.
At
\[\sigma_{b\mid c}=1,\]
\[M=M_N.\]
42. The Multiplicity Gap Is Still More Primitive Than the Area Ratio
The ratio
\[\sigma_{b\mid c}\]
is useful because it gives a compact geometric representation.
But it compares logarithmic count coordinates:
\[\sigma_{b\mid c}=\frac{S_b}{S_c}.\]
The true multiplicative count gap is
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}.\]
Thus Addendum 45 places the multiplicity gap before the areal ratio in the explanatory hierarchy.
43. The Nariai Mass Is Not the Primitive Saturation Condition
Conventional gravity may state the limit as
\[M=M_N.\]
R²D asks what boundary distinction is lost at that value.
The exact solution gives
\[M=M_N\]
at the same point where
\[r_b=r_c,\]
\[A_b=A_c,\]
\[S_b=S_c,\]
and
\[W_b=W_c.\]
Thus the mass is one physical coordinate of a deeper boundary marginality.
44. The Nariai Limit Is Not Caused by Area Equality
The R²D inversion should not simply replace
\[M\longrightarrow\text{Nariai}\]
with
\[A_b=A_c\longrightarrow\text{Nariai}.\]
Area is itself a physical projection.
The prior R²D statement is
\[\Delta S_{c\mid b}=0.\]
Under the common horizon calibration, this is read as area equality.
Thus the explanatory sequence remains
count distinction \(\longrightarrow\) areal distinction \(\longrightarrow\) mass coordinate.
45. This Is an Exact Projection Test of Addendum 44
Addendum 44 proposed, before invoking Schwarzschild–de Sitter,
gravitational saturation \(\Longleftrightarrow\Delta S_{E\mid H}=0\).
Schwarzschild–de Sitter independently gives a known maximum mass through the double-root condition.
At that exact maximum,
\[\Delta S_{c\mid b}=0.\]
The proposed R²D marginal condition and the independently known gravitational maximum coincide.
46. This Is Stronger Than Post-Hoc Reinterpretation
The count criterion was not obtained by first examining the Nariai solution and then defining saturation to match it.
The logical sequence was:
Addendum 44:
\[\Delta S_{E\mid H}=0\]
as the generic marginal condition.
Then Addendum 45:
\[f(r)=0,\qquad f'(r)=0\]
independently defines the known physical maximum.
Only afterward are they compared.
Their coincidence is therefore a genuine projection test.
47. The Result Does Not Prove the Whole R²D Ontology
The agreement establishes a narrower statement.
It shows that the exact Schwarzschild–de Sitter maximum-mass boundary possesses precisely the count structure predicted by the proposed subordinate–enclosing multiplicity-gap criterion.
It does not by itself prove:
• that all gravitational maxima arise this way; • that the Bekenstein–Hawking relation has been derived from primitive R²D counting; • that \(\Delta r=\ell_P\) is exact; • that cosmological horizons are ontologically identical to black-hole horizons; • or that R²D uniquely explains the Nariai geometry.
The result is an exact compatibility test of one independently proposed boundary condition.
48. The Gibbs-Type Interpretation Is Now Physically Realized
Before Nariai,
\[\Delta S_{c\mid b}>0.\]
The black-hole and cosmological branches are separately distinguishable.
At Nariai,
\[\Delta S_{c\mid b}=0.\]
Their defining multiplicity gap vanishes.
Formal continuation of the same nested two-horizon classification past that condition would attempt to retain two count classes after the distinction supporting them has disappeared.
Thus the Gibbs-type boundary problem proposed generically in Addendum 44 is realized concretely by root coalescence.
49. Nothing Requires Physical Annihilation
The disappearance of the old two-horizon distinction does not imply disappearance of all physical organization.
It means that the old Schwarzschild–de Sitter static-patch classification becomes marginal.
A different gravitational boundary classification may remain physically realizable.
Thus,
gap closure \(\ne\) physical annihilation.
It is a landscape-replacement condition.
50. Nariai Is Therefore a Boundary-Replacement Point
Before the limit:
\[B_b\quad\text{and}\quad B_c\]
are distinguishable through separate horizon reads.
At the limit:
\[r_b=r_c.\]
The multiplicity gap vanishes:
\[W_b=W_c.\]
The old two-boundary static classification therefore loses the distinction required to define its separate branches.
The Nariai geometry should consequently be interpreted as a replacement gravitational landscape rather than simply as the final member of an unchanged two-horizon family.
51. The Recurrence Question Must Remain Separate
Nothing in the multiplicity-gap argument determines the physical clock associated with the replacement landscape.
The disappearance of the old horizon distinction does not imply
\[\text{recurrence}=0.\]
Multiplicity, distinguishability, recurrence, and physical time calibration remain different relations.
The next addendum therefore asks what happens to the clock read when the old two-horizon boundary classification is replaced.
52. Principle — Coupled Horizon Multiplicity
Within the Schwarzschild–de Sitter two-horizon regime, define
\[S_b=\frac{A_b}{4\ell_P^2}\]
and
\[S_c=\frac{A_c}{4\ell_P^2}.\]
Their exact coupled areas imply
\[\Delta S_{c\mid b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
Thus the subordinate and enclosing multiplicity coordinates are not independent.
53. Principle — Horizon Multiplicity-Gap Closure
For the ordinary two-horizon regime,
\[\Delta S_{c\mid b}>0.\]
At the Nariai boundary,
\[\Delta S_{c\mid b}=0.\]
Thus the exact limiting geometry coincides with closure of the subordinate–enclosing horizon multiplicity gap.
54. Corollary — Areal Equality Is the Physical Projection of Gap Closure
Because both horizons share
\[S=\frac A{4\ell_P^2},\]
we have
\(\Delta S_{c\mid b}=0\;\Longleftrightarrow\;A_b=A_c\).
Within spherical projection,
\[A_b=A_c\quad\Longleftrightarrow\quad r_b=r_c.\]
Thus root coalescence is the radial geometric read of prior count marginality.
55. Corollary — Nariai Mass Is Downstream
The double-root condition gives
\[M_N=\frac{c^2}{3G\sqrt\Lambda}.\]
But this same condition occurs when
\[\Delta S_{c\mid b}=0.\]
Therefore,
multiplicity-gap closure \(\longrightarrow\) areal equality \(\longrightarrow\) root coalescence \(\longrightarrow M_N\).
The maximum mass comes last.
56. Logical Status
Proposed Independently in Addendum 44
The generic gravitational saturation criterion is
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}=0.\]
Under common areal calibration,
\[\Delta S_{E\mid H}=0\quad\Longrightarrow\quad A_E=A_H.\]
Conventional Exact Physical Input
Schwarzschild–de Sitter geometry is
\[f(r)=1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3}.\]
The physical horizons are positive roots of
\[f(r)=0.\]
The limiting Nariai solution is a double positive root.
Derived Exact Geometry
The two positive horizon roots satisfy
\[r_b^2+r_br_c+r_c^2=\frac3\Lambda.\]
Defining
\[q=\frac{r_b}{r_c},\]
gives
\[A_b=\frac{12\pi q^2}{\Lambda(1+q+q^2)},\]
\[A_c=\frac{12\pi}{\Lambda(1+q+q^2)},\]
and
\[\frac{A_b}{A_c}=q^2.\]
Derived Multiplicity Gap
Using the common horizon calibration,
\[\Delta S_{c\mid b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
Independently Derived Maximum
The double-root conditions give
\[r_N=\frac1{\sqrt\Lambda},\]
and
\[M_N=\frac{c^2}{3G\sqrt\Lambda}.\]
Exact Projection Test
At the independently determined Nariai maximum,
\[q=1,\]
and therefore
\[\Delta S_{c\mid b}=0.\]
Thus the exact physical maximum satisfies the independently proposed R²D marginal condition.
Not Established Here
This addendum does not derive:
• Schwarzschild–de Sitter geometry from primitive R²D counting; • the value of Λ; • the Bekenstein–Hawking calibration; • the microscopic horizon multiplicities; • the Nariai temperature; • the physical clock of the replacement landscape; • or the recurrence relation surviving horizon-branch coalescence.
Those recurrence questions are deliberately deferred.
57. The Central R²D Inversion
The conventional description is
\(M\uparrow\)
\(\Downarrow\)
\[r_b\uparrow,\qquad r_c\downarrow\]
\(\Downarrow\)
\[r_b=r_c\]
\(\Downarrow\)
Nariai.
R²D resolves the same physical relation one level earlier:
shared gravitational boundary relation
\(\Downarrow\)
\[W_b(M),\qquad W_c(M)\]
\(\Downarrow\)
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}.\]
\(\Downarrow\)
\[\Delta S_{c\mid b}\longrightarrow0\]
\(\Downarrow\)
\[A_b\longrightarrow A_c\]
\(\Downarrow\)
\[r_b\longrightarrow r_c\]
\(\Downarrow\)
double-root landscape
\(\Downarrow\)
\[M\longrightarrow M_N.\]
The mass limit is the last physical coordinate assigned to the transition.
58. Conclusion
Addendum 44 proposed that a gravitational multiplicity well reaches its enclosing limit when the multiplicity distinction separating subordinate and enclosing count boundaries vanishes.
The proposed count coordinate was
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
Its marginal condition was
\[\Delta S_{E\mid H}=0.\]
That proposal was made without invoking Schwarzschild–de Sitter geometry or the Nariai solution.
Schwarzschild–de Sitter therefore provides an independent exact physical test.
Its black-hole and cosmological horizons are positive roots of
\[f(r)=1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3}.\]
The two positive roots are coupled by
\[r_b^2+r_br_c+r_c^2=\frac3\Lambda.\]
Defining
\[q=\frac{r_b}{r_c},\]
their areas are
\[A_b=\frac{12\pi q^2}{\Lambda(1+q+q^2)},\]
and
\[A_c=\frac{12\pi}{\Lambda(1+q+q^2)}.\]
With the common horizon multiplicity calibration,
\[S_b=\ln W_b=\frac{A_b}{4\ell_P^2},\]
and
\[S_c=\ln W_c=\frac{A_c}{4\ell_P^2},\]
the exact multiplicity gap is
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
Throughout the ordinary two-horizon regime,
\[q<1,\]
so
\[\Delta S_{c\mid b}>0.\]
The subordinate and enclosing horizon classes remain multiplicatively distinct.
The independently known physical maximum occurs when the two positive roots form a double root:
\[f(r_N)=0,\]
\[f'(r_N)=0.\]
This gives
\[r_N=\frac1{\sqrt\Lambda},\]
and
\[M_N=\frac{c^2}{3G\sqrt\Lambda}.\]
At this exact maximum,
\[r_b=r_c,\]
so
\[q=1.\]
Therefore,
\[A_b=A_c,\]
\[S_b=S_c,\]
and most importantly,
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}=0.\]
The exact maximum-mass gravitational solution therefore reaches precisely the count-level marginal condition proposed independently in Addendum 44.
The enclosing cosmological boundary does not remain fixed while the black-hole multiplicity well grows.
Instead,
\[A_b\uparrow\]
while
\[A_c\downarrow.\]
At zero black-hole mass,
\[A_{\mathrm{dS}}=\frac{12\pi}{\Lambda}.\]
At Nariai,
\[A_b=A_c=\frac{4\pi}{\Lambda}.\]
Thus the enclosing area contracts to one third of its pure de Sitter value while the subordinate horizon grows to meet it.
The limit is therefore not a local well filling a fixed box.
It is coupled boundary reorganization.
At the point where the multiplicity gap disappears, the two positive horizon roots also coalesce.
The old two-horizon count classification has lost the distinction required to sustain it.
The physical system need not disappear.
A replacement gravitational landscape can remain.
What fails is the old classification.
The clock and recurrence associated with that replacement landscape are separate questions.
They come next.
The causal hierarchy established here is
shared gravitational relation → coupled horizon multiplicities → multiplicity-gap closure → areal equality → root… [PDF line clipped; author review required.]
Thus:
multiplicity-gap closure occurs before the Nariai limit.
And more specifically:
the Nariai maximum mass is the physical source-coordinate read of an exact gravitational boundary… [PDF line clipped; author review required.]