Addendum 46 — Boundary-Clock Replacement Occurs Before Nariai Temperature
Abstract
Addendum 45 established that the Schwarzschild–de Sitter black-hole and cosmological horizon multiplicity classes lose their distinction at the Nariai limit.
Their multiplicity gap,
\[\Delta S_{c\mid b}=S_c-S_b=\ln\frac{W_c}{W_b},\]
satisfies
\[\Delta S_{c\mid b}>0\]
throughout the ordinary two-horizon regime and approaches
\[\Delta S_{c\mid b}\longrightarrow0\]
when
\[r_b\longrightarrow r_c.\]
At that same physical boundary, the ordinary Schwarzschild–de Sitter horizon surface gravities vanish:
\[\kappa_b^{(0)}\longrightarrow0,\qquad\kappa_c^{(0)}\longrightarrow0.\]
Consequently the corresponding static-patch phase-frequency reads,
\[\omega_h^{(0)}=\frac{\kappa_h^{(0)}}{2\pi c},\]
also vanish.
Read without regard to boundary ownership, this might appear to imply that recurrence itself stops at Nariai.
R²D does not permit that inference.
A physical clock is a calibrated read belonging to a specified count boundary. When the distinction defining that boundary disappears, the physical clock belonging to that classification may itself become degenerate.
Schwarzschild–de Sitter provides an exact example.
The ordinary two-horizon static clock is normalized relative to a reference relation that also degenerates as the two horizon branches coalesce.
Let
\[r_g\]
denote the geodesic static reference radius satisfying
\[f'(r_g)=0.\]
Define the corresponding static normalization factor
\[N_{\mathrm{stat}}\equiv\sqrt{f(r_g)}.\]
As Nariai is approached,
\[N_{\mathrm{stat}}\longrightarrow0.\]
More strongly, with
\[q=\frac{r_b}{r_c}\]
and
\[\epsilon\equiv1-q,\]
the multiplicity gap, both ordinary surface gravities, and the clock normalization all vanish linearly:
\[\Delta S_{c\mid b}=O(\epsilon),\]
\[\kappa_b^{(0)}=O(\epsilon),\]
\[\kappa_c^{(0)}=O(\epsilon),\]
and
\[N_{\mathrm{stat}}=O(\epsilon).\]
The zero of the old clock therefore occurs at the same boundary at which the count distinction defining the old two-horizon landscape disappears.
But the ratio
\[\frac{\kappa_h^{(0)}}{N_{\mathrm{stat}}}\]
has a finite limiting value.
Under the Bousso–Hawking normalization,
\[\kappa_h^{\mathrm{BH}}=\frac{\kappa_h^{(0)}}{N_{\mathrm{stat}}},\]
and at Nariai,
\[\kappa_N=c^2\sqrt{\Lambda}.\]
The corresponding phase-frequency read is
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Only afterward does thermal dimensionalization give
\[k_BT_N=\hbar\omega_N=\frac{\hbar c\sqrt\Lambda}{2\pi}.\]
The finite quantity \(\omega_N\) should not be interpreted as the old static-patch clock recovering from zero.
The old clock satisfies
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
The finite quantity
\[\omega_N\]
belongs to the replacement Nariai landscape.
The old and new clock reads have different boundary owners.
Thus the correct R²D order is
two-horizon count distinction
\(\Downarrow\)
two-horizon clock calibration
\(\Downarrow\)
\[\Delta S_{c\mid b}\longrightarrow0\]
\(\Downarrow\)
\[\kappa_b^{(0)},\kappa_c^{(0)},N_{\mathrm{stat}}\longrightarrow0\]
\(\Downarrow\)
old clock classification becomes degenerate
\(\Downarrow\)
Nariai replacement landscape
\(\Downarrow\)
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}\]
\(\Downarrow\)
\[T_N.\]
Temperature comes last.
Therefore:
boundary-clock replacement occurs before Nariai temperature.
More precisely:
the vanishing Schwarzschild–de Sitter static-patch temperature is the degeneration of a boundary-s
1. Addendum 45 Established the Boundary-Replacement Point
For the ordinary Schwarzschild–de Sitter two-horizon regime,
\[r_b<r_c.\]
The corresponding horizon multiplicities satisfy
\[W_b<W_c.\]
Their logarithmic gap is
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}.\]
Addendum 45 showed
\[\Delta S_{c\mid b}>0\]
for
\[r_b<r_c,\]
while at the Nariai limit,
\[r_b=r_c,\]
we obtain
\[\Delta S_{c\mid b}=0.\]
The old two-horizon count classification therefore becomes marginal at Nariai.
2. The Next Question Is Not Another Area Question
Addendum 45 resolved the distinguishability side of the transition.
The horizon multiplicity distinction disappears.
But R²D keeps distinguishability and recurrence conceptually separate.
The next question is therefore:
What happens to the physical clock associated with the old two-horizon boundary when that boundary classification becomes degenerate?
This is the subject of the present addendum.
3. A Physical Clock Has a Boundary Owner
R²D does not treat physical time as a primitive universal coordinate existing independently of count boundaries.
A physical clock is a later read of recurrence under a specified boundary relation and calibration.
Therefore every physical clock must answer:
Which boundary defines this clock?
A clock cannot automatically be extrapolated unchanged after the boundary relation supplying its calibration has disappeared.
4. Begin With the Schwarzschild–de Sitter Static Patch
The static Schwarzschild–de Sitter metric is
\[ds^2=-f(r)c^2dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2,\]
with
\[f(r)=1-\frac{2GM}{c^2r}-\frac{\Lambda r^2}{3}.\]
The coordinate t is the static-patch time coordinate.
It belongs to the physical classification in which the black-hole and cosmological horizons remain distinct.
5. The Ordinary Horizon Surface Gravities
For the black-hole horizon,
\[\kappa_b^{(0)}=\frac{c^2}{2}\left(\frac1{r_b}-\Lambda r_b\right).\]
For the cosmological horizon, taking the positive magnitude,
\[\kappa_c^{(0)}=\frac{c^2}{2}\left(\Lambda r_c-\frac1{r_c}\right).\]
These are the ordinary static-patch surface-gravity reads.
6. Surface Gravity Is Already a Physical Clock Coordinate
Associate the phase-frequency read
\[\omega_h^{(0)}=\frac{\kappa_h^{(0)}}{2\pi c}.\]
Thus
\[\omega_b^{(0)}=\frac{\kappa_b^{(0)}}{2\pi c},\]
and
\[\omega_c^{(0)}=\frac{\kappa_c^{(0)}}{2\pi c}.\]
These are physical frequency reads.
They are not themselves primitive R²D recurrence counts.
7. The Ordinary Frequencies Vanish at Nariai
At Nariai,
\[r_b,r_c\longrightarrow\frac1{\sqrt\Lambda}.\]
Therefore,
\[\frac1{r_b}-\Lambda r_b\longrightarrow0,\]
and
\[\Lambda r_c-\frac1{r_c}\longrightarrow0.\]
Hence,
\[\kappa_b^{(0)},\kappa_c^{(0)}\longrightarrow0.\]
Consequently,
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
8. The Conventional Static Temperature Therefore Also Vanishes
Using the thermal calibration
\[k_BT_h^{(0)}=\frac{\hbar\kappa_h^{(0)}}{2\pi c},\]
or equivalently,
\[k_BT_h^{(0)}=\hbar\omega_h^{(0)},\]
we obtain
\[T_b^{(0)},T_c^{(0)}\longrightarrow0.\]
This is the familiar zero-temperature static-patch Nariai limit under this normalization.
9. Zero Physical Frequency Does Not Yet Establish Zero Primitive Recurrence
The inference
\[\omega_h^{(0)}\longrightarrow0\]
therefore establishes only
this specified physical frequency read vanishes.
It does not by itself establish
all recurrence has vanished. [Source rejects this inference.]
The distinction is essential because the clock is boundary calibrated.
10. The Old Clock Belongs to the Two-Horizon Classification
The ordinary static-patch coordinate is defined while
\[r_b<r_c.\]
That same regime possesses
\[\Delta S_{c\mid b}>0.\]
Thus the ordinary static clock belongs to the physical landscape in which the two horizon classes remain distinguishable.
Schematically,
\(\Delta S_{c\mid b}>0\longrightarrow\) two-horizon static clock.
The clock should therefore not be assumed to remain primitive after
\[\Delta S_{c\mid b}=0.\]
11. Schwarzschild–de Sitter Has No Unique Asymptotic Clock
Unlike asymptotically flat Schwarzschild geometry, Schwarzschild–de Sitter possesses no spatial infinity at which the timelike Killing vector can be assigned a unique asymptotic normalization.
Therefore surface gravity and horizon temperature require an explicit clock normalization.
This is not a minor technicality.
It identifies the reference relation through which the physical clock is being read.
12. Introduce the Geodesic Static Reference
Between the black-hole and cosmological horizons there is a static radius at which
\[f'(r_g)=0.\]
From
\[f'(r)=\frac{2GM}{c^2r^2}-\frac{2\Lambda r}{3},\]
the reference radius satisfies
\[\frac{GM}{c^2}=\frac{\Lambda r_g^3}{3}.\]
Thus
\[r_g^3=\frac{3GM}{\Lambda c^2}.\]
13. Express the Reference Radius Through the Two Horizons
The Schwarzschild–de Sitter root relations give
\[r_g^3=\frac12r_br_c(r_b+r_c).\]
Using
\[q\equiv\frac{r_b}{r_c},\]
we obtain
\[r_g=r_c\left[\frac{q(1+q)}2\right]^{1/3}.\]
The clock reference therefore changes as the two horizon branches reorganize.
It is not a fixed external observer independent of the gravitational landscape.
14. Define the Static Clock Normalization
Define
\[N_{\mathrm{stat}}\equiv\sqrt{f(r_g)}.\]
Because
\[f'(r_g)=0,\]
we have
\[\frac{2GM}{c^2r_g}=\frac{2\Lambda r_g^2}{3}.\]
Hence,
\[f(r_g)=1-\Lambda r_g^2.\]
Therefore,
\[N_{\mathrm{stat}}^2=1-\Lambda r_g^2.\]
15. Express the Normalization in Terms of q
Addendum 45 gave
\[r_c^2=\frac3{\Lambda(1+q+q^2)}.\]
Therefore,
\[\Lambda r_g^2=\frac3{1+q+q^2}\left[\frac{q(1+q)}2\right]^{2/3}.\]
Thus,
\[N_{\mathrm{stat}}^2=1-\frac3{1+q+q^2}\left[\frac{q(1+q)}2\right]^{2/3}.\]
This is the normalization of the old static clock.
16. The Old Clock Reference Vanishes at Nariai
At Nariai,
\[q\longrightarrow1.\]
Then
\[\left[\frac{q(1+q)}2\right]^{2/3}\longrightarrow1,\]
and
\[1+q+q^2\longrightarrow3.\]
Therefore,
\[N_{\mathrm{stat}}^2\longrightarrow1-\frac33=0.\]
Hence,
\[N_{\mathrm{stat}}\longrightarrow0.\]
The static frequencies vanish at the same time that the reference normalization defining their clock also vanishes.
17. This Creates a 0/0 Structure
At Nariai,
\[\kappa_h^{(0)}\longrightarrow0,\]
while
\[N_{\mathrm{stat}}\longrightarrow0.\]
Thus a boundary-normalized surface gravity has the form
\[\frac{\kappa_h^{(0)}}{N_{\mathrm{stat}}}\longrightarrow\frac00.\]
The numerator alone therefore cannot determine the limiting physical clock.
The normalization is disappearing with it.
18. Parameterize the Boundary Separation Define
\[\epsilon\equiv1-q.\]
The Nariai limit is
\[\epsilon\longrightarrow0.\]
This parameter directly measures the remaining separation of the two horizon-radius reads.
19. The Static Normalization Vanishes Linearly
Near Nariai,
\[N_{\mathrm{stat}}=\frac\epsilon2+O(\epsilon^2).\]
Thus the reference supporting the ordinary static clock disappears linearly with the horizon-branch distinction.
20. The Black-Hole Surface Gravity Vanishes at the Same Rate
Near Nariai,
\[\kappa_b^{(0)}=\frac{c^2\sqrt\Lambda}{2}\epsilon+O(\epsilon^2).\]
Therefore,
\[\kappa_b^{(0)}=O(\epsilon).\]
21. The Cosmological Surface Gravity Does Likewise
Similarly,
\[\kappa_c^{(0)}=\frac{c^2\sqrt\Lambda}{2}\epsilon+O(\epsilon^2).\]
Thus,
\[\kappa_c^{(0)}=O(\epsilon).\]
The two ordinary horizon clock reads vanish together.
22. The Multiplicity Gap Also Vanishes Linearly
From Addendum 45,
\[\Delta S_{c\mid b}=\frac{3\pi}{\Lambda\ell_P^2}\frac{1-q^2}{1+q+q^2}.\]
Near
\[q=1,\]
this becomes
\[\Delta S_{c\mid b}=\frac{2\pi}{\Lambda\ell_P^2}\epsilon+O(\epsilon^2).\]
Therefore,
\[\Delta S_{c\mid b}=O(\epsilon).\]
24. The Clock Degeneracy Tracks the Count-Boundary Degeneracy
The simultaneous scaling gives the physical sequence
\[\Delta S_{c\mid b}\longrightarrow0\]
together with
\[N_{\mathrm{stat}}\longrightarrow0\]
and
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
The old horizon distinction and the physical clock belonging to that distinction disappear together.
That is the R²D-relevant observation.
25. The Appropriate Question Is No Longer “Why Does Recurrence Stop?”
Nothing in the calculation establishes that primitive recurrence stops.
What it establishes is
the old boundary can no longer support the same clock read.
The question therefore changes from
Why does time stop at Nariai?
to
What physical clock, if any, belongs to the replacement boundary classification?
26. This Is a Clock-Version of the Gibbs Problem
Addenda 44 and 45 identified a Gibbs-type classification error:
remove the defining multiplicity distinction + retain the old state classification.
The clock analogue is
remove the clock-defining boundary + retain its clock unchanged.
Once
\[\Delta S_{c\mid b}=0,\]
the old two-horizon clock should not automatically be extrapolated beyond the classification that defined it.
27. A Replacement Landscape Is Required
The exact Nariai limit is not most naturally represented by merely setting
\[r_b=r_c\]
inside an otherwise unchanged static coordinate system.
A rescaled limiting geometry gives the nontrivial Nariai spacetime.
Conventionally this has the product structure
\[\mathrm{dS}_2\times S^2.\]
The old static-patch coordinate degenerates while a finite limiting geometry remains.
This is precisely what boundary replacement should look like physically.
28. Replacement Does Not Restore the Old Horizon Distinction
The Nariai landscape does not recreate
\[r_b<r_c.\]
The old multiplicity gap remains closed.
Thus replacement does not mean
recover old two-horizon classification.
It means
new readable organization after old classification becomes marginal.
29. Bousso–Hawking Normalization Supplies a Different Clock Read
Normalize the static Killing field relative to the geodesic static reference:
\[\xi_{\mathrm{BH}}=\frac1{N_{\mathrm{stat}}}\partial_t.\]
Surface gravity rescales accordingly:
\[\kappa_h^{\mathrm{BH}}=\frac{\kappa_h^{(0)}}{N_{\mathrm{stat}}}.\]
This is not the same clock normalization as the unnormalized static-patch read.
30. The Normalized Ratio Has a Finite Limit
Since both numerator and denominator vanish linearly,
\[\lim_{q\to1}\frac{\kappa_b^{(0)}}{N_{\mathrm{stat}}}\]
exists.
Likewise for the cosmological horizon.
Using the leading-order forms,
\[\frac{(c^2\sqrt\Lambda/2)\epsilon}{\epsilon/2}=c^2\sqrt\Lambda.\]
Therefore,
\[\lim_{q\to1}\kappa_b^{\mathrm{BH}}=\lim_{q\to1}\kappa_c^{\mathrm{BH}}=c^2\sqrt\Lambda.\]
31. Define the Replacement Nariai Surface Gravity
The common finite limiting value is
\[\kappa_N=c^2\sqrt{\Lambda}.\]
This belongs to the boundary-normalized Nariai read.
It is not equal to either old unnormalized surface gravity at the limiting point, because those satisfy
\[\kappa_b^{(0)},\kappa_c^{(0)}\longrightarrow0.\]
32. The Replacement Phase-Frequency Read Is Finite
Convert the normalized surface gravity to a phase-frequency read:
\[\omega_N=\frac{\kappa_N}{2\pi c}.\]
Therefore,
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
This is the finite physical recurrence calibration of the replacement Nariai landscape.
33. The New Clock Is Not the Old Clock Rescued From Zero
The old clock satisfies
\[\omega_b^{(0)}\longrightarrow0,\qquad\omega_c^{(0)}\longrightarrow0.\]
The replacement read satisfies
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}\ne0.\]
There is no contradiction because the clock reads have different boundary owners.
Thus:
\[\omega_b^{(0)}\ne\omega_N\]
and
\[\omega_c^{(0)}\ne\omega_N\]
as boundary-defined physical quantities.
34. A Limiting Ratio Does Not Imply Continuity of Clock Identity
The mathematical relation
\[\omega_N=\lim\frac{\omega_h^{(0)}}{N_{\mathrm{stat}}}\]
does not mean that the unnormalized old clock remains the same physical object throughout the boundary replacement.
The ratio defines the limiting clock under a different normalization.
R²D therefore distinguishes:
continuity of a limiting relation
from
identity of the boundary-defined physical coordinate.
35. The Replacement Geometry Has Its Own Scale
At Nariai,
\[r_N=\frac1{\sqrt\Lambda}.\]
The replacement clock is
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Therefore,
\[2\pi r_N\omega_N=c.\]
Equivalently,
\[r_N=\frac{c}{2\pi\omega_N}.\]
The replacement geometry and its physical phase-clock calibration are mutually consistent.
36. This Is a Physical Cycle–Phase Relation, Not a New Primitive R²D Law
The equation
\[2\pi r_N\omega_N=c\]
should not be reinterpreted as a primitive relation between R²D distinguishability and occurrence count.
It is already written in physical variables:
\[r_N,\quad\omega_N,\quad c.\]
It therefore belongs to the physical projection.
This is especially important because R²D does not contain a primitive law of the form
\[dr/d\lambda=2\pi\lambda.\]
No such relation is introduced here.
37. Nor Is This the Planck Radial-Distinction Relation
Addendum 43 proposed
\[\Delta r=\ell_P\]
as the physical calibration of one elementary radial macrostate distinction.
The Nariai relation
\[r_N=\frac{c}{2\pi\omega_N}\]
has a different meaning.
It relates the entire replacement horizon scale to a boundary-specific phase-frequency read.
Thus:
\[\Delta r=\ell_P\]
and
\[r_N=\frac{c}{2\pi\omega_N}\]
must remain distinct.
38. Temperature Is a Later Calibration Still
Only after the finite phase-frequency read is identified do we write
\[k_BT_N=\hbar\omega_N.\]
Therefore,
\[k_BT_N=\frac{\hbar c\sqrt\Lambda}{2\pi}.\]
And
\[T_N=\frac{\hbar c\sqrt\Lambda}{2\pi k_B}.\]
Temperature is the dimensional thermal read of the replacement clock.
It is not the primitive source of that clock.
39. Zero Old Temperature and Finite New Temperature Are Compatible
We therefore have
\[T_b^{(0)},T_c^{(0)}\longrightarrow0\]
while
\[T_N\ne0.\quad\text{[PDF callout clipped; author review required.]}\]
These statements belong to different physical clock normalizations and different boundary classifications.
Thus:
\[T^{(0)}\longrightarrow0\]
does not imply
\[T_N=0.\]
Nor does either equation by itself establish anything about primitive recurrence independently of its physical calibration.
40. The Conventional Normalization Ambiguity Becomes Informative
Schwarzschild–de Sitter admits different horizon-temperature prescriptions because it lacks a unique asymptotic clock and its horizons are not generally in ordinary thermal equilibrium.
R²D does not require one of these prescriptions to be declared the primitive clock of all states.
Instead, the ambiguity asks a more useful question:
Which boundary relation owns each clock?
The ordinary static clock belongs to the old two-horizon landscape.
The finite Nariai clock belongs to the replacement landscape.
41. What the Physical Calculation Actually Establishes
The physical calculation establishes:
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0,\]
while
\[N_{\mathrm{stat}}\longrightarrow0.\]
It also establishes the finite boundary-normalized limit
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Therefore,
zero old clock \(\not\Rightarrow\) zero every possible clock. [PDF glyph clipped; author review required.]
That is the experimentally and mathematically supportable statement.
42. What R²D Adds Is Boundary Ownership
R²D interprets the simultaneous disappearance of
\[\Delta S_{c\mid b},\]
\[\omega_b^{(0)},\]
\[\omega_c^{(0)},\]
and
\[N_{\mathrm{stat}}\]
as degeneration of one boundary-defined clock architecture.
The finite normalized Nariai clock is then assigned to the replacement boundary classification.
This is an interpretation of the physical structure.
It is not an independent derivation of the Nariai temperature from primitive R²D counting.
43. Principle — Clock Boundary Ownership
A physical clock is a recurrence read calibrated relative to a specified count boundary.
Therefore,
physical clock = boundary-owned physical read.
If the boundary relation supplying that clock ceases to define the same count domain, its physical frequency may vanish, diverge, or become undefined without establishing that every recurrence relation has vanished.
44. Principle — Clock Degeneracy
For the Schwarzschild–de Sitter two-horizon landscape,
\[\Delta S_{c\mid b}\longrightarrow0\]
occurs together with
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0\]
and
\[N_{\mathrm{stat}}\longrightarrow0.\]
The old clock degenerates with the distinction supporting its boundary classification.
45. Principle — Clock Replacement
The boundary-normalized clock relation
\[\omega_h^{\mathrm{BH}}=\frac{\omega_h^{(0)}}{N_{\mathrm{stat}}}.\]
has the finite Nariai limit
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
R²D assigns this finite physical recurrence read to the replacement Nariai landscape.
It is not the unchanged old static clock.
46. Corollary — Geometry and Clock Recalibrate Together
At Nariai,
\[r_N=\frac1{\sqrt\Lambda},\]
and
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
Therefore,
\[2\pi r_N\omega_N=c.\]
The replacement geometry and replacement physical clock belong to one mutually calibrated physical boundary read.
47. Corollary — Static-Patch Zero Temperature Is Boundary Specific
The ordinary static-patch temperatures satisfy
\[T_b^{(0)},T_c^{(0)}\longrightarrow0.\]
But the replacement Nariai read satisfies
\[T_N=\frac{\hbar c\sqrt\Lambda}{2\pi k_B}.\]
Therefore the phrase
“Nariai has zero temperature”
is incomplete unless the clock normalization and boundary owner are specified.
48. The Central Gibbs-Type Warning The forbidden extrapolation is
remove the count boundary + retain its clock unchanged.
A clock is not more boundary independent than the states, multiplicities, or spatial coordinates from which its physical meaning was constructed.
This is the temporal analogue of extending a count classification beyond the disappearance of its defining distinctions.
49. Boundary Replacement Does Not Require Conservation of Clock Value
Across the replacement,
\[\omega^{(0)}\]
does not remain numerically invariant.
Indeed,
\[\omega^{(0)}\longrightarrow0\]
while
\[\omega_N\ne0.\quad\text{[PDF callout clipped; author review required.]}\]
R²D does not require dimensional physical coordinates to remain fixed across a change of count boundary.
The next question must therefore be whether some deeper relation survives when the individual physical reads change.
50. That Invariance Question Is Deferred
The present addendum establishes clock replacement.
It does not yet claim what count relation is preserved across that replacement.
In particular, it does not yet derive or interpret any invariant involving
\[S\]
and
\[\omega.\]
That is the subject of the next addendum.
51. Logical Status
Established in Addendum 45
The Schwarzschild–de Sitter multiplicity gap satisfies
\[\Delta S_{c\mid b}>0\]
throughout the ordinary two-horizon regime and
\[\Delta S_{c\mid b}\longrightarrow0\]
at Nariai.
The old two-horizon count classification becomes marginal.
Conventional Schwarzschild–de Sitter Input
The ordinary horizon surface gravities are
\[\kappa_b^{(0)}=\frac{c^2}{2}\left(\frac1{r_b}-\Lambda r_b\right),\]
and
\[\kappa_c^{(0)}=\frac{c^2}{2}\left(\Lambda r_c-\frac1{r_c}\right).\]
They vanish as
\[r_b\longrightarrow r_c.\]
Conventional Reference Normalization
The geodesic static reference satisfies
\[f'(r_g)=0,\]
and the corresponding clock normalization is
\[N_{\mathrm{stat}}=\sqrt{f(r_g)}.\]
At Nariai,
\[N_{\mathrm{stat}}\longrightarrow0.\]
Derived Near-Nariai Scaling
For
\[\begin{aligned}\epsilon&=1-q,\\N_{\mathrm{stat}}&=\epsilon/2+O(\epsilon^2),\\\kappa_b^{(0)}&=(c^2\sqrt\Lambda/2)\epsilon+O(\epsilon^2),\\\kappa_c^{(0)}&=(c^2\sqrt\Lambda/2)\epsilon+O(\epsilon^2).\end{aligned}\]
and
\[\Delta S_{c\mid b}=\frac{2\pi}{\Lambda\ell_P^2}\epsilon+O(\epsilon^2).\]
Conventional Bousso–Hawking Limit
With
\[\kappa_h^{\mathrm{BH}}=\frac{\kappa_h^{(0)}}{N_{\mathrm{stat}}},\]
the Nariai limit is
\[\kappa_N=c^2\sqrt{\Lambda}.\]
Therefore,
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
And under thermal dimensionalization,
\[k_BT_N=\frac{\hbar c\sqrt\Lambda}{2\pi}.\]
New R²D Interpretation
The ordinary zero-frequency limit is interpreted as degeneration of the physical clock belonging to the disappearing two-horizon classification.
The finite normalized Nariai frequency is interpreted as a physical clock read belonging to the replacement landscape.
Not Established Here
This addendum does not establish:
• that \(\omega_N\) is primitive R²D recurrence; • that Bousso–Hawking normalization is the uniquely fundamental temperature prescription for every Schwarzschild–de Sitter state; • that primitive recurrence has been proven to persist across the transition; • that the old and new clocks are the same physical coordinate; • that the Nariai geometry is derived from primitive R²D counting; • or what invariant count relation, if any, survives the boundary replacement.
Those questions remain separate.
52. The Central R²D Inversion
The conventional description may begin with
\[\kappa_b^{(0)},\kappa_c^{(0)}\longrightarrow0\]
and conclude
\[T\longrightarrow0.\]
R²D asks first which boundary owns that clock.
The fuller sequence is
two distinguishable horizon multiplicity classes
\(\Downarrow\)
two-horizon static boundary
\(\Downarrow\)
static clock normalization \(N_{\mathrm{stat}}\)
\(\Downarrow\)
\[\omega_b^{(0)},\omega_c^{(0)}\]
while
\[\Delta S_{c\mid b}>0.\]
Then,
\[\Delta S_{c\mid b}\longrightarrow0\]
\(\Downarrow\)
\[N_{\mathrm{stat}}\longrightarrow0\]
and
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0\]
\(\Downarrow\)
old two-horizon clock becomes degenerate
\(\Downarrow\)
Nariai replacement boundary
\(\Downarrow\)
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}\]
\(\Downarrow\)
\[T_N=\frac{\hbar c\sqrt\Lambda}{2\pi k_B}.\]
The temperature is the final dimensional read.
53. Conclusion
The Nariai limit does not merely merge two horizon areas.
It also destroys the physical clock normalization belonging to the ordinary Schwarzschild–de Sitter two- horizon static patch.
Addendum 45 established that the multiplicity distinction between the black-hole and cosmological horizons vanishes:
\[\Delta S_{c\mid b}=\ln\frac{W_c}{W_b}\longrightarrow0.\]
At precisely the same boundary,
\[\kappa_b^{(0)},\kappa_c^{(0)}\longrightarrow0.\]
Thus the ordinary phase-frequency reads satisfy
\[\omega_b^{(0)},\omega_c^{(0)}\longrightarrow0.\]
But the reference defining that old static clock also degenerates.
For the geodesic static reference,
\[N_{\mathrm{stat}}=\sqrt{f(r_g)}\longrightarrow0.\]
Near Nariai, with
\[\epsilon=1-\frac{r_b}{r_c},\]
the four relations
\[\Delta S_{c\mid b},\]
\[\kappa_b^{(0)},\]
\[\kappa_c^{(0)},\]
and
\[N_{\mathrm{stat}}\]
all vanish linearly in
\[\epsilon.\]
The old horizon distinction and its old physical clock therefore become degenerate together.
The correct limiting object is not the numerator alone.
It is the boundary-normalized ratio
\[\frac{\kappa_h^{(0)}}{N_{\mathrm{stat}}}.\]
Its finite limit is
\[\kappa_N=c^2\sqrt{\Lambda}.\]
Therefore the replacement Nariai landscape possesses the finite phase-frequency read
\[\omega_N=\frac{c\sqrt\Lambda}{2\pi}.\]
This does not mean the old clock survives unchanged.
The old clock goes to zero.
The replacement boundary possesses a different clock calibration.
Only afterward does thermal dimensionalization produce
\[T_N=\frac{\hbar c\sqrt\Lambda}{2\pi k_B}.\]
The zero of the old static-patch temperature therefore cannot, by itself, establish disappearance of all recurrence.
It establishes the disappearance of a specified boundary-owned clock.
The physical lesson is the same one that R²D has repeatedly exposed elsewhere:
a projected coordinate can vanish because the boundary defining its calibration is vanishing with it.
The clock does not define the boundary.
The boundary defines the clock.
When the boundary classification changes, the clock may change with it.
The logical hierarchy is therefore
count distinction → clock boundary → physical recurrence read → boundary replacement → replacement clock → … [PDF line clipped; author review required.]
Thus:
boundary-clock replacement occurs before Nariai temperature.
And more specifically:
the vanishing ordinary Schwarzschild–de Sitter horizon temperature is the degeneration of the clock… [PDF line clipped; author review required.]