Addendum 44 — Entropic-Well Depth Occurs Before Maximum Gravitational Mass
Abstract
A maximum gravitational mass is ordinarily posed as a property of matter or of a gravitational solution.
Recursively Realized Distinguishability (R²D) begins earlier.
Addendum 43 established that a Schwarzschild horizon can be represented as a boundary-defined count domain containing radially distinguishable macrostates. Their multiplicities are
\[W_{H,i},\]
with logarithmic coordinates
\[S_{H,i}=\ln W_{H,i}.\]
The horizon boundary is physically projected through area,
\[B_H\longrightarrow A_H,\]
while distinguishability within the boundary is projected through radius,
\[\lambda_H\longrightarrow r_H.\]
Under the Bekenstein–Hawking physical calibration,
\[S_H=\frac{A_H}{4\ell_P^2},\]
and under the candidate Planck radial calibration,
\[r_i=i\ell_P.\]
The present addendum asks what it means for such a gravitational multiplicity landscape to become deep, and whether that depth can possess an intrinsic maximum.
Define the depth of a gravitational multiplicity well relative to a reference macrostate by
\[D_H\equiv S_H-S_0=\ln\frac{W_H}{W_0}.\]
For the convenient reference
\[W_0=1,\]
this becomes
\[D_H=S_H.\]
This does not identify horizon entropy with gravitational retention.
Retention and multiplicity remain distinct:
\[R_H\ne S_H.\]
The well depth measures separation in logarithmic multiplicity, not retained possibility.
For the Planck-distinguished radial horizon construction,
\[D_H=\pi N^2,\]
where
\[N=\frac{r_H}{\ell_P}.\]
Under the Schwarzschild mapping
\[r_H=\frac{2GM}{c^2},\]
this becomes
\[D_H=\frac{4\pi GM^2}{\hbar c}.\]
Thus physical gravitational mass is a downstream coordinate of multiplicity-well depth:
\[M=\sqrt{\frac{\hbar c}{4\pi G}}\sqrt{D_H}.\]
Nothing in this relation imposes a maximum value of DH or M .
A maximum gravitational mass therefore cannot be intrinsic to the subordinate well alone.
It requires an enclosing boundary.
Let the subordinate gravitational boundary have multiplicity
\[W_H\]
and let an enclosing gravitational count boundary have multiplicity
\[W_E.\]
Define the subordinate–enclosing multiplicity gap
\[\Delta S_{E\mid H}=S_E-S_H=\ln\frac{W_E}{W_H}.\]
As long as
\[\Delta S_{E\mid H}>0,\]
the subordinate and enclosing count classes remain distinguishable under the nested classification considered here.
The marginal condition is
\[\Delta S_{E\mid H}=0.\]
Then
\[S_E=S_H\]
and therefore
\[W_E=W_H.\]
The multiplicity distinction defining the subordinate/enclosing relation has vanished.
If both gravitational boundaries are read through the same areal multiplicity calibration,
\[S=\frac A{4\ell_P^2},\]
then
\[\Delta S_{E\mid H}=\frac{A_E-A_H}{4\ell_P^2}.\]
Therefore the count-level marginal condition projects physically as
\[A_E=A_H.\]
Maximum gravitational mass is then the later physical source coordinate assigned to this prior loss of subordinate–enclosing multiplicity distinction.
The logical order is therefore
multiplicity well → well depth → enclosing multiplicity gap → gap closure → areal equality → maximum mass.
A formal continuation of the original subordinate/enclosing classification after
\[W_E=W_H\]
would preserve a distinction no longer supported by the defining count relation.
This is the gravitational analogue of a Gibbs-type boundary-classification failure.
The present addendum does not identify the physical solution that realizes this marginal condition.
That is the purpose of Addendum 45.
Its central claim is narrower:
a maximum gravitational mass is not an intrinsic property of a local mass.
Rather,
gravitational mass saturation is the projected physical read of a prior enclosing multiplicity limit.
Thus:
entropic-well depth occurs before maximum gravitational mass.
1. Addendum 43 Separates Multiplicity From Retention
Addendum 42 established the completely retained Schwarzschild horizon condition
\[P_H=R_H.\]
Addendum 43 then established that horizon multiplicity is a different relation.
The horizon macrostate multiplicity is
\[W_H,\]
with logarithmic coordinate
\[S_H=\ln W_H.\]
Therefore,
\[R_H\ne S_H\]
in general.
The old shortcut
\[R_H=S_H\]
must not be used.
Retention measures how an enclosing directional relation is partitioned.
Multiplicity measures how many compatible realizations belong to a macrostate.
They occupy different logical positions.
2. A Gravitational Multiplicity Well Is Defined by W
Consider the horizon macrostate landscape
\[\{W_{H,i}\}.\]
Its logarithmic form is
\[\{S_{H,i}\},\]
where
\[S_{H,i}=\ln W_{H,i}.\]
A larger W means that more compatible realizations belong to that macrostate.
If the landscape is conventionally represented as
\[-S,\]
then greater S corresponds to a deeper position in the represented entropic well.
The well itself remains a count structure.
3. Define Multiplicity-Well Depth
Choose a reference macrostate 0.
Define the depth of macrostate i relative to that reference by
\[D_{H,i\mid0}=S_{H,i}-S_{H,0}.\]
Since
\[S=\ln W,\]
\[D_{H,i\mid0}=\ln\frac{W_{H,i}}{W_{H,0}}.\]
This is a dimensionless count difference.
No force, energy, mass, radius, or area is required to define it.
4. A Unit Reference Simplifies the Well
If the reference multiplicity is chosen as
\[W_{H,0}=1,\]
then
\[S_{H,0}=0.\]
Therefore,
\[D_{H,i}=S_{H,i}.\]
At the realized horizon macrostate N ,
\[D_H=S_H.\]
This equality is a reference choice within the multiplicity landscape.
It is not an identification of entropy with retention.
5. Well Depth Is Also a Relative Realization Burden
Under proportional realization,
\[\Delta\tau\approx\Delta S.\]
Therefore, across the full reference-to-horizon macrostate difference,
\[\Delta\tau_{H\mid0}\approx D_H.\]
In the older physical recurrence calibration,
\[T=t_cW=t_ce^S.\]
Thus
\[\frac{T_H}{T_0}\approx e^{D_H}.\]
A deep multiplicity well therefore possesses a large relative realization burden.
This is a physical realization of the count depth.
It does not define the depth.
6. Complete Retention Allows Depth to Be Read Through Occupancy
For a completely retained stationary macrostate relation,
\[\Delta P=\Delta R.\]
Hence
\[\Delta A=0.\]
The occupancy relation becomes
\[\Delta G\approx\Delta S.\]
Therefore,
\[\ln\frac{c_H}{c_0}\approx D_H.\]
Equivalently,
\[\frac{c_H}{c_0}\approx e^{D_H}=\frac{W_H}{W_0}.\]
The depth of the multiplicity well is therefore realized as an occupancy contrast at complete retention.
7. Addendum 43 Supplies the Radial Macrostate Construction
Under the candidate radial calibration,
\[r_i=i\ell_P.\]
The spherical boundary projection is
\[A_i=4\pi r_i^2.\]
Therefore,
\[A_i=4\pi i^2\ell_P^2.\]
Using the Bekenstein–Hawking physical calibration,
\[S_i=\frac{A_i}{4\ell_P^2},\]
gives
\[S_i=\pi i^2.\]
8. Each Radial Step Deepens the Multiplicity Well
For
\[i\longrightarrow i+1,\]
the logarithmic multiplicity increment is
\[\Delta S_i=S_{i+1}-S_i.\]
Therefore,
\[\Delta S_i=\pi(2i+1).\]
Each additional radial macrostate distinction adds a larger logarithmic multiplicity increment than the preceding one.
The radial coordinate grows linearly.
The accumulated multiplicity depth grows quadratically.
9. Sum the Well Depth
After N radial distinctions,
\[D_H=\sum_{i=0}^{N-1}\Delta S_i.\]
Therefore,
\[D_H=\pi\sum_{i=0}^{N-1}(2i+1).\]
Because
\[\sum_{i=0}^{N-1}(2i+1)=N^2,\]
we obtain
\[D_H=\pi N^2.\]
This is the horizon multiplicity-well depth.
10. Radial Distinguishability Gives the Physical Depth Scale
Since
\[r_H=N\ell_P,\]
we have
\[N=\frac{r_H}{\ell_P}.\]
Thus
\[D_H=\pi\frac{r_H^2}{\ell_P^2}.\]
The radial physical coordinate is therefore a projection of the prior count depth.
11. Boundary Area Gives the Equivalent Physical Read
Since
\[A_H=4\pi r_H^2,\]
we can also write
\[D_H=\frac{A_H}{4\ell_P^2}.\]
Again, area does not define the primitive depth.
It is the gravitational boundary read through which the depth is physically calibrated.
12. The Schwarzschild Mapping Introduces Mass
Only now introduce
\[r_H=\frac{2GM}{c^2}.\]
Substituting into
\[D_H=\pi\frac{r_H^2}{\ell_P^2}\]
gives
\[D_H=\pi\frac{4G^2M^2/c^4}{\hbar G/c^3}.\]
Therefore,
\[D_H=\frac{4\pi GM^2}{\hbar c}.\]
Mass appears after the multiplicity-well depth has already been defined.
13. Invert the Relation
Solving for M ,
\[M^2=\frac{\hbar c}{4\pi G}D_H.\]
Thus
\[M=\sqrt{\frac{\hbar c}{4\pi G}}\sqrt{D_H}.\]
Define the Planck mass
\[m_P=\sqrt{\frac{\hbar c}{G}}.\]
Then
\[M=\frac{m_P}{2\sqrt\pi}\sqrt{D_H}.\]
The gravitational mass coordinate is therefore proportional to the square root of multiplicity-well depth.
14. Mass Squared Is a Count-Depth Read
The familiar relation
\[S_H\propto M^2\]
is therefore inverted:
\[M^2\propto D_H.\]
Mass squared is the physical source-coordinate read of horizon multiplicity depth.
The count difference comes first.
The dimensional mass comes later.
15. Why the Quadratic Dependence Appears
The radial macrostate count satisfies
\[N\propto r_H.\]
Schwarzschild gives
\[r_H\propto M.\]
The multiplicity-well depth satisfies
\[D_H\propto N^2.\]
Therefore,
\[D_H\propto N^2\propto r_H^2\propto M^2.\]
The quadratic dependence does not require entropy to be a substance produced by mass.
It follows from the radial organization of the boundary count domain and its spherical projection.
16. There Is No Maximum in the Local Well-Depth Relation
The relation
\[D_H=\pi N^2\]
contains no upper limit on N .
Likewise,
\[M=\frac{m_P}{2\sqrt\pi}\sqrt{D_H}\]
contains no intrinsic maximum mass.
Therefore,
local multiplicity depth alone does not define \(M_{\max}\).
A maximum requires another relation.
17. Maximum Mass Requires an Enclosing Boundary
R²D quantities are boundary defined.
A subordinate gravitational count domain cannot define its own maximum relative to nothing.
Introduce an enclosing gravitational count boundary
\[B_E.\]
Let its relevant multiplicity be
\[W_E,\]
with logarithmic coordinate
\[S_E=\ln W_E.\]
The subordinate horizon boundary has
\[W_H,\qquad S_H=\ln W_H.\]
The limiting problem is now relational.
18. Define the Subordinate–Enclosing Multiplicity Gap
Define
\[\Delta S_{E\mid H}=S_E-S_H.\]
Equivalently,
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
This quantity asks whether the subordinate and enclosing gravitational count classes remain multiplicatively distinguishable.
19. Positive Gap Defines a Distinct Nested Classification
For the nested gravitational classification considered here, require
\[\Delta S_{E\mid H}>0.\]
Then
\[W_E>W_H.\]
The enclosing class remains distinguishable from the subordinate gravitational well.
The nested relation
\[B_H\subset B_E\]
continues to possess a positive multiplicity distinction.
20. Saturation Is Gap Closure
The marginal condition occurs when
\[\Delta S_{E\mid H}=0.\]
Then
\[S_E=S_H.\]
Since
\[S=\ln W,\]
it follows that
\[W_E=W_H.\]
The subordinate and enclosing states no longer differ by the multiplicity coordinate that defined their nested gravitational classification.
21. This Is the Primitive Saturation Condition
The primitive condition is therefore not initially
\[M=M_{\max}.\]
Nor is it initially
\[r_H=r_E.\]
Nor is it initially
\[A_H=A_E.\]
It is
\[\ln\frac{W_E}{W_H}=0.\]
The count distinction vanishes first.
22. Saturation Is Not Complete Retention
The condition
\[P_H=R_H\]
belongs to horizon retention.
The condition
\[W_E=W_H\]
belongs to subordinate–enclosing multiplicity saturation.
These must remain separate.
Thus
\[P_H=R_H\;\not\Rightarrow\;W_E=W_H.\]
A Schwarzschild horizon may be completely retained while remaining well below its enclosing multiplicity limit.
23. Saturation Is Not the Schwarzschild Horizon
The Schwarzschild horizon is a readability horizon.
The relevant exterior-directed recurrence becomes unreadable while the gravitational boundary remains realized.
Multiplicity saturation is different.
At saturation, the distinction supporting the old subordinate/enclosing count classification itself becomes marginal.
Therefore,
Schwarzschild horizon \(\ne\) enclosing marginality. [PDF callout clipped; author review required.]
The horizon supplies the subordinate well.
The enclosing relation determines whether that well remains separately classifiable.
24. Apply a Common Horizon-Area Calibration
Suppose both the subordinate and enclosing gravitational boundaries are represented by the same physical multiplicity calibration
\[S=\frac A{4\ell_P^2}.\]
Then
\[S_H=\frac{A_H}{4\ell_P^2},\]
and
\[S_E=\frac{A_E}{4\ell_P^2}.\]
Therefore,
\[\Delta S_{E\mid H}=\frac{A_E-A_H}{4\ell_P^2}.\]
25. Multiplicity Equality Projects Onto Area Equality
At saturation,
\[\Delta S_{E\mid H}=0.\]
Therefore,
\[A_E-A_H=0.\]
Hence,
\[A_E=A_H.\]
Area equality is therefore the physical gravitational projection of prior multiplicity-gap closure.
The area relation does not create the saturation.
It reads it.
26. Boundary Before Area Applies Again
The logical order is
\[W_E=W_H\]
followed under common calibration by
\[S_E=S_H\]
and then
\[A_E=A_H.\]
The equality of the projected areas therefore occurs after the loss of the count distinction they represent.
27. Radius Equality Is Still Later
For spherical projection,
\[A=4\pi r_A^2.\]
Therefore,
\[A_E=A_H\]
implies
\[r_E=r_H.\]
Thus the full order is
\[W_E=W_H\longrightarrow S_E=S_H\longrightarrow A_E=A_H\longrightarrow r_E=r_H.\]
The radius equality is the radial representation of prior boundary-count equality.
28. Maximum Mass Comes After the Enclosing Relation Is Specified
Only after the enclosing boundary has been physically identified can the marginal condition be translated into a mass coordinate.
Schematically,
\[A_H(M)=A_E.\]
Solving this relation produces some
\[M_{\max}.\]
Therefore,
\(M_{\max}\) = physical source coordinate of enclosing multiplicity saturation.
Maximum mass is not primitive.
29. There Is No Boundary-Independent Maximum Mass
Because
\[W_E\]
belongs to the enclosing boundary,
\[M_{\max}\]
must depend on which enclosing gravitational relation is being considered.
Therefore,
\(M_{\max}\) is boundary relative.
There is no generic R²D mass ceiling defined independently of enclosing structure.
30. The Enclosing Boundary Is Not a Fixed Container
Nothing in the count architecture requires
\[W_E\]
or
\[A_E\]
to remain fixed while the subordinate gravitational well changes.
The subordinate and enclosing boundaries may belong to one coupled realization.
Therefore the generic saturation condition is
\[W_E(M)=W_H(M)\]
or, under common areal calibration,
\[A_E(M)=A_H(M).\]
This distinction becomes essential in the next addendum.
31. A Fixed-Capacity Approximation Is Only Provisional
One may first approximate the enclosing capacity as fixed:
\[W_E\approx\text{constant}\]
or
\[A_E\approx\text{constant}.\]
Then increasing subordinate well depth eventually produces
\[W_H\longrightarrow W_E.\]
But if the enclosing gravitational boundary responds to the same physical realization, its own capacity may change simultaneously.
Therefore a physically exact maximum-mass solution must determine both boundaries together.
32. This Is a Gibbs-Type Boundary Problem
Consider formal continuation past
\[\Delta S_{E\mid H}=0.\]
The original nested classification would then require
\[\Delta S_{E\mid H}<0,\]
or
\[W_H>W_E,\]
while still calling H subordinate to E under the same multiplicity relation.
But the multiplicity distinction that supported that classification has already vanished at zero gap.
The old labels are being preserved after their defining count difference disappears.
That is the gravitational analogue of a Gibbs-type classification problem.
33. The Problem Is Not Too Much Substance in Too Little Space
The conventional intuition is
too much mass + too little space \(\longrightarrow\) collapse.
R²D begins instead with count classification:
subordinate multiplicity \(\longrightarrow\) enclosing multiplicity.
The limiting event is
loss of the distinction required to classify one as subordinate to the other.
The mass and geometry are later physical reads of that boundary event.
34. Landscape Replacement Can Follow Saturation
When
\[W_E=W_H,\]
R²D does not require the physical system to cease existing.
Rather, the previous subordinate/enclosing count classification may cease to be adequate.
A new boundary classification may replace it.
Therefore,
multiplicity-gap closure \(\not\Rightarrow\) annihilation.
It indicates marginality of the old count landscape.
35. This Is Distinct From a Readability Horizon
At a readability horizon, a recursively realized relation remains physically organized while one coordinate becomes unreadable from the selected observing boundary.
At multiplicity saturation, the old nested classification loses the count distinction required to define its two classes.
Thus:
readability horizon \(\ne\) landscape saturation.
This distinction must remain explicit in the gravitational program.
36. Define the Relative Multiplicity-Capacity Coordinate
A useful dimensionless ratio is
\[\sigma_{H\mid E}\equiv\frac{S_H}{S_E}.\]
when both quantities are positive.
Under the common areal calibration,
\[\sigma_{H\mid E}=\frac{A_H}{A_E}.\]
Then
\[0\le\sigma_{H\mid E}<1\]
describes a distinguishable subordinate well,
while
\[\sigma_{H\mid E}=1\]
is marginality.
This ratio is a convenient physical comparison coordinate.
The more primitive gap remains
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
37. Ratio and Gap Encode Different Count Questions
The ratio
\[\frac{S_H}{S_E}\]
compares logarithmic capacities.
The difference
\[S_E-S_H\]
measures the remaining logarithmic multiplicity gap.
Both reach their marginal values at the same condition:
\[\frac{S_H}{S_E}=1\]
and
\[S_E-S_H=0.\]
But they should not be conflated.
The difference has the direct multiplicative interpretation
\[S_E-S_H=\ln\frac{W_E}{W_H}.\]
38. The Multiplicity Gap Has an Occupancy Read at Complete Retention
If both relevant gravitational landscapes are realized under complete-retention proportional conditions, then relative occupancy can reproduce relative multiplicity.
Schematically,
\[\Delta G_{E\mid H}\approx\Delta S_{E\mid H}.\]
Therefore,
\[\frac{c_E}{c_H}\approx\frac{W_E}{W_H}.\]
As saturation is approached,
\[\frac{W_E}{W_H}\longrightarrow1,\]
so the corresponding relative logarithmic occupancy distinction also approaches zero.
This gives a realized interpretation of multiplicity-gap closure without identifying occupancy with multiplicity.
39. Gravitational Mass Is Therefore Relational Twice
Mass is already a physical calibration of a boundary coupling.
The present argument adds a second relational fact.
The maximum allowed value of that mass coordinate depends on the enclosing multiplicity domain.
Thus:
mass = boundary-calibrated local read,
while
maximum mass = enclosing-boundary-relative saturation read.
The latter cannot be assigned to the local mass alone.
40. Principle — Gravitational Multiplicity-Well Depth
For a gravitational count domain with reference multiplicity W0 and realized horizon multiplicity WH , define
\[D_H=\ln\frac{W_H}{W_0}.\]
For
\[W_0=1,\]
\[D_H=S_H.\]
Well depth is therefore a logarithmic multiplicity difference.
It is not gravitational retention.
41. Principle — Enclosing Multiplicity Capacity
Let an enclosing gravitational boundary possess multiplicity
\[W_E.\]
Define the subordinate–enclosing multiplicity gap
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
A positive gap supports the nested gravitational classification.
The marginal condition is
\[\Delta S_{E\mid H}=0.\]
42. Corollary — Areal Marginality
If subordinate and enclosing gravitational boundaries share the same areal multiplicity calibration,
\[S=\frac A{4\ell_P^2},\]
then
\[\Delta S_{E\mid H}=0\]
is equivalent to
\[A_E=A_H.\]
Thus areal equality is the gravitational projection of multiplicity-gap closure.
43. Corollary — Maximum Mass Is Downstream
If the subordinate horizon area depends on the physical mass coordinate,
\[A_H=A_H(M),\]
and the enclosing boundary is specified,
\[A_E=A_E(M,\ldots),\]
then the limiting mass is defined by
\[A_H(M_{\max})=A_E(M_{\max},\ldots).\]
Therefore,
multiplicity-gap closure \(\longrightarrow\) areal equality \(\longrightarrow M_{\max}\).
The mass limit comes last.
44. Logical Status
Canonical R²D
Multiplicity is
\[W,\]
with logarithmic coordinate
\[S=\ln W.\]
Multiplicity and retention are distinct.
State classifications are boundary defined.
A distinction that disappears cannot continue to support the same boundary-relative classification without redefinition.
Result Carried From Addendum 43
For the Schwarzschild horizon,
\[S_H=\frac{A_H}{4\ell_P^2}\]
is used as the established physical multiplicity calibration.
Under the candidate radial mapping,
\[r_i=i\ell_P,\]
the horizon landscape satisfies
\[S_i=\pi i^2.\]
Definition Introduced Here
Multiplicity-well depth is
\[D_H=S_H-S_0.\]
Derived Schwarzschild Mapping
For S0 = 0,
\[D_H=\frac{4\pi GM^2}{\hbar c}.\]
Thus
\[M=\sqrt{\frac{\hbar c}{4\pi G}}\sqrt{D_H}.\]
New Addendum-Level Saturation Criterion
For an enclosing gravitational multiplicity WE ,
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
The proposed marginal condition is
\[\Delta S_{E\mid H}=0.\]
Conditional Physical Projection
If both boundaries use the same horizon-area calibration,
\[\Delta S_{E\mid H}=0\quad\Longleftrightarrow\quad A_E=A_H.\]
Not Derived Here
This addendum does not derive:
• a specific enclosing gravitational background; • a specific value of Mmax ; • the Schwarzschild–de Sitter solution; • the Nariai mass; • cosmological horizon dynamics; • horizon temperature; • landscape replacement after marginality; • or a universal upper bound on gravitational mass.
These belong to the next physical projection test.
45. The Exact Test Is Deferred
The generic R²D proposal is now clear:
gravitational saturation \(\Longleftrightarrow\) loss of subordinate–enclosing multiplicity distinction.
Under common areal calibration,
\[W_E=W_H\quad\Longleftrightarrow\quad A_E=A_H.\]
A stringent physical test therefore exists.
Find an exact gravitational system containing:
• a subordinate black-hole horizon; • an enclosing gravitational horizon; • a continuously varying source mass; • and a limiting state at which the two horizon areas become equal.
If such a system exists, the mass at that equality should be the conventional maximum mass of that two- boundary gravitational classification.
That test is taken up in Addendum 45.
46. The Central R²D Inversion
The conventional explanatory order is
\(M\uparrow\)
\(\Downarrow\)
stronger gravity
\(\Downarrow\)
larger horizon
\(\Downarrow\)
maximum mass.
R²D instead reads
boundary-defined multiplicity well
\(\Downarrow\)
\[D_H=\ln\frac{W_H}{W_0}.\]
\(\Downarrow\)
enclosing multiplicity \(W_E\)
\(\Downarrow\)
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
\(\Downarrow\)
\[\Delta S_{E\mid H}\longrightarrow0\]
\(\Downarrow\)
under common gravitational calibration
\[A_H\longrightarrow A_E\]
\(\Downarrow\)
physical mass projection
\[M\longrightarrow M_{\max}.\]
The maximum mass is therefore not the primitive event.
Loss of the subordinate–enclosing multiplicity distinction is the prior count event.
47. Conclusion
Large gravitational mass corresponds to a deep horizon multiplicity well.
That statement can now be made without identifying multiplicity with retention.
The well depth is
\[D_H=\ln\frac{W_H}{W_0}.\]
For the unit reference,
\[W_0=1,\]
this becomes
\[D_H=S_H.\]
Under the Planck-distinguished radial construction of Addendum 43,
\[D_H=\pi N^2.\]
Under the Schwarzschild physical mapping,
\[N=\frac{2GM}{c^2\ell_P},\]
and therefore
\[D_H=\frac{4\pi GM^2}{\hbar c}.\]
Thus the physical mass coordinate satisfies
\[M=\sqrt{\frac{\hbar c}{4\pi G}}\sqrt{D_H}.\]
Mass is downstream of multiplicity depth.
But the local depth relation contains no maximum.
A maximum requires an enclosing count boundary.
Let the enclosing multiplicity be
\[W_E.\]
Then the remaining subordinate–enclosing distinction is
\[\Delta S_{E\mid H}=\ln\frac{W_E}{W_H}.\]
As long as
\[\Delta S_{E\mid H}>0,\]
the nested gravitational classification remains countably distinct.
At
\[\Delta S_{E\mid H}=0,\]
we obtain
\[W_E=W_H.\]
The multiplicity distinction defining the old subordinate/enclosing relation has vanished.
If both boundaries share the same gravitational horizon-area calibration,
\[S=\frac A{4\ell_P^2},\]
then this prior count equality projects as
\[A_E=A_H.\]
Only after the enclosing physical relation is specified can that equality be translated into a limiting source mass,
\[M_{\max}.\]
The causal order is therefore not
maximum mass \(\longrightarrow\) boundary saturation.
It is
multiplicity depth → enclosing multiplicity gap → gap closure → areal equality → maximum gravitational mass.
Formal continuation beyond gap closure would preserve a subordinate/enclosing classification after the multiplicity distinction that defined it had vanished.
That is the gravitational analogue of a Gibbs-type boundary-classification failure.
The physical system need not disappear.
The old count landscape may instead require replacement by a new one.
The exact gravitational realization of this proposal is not assumed here.
It is tested next.
Thus:
entropic-well depth occurs before maximum gravitational mass.
And more specifically:
gravitational mass saturation is the physical projection of prior closure of the multiplicity gap betwe… [PDF line clipped; author review required.]