A personal note

PIN starts from the premise that quantum representation is required for efficiency in a universe with finite informational resources. Its mathematics connects black-hole information capacity to mass, Hawking temperature, and rotational energy, and relates clock rates to processing headroom. These connections use the framework’s stated assumptions without adjustable fitting parameters.

I am spatially neurodivergent and eccentric on a good day. Here, I am most proud of the math.

The mathematics of PIN builds a bridge between quantum physics and general relativity, and I offer my sincere thanks to Dr. Vopson as I share this work.

R.P.S.

Principia Informationis Naturalis

Black-Hole Scaling, Gravity and Committed Reality

R. P. S.

24 October 2026

Foreword

PIN brings together a universe with finite informational resources and a physical history that is smooth, continuous and consistent. A committed frame anchors an established past. Ahead of that frame, quantum possibilities remain open. This relationship between an actual history and a possible future is central to the framework.

Finiteness makes the representation of physical information consequential. Exhaustive, independently exact specification of microscopic positions requires unbounded information as resolution becomes arbitrarily fine. PIN gives the wave function a computational role: amplitudes, phases and correlations carry the distinctions needed to calculate possible outcomes. Its initial efficiency lies in this predictive representation of physical possibilities.

Commitment addresses the next step. When resolving alternatives and continuing from an outcome requires fewer resources than carrying those alternatives forward, efficiency favors commitment. The resulting event joins the established history, and subsequent commitments must remain consistent with that past. Smoothness expresses this continuity; efficiency governs the use of finite resources as the future becomes actual.

The work of John Archibald Wheeler, Jacob Bekenstein and Ted Jacobson helped shape the questions behind PIN. Their connections among information, horizon area, entropy and gravity provide the intellectual starting points for the development that follows.

“all things physical are information-theoretic in origin and this is a participatory universe.”

John Archibald Wheeler, Information, Physics, Quantum, Sec. 19.2 of the linked reprint, p. 311 [9].

Wheeler places information and the establishment of physical outcomes at the centre of the inquiry.

“Most striking is the similarity in the behaviors of black-hole area and of entropy: Both quantities tend to increase irreversibly.”

Jacob D. Bekenstein, Black Holes and Entropy, abstract, p. 2333 [7].

Bekenstein’s area–entropy connection, together with Hawking’s temperature calculation, supplies the thermodynamic foundation for PIN’s saturated-area scaling.

“Viewed in this way, the Einstein equation is an equation of state.”

Ted Jacobson, Thermodynamics of Spacetime, abstract [26].

Jacobson connects gravitational geometry to local horizon thermodynamics. These influences guide the route developed here from the Planck-area normalization through black-hole mass and information retention to clocks and committed history. Susskind’s optical construction contributes a further geometric insight: gravitational lensing separates the screen images of black holes aligned in depth [11].

1 PWP and finite-information foundations

Principia Informationis Naturalis (PIN) is a finite-information framework connecting information capacity, gravitational mass and thermal emission through saturated black-hole scaling. It treats physical reality as information carried and processed by a finite physical substrate. The informational substrate is two-dimensional and locally flat. A saturated black hole is a disk of saturation within that substrate. The saturated disk has one Planck-area position per retained Planck-wavelength photon (PWP), whose reduced wavelength is the Planck length. Its one-dimensional rim and outward disk-to-sphere area correspondence of 1:4 organize the derivation from the single-area normalization to quasar central-black-hole scales. Along this scaling, PIN identifies gravity as the attractive component of entanglement, whose role is to reduce latency in information coordination. PIN connects clock comparisons to local processing headroom, and definite events to efficiency-driven commitment within one continuous, consistent history. The final section develops the vacuum-energy calculation and its connection to cosmological curvature, then recovers the local black-hole limit.

1.1 The Planck-wavelength photon and its area

Maxwell’s theory describes light as a propagating electromagnetic disturbance [1]. PIN associates each Planck-wavelength photon with a Planck-area position on the informational substrate. The photon supplies energy and momentum; the area sets the capacity; the quantum state describes the arrangement and correlations of the carriers. The standard Planck definitions in SI units are [2] $$\begin{aligned} \ell_{\mathrm P}^{2}&=\frac{\hbar G}{c^3}, & m_{\mathrm P}&=\sqrt{\frac{\hbar c}{G}},\\ E_{\mathrm P}&=m_{\mathrm P}c^2=\frac{\hbar c}{\ell_{\mathrm P}}, & t_{\mathrm P}&=\frac{\ell_{\mathrm P}}{c}. \end{aligned}\tag{1}$$(1)

Here $G$ is the gravitational constant, $c$ is the local vacuum speed of light, $h$ is Planck’s constant, and $\hbar=h/(2\pi)$ is the reduced Planck constant. The symbol $m_{\mathrm P}$ denotes the unreduced Planck mass.

The photon energy and momentum relations follow quantum radiation theory [3, §§2–3, 7]. PIN assigns the Planck-wavelength photon (PWP) one Planck energy in the specified substrate energy frame. Its reduced wavelength $\bar\lambda=\lambda/(2\pi)$ therefore equals $\ell_{\mathrm P}$; its ordinary wavelength spans a full cycle and is larger by a factor of $2\pi$: $$\begin{gathered} \bar\lambda_{\mathrm{PWP}}=\frac{\hbar}{p_{\mathrm{PWP}}}=\ell_{\mathrm P},\qquad \lambda_{\mathrm{PWP}}=2\pi\ell_{\mathrm P},\\ p_{\mathrm{PWP}}=m_{\mathrm P}c,\qquad E_{\mathrm{PWP}}=cp_{\mathrm{PWP}}=E_{\mathrm P},\\ E_{\mathrm{PWP}}^2-c^2p_{\mathrm{PWP}}^2=0,\qquad m_{0,\mathrm{PWP}}=0,\qquad E_{\mathrm{PWP}}/c^2=m_{\mathrm P}. \end{gathered}\tag{2}$$(2)

The PWP is massless: its invariant rest mass is zero. Its assigned energy, expressed as an energy-equivalent mass, is $m_{\mathrm P}$. Energy and wavelength transform with the observer; the null momentum relation holds in every inertial frame.

PIN takes the PWP to be the maximally energetic elementary carrier in the specified substrate frame. The minimum reduced wavelength and one-PWP-per-Planck-area capacity give $$\bar\lambda\geq\ell_{\mathrm P}\quad\Longrightarrow\quad E=\frac{\hbar c}{\bar\lambda}\leq E_{\mathrm P},\qquad \sigma_{E,\max}=\frac{E_{\mathrm P}}{\ell_{\mathrm P}^{2}}. \tag{3}$$(3)

The wavelength floor and occupancy limit are PIN’s physical premises; the energy ceiling follows from the photon dispersion relation. Here $\sigma_{E,\max}$ is the maximum local carrier-energy inventory per unit substrate area, in joules per square metre. At capacity, additional retained inventory requires additional occupied area.

Saturation means occupation of every permitted positional address in the region being counted. Let $N$ be the effective number of saturated PWP–area pairs. PIN assigns one nat of capacity to each occupied pair: $$S_H=k_{\mathrm B}N.\tag{4}$$(4)

Here $k_{\mathrm B}$ is Boltzmann’s constant, and $S_H$ has units of joules per kelvin. For equiprobable microstates, $S=k_{\mathrm B}\ln W$, where $W$ is their multiplicity [4, 5]. The symbol $\ln$ denotes the natural logarithm, so $S/k_{\mathrm B}$ is dimensionless entropy measured in nats [6, Introduction]. Assigning one nat per saturated pair is PIN’s separate physical rule. The Planck scales also satisfy $E_{\mathrm P}t_{\mathrm P}=\hbar$.

1.2 Saturation, the disk and its outward correspondence

Let $\Sigma$ be the black hole’s flat disk of saturation within the informational substrate, with radius $R>0$. Its area, capacity and locally assigned carrier-energy inventory are $$A_\Sigma=\pi R^2,\qquad N=\frac{A_\Sigma}{\ell_{\mathrm P}^{2}}=\frac{\pi R^2}{\ell_{\mathrm P}^{2}},\qquad E_\Sigma=NE_{\mathrm P}.\tag{5}$$(5)

The physical region is the whole two-dimensional disk. Its rim is the one-dimensional circle of circumference $2\pi R$. Growth at fixed saturation adds occupied pairs and area together, preserving the local inventory density $E_{\mathrm P}/\ell_{\mathrm P}^{2}$.

For equal-area cells tiling a plane, the regular hexagonal arrangement minimizes boundary length per unit area [51]. Hexagonal packing distinguishes positions in the saturated interior from positions on the one-dimensional rim. An interior position has six saturated side-sharing neighbours. At the rim, the notation $z+1$ counts $z$ saturated side-sharing neighbours and one outward-admissible relation to available substrate capacity. The $3+1$ and $4+1$ classes describe different local boundary environments, with $+1$ denoting the outward relation. A finite contour can also contain $5+1$ positions. These local neighbour counts describe the packing geometry. The disk area and rim circumference used here are its continuum description.

A regular triangulated realization of this packing gives an exact signed boundary count. Let $B_{\mathrm{rim}}$ count boundary vertices and $z_i$ their saturated neighbours. For a topological disk with one simple boundary and degree-six interior, $$\sum_{i\in\mathrm{rim}}(4-z_i)=6,\qquad \epsilon_{\mathrm{rim}}\equiv\frac{1}{B_{\mathrm{rim}}}\sum_{i\in\mathrm{rim}}(4-z_i)=\frac{6}{B_{\mathrm{rim}}}.\tag{5a}$$(5a) Euler’s relation and triangle–edge incidence counting give $E_{\mathrm{mesh}}=3V-B_{\mathrm{rim}}-3$, where $V$ and $E_{\mathrm{mesh}}$ count vertices and edges. Subtracting the degree-six interior contribution from the total degree $2E_{\mathrm{mesh}}$ yields Eq. (5a). This is the flat-interior specialization of discrete Gauss–Bonnet [56, Secs. 2.6–2.7]. The signed turn at a boundary vertex is $\theta_i=(\pi/3)(4-z_i)$; the sum is $2\pi$. Every signed contribution enters, including the negative contribution from $z_i=5$.

Let $a$ be the distance between neighbouring centres. The triangular lattice has area $\sqrt{3}a^2/2=\ell_{\mathrm P}^2$ per site. For a lattice-centred circular cutoff of radius $R_c$, write $m_{\mathrm{lat}}$ for the largest occupied axial lattice coordinate. Each of the three lattice-line families has $2m_{\mathrm{lat}}+1$ contiguous occupied rows. Counting their ends gives $U=6(2m_{\mathrm{lat}}+1)$ outward neighbour incidences. Equation (5a) also gives $U=2B_{\mathrm{rim}}+6$, so $B_{\mathrm{rim}}=6m_{\mathrm{lat}}$. Since $m_{\mathrm{lat}}=2R_c/(\sqrt{3}a)+O(1)$, $$B_{\mathrm{rim}}=\frac{4\sqrt{3}R_c}{a}+O(1),\qquad R=R_c+O(a),\qquad \epsilon_{\mathrm{rim}}=\frac{\sqrt{3}a}{2R}+O\!\left(\frac{a^2}{R^2}\right). \tag{5b}$$(5b)

Here $R=\ell_{\mathrm P}\sqrt{N/\pi}$ is the area-equivalent radius of the occupied cells, consistent with Eq. (5); $R_c$ specifies the lattice sampling. Increasing the disk radius spreads its fixed total turn across more rim sites. The resulting inverse-radius dependence gives a geometric connection to the thermal scale developed in Section 2.3.

In PIN’s spatial account of a saturated black hole, physical space within the occupied disk remains two-dimensional; the third spatial dimension emerges outward, beyond its rim. The disk’s one-dimensional edge is its physical horizon. The surrounding emergent geometry describes the black hole as it is encountered in space.

For the stationary, neutral, nonrotating black-hole construction, PIN assigns this whole disk an outward projection to a spherical surface of the same radius. In the Schwarzschild exterior, that sphere is a two-dimensional spatial section $H$ of the event horizon. The event horizon itself is the null hypersurface traced through spacetime. The disk and its physical rim describe the saturated state; the sphere and null hypersurface describe its exterior gravitational geometry. Their area correspondence is $$A_H=4\pi R^2=4A_\Sigma,\qquad N=\frac{A_\Sigma}{\ell_{\mathrm P}^2}=\frac{A_H}{4\ell_{\mathrm P}^2}. \tag{6}$$(6)

The count begins on the physical disk: one retained PWP per Planck area gives $N=A_\Sigma/\ell_{\mathrm P}^2$. Its outward spherical surface of the same radius has four times the area. Expressing this same count using $A_H$ therefore introduces the geometric ratio $A_\Sigma/A_H=\pi R^2/(4\pi R^2)=1/4$, giving $N=A_H/(4\ell_{\mathrm P}^2)$. PIN’s separate one-nat-per-pair rule then converts the count to entropy. With Eq. (4), $$S_H=\frac{k_{\mathrm B}A_\Sigma}{\ell_{\mathrm P}^2}=\frac{k_{\mathrm B}A_H}{4\ell_{\mathrm P}^2}. \tag{7}$$(7)

This reproduces the Bekenstein–Hawking area normalization. In the conventional thermodynamic account, Bekenstein connected black-hole entropy with area [7]; Hawking’s radiation temperature, together with the first law, fixes the coefficient at one quarter in Planck-area units [8]. Wheeler’s information-first account of physical description [9] provides the broader setting. The holographic formulations of ’t Hooft and Susskind relate the degrees of freedom of a spatial region to surface area [10, 11, 52]. Susskind’s formulation already uses a flat viewing screen to encode three-dimensional configurations [11].

PIN gives the flat disk a specific physical role: the whole saturated black hole occupies a two-dimensional region of the informational substrate, with retained PWPs occupying its area and a one-dimensional physical rim. Its correspondence runs outward from this disk to the spherical horizon section $H$ in the exterior geometry. Both the disk and $H$ are two-dimensional surfaces; their intrinsic geometry and physical roles differ. Equation (7) combines PIN’s one-nat-per-Planck-area rule with this outward, same-radius correspondence. On macroscopic scales $N$ is the continuous area count used by thermodynamics, with the Planck-area pair setting its normalization.

2 Black-hole scaling and thermodynamics

2.1 Gravitational mass from the projected exterior

We first establish the adopted gravitational response and its vacuum exterior, then use that exterior to relate the disk radius to gravitational mass.

Einstein’s field equation relates curvature to stress and energy [12, §§14, 16]. In four dimensions, in modern notation with signature $(-,+,+,+)$, define $$G_{ab}=\mathcal R_{ab}-\frac12\mathcal R g_{ab},\qquad \mathcal R=g^{ab}\mathcal R_{ab},\qquad \kappa=\frac{8\pi G}{c^4}.$$

Here $G_{ab}$ is the Einstein tensor, $\mathcal R_{ab}$ the Ricci tensor, $\mathcal R$ the scalar curvature and $G$ Newton’s constant. Including a cosmological term, the ordinary field equation and the Lorentz-invariant vacuum contribution take the form [13, Sec. II] $$G_{ab}+\Lambda_{\mathrm{bare}}g_{ab}=\kappa T_{ab}^{\mathrm{total}},\qquad T_{ab}^{\mathrm{total}}=T_{ab}^{\mathrm{excit}}-\varepsilon_{\mathrm{vac}}g_{ab}.$$

Here $T_{ab}^{\mathrm{excit}}$ denotes stress above the vacuum baseline. In the ordinary field equation, the constant baseline contributes $\kappa\varepsilon_{\mathrm{vac}}$ to the effective cosmological term.

PIN adopts a trace-free gravitational response to the vacuum baseline. Fixed-volume variation is an independent gravitational assumption. The factor $1/4$ in trace removal follows from $g^{ab}g_{ab}=4$: subtracting one quarter of a tensor’s scalar trace times $g_{ab}$ leaves a tensor with zero trace. The area ratio $A_\Sigma/A_H=1/4$ instead compares the flat disk with its same-radius sphere. These coefficients have different origins. The energy ratio $\gamma=E_{\mathrm{BH}}/E_\Sigma$ is a separate, radius-dependent quantity, derived in Eq. (24). The trace-free equation is the classical field response used to connect PIN’s physical energy account to geometry. With $x^0=ct$, an action with the usual Einstein normalization is $$\mathcal I[g,\psi]=\frac{1}{2\kappa c}\int d^4x\sqrt{-g}\,\mathcal R+\mathcal I_m[g,\psi],\qquad \frac{1}{2\kappa c}=\frac{c^3}{16\pi G}. \tag{8}$$(8)

Here $\psi$ denotes the matter fields and $\mathcal I$ has units of action. The stress convention is $$\delta\mathcal I_m=-\frac{1}{2c}\int d^4x\sqrt{-g}\,T_{ab}\delta g^{ab},\qquad \delta\mathcal I=\frac{1}{2\kappa c}\int d^4x\sqrt{-g}(G_{ab}-\kappa T_{ab})\delta g^{ab}. \tag{9}$$(9)

The matter fields satisfy their equations of motion; metric variations have compact support, so the surface term vanishes. Unrestricted metric variation yields $G_{ab}=\kappa T_{ab}$. PIN holds the volume density fixed during metric variation. This is the established variational formulation of unimodular gravity [14, Sec. II]: the four-dimensional exterior metric varies while its spacetime volume density $\sqrt{-g}$ stays fixed. Since $$\delta\sqrt{-g}=-\frac12\sqrt{-g}\,g_{ab}\delta g^{ab},\qquad \delta\sqrt{-g}=0\quad\Longrightarrow\quad g_{ab}\delta g^{ab}=0, \tag{10}$$(10)

only trace-free metric variations contribute. At each point, a symmetric tensor orthogonal to every trace-free variation is proportional to $g_{ab}$; its trace-free part therefore vanishes. The Einstein–Hilbert and matter variation is proportional to $\int d^4x\sqrt{-g}(G_{ab}-\kappa T_{ab})\delta g^{ab}$, up to the usual surface term. Taking the variations with compact support gives $$\begin{gathered} X_{ab}^{\mathrm{TF}}=X_{ab}-\frac14g_{ab}g^{cd}X_{cd},\qquad G_{ab}^{\mathrm{TF}}-\kappa T_{ab}^{\mathrm{TF}}=0,\\ \mathcal R_{ab}-\frac14\mathcal R g_{ab}=\kappa\left(T_{ab}-\frac14Tg_{ab}\right),\qquad T=g^{ab}T_{ab}. \end{gathered}\tag{11}$$(11)

The trace-free field equation follows from the restricted variations. Its classical relation to Einstein gravity is developed by Ellis and collaborators [15].

Equivalently, $(\mathcal R_{ab}-\kappa T_{ab})n^an^b=0$ for every null vector $n^a$. Padmanabhan and Paranjape obtain this condition from an entropy functional with Einstein normalization by varying null vectors with the metric fixed and taking the complete matter stress to be conserved [16, Sec. II, Eqs. (15)–(21)]. This distinct variational route gives the same classical response.

The vacuum term now cancels component by component: $$T^{\mathrm{vac}}=g^{ab}(-\varepsilon_{\mathrm{vac}}g_{ab})=-4\varepsilon_{\mathrm{vac}},\qquad (T^{\mathrm{vac}})_{ab}^{\mathrm{TF}}=-\varepsilon_{\mathrm{vac}}g_{ab}-\frac14(-4\varepsilon_{\mathrm{vac}})g_{ab}=0.\tag{12}$$(12)

Conservation of the complete matter stress, $\nabla^aT_{ab}=0$, supplies the integrability condition. The contracted Bianchi identity $\nabla^a\mathcal R_{ab}=\frac12\nabla_b\mathcal R$ turns the divergence of Eq. (11) into $$\frac14\nabla_b\mathcal R=-\frac\kappa4\nabla_bT,\qquad \nabla_b(\mathcal R+\kappa T)=0,\qquad \mathcal R+\kappa T=4\Lambda_{\mathrm{int}}.\tag{13}$$(13)

The last equality defines a constant on each connected solution domain. This reconstruction uses conservation of the complete matter stress as an independent condition. Substitute $\kappa T=-\mathcal R+4\Lambda_{\mathrm{int}}$ back into the trace-free equation: $$\begin{gathered} \mathcal R_{ab}-\frac14\mathcal R g_{ab}=\kappa T_{ab}+\frac14\mathcal R g_{ab}-\Lambda_{\mathrm{int}}g_{ab},\\ G_{ab}+\Lambda_{\mathrm{int}}g_{ab}=\kappa T_{ab}. \end{gathered}\tag{14}$$(14)

This reconstruction follows the classical trace-free formulation of Ellis and collaborators [15, Sec. 3, Eqs. (13)–(19)]. Consequently, a constant energy-density shift $\varepsilon_{\mathrm{shift}}$ is represented by $$\begin{gathered} T_{ab}\mapsto T_{ab}-\varepsilon_{\mathrm{shift}}g_{ab},\qquad \Lambda_{\mathrm{int}}\mapsto\Lambda_{\mathrm{int}}-\kappa\varepsilon_{\mathrm{shift}},\\ \varepsilon_{\mathrm{vac}}\mapsto\varepsilon_{\mathrm{vac}}+\varepsilon_{\mathrm{shift}},\qquad \Lambda_{\mathrm{eff}}=\Lambda_{\mathrm{int}}+\kappa\varepsilon_{\mathrm{vac}}\mapsto\Lambda_{\mathrm{eff}}. \end{gathered}\tag{15}$$(15)

A constant baseline leaves the geometry unchanged under this relabeling of the integration constant. For a varying term, $$\nabla^a[-\varepsilon_{\mathrm{shift}}(x)g_{ab}]=-\partial_b\varepsilon_{\mathrm{shift}},\qquad \nabla^aT_{ab}^{\mathrm{excit}}=\partial_b\varepsilon_{\mathrm{vac}}\quad\text{when}\quad\nabla^aT_{ab}^{\mathrm{total}}=0.\tag{16}$$(16)

Transitions and exchange with matter remain part of the conserved physical evolution. A physical transition has additional conserved stress and matching data, with which its evolution must be calculated.

Ellis and collaborators distinguish the vacuum baseline from integration data fixed by observation [15, Secs. 3–4]. Padilla and Saltas identify quantum radiative stability as a further question requiring a specified action and measure [14, Secs. II–III]. The stated equations establish constant-baseline cancellation under the adopted response and conservation law.

For the isolated, stationary, neutral, nonrotating exterior used here, select $\Lambda_{\mathrm{eff}}=0$ and no excitations, giving $G_{ab}=0$ and $\mathcal R_{ab}=0$. The positive-$\Lambda_{\mathrm{eff}}$ extension and the conditions for this local limit are derived in Eqs. (62)–(67). In coordinates $x^0=ct_{\mathrm{PG}},r,\theta,\phi$, write the ingoing Painlevé–Gullstrand form with dimensionless profile $u(r)>0$: $$ds^2=-(dx^0)^2+\left(dr+\sqrt{u(r)}\,dx^0\right)^2+r^2d\Omega^2,\qquad d\Omega^2=d\theta^2+\sin^2\theta\,d\phi^2.\tag{17}$$(17)

Here $r$ is the projected areal radius. A constant-$t_{\mathrm{PG}}$ slice has the three-dimensional metric $dr^2+r^2d\Omega^2$; PIN’s occupied disk is a saturated two-dimensional region of the physical substrate. This coordinate form remains regular at the black-hole null radius [17].

The inverse metric has $g^{rr}=1-u$. Computing the connection and curvature gives the mixed Einstein components $$\begin{gathered} G^0{}_0=G^r{}_r=-\frac{(ru)'}{r^2},\qquad G^\theta{}_\theta=G^\phi{}_\phi=-\frac{(ru)''}{2r},\\ G^a{}_b=0\quad(a\ne b). \end{gathered}\tag{18}$$(18)

A prime denotes differentiation with respect to $r$. In vacuum these components vanish. The first equation integrates to $$(ru)'=0\quad\Longrightarrow\quad u(r)=\frac{C_r}{r},\qquad C_r>0.\tag{19}$$(19)

The constant $C_r$ has units of length. This solution also satisfies the angular equations and gives a vanishing Ricci tensor throughout the vacuum region.

The invariant gradient of the areal radius supplies the spherical gravitational mass. The Misner–Sharp definition, in SI units, gives $$\begin{gathered} 1-\frac{2GM_{\mathrm{MS}}(r)}{c^2r}=g^{ab}\nabla_ar\nabla_br=1-u(r),\\ M_{\mathrm{MS}}(r)=\frac{c^2ru(r)}{2G}=\frac{c^2C_r}{2G}. \end{gathered}\tag{20}$$(20)

The vacuum mass is constant [18]. PIN identifies its projected spherical radius with the constant null radius $r=R$. The normal is lightlike when $g^{rr}(R)=1-u(R)=0$, fixing $C_r=R$. Radial light rays satisfy $dr/dx^0=-\sqrt u\pm1$; the outgoing branch is stationary there. On the selected stationary black-hole branch, $r=R$ is the future event horizon in the exterior spacetime description. PIN’s physical horizon remains the disk’s one-dimensional rim. The same radius that counts the occupied disk therefore fixes the gravitational mass: $$R=r_s=\frac{2GM}{c^2},\qquad E_{\mathrm{BH}}=Mc^2=\frac{c^4R}{2G}.\tag{21}$$(21)

Here $M>0$ is the Schwarzschild mass parameter, and $E_{\mathrm{BH}}$ uses the usual normalization of the stationary time generator [19]. The preceding construction connects the disk radius to this mass through the vacuum Einstein equation, the spherical mass definition and the projected null-radius assignment. The clock and test-body comparisons below use a reference observer at a specified finite radius.

2.2 Why black-hole mass grows as the square root of capacity

Substitute this radius into the occupied-area count: $$\begin{aligned} N&=\frac{\pi}{\ell_{\mathrm P}^2}\left(\frac{2GM}{c^2}\right)^2=\frac{4\pi GM^2}{\hbar c}\\ &=4\pi\left(\frac{M}{m_{\mathrm P}}\right)^2. \end{aligned}\tag{22}$$(22)

Solving for the positive mass gives the central scaling family: $$\boxed{\begin{gathered} R(N)=\ell_{\mathrm P}\sqrt{\frac N\pi},\qquad M(N)=\frac{m_{\mathrm P}}{2\sqrt\pi}\sqrt N,\\ E_\Sigma(N)=NE_{\mathrm P},\qquad E_{\mathrm{BH}}(N)=\frac{E_{\mathrm P}}{2\sqrt\pi}\sqrt N. \end{gathered}}\tag{23}$$(23)

Every member of this scaling family has the same saturated density: one retained PWP per Planck area. The occupied capacity grows as the disk area, $N\propto R^2$, while the exterior gravitational construction gives $M\propto R$. Consequently $M\propto\sqrt N$. Doubling the disk’s radius quadruples its area, occupied capacity and local PWP energy inventory, while doubling its gravitational mass. The exterior energy per occupied pair therefore halves, while the local saturation density stays fixed. $E_\Sigma=NE_{\mathrm P}$ is the local carrier-energy inventory in the specified substrate frame. $E_{\mathrm{BH}}=Mc^2$ is the collective exterior gravitational energy supplied by the exterior construction. Their different scaling with $N$ describes two distinct energy accounts. Their ratio is $$\boxed{\gamma\equiv\frac{E_{\mathrm{BH}}}{E_\Sigma}=\frac{1}{2\sqrt{\pi N}}=\frac{\ell_{\mathrm P}}{2\pi R}.}\tag{24}$$(24)

The same ratio can be expressed using the defined substrate rim circumference scale $\tau_\Sigma=2\pi R/c$: $$\gamma=\frac{t_{\mathrm P}}{\tau_\Sigma},\qquad \tau_\Sigma=\frac{2\pi R}{c}.\tag{25}$$(25)

This time-scale identity follows from the same area and mass relations. The ratio $\gamma$ compares the two energy accounts.

The single-Planck-area anchor is obtained by setting $N=1$: $$R_1=\frac{\ell_{\mathrm P}}{\sqrt\pi},\qquad M_1=\frac{m_{\mathrm P}}{2\sqrt\pi},\qquad M=m_{\mathrm P}\quad\Longrightarrow\quad N=4\pi.\tag{26}$$(26)

At this point the substrate area is $\ell_{\mathrm P}^2$ and the projected sphere has area $4\ell_{\mathrm P}^2$. The local carrier inventory is $E_{\mathrm P}$; the extrapolated exterior energy is $E_{\mathrm P}/(2\sqrt\pi)$. These values fix the normalization of the curve. Its semiclassical black-hole application occupies $R\gg\ell_{\mathrm P}$; the realization of the single-area endpoint is the Planck-scale microscopic question.

2.3 Temperature and finite desaturation energy

Temperature measures the marginal exterior energy associated with entropy. Along neighboring stationary black-hole states, the first law $dE_{\mathrm{BH}}=T_H\,dS_H$ gives $$\begin{gathered} k_{\mathrm B}T_H=\frac{dE_{\mathrm{BH}}}{dN}=\frac{E_{\mathrm P}}{4\sqrt{\pi N}}=\frac{\hbar c}{4\pi R},\\ T_H=\frac{\hbar c^3}{8\pi GMk_{\mathrm B}}. \end{gathered}\tag{27}$$(27)

The area entropy and energy curve recover the Hawking temperature [8, 20]. A larger black hole has greater capacity and lower temperature. The inverse-temperature time scale is $\tau_H=\hbar/(k_{\mathrm B}T_H)=4\pi R/c=2\tau_\Sigma$, twice the defined rim-circumference scale.

The same exterior geometry fixes a near-horizon boost scale. Let $t$ be Schwarzschild time with its standard asymptotic normalization and define $t_b=4GM/c^3$. Susskind’s near-horizon coordinates have boost parameter $\Theta=t/t_b$ [11, Sec. 4, Eqs. (4.3)–(4.4)]. Equations (21), (25), and (27) give $$t_b=\frac{2R}{c}=\frac{\tau_\Sigma}{\pi}=\frac{\tau_H}{2\pi}=\frac{\hbar}{2\pi k_{\mathrm B}T_H}.$$

These fixed ratios connect the rim and thermal scales to near-horizon kinematics.

The gravitational boundary formulation pairs the dimensionless near-horizon boost angle $\Theta=t/t_b$ with the horizon area. Choosing the positive-energy orientation, its conjugate momentum is [57, Eq. (7)] $$\begin{gathered} p_\Theta=\frac{c^3A_H}{8\pi G}=\frac{\hbar N}{2\pi},\qquad \dot\Theta=\frac1{t_b}=\frac{c}{2R},\\ \frac{dE_{\mathrm{BH}}}{dN}=\frac{\hbar\dot\Theta}{2\pi}=k_{\mathrm B}T_H,\qquad p_\Theta\dot\Theta=\frac{E_{\mathrm{BH}}}{2}. \end{gathered}\tag{27a}$$(27a)

The derivative uses the same infinity-normalized time and Schwarzschild energy curve as Eq. (27). The variable $p_\Theta$ has units of action, while $\dot\Theta$ is an inverse time. Integrating the thermal slope recovers the collective exterior energy with its existing mass normalization. These relations connect the entropy count, thermal derivative and near-horizon clock. The boost acts in the time–radial normal plane; PIN’s saturated disk occupies the spatial substrate.

Mean energy divides the existing collective exterior energy among the existing occupied pairs: $E_{\mathrm{BH}}/N$. Marginal energy measures the change in that collective energy along the stationary family per additional nat: $dE_{\mathrm{BH}}/dN$. Since $E_{\mathrm{BH}}=E_{\mathrm P}\sqrt N/(2\sqrt\pi)$, division by $N$ gives $E_{\mathrm P}/(2\sqrt{\pi N})$, while differentiation gives $E_{\mathrm P}/(4\sqrt{\pi N})$. The factor of two follows from the derivative of $\sqrt N$.

The same result follows by allowing the collective-energy ratio $\gamma=E_{\mathrm{BH}}/E_\Sigma$ to change as the disk grows: $$\frac{d}{dN}\left(NE_{\mathrm P}\gamma(N)\right)=E_{\mathrm P}\left(\gamma+N\frac{d\gamma}{dN}\right)=\frac{E_{\mathrm P}\gamma}{2}.$$

Thus the mean energy per existing pair is twice the marginal energy per added nat. Consequently, $$\frac{E_{\mathrm{BH}}}{N}=2k_{\mathrm B}T_H,\qquad E_{\mathrm{BH}}=2T_HS_H,\qquad C_H\equiv\frac{dE_{\mathrm{BH}}}{dT_H}=-2Nk_{\mathrm B}.\tag{28}$$(28)

The middle identity is the Schwarzschild specialization of the Smarr relation [21]; negative heat capacity means that losing energy along this branch raises the temperature. Padmanabhan’s horizon equipartition count is $N_{\mathrm{surf}}=A_H/\ell_{\mathrm P}^2=4N$, so $E_{\mathrm{BH}}=\frac12N_{\mathrm{surf}}k_{\mathrm B}T_H=2Nk_{\mathrm B}T_H$ on this branch [22]. The surface count and PIN’s entropy count therefore give the same energy with their respective normalizations.

For a finite capacity change between stationary endpoints $N_i>N_f>0$, subtraction and rationalization give $$\Delta E_{\mathrm{BH}}=\frac{E_{\mathrm P}}{2\sqrt\pi}\frac{N_i-N_f}{\sqrt{N_i}+\sqrt{N_f}}.\tag{29}$$(29)

This is the exterior energy difference between the stated endpoints; $k_{\mathrm B}T_H$ is its small-change limit per nat. In PIN, Hawking radiation is desaturation: the same retained PWP is released in the informational form in which it accreted. The energy decrease follows this collective mass curve; the preservation of the released information is developed in the information-paradox discussion.

For a one-cell reduction between stationary saturated endpoints, set $N_i=N$ and $N_f=N-1$, with $N>1$. Equations (24), (27) and (29) then give $$\frac{\Delta E_{\mathrm{BH}}^{(1)}(N)}{E_{\mathrm P}} =\frac{\gamma(N)}{1+\sqrt{1-1/N}}, \qquad \frac{\Delta E_{\mathrm{BH}}^{(1)}(N)}{k_{\mathrm B}T_H(N)} =\frac{2}{1+\sqrt{1-1/N}}.\tag{29a}$$(29a) The mean exterior energy per nat, the marginal thermal energy, and the finite one-cell release are thus related explicitly. The second ratio exceeds one for finite $N>1$ and approaches one as $N\to\infty$.

2.4 From one Planck area to a quasar-scale black hole

Choose black-hole mass benchmarks of $10$, $10^6$ and $10^9$ nominal solar masses. The last is a representative supermassive scale for a quasar’s central engine. Each benchmark uses the Schwarzschild relation between mass and radius. A quasar’s visible power comes from its accreting environment; the Hawking temperature below belongs to the black hole’s thermal emission [8, 23].

Use the IAU nominal mass parameter $(GM)_\odot^{\mathrm N}=1.3271244\times10^{20}\ \mathrm{m^3\,s^{-2}}$ [24]. For multiplier $q$, $GM=q(GM)_\odot^{\mathrm N}$. The numerical inputs use exact SI values of $c,h,k_{\mathrm B}$ and the CODATA central value $G=6.67430\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ [2]. Thus $$\begin{gathered} R=\frac{2q(GM)_\odot^{\mathrm N}}{c^2},\qquad T_H=\frac{\hbar c^3}{8\pi k_{\mathrm B}q(GM)_\odot^{\mathrm N}},\\ N=\frac{\pi R^2c^3}{\hbar G}. \end{gathered}\tag{30}$$(30)

Scale	$R$ (m)	$T_H$ (K)	$N$ (nats)
$10\,M_\odot^{\mathrm N}$	$2.95325\times10^4$	$6.17026\times10^{-9}$	$1.04889\times10^{79}$
$10^6\,M_\odot^{\mathrm N}$	$2.95325\times10^9$	$6.17026\times10^{-14}$	$1.04889\times10^{89}$
$10^9\,M_\odot^{\mathrm N}$	$2.95325\times10^{12}$	$6.17026\times10^{-17}$	$1.04889\times10^{95}$

The nominal solar mass parameter is an exact conversion reference. At these fixed nominal inputs, $R$ and $T_H$ are independent of the experimental uncertainty in $G$, while $N$ carries its relative uncertainty of about $2.2\times10^{-5}$.

Between $10$ and $10^9$ nominal solar masses, mass and radius grow by $10^8$, capacity by $10^{16}$, and temperature falls by $10^8$.

3 Rotating black holes, clocks and accretion

The saturated-area construction extends to rotating black holes by retaining angular momentum alongside capacity. Kerr geometry supplies the rotating exterior, and the area assignment $A_H=4A_\Sigma$ connects its horizon section to PIN’s saturated disk. The resulting mass relation links capacity, rotation, temperature, clocks and accretion in one mathematical account.

3.1 Conventions and additional notation

Equations (1), (2), (4) and (5) fix the Planck scales, reduced PWP wavelength, nat convention and saturated inventory. The same definitions apply here. The radius $R_\Sigma$ denotes the saturated disk radius called $R$ in the main derivation; $E_{\mathrm{BH}} =Mc^2$ is its collective gravitational energy and $E_\Sigma =N E_{\mathrm P}$ its assigned carrier-energy inventory. Saturation fixes $N/A_\Sigma= \ell_{\mathrm P} ^{-2}$ and $E_\Sigma /A_\Sigma= E_{\mathrm P} / \ell_{\mathrm P} ^2$. The rotating mass relation determines the gravitational energy for the specified capacity and angular momentum. The entropy and temperature normalizations retain Bekenstein’s area entropy and Hawking’s temperature calculation [7, 8].

SI units and signature $(-,+,+,+)$ apply except in the explicitly marked fluid equations. The principal square root is used. The stationary black hole is neutral, $M>0$, and $| a_* |<1$; extremal values are limits. Evolution equations compare neighboring Kerr states with slowly changing parameters. The exterior coordinate $t$ has standard Kerr stationary normalization; finite-reference clock comparisons specify their observer.

Symbol	Meaning
$j=J/\hbar, \mu_K=M/ m_{\mathrm P}$	Dimensionless angular momentum and mass; $N$ and $j$ are continuous macroscopic variables.
$r_g=GM/c^2, a_{\mathrm K} =J/(Mc)$	Geometric mass length and signed Kerr spin length.
$a_* = a_{\mathrm K} /r_g, s=\sqrt{1- a_* ^2}$	Dimensionless spin and positive subextremal spin factor.
$r_+, R_A , R_{\mathrm{eq}}$	Outer Boyer–Lindquist radius, equivalent-area radius and proper equatorial circumference divided by $2\pi$.
$\Sigma_{\mathrm K} ,\Delta, \mathcal{A}_{\mathrm K}$	Kerr metric functions; $\Sigma_{\mathrm K}$ is distinct from the disk label $\Sigma$.
$\alpha,\omega,\Omega$	Lapse, local frame-dragging angular velocity and observer angular velocity.
$\gamma_{ij}, p_i$	Induced constant-$t$ spatial metric and covariant spatial momenta, $i,j\in\{r,\theta,\phi\}$.
$\Omega_H , T_H$	Horizon angular velocity and Hawking temperature.
$\dot E_H,\dot J_H$	Signed net inward energy and angular-momentum fluxes at the horizon.

3.2 Kerr geometry and the area match

The Kerr parameters are $$r_g=\frac{GM}{c^2},\qquad a_{\mathrm K} =\frac{J}{Mc},\qquad a_* =\frac{ a_{\mathrm K} }{r_g}=\frac{cJ}{GM^2},\qquad r_\pm=r_g(1\pm s),\quad s=\sqrt{1- a_* ^2}. \tag{K1}$$(K1) The geometric functions are $$\Sigma_{\mathrm K} =r^2+ a_{\mathrm K} ^2\cos^2\theta,\quad \Delta=r^2-2r_gr+ a_{\mathrm K} ^2,\quad \mathcal{A}_{\mathrm K} =(r^2+ a_{\mathrm K} ^2)^2- a_{\mathrm K} ^2\Delta\sin^2\theta. \tag{K2}$$(K2) In Boyer–Lindquist coordinates the line element is [58, 59] $$\begin{aligned} \mathrm d s^2={}&-\left(1-\frac{2r_gr}{ \Sigma_{\mathrm K} }\right)c^2 \mathrm d t^2 -\frac{4r_g a_{\mathrm K} r\sin^2\theta}{ \Sigma_{\mathrm K} }\,c\, \mathrm d t\, \mathrm d \phi\\ &+\frac{ \Sigma_{\mathrm K} }{\Delta} \mathrm d r^2+ \Sigma_{\mathrm K} \mathrm d \theta^2 +\frac{ \mathcal{A}_{\mathrm K} \sin^2\theta}{ \Sigma_{\mathrm K} } \mathrm d \phi^2. \end{aligned} \tag{K3}$$(K3) The outer root satisfies $\Delta(r_+)=0$ and $r_+^2+ a_{\mathrm K} ^2=2r_gr_+$. Its intrinsic area element integrates to $$\mathrm d A_H=(r_+^2+ a_{\mathrm K} ^2)\sin\theta\, \mathrm d \theta\, \mathrm d \phi, \qquad A_H=4\pi(r_+^2+ a_{\mathrm K} ^2)=8\pi r_g^2(1+s). \tag{K4}$$(K4)

PIN retains the outward disk-to-exterior area assignment $A_\Sigma:A_H=1:4$ in the rotating sector through the matching condition $$\boxed{A_H=4A_\Sigma,\qquad N=\frac{A_\Sigma}{ \ell_{\mathrm P} ^2}=\frac{A_H}{4 \ell_{\mathrm P} ^2}.} \tag{K5}$$(K5) The flat disk therefore has area $A_H/4$. Its radius under this matching is the equivalent-area radius, $$R_\Sigma = R_A \equiv\sqrt{\frac{A_H}{4\pi}} =\sqrt{r_+^2+ a_{\mathrm K} ^2}=r_g\sqrt{2(1+s)} = \ell_{\mathrm P} \sqrt{\frac N\pi}. \tag{K6}$$(K6) Equation (K5) is PIN’s area-matching choice. Equation (K6) is its consequence. The outer Kerr section has its own intrinsic geometry, and its proper equatorial radius is $$R_{\mathrm{eq}} =\sqrt{g_{\phi\phi}(r_+,\pi/2)} =\frac{r_+^2+ a_{\mathrm K} ^2}{r_+}=2r_g,\qquad \frac{A_H}{\pi R_{\mathrm{eq}} ^2}=2(1+s). \tag{K7}$$(K7) Consequently, the physical disk radius in the area-matched description and the exterior equatorial circumference radius have different values at nonzero spin. A projection law relating the physical rim to the exterior equator must respect this distinction. For example, at $a_* =\sqrt3/2$, $R_A =\sqrt3r_g$ and $R_{\mathrm{eq}} =2r_g$: the area-matched disk retains the ratio $1:4$, while a Euclidean disk of radius $R_{\mathrm{eq}}$ has the ratio $1:3$. The difference is a concrete matching constraint.

3.3 The mass equation in PIN variables

Write $\mu_K=M/ m_{\mathrm P}$ and $j=J/\hbar$. The outer-horizon capacity becomes $$N=2\pi\mu_K^2\left(1+\sqrt{1-\frac{j^2}{\mu_K^4}}\right) =2\pi\left(\mu_K^2+\sqrt{\mu_K^4-j^2}\right). \tag{K8}$$(K8) Isolating the square root and squaring gives $$\left(\frac{N}{2\pi}-\mu_K^2\right)^2=\mu_K^4-j^2 \quad\Longrightarrow\quad \boxed{\frac{M^2}{ m_{\mathrm P} ^2}=\frac{N}{4\pi}+\frac{\pi j^2}{N}}, \qquad N\geq2\pi|j|. \tag{K9}$$(K9) The inequality selects the outer-area branch retained before squaring. The strict inequality describes subextremal states.

The first term is the area contribution to mass squared; the second is the rotational contribution to mass squared. Introducing Christodoulou’s irreducible mass makes the relation explicit [60]: $$M_{\mathrm{irr}}=\sqrt{\frac{c^4A_H}{16\pi G^2}} = m_{\mathrm P} \sqrt{\frac{N}{4\pi}},\qquad M^2=M_{\mathrm{irr}}^2+\frac{J^2c^2}{4G^2M_{\mathrm{irr}}^2}. \tag{K10}$$(K10) Thus PIN’s nonrotating square-root relation carries directly into the irreducible mass of Kerr. The full mass includes rotation through Eq. (K9); its algebra is the established Kerr relation expressed in the saturated capacity variables.

A useful spin-to-capacity ratio is $$x\equiv\frac{2\pi j}{N}=\frac{ a_{\mathrm K} }{r_+},\qquad a_* =\frac{2x}{1+x^2},\qquad s=\frac{1-x^2}{1+x^2},\qquad |x|<1. \tag{K11}$$(K11) It yields a compact geometric key: $$\boxed{\begin{gathered} R_\Sigma = R_A = \ell_{\mathrm P} \sqrt{N/\pi},\qquad M=M_{\mathrm{irr}}\sqrt{1+x^2},\\ R_{\mathrm{eq}} = R_A \sqrt{1+x^2},\qquad r_+=\frac{ R_A }{\sqrt{1+x^2}},\qquad a_{\mathrm K} =xr_+. \end{gathered}} \tag{K12}$$(K12) At fixed dimensionless spin $a_*$ (equivalently $x$), mass and every radius scale as $\sqrt N$. Doubling the matched disk radius quadruples capacity and doubles mass. The same curve supplies a formal single-area normalization at $N=1$; the semiclassical Kerr application uses areas large compared with $\ell_{\mathrm P} ^2$.

The gravitational-to-inventory energy ratio and a disk circumference scale are $$\gamma_K\equiv\frac{ E_{\mathrm{BH}} }{ E_\Sigma } =\frac{\sqrt{1+x^2}}{2\sqrt{\pi N}},\qquad \tau_\Sigma\equiv\frac{2\pi R_\Sigma }{c},\qquad \gamma_K=\sqrt{1+x^2}\,\frac{ t_{\mathrm P} }{\tau_\Sigma}. \tag{K13}$$(K13) Here $\tau_\Sigma$ is the defined light-circuit scale of the flat disk. The factor $\sqrt{1+x^2}$ is required by the rotating mass equation. Actual exterior clock rates follow the metric and the observer’s motion.

3.4 Temperature, rotation and capacity growth

Treat the energy as a function of the independent variables $N$ and $j$: $$E_{\mathrm{BH}} (N,j)= E_{\mathrm P} \mu_K(N,j),\qquad \mu_K(N,j)=\sqrt{\frac{N}{4\pi}+\frac{\pi j^2}{N}}. \tag{K14}$$(K14) At fixed angular momentum, its capacity derivative gives $k_{\mathrm B} T_H$; at fixed capacity, its $j$ derivative gives $\hbar \Omega_H$: $$\begin{aligned} \left(\frac{\partial E_{\mathrm{BH}} }{\partial N}\right)_j &=\frac{ E_{\mathrm P} }{8\pi\mu_K}(1-x^2)= k_{\mathrm B} T_H ,\\ \left(\frac{\partial E_{\mathrm{BH}} }{\partial j}\right)_N &=\frac{\pi E_{\mathrm P} j}{\mu_K N}=\hbar \Omega_H . \end{aligned} \tag{K15}$$(K15) Although $x$ shortens the result, $j$, rather than $x$, is held fixed in the first derivative. In geometric variables these quantities are $$\begin{aligned} \Omega_H &=\frac{c a_{\mathrm K} }{r_+^2+ a_{\mathrm K} ^2} =\frac{c^3 a_* }{2GM(1+s)}=\frac{cx}{ R_{\mathrm{eq}} },\\ k_{\mathrm B} T_H &=\frac{\hbar c(r_+-r_g)}{2\pi(r_+^2+ a_{\mathrm K} ^2)} =\frac{\hbar c^3s}{4\pi GM(1+s)} =\frac{\hbar c(1-x^2)}{4\pi R_{\mathrm{eq}} }. \end{aligned} \tag{K16}$$(K16) These are the Kerr angular velocity and Hawking temperature. The first law and the corresponding homogeneous mass identity are $$\boxed{ \mathrm d E_{\mathrm{BH}} = k_{\mathrm B} T_H \, \mathrm d N+ \Omega_H \, \mathrm d J},\qquad E_{\mathrm{BH}} =2 k_{\mathrm B} T_H N+2 \Omega_H J. \tag{K17}$$(K17) The normalization is consistent with the physical-process first law [8, 61]; the second equality is the neutral Smarr relation [21].

Using $E_{\mathrm{BH}}/N=E_{\mathrm P}\gamma_K$, the relations in Eqs. (K11)–(K15) also express the two marginal energies and the normalized Smarr terms as $$\begin{gathered} k_{\mathrm B}T_H=\frac{s}{2}E_{\mathrm P}\gamma_K, \qquad \hbar\Omega_H=\pi a_*E_{\mathrm P}\gamma_K,\\ \frac{2k_{\mathrm B}T_HN}{E_{\mathrm{BH}}} =s=\frac{1-x^2}{1+x^2},\\ \frac{2\Omega_HJ}{E_{\mathrm{BH}}} =xa_*=\frac{2x^2}{1+x^2},\qquad s+xa_*=1. \end{gathered}\tag{K17a}$$(K17a) These are fractions of the two terms in the Smarr identity; the irreducible-mass relation remains Eq. (K10).

The inverse-temperature time is $$\tau_H\equiv\frac{\hbar}{ k_{\mathrm B} T_H } =\frac{4\pi R_{\mathrm{eq}} }{c(1-x^2)} =\frac{2\tau_\Sigma\sqrt{1+x^2}}{1-x^2}. \tag{K18}$$(K18) Both the energy ratio and the thermal scale acquire a spin dependence. The zero-spin limits are $\gamma_K= t_{\mathrm P} /\tau_\Sigma$ and $\tau_H=2\tau_\Sigma$.

With one common time normalization for the net horizon fluxes, $$\boxed{\dot M=\frac{\dot E_H}{c^2},\qquad \dot{a_*}=\frac{c\dot J_H}{GM^2}-2 a_* \frac{\dot M}{M},\qquad \dot N=\frac{\dot E_H- \Omega_H \dot J_H}{ k_{\mathrm B} T_H }.} \tag{K19}$$(K19) The power $\Omega_H \dot J_H$ accounts for the rotational part of the balance. For classical flux satisfying the null energy condition, the horizon generator gives $\dot E_H- \Omega_H \dot J_H\geq0$ and hence $\dot N\geq0$ in this quasistationary regime [61].

The matched disk grows according to $$\dot{R_\Sigma}=\frac{ \ell_{\mathrm P} ^2}{2\pi R_\Sigma }\dot N,\qquad \frac{\dot{R_A}}{ R_A }=\frac{\dot M}{M} -\frac{ a_* \dot{a_*}}{2s(1+s)},\qquad \dot{R_{\mathrm{eq}}}=\frac{2G}{c^4}\dot E_H. \tag{K20}$$(K20) An increment $\mathrm d R_\Sigma$ adds the saturated annulus $\mathrm d A_\Sigma=2\pi R_\Sigma \, \mathrm d R_\Sigma$. The physical edge is a one-dimensional growth front for a two-dimensional capacity. Different behaviour of $R_\Sigma$ and $R_{\mathrm{eq}}$ during rotational-energy extraction is already contained in these equations. Equation (K19) uses $T_H >0$. At extremality, where $T_H =0$, the undivided first law in Eq. (K17) supplies the energy balance.

3.5 Gravity, rotation and local clocks

PIN interprets gravity through the spatial coordination of information. Bringing interacting regions closer can shorten the paths over which they exchange signals, while concentrating those interactions can increase the demand for local coordination. Signal travel time and local processing demand are distinct parts of this account. Rotation adds direction: the exterior geometry treats propagation with and against the spin differently. The Kerr metric supplies a precise relation among clock rates, directional transport and spatial distances. Completing the square in Eq. (K3) gives $$\begin{gathered} \alpha^2=\frac{ \Sigma_{\mathrm K} \Delta}{ \mathcal{A}_{\mathrm K} },\qquad \omega=\frac{2r_g a_{\mathrm K} r c}{ \mathcal{A}_{\mathrm K} },\qquad g_{\phi\phi}=\frac{ \mathcal{A}_{\mathrm K} \sin^2\theta}{ \Sigma_{\mathrm K} },\\ \mathrm d s^2=-\alpha^2c^2 \mathrm d t^2 +g_{\phi\phi}( \mathrm d \phi-\omega \mathrm d t)^2 +\frac{ \Sigma_{\mathrm K} }{\Delta} \mathrm d r^2+ \Sigma_{\mathrm K} \mathrm d \theta^2. \end{gathered} \tag{K21}$$(K21) The lapse $\alpha$ and frame-dragging rate $\omega$ follow from the same Kerr metric [59]. The local zero-angular-momentum observer rotates at $\omega$ and measures proper time $\alpha\, \mathrm d t$. In the exterior $r>r_+$, a circular observer at fixed $r,\theta$, with $\Omega= \mathrm d \phi/ \mathrm d t$, has $$Z(r,\theta;\Omega)\equiv\frac{ \mathrm d \tau}{ \mathrm d t} =\sqrt{\alpha^2-\frac{g_{\phi\phi}}{c^2}(\Omega-\omega)^2}. \tag{K22}$$(K22) The same clock relation separates the lapse from motion relative to that local frame: $$v_{\mathrm{loc}}=\frac{\sqrt{g_{\phi\phi}}}{\alpha}(\Omega-\omega), \qquad Z=\alpha\sqrt{1-\frac{v_{\mathrm{loc}}^2}{c^2}}. \tag{K23}$$(K23) Here $v_{\mathrm{loc}}$ is the signed local azimuthal velocity. Reversing $J$ at fixed $M,r,\theta$ leaves $\alpha$ and the induced spatial metric unchanged, while reversing $\omega$. The same saturated capacity can therefore accompany opposite directions of frame dragging. Both the clock rate and the direction of transport enter the rotating description.

Away from the rotation axis, timelike circular motion requires $$\omega-\frac{c\alpha}{\sqrt{g_{\phi\phi}}} <\Omega< \omega+\frac{c\alpha}{\sqrt{g_{\phi\phi}}}. \tag{K24}$$(K24) For zero-angular-momentum observers, $\Omega=\omega$ and $Z=\alpha$. As $r\to r_+^+$ this circular interval closes onto $\Omega_H$. Infalling clocks instead accumulate proper time on their moving worldlines, using the complete line element.

3.6 Energy and motion in the rotating exterior

The metric also gives a compact energy equation. Its induced spatial metric is diagonal, with $\gamma_{rr}= \Sigma_{\mathrm K} /\Delta$, $\gamma_{\theta\theta}= \Sigma_{\mathrm K}$ and $\gamma_{\phi\phi}=g_{\phi\phi}$. The symbol $\gamma^{ij}$ denotes its three-dimensional inverse. Let $m_b$ be a test body’s rest mass, $p_i$ its covariant spatial momenta, $E_b=-p_t$ its stationary energy, and $L_b=p_\phi$ its axial angular momentum. With $t$ in seconds, the inverse-metric mass shell becomes $$g^{\mu\nu}p_\mu p_\nu=-m_b^2c^2 \quad\Longleftrightarrow\quad -\frac{(E_b-\omega L_b)^2}{\alpha^2c^2} +\gamma^{ij}p_i p_j=-m_b^2c^2. \tag{K25}$$(K25) Selecting positive energy in the local zero-angular-momentum frame gives $$\boxed{E_b=\omega L_b+ \alpha\sqrt{m_b^2c^4+c^2\gamma^{ij}p_i p_j}},\qquad E_{\mathrm{loc}}=\frac{E_b-\omega L_b}{\alpha}>0. \tag{K26}$$(K26) The square root is $E_{\mathrm{loc}}$. The lapse relates that local energy to stationary time, and the angular shift contributes $\omega L_b$. The spatial metric fixes the physical momentum norm. This is a derivation from the imported Kerr metric, giving PIN’s informational dynamics a single energy relation to reproduce. In the ergoregion, oppositely directed angular momentum can make $E_b$ negative while $E_{\mathrm{loc}}$ remains positive.

For freely falling neutral test bodies, $E_b$ acts as the Hamiltonian for coordinate-time evolution: $$\frac{ \mathrm d x^i}{ \mathrm d t}=\frac{\partial E_b}{\partial p_i},\qquad \frac{ \mathrm d p_i}{ \mathrm d t}=-\frac{\partial E_b}{\partial x^i}. \tag{K27}$$(K27) The massless case $m_b=0$, with nonzero momentum, describes light rays in geometric optics. At a fixed position away from the axis, tangential null directions have $\Omega_\pm=\omega\pm c\alpha/\sqrt{g_{\phi\phi}}$. Their mean is $\omega$ and their separation is set by the local light cone. These are local propagation directions; a freely orbiting photon also has to satisfy the radial and polar equations of motion. The locally measured speed of light is $c$.

3.7 Finite-reference time and work

Choose a constant $\Omega_0$ and a finite reference position $P_0$, such that the Killing field $K=\partial_t+\Omega_0\partial_\phi$ is timelike along the comparison route. Let $Z_0=Z(P_0;\Omega_0)>0$. The clock comparison and the generator-normalized energy of a supported test body, comoving with $K$ and of fixed rest mass $m_b$, are $$h_K(P;P_0,\Omega_0)=\frac{Z(P;\Omega_0)}{Z_0},\qquad \mathcal E_{K,0}=\frac{E_b-\Omega_0L_b}{Z_0} =m_bc^2h_K. \tag{K28}$$(K28) Here $E_b$ and $L_b$ are the body’s stationary and axial Killing charges. Reversible relocation to $P_0$ has the generalized work cost $$W_{K,0}=m_bc^2(1-h_K),\qquad a_i=c^2\partial_i\ln Z\big|_{\Omega_0},\quad i\in\{r,\theta\}. \tag{K29}$$(K29) The body is comoving with $K$ at both endpoints. The work is expressed in $E_b-\Omega_0L_b$, so it includes the angular-momentum exchange with the co-rotating support system. The covariant spatial acceleration components $a_i$ follow from the norm of the same Killing field. PIN’s latency account uses these specified clock and work comparisons as its macroscopic target. This comparison retains the same constant $\Omega_0$ along the route. Local zero-angular-momentum observers follow the position-dependent $\omega(r,\theta)$; comparisons between those frames retain that variation explicitly.

In the nonrotating, static limit, for $r,R_0>2r_g$, the equations recover $$h_K=\sqrt{\frac{1-2r_g/r}{1-2r_g/R_0}},\qquad a_{\mathrm{proper}}=\frac{GM}{r^2\sqrt{1-2r_g/r}} \longrightarrow\frac{GM}{r^2}\quad(r\gg r_g). \tag{K30}$$(K30) For a count $n$ of events, the Kerr clock ratio also fixes the conversion of a local event rate: $$\nu_0=\frac{ \mathrm d n}{ \mathrm d \tau_0} =h_K\frac{ \mathrm d n}{ \mathrm d \tau}=h_K\nu_{\mathrm{proper}}. \tag{K31}$$(K31) In PIN’s account, the wave function computes possible future continuations, and the present is the most recently committed frame. That frame fixes a consistent past through which earlier positions can be traced continuously. Equation (K31) separates the proposed proper commitment rate from its reference-clock rate. A physical commitment criterion supplies $\nu_{\mathrm{proper}}$; geometry supplies the conversion.

3.8 Accretion and the energy–angular-momentum balance

The inflow is modeled with relativistic magnetized-fluid equations on a Kerr background. For the fluid equations and flux integrals below, set $G=c=1$, use $u^\mu u_\mu=-1$, and absorb electromagnetic $4\pi$ factors into the field normalization. With rest-mass density $\rho$, internal-energy density $e_{\mathrm{int}}$, pressure $p$, and fluid-frame magnetic four-vector $b^\mu$, ideal GRMHD gives [62] $$\nabla_\mu(\rho u^\mu)=0,\qquad \nabla_\mu T^{\mu\nu}=0,\qquad \nabla_\mu{}^*F^{\mu\nu}=0,\qquad F_{\mu\nu}u^\nu=0, \tag{K32}$$(K32) $$T^{\mu\nu}=(\rho+e_{\mathrm{int}}+p+b^2)u^\mu u^\nu +\left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu, \quad b^2=b^\mu b_\mu,\quad b^\mu u_\mu=0. \tag{K33}$$(K33) Here $F_{\mu\nu}$ is the electromagnetic field tensor and ${}^*F^{\mu\nu}$ its dual. An equation of state closes the fluid model. These equations evolve the exterior inflow, including magnetic torque and energy transport. For the accretion approximation the fluid’s gravitational backreaction is small compared with the black-hole background.

With $g=\det(g_{\mu\nu})$ in horizon-penetrating coordinates, consistently inward-positive signed fluxes through a constant-radius shell are $$\begin{aligned} \dot M_0&=-\int\rho u^r\sqrt{-g}\, \mathrm d \theta\, \mathrm d \phi,\\ \dot E_H&=\int T^r{}_t\sqrt{-g}\, \mathrm d \theta\, \mathrm d \phi,\\ \dot J_H&=-\int T^r{}_\phi\sqrt{-g}\, \mathrm d \theta\, \mathrm d \phi. \end{aligned} \tag{K34}$$(K34) The subscript $H$ applies when the shell is at the horizon. These energy and angular-momentum signs reverse the outward-positive signs in Dhruv et al. Eqs. (13)–(14) [62]. Physical units and one time normalization are restored before applying Eq. (K19). Rest-mass inflow $\dot M_0$ is distinct from $\dot E_H/c^2$.

To measure concentration near the spin equator, retain the angular distribution and define $$\mathcal F_{\mathrm{in}}=\max(-\rho u^r,0)\sqrt{-g},\qquad f_{\mathrm{eq}}(\delta;r,t)= \frac{\displaystyle\int_0^{2\pi}\!\int_{\pi/2-\delta}^{\pi/2+\delta} \mathcal F_{\mathrm{in}}\, \mathrm d \theta\, \mathrm d \phi} {\displaystyle\int_0^{2\pi}\!\int_0^\pi \mathcal F_{\mathrm{in}}\, \mathrm d \theta\, \mathrm d \phi}. \tag{K35}$$(K35) For $0\leq\delta\leq\pi/2$ and finite, strictly positive total incoming flux, this definition gives $0\leq f_{\mathrm{eq}}\leq1$. For a spherical radial-flow control, $\mathcal F_{\mathrm{in}}\propto\sin\theta$ and $f_{\mathrm{eq}}=\sin\delta$. This validates the angular measure; the primary physical benchmark is spinning Kerr.

The public prograde $a_* =1/2$ SANE (standard and normal evolution) run of Dhruv et al. provides an initial case; a same-spin MAD (magnetically arrested disk) run provides a comparison with stronger accumulated magnetic flux. The simulations hold the Kerr metric fixed, while their fluxes diagnose mass and spin tendencies. Density-floor flags and the shell radius must accompany each measurement [62]. Tilted simulations add a distinct test: Liska et al. find polar plunging streams, making the difference between spin-equatorial and gas-plane angles physically relevant [63].

3.9 The rim as a capacity front and an information interface

The area-growth relation specifies how much saturated capacity is added. A separate angular ledger specifies what arrives. For example, the azimuthal rate per unit matched rim length can be formed as $$q(\phi,t)=\frac1{ R_\Sigma }\int_0^\pi \mathcal F_{\mathrm{in}}(\theta,\phi,t)\, \mathrm d \theta,\qquad \int_0^{2\pi}q\, R_\Sigma \, \mathrm d \phi=\dot M_{\mathrm{in}}. \tag{K36}$$(K36) This is a flux diagnostic. Conservation of its integral alone leaves the latitude information unresolved. Here $\dot M_{\mathrm{in}}$ is the total gross inward rest-mass flux, counting inward motion only. Two incoming distributions, $$\mathcal F_\pm=\frac{f(\phi)}2\sin\theta (1\pm\epsilon\cos\theta),\qquad f\geq0,\quad0<\epsilon<1, \tag{K37}$$(K37) have the same $q=f/ R_\Sigma$, but different mean polar-direction cosines, $\langle\cos\theta\rangle_\pm=\pm\epsilon/3$. These normalized means assume $0<\int_0^{2\pi}f(\phi)\, \mathrm d \phi<\infty$. The physical rim coupling must retain the distinctions carried by the incoming state as well as satisfy its energy and angular-momentum budget. This is a necessary information-preservation check when specifying that coupling.

The rotating mathematical key can now be read in one line: the PWP assignment fixes saturated area per nat; Kerr connects that area and angular momentum to gravitational mass; the first law controls their exchange; and the exterior metric supplies clock, force and inflow comparisons. PIN’s physical disk and rim give the spatial interpretation to be matched to those relations.

Calculation and source access. The accompanying verification record provides the equation crosswalk, source locators and checks for this section. The accretion comparisons above refer to the published simulations cited there. The capacity, thermodynamic, clock and flux equations specify the quantities used to compare the rotating exterior with PIN’s physical disk and retained information.

4 Gravity, entanglement and emergent geometry

Section 2 related the capacity of a saturated disk to the mass of its exterior black-hole geometry. We now ask what gravity does in PIN’s physical account, and how that account connects to the clock rates, energy changes and attraction measured outside the black hole.

The argument begins with the physical system and PIN’s explanation of gravity as the organization of correlated information. We then identify what established thermodynamic and entanglement results contribute. Finally, using the stationary exterior from Section 2, we calculate compactness, relative clock rates, reversible lifting work and support force. The physical explanation and the exterior calculation have specific roles: PIN proposes why the information organizes, while the adopted geometry determines the gravitational effects calculated here.

4.1 The saturated disk and its exterior gravity

In PIN, physical space and its gravitational geometry emerge from the informational substrate. The substrate carries physical states and their correlations. PIN interprets the surrounding three-dimensional geometry as the macroscopic expression of that underlying organization. For the black-hole construction developed here, the saturated region is a flat two-dimensional disk. Its physical horizon is its one-dimensional rim, beyond which spatial emergence proceeds outward. The spherical horizon section belongs to this exterior description. The correspondence in Section 1 therefore connects two physical roles: the disk on which information is retained and the geometry through which the black hole is encountered in space.

The same-radius correspondence preserves the total capacity. A disk of area $\pi R^2$ carries $N=\pi R^2/\ell_{\mathrm P}^2$ nats; expressing that count on the spherical surface of area $4\pi R^2$ gives $N=A_H/(4\ell_{\mathrm P}^2)$. The quarter is the ratio of the two areas. All of the counted information remains in the account. This geometrical statement supplies the normalization used throughout the black-hole scaling argument.

Section 2 connects this capacity to an exterior gravitational mass. With the adopted trace-free response, conservation of the complete matter stress, and the isolated vacuum choice $\Lambda_{\mathrm{eff}}=0$, the spherically symmetric solution has $u(r)=R/r$. Assigning the disk radius to the exterior null radius gives $R=r_s=2GM/c^2$. Consequently, $N\propto R^2$ while $M\propto R$, so $M\propto\sqrt N$. The mass family in Eq. (23) combines the area count, the adopted gravitational response and the disk–exterior correspondence. Adding the stationary first law gives the temperature in Eq. (27), while the same exterior energy curve fixes the finite release budget in Eq. (29). These relations organize the later clock and information-return discussions.

Collective mass is compatible with massless constituents. In one inertial frame, an isolated collection satisfies $\mathcal M^2c^4=E_{\mathrm{tot}}^2-c^2|\mathbf P_{\mathrm{tot}}|^2$; two oppositely directed photons of equal energy $E$ therefore have invariant mass $2E/c^2$ [25]. Each photon still has zero rest mass. This example relates collective invariant mass to total energy and momentum. PIN’s black-hole gravitational mass follows from the exterior construction in Section 2, with $E_\Sigma=NE_P$ specifying its separate substrate energy inventory.

The scale dependence also separates local saturation from exterior gravity. Throughout the family, $N/A_\Sigma=\ell_{\mathrm P}^{-2}$ and $E_\Sigma/A_\Sigma=E_{\mathrm P}/\ell_{\mathrm P}^2$ remain fixed. A larger black hole contains more saturated area. Meanwhile, its exterior energy divided by the areal-radius comparison volume is $\bar\varepsilon_{\mathrm{BH}}=E_{\mathrm{BH}}/(4\pi R^3/3)=3c^4/(8\pi GR^2)$, which decreases as $R^{-2}$. This volume comparison characterizes the exterior mass scale; the physical disk retains the same energy inventory per unit area.

4.2 Gravity as the organization of correlated information

In PIN, gravity is the attractive component of entanglement. Its role is to reduce latency in information coordination. Here, latency means the delay involved in coordinating physical interactions. Entanglement describes correlations in the joint physical state, whose complete informational description includes both the constituents and their correlations. PIN places the arrangement of the carriers and their correlations within one physical account: gravity is the entropic pressure organizing this information in the two-dimensional substrate.

In PIN, physical interactions are organized under finite local resources. Establishing, changing and coordinating an interacting state requires causal physical processes limited by $c$. Arrangement can affect the distances those processes span and the demand concentrated in a region. Entanglement correlations satisfy the no-signalling condition [33]. The identification of an attractive component with gravity is PIN’s physical hypothesis; its quantitative exterior expression is supplied by the adopted gravitational response.

A shorter interaction path gives a concrete meaning to reduced latency. For a specified local proper path length $L$, propagation at the local causal speed has the travel-time scale $L/c$. Reducing that length can reduce the propagation interval needed by an interaction. Paths spanning regions with different clock rates require the corresponding spacetime propagation calculation.

For a concrete exterior comparison, consider static endpoints at proper distance $\rho_0$ from the horizon and transverse separation $L$, with $\rho_0,L\ll R$. Taking emission at $t=0$, combining Susskind’s local coordinate and light-cone equations [11, Sec. 4, Eqs. (4.3), (4.11)–(4.15)] gives the transverse causal-front radius and its arrival time: $$\begin{aligned} r_{\perp,\max}^{2}(t)&=2\rho_0^2\left[\cosh\!\left(\frac{t}{t_b}\right)-1\right],\\ t_{\mathrm{reach}}(L)&=2t_b\operatorname{arsinh}\!\left(\frac{L}{2\rho_0}\right). \end{aligned}$$

Here $t_{\mathrm{reach}}$ is the earliest arrival time in this local exterior model. The connecting null ray leaves the constant-$\rho_0$ surface and returns to it, reaching $\rho_{\max}=\sqrt{\rho_0^2+L^2/4}$. For $\rho_0\ll L\ll R$, the arrival time is approximately $2t_b\ln(L/\rho_0)$. This supplies an exterior causal benchmark for PIN’s latency account.

Bringing interactions into a smaller region can increase the local processing workload. At fixed processing capacity, using a larger share for that workload leaves less headroom for further distinguishable changes. Shorter communication paths and reduced processing headroom can therefore occur together: the arrangement improves the coordination of interactions while their concentration demands more local processing. In PIN this connects gravitational organization with time dilation. A region can require more processing while advancing through fewer additional distinguishable states relative to an exterior comparison clock. Here, “entropic pressure” expresses PIN’s tendency for correlated information to arrange itself so that the physical interactions needed to coordinate it use the available resources more efficiently. This organization involves a portion of the self-gravitating system’s own energy. Its energetic cost depends on the system’s state and dynamics. The exterior calculation below supplies the clock, energy and force relations that constrain this account.

PIN describes actual history through quantum frames committed by collapse. The wave function is the computational description of unresolved possibilities; a committed frame is an actual outcome from which further possibilities are conditioned. Successive frames form a consistent actual history, and their stacking and processing underlie experienced time. The workload discussed above matters because it leaves less processing capacity available for further local frame progression. At saturation, all available local processing retains the current informational state; no capacity remains for further local frame progression. Section 6 develops the initial economy of retaining unresolved possibilities and the efficiency crossover that favours commitment. Section 7 develops the saturated limit and the return of the retained information through desaturation at the physical rim.

4.3 What established gravity results contribute

PIN connects the organization of information to the geometry of the surrounding space. Thermodynamic and entanglement approaches provide specific examples of a connection between entropy, energy and the Einstein equation. Their role here is to explain the established basis for relating gravitational geometry to information, and to identify the assumptions under which those results hold.

Jacobson’s 1995 argument begins with a local causal horizon: a boundary of the region accessible to a locally accelerated observer. He assigns entropy in proportion to its area and uses the Unruh temperature $T_U$ associated with the observer’s acceleration. Here $\delta Q$ is the heat flow through the horizon and $dS$ is its entropy change. Requiring the equilibrium balance $\delta Q=T_UdS$ for local horizons in every lightlike direction at each spacetime point, together with local conservation of stress and energy, gives the Einstein equation [26]. The importance for PIN is that a gravitational field equation can follow from a thermodynamic condition. Eling, Guedens and Jacobson’s 2006 extension allows the horizon entropy to depend on the Ricci scalar, which measures spacetime curvature. That extension requires internal entropy production as well as heat flow; the paper also discusses shear-related entropy production in Einstein gravity [53]. It distinguishes equilibrium balance from internal entropy production in that horizon model.

Jacobson’s 2016 argument starts instead from vacuum entanglement. Consider a small geodesic ball in a spatial slice and small changes in its geometry and quantum state about a locally maximally symmetric vacuum. At fixed spatial volume, stationarity of the total entanglement entropy means that its first-order change vanishes. With a finite universal area-entropy density, this condition leads to the semiclassical Einstein equation for conformal fields; the extension to nonconformal fields requires an additional conjecture [54]. This result connects entanglement to gravitational response. Its volume condition holds the three-dimensional volume of the small ball fixed. PIN’s Section 2 instead restricts variation of the four-dimensional spacetime volume density.

Verlinde relates gravitational work to entropy change and displacement using an information-carrying surface, called a screen, an assigned temperature, and an equal average energy assignment to the screen’s microscopic degrees of freedom (equipartition) [27]. This is relevant to PIN’s entropic-pressure account because it gives a mathematical example of the connection between entropy and force. PIN’s own proposal specifies a physical substrate, a saturated disk and an outward correspondence to the exterior geometry.

The calculation that follows starts from the stationary exterior obtained in Section 2. It determines compactness, relative clock rates, reversible lifting work and the local force needed to support a body. These results specify the gravitational behaviour that PIN’s proposed organization of information is intended to explain.

4.4 Compactness: the gravitational scale at a given radius

At an exterior areal radius $r$, the Bekenstein energy–size expression in nats can be compared with the capacity of a disk of that radius: $$\chi(r)=\frac{2\pi rMc/\hbar}{\pi r^2/\ell_{\mathrm P}^2}=\frac{2GM}{rc^2}=\frac{r_s}{r}.\tag{31}$$(31)

The numerator is Bekenstein’s energy–size expression, whose entropy-bound interpretation applies to bounded, weakly self-gravitating systems [28]. Using $\ell_P^2=\hbar G/c^3$ makes its ratio to area capacity exactly the Schwarzschild compactness. For the selected exterior, $\chi(r)=u(r)$. At $r=r_s$, the two expressions coincide algebraically, giving $\chi=1$.

The denominator is a comparison capacity at radius $r$, and the numerator is Bekenstein’s energy–size expression. Their ratio $\chi$ is fixed by the exterior mass and radius. Section 5 assigns this compactness to processing utilization in PIN’s clock account.

This distinction matters along the full scaling curve. Every member has $\chi=1$ at its own horizon, while the exterior-to-inventory ratio in Eq. (24), $\gamma=\ell_{\mathrm P}/(2\pi R)$, decreases with black-hole size. The horizon compactness and the collective energy per unit assigned inventory therefore answer different questions even at complete saturation.

4.5 From the exterior geometry to clocks, work and force

We can now calculate how clocks at different radii compare and how much work is required to lift a body between them. Both results follow from the same adopted Einstein geometry [12, 19]. Consider the stationary exterior, $r>r_s$, and set $f(r)=1-r_s/r$. Introducing static time $t$ in the ingoing Painlevé–Gullstrand metric gives $$\begin{aligned} c\,dt_{\mathrm{PG}}&=c\,dt+\frac{\sqrt{r_s/r}}{f(r)}\,dr,\\ ds^2&=-f(r)c^2dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2. \end{aligned}\tag{32}$$(32)

Substitution in Eq. (17) cancels the mixed time–radial term [17, Sec. II]. A clock held at fixed radius has $d\tau=\sqrt f\,dt$. The proper radial distance on a static spatial slice is $d\ell=dr/\sqrt f$. Both factors follow from the same metric.

Choose a supported test body of rest mass $m$ with negligible backreaction, and a stationary reference observer at finite radius $R_0$. For $f_0=f(R_0)>0$ and $r_s<r\leq R_0$, comparison over the same stationary-coordinate interval gives $$\begin{gathered} h_{\mathrm{clk}}(r;R_0)=\frac{d\tau(r)}{d\tau_0}=\sqrt{\frac{1-r_s/r}{1-r_s/R_0}},\\ E_0(r)=mc^2h_{\mathrm{clk}}(r;R_0),\qquad W_0(r\longrightarrow R_0)=mc^2[1-h_{\mathrm{clk}}(r;R_0)]. \end{gathered}\tag{33}$$(33)

The energy $E_0$ uses the stationary time-translation generator normalized to the reference observer’s proper time $\tau_0$. This normalization assigns energies at different radii to one comparison. A local observer beside the supported body measures its rest energy $mc^2$; the reference-normalized energy is smaller when $r<R_0$. Reversible lifting supplies the difference $W_0$ at that reference. Reversible lowering releases the same difference. For a freely moving test body, the energy associated with the same stationary generator is conserved.

The radial energy gradient also determines the force needed to hold the body in place. Since $dE_0=h_{\mathrm{clk}}\,\delta W_{\mathrm{local}}$ for a quasistatic lifting increment, the local outward support force is $$F_{\mathrm{support}}^{\mathrm{local}}=\frac{1}{h_{\mathrm{clk}}}\frac{dE_0}{d\ell}=ma_{\mathrm{support}}=\frac{GMm}{r^2\sqrt{f(r)}}.\tag{34}$$(34)

The reference normalization cancels from this local measurement. Moving the supported body outward raises $E_0$; allowing it to move inward lowers the static energy associated with its position. In the weak-field limit the supporting force approaches $GMm/r^2$, and the inward gravitational pull has the Newtonian form $-GMm\hat{\mathbf r}/r^2$ [29]. The corresponding lifting work approaches $GMm(1/r-1/R_0)$.

These relations tie attraction, clock comparison and reversible work to a single exterior geometry. As $r$ approaches $r_s$ from outside, the supported clock ratio tends to zero and the required support force diverges. Static timelike observers occupy $r>r_s$. In the regular Schwarzschild geometry, a freely falling clock can cross the horizon on a timelike trajectory while accumulating proper time. PIN’s saturated-state retention rule concerns the substrate; the exterior calculation describes the limiting behaviour of supported clocks.

At its specified reference, $W_0$ is test-body lifting work. Its ratio to the test body’s rest energy is $1-h_{\mathrm{clk}}$. The factor $1-\gamma$ expresses the fractional difference between the exterior and inventory energy accounts relative to the inventory. PIN’s internal coordination energy is a portion of the self-gravitating system’s own energy whose value depends on its physical state and dynamics. These distinct energy accounts let the exact exterior relations constrain PIN’s physical explanation.

Across the black-hole family, fixed saturated density, growing area capacity and collective gravitational mass describe the same physical system. Its exterior geometry also connects clock rates, gravitational energy and support force. PIN’s emergence joins this spatial organization to retained information and committed change.

The next question is how PIN interprets the clock-rate difference in terms of physical processing. Section 5 develops this connection by distinguishing the information retained, the processing already in use and the capacity available for further changes. Its starting point is the frame account introduced above: the stacking and processing of committed frames underlie experienced time, and less available processing headroom means slower local progression in PIN’s account.

5 Time, causality and processing limits

In PIN, each collapse establishes an actual frame of reality, and subsequent processing calculates possible successors from that committed state. The stacking of committed frames establishes the actual past; their processing underlies the passage of time. The physical question is how this succession relates to causal propagation, clock readings and the finite capacity for further change. Section 6 develops the commitment account. Here the exterior geometry established in Sections 2, 4 supplies the clock comparison, and PIN connects that comparison to processing headroom.

Light establishes the local causal information-speed limit. PIN’s universe-clock concept connects this limit to a Planck-length update scale: $t_{\mathrm P}=\ell_{\mathrm P}/c$ and $c/\ell_{\mathrm P}=1/t_{\mathrm P}$ give the corresponding causal refresh scale in inverse seconds. This is PIN’s maximum-refresh interpretation of the causal speed. The frequency of universal commitments additionally depends on the physical timing prescription and which relations each event resolves.

The PWP relation $E_{\mathrm P}t_{\mathrm P}=\hbar$ provides an angular phase frequency $1/t_{\mathrm P}$ and an ordinary optical-cycle frequency $1/(2\pi t_{\mathrm P})$. A physical clock converts changing phase relationships, positions, or correlations into distinguishable readings. Those readings belong to a timelike system; the proper interval on a photon’s null path is zero.

The proper time accumulated by an exterior clock is, in signature $(-,+,+,+)$ [12, §4], $$c^2d\tau^2=-g_{\mu\nu}dx^\mu dx^\nu.\tag{35}$$(35)

Holding Schwarzschild observers at their respective radii and comparing equal stationary-coordinate intervals gives Eq. (33).

To express this rate comparison in PIN’s processing terms, let $u_p$ denote utilization and $\eta_p=1-u_p$ the remaining headroom, with $0\leq u_p\leq1$. At fixed processing capacity, greater information-processing workload leaves less reserve for further distinguishable changes.

Processing bandwidth denotes a rate capacity for distinguishable updates, expressed in the units used for the processing budget in Eq. (42). Near saturation, limited bandwidth appears as diminishing processing headroom: an increasing share of the finite processing capacity is already occupied. PIN expresses the associated dilation through the utilization assignment and relative clock rate below.

Equation (33) has already fixed the clock comparison in the chosen exterior. PIN connects this result to processing by assigning compactness to utilization: $$u_p(r)=\chi(r)=\frac{r_s}{r},\qquad \eta_p(r)=1-\frac{r_s}{r}.$$

Substitution into the exterior clock comparison gives $$h_{\mathrm{clk}}(r;R_0)=\sqrt{\frac{\eta_p(r)}{\eta_p(R_0)}}=\sqrt{\frac{1-r_s/r}{1-r_s/R_0}},\qquad r_s<r\leq R_0.\tag{36}$$(36)

The assignment connects processing headroom to geometry; the substitution expresses its consequence for these static clocks.

The rotating comparison uses the lapse and local-velocity factorization in Eq. (K23), together with the common-Killing-generator normalization in Eq. (K28). In the static zero-spin limit, $a_{\mathrm K}=0$ and $\Omega_0=0$ give $\alpha^2=1-r_s/r$ and $v_{\mathrm{loc}}=0$, recovering Eq. (36).

Stored capacity has a different radial ratio. Dividing the central black-hole entropy count by the disk capacity at $r$ gives $\sigma_A(r)=(r_s/r)^2=\chi(r)^2$, whereas processing utilization is assigned as $\chi(r)$. Both equal one at $r=r_s$; their different radial powers distinguish the storage-capacity ratio from the processing assignment.

Collapse frequency introduces a third quantity: the count of specified events per unit time. For the same events sampled in a region, let $\Gamma_{\mathrm{loc}}$ be their intensity per unit local proper time and $\Gamma_0$ the intensity referred to $\tau_0$. The event count is the same in either clock, so $$\Gamma_0\,d\tau_0=\Gamma_{\mathrm{loc}}\,d\tau,\qquad \Gamma_0=h_{\mathrm{clk}}\Gamma_{\mathrm{loc}}.\tag{37}$$(37) This converts the count of the same events between clocks; the commitment dynamics must determine the local rate and which relations each frame resolves.

As $r\to r_s^+$, this static observer family has vanishing clock ratio and assigned headroom, and divergent support force. Freely falling clocks continue to accumulate proper time.

6 The measurement problem and committed reality

The frame account of time leads to a physical question: how does a calculation of possible futures establish the next actual event? PIN answers through the computational economy of the wave function and the resource advantage of commitment. The mathematical account below distinguishes the representation of alternatives, the resources they require, the comparison of continuation costs, and the operation that records an outcome.

6.1 The wave function and computational economy

In PIN, the wave function calculates possible futures through amplitudes, phases and correlations that carry the distinctions needed for physical predictions. Its primary computational economy is to carry this predictive representation in place of an exhaustive, continuously definite classical inventory of every particle’s position, spin and other properties at every instant. The representation cost of a particular quantum state remains state-dependent.

The finite-information concern is the cost of exhaustive, independently specified microscopic values. A classical description that must distinguish every arbitrary position assignment for $n$ labelled particles, with $q$ bins along each of three coordinates, has $q^{3n}$ possible assignments and requires capacity of at least $\ln(q^{3n})=3n\ln q$ nats. For fixed $n>0$, this requirement is unbounded as $q\to\infty$. Exact arbitrary continuum specification therefore exceeds any finite capacity. Smooth macroscopic evolution can still have a concise mathematical description; the resource demand comes from separately specifying every independent microscopic value. PIN’s wave-function account preserves the distinctions needed for physical predictions within the substrate’s finite resource budget.

Collapse addresses the separate question of when a definite event is committed. It occurs when resolving alternatives and continuing from the result becomes more efficient than carrying those alternatives forward. Each collapse of the universal wave function establishes a frame of reality. The next calculation begins from that committed frame, with the actual past already fixed.

Let $F_n$ denote the $n$th committed frame and $|\Psi_n\rangle$ the computational state for its possible successors. At the declared physical resolution $\epsilon_{\mathrm{res}}$ used in the resource accounting below, choose a finite orthonormal set of alternatives relevant to the interaction and write $$|\Psi_n\rangle=\sum_{k=1}^{d}c_k|k\rangle,\qquad\sum_{k=1}^{d}|c_k|^2=1.\tag{38}$$(38)

The coefficients retain amplitudes, relative phases and correlations. The wave function is purely computational in PIN’s account: it represents the unresolved calculation carried by the physical substrate. $F_n$ names the actual commitment from which that calculation proceeds. The ordinary composition of quantum amplitudes illustrates how predictions are carried forward by combining path contributions, as in Feynman’s propagator formulation [30, Secs. 5–7]. For unitary steps on the relevant joint state space, $$\psi_{r+1}(x)=\sum_y U_r(x,y)\psi_r(y),\qquad U_r^\dagger U_r=I.\tag{39}$$(39)

Expanding this expression produces products over intermediate paths. Evaluating it step by step combines contributions that lead to the same joint state. At each step, the summed amplitude for each joint state carries the path contributions needed for the next update. If interactions leave distinguishable records elsewhere, those records belong in the joint state being updated. The economy must preserve the distinctions needed for subsequent predictions.

Continuing the computation must preserve amplitudes, phases and correlations wherever they can affect a later event.

6.2 Efficiency and commitment

The resource budget asks whether the required storage and processing fit what is available; Eq. (42) states those limits. Within the allowed possibilities, the cost comparison asks whether resolving alternatives and continuing is less costly than retaining them. Equation (43) identifies the distinct case in which retaining them is infeasible. The outcome rule then describes what is committed.

PIN places the transition to commitment at the resource crossover $$C_{\mathrm{keep}}>C_{\mathrm{commit}}.\tag{40}$$(40)

Both costs refer to the same present state, physical task, precision and future interval, and use the same specified cost unit. The keep cost includes maintaining the live amplitudes and their correlations. The commitment cost includes establishing the event, trimming unrealized continuations, retaining the actual history, and carrying out the next calculation. Decision and transition overheads belong in the comparison once each.

For outcome-dependent costs, a quantitative realization also specifies how those costs enter the comparison. The existence of one cheaper supported outcome and a Born-weighted mean saving are distinct criteria. The physical cost law must specify which comparison determines commitment.

Finite computational resources also motivate collapse in Srikanth’s model, where an information transition restores a manageable state [31]. In PIN, capacity sets the admissible possibilities; the cost crossover can favor commitment before capacity is exhausted, removing unrealized continuations from the next calculation.

The cost comparison specifies the proposed criterion for commitment; the probabilities and state change require an outcome rule. For a projective representation of an individual commitment, take a normalized pre-event state $|\Psi^-\rangle$ and an exhaustive family of mutually orthogonal record projectors. For the selected outcome $j$ with $p_j>0$, $$\begin{gathered} P_j^\dagger=P_j,\qquad P_jP_k=\delta_{jk}P_j,\qquad\sum_jP_j=I,\qquad p_j=\langle\Psi^-|P_j|\Psi^-\rangle,\\ |\Psi_j^+\rangle=\frac{P_j|\Psi^-\rangle}{\sqrt{p_j}},\qquad F_{n+1}=\operatorname{Commit}(F_n,j). \end{gathered}\tag{41}$$(41) The first line adopts the Born probabilities and the second the normalized projective update associated with Lüders [32]. Here $P_j$ selects a quantum subspace; this use of projection is distinct from the saturated disk’s outward spatial correspondence. PIN’s commitment records the actual event. The cost criterion concerns its timing; unresolved degrees of freedom supply the conditional calculation for what follows. PIN’s physical principle is efficiency-driven commitment and emergent reality.

For a macroscopic object, Eq. (41) assigns an outcome probability using the joint state, including the correlations that constrain its possible continuations. Products of constituent probabilities apply when the state and the selected alternatives factorize. Each selected outcome joins the retained event record; subsequent commitments must fit that same established history.

The physical commitment operation includes the interacting system, accessible substrate, physical records and their correlations. Under the ordinary trace rule and no-signalling assumptions, its outcome-averaged map $\Phi$ over a fixed clock interval must be affine, trace preserving and completely positive, including when an untouched reference is present [33]. The average includes activation (the triggering of a commitment), outcome selection and continuation without commitment. Individual normalized conditional states can depend nonlinearly on their pre-event amplitudes.

A quantum instrument resolves this operation into completely positive record-conditioned maps whose sum is $\Phi$ [34, Secs. 4, 7–8]. Activation can carry information about the alternatives through correlations with load and control records. The Born probabilities in Eq. (41) apply to the normalized state at resolution, including the physical conditioning introduced by activation. The operation must specify activation and outcome selection together.

Substrate, control and record energies enter the same account, including the record’s initialized state and its written outcome. For a joint input $\rho$ and observable $O_{\mathrm{obs}}$, the trace-dual map is defined by $\operatorname{Tr}[O_{\mathrm{obs}}\Phi(\rho)]=\operatorname{Tr}[\Phi^\dagger(O_{\mathrm{obs}})\rho]$. For a closed joint system with fixed total Hamiltonian $H_{\mathrm{tot}}$, conservation of mean energy for every joint input is expressed by $$\Phi^\dagger(H_{\mathrm{tot}})=H_{\mathrm{tot}}.$$

With external driving, the supplied work enters the balance.

The complete black-hole retention-and-release operation, including its commitments, must satisfy the complete-return condition in Eq. (45). A projective reduction generally removes coherence between the alternatives it distinguishes. Returning the full quantum information requires relative phases and correlations together with the outcome probabilities. The admissible information sector and the complete physical operation must satisfy the commitment description and the correlated-return condition together. Equation (41) describes the selected commitment, while Eq. (45) specifies complete information return.

The index $n$ orders universal commitments. Observable duration is read from physical clocks, as in the finite-reference comparison above. Relating the event order to those clocks requires a physical timing and causal-order prescription. Penrose’s discussion of quantum observation and time-asymmetry identifies this ordering question as part of the interface between quantum states and spacetime [35].

The physical representation and its commitment operation must both fit the substrate’s resources. At resolution $\epsilon_{\mathrm{res}}$, let $K$ denote storage demand in nats under a specified encoding and $B_{\mathrm{proc}}$ the available processing rate in specified updates per second. Then $$K_{\mathrm{history}}+K_{\mathrm{live}}+K_{\mathrm{control}}\leq K_{\max},\qquad J_{\mathrm{demand}}\leq B_{\mathrm{proc}}.\tag{42}$$(42)

The storage terms account for disjoint uses of the same capacity; $J_{\mathrm{demand}}$ is a rate. The demanded and available rates use the same clock and update convention. Lloyd’s physical-computation analysis connects available memory to accessible degrees of freedom and processing speed to available energy [36]. In PIN, these constraints apply to the substrate implementing the calculation.

These bounds distinguish feasibility from the cost advantage in Eq. (40). If keeping and committing are the two physically allowed continuations, all other constraints are met, and their peak storage demands satisfy $$K_{\mathrm{keep}}>K_{\max}\geq K_{\mathrm{commit}},\tag{43}$$(43)

commitment is the feasible continuation. Peak demand includes transition workspace, and every supported outcome must fit. This capacity condition sets feasibility alongside the cost criterion for an economical event.

6.3 The smooth committed past

A committed frame gives the present an anchor: looking backward follows one continuous, consistent history; looking forward evaluates possibilities carried by amplitudes, phases and correlations. Commitment makes an event actual and joins it to the past against which later events must fit.

Smoothness applies to every commitment, including events far from black-hole saturation. The macroscopic dynamics supplies the continuous trajectory consistent with the committed events. The finite-information count in Section 6.1 concerns the capacity required to distinguish arbitrary independent microscopic assignments; the history account concerns the actual sequence and its retained records.

Let $h_n=(e_1,\ldots,e_n)$ represent the ordered committed events, with sufficient recorded content for the intended retrospective questions at resolution $\epsilon_{\mathrm{res}}$. History extends by $$h_{n+1}=h_n\mathbin{\Vert}e_{n+1},\qquad \operatorname{prefix}_n(h_{n+1})=h_n.\tag{43a}$$(43a)

Here $\Vert$ denotes ordered append. Each new frame preserves the same earlier events and their order. A losslessly compressed representation is sufficient. Its actual storage, continuing support and extension workspace enter Eq. (42). Retained records and the dynamics together supply the backward account at the stated resolution.

7 The information paradox and Hawking radiation

In PIN, Hawking radiation is desaturation. The PWP retained in the saturated black-hole disk is the same PWP released, in the exact informational form in which it accreted. Accretion retains the incoming information; at saturation, all available local processing maintains the retained state, leaving no capacity for further local frame progression; desaturation restores its participation in the calculation of what comes next. The physical carrier and its retained information persist through this sequence.

Desaturation ends this saturated retention interval. At the one-dimensional rim, the saturated disk meets available substrate capacity, and the retained PWP becomes available to projected spacetime again. As the disk shrinks, release of an accessible layer exposes the next. The returned information belongs to the actual history; the renewed wave function calculates its possible future.

The physical claim concerns the origin and informational content of the emitted radiation. Hawking’s semiclassical calculation treats quantum fields on a spacetime containing gravitational collapse. The field modes used to define the initial vacuum and the outgoing particles have different positive-frequency decompositions, giving a nonzero outgoing particle population [8, Sec. 2]. The resulting thermal occupation factors are modified, on propagation to infinity, by the transmission of the exterior geometry. These frequency- and angular-momentum-dependent transmission factors are usually called greybody factors. The familiar near-horizon pair picture is a heuristic illustration of the field calculation [8, p. 202]. The comparison with PIN concerns that calculation and its physical interpretation.

PIN’s desaturation account uses the Schwarzschild temperature. Equation (27) recovers $T_H$ from the stationary energy curve $E_{\mathrm{BH}}(N)$ and the capacity entropy $S_H=k_{\mathrm B}N$; Eq. (29) gives the exterior energy released between two specified capacity states. These thermodynamic relations constrain desaturation. PIN identifies the outgoing content with the PWPs retained during accretion, carrying the informational distinctions that entered with them, including their relative phases and correlations. The physical interaction that makes this retained content available as outgoing field states must satisfy both informational return and exterior energy balance.

The distinction between local and distant measurements is essential to interpreting the radiation. The temperature $T_H$ in Eq. (27) uses the standard normalization of the exterior time coordinate to a stationary clock at infinity. A detector held at a fixed radius $r$ outside a neutral, nonrotating hole uses a different proper time. Write $r_s=2GM/c^2=R$ and $f(r)=1-r_s/r$, with $r>r_s$. In the stationary exterior, the clock comparison already used in Eq. (33) and the corresponding gravitational frequency shift give $$\begin{gathered} d\tau_r=\sqrt{f(r)}\,dt_\infty,\quad E_{\mathrm{loc}}(r)=\frac{E_\infty}{\sqrt{f(r)}},\\ T_{\mathrm{loc}}(r)=\frac{T_H}{\sqrt{f(r)}}. \end{gathered}$$

Here $E_\infty$ and $E_{\mathrm{loc}}$ refer to the same propagating quantum, measured in the distant and local static frames. The last expression is the redshifted Hawking temperature scale. Thus a low distant temperature can correspond to a much higher local spectral energy scale for a detector held close to the horizon. This gravitational blueshift is part of the standard exterior description and supplies the comparison that PIN must respect.

The observer and the field state also determine how that scale appears in an actual detector. For a static detector coupled to a massless scalar field, the Hartle–Hawking equilibrium state gives a thermal response at $T_{\mathrm{loc}}$. In the Unruh state used to describe outgoing radiation without an incoming thermal bath, the response becomes approximately thermal at this temperature in the limit of a large detector energy gap [55, Secs. 5.1, 5.3 and 8.1]. The absorbed power depends on the field correlations, detector coupling and energy flux. The net energy carried away to infinity determines the black hole’s exterior mass loss.

For an ideal static test detector near a hole with $GM=10^8(GM)_\odot^N$, using the nominal solar mass parameter of Section 2 [24], take the areal-radius offset $r-r_s=100\ell_P$. The clock comparison gives approximately $4.28\times10^{14}$ distant years per local second, and the local Hawking temperature scale is approximately $8.34\times10^6$ K. At a distant radius of 25,000 light-years the relative correction to the asymptotic clock comparison is less than $10^{-9}$. The areal-radius offset corresponds to a proper radial distance on a static slice of approximately $2\sqrt{r_s(r-r_s)}=4.37\times10^{-11}$ m. These static test-observer comparisons describe the strong local blueshift in a quasi-stationary exterior, with the support apparatus and its backreaction neglected. A freely falling detector requires the clock and response calculation for its own trajectory.

The informational question concerns the complete outgoing state. Hawking’s later information-loss analysis asks whether that state preserves the initial quantum distinctions after evaporation [37]. An emitted part with a thermal description can remain correlated with the rest of the system and with other emitted parts. PIN’s proposal returns the retained carriers themselves with those distinctions intact.

The central requirement is complete return of the accreted information, including its relative phases and correlations with other systems. Equation (45) states this requirement for the full accretion–retention–release process. The calculation below treats retention of one specified state during a closed saturated interval; the following tests examine radiation entropy, the interior-partner premise, and the exterior energy budget.

7.1 Retention during saturation

Retention preserves the informational state through continuing processing, while further local frame progression is arrested.

The spatial restriction is complete occupation of the permitted positional addresses. In a finite region with occupancy $n_i\in\{0,1\}$ at each positional address, let $H_{\mathrm{hop}}$ contain only internal moves that require a vacant destination. At complete occupation every such move is blocked: $H_{\mathrm{hop}}\Pi_{\mathrm{sat}}=0$, where $\Pi_{\mathrm{sat}}$ projects onto the fully occupied positional sector. A vacancy at an allowed destination reopens that positional move. This calculation represents PIN’s hard capacity rule for vacancy-mediated positional moves. Internal and collective transitions require the separate retained-state closure condition below.

For a specified normalized retained state $|*\rangle$ and its projector $P_*=|*\rangle\langle*|$, take the effective generator $H_s$, with units of energy, to be self-adjoint and time independent during a closed saturated interval. Impose the retained-state closure condition below. With $t_{\mathrm{ret}}$ the evolution parameter of this retained-sector description and $U_s(t_{\mathrm{ret}})=\exp(-iH_st_{\mathrm{ret}}/\hbar)$, $$\begin{gathered} (I-P_*)H_sP_*=0\quad\Longrightarrow\quad H_s|*\rangle=E_*|*\rangle,\\ U_s(t_{\mathrm{ret}})|*\rangle=e^{-iE_*t_{\mathrm{ret}}/\hbar}|*\rangle,\\ U_s(t_{\mathrm{ret}})P_*U_s^\dagger(t_{\mathrm{ret}})=P_*. \end{gathered}\tag{44}$$(44)

Here $E_*$ is the real eigenvalue of the specified state, and the timing prescription relates the time parameter $t_{\mathrm{ret}}$ to a physical clock. Closure of this state under $H_s$ makes it acquire a global phase while preserving its informational content. Full occupancy exhausts the permitted positional addresses; the generator governs preservation of the stored distinctions.

Equation (44) gives the stationary ray of one specified state. For a finite-dimensional space of admissible retained records, let $P_{\mathrm{rec}}$ be its orthogonal projector. With the same self-adjoint, time-independent generator $H_s$, preservation of every supported density matrix throughout a nonzero interval beginning at $t_{\mathrm{ret}}=0$ is equivalent to one common energy across that space: $$\begin{gathered} U_s(t_{\mathrm{ret}})\rho U_s(t_{\mathrm{ret}})^\dagger=\rho\\ \text{for every }\rho=P_{\mathrm{rec}}\rho P_{\mathrm{rec}}\text{ and every }t_{\mathrm{ret}}\in[0,T_{\mathrm{ret}}],\qquad T_{\mathrm{ret}}>0,\\ \Longleftrightarrow\quad H_sP_{\mathrm{rec}}=E_sP_{\mathrm{rec}},\qquad U_s(t_{\mathrm{ret}})P_{\mathrm{rec}}=e^{-iE_st_{\mathrm{ret}}/\hbar}P_{\mathrm{rec}}. \end{gathered}\tag{44a}$$(44a)

The common phase preserves relative phases within the record space. Conversely, differentiating the first line at $t_{\mathrm{ret}}=0$ shows that $H_s$ commutes with every supported density matrix. Applying this to basis-state projectors makes their vectors eigenvectors; applying it to pairwise superpositions makes all their eigenvalues equal. The common phase also cancels in any joint state with an untouched reference, preserving its correlations.

This criterion describes unchanged retention throughout the selected interval. A complete process can also return an input by reversing a known unitary evolution before release. Equation (45) tests the complete accretion–retention–release operation. The physical retention dynamics must also specify the processing workload that maintains the state.

7.2 The complete return condition

Let $r_{\mathrm{rec}}$ denote the complete informational record present at accretion. PIN’s preservation rule is $\operatorname{Release}(\operatorname{Retain}(r_{\mathrm{rec}}))=r_{\mathrm{rec}}$. For the quantum-information question, completeness includes relative phases and correlations among the accreted PWPs and with other systems. Let $A$ denote the admissible incoming informational degrees of freedom, $X$ an external reference left untouched by the black-hole process, and $\mathcal T$ the complete accretion–retention–release operation. After identifying the returned degrees of freedom with their incoming counterparts, complete return means $$(\mathcal T\otimes\operatorname{id}_X)(\rho_{AX})=\rho_{AX}\qquad\text{for every admissible joint input }\rho_{AX}.\tag{45}$$(45)

This states the informational requirement on the physical operation. Any specified reversible change of transport description is accounted for before comparing the states. Including an untouched reference tests recovery of correlations as well as the incoming system’s own state, as in Hayden and Preskill’s recovery criterion [38, Sec. 3].

The information to be preserved is the information present at accretion, including its phases and correlations. The complete operation $\mathcal T$, including every commitment, must satisfy Eq. (45); the committed past supplies the record from which the future is computed.

For projective commitment in Eq. (41), the admissible input sector must have its information preserved by the complete operation. An outcome-selected projection generally removes coherence between the alternatives it distinguishes. The admitted state sector and complete physical operation must therefore satisfy Eqs. (41) and (45) together, preserving the required coherence as well as outcome probabilities and classical records. A microscopic release model must implement this complete-return action.

The capacity entropy $S_H=k_{\mathrm B}N$ counts saturated storage. A committed event has a definite recorded outcome, and that record continues to occupy physical capacity.

7.3 Radiation entropy and correlations

The first entropy test explains how locally mixed radiation can be compatible with information in the complete state. In an ordinary bipartite quantum description, let $B$ be the retained system and $O$ the emitted radiation. A normalized pure joint state has a Schmidt representation $$\begin{aligned} |\Psi_{BO}\rangle&=\sum_\alpha\sqrt{p_\alpha}|\alpha\rangle_B|\alpha\rangle_O,\qquad p_\alpha\geq0,\qquad\sum_\alpha p_\alpha=1,\\ \rho_O&=\operatorname{Tr}_B|\Psi_{BO}\rangle\langle\Psi_{BO}|=\sum_\alpha p_\alpha|\alpha\rangle_O\langle\alpha|,\\ s_O&=-\operatorname{Tr}(\rho_O\ln\rho_O)=-\sum_\alpha p_\alpha\ln p_\alpha. \end{aligned}\tag{46}$$(46)

The Schmidt states are orthonormal and $0\ln0$ is taken as zero. The radiation can be mixed while the joint state is pure. The information-loss problem arises when the initially pure joint system finishes evaporation as mixed radiation after the retained system disappears. PIN assigns the transfer of retained information to desaturation of the PWPs themselves.

Page’s analysis distinguishes the fine-grained entropy of the joint radiation from thermal descriptions of its parts [39]. Fine-grained gravitational entropy calculations likewise recover curves compatible with unitary evaporation in the settings studied [40]. Thermal-looking emission can coexist with information in correlations among the emitted parts.

Equation (46) measures entropy in nats; $N$ counts the remaining saturated capacity. For a closed, pure $BO$ state, $s_O=s_B$. If the retained state space has dimension $d_B$ with $\ln d_B\leq N$, then $s_O\leq N$. As the remaining information-bearing system empties, complete return places its correlations in the outgoing state. When the input is entangled with an external reference, that reference must be included in this purity comparison. A specific Page curve additionally requires an evaporation dynamics; Page’s typicality assumptions are a particular statistical construction.

Small corrections to approximately information-free Hawking pair creation leave the entanglement problem in place, as shown by Mathur [41, Secs. 3, 5–6]. PIN’s same-PWP return assigns the accreted information to the emitted content itself. Under Eq. (45), evaporation preserves the incoming informational distinctions as the saturated disk’s content returns.

7.4 The interior-partner premise

The interior-partner test concerns the assignment of physical degrees of freedom. In the usual comparison, a late outgoing mode has a quantum partner in an independent interior subsystem, with the pair close to the entanglement structure of the smooth-horizon vacuum. In PIN, the physical saturated disk supplies the retained degrees of freedom, and the third spatial dimension emerges outward. This disk is the retained subsystem in PIN’s entanglement account; the complete retention-and-release operation must satisfy Eq. (45).

AMPS expresses the tension between late-time purification and an independent smooth-horizon interior partner [42, Sec. 2]. Use the subsystem labels early for early radiation, late for a late outgoing mode, and int for its independent interior partner. Strong subadditivity, with dimensionless entropies, gives $$\begin{aligned} s_{\mathrm{early,late}}+s_{\mathrm{late,int}}&\geq s_{\mathrm{late}}+s_{\mathrm{early,late,int}},\\ s_{\mathrm{late,int}}=0&\quad\Longrightarrow\quad s_{\mathrm{early,late,int}}=s_{\mathrm{early}},\\ &\quad\Longrightarrow\quad s_{\mathrm{early,late}}\geq s_{\mathrm{early}}+s_{\mathrm{late}}. \end{aligned}\tag{47}$$(47)

For $s_{\mathrm{late}}>0$, this contradicts the purification condition $s_{\mathrm{early,late}}<s_{\mathrm{early}}$ for the chosen late mode. The pure-pair condition $s_{\mathrm{late,int}}=0$ is the idealized smooth-vacuum premise in this comparison. PIN assigns the retained subsystem to its physical flat, two-dimensional saturated disk, bounded by its one-dimensional rim, with the third spatial dimension emerging outward. Its retention and release description must track the disk degrees of freedom and their correlations with radiation; Eq. (45) specifies the required information-bearing output.

7.5 Exterior energy balance and finite-density saturation

For quasi-stationary neutral, nonrotating endpoints $N_i>N_f>0$, Eq. (29) gives the positive decrease $E_{\mathrm{BH}}(N_i)-E_{\mathrm{BH}}(N_f)$, with radiation, recoil and any other exchanges included in the balance. Its small-change limit per lost capacity nat is $dE_{\mathrm{BH}}/dN=k_BT_H$. The returned PWP’s informational identity and the exterior energy of its release have separate roles: the substrate inventory assigns $E_P$ per occupied pair, while the collective mass curve fixes the exterior energy change. Equations (45) and (29) jointly constrain the same physical accretion–retention–release operation: it must return the admitted informational distinctions and correlations while respecting the exterior energy balance. A state-resolved realization would determine the outgoing correlations and radiation spectrum.

For a one-cell reduction between saturated endpoints, $N_i=N$ and $N_f=N-1$ with $N>1$, the same finite energy relation reads $$\Delta E_{\mathrm{BH}}^{(1)}=\frac{E_P}{2\sqrt{\pi}\bigl(\sqrt N+\sqrt{N-1}\bigr)}.$$

This is the exterior energy difference associated with one lost PWP–area capacity pair on the stationary branch. At large $N$ it approaches $k_BT_H(N)$, the marginal energy per nat. The expression compares two saturated endpoints; a temporary vacancy at fixed disk area is an intermediate configuration. An available positional destination permits the move described in the retention calculation. The emission interaction determines its conversion into outgoing modes, including the waiting time, photon multiplicity and spectral distribution. The endpoint energy difference supplies the release-energy budget. A time-dependent continuation must connect the intermediate configurations to the stationary endpoints while conserving the complete energy and returning the specified information.

The same capacity rule also governs how the saturated state accommodates further information. In PIN’s account of the singularity problem, that state occupies an extended disk at capacity density $1/\ell_P^2$ and assigned inventory-energy density $E_P/\ell_P^2$. Additional retained PWPs enlarge the occupied area at fixed density, following Eqs. (5) and (23). Rovelli and Vidotto’s Planck-star proposal also invokes a finite density [43]. Their criterion is volumetric, while PIN’s inventory-energy density is areal.

Additional retained PWPs occupy additional area at finite areal capacity. A continuum continuation across saturation must connect incoming matter, retained records and subsequent release. Curvature and causal geodesic extendibility test its regularity wherever a continuum spacetime description applies [44].

8 The vacuum catastrophe and cosmological curvature

The black-hole calculation relates area-counted capacity to exterior gravitational energy. Its cosmological extension asks how the vacuum source and horizon capacity relate to curvature. The argument has three parts. First, a regulated scalar-field calculation determines the energy and pressure of a specified set of modes. Second, the independently adopted trace-free response in Eqs. (10)–(16) removes a constant metric-proportional vacuum baseline from the local equation. With conserved complete stress, reconstruction of the field equation leaves an integration constant. Third, for a supplied positive de Sitter state, its horizon geometry, area capacity and collective energy give the inverse-capacity relations in Eq. (61).

The mode sum makes the vacuum-energy scale and the stress-tensor issue explicit. The capacity relation expresses the curvature of the supplied cosmological state, whose capacity or cosmological constant enters as state data. The local Schwarzschild–de Sitter calculation connects the result back to the black-hole construction. In Eq. (67), the local mass-parameter energy and the cosmological contribution enter the same spherical energy account.

Three energy densities therefore remain distinct throughout:

• $\varepsilon_0$ is the regulated scalar mode sum.

• $\varepsilon_{\mathrm{vac}}$ is the vacuum energy density in the metric-proportional stress tensor.

• $\varepsilon_\Lambda=\Lambda_{\mathrm{eff}}/\kappa$ is the energy density associated with the effective cosmological constant.

Energy density $\varepsilon$ and pressure $p$ have units $\mathrm{J\,m^{-3}}$; energy density divided by $c^2$ is mass density. The scalar field has mass parameter $m$, the black hole has mass $M$, and the PWP has zero rest mass.

8.1 The quantum-field vacuum calculation

The first question is what energy a specified collection of vacuum modes contributes before deciding how that contribution enters the gravitational source. The standard free-scalar calculation makes the cutoff and stress conventions explicit [45, 13]. Consider one free, real bosonic degree of freedom of mass $m\geq0$ in a large periodic box of volume $V$. Writing $\mu=mc/\hbar$, its mode frequency, ground-state energy and continuum mode measure are $$\omega_{\mathbf{k}}=c\sqrt{k^2+\mu^2},\qquad E_{0,\mathbf{k}}=\frac{\hbar\omega_{\mathbf{k}}}{2},\qquad \frac1V\sum_{\mathbf{k}}\longrightarrow\int\frac{d^3k}{(2\pi)^3}.\tag{48}$$(48)

A sharp wave-number cutoff $k_c>0$ corresponds to momentum $\hbar k_c$. Angular integration gives $$\begin{aligned} \varepsilon_0(k_c,m)&=\frac{\hbar}{2}\int_{|\mathbf{k}|\leq k_c}\frac{d^3k}{(2\pi)^3}\,\omega_{\mathbf{k}}\\ &=\frac{\hbar c}{2}\frac{4\pi}{(2\pi)^3}\int_0^{k_c}k^2\sqrt{k^2+\mu^2}\,dk=\frac{\hbar c}{4\pi^2}\int_0^{k_c}k^2\sqrt{k^2+\mu^2}\,dk. \end{aligned}\tag{49}$$(49)

For $m>0$, this integral is [13, Sec. IV.A, Eq. (74)] $$\varepsilon_0(k_c,m)=\frac{\hbar c}{32\pi^2}\left[k_c\sqrt{k_c^2+\mu^2}(2k_c^2+\mu^2)-\mu^4\ln\left(\frac{k_c+\sqrt{k_c^2+\mu^2}}{\mu}\right)\right].\tag{50}$$(50)

The logarithm is natural and its argument is dimensionless. For $m=0$, $\varepsilon_0=\hbar ck_c^4/(16\pi^2)$, also the continuous massless limit of Eq. (50). There is one such contribution per real bosonic degree of freedom; species and polarization counts require an explicit field inventory. This oscillator ground-state sum differs from PIN’s saturated PWP occupancy in Eq. (5); identifying their inventories requires a physical mapping. At the illustrative cutoff $k_c=\ell_{\mathrm P}^{-1}$, $$\varepsilon_{\mathrm P}\equiv\frac{E_{\mathrm P}}{\ell_{\mathrm P}^3}=\frac{\hbar c}{\ell_{\mathrm P}^4}=\frac{c^7}{\hbar G^2},\qquad\varepsilon_0(\ell_{\mathrm P}^{-1},0)=\frac{\varepsilon_{\mathrm P}}{16\pi^2}.\tag{51}$$(51)

8.2 Pressure, Lorentz symmetry and the vacuum source

The pressure distinguishes the regulated mode sum from a Lorentz-invariant vacuum source. The canonical free-scalar stress gives [45, Eqs. (4), (9)] $$\begin{aligned} p_0(k_c,m)&=\frac{\hbar c}{12\pi^2}\int_0^{k_c}\frac{k^4}{\sqrt{k^2+\mu^2}}\,dk\\ &=\frac{\hbar ck_c^3\sqrt{k_c^2+\mu^2}}{12\pi^2}-\varepsilon_0(k_c,m),\\ p_0(k_c,0)&=\frac{\hbar ck_c^4}{48\pi^2}=\frac{\varepsilon_0(k_c,0)}{3}. \end{aligned}\tag{52}$$(52)

The massless result $p_0=\varepsilon_0/3$ reflects the frame selected by the wave-number cutoff. Lorentz invariance instead requires the same isotropic stress in every inertial frame, giving $p_{\mathrm{vac}}=-\varepsilon_{\mathrm{vac}}$. In the $(-,+,+,+)$ convention, $$\begin{gathered} T_{ab}^{\mathrm{vac}}=-\varepsilon_{\mathrm{vac}}g_{ab},\qquad T_{\hat a\hat b}^{\mathrm{vac}}=\operatorname{diag}(\varepsilon_{\mathrm{vac}},-\varepsilon_{\mathrm{vac}},-\varepsilon_{\mathrm{vac}},-\varepsilon_{\mathrm{vac}}),\\ p_{\mathrm{vac}}=-\varepsilon_{\mathrm{vac}}. \end{gathered}\tag{53}$$(53)

The gravitational source is the renormalized stress tensor, with its symmetries and counterterms treated consistently [45, 13]. Equation (53) uses its Lorentz-invariant vacuum component, with $p_{\mathrm{vac}}=-\varepsilon_{\mathrm{vac}}$.

With the curvature convention $G_{ab}+\Lambda g_{ab}=\kappa T_{ab}$ already used above, positive $\Lambda$ on the empty maximally symmetric branch gives de Sitter spacetime with Ricci scalar $\mathcal R=4\Lambda$. Separating vacuum and excitations in the ordinary field equation gives [13, Sec. II] $$\begin{gathered} G_{ab}+\Lambda_{\mathrm{bare}}g_{ab}=\kappa(T_{ab}^{\mathrm{excit}}-\varepsilon_{\mathrm{vac}}g_{ab}),\\ G_{ab}+\Lambda_{\mathrm{eff}}g_{ab}=\kappa T_{ab}^{\mathrm{excit}},\\ \Lambda_{\mathrm{eff}}=\Lambda_{\mathrm{bare}}+\kappa\varepsilon_{\mathrm{vac}},\qquad\varepsilon_\Lambda\equiv\frac{\Lambda_{\mathrm{eff}}}{\kappa}=\frac{\Lambda_{\mathrm{eff}}c^4}{8\pi G}. \end{gathered}\tag{54}$$(54) In the renormalized equation, $\Lambda_{\mathrm{bare}}$ and the regulated vacuum term are replaced by their consistently matched finite combination. Massive thresholds and changes of the effective matter description can change the separate vacuum contribution. The cosmological constant problem concerns the small physical combination and its stability under those changes [14].

For a positive reference $\Lambda_{\mathrm{ref}}$, the two size comparisons are $$\frac{\varepsilon_{\mathrm P}}{\varepsilon_{\Lambda,\mathrm{ref}}}=\frac{8\pi}{\Lambda_{\mathrm{ref}}\ell_{\mathrm P}^2},\qquad\frac{\varepsilon_0(\ell_{\mathrm P}^{-1},0)}{\varepsilon_{\Lambda,\mathrm{ref}}}=\frac{1}{2\pi\Lambda_{\mathrm{ref}}\ell_{\mathrm P}^2}.\tag{55}$$(55)

For the illustrative reference value $\Lambda_{\mathrm{ref}}=1.1\times10^{-52}\,\mathrm{m^{-2}}$, $\varepsilon_{\Lambda,\mathrm{ref}}\simeq5.30\times10^{-10}\,\mathrm{J\,m^{-3}}$. The ratios are approximately $8.75\times10^{122}$ and $5.54\times10^{120}$, respectively. This input scale is consistent with the Planck base-$\Lambda$CDM parameters [46, Table 2].

PIN treats the constant baseline through the adopted gravitational response. Equation (12) gives the trace-free cancellation of a metric-proportional vacuum term; Eqs. (13)–(16) give its reconstruction, constant-baseline relabeling and the conserved exchange required when the vacuum contribution varies. These results determine the response to the baseline. The following geometric calculation relates a supplied positive cosmological curvature to capacity.

8.3 Cosmological curvature and information capacity

The question is now how a supplied cosmological geometry relates its horizon capacity to its collective energy and curvature. The same spatial accounting used for the saturated black hole supplies this comparison. Consider the positive-curvature de Sitter branch, with cosmological radius $R_\Lambda$ and Hubble rate $H_\Lambda$ [15, Sec. II]: $$H_\Lambda^2=\frac{\Lambda_{\mathrm{eff}}c^2}{3},\qquad R_\Lambda=\frac{c}{H_\Lambda}=\sqrt{\frac3{\Lambda_{\mathrm{eff}}}},\qquad\Lambda_{\mathrm{eff}}>0.\tag{56}$$(56)

The positive, maximally symmetric vacuum satisfies $\mathcal R_{ab}=\Lambda_{\mathrm{eff}}g_{ab}$ and $\mathcal R=4\Lambda_{\mathrm{eff}}$. In the spatially flat de Sitter patch, the Friedmann equation is $3H_\Lambda^2=\Lambda_{\mathrm{eff}}c^2$, which gives Eq. (56). The static-patch metric can be written $$ds^2=-f_\Lambda(r)c^2dt^2+\frac{dr^2}{f_\Lambda(r)}+r^2d\Omega^2,\qquad f_\Lambda(r)=1-\frac{r^2}{R_\Lambda^2}.\tag{57}$$(57)

The horizon is the zero of $f_\Lambda$, at $r=R_\Lambda$. The spherical Misner–Sharp energy is defined geometrically by $1-2GE_{\mathrm{MS}}/(c^4r)=g^{ab}\nabla_ar\nabla_br$ [18]. Since the right-hand side is $f_\Lambda(r)$, $$E_{\mathrm{MS}}(r)=\frac{c^4r}{2G}[1-f_\Lambda(r)]=\frac{c^4r^3}{2GR_\Lambda^2},\qquad E_\Lambda\equiv E_{\mathrm{MS}}(R_\Lambda)=\frac{c^4R_\Lambda}{2G}.\tag{58}$$(58)

These are collective gravitational energies, evaluated using the specified spherical geometry.

The de Sitter horizon section has area $A_\Lambda=4\pi R_\Lambda^2$. Define $A_{\Sigma,\Lambda}$ as its quarter-area measure, so that the cosmological entropy count uses the same area normalization as the black-hole calculation: $$\begin{gathered} A_{\Sigma,\Lambda}=\pi R_\Lambda^2,\qquad A_\Lambda=4A_{\Sigma,\Lambda},\\ N_\Lambda=\frac{A_{\Sigma,\Lambda}}{\ell_{\mathrm P}^2}=\frac{3\pi}{\Lambda_{\mathrm{eff}}\ell_{\mathrm P}^2},\qquad S_\Lambda=k_{\mathrm B}N_\Lambda. \end{gathered}\tag{59}$$(59)

The entropy–curvature relation has the same normalization as the de Sitter entropy used in the finite-capacity analysis of Bousso [47]. Here $A_{\Sigma,\Lambda}$ is an equivalent area used to compare capacity counts. The physical saturated-disk assignment concerns the black hole; the cosmological calculation specifies this de Sitter state’s area, entropy and curvature.

The spherical Misner–Sharp energy at this cosmological radius is $E_\Lambda=c^4R_\Lambda/(2G)$ [48]. Divide it by the areal volume $V_\Lambda=4\pi R_\Lambda^3/3$: $$\varepsilon_\Lambda=\frac{E_\Lambda}{V_\Lambda}=\frac{c^4R_\Lambda}{2G}\frac{3}{4\pi R_\Lambda^3}=\frac{3c^4}{8\pi GR_\Lambda^2}=\frac{\Lambda_{\mathrm{eff}}c^4}{8\pi G}.\tag{60}$$(60)

This is the spherical areal-volume energy account; a generic curved spatial slice has its own proper-volume measure. The same inverse-area density normalization appears in Li’s spherical energy comparison [49, Eq. (10)]. Substitution of $R_\Lambda^2=N_\Lambda\ell_{\mathrm P}^2/\pi$ and $\ell_{\mathrm P}^2=\hbar G/c^3$ gives the inverse-capacity relation: $$\frac{\varepsilon_\Lambda}{\varepsilon_{\mathrm P}}=\frac{3c^4}{8\pi GR_\Lambda^2}\frac{\ell_{\mathrm P}^4}{\hbar c}=\frac{3\ell_{\mathrm P}^2}{8\pi R_\Lambda^2}=\frac3{8N_\Lambda},\qquad\Lambda_{\mathrm{eff}}=\frac{3\pi}{N_\Lambda\ell_{\mathrm P}^2}.\tag{61}$$(61)

The collective energy grows as $R_\Lambda$, the areal volume as $R_\Lambda^3$, and the capacity as $R_\Lambda^2$, giving the reciprocal entropy–curvature relation discussed by Bousso [47, Sec. 1.1]. The reference value above corresponds to $N_{\Lambda,\mathrm{ref}}\simeq3.28\times10^{122}$.

For the single massless degree of freedom in Eq. (51), the corresponding cutoff comparison is $\varepsilon_\Lambda/\varepsilon_0(\ell_P^{-1},0)=6\pi^2/N_\Lambda$. With $N_\Lambda$ supplied as state data, Eq. (61) fixes the corresponding curvature.

The nonrotating black-hole comparison density in Section 4.1 has the same inverse-capacity normalization. Dividing that density by $\varepsilon_{\mathrm P}$ from Eq. (51), and using Eq. (5), gives $$\frac{\bar{\varepsilon}_{\mathrm{BH}}}{\varepsilon_{\mathrm P}} =\frac{3\ell_{\mathrm P}^2}{8\pi R^2} =\frac{3}{8N}.\tag{61a}$$(61a) Here $\bar{\varepsilon}_{\mathrm{BH}}$ uses the spherical areal-radius comparison volume $4\pi R^3/3$, while Eq. (61) uses the specified de Sitter state and its capacity $N_\Lambda$. The matching normalization preserves these distinct physical assignments.

8.4 The local black-hole limit

The remaining question is how this cosmological state connects to the local black-hole scaling. The two meet in the stationary Schwarzschild–de Sitter comparison [50, Sec. 3.1]: $$\begin{gathered} ds^2=-f(r)c^2dt^2+f(r)^{-1}dr^2+r^2d\Omega^2,\qquad f(r)=1-\frac{r_s}{r}-\frac{\Lambda_{\mathrm{eff}}r^2}{3},\\ r_s=\frac{2GM}{c^2},\qquad M>0,\qquad\Lambda_{\mathrm{eff}}>0. \end{gathered}\tag{62}$$(62)

Horizons are positive roots of $f$. Its positive stationary point is $r_*=(3r_s/(2\Lambda_{\mathrm{eff}}))^{1/3}$, found from $f'(r_*)=0$. At that point $r_s/r_*=2\Lambda_{\mathrm{eff}}r_*^2/3$, so $f(r_*)=1-\Lambda_{\mathrm{eff}}r_*^2$. The condition for two distinct positive roots is consequently $$f(r_*)>0\quad\Longleftrightarrow\quad0<\frac94\Lambda_{\mathrm{eff}}r_s^2=\frac{9G^2M^2\Lambda_{\mathrm{eff}}}{c^4}<1.\tag{63}$$(63)

The smaller root $R_b$ is the black-hole horizon, and the larger root the cosmological horizon. Static observers lie between them where $f>0$. At the smaller root, multiplication of $f(R_b)=0$ by $R_b$ gives $$r_s=R_b\left(1-\frac{\Lambda_{\mathrm{eff}}R_b^2}{3}\right),\qquad M=\frac{c^2R_b}{2G}\left(1-\frac{\Lambda_{\mathrm{eff}}R_b^2}{3}\right).\tag{64}$$(64)

This explicitly recovers the isolated Schwarzschild mass relation as $\Lambda_{\mathrm{eff}}\to0$ at fixed $M$.

To find the leading radius correction, define $a_\Lambda=\Lambda_{\mathrm{eff}}r_s^2/3$ and write $R_b=r_s(1+\delta)$. The root equation reduces to $$\delta=a_\Lambda(1+\delta)^3,\qquad\delta=a_\Lambda+3a_\Lambda^2+O(a_\Lambda^3).\tag{65}$$(65)

For $a_\Lambda\ll1$, the fractional correction is therefore $(R_b-r_s)/r_s\simeq\Lambda_{\mathrm{eff}}r_s^2/3$. Using $\Lambda_{\mathrm{ref}}=1.1\times10^{-52}\,\mathrm{m^{-2}}$ and $M=10^9M_\odot^{\mathrm N}$ gives $3.20\times10^{-28}$. The nominal solar mass here means $GM_\odot^{\mathrm N}=1.3271244\times10^{20}\,\mathrm{m^3\,s^{-2}}$ under the IAU nominal conversion [24].

Static clock comparisons follow directly from the metric: $$\frac{d\tau(r)}{d\tau(r_0)}=\sqrt{\frac{f(r)}{f(r_0)}},\qquad f(r)>0,\qquad f(r_0)>0,\tag{66}$$(66)

where the clocks span equal intervals of the stationary coordinate $t$. The Schwarzschild expression is accurate when $\Lambda_{\mathrm{eff}}r^2/3\ll1-r_s/r$ over the whole comparison region, including the reference radius $r_0$. The horizon expansion and this clock-domain condition are separate approximations.

Finally, the same spherical-energy definition used above gives $$E_{\mathrm{MS}}(r)=\frac{c^4r}{2G}[1-f(r)]=Mc^2+\frac{\Lambda_{\mathrm{eff}}c^4r^3}{6G}=Mc^2+\varepsilon_\Lambda\frac{4\pi r^3}{3}.\tag{67}$$(67)

At either horizon $E_{\mathrm{MS}}=c^4r/(2G)$. The local mass-parameter energy and the cosmological contribution therefore belong to the same spherical energy account, while retaining their separate definitions.

8.5 Connection to the black-hole scaling

Equation (61) connects the inverse-area density to PIN’s information-capacity account. Equation (67) places the cosmological contribution and local mass-parameter energy in one spherical energy account, and the limit in Eq. (64) recovers the black-hole mass relation used in the scaling construction. The mode calculation supplies the regulated field comparison; the adopted trace-free response supplies the constant-baseline result; the specified geometry supplies the capacity relation.

The cosmological state is specified by $N_\Lambda$, or equivalently $\Lambda_{\mathrm{eff}}$. Assessing the stability of a small effective cosmological constant under quantum corrections requires a specified quantum formulation with conserved matching across matter transitions. PIN’s saturated flat disk and outward correspondence describe the black hole. The universe’s global geometry remains an open physical question.

References

[1] J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Philosophical Transactions of the Royal Society of London 155, 459–512 (1865). doi:10.1098/rstl.1865.0008. Part VI, article 97.

[2] P. J. Mohr, D. B. Newell, B. N. Taylor and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Reviews of Modern Physics 97, 025002 (2025). doi:10.1103/RevModPhys.97.025002; NIST values.

[3] A. Einstein, “Zur Quantentheorie der Strahlung,” Physikalische Zeitschrift 18, 121–128 (1917); first printed in Mitteilungen der Physikalischen Gesellschaft Zürich, No. 18 (1916). English translation: On the quantum theory of radiation, §§2–3 and 7; Eq. (9).

[4] L. Boltzmann, “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht,” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe 76, 373–435 (1877). English translation: Sharp and Matschinsky [5].

[5] K. Sharp and F. Matschinsky, “Translation of Ludwig Boltzmann’s paper ‘On the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium’,” Entropy 17, 1971–2009 (2015). doi:10.3390/e17041971.

[6] C. E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal 27, 379–423, 623–656 (1948). doi:10.1002/j.1538-7305.1948.tb01338.x (Part I); doi:10.1002/j.1538-7305.1948.tb00917.x (Part II). Introduction and Sec. 6, Theorem 2; corrected reprint.

[7] J. D. Bekenstein, “Black holes and entropy,” Physical Review D 7, 2333–2346 (1973). doi:10.1103/PhysRevD.7.2333.

[8] S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics 43, 199–220 (1975); erratum 46, 206 (1976). doi:10.1007/BF02345020.

[9] J. A. Wheeler, “Information, physics, quantum: The search for links,” in Complexity, Entropy and the Physics of Information, W. H. Zurek, ed. (Addison–Wesley, 1990), pp. 3–28. Differently paginated digitized reprint, opening and Sections 19.1–19.2. The linked reprint credits the 1989 Tokyo proceedings, pp. 354–368; its pagination differs from the 1990 book edition listed here.

[10] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conference Proceedings C 930308, 284–296 (1993). arXiv:gr-qc/9310026, v2.

[11] L. Susskind, “The world as a hologram,” Journal of Mathematical Physics 36, 6377–6396 (1995). doi:10.1063/1.531249; arXiv:hep-th/9409089, Secs. 1–2 and 4.

[12] A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Annalen der Physik 354, 769–822 (1916). doi:10.1002/andp.19163540702. §14, Eq. (47), and §16, Eq. (53).

[13] J. Martin, “Everything you always wanted to know about the cosmological constant problem (but were afraid to ask),” Comptes Rendus Physique 13, 566–665 (2012). doi:10.1016/j.crhy.2012.04.008; arXiv:1205.3365.

[14] A. Padilla and I. D. Saltas, “A note on classical and quantum unimodular gravity,” European Physical Journal C 75, 561 (2015). doi:10.1140/epjc/s10052-015-3767-0; arXiv:1409.3573.

[15] G. F. R. Ellis, H. van Elst, J. Murugan and J.-P. Uzan, “On the trace-free Einstein equations as a viable alternative to general relativity,” Classical and Quantum Gravity 28, 225007 (2011). doi:10.1088/0264-9381/28/22/225007; arXiv:1008.1196.

[16] T. Padmanabhan and A. Paranjape, “Entropy of null surfaces and dynamics of spacetime,” Physical Review D 75, 064004 (2007). doi:10.1103/PhysRevD.75.064004; arXiv:gr-qc/0701003, Sec. II, Eqs. (15)–(21).

[17] K. Martel and E. Poisson, “Regular coordinate systems for Schwarzschild and other spherical spacetimes,” American Journal of Physics 69, 476–480 (2001). doi:10.1119/1.1336836; arXiv:gr-qc/0001069, Sec. II.

[18] S. A. Hayward, “Gravitational energy in spherical symmetry,” Physical Review D 53, 1938–1949 (1996). doi:10.1103/PhysRevD.53.1938; arXiv:gr-qc/9408002, Eq. (4) and Proposition 4.

[19] K. Schwarzschild, “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 189–196 (1916), especially Eq. (14). English translation: arXiv:physics/9905030.

[20] J. M. Bardeen, B. Carter and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics 31, 161–170 (1973). doi:10.1007/BF01645742.

[21] L. Smarr, “Mass formula for Kerr black holes,” Physical Review Letters 30, 71–73 (1973); erratum 30, 521 (1973). doi:10.1103/PhysRevLett.30.71.

[22] T. Padmanabhan, “Equipartition of energy in the horizon degrees of freedom and the emergence of gravity,” Modern Physics Letters A 25, 1129–1136 (2010). doi:10.1142/S021773231003313X; arXiv:0912.3165, Sec. 2, Eqs. (7)–(12).

[23] D. J. Mortlock et al., “A luminous quasar at a redshift of $z=7.085$,” Nature 474, 616–619 (2011). doi:10.1038/nature10159; arXiv:1106.6088. The central-black-hole mass and accretion interpretation are discussed on pp. 1–2 of the preprint.

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[27] E. P. Verlinde, “On the origin of gravity and the laws of Newton,” Journal of High Energy Physics 04, 029 (2011). doi:10.1007/JHEP04(2011)029; arXiv:1001.0785, Secs. 3.1–3.2, Eqs. (3.6)–(3.13).

[28] J. D. Bekenstein, “Universal upper bound on the entropy-to-energy ratio for bounded systems,” Physical Review D 23, 287–298 (1981). doi:10.1103/PhysRevD.23.287.

[29] I. Newton, The Mathematical Principles of Natural Philosophy, English edition (1846), Book III, Proposition VII and Corollaries, p. 397. Digitized primary text.

[30] R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Reviews of Modern Physics 20, 367–387 (1948). doi:10.1103/RevModPhys.20.367. Secs. 5–7, Eqs. (17)–(18), (35); digitized reprint.

[31] R. Srikanth, “A computational model for quantum measurement,” Quantum Information Processing 2, 153–199 (2003). doi:10.1023/B:QINP.0000004123.82268.f4; arXiv:quant-ph/0302160, Secs. VI–VII.

[32] G. Lüders, “Über die Zustandsänderung durch den Meßprozeß,” Annalen der Physik (Leipzig) 8, 322–328 (1951). English translation by K. A. Kirkpatrick, Annalen der Physik 15, 663–670 (2006). doi:10.1002/andp.20065180904; arXiv:quant-ph/0403007.

[33] C. Simon, V. Bužek and N. Gisin, “No-signaling condition and quantum dynamics,” Physical Review Letters 87, 170405 (2001). doi:10.1103/PhysRevLett.87.170405; arXiv:quant-ph/0102125, Eqs. (1)–(5) and footnote 14.

[34] M. Ozawa, “Quantum measurement, information, and completely positive maps,” in Quantum Communication, Computing, and Measurement 3, P. Tombesi and O. Hirota, eds. (Kluwer Academic/Plenum, New York, 2001), pp. 97–106. arXiv:quant-ph/0107090, Secs. 4, 7 and 8, Eq. (17) and Theorem 8.

[35] R. Penrose, “Singularities and time-asymmetry,” in General Relativity: An Einstein Centenary Survey, S. W. Hawking and W. Israel, eds. (Cambridge University Press, 1979), pp. 581–638. Sections 12.1–12.2; digitized chapter.

[36] S. Lloyd, “Ultimate physical limits to computation,” Nature 406, 1047–1054 (2000). doi:10.1038/35023282; arXiv:quant-ph/9908043.

[37] S. W. Hawking, “Breakdown of predictability in gravitational collapse,” Physical Review D 14, 2460–2473 (1976). doi:10.1103/PhysRevD.14.2460.

[38] P. Hayden and J. Preskill, “Black holes as mirrors: Quantum information in random subsystems,” Journal of High Energy Physics 09, 120 (2007). doi:10.1088/1126-6708/2007/09/120; arXiv:0708.4025, Sec. 3, Eqs. (2)–(7).

[39] D. N. Page, “Information in black hole radiation,” Physical Review Letters 71, 3743–3746 (1993). doi:10.1103/PhysRevLett.71.3743; arXiv:hep-th/9306083, v2, Eqs. (3)–(8).

[40] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “The entropy of Hawking radiation,” Reviews of Modern Physics 93, 035002 (2021). doi:10.1103/RevModPhys.93.035002; arXiv:2006.06872, Secs. 4, 8 and 11.2.

[41] S. D. Mathur, “The information paradox: A pedagogical introduction,” Classical and Quantum Gravity 26, 224001 (2009). doi:10.1088/0264-9381/26/22/224001; arXiv:0909.1038, v2, Secs. 5.1–5.2 and 6, Eqs. (37), (55).

[42] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, “Black holes: Complementarity or firewalls?” Journal of High Energy Physics 02, 062 (2013). doi:10.1007/JHEP02(2013)062; arXiv:1207.3123, Sec. 2, Eq. (2.3).

[43] C. Rovelli and F. Vidotto, “Planck stars,” International Journal of Modern Physics D 23, 1442026 (2014). doi:10.1142/S0218271814420267; arXiv:1401.6562, pp. 1–2, Eqs. (1)–(3).

[44] R. Penrose, “Gravitational collapse and space-time singularities,” Physical Review Letters 14, 57–59 (1965). doi:10.1103/PhysRevLett.14.57, pp. 58–59, assumptions (ii)–(iii).

[45] E. Kh. Akhmedov, “Vacuum energy and relativistic invariance” (2002). arXiv:hep-th/0204048, v2.

[46] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astronomy and Astrophysics 641, A6 (2020); corrigendum 652, C4 (2021). doi:10.1051/0004-6361/201833910; corrigendum; arXiv:1807.06209, Table 2, base-ΛCDM TT,TE,EE+lowE+lensing column.

[47] R. Bousso, “Positive vacuum energy and the N-bound,” Journal of High Energy Physics 11, 038 (2000). doi:10.1088/1126-6708/2000/11/038; arXiv:hep-th/0010252.

[48] R.-G. Cai and S. P. Kim, “First law of thermodynamics and Friedmann equations of Friedmann–Robertson–Walker universe,” Journal of High Energy Physics 02, 050 (2005). doi:10.1088/1126-6708/2005/02/050; arXiv:hep-th/0501055.

[49] M. Li, “A model of holographic dark energy,” Physics Letters B 603, 1–5 (2004). doi:10.1016/j.physletb.2004.10.014; arXiv:hep-th/0403127, Eqs. (1) and (10).

[50] M. Urano, A. Tomimatsu and H. Saida, “The mechanical first law of black hole spacetimes with a cosmological constant and its application to the Schwarzschild-de Sitter spacetime,” Classical and Quantum Gravity 26, 105010 (2009). doi:10.1088/0264-9381/26/10/105010; arXiv:0903.4230, Sec. 3.1, Eqs. (3.1)–(3.8).

[51] T. C. Hales, “The honeycomb conjecture,” Discrete & Computational Geometry 25, 1–22 (2001). arXiv:math/9906042, Sec. 2, Theorem 1-A.

[52] R. Bousso, “The holographic principle,” Reviews of Modern Physics 74, 825–874 (2002). doi:10.1103/RevModPhys.74.825; arXiv:hep-th/0203101, Secs. III and VIII.

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[54] T. Jacobson, “Entanglement Equilibrium and the Einstein Equation,” Physical Review Letters 116, 201101 (2016). doi:10.1103/PhysRevLett.116.201101.

[55] L. Hodgkinson, J. Louko and A. C. Ottewill, “Static detectors and circular-geodesic detectors on the Schwarzschild black hole,” Physical Review D 89, 104002 (2014). doi:10.1103/PhysRevD.89.104002; arXiv:1401.2667v3, Secs. 5.1, 5.3 and 8.1.

[56] L. De Luca and G. Friesecke, “Crystallization in two dimensions and a discrete Gauss–Bonnet theorem,” arXiv:1605.00034v1 (2016), Secs. 2.6–2.7, Eqs. (2.7), (2.11). https://arxiv.org/pdf/1605.00034.

[57] E. Bianchi and W. Wieland, “Horizon energy as the boost boundary term in general relativity and loop gravity,” arXiv:1205.5325 (2012), Eqs. (7)–(9). https://arxiv.org/pdf/1205.5325.

[58] R. P. Kerr, “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,” Physical Review Letters 11, 237–238 (1963). doi:10.1103/PhysRevLett.11.237. The angular-coordinate orientation used in this key follows Ref. [59].

[59] M. Visser, “The Kerr spacetime: A brief introduction,” arXiv:0706.0622, version 3 (2008). Boyer–Lindquist metric: Eq. (57); horizon angular velocity: Eq. (112); intrinsic horizon geometry and area: Eqs. (117)–(118).

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[61] S. Gao and R. M. Wald, “The ‘physical process’ version of the first law and the generalized second law for charged and rotating black holes,” Physical Review D 64, 084020 (2001). arXiv:gr-qc/0106071. Horizon flux and area balance: Eqs. (45), (49).

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[63] M. Liska et al., “Formation of precessing jets by tilted black hole discs in 3D general relativistic MHD simulations,” Monthly Notices of the Royal Astronomical Society: Letters 474, L81–L85 (2018). doi:10.1093/mnrasl/slx174. arXiv:1707.06619, version 4 (2022). Tilted inner flow and polar plunging streams: Sec. 4.

ASCII Math Key

PIN: ASCII MATH KEY
Companion to the approved updated manuscript, 26 October 2026

SOURCE AND COVERAGE
Source: PIN_UPDATED.pdf; manuscript cover date 24 October 2026.
Updated version: 26 October 2026.
Source size: 221442 bytes; 43 PDF pages.
Source SHA-256:
d8c0ed5b1e6a399ea567099c9020983181cd53e33c8f44e4ad73fb46feb3fa40
Coverage: all 112 numbered equation groups, in manuscript order.
The 109 existing groups retain their labels; additions are (29a), (K17a),
and (61a).
Equation labels and source page locations refer to this exact source version.
PDF page numbers are one-based; printed page numbers follow the manuscript.
Unnumbered formulas are outside this numbered-equation key unless separately
identified. No original equation is renumbered. This key transcribes the
source notation and records definitions, units and domains; it does not add
physical mechanisms or alter the manuscript's premises.

ASCII CONVENTIONS
This is an ASCII mathematical transcription with the conventions below,
not executable source code for a particular computer-algebra language.
* denotes multiplication or ordered operator composition as determined by
operand type; / denotes division. Numeric powers use ^. Parentheses enclose
complete compound numerators, denominators, exponents and function arguments.
sqrt(z) is the principal square root; ln(z) is the natural logarithm;
exp(z), sin(z), cos(z), abs(z), max(a,b), and diag(...) have their usual meanings.
A tensor superscript is an upper index, not a power: T^(a b) is contravariant,
T_(a b) is covariant, and T^a_b is mixed. Likewise x^0 and x^i are coordinate
components, and dx^0 and dx^i are their differentials, not numeric powers.
Repeated abstract tensor indices appearing once upper and once lower are
contracted. Explicit component equations such as G^0_0 or G^r_r in (18)
have no implicit sum over the named component. Quantum labels, mode labels,
state-space indices and sequence indices are summed only by an explicit sum.
Superscripts vac, excit, total and TF label tensor parts, not powers.
The superscript (1) on DeltaE_BH labels a one-cell step, not a power
or derivative. The + and - labels on Psi distinguish post-/pre-event states.
delta(X) is a variation; delta_(j k) is the Kronecker delta. The unindexed
delta in K35 is an angular half-width; in (65) it is a radius correction.
partial_i is a coordinate partial derivative; nabla_i is the metric-compatible
covariant derivative. A raised derivative index is raised with the metric.
dot(X) means the derivative in the time parameter specified for that equation.
A prime on a radial function means d/dr; a double prime means d^2/dr^2.
In K15 the subscript after the derivative specifies the held-fixed variable.
Tr is the operator trace; Tr_B is the partial trace over subsystem B.
dagger(X) is the Hermitian adjoint. I is the identity operator and id_X is the
identity map on subsystem X. |psi> is a ket, <psi| its bra, and |psi><phi|
is an outer product. Products of operators retain their displayed order.
tensor denotes a tensor product. || in (43a) denotes ordered append, not norm.
In K32, (*F) denotes the dual electromagnetic tensor; the star there is not
multiplication. In P_star, E_star and |star>, star is a state label.
In line elements ds^2 means the squared spacetime interval and dOmega^2 means
the unit-sphere line element; neither uses the Kerr spin s or angular rate
Omega. dtheta^2, dr^2 and dt^2 mean squares of coordinate differentials.
For (48)-(49), k_vec denotes the printed bold wave vector and k=abs(k_vec)
its nonnegative magnitude; vector mode labels and scalar radial integrals are
distinct. The continuum replacement applies to the whole mode-sum measure.
O(z) denotes the stated asymptotic remainder in the stated limit.
=> denotes implication; <=> denotes equivalence. In (15), -> is the explicitly
stated constant-baseline relabeling; in (48) it denotes a continuum-measure
limit; in K30 it denotes the stated weak-field asymptotic comparison.
SI units and signature (-,+,+,+) apply except to the fluid equations and flux
definitions beginning at K32. K32-K35 use G=c=1 with electromagnetic 4*pi
absorbed into the field normalization. K36-K37 reuse the same Fcal_in and
inherit its flux normalization; they do not silently restore SI. Physical
units and one common time normalization are restored before using K19.
Coordinates and tensor components carry basis-dependent units as stated in
the registry. No blanket dimensionless assignment to metric or velocity
components is intended.

NOTATION AND UNITS REGISTRY
1 denotes dimensionless units. P denotes a printed page of PIN_UPDATED.pdf;
PDF page = P+1. Registry page ranges include any continuation page.

CONSTANTS AND BASIC SCALES
h: Planck constant [J*s]; hbar=h/(2*pi) [J*s]. P2, (1)-(2).
c: vacuum light speed [m/s]; G: Newton constant [m^3/(kg*s^2)];
k_B: Boltzmann constant [J/K]. P2-3, (1)-(4).
pi: circle constant [1]; i in quantum phases: sqrt(-1), distinct from index i.
ell_P=sqrt(hbar*G/c^3) [m]; t_P=ell_P/c [s];
m_P=sqrt(hbar*c/G), unreduced Planck mass [kg]; E_P=m_P*c^2 [J]. P2, (1).
lambda_bar=lambda/(2*pi) [m]; lambda [m]; lambda_bar_PWP=ell_P;
lambda_PWP=2*pi*ell_P; p_PWP=m_P*c [kg*m/s]; E_PWP=E_P [J];
m_0_PWP=0 [kg], invariant rest mass. P2, (2).
sigma_E_max=E_P/ell_P^2 [J/m^2], assigned areal inventory density. P3, (3).

AREA, CAPACITY, ENERGY AND LATTICE
Sigma: physical disk, not a scalar variable; A_Sigma [m^2]; R [m], its radius.
A_H [m^2]: exterior horizon-section area. P3-4, (5)-(7).
N=A_Sigma/ell_P^2 [1, counted in nats]; S_H=k_B*N [J/K]. P3, (4)-(5).
E_Sigma=N*E_P [J], assigned local inventory; E_BH=M*c^2 [J], exterior energy.
They are distinct energy accounts. P3,7-8, (5),(21)-(24).
gamma=E_BH/E_Sigma [1]; gamma_K analogous Kerr ratio [1]. P8,13-14, (24),(K13).
B_rim,z_i,m_lat: dimensionless counts; epsilon_rim=6/B_rim [1].
a: neighboring-centre spacing [m]; R_c: circular cutoff [m].
sqrt(3)*a^2/2=ell_P^2 is the source lattice convention. P3-4, (5a)-(5b).
The index i in (5a) enumerates rim vertices; it is not an Einstein contraction.

GRAVITY, COORDINATES AND DIMENSIONAL TENSORS
x^a,x^mu: coordinate components, with units attached to the coordinate chart.
x^0=c*t or c*t_PG where specified [m]; r,R,r_s,C_r,R_0,r_0,ell [m].
t,t_PG,tau,tau_0 [s]; theta,phi [1, radians]. P5,7,22-24.
ell in (34) is proper radial length, not the Planck length ell_P.
ds^2 [m^2]; dOmega^2=dtheta^2+sin(theta)^2*dphi^2 [1]. P7, (17).
g_(a b): metric; g^(a b): inverse metric; g=det(g_(a b)).
If coordinate x^a has unit U_a, [g_(a b)]=m^2/(U_a*U_b) and
[g^(a b)]=U_a*U_b/m^2. Thus angular metric components carry m^2, while
in a t-seconds chart g_tt carries m^2/s^2. Determinant units follow the chart.
gamma_(i j),gamma^(i j): induced spatial metric/inverse with the same rule.
P5,7,15-16, (8)-(18),(K21),(K25).
Rcal: Ricci scalar [1/m^2]; Rcal_(a b),G_(a b): Ricci/Einstein tensors,
component units 1/(U_a*U_b). G_(a b)=Rcal_(a b)-(1/2)*Rcal*g_(a b).
X_(a b): generic symmetric tensor; TF subtracts one quarter of its metric
trace in four dimensions. P5-6, (11).
T_(a b): stress-energy tensor. Local orthonormal components have [J/m^3];
coordinate covariant components have [J/m^3]*m^2/(U_a*U_b). T=g^(a b)*T_(a b)
is its scalar trace [J/m^3]. vac,excit,total are physical-part labels. P5-6,35.
kappa=8*pi*G/c^4 [m/J]; Lambda_int,Lambda_bare,Lambda_eff,Lambda_ref [1/m^2].
Ical,Ical_m [J*s]: gravitational-plus-matter and matter actions;
psi in Ical denotes matter fields with field-dependent units, not psi_r in (39).
P5-6, (8)-(16).
M,M_MS,M_1,M_irr,m,m_b [kg]: black-hole, Misner-Sharp, single-area,
irreducible, or test-body/field masses as stated. P7-8,13-16,22,34.
u(r) [1]: PG profile; chi=r_s/r [1]: compactness. P7,22, (17),(31).
T_H [K]; C_H=dE_BH/dT_H [J/K]; DeltaE_BH [J]; N_i,N_f [1]. P9-11.
tau_Sigma=2*pi*R/c [s], or 2*pi*R_Sigma/c for Kerr; tau_H=hbar/(k_B*T_H) [s].
t_b=2*R/c [s]; Theta=t/t_b [1, boost angle]; dot(Theta) [1/s];
p_Theta [J*s], its conjugate momentum. P8-9,13-15, (25),(27a),(K13),(K18).
GM_sun_nom=1.3271244e20 [m^3/s^2], exact nominal mass parameter;
q in (30) [1], nominal-mass multiplier, not q(phi,t) of K36. P10-11.

KERR GEOMETRY, CLOCKS, MOMENTA AND FLUXES
J [J*s]: angular momentum; j=J/hbar [1]; mu_K=M/m_P [1]. P11-14.
r_g=G*M/c^2,a_K=J/(M*c),r_plus,r_minus,R_Sigma,R_A,R_eq [m].
a_star=c*J/(G*M^2), x=2*pi*j/N, s=sqrt(1-a_star^2) [1].
R_A=sqrt(A_H/(4*pi)); R_eq is proper equatorial circumference/(2*pi).
P11-14, (K1)-(K14).
Sigma_K,Delta [m^2]; Acal_K [m^4]. P12-13, (K2).
alpha,Z,Z_0,h_K,h_clk [1]: lapse, clock factors and clock ratios.
omega,Omega,Omega_0,Omega_H [1/s, angular rates]; v_loc [m/s], signed local
azimuthal velocity. P14-17, (K16),(K21)-(K31).
E_b=-p_t,E_loc,Ecal_K_0,W_K_0 [J]. L_b=p_phi [J*s].
p_mu or p_i is canonical covariant momentum, with unit (J*s)/U_mu or
(J*s)/U_i; therefore in the t-seconds,r,theta,phi chart p_t [J],
p_r [kg*m/s], p_theta,p_phi [J*s]. P15-17, (K25)-(K29).
a_i=c^2*partial_i ln(Z) [m^2/(s^2*U_i)]: covariant acceleration components;
a_r [m/s^2] but a_theta [m^2/s^2] for a dimensionless angular coordinate.
a_proper,a_support [m/s^2] are norms/local physical accelerations.
F_support_local [N = kg*m/s^2]; E_0,W_0 [J]. P16-17,23.
P,P_0: spatial positions, not projectors; K=partial_t+Omega_0*partial_phi:
Killing field, not storage K. P16-17, (K28)-(K29).
nu_0,nu_proper,Gamma_0,Gamma_loc [1/s] for the specified physical clocks;
n [1] counts events. P16-17,24, (K31),(37).
dot(M),dot(M_0),dot(M_in) after physical-unit restoration [kg/s];
dot(E_H) [J/s]; dot(J_H) [J*s/s]; dot(N) [1/s]; dot(a_star) [1/s];
dot(R_Sigma),dot(R_A),dot(R_eq) [m/s]. P14-15,17-18.
The raw K34 integrals instead use the source's G=c=1 normalization.

GEOMETRIZED FLUID BLOCK AND ITS INHERITED FLUX DIAGNOSTICS
For K32-K37 let L denote the chosen geometric length unit; time and mass
are expressed as lengths. rho: proper rest-mass density; e_int: internal
energy density; p: pressure; b^2: fluid-frame magnetic squared magnitude.
Each has [L^-2]. Local orthonormal b and normalized F components have [L^-1];
electromagnetic 4*pi is absorbed. The source does not select a numerical
simulation density scale, so no numerical conversion to SI is implied.
u^mu=dx^mu/dtau_geo; u_mu=g_(mu nu)*u^nu; u_mu*u^mu=-1.
Their coordinate units are U_mu/L and L/U_mu, respectively. Only appropriate
orthonormal/homogeneous-length components are dimensionless. P17-18, (K32)-(K33).
F_(mu nu): electromagnetic tensor; (*F)^(mu nu): its dual, with units from
the tensor basis and adopted geometrized normalization. P17-18, (K32).
For the usual (t_geo,r,theta,phi) shell coordinates, sqrt(-g) has [L^2];
Fcal_in=max(-rho*u^r,0)*sqrt(-g) has geometric mass-rate units [1].
Raw dot(M_0),dot(M_in),dot(E_H) have [1], and raw dot(J_H) has [L].
These are not SI kg/s, J/s, J. Restore G,c and any chosen simulation scaling
before applying the SI evolution equations. P17-18, (K34)-(K36).
f_eq [1]: equatorial-band fraction; delta in K35 [1,radians].
q(phi,t)=integral Fcal_in dtheta/R_Sigma has [L^-1] in these units, or
[kg/(s*m)] after consistently restoring the mass flux and rim length.
Fcal_+,Fcal_-,f(phi) share Fcal_in's flux units; epsilon in K37 [1].
0<integral f dphi<infinity is required for the stated normalized polar means.
P17-18, (K35)-(K37).

PROCESSING AND QUANTUM INFORMATION
u_p,eta_p=1-u_p [1]: utilization/headroom; u_p=chi for the specified static
comparison. They are not fluid u^mu. P24, (36).
F_n: committed frame; h_n: ordered event history; e_n: event record;
prefix_n: history-prefix operation; n is a sequence index, not elapsed time.
These record objects have no fixed SI scalar unit. P25-28, (38)-(43a).
epsilon_res: declared physical resolution used in resource accounting;
its measurement convention/units are not assigned universally in the source.
Psi_n,Psi^-,Psi_j^+,Psi_BO,psi_r(x),c_k: normalized discrete-state amplitudes
or state vectors in their stated bases [1]. d: finite state-space dimension [1].
x,y in (39) label joint states; r there is an update index, not radial distance.
U_r,U_s,I,P_j,P_star,P_rec,rho,rho_O,rho_AX [1] as operators in normalized
bases; rho denotes density operators (positive, trace one), not fluid density.
P_rec is an orthogonal retained-record-space projector. p_j,p_alpha [1]
are probabilities. alpha as Schmidt label is not the Kerr lapse alpha.
P25-26,30-32, (38)-(41),(44)-(46).
C_keep,C_commit: same declared cost unit, not automatically energy or time.
K_history,K_live,K_control,K_max,K_keep,K_commit [1,nats]: storage demands
or capacity. J_demand,B_proc: specified updates per second using the same
clock and update definition; J_demand is not angular momentum J. P26-28.
H_s,H_tot,H_hop,E_star,E_s [J]: Hamiltonians/eigenvalues.
t_ret,T_ret [s-dimension time parameters]; a separate physical timing
prescription relates t_ret to observable clocks. P27,30-31, (44)-(44a).
Phi,Tcal: maps on density operators, with no SI scalar unit; id_X: identity
map on untouched reference X. A,X,B,O: subsystem labels. P27,31-32.
s_O,s_B and all s_(subsystem-list) [1,nats]: von Neumann entropies; unlike
S_H they do not include k_B. P32, (46)-(47).

VACUUM, COSMOLOGY AND OTHER CONTEXT VARIABLES
k_vec [1/m]: wave vector; k=abs(k_vec),k_c,mu=m*c/hbar [1/m].
omega_(k_vec) [1/s, angular frequency]; E_(0,k_vec) [J]; V [m^3], periodic
box volume. m>=0 [kg] here is scalar-field mass, not black-hole M. P34.
epsilon_0,epsilon_vac,epsilon_shift,epsilon_P,epsilon_Lambda,
epsilon_Lambda_ref,epsilon_bar_BH [J/m^3]: respectively regulated mode sum,
metric-proportional vacuum baseline, baseline shift, Planck volumetric scale,
effective cosmological density, reference density, exterior comparison density.
p_0,p_vac [J/m^3 = Pa]. These are not the fixed disk density sigma_E_max.
P20,34-36, (48)-(55).
H_Lambda [1/s]: positive de Sitter Hubble rate, not a Hamiltonian;
R_Lambda [m]; f_Lambda(r),f(r) [1]: metric functions, with f defined anew in
(62), not f(phi) in K37. E_MS,E_Lambda [J]. P36-38, (56)-(67).
A_Sigma_Lambda,A_Lambda [m^2]: quarter-area measure and de Sitter horizon
area; N_Lambda [1,nats]; S_Lambda [J/K]; V_Lambda=4*pi*R_Lambda^3/3 [m^3].
The quarter-area measure does not assert a physical cosmological flat disk.
P36-38, (59)-(61).
r_star,R_b [m]: stationary radius of f and smaller positive horizon root.
a_Lambda=Lambda_eff*r_s^2/3 and delta=(R_b-r_s)/r_s [1]. P37-38, (63)-(65).
rho_0,rho_max,L [m] in the Section 4.2 latency comparison are proper distances,
not fluid or quantum densities. P20. They occur in the registry/context,
not in a numbered group. H_s and H_Lambda likewise must remain distinct.

SOURCE DOMAINS AND HELD-FIXED CONDITIONS
These conditions specify the domains and conventions of the source equations.
- Kerr sector: neutral stationary Kerr family, M>0, abs(a_star)<1, s>0;
  extremality is a limit. N>0 and j=J/hbar are continuous macroscopic variables.
  K19-K20 compare neighboring slowly changing Kerr states using one time.
  Source PDF12-13/printed11-12; K9 outer branch PDF13-14/printed12-13.
- K22-K27: r>r_plus; alpha>0; circular clock formulas hold at fixed r,theta.
  Timelike clocks require Z>0 (abs(v_loc)<c). K24 excludes the rotation axis.
  K25-K27 use t in seconds, E_b=-p_t, L_b=p_phi, i,j in {r,theta,phi}.
  K27 concerns freely falling neutral test bodies; m_b=0 requires nonzero
  momentum. Source PDF16-18/printed15-17.
- K28-K29: K=partial_t+Omega_0*partial_phi is timelike along the entire route,
  not just at P_0. The supported body is comoving with K and has fixed m_b.
  Omega_0 remains constant; Z=Z(P;Omega_0). Work includes angular-momentum
  exchange. a_i are covariant components, not the proper acceleration norm.
  Source PDF17-18/printed16-17.
- K32-K34: u_mu*u^mu=-1. g=det(g_(mu nu)) in horizon-penetrating coordinates;
  the angular integrals are over a constant-radius shell (theta 0..pi,
  phi 0..2*pi). H denotes horizon evaluation. Source PDF18-19/printed17-18.
- (8)-(16): four spacetime dimensions; g=det(g_(a b)); x^0=c*t in the action.
  Matter is on shell; metric variations have compact support; reconstruction
  assumes conserved complete matter stress on a connected solution domain.
  (15) requires constant epsilon_shift. Source PDF6-7/printed5-6.
- (17)-(20): u(r)>0; prime=d/dr; the final equality M_MS=c^2*C_r/(2G) uses
  the vacuum solution (19), not a general u. Source PDF8/printed7.
- (21)-(30): isolated neutral nonrotating stationary family, M,R,N>0;
  E_BH uses standard asymptotic stationary-time normalization. N=1 is formal.
- (27a): dot(Theta)=dTheta/dt with Schwarzschild infinity-normalized time t;
  Theta=t/t_b and t_b=2*R/c=4*G*M/c^3. Source PDF10/printed9.
- (32)-(34): exterior static comparison r>r_s; d_ell=dr/sqrt(f(r)) on a
  static spatial slice. In (34), dE_0/dell means derivative with respect to
  this proper radial length, at fixed black-hole/reference data, not ell_P.
  Source PDF23-24/printed22-23.
- (38): finite orthonormal alternatives at declared resolution epsilon_res.
- (41): normalized pre-event state and exhaustive orthogonal record
  projectors; selected outcome p_j>0. Source PDF27/printed26.
- (43): every supported outcome must fit; peak demands include transition
  workspace. Source PDF29/printed28.
- (44): |star> is normalized; P_star=|star><star|; E_star is real. t_ret is a
  retained-sector evolution parameter, not an already derived external clock.
- (44a): rho denotes density matrices (positive, trace one), not arbitrary
  untyped matrices; P_rec is an orthogonal projector. The quantifiers cover
  every supported density matrix and every t_ret in the whole nonzero interval.
  Source PDF31-32/printed30-31.
- (46)-(47): dimensionless von Neumann entropies in nats. Subsystem suffixes
  denote joint subsystems, not products. In (47) the last implication also uses
  the strong-subadditivity inequality on its first line.
- (55): Lambda_ref>0; epsilon_Lambda_ref=Lambda_ref/kappa. Source PDF37/printed36.
- (56)-(61): positive de Sitter branch; H_Lambda>0 and R_Lambda>0 as selected
  by R_Lambda=c/H_Lambda=sqrt(3/Lambda_eff). V_Lambda=4*pi*R_Lambda^3/3 is
  areal-volume bookkeeping, not the proper volume of an arbitrary slice.
- (63)-(65): R_b is the smaller positive root on 0<9*G^2*M^2*Lambda_eff/c^4<1;
  (65) is the branch delta->0 as a_Lambda->0, expanded for a_Lambda<<1.
  Source PDF38-39/printed37-38.

NUMBERED EQUATIONS

EQ (1) [PDF page 3; printed page 2]
ell_P^2 = hbar*G/c^3
m_P = sqrt(hbar*c/G)
E_P = m_P*c^2 = hbar*c/ell_P
t_P = ell_P/c

EQ (2) [PDF page 3; printed page 2]
lambda_bar_PWP = hbar/p_PWP = ell_P
lambda_PWP = 2*pi*ell_P
p_PWP = m_P*c
E_PWP = c*p_PWP = E_P
E_PWP^2 - c^2*p_PWP^2 = 0
m_0_PWP = 0
E_PWP/c^2 = m_P

EQ (3) [PDF page 4; printed page 3]
lambda_bar >= ell_P  =>  E = hbar*c/lambda_bar <= E_P
sigma_E_max = E_P/ell_P^2

EQ (4) [PDF page 4; printed page 3]
S_H = k_B*N

EQ (5) [PDF page 4; printed page 3]
A_Sigma = pi*R^2
N = A_Sigma/ell_P^2 = pi*R^2/ell_P^2
E_Sigma = N*E_P

EQ (5a) [PDF page 4; printed page 3]
sum_(i in rim) (4-z_i) = 6
epsilon_rim = (1/B_rim)*sum_(i in rim) (4-z_i) = 6/B_rim
Source domain: topological disk, one simple boundary, degree-six interior.

EQ (5b) [PDF page 5; printed page 4]
B_rim = 4*sqrt(3)*R_c/a + O(1)
R = R_c + O(a)
epsilon_rim = sqrt(3)*a/(2*R) + O(a^2/R^2)
Source convention: sqrt(3)*a^2/2 = ell_P^2; the cutoff is lattice-centred
and circular. This is an asymptotic boundary relation, not an exact count
for arbitrary shapes.

EQ (6) [PDF page 5; printed page 4]
A_H = 4*pi*R^2 = 4*A_Sigma
N = A_Sigma/ell_P^2 = A_H/(4*ell_P^2)

EQ (7) [PDF page 5; printed page 4]
S_H = k_B*A_Sigma/ell_P^2 = k_B*A_H/(4*ell_P^2)

EQ (8) [PDF page 6; printed page 5]
Ical[g,psi] = (1/(2*kappa*c))*integral d^4x sqrt(-g)*Rcal
              + Ical_m[g,psi]
1/(2*kappa*c) = c^3/(16*pi*G)

EQ (9) [PDF page 6; printed page 5]
delta(Ical_m) = -(1/(2*c))*integral d^4x sqrt(-g)*T_(a b)*delta(g^(a b))
delta(Ical) = (1/(2*kappa*c))*integral d^4x sqrt(-g)
              *(G_(a b)-kappa*T_(a b))*delta(g^(a b))

EQ (10) [PDF page 7; printed page 6]
delta(sqrt(-g)) = -(1/2)*sqrt(-g)*g_(a b)*delta(g^(a b))
delta(sqrt(-g)) = 0  =>  g_(a b)*delta(g^(a b)) = 0

EQ (11) [PDF page 7; printed page 6]
X_(a b)^TF = X_(a b) - (1/4)*g_(a b)*g^(c d)*X_(c d)
G_(a b)^TF - kappa*T_(a b)^TF = 0
Rcal_(a b) - (1/4)*Rcal*g_(a b)
  = kappa*(T_(a b) - (1/4)*T*g_(a b))
T = g^(a b)*T_(a b)

EQ (12) [PDF page 7; printed page 6]
T^vac = g^(a b)*(-epsilon_vac*g_(a b)) = -4*epsilon_vac
(T^vac)_(a b)^TF
  = -epsilon_vac*g_(a b) - (1/4)*(-4*epsilon_vac)*g_(a b) = 0

EQ (13) [PDF page 7; printed page 6]
(1/4)*nabla_b(Rcal) = -(kappa/4)*nabla_b(T)
nabla_b(Rcal+kappa*T) = 0
Rcal+kappa*T = 4*Lambda_int

EQ (14) [PDF page 7; printed page 6]
Rcal_(a b) - (1/4)*Rcal*g_(a b)
  = kappa*T_(a b) + (1/4)*Rcal*g_(a b) - Lambda_int*g_(a b)
G_(a b) + Lambda_int*g_(a b) = kappa*T_(a b)

EQ (15) [PDF page 7; printed page 6]
T_(a b) -> T_(a b) - epsilon_shift*g_(a b)
Lambda_int -> Lambda_int - kappa*epsilon_shift
epsilon_vac -> epsilon_vac + epsilon_shift
Lambda_eff = Lambda_int + kappa*epsilon_vac -> Lambda_eff
Here the arrows denote the stated constant-baseline relabeling.

EQ (16) [PDF page 7; printed page 6]
nabla^a[-epsilon_shift(x)*g_(a b)] = -partial_b(epsilon_shift)
nabla^a(T_(a b)^excit) = partial_b(epsilon_vac)
  when nabla^a(T_(a b)^total) = 0

EQ (17) [PDF page 8; printed page 7]
ds^2 = -(dx^0)^2 + (dr+sqrt(u(r))*dx^0)^2 + r^2*dOmega^2
dOmega^2 = dtheta^2 + sin(theta)^2*dphi^2
x^0 = c*t_PG

EQ (18) [PDF page 8; printed page 7]
G^0_0 = G^r_r = -(r*u)'/r^2
G^theta_theta = G^phi_phi = -(r*u)''/(2*r)
G^a_b = 0 for a != b

EQ (19) [PDF page 8; printed page 7]
(r*u)' = 0  =>  u(r) = C_r/r
C_r > 0

EQ (20) [PDF page 8; printed page 7]
1 - 2*G*M_MS(r)/(c^2*r) = g^(a b)*nabla_a(r)*nabla_b(r) = 1-u(r)
M_MS(r) = c^2*r*u(r)/(2*G) = c^2*C_r/(2*G)

EQ (21) [PDF page 8; printed page 7]
R = r_s = 2*G*M/c^2
E_BH = M*c^2 = c^4*R/(2*G)

EQ (22) [PDF page 9; printed page 8]
N = (pi/ell_P^2)*(2*G*M/c^2)^2
  = 4*pi*G*M^2/(hbar*c)
  = 4*pi*(M/m_P)^2

EQ (23) [PDF page 9; printed page 8]
R(N) = ell_P*sqrt(N/pi)
M(N) = m_P*sqrt(N)/(2*sqrt(pi))
E_Sigma(N) = N*E_P
E_BH(N) = E_P*sqrt(N)/(2*sqrt(pi))

EQ (24) [PDF page 9; printed page 8]
gamma = E_BH/E_Sigma = 1/(2*sqrt(pi*N)) = ell_P/(2*pi*R)

EQ (25) [PDF page 9; printed page 8]
gamma = t_P/tau_Sigma
tau_Sigma = 2*pi*R/c

EQ (26) [PDF page 9; printed page 8]
R_1 = ell_P/sqrt(pi)
M_1 = m_P/(2*sqrt(pi))
M = m_P  =>  N = 4*pi
This is the source's formal single-area normalization, not a demonstrated
microscopic endpoint transition.

EQ (27) [PDF page 10; printed page 9]
k_B*T_H = dE_BH/dN = E_P/(4*sqrt(pi*N)) = hbar*c/(4*pi*R)
T_H = hbar*c^3/(8*pi*G*M*k_B)

EQ (27a) [PDF page 10; printed page 9]
p_Theta = c^3*A_H/(8*pi*G) = hbar*N/(2*pi)
dot(Theta) = 1/t_b = c/(2*R)
dE_BH/dN = hbar*dot(Theta)/(2*pi) = k_B*T_H
p_Theta*dot(Theta) = E_BH/2

EQ (28) [PDF page 11; printed page 10]
E_BH/N = 2*k_B*T_H
E_BH = 2*T_H*S_H
C_H = dE_BH/dT_H = -2*N*k_B

EQ (29) [PDF page 11; printed page 10]
DeltaE_BH = (E_P/(2*sqrt(pi)))*(N_i-N_f)/(sqrt(N_i)+sqrt(N_f))
N_i > N_f > 0

EQ (29a) [PDF page 11; printed page 10]
DeltaE_BH^(1)(N)/E_P = gamma(N)/(1+sqrt(1-1/N))
DeltaE_BH^(1)(N)/(k_B*T_H(N)) = 2/(1+sqrt(1-1/N))
DeltaE_BH^(1)(N) = E_BH(N)-E_BH(N-1) [J]; E_P [J];
gamma(N)=E_BH(N)/(N*E_P) [1]; k_B*T_H(N) [J].
Domain: one-cell stationary saturated Schwarzschild endpoints N and N-1,
with N>1. Both displayed ratios are dimensionless. The second exceeds one
at finite N and approaches one as N tends to infinity.

EQ (30) [PDF page 11; printed page 10]
R = 2*q*GM_sun_nom/c^2
T_H = hbar*c^3/(8*pi*k_B*q*GM_sun_nom)
N = pi*R^2*c^3/(hbar*G)
GM_sun_nom = 1.3271244e20 m^3/s^2; q is the chosen nominal-mass multiplier.

EQ (K1) [PDF page 13; printed page 12]
r_g = G*M/c^2
a_K = J/(M*c)
a_star = a_K/r_g = c*J/(G*M^2)
r_plus = r_g*(1+s); r_minus = r_g*(1-s)
s = sqrt(1-a_star^2)

EQ (K2) [PDF page 13; printed page 12]
Sigma_K = r^2 + a_K^2*cos(theta)^2
Delta = r^2 - 2*r_g*r + a_K^2
Acal_K = (r^2+a_K^2)^2 - a_K^2*Delta*sin(theta)^2

EQ (K3) [PDF page 13; printed page 12]
ds^2 = -(1-2*r_g*r/Sigma_K)*c^2*dt^2
       -(4*r_g*a_K*r*sin(theta)^2/Sigma_K)*c*dt*dphi
       +(Sigma_K/Delta)*dr^2 + Sigma_K*dtheta^2
       +(Acal_K*sin(theta)^2/Sigma_K)*dphi^2

EQ (K4) [PDF page 13; printed page 12]
dA_H = (r_plus^2+a_K^2)*sin(theta)*dtheta*dphi
A_H = 4*pi*(r_plus^2+a_K^2) = 8*pi*r_g^2*(1+s)

EQ (K5) [PDF page 13; printed page 12]
A_H = 4*A_Sigma
N = A_Sigma/ell_P^2 = A_H/(4*ell_P^2)

EQ (K6) [PDF page 13; printed page 12]
R_Sigma = R_A = sqrt(A_H/(4*pi)) = sqrt(r_plus^2+a_K^2)
  = r_g*sqrt(2*(1+s)) = ell_P*sqrt(N/pi)

EQ (K7) [PDF page 13; printed page 12]
R_eq = sqrt(g_(phi phi)(r_plus,pi/2))
     = (r_plus^2+a_K^2)/r_plus = 2*r_g
A_H/(pi*R_eq^2) = 2*(1+s)

EQ (K8) [PDF page 14; printed page 13]
N = 2*pi*mu_K^2*(1+sqrt(1-j^2/mu_K^4))
  = 2*pi*(mu_K^2+sqrt(mu_K^4-j^2))

EQ (K9) [PDF page 14; printed page 13]
(N/(2*pi)-mu_K^2)^2 = mu_K^4-j^2
  => M^2/m_P^2 = N/(4*pi) + pi*j^2/N
N >= 2*pi*abs(j)

EQ (K10) [PDF page 14; printed page 13]
M_irr = sqrt(c^4*A_H/(16*pi*G^2)) = m_P*sqrt(N/(4*pi))
M^2 = M_irr^2 + J^2*c^2/(4*G^2*M_irr^2)

EQ (K11) [PDF page 14; printed page 13]
x = 2*pi*j/N = a_K/r_plus
a_star = 2*x/(1+x^2)
s = (1-x^2)/(1+x^2)
abs(x) < 1

EQ (K12) [PDF page 14; printed page 13]
R_Sigma = R_A = ell_P*sqrt(N/pi)
M = M_irr*sqrt(1+x^2)
R_eq = R_A*sqrt(1+x^2)
r_plus = R_A/sqrt(1+x^2)
a_K = x*r_plus

EQ (K13) [PDF page 14; printed page 13]
gamma_K = E_BH/E_Sigma = sqrt(1+x^2)/(2*sqrt(pi*N))
tau_Sigma = 2*pi*R_Sigma/c
gamma_K = sqrt(1+x^2)*t_P/tau_Sigma

EQ (K14) [PDF page 15; printed page 14]
E_BH(N,j) = E_P*mu_K(N,j)
mu_K(N,j) = sqrt(N/(4*pi)+pi*j^2/N)

EQ (K15) [PDF page 15; printed page 14]
(partial E_BH/partial N)_j = E_P*(1-x^2)/(8*pi*mu_K) = k_B*T_H
(partial E_BH/partial j)_N = pi*E_P*j/(mu_K*N) = hbar*Omega_H
The first derivative holds j fixed, not x.

EQ (K16) [PDF page 15; printed page 14]
Omega_H = c*a_K/(r_plus^2+a_K^2)
        = c^3*a_star/(2*G*M*(1+s)) = c*x/R_eq
k_B*T_H = hbar*c*(r_plus-r_g)/(2*pi*(r_plus^2+a_K^2))
        = hbar*c^3*s/(4*pi*G*M*(1+s))
        = hbar*c*(1-x^2)/(4*pi*R_eq)

EQ (K17) [PDF page 15; printed page 14]
dE_BH = k_B*T_H*dN + Omega_H*dJ
E_BH = 2*k_B*T_H*N + 2*Omega_H*J

EQ (K17a) [PDF page 15; printed page 14]
k_B*T_H = (s/2)*E_P*gamma_K
hbar*Omega_H = pi*a_star*E_P*gamma_K
2*k_B*T_H*N/E_BH = s = (1-x^2)/(1+x^2)
2*Omega_H*J/E_BH = x*a_star = 2*x^2/(1+x^2)
s+x*a_star = 1
Definitions: gamma_K=E_BH/(N*E_P), j=J/hbar, x=2*pi*j/N,
a_star=2*x/(1+x^2), and s=(1-x^2)/(1+x^2).
Domain: the source's neutral subextremal Kerr branch, N>2*pi*abs(j),
equivalently N>0 and abs(x)<1. The marginal energies k_B*T_H and
hbar*Omega_H have units J. The ratios and spin factors are dimensionless.
These are fractions of the two Smarr terms in (K17); the irreducible-mass
relation remains (K10).

EQ (K18) [PDF page 15; printed page 14]
tau_H = hbar/(k_B*T_H)
      = 4*pi*R_eq/(c*(1-x^2))
      = 2*tau_Sigma*sqrt(1+x^2)/(1-x^2)

EQ (K19) [PDF page 15; printed page 14]
dot(M) = dot(E_H)/c^2
dot(a_star) = c*dot(J_H)/(G*M^2) - 2*a_star*dot(M)/M
dot(N) = (dot(E_H)-Omega_H*dot(J_H))/(k_B*T_H)
The divided capacity equation is subextremal: T_H > 0.

EQ (K20) [PDF page 16; printed page 15]
dot(R_Sigma) = ell_P^2*dot(N)/(2*pi*R_Sigma)
dot(R_A)/R_A = dot(M)/M - a_star*dot(a_star)/(2*s*(1+s))
dot(R_eq) = (2*G/c^4)*dot(E_H)

EQ (K21) [PDF page 16; printed page 15]
alpha^2 = Sigma_K*Delta/Acal_K
omega = 2*r_g*a_K*r*c/Acal_K
g_(phi phi) = Acal_K*sin(theta)^2/Sigma_K
ds^2 = -alpha^2*c^2*dt^2 + g_(phi phi)*(dphi-omega*dt)^2
       +(Sigma_K/Delta)*dr^2 + Sigma_K*dtheta^2

EQ (K22) [PDF page 16; printed page 15]
Z(r,theta;Omega) = dtau/dt
                = sqrt(alpha^2 - g_(phi phi)*(Omega-omega)^2/c^2)

EQ (K23) [PDF page 16; printed page 15]
v_loc = sqrt(g_(phi phi))*(Omega-omega)/alpha
Z = alpha*sqrt(1-v_loc^2/c^2)

EQ (K24) [PDF page 16; printed page 15]
omega - c*alpha/sqrt(g_(phi phi)) < Omega
Omega < omega + c*alpha/sqrt(g_(phi phi))
This interval is for timelike circular observers away from the axis.

EQ (K25) [PDF page 17; printed page 16]
g^(mu nu)*p_mu*p_nu = -m_b^2*c^2
  <=> -(E_b-omega*L_b)^2/(alpha^2*c^2)
      + gamma^(i j)*p_i*p_j = -m_b^2*c^2

EQ (K26) [PDF page 17; printed page 16]
E_b = omega*L_b + alpha*sqrt(m_b^2*c^4+c^2*gamma^(i j)*p_i*p_j)
E_loc = (E_b-omega*L_b)/alpha > 0

EQ (K27) [PDF page 17; printed page 16]
dx^i/dt = partial E_b/partial p_i
dp_i/dt = -partial E_b/partial x^i

EQ (K28) [PDF page 17; printed page 16]
h_K(P;P_0,Omega_0) = Z(P;Omega_0)/Z_0
Ecal_K_0 = (E_b-Omega_0*L_b)/Z_0 = m_b*c^2*h_K
Z_0 = Z(P_0;Omega_0) > 0; the same constant Omega_0 is used along the route.

EQ (K29) [PDF page 17; printed page 16]
W_K_0 = m_b*c^2*(1-h_K)
a_i = c^2*(partial_i ln(Z))_(Omega_0 fixed), i in {r,theta}
The a_i are covariant spatial acceleration components, not a proper norm.

EQ (K30) [PDF page 18; printed page 17]
h_K = sqrt((1-2*r_g/r)/(1-2*r_g/R_0))
a_proper = G*M/(r^2*sqrt(1-2*r_g/r))
         -> G*M/r^2 as r/r_g -> infinity
Source domain: nonrotating static comparison, r and R_0 greater than 2*r_g.

EQ (K31) [PDF page 18; printed page 17]
nu_0 = dn/dtau_0 = h_K*dn/dtau = h_K*nu_proper

EQ (K32) [PDF page 18; printed page 17]
nabla_mu(rho*u^mu) = 0
nabla_mu(T^(mu nu)) = 0
nabla_mu((*F)^(mu nu)) = 0
F_(mu nu)*u^nu = 0
Here *F is the dual electromagnetic tensor, not multiplication by F.

EQ (K33) [PDF page 18; printed page 17]
T^(mu nu) = (rho+e_int+p+b^2)*u^mu*u^nu
           +(p+b^2/2)*g^(mu nu) - b^mu*b^nu
b^2 = b^mu*b_mu
b^mu*u_mu = 0

EQ (K34) [PDF page 18; printed page 17]
dot(M_0) = -integral rho*u^r*sqrt(-g) dtheta dphi
dot(E_H) =  integral T^r_t*sqrt(-g) dtheta dphi
dot(J_H) = -integral T^r_phi*sqrt(-g) dtheta dphi
Flux signs are inward-positive. Restore physical units and the common time
normalization before using K19. Rest-mass inflow is not dot(E_H)/c^2.

EQ (K35) [PDF page 18; printed page 17]
Fcal_in = max(-rho*u^r,0)*sqrt(-g)
f_eq(delta;r,t)
  = [integral_(phi=0)^(2*pi) integral_(theta=pi/2-delta)^(pi/2+delta)
       Fcal_in dtheta dphi]
    /[integral_(phi=0)^(2*pi) integral_(theta=0)^pi Fcal_in dtheta dphi]
0 <= delta <= pi/2; the denominator is finite and strictly positive.

EQ (K36) [PDF page 19; printed page 18]
q(phi,t) = (1/R_Sigma)*integral_(theta=0)^pi Fcal_in(theta,phi,t) dtheta
integral_(phi=0)^(2*pi) q*R_Sigma dphi = dot(M_in)
dot(M_in) is gross inward rest-mass flux, not the signed net energy flux.

EQ (K37) [PDF page 19; printed page 18]
Fcal_+(theta,phi) = (f(phi)/2)*sin(theta)*(1+epsilon*cos(theta))
Fcal_-(theta,phi) = (f(phi)/2)*sin(theta)*(1-epsilon*cos(theta))
f >= 0; 0 < epsilon < 1
Source context: 0 < integral_0^(2*pi) f(phi) dphi < infinity.
Both have q=f/R_Sigma; their normalized polar means are +/-epsilon/3.

EQ (31) [PDF page 23; printed page 22]
chi(r) = (2*pi*r*M*c/hbar)/(pi*r^2/ell_P^2)
       = 2*G*M/(r*c^2) = r_s/r

EQ (32) [PDF page 23; printed page 22]
c*dt_PG = c*dt + sqrt(r_s/r)*dr/f(r)
ds^2 = -f(r)*c^2*dt^2 + dr^2/f(r) + r^2*dOmega^2
f(r) = 1-r_s/r

EQ (33) [PDF page 24; printed page 23]
h_clk(r;R_0) = dtau(r)/dtau_0 = sqrt((1-r_s/r)/(1-r_s/R_0))
E_0(r) = m*c^2*h_clk(r;R_0)
W_0(r -> R_0) = m*c^2*(1-h_clk(r;R_0))
Source domain: r_s < r <= R_0; clocks span the same stationary-time interval.

EQ (34) [PDF page 24; printed page 23]
F_support_local = (1/h_clk)*(dE_0/dell)
                = m*a_support = G*M*m/(r^2*sqrt(f(r)))

EQ (35) [PDF page 25; printed page 24]
c^2*dtau^2 = -g_(mu nu)*dx^mu*dx^nu

EQ (36) [PDF page 25; printed page 24]
h_clk(r;R_0) = sqrt(eta_p(r)/eta_p(R_0))
            = sqrt((1-r_s/r)/(1-r_s/R_0))
r_s < r <= R_0
Source assignment: u_p(r)=chi(r)=r_s/r; eta_p(r)=1-u_p(r).

EQ (37) [PDF page 25; printed page 24]
Gamma_0*dtau_0 = Gamma_loc*dtau
Gamma_0 = h_clk*Gamma_loc
This converts counts of the same events; it does not supply Gamma_loc.

EQ (38) [PDF page 26; printed page 25]
|Psi_n> = sum_(k=1)^d c_k*|k>
sum_(k=1)^d abs(c_k)^2 = 1

EQ (39) [PDF page 27; printed page 26]
psi_(r+1)(x) = sum_y U_r(x,y)*psi_r(y)
dagger(U_r)*U_r = I

EQ (40) [PDF page 27; printed page 26]
C_keep > C_commit
The costs use the same present state, task, precision, future interval,
and specified cost unit. This is the source's resource-crossover premise.

EQ (41) [PDF page 27; printed page 26]
dagger(P_j) = P_j
P_j*P_k = delta_(j k)*P_j
sum_j P_j = I
p_j = <Psi^-|P_j|Psi^->
|Psi_j^+> = P_j*|Psi^->/sqrt(p_j)
F_(n+1) = Commit(F_n,j)
The selected outcome requires p_j > 0.

EQ (42) [PDF page 29; printed page 28]
K_history + K_live + K_control <= K_max
J_demand <= B_proc
Storage terms are disjoint capacity uses in nats. Rates use the same clock
and update convention; processing demand is not angular momentum J.

EQ (43) [PDF page 29; printed page 28]
K_keep > K_max >= K_commit
The source's feasibility conclusion also assumes two allowed continuations,
all other constraints satisfied, and peak demand including transition space.

EQ (43a) [PDF page 29; printed page 28]
h_(n+1) = h_n || e_(n+1)
prefix_n(h_(n+1)) = h_n
This is ordered history extension. The surrounding smooth-history principle
is preserved; this algebra alone is not a differential smoothness theorem.

EQ (44) [PDF page 31; printed page 30]
(I-P_star)*H_s*P_star = 0  =>  H_s*|star> = E_star*|star>
U_s(t_ret)*|star> = exp(-i*E_star*t_ret/hbar)*|star>
U_s(t_ret)*P_star*dagger(U_s(t_ret)) = P_star
Source definitions: P_star=|star><star|, U_s=exp(-i*H_s*t_ret/hbar);
H_s is self-adjoint and time independent during the closed interval.

EQ (44a) [PDF page 32; printed page 31]
U_s(t_ret)*rho*dagger(U_s(t_ret)) = rho
  for every rho=P_rec*rho*P_rec and every t_ret in [0,T_ret], T_ret>0
  <=> H_s*P_rec = E_s*P_rec
U_s(t_ret)*P_rec = exp(-i*E_s*t_ret/hbar)*P_rec
The source uses a finite-dimensional retained space and the same closed,
self-adjoint, time-independent generator as Eq. (44).

EQ (45) [PDF page 32; printed page 31]
(Tcal tensor id_X)(rho_AX) = rho_AX
  for every admissible joint input rho_AX
Returned and incoming degrees of freedom are identified as stated in the
source. Tcal is the complete process, including commitments. This equation
states its required informational action; it is not a supplied release law.

EQ (46) [PDF page 33; printed page 32]
|Psi_BO> = sum_alpha sqrt(p_alpha)*|alpha>_B tensor |alpha>_O
p_alpha >= 0; sum_alpha p_alpha = 1
rho_O = Tr_B(|Psi_BO><Psi_BO|)
      = sum_alpha p_alpha*|alpha>_O<alpha|_O
s_O = -Tr(rho_O*ln(rho_O)) = -sum_alpha p_alpha*ln(p_alpha)
Schmidt states are orthonormal; 0*ln(0) is interpreted as zero.

EQ (47) [PDF page 33; printed page 32]
s_(early,late) + s_(late,int) >= s_late + s_(early,late,int)
s_(late,int) = 0  =>  s_(early,late,int) = s_early
                 =>  s_(early,late) >= s_early + s_late

EQ (48) [PDF page 35; printed page 34]
omega_(k_vec) = c*sqrt(k^2+mu^2)
E_(0,k_vec) = hbar*omega_(k_vec)/2
(1/V)*sum_(k_vec) -> integral d^3k_vec/(2*pi)^3
k = abs(k_vec); mu = m*c/hbar
Source domain: one free real bosonic degree of freedom, m>=0, large
periodic box. The limit is of the mode measure, not of an individual mode.

EQ (49) [PDF page 35; printed page 34]
epsilon_0(k_c,m)
  = (hbar/2)*integral_(abs(k_vec)<=k_c)
      [d^3k_vec/(2*pi)^3]*omega_(k_vec)
  = (hbar*c/2)*(4*pi/(2*pi)^3)
      *integral_0^k_c k^2*sqrt(k^2+mu^2) dk
  = hbar*c/(4*pi^2)*integral_0^k_c k^2*sqrt(k^2+mu^2) dk
Source cutoff: k_c > 0; k >= 0 in the radial integrals.

EQ (50) [PDF page 36; printed page 35]
epsilon_0(k_c,m) = hbar*c/(32*pi^2)
  *[k_c*sqrt(k_c^2+mu^2)*(2*k_c^2+mu^2)
    -mu^4*ln((k_c+sqrt(k_c^2+mu^2))/mu)]
This displayed form has m>0; the source treats m=0 by its continuous limit.

EQ (51) [PDF page 36; printed page 35]
epsilon_P = E_P/ell_P^3 = hbar*c/ell_P^4 = c^7/(hbar*G^2)
epsilon_0(ell_P^(-1),0) = epsilon_P/(16*pi^2)

EQ (52) [PDF page 36; printed page 35]
p_0(k_c,m) = hbar*c/(12*pi^2)
             *integral_0^k_c k^4/sqrt(k^2+mu^2) dk
           = hbar*c*k_c^3*sqrt(k_c^2+mu^2)/(12*pi^2)
             -epsilon_0(k_c,m)
p_0(k_c,0) = hbar*c*k_c^4/(48*pi^2) = epsilon_0(k_c,0)/3

EQ (53) [PDF page 36; printed page 35]
T_(a b)^vac = -epsilon_vac*g_(a b)
T_(a_hat b_hat)^vac
  = diag(epsilon_vac,-epsilon_vac,-epsilon_vac,-epsilon_vac)
p_vac = -epsilon_vac

EQ (54) [PDF page 36; printed page 35]
G_(a b)+Lambda_bare*g_(a b)
  = kappa*(T_(a b)^excit-epsilon_vac*g_(a b))
G_(a b)+Lambda_eff*g_(a b) = kappa*T_(a b)^excit
Lambda_eff = Lambda_bare+kappa*epsilon_vac
epsilon_Lambda = Lambda_eff/kappa = Lambda_eff*c^4/(8*pi*G)

EQ (55) [PDF page 37; printed page 36]
epsilon_P/epsilon_Lambda_ref = 8*pi/(Lambda_ref*ell_P^2)
epsilon_0(ell_P^(-1),0)/epsilon_Lambda_ref
  = 1/(2*pi*Lambda_ref*ell_P^2)

EQ (56) [PDF page 37; printed page 36]
H_Lambda^2 = Lambda_eff*c^2/3
R_Lambda = c/H_Lambda = sqrt(3/Lambda_eff)
Lambda_eff > 0

EQ (57) [PDF page 37; printed page 36]
ds^2 = -f_Lambda(r)*c^2*dt^2 + dr^2/f_Lambda(r) + r^2*dOmega^2
f_Lambda(r) = 1-r^2/R_Lambda^2

EQ (58) [PDF page 37; printed page 36]
E_MS(r) = (c^4*r/(2*G))*(1-f_Lambda(r))
        = c^4*r^3/(2*G*R_Lambda^2)
E_Lambda = E_MS(R_Lambda) = c^4*R_Lambda/(2*G)

EQ (59) [PDF page 38; printed page 37]
A_Sigma_Lambda = pi*R_Lambda^2
A_Lambda = 4*A_Sigma_Lambda
N_Lambda = A_Sigma_Lambda/ell_P^2 = 3*pi/(Lambda_eff*ell_P^2)
S_Lambda = k_B*N_Lambda
The quarter-area comparison does not assign a physical flat disk to the
universe. That assignment in the source concerns the saturated black hole.

EQ (60) [PDF page 38; printed page 37]
epsilon_Lambda = E_Lambda/V_Lambda
  = (c^4*R_Lambda/(2*G))*(3/(4*pi*R_Lambda^3))
  = 3*c^4/(8*pi*G*R_Lambda^2) = Lambda_eff*c^4/(8*pi*G)

EQ (61) [PDF page 38; printed page 37]
epsilon_Lambda/epsilon_P
  = 3*c^4*ell_P^4/(8*pi*G*R_Lambda^2*hbar*c)
  = 3*ell_P^2/(8*pi*R_Lambda^2) = 3/(8*N_Lambda)
Lambda_eff = 3*pi/(N_Lambda*ell_P^2)
N_Lambda, equivalently Lambda_eff, is supplied cosmological state data.

EQ (61a) [PDF page 38; printed page 37]
epsilon_bar_BH/epsilon_P = 3*ell_P^2/(8*pi*R^2) = 3/(8*N)
Definitions: epsilon_bar_BH=E_BH/(4*pi*R^3/3),
epsilon_P=E_P/ell_P^3, and N=pi*R^2/ell_P^2.
Domain: the nonrotating black-hole comparison with R>0; the densities
have units J/m^3 and their ratio is dimensionless. The denominator
4*pi*R^3/3 is the spherical areal-radius comparison volume. Equation (61)
instead uses the specified de Sitter state and its capacity N_Lambda.

EQ (62) [PDF page 38; printed page 37]
ds^2 = -f(r)*c^2*dt^2 + f(r)^(-1)*dr^2 + r^2*dOmega^2
f(r) = 1-r_s/r-Lambda_eff*r^2/3
r_s = 2*G*M/c^2
M > 0; Lambda_eff > 0

EQ (63) [PDF page 38; printed page 37]
f(r_star) > 0
  <=> 0 < (9/4)*Lambda_eff*r_s^2
         = 9*G^2*M^2*Lambda_eff/c^4 < 1
r_star = (3*r_s/(2*Lambda_eff))^(1/3) in the source context.

EQ (64) [PDF page 39; printed page 38]
r_s = R_b*(1-Lambda_eff*R_b^2/3)
M = (c^2*R_b/(2*G))*(1-Lambda_eff*R_b^2/3)

EQ (65) [PDF page 39; printed page 38]
delta = a_Lambda*(1+delta)^3
delta = a_Lambda + 3*a_Lambda^2 + O(a_Lambda^3)
Source definitions: a_Lambda=Lambda_eff*r_s^2/3; R_b=r_s*(1+delta).

EQ (66) [PDF page 39; printed page 38]
dtau(r)/dtau(r_0) = sqrt(f(r)/f(r_0))
f(r) > 0; f(r_0) > 0
Clocks span equal intervals of the stationary coordinate t.

EQ (67) [PDF page 39; printed page 38]
E_MS(r) = (c^4*r/(2*G))*(1-f(r))
        = M*c^2 + Lambda_eff*c^4*r^3/(6*G)
        = M*c^2 + epsilon_Lambda*(4*pi*r^3/3)

END OF NUMBERED EQUATION KEY
This companion covers the 112 numbered groups in the identified source.
Future manuscript versions require a new source fingerprint and a checked
equation-to-key crosswalk.