Restricted reversible horizon entropy bridge

Use the connected strict subextremal parameter domain $\mathcal{P}:=\{(m,a,q)\in {\mathbb{R}}^3:m>0,m^2>a^2+q^2\},$ with its physical Kerr–Newman family and SI conversion as in Physical KN family work and constancy. Fix positive constants $G_N$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $\mathrm{\hslash }$ in $\mathrm{J}\mathrm{s}$, and $k_B$ in $\mathrm{J}{\mathrm{K}}^{-1}$. Write $E_{\mathrm{p}\mathrm{a}\mathrm{r}}:\mathcal{P}\to \mathrm{J}$, $A_H:\mathcal{P}\to {\mathrm{m}}^2$, $J_{\mathrm{S}\mathrm{I}}:\mathcal{P}\to \mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, $Q_e:\mathcal{P}\to \mathrm{C}$, ${\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}:\mathcal{P}\to {\mathrm{s}}^{-1}$, and ${\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}:\mathcal{P}\to \mathrm{V}$ for the smooth parameter functions supplied by Physical KN family work and constancy.

Define the work-subtracted reversible heat one-form on $\mathcal{P}$ by $\delta Q_{\mathrm{r}\mathrm{e}\mathrm{v}}:=d{E}_{\mathrm{p}\mathrm{a}\mathrm{r}}-{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}d{J}_{\mathrm{S}\mathrm{I}}-{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}d{Q}_e\mathrm{.}$ Here a one-form assigns a linear functional to each tangent direction in the family parameter domain; each term has units of joules. Let $S_H:\mathcal{P}\to \mathbb{R}$ be a $C^1$ function with values measured in $\mathrm{J}{\mathrm{K}}^{-1}$. It is a state function here in the explicit sense that its value is assigned to each family parameter point and does not depend on a path used to reach that point.

Assume the mathematical premise ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) required by the selected spectral supplier. This is not a physical postulate.

For $x=(m,a,q)\in \mathcal{P}$, define $s(x):=\sqrt{m^2-a^2-q^2},r_{+}(x):=m+s(x),D(x):=r_{+}(x{)}^2+a^2\mathrm{.}$ On the selected smooth ingoing Kerr–Newman spacetime belonging to the family member $x$, write ${\chi }_x$ for the selected horizon generator with the length-flow parameter and reference normalization of Physical KN family work and constancy. Let ${\mathrm{\nabla }}^{(x)}$ denote the Levi-Civita connection of that member's metric. By that supplier and the conditional geometric definition in Thermodynamics-GR fields, currents, reference normalization and units, ${\mathrm{\nabla }}_{{\chi }_x}^{(x)}{\chi }_x={\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x){\chi }_x,{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}:\mathcal{P}\to (0,\mathrm{\infty }),{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x):=\frac{s(x)}{D(x)},$ where the connection is taken on the spacetime indexed by the family point, not on the parameter domain $\mathcal{P}$. This is a smooth positive family map with values in ${\mathrm{m}}^{-1}$. Put ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}(x):=c^2{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x)$, in $\mathrm{m}{\mathrm{s}}^{-2}$, as in the SI family theorem.

For each fixed $x\in \mathcal{P}$, use a reduced exponential-current branch instance of Reduced quantum horizon temperature calibration whose Killing flow field is identified with ${\chi }_x$ and whose length-flow normalization is the family member's. The reference event is fixed within that instance as required by the PHS02 branch model. The branch instance satisfies the supplier's positive energy-gap, positive ${\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}$, and two-state-rate hypotheses; its scalar inverse-length parameter is held constant at $\kappa ={\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x)$. Let $T_{\mathrm{r}\mathrm{e}\mathrm{d}}(x)$ denote that instance's effective reference temperature. The supplier gives $T_{\mathrm{r}\mathrm{e}\mathrm{d}}(x)=\frac{\mathrm{\hslash }c{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x)}{2\pi k_B}\mathrm{.}$ Adopt the additional reference-temperature identification $T_K:\mathcal{P}\to (0,\mathrm{\infty })$, $T_K(x):=T_{\mathrm{r}\mathrm{e}\mathrm{d}}(x)$, so $T_K(x)=\frac{\mathrm{\hslash }c{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x)}{2\pi k_B}=\frac{\mathrm{\hslash }{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}(x)}{2\pi c{k}_B}\mathrm{.}$ Thus $T_K$ is a smooth family map with values in kelvins. At a fixed family point $x$, the unparameterized symbol $T_K$ denotes the value $T_K(x)$. It names only the selected reference scale in the reversible relation; it is not a local proper-temperature field and does not assert a stationary bath, a KMS state, or an observed detector reading.

Adopt the reversible thermodynamic identification $\delta Q_{\mathrm{r}\mathrm{e}\mathrm{v}}=T_{K}d{S}_H$ as an equality of one-forms on $\mathcal{P}$: at each $x\in \mathcal{P}$ and $v\in T_x\mathcal{P}$, it means $\delta Q_{\mathrm{r}\mathrm{e}\mathrm{v}}(x)[v]=T_K(x)d{S}_H(x)[v]$. This physical identification is an additional assumption. The geometric family identity and the reduced effective-temperature model do not by themselves assert it. No global entropy zero or identification with a quantum-gravity microstate count or an unrenormalized outside-state entropy is adopted.

Use the physical Kerr–Newman family parameterized by the connected strict subextremal domain $\mathcal{P}:=\{(m,a,q)\in {\mathbb{R}}^3:m>0,m^2>a^2+q^2\},$ with its current SI parameter map and selected horizon conventions from Physical KN family work and constancy. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), the mathematical premise used by the current Kerr–Newman and reduced spectral supplier proofs. Let $S_H:\mathcal{P}\to \mathbb{R}$ be the $C^1$ entropy state function adopted in Restricted reversible horizon entropy identification, measured in $\mathrm{J}{\mathrm{K}}^{-1}$. Assume that postulate's reversible-heat identity $\delta Q_{\mathrm{r}\mathrm{e}\mathrm{v}}=d{E}_{\mathrm{p}\mathrm{a}\mathrm{r}}-{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}d{J}_{\mathrm{S}\mathrm{I}}-{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}d{Q}_e=T_{K}d{S}_H$ on $\mathcal{P}$. Also assume its pointwise temperature identification: for each $x=(m,a,q)\in \mathcal{P}$, the reduced branch parameter is held fixed at $\kappa ={\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x)=s(x)\mathrm{/}D(x)>0$, and the smooth reference-temperature field is $T_K:\mathcal{P}\to (0,\mathrm{\infty })$ with $T_K(x)=T_{\mathrm{r}\mathrm{e}\mathrm{d}}(x)=\mathrm{\hslash }c{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(x)\mathrm{/}(2\pi k_B)$. Then there is a constant $s_{\ast }\in \mathbb{R}$, with units $\mathrm{J}{\mathrm{K}}^{-1}$, such that throughout $\mathcal{P}$ $S_H=\frac{k_B}{4{\mathrm{\ell }}_P^2}{A}_H+s_{\ast }=\frac{k_B{c}^3}{4G_N\mathrm{\hslash }}{A}_H+s_{\ast },{\mathrm{\ell }}_P^2:=\frac{G_N\mathrm{\hslash }}{c^3}\mathrm{.}$ The deduction determines the area coefficient within this adopted model and leaves the entropy zero unselected. It does not identify $S_H$ with a statistical microstate count or an outside-state entropy.

The reduced-temperature result is used only at its specified effective-model interface, including its explicit Countable Choice premise. The separate reversible heat identification is essential: the theorem makes no claim that the reduced model describes every black hole or that the postulated $S_H$ is a microscopic state count.