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Conditional area entropy coefficient with free offset

Physical scope

Conditional deduction on the selected connected, strictly subextremal, zero-magnetic-flux stationary Kerr–Newman family, using the exact SI family differential, the explicitly adopted reversible heat and entropy state function, and the pointwise reduced exponential-current reference-temperature field obtained by matching each branch parameter to that member's same-normalized geometric surface gravity. It does not assert a general Hawking or KMS theorem, a dynamical-process first law, an empirical temperature, a microstate count, or an absolute entropy zero.

Statement

Use the physical Kerr–Newman family parameterized by the connected strict subextremal domain $P : = {(m, a, q) \in R^{3} : m > 0, m^{2} > a^{2} + q^{2}},$ with its current SI parameter map and selected horizon conventions from Physical KN family work and constancy. Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ), the mathematical premise used by the current Kerr–Newman and reduced spectral supplier proofs. Let $S_{H} : P \to R$ be the $C^{1}$ entropy state function adopted in Restricted reversible horizon entropy identification, measured in $J K^{- 1}$ . Assume that postulate's reversible-heat identity $δ Q_{r e v} = d E_{p a r} - Ω_{S I} d J_{S I} - Φ_{S I} d Q_{e} = T_{K} d S_{H}$ on $P$ . Also assume its pointwise temperature identification: for each $x = (m, a, q) \in P$ , the reduced branch parameter is held fixed at $κ = κ_{g e o m} (x) = s (x) / D (x) > 0$ , and the smooth reference-temperature field is $T_{K} : P \to (0, \infty)$ with $T_{K} (x) = T_{r e d} (x) = ℏ c κ_{g e o m} (x) / (2 π k_{B})$ . Then there is a constant $s_{*} \in R$ , with units $J K^{- 1}$ , such that throughout $P$ $S_{H} = \frac{k_{B}}{4 ℓ_{P}^{2}} A_{H} + s_{*} = \frac{k_{B} c^{3}}{4 G_{N} ℏ} A_{H} + s_{*}, ℓ_{P}^{2} : = \frac{G_{N} ℏ}{c^{3}} .$ 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.

Facts & Assumptions

Given: The connected strict subextremal family, its exact physical and mathematical first-law interfaces, and the additional thermodynamic and reduced-temperature assumptions stated below.

[F1]

Thermodynamics-GR fields, currents, reference normalization and units: $G_{N}, c, ℏ, k_{B} > 0$ have the stated SI units, and on a supplied null generator $κ_{a c c} = c^{2} κ_{g e o m}$ , with $κ_{g e o m}$ in $m^{- 1}$ and $κ_{a c c}$ in $m s^{- 2}$ . The entropy formula defines $ℓ_{P}^{2} : = G_{N} ℏ / c^{3}$ explicitly.

[F2]

Physical KN family work and constancy: on the selected strict subextremal, zero-magnetic-flux physical family, with $G_{N}, c, μ_{0}$ fixed, the exact SI identity is $d E_{p a r} = \frac{κ_{a c c} c^{2}}{8 π G_{N}} d A_{H} + Ω_{S I} d J_{S I} + Φ_{S I} d Q_{e},$ where $E_{p a r}$ is in joules, $A_{H}$ in $m^{2}$ , $J_{S I}$ in $k g m^{2} s^{- 1}$ , $Q_{e}$ in coulombs, $Ω_{S I}$ in $s^{- 1}$ , and $Φ_{S I}$ in volts; $κ_{a c c} > 0$ on this strict domain.

[F3]

Verified Kerr-Newman family first law and Smarr identity: on $P$ , with $A = 4 π (r_{+}^{2} + a^{2})$ , $j = m a$ , $κ_{K N} = \sqrt{m^{2} - a^{2} - q^{2}} / (r_{+}^{2} + a^{2})$ , $Ω_{K N} = a / (r_{+}^{2} + a^{2})$ , and $Φ_{K N} = q r_{+} / (r_{+}^{2} + a^{2})$ , the mathematical family differential is $d m = \frac{κ_{K N}}{8 π} d A + Ω_{K N} d j + Φ_{K N} d q .$

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence $(X_{n})_{n \in N}$ of nonempty sets has a choice function $f$ with $f (n) \in X_{n}$ for every $n$ . The full current spectrum supplier explicitly assumes this axiom.

[L1]

Reduced quantum horizon temperature calibration: for each reduced exponential-current branch with $E > 0$ , constant parameter $κ > 0$ , positive $γ_{r e d}$ , and the adopted two-state rates, the effective reference temperature defined by the stationary excited-to-ground ratio is $T_{r e d} = ℏ c κ / (2 π k_{B})$ . At each $x \in P$ , the postulate uses the branch instance whose Killing flow and reference normalization match $χ_{x}$ , with branch parameter fixed at $κ = κ_{g e o m} (x)$ ; it yields the pointwise field $T_{K} (x) = ℏ c κ_{g e o m} (x) / (2 π k_{B})$ . The supplier's Statement explicitly assumes Countable Choice, retained here in [A2], as required by its cited spectrum supplier. This use retains the selected reduced model, positive gap, positive rate, same-generator identification and reference normalization; it does not use the supplier's optional stationary-bath clause.

[F4]

Classical first-law product leaves entropy-temperature calibration free: a connected mathematical family first-law product does not select a unique positive temperature scale or absolute entropy offset; every stated offset leaves the differential unchanged.

[F5]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : for totally differentiable maps between Euclidean spaces, the derivative of a composite is the composite of the derivatives.

[F6]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : if a real function is continuous on a closed interval and differentiable in its interior, its endpoint difference equals its derivative at an interior point times the interval length.

[A1]

Restricted reversible horizon entropy identification: on $P$ , the one-form $δ Q_{r e v} = d E_{p a r} - Ω_{S I} d J_{S I} - Φ_{S I} d Q_{e}$ satisfies $δ Q_{r e v} = T_{K} d S_{H}$ , where $T_{K} : P \to (0, \infty)$ and, at every $x \in P$ , the reduced branch parameter is held fixed at $κ = κ_{g e o m} (x)$ and $T_{K} (x) = T_{r e d} (x) = ℏ c κ_{g e o m} (x) / (2 π k_{B})$ .

Proof

1.1givenA2F2A1algebra

Under the mathematical premise [A2], by [F2], the work-subtracted one-form $δ Q_{r e v}$ defined in [A1] is exactly $\frac{κ_{a c c} c^{2}}{8 π G_{N}} d A_{H}$ on $P$ : the angular-momentum and electric-charge work terms cancel with the same terms in its definition. The physical postulate [A1] identifies this one-form with the product of the family field $T_{K}$ and $d S_{H}$ . Hence $T_{K} d S_{H} = \frac{κ_{a c c} c^{2}}{8 π G_{N}} d A_{H} .$ This is conditional on the selected stationary-family and thermodynamic assumptions.

1.2givenA2F1L1A1algebra

For each $x \in P$ , apply [L1] to the reduced branch whose constant parameter is $κ = κ_{g e o m} (x)$ , as specified in [A1]. Thus $T_{K} (x) = T_{r e d} (x) = ℏ c κ_{g e o m} (x) / (2 π k_{B})$ . By [F1], $κ_{a c c} (x) = c^{2} κ_{g e o m} (x) > 0$ , so $T_{K} (x) = ℏ κ_{a c c} (x) / (2 π c k_{B}) > 0$ at every family point.

2.1step 1.1step 1.2F1algebra

At each $x \in P$ , evaluate the one-form equality in step 1.1 on an arbitrary tangent vector and divide by the positive scalar $T_{K} (x)$ from step 1.2. Using [F1] gives the one-form identity $d S_{H} = \frac{c^{2} κ_{a c c}}{8 π G_{N}} \frac{2 π c k_{B}}{ℏ κ_{a c c}} d A_{H} = \frac{k_{B} c^{3}}{4 G_{N} ℏ} d A_{H} = \frac{k_{B}}{4 ℓ_{P}^{2}} d A_{H} .$ The coefficient has units $J K^{- 1} m^{- 2}$ , so the right-hand side has entropy units $J K^{- 1}$ .

3.1F3step 2.1F5F6construct

For arbitrary $x_{0} = (m_{0}, a_{0}, q_{0})$ and $x_{1} = (m_{1}, a_{1}, q_{1})$ in $P$ , put $v_{i} = (a_{i}, q_{i})$ with its Euclidean norm and let $γ (t) = (1 - t) x_{0} + t x_{1}$ for $0 \leq t \leq 1$ . Since $m_{i} > ∥ v_{i} ∥_{2}$ , for every such $t$ , $m (t) = (1 - t) m_{0} + t m_{1} > (1 - t) ∥ v_{0} ∥_{2} + t ∥ v_{1} ∥_{2} \geq ∥ (1 - t) v_{0} + t v_{1} ∥_{2} = \sqrt{a (t)^{2} + q (t)^{2}} .$ Thus the segment stays in $P$ . Set $R : = S_{H} - \frac{k_{B}}{4 ℓ_{P}^{2}} A_{H}$ ; step 2.1 gives $d R = 0$ . The path $γ$ is smooth; since $P$ is open, it maps a neighborhood of every $t \in (0, 1)$ into $P$ . The chain rule [F5] gives $(R \circ γ)^{'} (t) = d R_{γ (t)} (x_{1} - x_{0}) = 0$ . Also $R \circ γ$ is continuous on $[0, 1]$ and differentiable on $(0, 1)$ , so [F6] yields $R (x_{1}) = R (x_{0})$ .

4.1A1F4step 3.1∎

Since $x_{0}, x_{1}$ were arbitrary, $R$ has one constant value $s_{*}$ on the connected family $P$ , proving the displayed formula. No reference value of $S_{H}$ was adopted; adding any constant to it leaves [A1] unchanged, in accord with the mathematical calibration freedom [F4]. This fixes the conditional area coefficient but not an entropy zero.

Remarks

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.

Conditional area entropy coefficient with free offset

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources