Em03-A A Fixed-Background Hawking Emission Theorem An explicit horizon-state calculation, Schwarzschild scattering, and a non-transfer audit

We give an audited formulation of the standard free-scalar emission law on a fixed Schwarzschild background. The geometry, time normalization, field, state, observable, and scattering problem are specified separately. Surface gravity is calculated from the metric. A vacuum defined by an affine horizon coordinate is pulled back to Killing time, and its distributional Fourier transform is evaluated without inserting a temperature or an occupation table. Schwarzschild transmission probabilities are constructed from the radial wave equation. For every normalized positive-frequency wavepacket supported in a compact band away from zero, the resulting number expectation is the greybody-weighted Bose occupation integral. A trace-class time–frequency window gives the corresponding number-rate normalization.

The boundary calculation and its regulator removal are proved here. The bulk existence and Hadamard property of the specified Unruh state are a precisely declared external input, not a result attributed to an earlier manuscript in the series. We also prove a compact-test late-time estimate under explicit peeling and state-regularity hypotheses. This estimate is not silently promoted to a collapse wavepacket theorem. The supplied manuscript record is audited with its retrieval limitations retained. No quantum-gravity correction, backreaction, evaporation law, island construction, or reopening of an inherited admission gate is claimed.

Hawking's original result is a semiclassical field-theory calculation, not a theorem of nonperturbative quantum gravity Hawking1975. The present paper is an explicit identification and calculation of one fixed-background observable. It is not a novelty claim for the Hawking effect.

The mathematical release is:

The temperature is not supplied as calibration data. It emerges from the affine-to-Killing coordinate relation after surface gravity has been computed. The greybody factor is not supplied as a numerical table. It is defined constructively by the same metric's radial scattering equation.

The choice of state is nevertheless essential. This paper does not claim that a metric and a classical wave equation uniquely determine a quantum emission law.

The internal record supplied with the problem consists of excerpts and summaries associated with corpus identifier ca99b12b, retrieval identifier retrieval-3 SuppliedRecord. It explicitly describes a local retrieval rather than a complete scan of the series.

Accordingly, the documentary conclusion below is restricted to the supplied record. Statements about Em02-E, En02-C, En02-D, and XVI-P that are available only as summaries are identified as such. Their full manuscripts have not been independently checked here. An internal label is a source locator, not a theorem proved by this paper.

In particular, “not constructed” and “not admitted” are not replaced by “mathematically nonexistent.” Similarly, “XV-B remains closed” means that its downstream admission condition remains undischarged; it does not mean that a manuscript named XV-B is absent.

We use units

The background is the right Schwarzschild exterior

with metric

The Killing vector is $\chi ={\partial }_t$, normalized by $\mathrm{g}(\chi ,\chi )\to -1$ as $r\to \infty$.

The field is a real, massless, minimally coupled scalar:

Schwarzschild is Ricci flat, so the massless conformally coupled and minimally coupled wave operators coincide on this background.

Let

The additive constant in $x$ is immaterial. We use real orthonormal spherical harmonics

This avoids imposing an independent reality convention on complex $m$-modes. Every result below is for one fixed $(\ell ,\alpha )$.

We prescribe independent incoming boundary states:

1. on the complete past Kruskal horizon, the vacuum defined by its affine coordinate $U$;

2. on past null infinity ${\mathcal{I}}^{-}$, the vacuum defined by positive Killing frequency in $v$.

The product of these states is the Unruh boundary prescription Unruh1976. On the exterior part of the past horizon,

Here $\kappa$ is calculated from (eq:metric) in Lemma lem:geometry; its value is not an independent parameter.

For the massless scalar on Schwarzschild, the past-horizon and past-null-infinity characteristic maps give a symplectic bulk-to-boundary embedding. Pulling back the product of the affine horizon vacuum and the Killing-frequency vacuum on ${\mathcal{I}}^{-}$ gives a quasifree Unruh state. This state is Hadamard on the exterior and on its stated extension through the future horizon.

The external input is the construction and regularity result of Dappiaggi, Moretti, and Pinamonti, specifically their bulk-to-boundary results and Theorem 4.1 DMP2011. We do not reproduce their global microlocal proof. Thus the calculation below is self-contained at the radiation-boundary level, while its identification with a regular bulk state uses this declared theorem.

The normalized horizon field for a partial wave is $r_{\mathrm{h}}\varphi$; the normalized field at null infinity is $r\varphi$. Both have the one-dimensional canonical normalization used below.

The input is positive affine frequency on the horizon and positive Killing frequency on ${\mathcal{I}}^{-}$. No Bose occupation function is included in this input. Its appearance is a conclusion of the Fourier calculation in Section sec:thermal. Choosing a state is necessary physical data, not a claim that thermality follows from the classical metric alone.

For a compact frequency interval

let $h\in {C^{\infty }}_c((0,\infty ))$ satisfy

Define the outgoing packet annihilator and number observable by

Their construction and finite expectation are proved below.

For the geometry, field, state, and normalization of Section sec:scope, the following statements hold.

1. The normalized Schwarzschild Killing horizon has

   $$\kappa =\frac{1}{4GM},{\beta }_{\mathrm{H}}=\frac{2\pi }{\kappa }=8\pi GM.$$

2. The incoming horizon current, expressed in Killing coordinate $u$, has two-point distribution

   $$C_{\kappa }(s)=-\frac{{\kappa }^2}{16\pi }{\mathrm{csch}}^2\left(\frac{\kappa (s-\mathrm{i}0)}{2}\right),s=u-u^{\prime },$$

   whose Fourier transform is

   $${\hat{C}}_{\kappa }(\nu )=\frac{\nu }{2(1-e^{-{\beta }_{\mathrm{H}}\nu })},{\hat{C}}_{\kappa }(0)=\frac{1}{2{\beta }_{\mathrm{H}}}.$$

3. The radial scalar equation constructs a unitary two-channel scattering matrix. Its horizon-to-infinity transmission coefficient ${\tau }_{\ell }(\omega )$ defines

   $${\Gamma }_{\ell }(\omega )=|{\tau }_{\ell }(\omega ){|}^2,0<{\Gamma }_{\ell }(\omega )\le 1(\omega >0).$$

4. The outgoing number covariance is the distributional identity

   $${\varpi }_{\mathrm{U}}(a_{\mathrm{out}}(\omega {)}^{\ast }{a}_{\mathrm{out}}({\omega }^{\prime }))=\frac{{\Gamma }_{\ell }(\omega )}{e^{{\beta }_{\mathrm{H}}\omega }-1}\delta (\omega -{\omega }^{\prime }),$$

   and consequently

   $$\boxed{{\varpi }_{\mathrm{U}}(N_h)={{\int }_0}^{\infty }|h(\omega ){|}^2\frac{{\Gamma }_{\ell }(\omega )}{e^{8\pi GM\omega }-1}\mathrm{d}\omega .}$$

5. With the time–frequency counting convention of Proposition prop:rate, the number rate per angular mode is

   $$\boxed{f_{\ell }(\omega ;M)=\frac{1}{2\pi }\frac{{\Gamma }_{\ell }(\omega )}{e^{8\pi GM\omega }-1}.}$$

   This formula is derived from the boundary state and the radial scattering equation, not assumed as calibration data.

The proof is assembled at the end of Section sec:observables. The result is exact in the declared stationary free-field model. No late-time approximation is used in (eq:packetlaw). Section sec:peeling treats a different, explicitly conditional late-time limit.

There is no normalized one-particle vector concentrated at one continuum frequency, and $\delta (0)$ is not a number expectation. Equation (eq:numberdensity) is a covariance density; equation (eq:packetlaw) is a finite observable expectation. The frequently written expression

must be understood with a packet, box, or spectral-density normalization specified.

For (eq:metric), surface gravity is

The coordinate $U$ in (eq:U) is affine on the past Kruskal horizon, and its exterior restriction satisfies

Proof. In ingoing Eddington–Finkelstein coordinates,

The relevant connection coefficients are

At $r=r_{\mathrm{h}}$, therefore,

Since $F^{\prime }(r)=r_{\mathrm{h}}/r^2$, the stated value follows. The normalization of $\chi$ at infinity removes a rescaling ambiguity.

Set

Their product is

and $2\kappa =1/r_{\mathrm{h}}$. Hence the radial metric extends across the horizons in the form

with $A$ smooth and nonzero at $UV=0$. On the past horizon $V=0$, the coefficient is constant along $U$, so the horizon generators have vanishing affine acceleration in coordinate $U$. Finally,

which gives $\chi (U)=-\kappa U$.

Writing

reduces ${\square }_{\mathrm{g}}\varphi =0$ to

where

The potential is nonnegative, smooth, and integrable:

Proof. In Schwarzschild coordinates,

Insert $r^{-1}\psi Y_{\ell \alpha }$, use ${\Delta }_{S^2}{Y}_{\ell \alpha }=-\ell (\ell +1)Y_{\ell \alpha }$, and use ${\partial }_x=F{\partial }_r$. Multiplication by $rF$ gives (eq:radialwave).

All terms in (eq:potential) are nonnegative. Since $\mathrm{d}x=F^{-1}\mathrm{d}r$,

which evaluates to (eq:potentialL1).

For every fixed $\ell$ and $\omega >0$, the radial equation

has uniquely normalized Jost solutions $f_{+}$ and $f_{-}$ with

including the corresponding derivative asymptotics.

They construct a unitary scattering matrix $S_{\ell }(\omega )$. The horizon-to-infinity coefficient is

where

In particular,

All these coefficients are continuous on compact subsets of $(0,\infty )$.

Proof. The outgoing right Jost solution is constructed by the Volterra equation

Successive iteration converges absolutely. Indeed, the $n$-th iterated correction is bounded by

and consequently

The same Volterra estimate applied to a difference proves uniqueness. Differentiation of (eq:Volterra) gives (eq:ODE) and the derivative asymptotics at $+\infty$.

Expanding the sine in (eq:Volterra) gives, at the other end,

The omitted tails tend to zero because $V_{\ell }{f}_{+}\in L^1$. Differentiating the same expressions gives the left derivative asymptotics.

For every solution of (eq:ODE), the current

is constant. For $f_{+}$, its right-end value is $\omega$, whereas its left-end value is

Thus $a_{\ell }\ne 0$, and the horizon-incident solution $q_{\mathrm{H}}=f_{+}/a_{\ell }$ has asymptotics

This proves the transmission formula and its bounds.

The left Jost solution is obtained from the analogous Volterra equation integrated from $-\infty$. Its right incoming coefficient is likewise nonzero by current conservation, so it constructs the solution $q_{\mathrm{I}}$ incident from infinity.

For arbitrary incoming amplitudes $A_{\mathrm{H}},A_{\mathrm{I}}$, the corresponding combination of $q_{\mathrm{H}}$ and $q_{\mathrm{I}}$ has outgoing amplitudes

Current conservation gives

for every pair of incoming amplitudes. Therefore $S_{\ell }^{\ast }{S}_{\ell }=I$, and the square matrix $S_{\ell }$ is unitary.

On a compact positive-frequency interval, the Volterra series have uniform majorants. Their terms and the coefficient integrals depend continuously on $\omega$. Dominated convergence proves the last assertion.

The potential (eq:potential) has not been discarded. The transmission probability in (eq:gamma) is exact for this ODE. This is the separation between the horizon occupation factor and exterior scattering. Numerical approximations to the ODE may be used later, but they require their own error bound.

For a normalized radiation field $\psi (s)$, write $j(s)={\partial }_s\psi (s)$. The positive-frequency convention is

This yields

The vacuum current covariance is the tempered distribution

In particular, for any complex compact test $g$,

This formula also proves finite covariance on compact smooth tests.

We work with currents rather than an unsmeared massless scalar zero mode. The number observables used below have frequency support bounded away from zero.

Let $j_U$ have covariance (eq:vacuumcurrent) in the affine coordinate $U$. On $U<0$, set

This defines a positive quasifree current state with the canonical commutator (eq:currentCCR). Its covariance is (eq:thermalcurrent). Moreover,

Proof. For $g\in {C^{\infty }}_c(\mathbb{R})$, change variables from $u$ to $U$:

The transformed test is smooth and compactly supported in $U<0$; its extension by zero is a valid test on the affine line. Positivity therefore follows from (eq:vacuumpositive). The antisymmetric covariance is preserved by this change of variable, which proves the current commutator.

The covariance is the pullback distribution

The boundary-value prescription is well defined because $U$ is a smooth strictly increasing coordinate.

Away from $u=u^{\prime }$, elementary substitution gives

At the diagonal, the pullback keeps the same positive-frequency boundary orientation. Equivalently, factor $U(u)-U(u^{\prime })=(u-u^{\prime })D(u,u^{\prime })$ with $D>0$, and use the boundary-value distribution in the variable $u-u^{\prime }$. This gives exactly (eq:thermalcurrent), including its distributional extension.

Finally,

at $z=0$. The only real singularity of $C_{\kappa }$ is its diagonal singularity, which coincides with that of $C_0$. Their difference therefore extends smoothly, with the value in (eq:subtraction).

For the boundary current calculation, the singular reference distribution is explicitly $C_0$, and (eq:subtraction) proves the required local subtraction. For a local bulk observable, the appropriate object would instead be the four-dimensional Hadamard parametrix associated with $\mathrm{g}$. The imported Hadamard theorem guarantees its applicability to the selected bulk state; this paper does not identify the one-dimensional boundary subtraction with a bulk stress-tensor renormalization prescription. These distinctions are standard in curved-spacetime QFT HollandsWald2015.

Our Fourier convention is

The distribution (eq:thermalcurrent) has Fourier transform (eq:thermalfourier). It has the spectral representation

understood in ${\mathcal{S}}^{\prime }(\mathbb{R})$. In particular,

Proof. Let

It is analytic in the strip $-{\beta }_{\mathrm{H}}<\mathrm{Im}z<0$. Its poles are $z=\mathrm{i}n{\beta }_{\mathrm{H}}$, $n\in \mathbb{Z}$, and its principal part at every pole is

For $0<\eta <{\beta }_{\mathrm{H}}$, consider

The integrand decays exponentially as $|\mathrm{Re}z|\to \infty$.

For $\nu >0$, close the contour upwards. The double-pole residue at $z=\mathrm{i}n{\beta }_{\mathrm{H}}$ is

The poles enclosed are $n\ge 0$. Taking rectangles whose horizontal edges stay halfway between consecutive poles, and then taking their limits, gives

For $\nu <0$, close downwards with clockwise orientation. The enclosed poles are $n\le -1$, so

The horizontal edges vanish because of the sign of $\nu$, and the vertical edges vanish by the real-direction exponential decay.

Putting $z=s-\mathrm{i}\eta$ shows that

For fixed $\eta$, the left-hand Fourier transform is continuous, so the value at $\nu =0$ is the continuous limit $1/(2{\beta }_{\mathrm{H}})$. As $\eta \downarrow 0$, the right-hand side converges in ${\mathcal{S}}^{\prime }(\mathbb{R})$: on positive frequencies it has at most linear growth; on negative frequencies it decays exponentially, uniformly for $0<\eta \le {\beta }_{\mathrm{H}}/2$. This proves the regulator removal and (eq:thermalfourier).

Separating positive and negative frequencies in the inverse Fourier transform yields (eq:thermalspectral). Formula (eq:detailedbalance) follows directly.

Equation (eq:detailedbalance) is the two-point KMS detailed-balance relation for the horizon current in Killing time. The full state includes a second incoming channel in its vacuum, and the greybody-weighted outgoing state is not generally a KMS state at the same temperature. We therefore do not transfer horizon KMS thermality to every detector or every exterior observable.

For positive-frequency tests supported in a compact subset of $(0,\infty )$, the horizon state has

The infinity input is independent and has

These statements define finite packet covariances without requiring the stationary state to be a density matrix in the outgoing vacuum Fock representation.

Proof. Comparing (eq:currentexpansion) with (eq:thermalspectral) identifies the coefficients of the positive and negative frequency terms. Stationarity excludes an anomalous covariance supported at $\omega +{\omega }^{\prime }=0$ on the strictly positive-frequency test domain. This gives (eq:Hnumber)–(eq:Hanomalous).

For an explicit realization on any compact positive-frequency band, take two independent vacuum oscillator fields $b_1,b_2$ and set

Its commutator is $\delta (\omega -{\omega }^{\prime })$, and its vacuum covariances are precisely (eq:Hnumber) and (eq:Hanomalous). All multiplication factors are bounded on the chosen band. A third independent vacuum oscillator field realizes $a_{\mathrm{I}}$ and (eq:Inumber). The horizon occupation function in this realization is the function already derived in Lemma lem:fourier, not an additional input.

Let the row of the scattering matrix corresponding to future null infinity be

Define

Because the classical scattering is stationary, it preserves Killing frequency and does not mix annihilation with creation in this equation. The affine-to-Killing change has already been accounted for in the horizon state.

The definition (eq:quantumscatter) gives canonical outgoing commutators. Its number covariance is (eq:numberdensity), and $N_h$ has the finite expectation (eq:packetlaw).

Proof. The two incoming channels commute, and unitarity gives

Hence the outgoing commutator is canonical. The input state is a product state with vanishing means and no cross-channel covariance. Lemma lem:horizonoccupation therefore gives

This is (eq:numberdensity). Smearing proves (eq:packetlaw). The integral is finite because ${\Gamma }_{\ell }\le 1$, $\omega \ge {\omega }_{-}>0$ on the support, and $h\in L^2$.

The full outgoing current has spectral covariance

Its negative-to-positive spectral ratio is

not generally $e^{-{\beta }_{\mathrm{H}}\omega }$. The appropriate statement is a greybody-weighted emission law, not global thermal equilibrium.

On the one-particle space $L^2((0,\infty ),\mathrm{d}\omega )$, let $P_B$ denote multiplication by the indicator of a compact positive-frequency interval $B$. For $T>0$, define

and

The expected number in this mode window is

where $M_{n_{\mathrm{out}}}$ is the multiplication operator. Equivalently, this is the monotone expectation of finite-rank positive number-functionals approximating $K_{B,T}$.

The operator $K_{B,T}$ is positive and trace class, and

Thus the rate density under Definition def:window is (eq:rate).

Proof. The integral kernel of $A_{B,T}$ is

It is Hilbert–Schmidt, with

Consequently $K_{B,T}=A_{B,T}^{\ast }{A}_{B,T}$ is positive and trace class.

The function $n_{\mathrm{out}}$ is nonnegative and bounded on $B$. Therefore

Substitution of its proved value gives (eq:windowcount). Dividing by $T$ identifies the density.

The positive mode-window functional is not asserted to be a sharply localized detector count or a finite measurement instrument. A detector implementation requires a coupling, switching, input class, and error estimate. Those are separate from this asymptotic number-rate normalization.

Equation (eq:rate) is per angular mode. Summing over the $2\ell +1$ real harmonics at fixed $\ell$ introduces that degeneracy. No all-$\ell$ luminosity theorem is asserted here.

Let $h$ be as in Theorem thm:main.

If an approximation ${\tilde{\Gamma }}_{\ell }$ satisfies

then

Writing $q(\omega )={\Gamma }_{\ell }(\omega )/(e^{{\beta }_{\mathrm{H}}\omega }-1)$, for every ${\omega }_0\in B$,

Thus normalized packets in bands shrinking to a fixed ${\omega }_0>0$ have expectation tending to $q({\omega }_0)$.

Proof. For (eq:gammaerror), integrate the absolute error against $|h{|}^2$, use the stated uniform transmission error, and use

For (eq:banderror), subtract $q({\omega }_0)$ inside (eq:packetlaw) and use $\int |h{|}^2=1$. Continuity follows from Lemma lem:scattering.

Proof. Lemma lem:geometry proves the surface gravity and its geometric normalization. Lemmas lem:pullback and lem:fourier prove the horizon covariance, its Fourier transform, and the distributional regulator removal. Lemmas lem:radial and lem:scattering construct the exact radial scattering coefficients. Proposition prop:packet proves the outgoing number covariance and packet formula. Proposition prop:rate proves the counting normalization.

The declared external theorem identifies the boundary prescription with the Hadamard bulk Unruh state on the stated Schwarzschild domain. No earlier internal manuscript is used as a substitute for that theorem. The exponent in the emission law is ${\beta }_{\mathrm{H}}\omega =8\pi GM\omega$, calculated rather than calibrated.

The exact stationary theorem does not by itself prove that a specified collapse state converges to the Unruh state. This section gives a useful local transfer theorem with every additional assumption displayed.

Let $p$ be a smooth strictly increasing function for $u\ge u_0$. For constants $A,\kappa ,\lambda >0$, suppose

Suppose a positive current state has affine-coordinate covariance

where $R$ is smooth and bounded in a neighborhood of $(U_{\mathrm{H}},U_{\mathrm{H}})$.

The pulled-back current covariance is

Positivity in this assumption is not inferred from an arbitrary choice of $R$. The statement concerns a state whose covariance already has the displayed form.

Under Assumption ass:peeling, fix $L>0$. There are constants $t_L,C_L>0$ such that, for $t\ge t_L$,

is represented on $[-L,L{]}^2$ by a continuous kernel $E_t(x,y)$ satisfying

Consequently, for $f,g\in {C^{\infty }}_c((-L,L))$,

This proves a local distributional thermal limit for one translated current covariance.

Proof. Affine translation and positive affine rescaling leave the singular ratio in (eq:Cp) unchanged. Define

Equation (eq:peeling) gives

For sufficiently large $t$, both $q_t^{\prime }$ and $q_{\infty }^{\prime }$ are bounded below by a positive constant on this fixed interval.

Set

Then $q(x)-q(y)=(x-y)D_q(x,y)$, and the singular pullback is

Write $m=(x+y)/2$ and $s=x-y$. The function $A_q$, expressed in $(m,s)$, is even in $s$, satisfies $A_q(m,0)=1$, and has ${\partial }_s{A}_q(m,0)=0$. Taylor's formula gives

The map $q\mapsto A_q$ is locally Lipschitz from $C^3$ to $C^2$ when $q^{\prime }$ has a common positive lower bound. Indeed, its derivatives through order two are finite sums of products of derivatives of $q$ through order three and inverse powers of $D_q$; the denominators are uniformly bounded away from zero. Thus (eq:qconvergence) implies

The distributional identity

now shows that the difference of the two singular pullbacks is the continuous kernel

The $q_{\infty }$ pullback is $C_{\kappa }$ by Lemma lem:pullback. This proves the first error term.

The smooth remainder contributes

On the fixed square, $p^{\prime }(t+x)=O_L(e^{-\kappa t})$, and $R$ is bounded near the limiting affine point. This contribution is $O_L(e^{-2\kappa t})$, proving (eq:localerror). Integrating the continuous kernel against $\overline{f}\otimes g$ proves (eq:smearederror).

Theorem thm:peeling assumes the peeling estimates and the affine regularity statement. It does not derive them from a stellar collapse metric. It also controls compact time tests, whereas a strictly positive-frequency packet generally has noncompact time support. Passing from (eq:smearederror) to a collapse packet number limit requires additional tail estimates and a controlled identification of the propagated observable.

Likewise, a generic statement that a bulk state is Hadamard does not, without a characteristic restriction theorem, supply (eq:affineremainder). These are genuine hypotheses, not consequences of terminology.

The distinction between a local scaling calculation and the full collapse observable is consistent with the role of scaling limits in the Fredenhagen–Haag approach FredenhagenHaag1990. Their collapse analysis is an external comparison, not an unproved internal premise of this paper.

A complete certificate for the observable studied here needs:

1. a named background and normalized time;

2. a named quantum field;

3. a specified state with an admissible two-point distribution;

4. a defined number or detector observable;

5. a scattering or propagation identification;

6. an evaluated expectation and any required regulator or late-time limit.

The supplied record does not exhibit this certificate in any of the pre-existing landmarks discussed below. This is a restricted documentary conclusion. It does not assert that no future construction, or no unexamined manuscript, can provide it.

{}1.14 {@{}p{0.23}p{0.70}@{}} Landmark or cluster & Supplied result and the non-transfer boundary
Landmark or cluster & Supplied result and the non-transfer boundary

I–V; I thm:main & The excerpts specify finite Born semantics and distinct representation or instrument problems. They do not construct the Schwarzschild wave equation, the affine horizon state, or its number expectation. A measurement rule consumes an observable and channel; it does not select the field state.

VI–VIII; VII-F prop:spacelike; VIII-B thm:dictionary; VIII-C thm:C5 & The supplied theorems concern relational descent, a specified conical family, spacelike slices, and support for prescribed block channels. The cited constructions select neither the black-hole horizon nor the state used in this paper. A backward channel-support set is not identified with a spacetime causal past or an island.

IX–X & According to the supplied summary, these are sharp-product and ramp-convolution models. Their remainder coefficients are not identified with an asymptotic scalar number covariance.

XI-A lem:compatibility & The excerpt proves the classical necessary condition $({\partial }_{\mu }{g}_{\mathrm{XI}}){T_0}^{\mu \nu }=0$. Here $g_{\mathrm{XI}}$ denotes the inherited source coefficient, not the metric $\mathrm{g}$ of (eq:metric). The condition is not a state, a Bogoliubov calculation, or a stress-tensor expectation.

XII–XV; XIV-I, XIV-J; XV-B & The excerpts distinguish defined internal kernels, partial off-characteristic references, and the unresolved full characteristic map. XIV-J stops at J2 without quantum admission. XV-B exists as a technical interface manuscript but does not supply the inherited full map or its admission. None of these statements constructs (eq:numberdensity).

XVI-N; XVI-O & The supplied obstruction in XVI-N is conditional on its declared prepared-source class. XVI-O constructs a kernel subsector under a specified prescription while retaining bulk and insertion limits. A constructed kernel Ward subsector is not bulk symmetry restoration, and a counterterm is not the present emission observable. O.1 remains NOT_CLOSED in the supplied record.

XVI-P & The supplied summary separates classical vertex and gauge-map verification from later quantum evaluation. Its full text is not verified here. The separation is a useful correction-program discipline, not a free-field emission theorem.

Em02-D thm:main & The excerpt gives a fixed seven-row inverse-cubic distributional asymptotic and a separated divergence witness. Its model is not identified with the state and scattering problem of this paper.

Em02-E thm:release & The supplied summary identifies a classical quadratic bridge on a power-law inflationary background and an incomplete quantum audit. Its full theorem is not independently checked here. Neither a relational lapse nor a classical quadratic kernel specifies the Unruh state.

En02-C & The supplied summary distinguishes recovery from a restricted streaming error and refuses a Page-curve interpretation. The full manuscript is not independently checked here. Recovery and entropy claims do not determine a frequency occupation density.

En02-D thm:D1, thm:D2 & The supplied summary describes energy-sector isometries and a quadrature estimate against an external rate. Those theorem texts are not reproduced in the supplied excerpts. The rate is an input there; in (eq:rate) it is the output of a field-state and scattering calculation.

The fixed geometry, free wave equation, and stationary scattering matrix do not uniquely determine the outgoing occupation density.

Proof. Keep the metric, wave equation, scattering matrix, and infinity-input vacuum fixed. On any compact positive-frequency band, one admissible incoming horizon state is the Killing-frequency vacuum. For that choice, both incoming number covariances vanish, and (eq:quantumscatter) gives zero outgoing number covariance.

A second choice is the affine horizon vacuum restricted to the exterior half-horizon. Lemmas lem:fourier and lem:horizonoccupation give its positive occupation $n_{\mathrm{H}}(\omega )$, and Proposition prop:packet gives ${\Gamma }_{\ell }(\omega )n_{\mathrm{H}}(\omega )>0$.

Thus the same classical scattering data admit different outgoing occupations. The first choice is not asserted to have the same future horizon regularity as the second. That difference is precisely why a state and its regularity requirements must be part of a derivation.

This proposition excludes the inference from “a classical scalar action exists” to “the Hawking flux has been proved.” It does not exclude a derivation after the missing quantum state data are supplied.

Energy preservation and the isometry condition alone do not select a Hawking occupation law or a dimensional emission rate.

Proof. Fix a source energy $E$, a source degeneracy label $a=1,\dots ,d$, and permitted radiation energies ${\omega }_1,\dots ,{\omega }_k\le E$. Choose orthonormal daughter–radiation states