I-B-VP-2: The exact global wavefront relation of the conical Friedrichs ground state Geometric propagation, apex diffraction, and positive frequency on the prescribed regular Enter spacetime

We determine the ordinary global wavefront set of the massless holonomy-twisted Friedrichs spectral ground-state covariance on the regular conical spacetime prescribed in I-B. The result is an equality, valid for every opening $0<\alpha <1$ and fractional holonomy $0\le \beta <1$, including the angular transition where a geometric front meets the apex-diffracted front.

Let ${\mathcal{G}}^{+}$ be the closure of the positive-frequency relation of regular null geodesics, including its diagonal limit, and let ${\mathcal{D}}_{\mathrm{ap}}^{+}$ be the positive conormal relation of

Then

where $N\ge 2$ and $p=0,\dots ,N-1$. Thus the full strictly diffractive relation disappears exactly in the finite-character quotient sectors. In other sectors it remains in the joint wavefront set even at isolated angular zeros of its leading coefficient.

The proof sums the Euclidean angular modes before continuation, uses a two-variable meromorphic expression to control every angular transition, and combines its reduced-variable structure with the bisolution equations. It also gives the explicit inverse-square-root diffractive coefficient and its half-order regularity improvement away from transition angles.

The contribution is a convention-complete global analysis of the specified covariance within the companion framework, not a claim of priority for conical diffraction. It distinguishes local Hadamard subtraction from a global geodesic-only spectrum condition. In particular, supported Green-operator agreement does not identify vacuum covariances, and the charged commutator is not the pointwise imaginary part of one complex bundle kernel. The result resolves V2 for this state and extension at the level of the ordinary two-point wavefront set. It neither restores global hyperbolicity nor constructs an apex field algebra, an interacting product, a measurement acquisition of wavefront data, or a gravitational source law.

Cheung's I-B prescribes a regular conical spacetime together with a separate finite operational interface CheungIB. I-B-G constructs intrinsic scalar propagation after a spatial self-adjoint extension has been supplied CheungIBG. I-B-VP-v2 constructs the distinguished massless Friedrichs spectral ground state and establishes its local Hadamard form CheungIBVP. I-B-VP-0 evaluates its charged current CheungIBVPZero; I-B-VP-1 constructs prescribed extensions of selected one-point distributions across the apex in an additional smooth completion CheungIBVPOne.

The present object is different: the global covariance on the regular spacetime $M$, with both endpoints satisfying $r,r^{\prime }>0$. No smooth structure at the apex and no apex-supported one-point counterterm enters the proof.

analytical background. Sommerfeld-type cone kernels and cosmic-string Euclidean Green functions have a substantial history Dowker,Linet. Geometric and diffractive propagation on cones and edges are treated by Cheeger–Taylor, Melrose–Wunsch, and Melrose–Vasy–Wunsch CheegerTaylorI,CheegerTaylorII,MelroseWunsch,MelroseVasyWunsch. The magnetic Aharonov–Bohm diffraction problem provides another relevant model Yang. We use these works for context, not as substitutes for checking the exact holonomy, state, and transition formulas below.

The global statement is proved directly from an explicitly summed kernel. The standard tools needed are distributional boundary values, the wavefront pullback and product theorems, microlocal elliptic regularity, and propagation of singularities for a smooth real-principal-type operator on the regular host HormanderI,HormanderIV.

The proposed V2 route requires the following corrections.

1. A retarded transfer theorem is not a state-transfer theorem. Equality of supported retarded or advanced operators in a laboratory window does not imply equality of positive-frequency covariances. The appropriate comparison is equality of the local singular structure, or smoothness of a specified covariance difference, not equality of states.

2. The global relation must include limiting geometric rays. A regular null geodesic can approach a ray through the removed apex. Its limiting covector pair belongs to a closed wavefront set, even though the limiting ray is not a geodesic in $M$. We therefore use the closure ${\mathcal{G}}^{+}$, rather than only the relation of actual regular null geodesics.

3. A diffraction bound must allow exact disappearance. Some cone–character pairs admit a finite image formula with no strictly diffractive relation. Conversely, a zero of one leading coefficient at one angular value does not remove a covector from the joint wavefront set if nearby angular values retain that covector in the limit.

4. The charged commutator is not obtained by taking a pointwise imaginary part. The spatial bundle kernel can be complex before any Lorentzian time prescription is introduced. The correct commutator uses opposite operator ordering and the conjugate charge sector. The operator-valued time-odd part, not the pointwise imaginary part of $W_{\beta }$, is fixed by the causal propagator.

These are corrections to the logical route, not changes to the prescribed geometry, Friedrichs extension, or spectral state.

The Philosophy of Enter distinguishes destructive readout from quantum recovery, a classical record from a decoder, and an analogy from an identification CheungPhilosophy. I-B and En09-A implement a finite destructive instrument returning a classical code and a surviving residual, not the measured quantum clock register CheungIB,CheungEn09. That interface does not select the continuum field, apex extension, vacuum, or wavefront relation.

The background, worldsheet defect, extension, and state remain inputs. A finite conditional density matrix is not the bidistribution analyzed here. A wavefront condition is not an Einstein equation or a derivation of the metric CheungIB (:not-source) CheungXAST (:noeinstein). The philosophical discussion fixes interpretation; it is not a premise of the continuum proof.

of the new result. We prove an exact ordinary $C^{\infty }$-wavefront theorem on $M\times M$, not a theorem about an unspecified boundary wavefront calculus. We do not claim a complete symbol calculus at the geometric–diffractive transition, a classification of all ground states, or an interacting microlocal algebra. The exact wavefront equality does not require any of those stronger claims.

Fix

The spacetime and metric are

The future orientation is that of ${\partial }_t$. The completion adds the worldsheet

It is not part of $M$.

A lifted parallel representation of the charged line bundle is fixed by

Changing a local parallel frame multiplies the covariance by a smooth nowhere-zero phase, and therefore does not change its wavefront set.

Write

Let $B_{\beta }$ be the nonnegative Friedrichs extension of $-{\Delta }_{q_N}$ on the twisted spatial Hilbert space. The extension is an input, as in I-B-G CheungIBG,KayStuder. No distributional curvature-coupling interaction at $\mathcal{W}$ is introduced.

The function $t$ is temporal, but $M$ is not globally hyperbolic.

Proof. The inverse metric gives $g^{-1}(\mathrm{d}t,\mathrm{d}t)=-1$. Choose $R>0$, $T_0>R$, and fixed ${\phi }_0,z_0$. The points

have a causal diamond containing

the radial spatial travel length from either endpoint is less than $R<T_0$. This sequence has no convergent subsequence in $M$. Thus the diamond is not compact.

This reproduces the relevant geometric obstruction of I-B-G :nonGH without importing a global Cauchy theorem. A self-adjoint extension supplies the evolution used here; it does not make the regular slices Cauchy surfaces for all causal curves.

Put $H_{\beta }=\sqrt{B_{\beta }}$. The covariance is

For compact spacetime sections,

For a real neutral scalar take $\beta =0$. For a charged scalar use the particle and antiparticle sectors on the bundle and its conjugate, as in I-B-VP-v2 :state CheungIBVP.

For completeness, the estimate needed at zero energy is recalled.

Every compact smooth spatial section belongs continuously to $\mathrm{dom}({B_{\beta }}^{-1/4})$ in its fixed-support test topology. Consequently (eq:W) defines a continuous positive covariance on compact spacetime tests.

Proof. With

the heat kernel is

It follows from the Friedrichs Bessel transform and Weber's integral Watson,CheungIBG. On a compact regular spatial set and for $s\ge 1$, the Bessel power series gives

For a compact spatial section $v$,

Use the contraction bound for $0<s<1$, and the displayed compact heat bound with ${\Vert v\Vert }_{L^1}^2$ for $s\ge 1$. The latter integrand is $O(s^{-2-{\nu }_{\ast }})$. The bounds are continuous in fixed-support test seminorms. Time integration then defines ${\mathcal{K}}_{\beta }$, and its Gram form proves positivity.

The spectral construction is quasifree and has nonnegative one-particle Hamiltonian $H_{\beta }$. It is the specified distinguished spectral state, not a uniqueness theorem among all algebraic ground states.

On regular charts,

These equations follow by spectral calculus and integration by parts on compact tests. Their principal symbol is

up to an irrelevant overall sign.

All wavefront sets below are ordinary, unprimed wavefront sets:

means the covector of the bidistribution on $M\times M$. A conventional canonical-relation notation would replace ${\kappa }^{\prime }$ by $-{\kappa }^{\prime }$.

We use the local Fourier convention

The factor $e^{-\mathrm{i}\omega (t-t^{\prime })}$, $\omega >0$, therefore has first temporal covector ${\kappa }_t=-\omega <0$. For signature $(-+++)$, such a null covector is future directed after raising its index. This is our positive-frequency orientation.

Use the differences

in a chosen angular lift. For $\epsilon >0$, define

where the branch has $\mathrm{Re}{\eta }_{\epsilon }>0$. For real $T\ne 0$, $Q_{\epsilon }$ approaches the real axis from the side $\mathrm{sgn}T$.

Define

For $a=\eta /\alpha$,

The formulas include $\beta =0$. In particular, the meromorphic continuation of $S_{\beta }$ is odd in $\eta$.

Proof. Separate $n\ge 0$ and $n\le -1$ and sum the two convergent geometric series. Putting their terms over a common denominator gives (eq:Sclosed). Its numerator is odd in $a$ and its denominator is even.

For $\epsilon >0$,

Its distributional boundary value as $\epsilon \downarrow 0$ is $W_{\beta }$.

Proof. For a real Euclidean separation $a_0>0$, the spectral identity

and (eq:heat) give

The heat integral is justified at large $s$ by Lemma lem:IR. The change of variable $\zeta =r{r}^{\prime }/(2s)$ uses

Watson. This is also the Euclidean mode kernel of I-B-VP and I-B-VP-0 CheungIBVP,CheungIBVPZero.

Both sides continue holomorphically in $a_0$ to $\mathrm{Re}{a}_0>0$. The denominators in (eq:Sgeom) are nonzero there under (eq:Qeta), since $\mathrm{Re}\eta >0$. Set $a_0=\epsilon +\mathrm{i}T$. Spectral dominated convergence on the compact-test form of Lemma lem:IR proves the boundary assertion.

Let $F(z,y)$ be holomorphic for $\mathrm{Im}z>0$, smooth in real parameters $y$, and locally bounded by a power of $(\mathrm{Im}z{)}^{-1}$, together with each fixed finite number of $y$-derivatives. Then it has a distributional boundary value on compact real coordinate patches, and

For a lower-half-plane boundary value, $\lambda \le 0$.

The assertion is local and remains valid after a real-analytic change of the complex transverse coordinate with the corresponding half-plane orientation.

Proof. This is the distributional boundary-value theorem with parameters HormanderI. We recall the two ingredients needed here. Sufficiently many holomorphic primitives in $z$ have controlled boundary limits; differentiating those limits defines the original distributional boundary value. The estimates with parameters make this construction continuous on joint tests.

For the wavefront assertion, localize and Fourier transform. In a cone where $\lambda <-c|(\lambda ,\nu )|$, shift the $z$ contour upward. An almost-analytic extension of the real cutoff makes its $\bar{\partial }$-error vanish to arbitrarily high order at the boundary. That order can be chosen larger than the fixed polynomial growth order of $F$. The shifted term decays exponentially in the negative $\lambda$-cone; the cutoff errors decay to every inverse power. This proves (eq:BVorientation). Reflection proves the lower-half-plane case. The same contour argument in an analytic transverse coordinate proves the last assertion.

For the function

the required polynomial bounds hold near $Q=-1$, uniformly on compact real angular intervals. Indeed, $\mathrm{Re}\mathrm{arcosh}Q\ge c\mathrm{Im}Q$ on a sufficiently small upper-half-plane compact patch, the geometric denominators in (eq:Sgeom) are bounded below by a constant times $\mathrm{Re}\mathrm{arcosh}Q$, and $\sinh (\mathrm{arcosh}Q)$ has only a square-root zero. Angular differentiation changes the polynomial bound, but not its existence.

Consequently, near a point with $T\ne 0$, the covariance has the reduced form

The analytic change from $T$ to $Q$ is regular there, since ${\partial }_{T}Q=-T/(r{r}^{\prime })\ne 0$. This observation is particularly useful at an angular transition: no moving image indicator is differentiated.

On a sufficiently small developing neighbourhood of each regular point, in a local parallel frame,

with $H_{\beta }$ smooth; in fact the local regular remainder is real-analytic in the corresponding real variables.

Proof. Take a lift with $\delta$ near zero and $Q$ near $1$. The quotient in (eq:reducedF) is meromorphic in $(Q,\delta )$, because both its numerator divided by $\sinh \eta$ and its denominator are even analytic functions of $\eta$ near $\eta =0$. Its denominator factors as

For $\delta \ne 0$ sufficiently small, either term in (eq:Sgeom) computes the pole residue. It gives

analytic across the pole. Analytic division extends the same statement through $(Q,\delta )=(1,0)$. Multiplication by the prefactor in (eq:Wclosed) gives the direct kernel in (eq:localHadamard), because

The remaining analytic function has a smooth Lorentzian boundary value. The direct term is the ordinary Minkowski positive-frequency distribution in the developing chart.

This independently recovers the local statement used in I-B-VP and I-B-VP-0. The global proof below is not an application of a globally hyperbolic local-to-global theorem.

For a chosen lift define

Only finitely many $k$ satisfy $|{\theta }_k|\le \pi$ on a fixed compact lifted angular patch.

Let ${\mathcal{G}}_{\mathrm{reg}}^{+}$ be the ordinary positive-frequency covector relation of null geodesic segments entirely in $M$. It includes

when ${\kappa }^{\prime }$ is parallel transported to $\kappa$ along that segment and $\kappa$ is future directed. Define

The closure includes the standard diagonal null relation.

Away from the diagonal, the actual regular geodesic relation is locally described by

Its closure allows $|{\theta }_k|=\pi$.

Formula (eq:Gcoordinates) describes all regular null geodesic segments. The additional off-diagonal limits in ${\mathcal{G}}^{+}$ occur at $|{\theta }_k|=\pi$.

Proof. A geodesic away from the apex develops into a straight line in the transverse plane, together with an affine longitudinal coordinate. A straight transverse segment avoiding the origin has a continuous angular change strictly between $-\pi$ and $\pi$. Its length is $R_k$ for the corresponding lift. The null condition is $|T|=R_k$. Differentiating its squared separation gives the parallel endpoint covectors in (eq:Gcoordinates).

A limit with angular change $\pi$ or $-\pi$ passes through the transverse origin. Conversely, those segments are limits of straight segments avoiding the origin. The local finiteness of the allowable lifts excludes other off-diagonal limits.

Put

Define

No angular covector occurs in this relation. The angular transition set is

Both endpoint covectors in (eq:Drelation) are nonzero and null. For example,

The first raised covector is future directed precisely when $\lambda T>0$.

For $T=L_{\mathrm{ap}}>0$, the relation ${\mathcal{D}}_{\mathrm{ap}}^{+}$ is the endpoint relation of an incoming radial null segment and an outgoing radial null segment meeting the removed worldsheet at

The energy and longitudinal momentum agree across that meeting. The incoming and outgoing angular positions need not agree. Moreover,

consists exactly of these radial covector pairs at the angular transition $\delta \in {\mathcal{B}}_{\alpha }$.

Proof. Minimize the length of a two-segment spatial path through the worldsheet:

Equality of the two longitudinal slopes gives (eq:scatterpoint) and total length $L_{\mathrm{ap}}$. The corresponding null segments have common longitudinal velocity $Z/T$ and transverse speed $(r+r^{\prime })/T$. The covectors in (eq:Drelation) record the same energy and longitudinal momentum.

For a limiting geometric ray to have this length,

With $|{\theta }_k|\le \pi$, this means ${\theta }_k=\pi$ or $-\pi$, equivalently $\delta \in {\mathcal{B}}_{\alpha }$. At that point $\mathrm{d}{Q}_k=\mathrm{d}{F}_{\mathrm{ap}}$.

The meeting point is not an added field-theoretic degree of freedom. It is a description of the endpoint relation selected by the Friedrichs kernel.

For every $0<\alpha <1$ and $0\le \beta <1$,

In particular, there are no one-zero-endpoint covectors and no additional pure angular wavefronts at an image boundary.

Proof. The proof separates the real values of $Q$ in (eq:reducedboundary).

{1. The regions $Q>1$ and $Q<-1$.} For $Q>1$, $\eta =\mathrm{arcosh}Q$ is real and positive, and the summed expression is smooth. For $Q<-1$, its two boundary values have

The geometric denominators in (eq:Sgeom) are nonzero because $e^{-v/\alpha }<1$. The factor $\sinh \eta$ is also nonzero. Thus each boundary value is smooth.

{2. The interval $-1<Q<1$.} Write

The only poles in (eq:Sgeom) occur when

They are precisely $Q_k=0$. Each of the two geometric fractions has a simple pole in a smooth defining function for the corresponding front. Even where two fronts cross in the base, the two fractions provide the separate conormal descriptions. Consequently the wavefront is contained in their conormals.

The half-plane boundary-value orientation in Lemma lem:boundaryvalue gives a nonnegative coefficient of $\mathrm{d}Q$ for $T>0$, and a nonpositive one for $T<0$. Since

the first temporal covector is negative. This is exactly the orientation $\lambda T>0$ in (eq:Gcoordinates).

{3. The branch point $Q=1$.} At an angular value not congruent to zero, the evenness in $\eta$ makes (eq:reducedF) analytic in $Q$. Near $\delta =0(\mathrm{mod}2\pi )$, the factorization in the proof of Lemma lem:local gives the direct geometric conormal. At a diagonal point the complete local wavefront is the usual positive-frequency diagonal null relation by (eq:localHadamard).

{4. The branch point $Q=-1$, including $\delta \in {\mathcal{B}}_{\alpha }$.} Here $T\ne 0$. The map

is a submersion, and (eq:reducedboundary) expresses the kernel as the pullback of a two-variable distribution, multiplied by the smooth factor $1/(8{\pi }^2\alpha r{r}^{\prime })$. The pullback theorem therefore gives

for every possible wavefront covector at this branch point. There is no contribution from passive coordinates.

On $F_{\mathrm{ap}}=0$,

The first endpoint part of $\mathrm{d}Q$ is null and has no angular component. Hence

The bisolution equation (eq:bisolution) and microlocal elliptic regularity force $p(X,\kappa )=0$, so $\nu =0$. The total covector is nonzero, therefore $\lambda \ne 0$; both endpoint covectors are nonzero. Lemma lem:boundaryvalue fixes its positive-frequency orientation. This is precisely ${\mathcal{D}}_{\mathrm{ap}}^{+}$.

This last argument applies to the total summed kernel. It does not assign or differentiate separate values of moving image-sector indicators.

5. Exhaustion. At $T=0$, the only possible singular points are on the diagonal: indeed $Q\ge 1$, with equality only when $r=r^{\prime }$ and $Z=0$, and a singular angular value then requires $\delta =0(\mathrm{mod}2\pi )$. The preceding cases cover all of $M\times M$.

The proof uses a reduced-variable pullback theorem, an oriented boundary-value theorem, and the characteristic equations. Neither finite propagation nor a finite-order estimate alone would prove (eq:upper). This is the distinction between distributional control and a wavefront theorem also emphasized in the supplied Em01-D material CheungEmOneD.

For all parameters,

Proof. Lemma lem:local gives every positive diagonal null covector. Fix a regular null geodesic segment starting at $X^{\prime }$. Propagation of singularities for $P_X$, with the primed variables retained as parameters, transports the first covector along that entire segment HormanderIV. The path remains in the smooth regular host; no passage through the apex is made in this argument. Thus every element of ${\mathcal{G}}_{\mathrm{reg}}^{+}$ occurs. The wavefront set is closed in the punctured cotangent bundle, so it contains ${\mathcal{G}}^{+}$.

One can also read the nonzero coefficient directly from (eq:Sgeom): a geometric branch has principal term

Its character factor is never zero. Propagation avoids any need to select a leading term at a crossing of geometric fronts.

Set

On the complement of the angular transition set, define

This is a complex, real-analytic angular function in each transition-free angular interval.

Near a point with $F_{\mathrm{ap}}=0$ and $\delta \notin {\mathcal{B}}_{\alpha }$, there are smooth functions $A_{\ast }$ and $B_{\ast }$, analytic in the local real variables, such that

and

The square root is the principal boundary value.

Proof. Take first $T>0$. Near $Q=-1$, write $\eta =\mathrm{i}\pi +w$. Since

one has

The angular denominator in (eq:Sclosed) is nonzero at $w=0$, and

Expand $S_{\beta }(\mathrm{i}\pi +w,\delta )/\sinh (\mathrm{i}\pi +w)$ in its convergent Laurent series in $w$. Its even and odd powers separate into an analytic function of $F_{\mathrm{ap}}$ plus $(F_{\mathrm{ap}}+\mathrm{i}0{)}^{-1/2}$ times another analytic function. Multiplication by the prefactor in (eq:Wclosed) gives (eq:diffcoefficient).

For $T<0$, expand at $\eta =-\mathrm{i}\pi$. Oddness of $S_{\beta }$ and of $\sinh \eta$ gives the same coefficient and the lower boundary value.

At $s=0$,

and the lower boundary value has $\lambda <0$. Multiplication by a smooth coefficient nonzero on $s=0$ preserves that conormal wavefront.

Proof. The representation

is understood as a damped oscillatory integral. It gives the one-sided wavefront and an elliptic conormal amplitude at every positive $\lambda$. The lower boundary value is obtained by conjugating the scalar model. Smooth elliptic multiplication preserves the wavefront HormanderI.

Define

The function ${\mathcal{A}}_{\alpha ,\beta }$ vanishes identically on a nonempty transition-free angular interval if and only if $(\alpha ,\beta )\in \mathcal{Q}$.

If $(\alpha ,\beta )\notin \mathcal{Q}$, its nonzero angular values are dense in the complement of ${\mathcal{B}}_{\alpha }$, and

Proof. Multiplying the numerator in (eq:A) by $e^{-\mathrm{i}\beta \delta }$ gives

It vanishes on an interval precisely when both constants vanish:

Their sum gives $1/\alpha =N\in \mathbb{N}$, $N\ge 2$, and then $\beta =p/N$ with $0\le p<N$. The converse is immediate.

Outside this exceptional case, the analytic numerator has no open zero interval. At every nonzero coefficient value, Lemmas lem:diffnormal and lem:sqrtWF give all of ${\mathcal{D}}_{\mathrm{ap}}^{+}$ over that angular value. Closedness of the joint wavefront set supplies the same radial covectors at its angular zeros and at the transition angles. This proves (eq:difflower).

At a special fixed value of $\delta$, a restricted scalar coefficient can vanish. That is not a neighbourhood on $M\times M$. The lower bound above is a joint statement in the angular and remaining variables. No inference from a single angular slice is used.

For orientation and normalization, define

The notation here uses $\delta$, rather than the developing angle $\alpha \delta$, as its second argument.

The Bessel angular identity gives, off the transition angles,

One obtains this formula by applying the heat-kernel angular summation before integrating the Euclidean time heat factor, and then continuing its total kernel. It is the covariance counterpart of the Sommerfeld–Carslaw–Dowker formulas in I-B-G and I-B-VP CheungIBG,CheungIBVP. At transition angles the joint distribution, not separate pointwise sector values, is the object.

The coefficient in (eq:A) can be checked without assigning a value to a divergent angular series:

Its transition-free limit is therefore

Since

the endpoint model

recovers (eq:diffcoefficient). The full wavefront proof at the angular transition was instead given by the reduced-variable argument of Proposition prop:upper; this endpoint check is not used to differentiate a singular moving image sector.

At a nonzero diffractive coefficient and away from geometric fronts, the transverse Fourier amplitude of $W_{\beta }$ has order $-1/2$. At a single noncoincident geometric front away from other fronts it has order $0$. Thus the strictly diffracted covariance is one-half derivative smoother in its normal variable.

Equivalently, at such elliptic points the local normal Sobolev thresholds are $s<0$ for the diffracted front and $s<-1/2$ for the geometric front. No uniform half-order assertion at their transition is made.

Proof. The square-root model has Fourier amplitude ${\lambda }^{-1/2}$ on a half-line. The simple-pole geometric model has amplitude of order $0$. After localization, the corresponding one-dimensional Sobolev integrals have thresholds

and

The nonzero leading coefficients make these thresholds sharp.

If $\alpha =1/N$ and $\beta =p/N$, then

where

${\mathcal{R}}_k$ rotates the covering transverse plane by $2\pi k/N$, and

Consequently,

Proof. Apply the character-weighted image sum first to the flat heat kernel. Its angular Fourier sum projects covering modes $m\in \mathbb{Z}$ to

because

The projected angular factor is $e^{\mathrm{i}(n+p/N)(\phi -{\phi }^{\prime })}$, and the Bessel order is $|Nn+p|=|n+\beta |/\alpha$. The factor $N$ supplies precisely the quotient normalization $1/\alpha$ relative to the conical volume. Thus the result is the Friedrichs heat kernel (eq:heat). Euclidean integration and spectral continuation give (eq:finiteimages).

Every image has the ordinary flat null wavefront relation. If its straight segment avoids the origin, it gives a regular geodesic on the quotient. If it crosses the origin, its endpoint covector is a limit of such geodesics. Thus the finite sum has wavefront contained in ${\mathcal{G}}^{+}$. Proposition prop:geolower gives the reverse inclusion. This argument uses the full finite image sum, including boundary representatives when $N$ is even.

For the massless Friedrichs spectral covariance (eq:W) on the regular host (eq:host),

The equality holds on all of $M\times M$, including the diagonal, crossings of geometric fronts, and every geometric–diffractive angular transition.

Every first endpoint covector is future null, every second endpoint covector is past null, and neither endpoint covector is zero.

Proof. Proposition prop:upper gives the global upper bound. Proposition prop:geolower gives the geometric lower bound. Outside $\mathcal{Q}$, Proposition prop:criterion gives the entire diffracted lower bound. Inside $\mathcal{Q}$, Proposition prop:quotient removes every strictly diffractive covector outside ${\mathcal{G}}^{+}$. The orientation and nonzero-endpoint statements were established in Proposition prop:upper and the definitions of the two relations.

The singular support is the projection to the base of the relation in (eq:main). In particular:

1. off the diagonal, geometric singularities occur at the regular or limiting geometric travel times $T=\pm R_k$;

2. in nonquotient sectors, the additional singular front is $T=\pm L_{\mathrm{ap}}$;

3. the open region $T^2>{L_{\mathrm{ap}}}^2$ is smooth.

The third statement does not say that the covariance or causal kernel vanishes there.

Proof. The first two statements follow from the exact wavefront equality. For the third, $Q<-1$, where the summed boundary expression is smooth by the first step of Proposition prop:upper.

The exact equality, the treatment of all angular transitions, and the holonomy criterion for disappearance of the strictly diffractive relation are the mathematical release of this companion. Cone diffraction and quotient image methods are established subjects CheegerTaylorI,CheegerTaylorII,MelroseWunsch,Yang; the present paper does not claim that these phenomena originate in the Enter framework.

Define

The bundle parameter $-\beta$ below is understood modulo one.

At the operator-kernel level,

For the charged spacetime kernels, the second term in (eq:odd) is the reversed opposite-charge ordering:

Thus the state has the extension-selected canonical commutator. The pointwise real and imaginary parts of the scalar expression for $W_{\beta }$ in a parallel frame are not, in general, the two operators in (eq:even)–(eq:odd).

Proof. The first two identities are the elementary spectral cosine and sine identities for (eq:W). Complex conjugation of the spatial bundle character, followed by endpoint reversal, identifies

This can be checked directly by reindexing the angular modes. The antiparticle Fock construction gives precisely this reversed operator ordering, proving (eq:chargedCCR).

In a twisted parallel representation, the angular factors $e^{\mathrm{i}(n+\beta )\delta }$ can make the time-even spatial kernel complex. Therefore scalar complex conjugation at the same ordered pair is not the operation $T\mapsto -T$ with spatial endpoints retained.

For the neutral real scalar, this reduces to

The sign agrees with the spectral convention in the supplied I-B-G and I-B-VP records CheungIBG,CheungIBVP.

Let ${\mathcal{G}}^{-}=-{\mathcal{G}}^{+}$ and ${\mathcal{D}}_{\mathrm{ap}}^{-}=-{\mathcal{D}}_{\mathrm{ap}}^{+}$, where the minus sign negates cotangent fibers without changing base points.

The wavefront of $E_{\beta }$ is the union of both frequency orientations of (eq:main):

Proof. Equation (eq:odd) writes $E_{\beta }$ as the difference of two kernels with opposite, disjoint temporal orientations. Theorem thm:main, applied to time reversal, identifies the two sets. A microlocal singularity in one temporal orientation cannot be cancelled by a kernel smooth in that orientation.

Finite propagation of the Friedrichs evolution gives causal commutator support for the intrinsic spatial length metric, as in I-B-G :speed CheungIBG. This support statement is compatible with the diffracted relation; it is not a support statement for $W_{\beta }$ itself.

Radzikowski's theorem relates the Hadamard condition and an oriented geodesic wavefront condition under its globally hyperbolic free-field hypotheses Radzikowski. Here one can still define the relation of ordinary regular null geodesics, but the hypotheses of that global theorem do not hold.

The covariance is locally of Hadamard form at every regular point. Nevertheless:

1. outside $\mathcal{Q}$, its global wavefront contains all strictly diffractive pairs in ${\mathcal{D}}_{\mathrm{ap}}^{+}\setminus {\mathcal{G}}^{+}$, which are not linked by a regular null geodesic with the displayed endpoint covectors;

2. even in a finite-character quotient sector, the closed geometric relation can contain limiting rays through the removed apex that are not actual regular geodesics.

Proof. Local Hadamard form is Lemma lem:local. For (i), choose $\delta \notin {\mathcal{B}}_{\alpha }$ on the apex front. Proposition prop:broken shows that its radial covector pair is outside ${\mathcal{G}}^{+}$, while Theorem thm:main places it in $\mathrm{WF}(W_{\beta })$. A regular geodesic carrying that radial covector cannot avoid the apex and connect the two endpoints.

For (ii), the finite image kernel includes a flat null segment whose transverse projection crosses the covering origin whenever the corresponding image endpoints are antipodal. It is the limit of regular quotient geodesics and hence belongs to ${\mathcal{G}}^{+}$, but its limiting radial segment is not in $M$.

The exact result is therefore a positive-frequency spectrum condition for a specified geometric-plus-apex relation. Calling it a geodesic-only globally Hadamard theorem would discard part of the actual wavefront. Conversely, its additional global singularities do not invalidate local Hadamard point splitting.

Let $W_{{\beta }_1},\dots ,W_{{\beta }_m}$ be covariances of the form considered here, with the same time orientation. Their product as kernels on $M\times M$, with the appropriate tensor-product bundle, is defined by the ordinary wavefront product theorem. The same holds after any fixed finite differentiations of the factors.

This assertion does not supply a product between arbitrary oppositely oriented kernels or an interacting algebra.

Proof. Every nonzero wavefront covector of every factor has a future-null first component. A finite sum of nonzero future-causal covectors cannot vanish. Thus no selection of wavefront covectors from the factors can sum to the zero covector on $M\times M$. The product criterion applies HormanderI. Differentiation does not enlarge a wavefront set.

For example, the ordinary sufficient criterion fails for $W_{\beta }\overline{W_{\beta }}$ at an elliptic geometric null covector: the two factors have opposite covectors there. That criterion failure is not a universal nonexistence theorem for every renormalized prescription.

The examples in VII-H and the declared polynomial domains in XI-E are methodological comparisons CheungVIIH,CheungXIE. Their observables and products are not identified with the present conical bundle kernels.

The equal-time restriction of $W_{\beta }$ exists and equals the spatial covariance $1/(2H_{\beta })$. For any smooth timelike curve $\gamma$ in $M$, the pullback

also exists as a bidistribution on its parameter domain.

Proof. A covector normal to an equal-time embedding has only temporal components at its endpoints. A nonzero null covector cannot be purely temporal, so the normal set is disjoint from $\mathrm{WF}(W_{\beta })$. The spectral restriction is therefore legitimate and gives $1/(2H_{\beta })$.

A nonzero null covector cannot annihilate a timelike tangent. Hence neither endpoint of a wavefront pair can lie in the annihilator required to obstruct the timelike pullback. The wavefront pullback theorem applies HormanderI.

The unrenormalized diagonal pullback of $W_{\beta }$ fails the ordinary wavefront criterion. In contrast, the local remainder $H_{\beta }$ in (eq:localHadamard) has well-defined diagonal jets of every finite order. Consequently the Wick square, stress tensor, and current prescriptions of the regular-part companions remain well-defined at every point of $M$.

Proof. The diagonal wavefront contains

These covectors are normal to the diagonal embedding. Thus one cannot justify the unrenormalized pullback by the ordinary criterion. The subtracted remainder is smooth in a neighbourhood of each regular diagonal point, so all its required local diagonal derivatives exist.

The apex front cannot meet the spacetime diagonal: at $X=X^{\prime }$, $F_{\mathrm{ap}}=4r^2>0$. The additional global fronts therefore do not alter this local subtraction argument.

For the charged current, one must retain the complex first jet of $H_{\beta }$, as in I-B-VP-0 CheungIBVPZero (:imaginary, :current). The global theorem does not turn that prescription into the real-part stress prescription.

The apex finite parts of I-B-VP-1 concern one-point distributions in a separately declared completion CheungIBVPOne. Their contact parameters neither change $W_{\beta }$ on $M\times M$ nor provide an extension of its two-point kernel to $\mathcal{W}$.

The global relation has a direct consequence for an already well-defined restriction, without constructing a detector model.

Fix a static worldline

Its proper-time parameter is $u$.

Suppose

Then, near $u-u^{\prime }=\pm 2r_0$,

The restricted covariance therefore has a genuine delayed singularity at each of those nonzero proper-time separations.

For example, $\alpha =2/3,\beta =0$ gives ${\mathcal{A}}_{\alpha ,\beta }(0)=1$, and the hypotheses hold.