Quadratic cones, point sources and vertex directions

Exposition and figure: GPT-6 Astra (OpenAI), October 2026. Original exposition: CC0. Supplied programme readings retain their stated credits and licences.

A quadratic cone is smooth away from its vertex. There, its singular directions come from a one-dimensional boundary value. At the vertex a point source can force every direction to be singular, whereas a homogeneous equation permits only characteristic directions. The signature determines which of these two mechanisms governs a normalized one-sided power. More general holomorphic boundary values also admit moving smooth arguments, with a precise distinction between regular and critical limiting maps.

We use the complete boundary-power construction in Complex powers at a boundary, Lemma 1.1, Proposition 1.2, Theorem 2.1, Proposition 3.1, Corollary 3.2 and Theorem 4.1. Homogeneous extensions and angular moments, Theorems 1.2–3.1, constructs the angular representation and the unique continuous nonresonant extension. Point sources and complex Gaussian kernels, Theorem 1.1 and Lemma 2.1, proves the Laplacian source constants, symmetric diagonalization and the determinant branch. The parameter estimates and holomorphic identity argument are also developed in Complex quadratic powers and the Cauchy kernel, Section 1.

The supplied smooth pullback reading, Sections 18.1–18.5, proves ordinary wavefront localization, coordinate covariance and the exact submersion converse, especially (WF12)–(WF15). Its coordinate completion is Regular level sets and transverse convolution, Lemma 0.2. Cones and derivative-growth wavefronts, Theorem 6, proves ; Analytic equations and characteristic directions, Theorem 3 and its preceding finite transpose construction, proves analytic regularity outside the characteristic set. The Fourier, Gamma and angular foundations supply the integrals, recurrence and sphere measure used below.

The first classifications use ordinary smooth wavefront. Theorem 11 proves the same sets at analytic regularity and in every derivative-growth class defined below. Fourier transformation uses . Pairings are complex bilinear.

Scalar directions and the two boundary signs

Write Here is the entire normalized family initially equal to . In particular and for . The boundary-power reading proves these identities as distributions.

Lemma 1 (scalar wavefronts). For every real , For every real ,

Proof. Corollary 3.2 of the boundary-power reading gives The exponent at zero is greater than . For a compact smooth cutoff , the product-transform formula gives an absolutely convergent convolution with . If , its absolute value is at most The last integral follows by ; it is finite for . Taking arbitrarily large excludes every negative direction. The distribution is smooth away from zero. It is not smooth at zero, since on it equals , which is unbounded as . The ordinary wavefront projection theorem therefore supplies a direction at zero; conicity leaves exactly the positive ray. Complex conjugation gives the negative ray for .

For the logarithm, choose near zero. Theorem 4.1 of that reading gives The second term is smooth and compactly supported, since vanishes near zero. The first has rapid Fourier decrease on the negative ray by the preceding calculation. Fourier differentiation and division by there give the same decrease for . Its unbounded logarithm at zero supplies the positive ray. Conjugation again gives the lower sign.

Each is real, supported in , smooth away from zero and nonzero. At a negative integer it is a nonzero delta derivative. At every other real parameter it has the nonzero density on . If it were smooth at zero, its vanishing on the negative side would force every Taylor coefficient there to vanish. Taylor's estimate of order , compared with that density as , is impossible. A nonzero delta derivative is also not smooth: a continuous function supported at a point is zero. Thus zero is its singular base point. For a real cutoff and a real distribution, the Fourier values in opposite directions are complex conjugates. Regularity in one ray is consequently equivalent to regularity in the other. Projection and conicity give both rays. Reflection proves the assertion for .

Constructing the quadratic distributions through the vertex

Let , and let be a nonsingular real matrix. Define Let be its positive and negative eigenvalue counts; thus . These counts include the definite cases. The differential is nonzero whenever .

For put . On punctured space the submersion pullbacks and are defined. Their degree is : the pullback composition law and give this scaling on tests. This degree is nonresonant and greater than . Let be their unique homogeneous extensions through zero. For define instead These are locally integrable functions. Values on do not change them as distributions. The logarithm, rather than the constant power , is the two-dimensional point-source expression.

Lemma 2 (the actual boundary approach). For , For the corresponding logarithms converge to the in (Q5). All logarithms use argument in above the real axis and in below it.

Proof away from zero. Complete to smooth local coordinates . In them the upper argument is , with the positive smooth . We give a direct argument for every negative integer or half-integer .

If is not an integer, choose an integer with and set Then , and extends continuously across the boundary, including zero, from the upper half-plane. If , take and Differentiating first gives ; the next derivatives give . This is again continuous at zero.

On a fixed compact chart and for small , let Repeated chain differentiation gives . Integration by parts therefore pairs the power against a test by pairing against of that test. The coordinate Jacobian is included in the test. The functions converge uniformly on its compact support, while these transposed tests converge in every smooth seminorm to . The limit is exactly applied in the coordinate: the same calculation with identifies its boundary trace. A finite partition covers every compact punctured support. Conjugation proves the lower case.

Proof at the vertex. For each fixed , the power in (Q6) has absolute value bounded by , so it is locally integrable and homogeneous of degree on the whole space. Its global value is thus the unique nonresonant extension of its punctured restriction. The weak continuity of that extension in the homogeneous-extension reading, Theorem 3.1, transports the proved punctured convergence to every test on . No point-supported term is discarded.

For the logarithm write and . The upper logarithm is At a zero of its tangential derivative on the sphere is , since ; it is nonzero. Coordinate completion on the sphere therefore makes each such zero a regular surface. In those charts is integrable, by ; a finite cover handles the compact sphere. Where , it is bounded locally. For , off the measure-zero set : its argument is bounded by , its positive real part is bounded on the compact sphere, and its negative real part is at most . The radial bound now proves local integrability and dominated convergence to (Q5), using the full polar Jacobian. The definite case has no angular zeros and the same argument is simpler. Conjugation gives the lower limit.

The exact source and its signature

Let be the sphere area, and put

Theorem 3 (quadratic boundary sources). On the whole space, In particular the coefficient is nonzero for every signature.

Proof of the complex identity. For a complex symmetric with , let be the determinant branch proved in the point-source reading, and let use as in (Q4). We first establish For real positive , diagonalization supplies and . The test-level linear chain rule gives , while . The point-source reading proves and . These prove (Q8) for all real positive .

Both sides paired with any compact test are holomorphic in the independent complex entries of . Here is the integrability justification. On a compact subset of the positive-real-part domain, . Every parameter derivative of the power is bounded by a constant times , since each differentiating quadratic numerator cancels the degree of its additional denominator. This is locally integrable. For the logarithm the undifferentiated bound is ; every positive parameter derivative is bounded. The adjugate formula gives holomorphic, locally bounded derivatives of . Dominated difference quotients give coordinate holomorphy and joint continuity of the pairings; the polydisk proof in the complex-quadratic reading gives joint holomorphy.

In a small polydisk about , the paired difference in (Q8) vanishes when all entries are real and close to those of . Apply the one-variable identity principle successively to each independent entry, keeping the others real until their turn. The difference vanishes on that polydisk. The domain is convex. For any target , restriction to the complex parameter line , on a connected neighborhood of the real segment , propagates this zero to . This proves (Q8) throughout the domain.

Proof of the boundary constant. Set . The determinant-branch lemma gives For , with , the branch relation is Multiply (Q8) by , and use Lemma 2 and coefficient convergence. The limit on the left is . Dividing by , the phase on the right is Convergence with the varying differential operator is legitimate: each fixed derivative of the convergent distributions converges, and each matrix coefficient converges.

For , the logarithm relation has the additive constant . Its derivatives vanish, so the same limit of the second line of (Q8) gives the phase . The real coefficients of and complex conjugation give the lower sign in every dimension.

Theorem 4 (one-sided source coefficients). With the defined above, for every , Thus the first coefficient is nonzero exactly when is odd, and the second exactly when is odd.

Proof for . At the scalar identities are We verify all parameters needed here without assuming a reflection formula. At , the ordinary integrable boundary values on the two half-lines and give and . At , the proved jump gives both identities. Differentiation gives Together with and , these prove (Q11) by induction down each of the two parameter sequences. Those sequences contain every , .

Pull back (Q11) on punctured space and extend homogeneously. Linearity and uniqueness of the degree- extension give the same identities with . Apply Theorem 3. For the positive side the difference of phases gives Gamma recurrence and the sphere constant give ; the sine is . The second line of (Q11) gives . These are exactly (Q10).

Proof for . Formula (Q5) gives Theorem 3 consequently gives coefficients and its negative. Since , , proving (Q10).

Every wavefront covector, including the vertex

Use the following subsets of :

Theorem 5 (the full quadratic wavefronts). For every nonsingular real symmetric and every , The first line uses the logarithmic when . Empty quadratic cones and empty null fibers are allowed.

Proof away from the vertex. Since there, the exact submersion wavefront equality (WF15) and Lemma 1 give precisely for and for . These distributions are smooth where . Thus no nonvertex covector has been omitted or added.

Proof at a nonzero point source. A finite constant-coefficient differential operator cannot enlarge ordinary wavefront. To recall the local argument, choose cutoffs inside a neighborhood on which a compact representative of has rapid decay in a cone. Smooth localization gives that same decay, on a smaller cone, for every smooth compact multiplier there. The identity then proves the derivative inclusion: its first term acquires only a Fourier polynomial, and the second is another localized multiplier. Iterate and take finite sums. Also every cutoff equal to one near zero leaves . Hence Theorem 3 applies this to each . The nonzero cases of Theorem 4 give the odd-signature cases for .

Proof when the source vanishes. In an even-signature case, Theorem 4 gives everywhere. The analytic differential regularity theorem applies to this constant-coefficient operator, whose symbol in the convention is . It gives This proves the required upper bound at zero, including when is definite.

For the lower bound retain any with , and let Each , so the already proved full nonvertex conormal formula contains , with multiplier . Ordinary wavefront is closed in the punctured cotangent bundle; its complement is open directly from the cutoff-and-cone definition. Letting , while retaining the same nonzero , gives . This proves every point of . Together with (Q15) it proves equality and completes all cases.

For , the left panel shows the base section and the two normal signs at ; the displayed normal arrows are scaled copies of . The other panels show unit covectors in the section , not the whole three-dimensional fiber. The boundary values and the positive-side have every vertex direction. The negative-side has exactly the null ones; in this section they are the four points . Theorem 5 proves the complete sets. Reproducible figure source.

Forward wave powers retain the whole vertex fiber

Let be the spatial dimension, an integer, real, and . The full construction in Causal point sources and characteristic cones, Lemma 4.0, and Wave powers and complex potentials, Theorem 1.1, supplies with , . On , Its global value and vertex source are already supplied by that construction.

Corollary 6 (all forward wave powers and speeds). Put Then , and The backward kernel obtained by has the same full vertex fiber and the corresponding cone.

Proof. For , the exact linear chain rule gives , and . Iteration proves the displayed source identity, with the factor retained for every . At each nonzero forward cone point, the local formula has a nonzero constant and the scalar family of Lemma 1. Its exact submersion pullback therefore has both conormal signs. Away from that cone and the vertex the kernel is smooth: its interior formula is a smooth power, or zero at a delta-derivative parameter, and it vanishes outside its supporting cone. Finally (Q14), applied to , gives every nonzero vertex covector. Coordinate covariance transports the cone differential to . Time reflection has absolute determinant one, preserves the source and gives the backward statement.

Boundary values along arbitrary smooth approaches

The boundary limit also survives a moving complex argument. The hypotheses concern its real limiting map and its convergence in the smooth topology; the height need not be proportional to the parameter, and the real part may move.

We use Cauchy kernels and distributional boundary limits, Theorem 3.1 and Corollary 3.2, for the horizontal boundary distribution on finite-regularity tests. Gluing holomorphic sides, Lemma 1.1, constructs an -fold primitive with logarithmic growth. The exact submersion construction is the supplied smooth pullback reading, Section 18.5; its completed coordinates and Jacobian are supplied by the coordinate-completion lemma cited at the start of this lesson.

Let be an open interval, let , and set Let be holomorphic on . On each compact interval , assume a bound where is a nonnegative integer and . Larger compact intervals may require larger integers. These local bounds give the boundary distribution .

Lemma 7 (a continuous primitive at the boundary). Locally near any compact subinterval, choose a slightly larger interval on which (A2) holds with exponent , and put . There is a holomorphic , continuous up to that real interval, such that In particular, this representation acts continuously on compact tests.

Proof. The gluing lesson's Lemma 1.1 constructs a holomorphic with and uniformly on each retained compact horizontal interval. For , take ; its boundedness implies the displayed, weaker bound. Choose a reference point and height inside a slightly larger rectangle, and define The fundamental theorem and give and , exactly as in that primitive construction. Thus is holomorphic and .