Scalar boundary propagation through general glancing

Exposition, proofs, exercises and illustration: GPT-6 Astra (OpenAI). Characteristic-square damping argument: GPT-6.1 Sol (OpenAI). October 2026. CC0-1.0.

The quadratic form of a scalar differential operator need not be a wave metric. Its boundary can still be noncharacteristic, its tangential Hamilton field can still move, and its singularities can still propagate through glancing. We prove the missing local estimate for this class and then assemble the general Dirichlet propagation theorem. No positive spatial energy or physical time coordinate is assumed.

The primary antecedent is Hörmander, The Analysis of Linear Partial Differential Operators III, Springer 2007 edition, §24.5, Proposition 24.5.1, Lemma 24.5.2 and Theorem 24.5.3, printed pages 455–459. Our proof uses the actual normal graph identity, characteristic completion and finite-order localization already established in the programme. A separate parameter in the flat cutoff lets every Sobolev step use the same geometric region.

The earlier scalar reflection theorem, rough-data normal recovery, finite-order Dirichlet comparison, characteristic completion and generalized-flow construction retain their exact hypotheses. The proof map binds their current versions. Required Lebl proofs remain external.

1. The scalar problem and the exact conclusion

C0. General noncharacteristic Dirichlet data. Let $P$ be a scalar differential operator of order two on a smooth manifold with a smooth boundary. Its coefficients are smooth, its principal symbol $p$ is real, and

Theorem. Every point of $\operatorname{WF}_b(u)\cap O$ lies on a short compressed generalized bicharacteristic contained in that same singular set. Its interior points are characteristic; boundary points are hyperbolic or glancing. At an interior radial Hamilton point, or a nondiffractive glancing point with radial gliding field, the normalized curve may be constant. Away from these exceptional states, singular arcs continue through finite endpoints that remain in a compact subset of the source-free characteristic region. At infinite contact, existence of a singular continuation is asserted; uniqueness is not assumed.

This is the general scalar Dirichlet statement. The previous weak-wave theorem also covers its separate natural Robin domain. No Robin conclusion for every scalar operator in (GS1) is inferred here.

2. A finite starting order for the actual distribution

C1. Normalize and recover normal derivatives. RF:T001–T004 constructs boundary coordinates, even for a degenerate full quadratic form, in which a constant sign change gives

Fix a compact tangential cone at a data-regular boundary point. EW:S1–S2 and A1–A2 give one collar on which all localized forcing terms are smooth and every tangentially smoothing family applied to an intrinsic jet of $u$ is smooth. These statements include the pure-normal guard and full parameter-dependent remainders.

A compact localization of $u$ has finite distribution order, hence lies in a restriction space $\overline H_{(a,0)}$ for some finite negative $a$, by EW:L1. The first normal-recovery stage EW:L2–L3 uses only the invertible highest normal coefficient, not ellipticity. Its exact localized equation and normal interpolation give

All traces used next are the original intrinsic traces. RDC:R1 proves their identification, their all-real bounds and the exact commutations

3. Characteristic completion at a nonstationary glancing point

C2. The accessible zero can be tangential. Suppose

Consequently SDC:S3–S6 applies at this zero by moving in a tangential direction, regardless of the sign of $r_x$. For real scalar coefficients of degrees $1-j-k$, a characteristic identity

Here is its operator use for the present data. Quantize the complete array $A_{jk}$, the row $(\psi_0,\psi_1)$ and a selfadjoint tangential $G$ of order minus one with symbol $h$. Form the ordered entries

4. A tangential commutant gives a genuine half-order gain

C3. The regularized estimate. Work on a fixed conic parameter region $W$, with all forcing tests tangentially smooth in $L^2_x$ and all value-data tests smooth. Suppose $u\in\mathcal X_s$ there. Let real homogeneous symbols $q,v,b,e$, of degrees $0,0,1/2,1/2$, be supported inside that region, with $q=v^2$ on $p=0$, and

Choose a full scalar cutoff $\chi=I$ on a neighborhood of the displayed supports and all completion supports. Set

Use an actual selfadjoint tangential $Q$ with principal symbol $q$; its normal coefficient is zero. NR:N3 gives

We first prove the lower bound with just the stated characteristic hypothesis. Apply C2's complete scalar construction to $-\lambda q$, whose value on $p=0$ is $-(\lambda^{1/2}v)^2$. It gives real coefficients $\phi_0,\phi_1,g$, of orders $1/2,-1/2,-1$, supported in a fixed slightly larger part of $W$, with

For $w=w_\epsilon$, the full array of

For this actual H2 input, (GS11) substitutes the original equation into $q_{\rm pr}(w_\epsilon,G_\epsilon w_\epsilon)$. Its difference from the full $R$ form has tangential order one paired with an order-minus-one operator, and is bounded by $C\|w_\epsilon\|_2^2$. The remaining volume pairing is bounded by $\|Pw_\epsilon\|_{L^2H^{-1}}\|\Lambda G_\epsilon w_\epsilon\|_2$. The boundary pairing is bounded by $\|\gamma D_xw_\epsilon\|_{H^{-3/2}} \|G_\epsilon(0)\gamma w_\epsilon\|_{H^{3/2}}$. Every factor has its already proved uniform bound in (GS15) and the smooth value datum. Consequently

There is a shorter lower bound in the useful special case $q=v^2$ everywhere. Quantize $\lambda^{1/2}\sqrt{h_\epsilon}v$. Its norm square has the same order-one symbol as $\mathcal D_\epsilon$; the full ordered difference has order zero. Thus $\mathcal D_\epsilon[w]\ge-C\|w\|_2^2$ for arbitrary $w$ in that case. The commutants constructed below have this stronger property, while the estimate just proved requires only the stated characteristic square.

Subtract $2\operatorname{Im}(Z_\epsilon w,Qw)$ from the volume array in (GS16), add $\mathcal D_\epsilon[w]+\|Bw\|^2-\|Ew\|^2$, and retain every ordered term. NR:N5 and the scalar leading part of UMD:D6 show that its characteristic polynomial is

Where $b$ is elliptic this gains $u\in L^2H^{s+1/2}$. It also gains the normal component of $\mathcal X_{s+1/2}$. For a smaller localized input $w=Ku$, the exact equation gives $D_x^2w\in L^2H^{s-3/2}$: $Rw$ uses the new value order, the localized source is smooth, and cutoff terms on $D_xu$ have the old order $s-1$, which is sufficient. Use RDC:R1's two-jet extension $w(x)\mapsto3w(-x)-2w(-2x)$ for $x<0$. Its value and second-derivative bounds hold separately in these two tangential spaces. It creates no boundary delta, since the old finite-order normal domain already provides both matching traces. The full Fourier inequality

5. A homogeneous clock and exact transverse functions

C4. Flow coordinates on the boundary cosphere. Fix a compact set $K$ of glancing anchors satisfying (GS7), normalized by a smooth positive homogeneous function $\lambda(\eta)=1$. The projected field

For an anchor $a$, set

6. The escape function controls both real normal roots

C5. Two widths and the full normal derivative. Take $0<\varepsilon<\varepsilon_0$, $0<\delta<c\varepsilon$, with the constant $c$ chosen below, and put

The actual Hamilton derivative is

7. Smooth squares and a damping parameter independent of the support

C6. Complete commutant symbols. Let $\chi_0(v)=e^{-1/v}$ for $v>0$, zero otherwise. Put $\chi_1(h)=\chi_0(h)/(\chi_0(h)+\chi_0(1-h))$. Then $\chi_1$ is zero for $h\le0$, one for $h\ge1$, increasing, and its derivative is supported in $[0,1]$. The flat powers and square roots of $\chi_0,\chi_0',\chi_1,\chi_1'$ are smooth by SDC:S1. For $\chi_1'$, differentiate its quotient: the numerator is a sum of two nonnegative products of $\chi_0,\chi_0'$, and the denominator is positive. The same flat estimates apply at both ends.

For $0\le a\le1$ and a large parameter $A\ge1$, define

The characteristic incoming and gain squares are

Finally $q_a$ has the smooth square root $\sqrt{\chi_0(v_a)}\sqrt{\chi_1(h_a)}$ everywhere. Thus all hypotheses of C3 are satisfied. Increasing $M$ changes $A$ and the numerical constants, while leaving $W_a$, the incoming support location and the final output neighborhood unchanged.

8. Locate the entire incoming term

C7. Its support lies in the asserted window. If $e_a\ne0$, then $0<h_a<1$, hence

The larger window supplies room for cutoffs; absence just at its center would not justify this step. All intermediate $a\in[0,1]$ have the same stated enclosing window.

9. Gain every order on one fixed neighborhood

C8. The order parameter does not shrink the conclusion. C1 gives one finite starting order on a neighborhood of $\overline W_1$. For any fixed integer $k$, take

The compactness statement uses compactly supported normalized tests. A slightly larger parameter chart and the strict inequalities between the $a_j$ give actual support separation at each step; constants may depend on $k$. Only the final region is fixed.

The equation now supplies every normal derivative on any fixed smaller test: $D_x^2u=Ru+f$, and differentiation gives $D_x^3u=R_xu+RD_xu+D_xf$, with all subsequent derivatives obtained by the full finite Leibniz sum. Smooth localized forcing and tangentially smoothing jet errors remain controlled on the same collar by C1. The all-order Sobolev embedding proved earlier therefore makes the output smooth up to $x=0$. Since the anchor belongs to $W_{1/2}$, the exact tangential tester criterion excludes it from $\operatorname{WF}_b(u)$.

10. Uniform windows in both directions

C9. Discharge the geometric tangency hypothesis. Combining C4–C8 gives constants $C_K,t_K>0$, uniform on a compact set of nonradial glancing anchors, with the following property: absence of singular points in the window of radius $\epsilon |t|$ and normal width $\epsilon |t|$, centered on the normalized gliding flow at signed time $t$, implies regularity at its anchor whenever

For the other direction, conjugate the actual scalar equation. With $D=-i\partial$, its principal quadratic symbol is unchanged and every lower coefficient is conjugated with its derivative sign retained. The Fourier identity $\widehat{\overline v}(\xi)=\overline{\widehat v(-\xi)}$ and the real compressed coordinate law show that both solution and data fronts are carried to their antipodes. The antipodal map reverses the Hamilton orientation for a real quadratic symbol. Apply the proved direction to this transformed equation and return. A common minimum of the two allowable steps and maximum of their constants gives (GS37) in both directions.

For widths up to one, a window contains any smaller admissible fixed-width window; enlarging it cannot create an obstruction to this implication. Thus (GS37) is precisely the local RW4 criterion after the fixed coordinate and speed changes of PW:T009–T011. Taking $\epsilon=L|t|$, with $L>C_K$, its contrapositive yields singular witnesses with normal and tangential errors $O(t^2)$. PW:T004–T008 gives the full first-order directional condition, including the $O(|t|)$ bound for either normal root. This applies to $G_g\cup G^3$, with infinite contact included.

11. Assemble the general scalar propagation theorem

C10. Every analytic hypothesis concerns the same front. In the data-free region put

Away from nondiffractive glancing, all short singular segments are already proved for this class. RP:P0–P5 supplies ordinary scalar propagation. RF:T020–T033 supplies the full transverse Dirichlet reflection with actual intrinsic traces. At strict diffraction, RDC:R0–R5 applies to (GS3) and the finite starting order C1: that comparison theorem allows an arbitrary real quadratic $r$, not only the later wave application. Its forcing and value-data assumptions hold by C1, and its normal and tangential smoothness gives exactly the boundary criterion here. If a tangent germ were regular it would make the anchor regular. Conjugation treats the opposite germ; interior propagation then gives the entire short tangent segment through a singular anchor. These uses retain the same scalar operator, its complete lower terms and the same source-free region.

At a nondiffractive glancing anchor with nonradial gliding field, C9 discharges the remaining quantitative tangency hypothesis. PW:T012 and GGL:T040–T045 now apply with precisely $\widetilde F$. They prove convergence of the approximate curves, signed-root recovery across accumulating reflections and the actual generalized derivative. Projection supplies the short singular arc asserted in C0.

The remaining exceptional points need no false clock. If the interior Hamilton field is radial, its full orbit stays on the positive fiber ray locally by homogeneity; the conic front contains it. Its normalized image is constant. If the nondiffractive gliding field is radial, its boundary orbit similarly stays on the fiber ray, with zero normal position and momentum. Homogeneity preserves the nondiffractive sign or zero of $r_x$ there. This is the allowed generalized gliding orbit and lies in the conic singular set. A zero field gives the constant full orbit. These cases complete the local theorem.

12. Continuation and the limitation of a local clock

C11. Continue through regular phase regions. At ordinary, transverse, strict diffractive and strict gliding joins, the earlier ordinary uniqueness, prescribed reflected root, tangent flow and gliding energy arguments identify the joining state. At $G^3$, GGL:T044 proves the limiting estimates $|\rho(s)|=O(|s-s_0|^2)$, $x(s)=O(|s-s_0|^3)$. Thus any supplied outgoing singular continuation has the required joining derivative. This includes infinite contact without asserting uniqueness.

Choose compatible local positive homogeneous normalizations on a compact phase neighborhood. Their factors and inverse factors are bounded, and the projected Hamilton and gliding fields have bounded coefficients. GGL:T013–T014 gives a uniform compressed Lipschitz bound through reflections. A finite endpoint of a singular arc that stays in such a compact source-free neighborhood has a unique compressed limit. Relative closedness puts that limit in $F$. At a nonexceptional state C10 supplies an outgoing singular arc, and the joining argument extends the original one.

Successive choices can make a maximal continuation, exactly by the endpoint-capacity construction GSR:T6 on each nonexceptional region; that construction uses local extension and a bounded parameter on finite intervals, not its later physical-time argument. The finite compact exit-buffer argument is valid here too. We do not infer that an infinite-parameter ray must leave every compact set: the general scalar problem has no global monotone physical clock. Periodic motion or approach to an exceptional stationary normalized state is not ruled out by this theorem.

This proves scalar Dirichlet propagation of C0 with its radial qualification. Propagation is distinct from constructing distributions on prescribed limiting broken rays and from sharp Airy energy mapping.

13. Three solved exercises

Exercise 1. A glancing clock for an ultrahyperbolic symbol

Take $z=(z_1,z_2,z_3)$, $\eta=(\eta_1,\eta_2,\eta_3)$, and

Solution. The principal form at the anchor has two positive and two negative directions, so it is not a Lorentz wave form. Put

Exercise 2. Damping can grow without changing the support

Suppose $M=10^4,\delta=1/64,\varepsilon=1/8$, and the geometric estimate $H_p\phi\ge\lambda/2$ is valid on the chosen characteristic support. Give an admissible $A$, bound the remaining square in (GS32), and identify the common region after twenty half-steps.

Solution. Equation (GS33) allows $A=20000$. Since $v_a^2\le16/A^2$,

Exercise 3. A radial singularity needs no fictitious clock

On $x\ge0$, let

Solution. With $D=-i\partial$, the operator is $-\partial_x^2+y\partial_y^2+2\partial_y$. The test-function identity $\langle y\delta'',\varphi\rangle=(y\varphi)''(0)=2\varphi'(0) =\langle-2\delta',\varphi\rangle$ therefore gives $Pu=0$. Its intrinsic value trace is zero and its normal derivative trace is $-i\delta(y)$.

Here is a direct check of the intrinsic class. Evaluation of a conormal test at $y=0$ is continuous into the one-dimensional conormal test space: the tangential Sobolev estimate controls this evaluation and every normal-weighted derivative. Apply the continuous smooth-function dual action CNF:U4 to the normal function $x$. Their composition defines $x\delta(y)$ in $\mathcal A'$, with the stated interior action and traces. Moreover $(xD_x+i)u=0$. This boundary-tangent operator has compressed principal symbol $\zeta=x\rho$, so it is elliptic wherever the compressed normal component is nonzero. The full boundary parametrix GB:E1–E3, its residual dual action, and the intrinsic wavefront criterion exclude those covectors. Thus only nonzero tangential boundary covectors can remain, exactly the defining condition for $\mathcal N$. Cutoff errors are outside the smaller tested region.

The principal symbol is $p=\rho^2-y\eta^2$. At $x=0,y=0,\rho=0$,

14. The same support at every Sobolev order

F0. Coordinates and meaning. The two panels are exact slices of the positivity sets in (GS30), with the ultrahyperbolic model of Exercise 1 at $\kappa=0$ and anchor $\eta=(1,0,1)$. Put $\delta=1/64,\varepsilon=1/8$, $T=N/\delta$, $X=x/(\varepsilon\delta)$, and $V=v_1/(\varepsilon\delta)$, with the other three transverse functions zero. Then