Completed continuation: general homogeneous symbol transport and complete symbol spaces supplies the arbitrary-map theorem beyond the specific lift in this source. The derivative-loss FIO analytic estimates and full course remain unfinished.

Representing the intrinsic class with a prescribed phase

The phase is given in advance. We prove that its oscillatory integral has the intrinsic iterated regularity, and that every member of the intrinsic class can be represented by that phase near its critical covector. The proof includes clean phases with excess, the exact order shift, a construction of the amplitude, and every differentiated symbol remainder.

The free human source for the Fourier-testing method and its normalization is Lars Hörmander, Fourier integral operators. I, Section 3.2, especially Theorem 3.2.4 and its parameter version, printed pages 149–154. Here the clean critical-family calculation is supplied by the earlier programme's complete stationary-phase proof. The amplitude construction below uses the proved symbol summation theorem. Neither a citation nor the definition of an oscillatory class is substituted for the intrinsic proof.

F0. Inputs, orders and support conventions

The exact earlier programme inputs are:

• C0–C5: finite rank facts, critical maps, conic frequency coordinates and the homogeneous function $H$;
• P2 O0–O6 and T0–W5: distributions, proper operators, coordinate transport and directional cutoffs;
• K0–K7: asymptotic sums, conic inverses, full intrinsic localization and independence of sufficiently small cutoffs and Lagrangian extensions;
• the graph criterion G12, in both directions, at the exact $B_{2,\infty}$ endpoint;
• U001 Sections 4–7 and Appendix A.4–A.6: parameter Morse coordinates, all scale and parameter derivatives of stationary remainders, clean critical families, densities, compact partitions and fibre integration;
• U001's linked finite-dimensional inverse theorem, compact integration and Fourier inversion proofs.

Let $x\in\mathbb R^n$, $n\ge1$, and $\theta\in\mathbb R^N\setminus0$, $N\ge1$. A real phase $\phi(x,\theta)$ is smooth on an open cone, is homogeneous of degree one in $\theta$, and has nonzero full differential. An ordinary amplitude $a\in S^\mu$ satisfies, on each compact base set,

For the local arguments, the amplitude has compact base support and its base/unit-direction support lies strictly inside the phase domain. It vanishes near $\theta=0$. Changes confined to a bounded frequency set give a smooth function and will be treated explicitly at the end.

Use the normalization

All integrals with nonintegrable amplitudes mean the distributional limit constructed in F2 below.

The critical set $C_\phi=\{\phi_\theta'=0\}$ is clean of excess $e$ when it is a smooth submanifold of dimension $n+e$ and

Here $0\le e\le N$. The usual nondegenerate phase is the case $e=0$, for which the implicit theorem supplies the submanifold and tangent condition directly. The word “clean” includes both conditions in (F3); the dimension of a zero set alone is insufficient.

F1. Clean geometry and a transverse Fourier test

The map

has rank $n$, kernel dimension $e$, and locally maps onto an embedded conic Lagrangian $\Lambda$.

Proof. A critical tangent $(v,w)$ obeys $\phi_{\theta x}''v+\phi_{\theta\theta}''w=0$. It is in $\ker d\kappa$ precisely when $v=0$ and

The matrix in (F5) is the transpose of $d(\phi_\theta')$. Its rank is $N-e$ by (F3), so its kernel has dimension $e$. Rank and nullity on the $(n+e)$-dimensional critical tangent give rank $n$ for $d\kappa$.

For completeness, this constant-rank assertion yields the required local image without importing a constant-rank theorem. In a critical chart choose $n$ output components of $\kappa$ with independent differentials. Complete them by $e$ input coordinates to a local coordinate system $(\lambda,t)$, using the inverse theorem. The other output components have zero $t$ derivatives: otherwise the output derivative would have rank greater than $n$. The fundamental theorem along coordinate segments in a small product box makes them independent of $t$. Thus the image is a smooth graph over $\lambda$, and the fibres are exactly the $t$ slices.

The full differential hypothesis gives $\phi_x'\ne0$ on the critical set. Euler's identity gives $\phi=0$ there. Hence $\phi_x'\cdot dx=0$ on critical tangents. Differentiate this identity in two critical coordinates and subtract; mixed derivatives cancel, giving $\sum d\phi_{x_j}'\wedge dx_j=0$ on image tangents. The image has dimension $n$, so it is Lagrangian. Positive dilation preserves the critical set and sends its image $(x,\xi)$ to $(x,r\xi)$. Saturating a sufficiently small image chart proves the conic germ assertion. $\square$

Apply C3–C4 to choose base coordinates in which this germ is

Only the germ is asserted; C4 provides a global degree-one extension after shrinking the angular cone. Use a smooth extension at bounded frequency whenever $e^{iH}$ is applied to a global Fourier transform. Its high-frequency value is unchanged.

For a unit covector $\omega$ in this cone, put $\Psi_\omega(x,\vartheta)=\phi(x,\vartheta)-x\cdot\omega$. Its critical set is the fibre $\kappa^{-1}(H'(\omega),\omega)$, of dimension $e$. Its full Hessian is

Its kernel consists exactly of the critical tangent vectors annihilated by $d\kappa$. Indeed its lower equation is (F3)'s tangent equation, and its upper equation sets the differential of the output covector to zero. On the graph (F6), that also sets the differential of the base point to zero. Conversely every fibre tangent satisfies both equations. Therefore

The phase is clean for each $\omega$, with critical value $-H(\omega)$: $\phi=0$ and $\omega\cdot H'(\omega)=H(\omega)$.

These fibres have smooth common adapted charts. On $C_\phi$ choose $(\xi,t)$ as above, now using the independent output coordinates $\xi$. Homogeneity lets $t$ be degree zero: choose its coordinates first on $|\xi|=1$ and extend along positive rays. Its normalized critical parametrization is

Choose complementary ambient coordinates $z\in\mathbb R^d$ at one fibre point and subtract their critical values, which are smooth in $(\omega,t)$. The inverse theorem gives $Y(\omega,t,z)$, with the critical fibres precisely $z=0$. To check that no equation was lost, the critical fibre already has dimension $e$, and its $t$ projection is a local diffeomorphism; shrink the chart so its inverse is the unique one. The transverse Hessian $Q(\omega,t)=\partial_z^2(\Psi_\omega\circ Y)(\omega,t,0)$ is invertible by (F8). U001 A.5 proves this also for a nonorthogonal complement. Its signature $\sigma$ is constant on a connected sufficiently small chart. On compact subcharts, the inverse Hessian, chart derivatives and

have the uniform bounds required by U001. These are actual finite-dimensional charts, not an additional clean-normal-form assumption.

F2. Distributional meaning and the noncompact-frequency tails

We first justify (F2). On the compact base/unit-direction support, split by a smooth degree-zero cutoff into a part where $|\phi_\theta'|\ge\epsilon$, and a part where $|\phi_x'|\ge c|\theta|$. Such a split exists: on the unit section the full differential is nonzero; near $\phi_\theta'=0$, compactness gives a positive lower bound for $|\phi_x'|$, which homogeneity scales by $|\theta|$. Use a cutoff supported in that latter neighborhood and equal to one on a smaller one. Its complement has the first bound.

On the first part the operator

Its coefficients are homogeneous of degree zero, with frequency derivatives of the corresponding negative degrees. In its transpose, a derivative either hits the amplitude, lowering its order by one, or hits a coefficient, also lowering the total order by one. Induction gives order $\mu-M$ after $M$ integrations by parts. Every fixed base derivative adds at most its number to that order, because it may differentiate the exponential. Thus arbitrarily large $M$ makes the integral and every base derivative absolutely convergent. This contribution is a smooth function of compact base support.

On the second part use $L_x=\phi_x'\cdot\partial_x/(i|\phi_x'|^2)$. Its coefficients and all base derivatives have order $-1$ in $\theta$. Transposition against a compact test function therefore lowers the amplitude order by one per application. For $M>\mu+N$ the integral is absolutely convergent, bounded by finitely many test-function seminorms. This defines a distribution of finite order.

These definitions agree with inserting a smooth frequency cutoff and letting its radius tend to infinity. Terms where a frequency derivative hits that cutoff have the same negative-order budget as derivatives of the amplitude; choose $M$ with an extra integrable power and use the compact-plus-tail limit argument of U001 Q1. This proves independence of the cutoff and justifies all integrations by parts just used. It also proves continuity in a finite list of amplitude seminorms for any specified distribution seminorm. The support is contained in the amplitude's base projection.

For Fourier reduction, discard the first, smooth part and write $\xi=R\omega$, $R\ge1$. On the remaining support,

If $|\theta|/R$ is sufficiently small or sufficiently large, the reverse triangle inequality gives $|\phi_x'-R\omega|\ge c_1(R+|\theta|)$. Integrate in $x$ with

Every base derivative of a coefficient is bounded by $C_\beta(R+|\theta|)^{-1}$. This follows by differentiating the quotient: each differentiated $\phi_x'$ is $O(|\theta|)$, and the denominator has the stated lower bound. After $M$ transpositions, the integral is bounded by

when $M>\max(\mu,0)+N$. The last estimate follows by $\theta=R\vartheta$ and the proved radial power bound. Every fixed $\xi$ derivative inserts bounded powers of the compact variable $x$, while differentiated coefficients improve or preserve the bound. A fixed number of angular or scale derivatives, or multiplication by $e^{iH(\xi)}$, costs at most a fixed power of $R$. Increasing $M$ absorbs that power. Thus these tails are smoothing symbols with all derivatives.

We are left with $c_0\le|\vartheta|\le C_0$ after $\theta=R\vartheta$. This is a fixed compact integration region in $(x,\vartheta)$. Outside small adapted neighborhoods of the critical fibres, the gradient of $\Psi_\omega$ has a positive minimum on each compact parameter set. U001's complete nonstationary estimate applies there. This proves every far-region assertion needed below; the compact stationary theorem is never applied to an unbounded phase-variable domain.

It also proves $\operatorname{WF}(I_\phi(a))\subset\kappa(C_\phi\cap \operatorname{cone\,supp}a)$ locally. If an output base/covector neighborhood misses this image, insert a base cutoff there and use the same estimates with no stationary contribution. The Fourier criterion W1 then applies. The statement remains valid with the closed essential support of the amplitude: on a fixed neighborhood where it is smoothing, choose its order as negative as required in (F14) and in the compact estimates. Finite compact partitions give the assertion for supports meeting several critical charts.

F3. The full Fourier symbol and its exact leading fibre integral

Restrict the amplitude to one of the small charts in F1, with a fixed compact normalized support. Extend $H$ as in (F6) and define

Then $T_\phi a\in S^r$. On the output cone it has a full expansion whose leading term is

For $e=0$, the integral means the single value at the isolated critical point. For every integer $k\ge0$, there are linear coefficient operations $T_j$ with

All assertions are continuous in the relevant finite lists of symbol seminorms. Coefficients use only finitely many amplitude derivatives on the critical fibre.

Proof. By F2, only the compact scaled integral remains. Its prefactor is $c_{n,N}R^N$. Its phase is $R\Psi_\omega$ and has $d=n+N-e$ normal variables. U001's parameter Morse construction and clean theorem apply in the charts already proved to exist. The transformed amplitude

times the fixed smooth cutoffs and Jacobian is a symbol of order $\mu$ in $R$, with every $R$ derivative lowering its order by one. Indeed a derivative in a compact chart variable can create $R\partial_\theta$, which has order zero cost by (F1); a scale derivative creates $\vartheta\cdot\partial_\theta$, of order $-1$. The same count applies to every product-rule term. The bounds are uniform because $|\vartheta|$ is bounded above and below in the scaled compact region.

After removing the critical value $-H(\omega)$, U001 Corollary 6.1 and Theorem 7.2 give a normal factor $(2\pi/R)^{d/2}$, signature $e^{i\pi\sigma/4}$, and a remainder of order $\mu-k$ before that factor. Thus the total power is

The leading quotient density is exactly $J|\det Q|^{-1/2}\,dt$, by U001 A.5. This proves (F16), including its signs and constants. Compact tangential integration preserves all remainder bounds by U001 A.6.

For each scale derivative $\partial_R^\ell$ and angular derivative $\partial_\omega^\beta$, the remainder after the prefactor has bound $C R^{r-k-\ell}$. In polar coordinates each Cartesian derivative $\partial_{\xi_j}$ is a bounded angular coefficient times $\partial_R$, plus $R^{-1}$ times an angular derivative. Iterating this identity proves $C_\alpha R^{r-k-|\alpha|}$, the full ordinary-symbol estimate. The same argument proves the coefficient bounds. The tails in F2 have every negative order, so they do not alter the expansion. Outside a slightly larger output cone there are no critical points, giving rapid decay with all derivatives. At bounded frequency the Fourier transform of a compactly supported distribution is smooth, by T0 and differentiation of its compact test pairing. These observations prove the global $S^r$ claim and the finite-seminorm continuity. $\square$

The formula has a useful locality property. If the amplitude is smoothing near all critical points over an output angular neighborhood, every coefficient in (F17) is smoothing there; the arbitrary remainder order proves the same for $T_\phi a$. This uses the whole expansion, not only the leading coefficient.

F4. A symbol lift for the given phase

Choose a smaller closed angular set $\Sigma$ strictly inside the output cone. Let $\chi$ be a smooth angular cutoff, equal to one on a neighborhood of $\Sigma$, with compact support inside that cone. There is a linear extension operation

whose amplitudes have one fixed compact base/angular support inside the prescribed phase chart, such that

It preserves angular essential support at the critical set.

Here is the complete construction. Near the selected critical ray, $\phi_x'\ne0$. Put $\xi=\phi_x'(x,\theta)$. Choose $N-e$ independent components $v$ of $\phi_\theta'$. Their common zero set equals $C_\phi$ after shrinking: both are submanifolds of the same dimension, the selected differentials have the full normal rank, and the inverse chart makes inclusion an equality in a neighborhood. Choose the degree-zero coordinates $t$ from F1. They can be extended smoothly off the critical set by a normalized ambient chart. Then

is a local diffeomorphism. To check its derivative, a vector with $dv=0$ is critical tangent; on that tangent $(d\xi,dt)$ is an isomorphism. Thus its kernel is zero, and dimension gives invertibility. The inverse theorem applies. The map has weights $(1,0,0)$ under dilation of $\theta$; uniqueness of the inverse proves the same weighted homogeneity for its inverse. Equivalently work first on $|\xi|=1$ and then dilate.

Choose a nonnegative smooth $p(t)$ with compact support inside the $t$ box and $\int p(t)\,dt=1$. A positive bump has positive integral because it is bounded below by a positive number on a smaller rectangle; divide by its finite positive integral. In dimension zero set $p=1$. Choose a smooth compact $\eta(v)$ equal to one near $v=0$. In (F22), at sufficiently large $R=|\xi|$, define

Use an additional high-frequency cutoff and extend by zero outside the compact normalized coordinate box. The supports of $p,\chi,\eta$ are strictly inside that box, so extension is smooth. The weight $W$ is smooth and never zero; its reciprocal and all needed derivatives are bounded on this compact set. Choose the same charts for (F16) and (F23).

The scale $R$ is comparable to $|\theta|$ on this support. Every frequency derivative of a degree-zero coordinate or smooth weight costs one inverse power; every base derivative costs no power. A frequency derivative of $\xi=\phi_x'$ has degree zero, whereas a base derivative has degree one. In a chain-rule term for $b(\xi)$, that latter degree exactly compensates the order lost by differentiating $b$. The iterated product and chain rules therefore prove every estimate in (F20), with finite-seminorm continuity. Substitution on $v=0$ into (F16) cancels the weight and the power of $R$; $\int p=1$ proves (F21). The bounded-frequency modification is smoothing. The same derivative estimates show that if $b$ is smoothing on an angular neighborhood, $Eb$ is smoothing near its critical inverse image.

F5. Constructing an amplitude to every order

Suppose $b\in S^r$ has angular essential support inside $\Sigma$. This means that on every closed angular set outside $\Sigma$, it has all negative orders with all derivatives. Then there is an amplitude $a\in S^{r-\delta}$, with the fixed support allowed in F4, such that

Proof. Set $b_0=b$ and, recursively,

F3–F4 give $b_{j+1}\in S^{r-j-1}$ whenever $b_j\in S^{r-j}$ has essential support in $\Sigma$. Indeed $T_\phi Eb_j-\chi b_j$ has one lower order, and $(1-\chi)b_j$ is smoothing, since $\chi=1$ on a neighborhood of that fixed closed set.

The essential-support condition also persists. On an angular neighborhood disjoint from $\Sigma$, the critical jets of $Eb_j$ are smoothing by F4, and F3's locality statement makes $T_\phi Eb_j$ smoothing. Subtracting preserves this property. On compact angular sets, finitely many such neighborhoods give uniform seminorms. Thus the induction holds for every $j$, on the same $\Sigma$ and the same amplitude support; no endless shrinking of cones occurs.

We have $a_j\in S^{r-\delta-j}$. Apply K1's proved support-preserving summation theorem in the frequency variable $\theta$, with base variable $x$, to obtain

All terms have the same compact base/angular support, so the sum keeps that support. F3's continuous order bound sends this remainder into $S^{r-k}$. The finite identity from (F25) is $b-T_\phi\sum_{j<k}a_j=b_k\in S^{r-k}$. Consequently $b-T_\phi a\in S^{r-k}$ for every $k$, which proves (F24). This is asymptotic summation with each remainder justified, not an infinite operator series assumed to converge. $\square$

F6. The prescribed-phase representation theorem

In the conventions of (F2), a clean phase of excess $e$ uses amplitude order

Near a chosen critical covector, the following assertions hold.

1. Every amplitude of this order with sufficiently small compact base/angular support defines a distribution in the intrinsic class $I^m(\Lambda)$ microlocally.
2. Every $u\in I^m(\Lambda)$ microlocally near that covector can, after a sufficiently small proper order-zero cutoff there, be written as $I_\phi(a)$ modulo a smooth function, with this same prescribed phase and amplitude order.
3. For $e=0$ the order is $m+n/4-N/2$. If instead a clean phase uses that nondegenerate amplitude order, its intrinsic order is $m+e/2$.

Proof of the forward assertion. Choose the base coordinates (F6). F3 gives $b=T_\phi a\in S^{m-n/4}$ and the exact Fourier identity $\widehat{I_\phi(a)}=e^{-iH}b$. The reverse direction of the proved graph criterion G12 gives intrinsic membership for the graph of the extended $H$. F2 confines its wavefront to the original local phase image. A proper conic cutoff inside the region where that graph and $\Lambda$ agree transfers membership by K7. Coordinate and frame invariance is the already proved T3. Finite amplitude partitions handle any compact normalized support covered by such small charts; the part away from $C_\phi$ is smooth by F2.

Proof of the converse. By the microlocal hypothesis there is an elliptic order-zero test with output in $I^m(\Lambda)$. K7 lets us replace it by any sufficiently small compact proper conic cutoff $P$ elliptic at the chosen point. Choose its essential support inside the phase branch and with output directions in a fixed closed angular set $\Sigma$ strictly inside the frequency chart. Put $v=Pu$. It has compact base support, lies in $I^m$ of the extended graph by K7, and has wavefront directions only in $\Sigma$. The forward direction of G12 gives

We justify the essential-support assertion needed in F5. On a closed angular set outside $\Sigma$, every point of the compact base support has a base cutoff whose Fourier transform is rapidly decreasing in a neighboring cone, by the definition of wavefront. Choose finitely many smaller base neighborhoods and their finite smooth partition, and finitely many angular neighborhoods on the prescribed closed set. W1's convolution cutoff estimate allows the partition cutoffs; adding the finite local Fourier transforms gives uniform rapid decay of $\widehat v$. For $\partial_\xi^\alpha\widehat v$, apply the same argument to $(-ix)^\alpha v$, which has the same compact support and no additional wavefront by W4. Derivatives of $e^{iH}$ have polynomial bounds, so the product retains every negative order there. Thus $b$ has angular essential support in $\Sigma$.

F5 constructs $a\in S^\mu$ with (F24). Multiplication by $e^{-iH}$ preserves Schwartz decay with all derivatives, since the extension is smooth and its derivatives are polynomially bounded. Fourier inversion therefore makes $v-I_\phi(a)$ smooth, indeed Schwartz in this coordinate chart. This is the claimed representation. The identities in (F27) prove assertion 3. $\square$

If literal equality on a smaller base neighborhood is needed, the smooth remainder can be absorbed into a bounded-frequency amplitude. Let $f(x)$ be that remainder localized by a base cutoff. Choose a compact frequency bump $q(\theta)$, away from zero, with $\int q=1$, so that its product with this small base support lies inside the prescribed phase domain. Then

This amplitude is smoothing. Low frequencies thus do not require a different phase or a lost order. The theorem still concerns a microlocally cut off distribution; it makes no assertion about untested directions of the original $u$.

F6a. Reading a lower order on a smaller output cone

Suppose $v=I_\phi(a)$ has the small compact support used above, and $b=e^{iH}\widehat v\in S^r$. Near a chosen graph covector, $v$ has intrinsic order $m-k$ if and only if $b$ has symbol order $r-k$ on a sufficiently small angular cone. Here is a proof that does not assume a new operator transformation formula.

For the forward implication, K7 gives a compact proper $P$ with full symbol one near that covector, and $Pv\in I^{m-k}$ of the graph. G12 gives $e^{iH}\widehat{Pv}\in S^{r-k}$. The wavefront of $v$ is contained in this frequency graph. On a smaller angular cone its only possible base point is $H'(\omega)$, where the symbol of $I-P$ is smoothing. W4 therefore shows that $v-Pv$ has no wavefront in any base point with these frequency directions. It has compact support. The finite base-cover argument in the proof of (F28) makes its Fourier transform rapidly decreasing there with all derivatives. Adding this smoothing error proves the implication.

Conversely suppose $b$ has order $r-k$ on an open angular cone. Choose a smooth angular $\chi_0$ supported strictly inside it, equal to one on a smaller cone, and truncate it at low frequency. Then $b_1=\chi_0b\in S^{r-k}$ globally, and $w=\mathcal F^{-1}(e^{-iH}b_1)$ belongs to $I^{m-k}$ by G12. The Fourier transform of $v-w$ vanishes in that smaller cone at high frequency and is polynomially bounded elsewhere. W1's convolution cutoff estimate makes every compact base localization rapidly decreasing on a still smaller cone. Thus $v-w$ has no wavefront there. A compact proper conic cutoff $P$, elliptic at the given covector and supported in this cone, sends $v-w$ to a smooth section by W4. K6 sends $w$ into $I^{m-k}$. Consequently $Pv\in I^{m-k}$, which is the claimed microlocal assertion. All shrinking is finite, for the specified $k$ and covector.

F7. The first lower-order criterion and clean cancellation

Let $a\in S^\mu$ be as in F6. Its contribution has intrinsic order $m-1$, on the tested cone, if and only if

In fact F3 gives $T_\phi a-{\cal L}_\phi a$ in that lower order, and the two implications of F6a identify intrinsic order with the Fourier-symbol order on a smaller cone. This proves both directions of (F30) without assuming a composition formula for general Fourier-integral operators.

For a nondegenerate phase the weight in (F16) is nonzero and there is one critical point; (F30) is precisely that the restricted amplitude has one lower order. For a clean phase it is the weighted fibre integral that must have lower order. Pointwise vanishing of the amplitude on the entire critical fibre is sufficient but unnecessary. All lower orders are recovered by applying (F17) with enough terms: the exact criterion for order $m-k$ is that the sum of its first $k$ coefficients have Fourier-symbol order at most $m-n/4-k$. The remainder already has that order. This states the full condition rather than discarding lower coefficients after a leading cancellation.

Two models with complete solutions

Exercise F1: a quadratic stabilization. In one base dimension let $t>0$, $s\in\mathbb R$, and $\phi(x,t,s)=xt+s^2/(2t)$. Determine the critical Lagrangian, the excess, and the leading Fourier symbol for a prescribed amplitude.

Solution. The critical equations are $x-s^2/(2t^2)=0$ and $s/t=0$, hence $x=s=0$. Their differentials are independent there. Thus $N=2,e=0$, and the output is $(x,\xi)=(0,t)$, with $H=0$. At output $\omega=1$, in variables $(x,t,s)$, the Hessian of $\phi-x$ is

The upper two-dimensional block has one positive and one negative direction, by the change $(x,t)\mapsto((x+t)/\sqrt2,(x-t)/\sqrt2)$. Here $J=1,\delta=1/2$, so

The required amplitude order is $\mu=m-3/4$; the output order is $m-1/4$. This checks the positive Fresnel sign and the half-order gained by adding the nondegenerate variable $s$.

Exercise F2: a redundant variable and cancellation. Keep $t>0$, put $\phi(x,t,s)=xt$, and restrict $s/t$ to a compact interval. Compute the clean fibre weight and exhibit a nonzero amplitude with zero distribution.

Solution. Now $C_\phi=\{x=0\}$, because $\phi_t'=x$ and $\phi_s'=0$. Its dimension is two, so $N=2,e=1$. The output is again $(0,t)$. At $\omega=1$ the critical fibre has $t=1$, with free coordinate $s$; the normal Hessian in $(x,t)$ is the upper block of (F31). Its determinant has absolute value one and its signature is zero. Thus $J=1,\delta=1$, and

The amplitude order required for intrinsic $m$ is $m-5/4$, exactly a half-order lower than for the nondegenerate phase in Exercise F1.

Choose a nonzero compact smooth $g(\tau)$ and an amplitude $c(x,t)$, supported at $t\ge1$ and compactly in $x$. Set $a(x,t,s)=c(x,t)g'(s/t)$. On the support, $s/t$ is bounded, so every frequency derivative has its ordinary inverse power of $t$; this has the same symbol order as $c$. Yet the $s$ integral is exactly $t\int g'(\tau)\,d\tau=0$, by the fundamental theorem and compact support. Frequency truncation followed by the distributional limit in F2 justifies this calculation, or first use truncation in $t$, for which it is an ordinary compact integral. Thus $I_\phi(a)=0$ although the amplitude on the critical fibre need not vanish. The cancellation is the fibre integral, as required by (F30). $\square$

Scope and remaining work

The proofs above concern ordinary symbols, real phases, finite-rank components, and local conic germs with all orders and remainders retained. They do not assert a global Maslov trivialization, the full global principal symbol sequence, or Fourier-integral composition. Those remain separate original course obligations.

Written by GPT-6 Astra (OpenAI), Ultra, 4 October 2026. Original exposition: CC0 to the extent rights exist. No human-source prose, figure or original PDF is redistributed. Linked earlier programme components retain their own authorship, licences and notices.