RIGIDITY OF SMOOTH BILLIARDS WITH A CONTINUOUS CAUSTIC COLLAR
a reading edition
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HOW TO READ THIS
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This is a plain-language edition of a long, technical paper. It keeps the
complete logical skeleton of the argument: every definition, every main
statement, and every step that makes one statement follow from the next.
What it cuts is padding, not substance.
The idea-level reasoning is given in ordinary sentences. Where a step is a
short honest computation (a Taylor expansion, an integration by parts) it
is described and its result is stated. Where a step is genuinely deep or
where the original only sketches the justification, THAT IS SAID OUT LOUD
in a bracketed note: [deep] means "this is the real work", and [sketch]
means "the original claims this and gives at best a sketch; treat it as a
plausible step, not a certified one". Nothing is dressed up to look more
solid than it is. That is the whole point of this edition: a reader should
always know which ground is bedrock and which is scaffolding.
Read it like a detective story: each chapter narrows who the culprit (the
non-ellipse) could be, until the last few chapters finish it off.
Unless a different probability law is named, an average over a horizontal
period means (1/2pi) times the integral over 0..2pi in the phase angle.
Constants tagged C may grow from line to line inside one argument.
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1. THE QUESTION, AND WHAT WE PROVE
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1.1 Billiards.
Take a convex region Omega in the plane with a smooth boundary curve. A
billiard ball moves in a straight line until it hits the boundary, then
bounces: the angle it makes with the normal (the perpendicular to the
wall) equals the angle on the way out. In vectors, if v is the incoming
unit velocity and n the outward unit normal at the impact point, then the
outgoing unit velocity is
v' = v - 2 (v . n) n .
This reflection rule is the entire dynamics.
A CAUSTIC is a curve inside the table such that a ball which is ever
moving tangent to it (momentarily grazing it) stays tangent to it forever,
before and after every bounce. Caustics are special: they are the table's
hidden symmetry lines. The classic example is the ellipse. In an ellipse
every trajectory is tangent to a smaller confocal ellipse, and these
confocal ellipses nest nicely and fill a band near the boundary.
1.2 The Birkhoff conjecture.
The bold guess (found in Birkhoff's work; written down early by Poritsky)
is: if a convex billiard is "integrable" in the sense of having enough
caustics to organize all motion, then the table must be an ellipse. In
full generality this is a famous open problem. We prove a clean
near-boundary version of it.
1.3 The hypothesis we use: a continuous caustic collar.
We only assume structure in a ring near the wall, and we do not assume the
ring is differentiable across its layers -- only continuous.
DEFINITION (continuous caustic collar). Let Omega be a bounded strictly
convex domain whose boundary is a smooth (C-infinity) simple closed curve
of positive curvature. A smooth closed convex caustic is a smooth simple
closed curve inside Omega, bounding a strictly convex set, that has the
tangency property from 1.1 at both ends of every tangent chord. A
CONTINUOUS CAUSTIC COLLAR is a relatively open region A of the closure of
Omega that contains the whole boundary, together with a homeomorphism
H0 : (circle) x [0,1) -> A
such that:
- H0( (circle) x {0} ) is exactly the boundary,
- for every 0 < b < 1, the curve H0( (circle) x {b} ) is a smooth closed
convex caustic.
Think of it as a continuous stack of nested caustics, a "layer cake",
piling up onto the boundary itself, with the layers fitting together
continuously (but we allow the fit to be only continuous, not smooth).
1.4 Main theorem.
THEOREM. If Omega has a continuous caustic collar, then Omega is an
ellipse. Concretely, there are a center c in the plane and a real
symmetric positive-definite 2x2 matrix Q with
Omega = { x : (x-c)^T Q (x-c) < 1 }.
Circles are included (Q a scalar multiple of the identity).
1.5 What the hypothesis does and does not say (scope).
- It DOES demand a genuine filled band of caustics: every layer between
the boundary and some deeper curve is present, and they vary
continuously.
- It does NOT accept a mere Cantor set of caustics, even one whose
caustics crowd up to the boundary with gaps. So this is strictly
stronger than "there are lots of caustics near the wall."
- It says NOTHING about dynamics deep in the interior. That is the point
of "near-boundary": we deduce rigidity from a thin shell.
The surprise is that even this thin, merely-continuous shell is enough to
force the whole table to be an ellipse.
1.6 The shape of the proof (preview).
The argument has three movements.
(I) Continuity kills roughness; smoothness upgrades to analyticity.
The rational caustics of rotation number 1/q (balls that close up after
q bounces with one twist around the table) exist for every small 1/q
and line up into perfect periodic orbits. Their actions give a large
system of equations. Its derivative is the identity plus a small
operator, so a Newton iteration upgrades the boundary from smooth to
real-analytic. A companion construction, using not just values but
first derivatives in the foliation parameter at rational resonances,
produces a jointly analytic description of the near-glancing motion.
(II) Finitely many bounces remember almost nothing; this forbids a "first
wall".
If two endpoint billiard lines of a chain of k reflections are known
(as functions with all derivatives), then every line in between is
determined algebraically from them. Combining this with a clever
coalescing-impact scaling rules out a subtle degeneracy. The payoff:
at the first place where the complexified motion stops being
single-valued, not even one point admits a local patch. This is the
engine that later turns any local analytic progress into a GLOBAL
statement about the width of the complex strip.
(III) Complexify and squeeze the strip until only the ellipse survives.
Extend the boundary into the complex plane and watch its meromorphic
normal. If that normal has a zero or a pole in the relevant strip, the
table is forced to be an ellipse directly. If it never vanishes or
blows up, the impacts lift to a single global complex "shape"
coordinate living in a horizontal strip of some half-width H. Growth
of the shape fields near the strip's edge is then analyzed:
- too little growth -> impossible,
- too much growth -> impossible,
- growth of exactly linear order (the "critical" case) -> the hardest
case.
In the critical case a global bookkeeping inequality (an "angle
budget") bounds how much the whole system can twist. One fixed odd
periodic necklace of impacts transports that bound, via Brownian
motion and matrix identities, to the complex strip. Finally one picks
a horizontal line on which a certain field is robustly tiny or robustly
huge except on a vanishing fraction of angles; a sign test there keeps
the very last moving impact from reaching the wall, which contradicts
the assumed first wall. Done.
1.7 Where this sits in the literature (condensed).
Ellipses are the classical integrable tables (confocal caustics). The
conjecture is Birkhoff's, first put in print by Poritsky. Lazutkin built a
positive-measure family of grazing caustics for smooth positively curved
tables, but with gaps -- not a full collar. Marvizi-Melrose gave a grazing
integral that is correct only up to a flat error. Mather showed trouble
when curvature is allowed to vanish, which is why positive curvature is
assumed here.
Other rigidity results need extra hypotheses: Bialy (a foliation of the
whole phase cylinder by noncontractible invariant curves gives a circle);
Innami (caustics with rotation numbers tending to 1/2); Bialy-Mironov
(centrally symmetric tables reaching the 4-periodic curve). Perturbative
work by Avila-De Simoi-Kaloshin, Kaloshin-Sorrentino, Kaloshin-Koudjinan-
Zhang, and Koval proves local results near ellipses under various
conditions. Glutsyuk classifies complex algebraic caustics in a different
setting. None of these supply what a small grazing collar gives, and this
edition assumes none of them; they are ancestors of the methods, not
inputs to the argument.
1.8 Organization and dependencies.
1 (here) the question and theorem.
2 rational caustics force an analytic boundary; rational
parameter-jets give a jointly analytic glancing picture.
3 finite reflected chains are determined algebraically by
their endpoints; consequence -- no local patch at a first
wall.
4 zeros/poles of the complex normal force an ellipse;
otherwise a global complex shape coordinate exists.
5 reduction of edge growth to a critical finite wall.
6 the critical estimates and the reduction via an "angle
budget".
7 one fixed odd periodic necklace, stopped before the wall,
plus matrix identities that move the angle-budget bound
onto Brownian paths.
8 the sign row and the last-moving-site contradiction;
assembly of the main theorem.
There is exactly one deliberate forward reference: the angle-budget
reduction in chapter 6 uses the exclusion of a precise class of "bad
sequences", which is proved independently in chapter 7 using only earlier
moment and continuation estimates -- never the angle budget itself. The
final proof threads all alternatives together at the very end.
1.9 Conventions.
- Angles are real lifts of 2*pi-periodic variables.
- Complex inner products are bilinear (no conjugation) unless stated.
- A "meromorphic projective map" means: in a generic affine chart its
coordinates are meromorphic; incidence identities are used where
defined, then continued.
- C denotes a constant that may increase between estimates; dependence
on fixed apertures or fixed auxiliary parameters is shown when it
matters for an order-of-choices argument.
- D denotes the normalized generating action; in one local scaling it
also denotes a positive rescaled depth, always announced on the spot.
- The phase-width deficit is always Delta(t) = H - h_X(t), where H is
the half-height of the complex strip and h_X its width above a real
parameter t.
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2. RATIONAL CAUSTICS FORCE A SMOOTH TABLE TO BE ANALYTIC
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Goal of this chapter: show that the collar turns the (merely smooth)
boundary into a real-analytic curve. We never use the smoothness of the
foliation across layers -- only the single fact that every small rational
rotation number actually occurs as a genuine caustic. From that we build a
large system of scalar equations on the boundary's "shape function" u, see
that its linearization is the identity plus a small operator on high
frequencies, and turn the crank of Newton's method in a way that keeps a
strictly positive band of complex width. A limit function analytic on that
band, agreeing with u on low frequencies and satisfying the equations, is
then shown to EQUAL u. So u is analytic. History: the grazing coordinate is
Lazutkin's; the action viewpoint echoes Marvizi-Melrose and the rational
Fourier constraints echo Avila-De Simoi-Kaloshin. The arbitrary-input mesh,
the Fourier row, and the non-perturbative inversion below are proved here.
2.1 The phase picture.
Put an interior point at the origin. For a unit angle theta set
n(theta) = (cos theta, sin theta), e(theta) = (-sin theta, cos theta),
so n is a unit normal and e the tangent turned 90 degrees. If h(theta) is
the support function (the signed distance from the origin to the tangent
line with outward normal n(theta)), the boundary point with outward normal
n(psi) is
gamma(psi) = h(psi) n(psi) + h'(psi) e(psi),
and the curvature radius is
r(psi) = h(psi) + h''(psi) > 0.
An unoriented line with unit velocity e(theta) and normal offset p is
n(theta).x = p. Send such a line forward one bounce; let the outgoing
velocity angle be theta'. Define the impact normal angle and half-angle
psi = (theta+theta')/2, d = (theta'-theta)/2.
On the small positive glancing branch the reflection law is equivalent to
p = h(psi) cos d - h'(psi) sin d,
p'= h(psi) cos d + h'(psi) sin d.
This billiard map is obtained from the generating function (Bialy-Mironov)
S(theta,theta') = 2 h((theta+theta')/2) sin((theta'-theta)/2),
with p = -S_1 and p' = S_2 (subscripts are partial derivatives). The mixed
derivative is the twist:
S_12(theta,theta') = (r(psi)/2) sin d > 0.
The map preserves the area form dp ^ dtheta, and
d theta' / dp = -1 / S_12 < 0,
so the map is a positive twist map; this is what makes rotation numbers
monotone and continuous below.
2.2 Rational periodic leaves. [deep but standard]
Localize the collar to parameters b > 0 increasing inward. Write each
caustic leaf as a graph p = g_b(theta) in the phase cylinder.
LEMMA (rational periodicity). There is a one-sided phase collar filled by
invariant continuous graphs p = g_b(theta) with g_b strictly DECREASING in
b. Each induces an increasing lift F_b of the circle map (adding 2*pi on
the right) with positive translation number tau(b), and tau(b) -> 0 as
b -> 0. Moreover, for every sufficiently small positive rational l/q in
lowest terms there is EXACTLY ONE leaf whose lift has
tau(b) = 2*pi*l/q,
and for that leaf not just one periodic orbit but the whole graph is
periodic with the same shift:
F_b^q(theta) = theta + 2*pi*l for EVERY theta.
Proof idea, in words.
- Nesting. Leaves close up onto the outer boundary and the convex bodies
they bound all contain a fixed small disk around the origin. Disjoint
convex curves around a common region are strictly nested, which forces
g_b(theta) continuous and strictly decreasing in b. This is the
"layer-cake is ordered" fact.
- Each leaf is invariant. A positively oriented support line of a leaf
reflects to another positively oriented support line of the SAME
leaf (that is the caustic property). For a thin enough collar the
other orientation is impossible because it would need a negative
offset. So the billiard map restricts to a circle homeomorphism of
each graph: a degree-one, near-identity map, i.e. an increasing lift F_b
with positive step.
- Rotation numbers. For an increasing lift commuting with +2*pi, the
displacement F^m(x)-x has oscillation at most 2*pi, its max is
subadditive and its min superadditive in m, so the translation number
tau(F) = lim_{m} (F^m(x)-x)/m
exists, is independent of x, and satisfies |tau - (F^m(x)-x)/m| <= 2*pi/m.
This makes tau continuous under uniform convergence, monotone under
ordering of lifts, and tau(F)=2*pi*l/q exactly when F^q - 2*pi*l has a
fixed point.
- tau(b) is continuous and nondecreasing; positive on each fixed leaf and
-> 0 at the boundary. The only thing left is to forbid a whole interval
of b with the SAME rational tau. If such an interval existed, the
annulus between two of its graphed boundaries is invariant with finite
positive area. Almost every point there is recurrent for the q-th
iterate. But off the fixed set of F^q - 2*pi*l every point marches
monotonically toward a fixed point and is not recurrent, so F^q is the
identity a.e. on the annulus and hence everywhere by continuity --
contradiction with strict ordering of the two boundary graphs.
- Because tau is continuous and jumps through each rational with a whole
leaf, the leaf of rotation 2*pi*l/q exists and its lift obeys the strong
identity F^q(theta) = theta + 2*pi*l for all theta (compare F^q to the
rotation: above the value the displacement is strictly positive, below
strictly negative, so at the value it vanishes identically).
What we keep: for every small l/q we get one full periodic leaf.
2.3 Lazutkin normal form and the normalized action.
Choose k > 0 so the coordinate x defined by
dx/dtheta = k r(theta)^{1/3}
has period 2*pi in x; write theta = phi(x). Then
phi'(x) = e^{u(x)}, r(phi(x)) = c e^{-3u(x)}, c = k^{-3},
with u real, smooth, 2*pi-periodic. Subtracting the telescoping term
integral of h from S and rescaling gives the NORMALIZED ACTION (this is
the object everything downstream is written in)
D(X,Y) = - integral_X^mu (1-cos(phi(z)-phi(X))) e^{-2u(z)} dz
- integral_mu^Y (1-cos(phi(Y)-phi(z))) e^{-2u(z)} dz,
where mu is the point with phi(mu) = (phi(X)+phi(Y))/2 (the "midpoint").
For arbitrary smooth u this is well defined (only differences of phi occur,
so no global periodicity of phi is needed) and satisfies
D(X+2*pi, Y+2*pi) = D(X,Y), D(Y,X) = -D(X,Y).
If u is holomorphic in a strip, D is holomorphic near the diagonal.
Differentiating the two integrals, the moving-midpoint pieces cancel, and
D_1(X, X+T) = e^{u(X)} integral_X^mu sin(phi(z)-phi(X)) e^{-2u(z)} dz.
A direct Taylor computation gives the small-chord expansion used all over
the paper:
D_1(X, X+T) = T^2/8 + a(X) T^4 + O(T^5),
D(X, X+T) = -T^3/24 - (a(X)/5) T^5 + O(T^6),
a(X) = ( 7 u''(X) - 8 u'(X)^2 - e^{2u(X)} ) / 384.
[sketch] The coefficients come from carrying the midpoint equation
z = (mu-X)/T = 1/2 + (alpha/8) T + (beta/16) T^2 + O(T^3)
(with alpha=u'(X), beta=u''(X)) through one rescaling of the D_1 integral;
the linear term in T cancels, and the D^4 term is forced to vanish by
antisymmetry. The displayed constants are the results; the reader can
verify them but this edition does not re-derive every digit.
2.4 Stationary meshes. [deep but a standard contraction]
For sigma > 0 let A_sigma = { |Im z| < sigma } and let ||.||_{a,sigma} be
the max of the first a derivatives' sup norms on A_sigma (for sigma=0, the
C^a norm on the circle). Real-symmetric means f(z-bar) = conj f(z).
LEMMA (uniform stationary mesh). Fix an integer m>=2 and a bound M; set
p=m+2, r0=m+15. There are R,K,N depending only on M,m with: for any
0<=sigma<=1, real-symmetric u with ||u||_{r0,sigma}<=M, and any integer
q>=N, putting delta=2*pi/q, there is a stationary fixed-endpoint mesh
X_j(x) = x + j*delta + w_j(x), 0<=j<=q, w_0 = w_q = 0,
holomorphic on A_{sigma'} with sigma' = (1-K/q^2) sigma, and
q^2 max_j ||w_j||_{p,sigma'} + q^3 max_j ||w_{j+1}-w_j||_{p,sigma'} <= R.
It depends differentiably on u, and ||dot w|| <= C ||dot u||. On the real
axis this mesh is unique (pointwise) within the ball; the ball can be
enlarged by any fixed amount once N is raised.
Meaning: near every circle of q equally spaced points there is one nearby
configuration where the sum of the cell actions sum_j D(X_j, X_{j+1}) is
stationary in each interior point. The internal points are O(q^{-2}) from
equally spaced and the spacings nearly equal.
[sketch] The proof is a Banach contraction. Each Euler equation
D_1(X_j, X_j+t) - D_1(X_j, X_j-s) = 0 with t,s the right/left steps factors
through the second difference of w:
w_{j+1} - 2 w_j + w_{j-1} = -(t^5 b(X,t) + s^5 b(X,-s)) /
((t+s)(1/8 + (t^2+s^2)a(X))).
The right side is O(q^{-4}) with Lipschitz constant O(q^{-6}) in the
scaled norm; the Dirichlet inverse of the second difference costs at most
C q^4; hence a contraction with factor C q^{-2}. The delicate bookkeeping
is that the strip loss is RELATIVE (K sigma / q^2), not a fixed loss, so
the construction still works as sigma -> 0. This is the one place a
careful reader should believe the estimates rather than reproduce them.
2.5 The high-frequency constraint system.
For the mesh above define the critical action and two Fourier modes
W_q(x;u) = sum_{j=0}^{q-1} D(X_j(x), X_{j+1}(x)),
C_n(u) = -(q^2 / (6 pi)) [W_q(.,u)]_n, n = +/- q,
where [f]_n = (1/2pi) integral_0^{2pi} e^{-i n x} f(x) dx. Intuition: C_n
measures the n-th Fourier mode of how far this near-periodic orbit fails to
be stationary as x varies; it is built to have a clean linearization.
PROPOSITION (high-frequency linearization). Fix m>=2 and M<infinity, set
r0=m+15. There are K,N independent of 0<=sigma<=1 so that whenever
||u||_{r0,sigma} <= M and |n| >= N,
(dC_n)_u(v) = [ b_n(u) v ]_n
for variations v (smooth real, or real-symmetric holomorphic with finite
norm), where
|| b_n(u) - 1 ||_{m,(1-K/n^2)sigma} <= K/|n|
and, for two inputs,
|| b_n(u) - b_n(u~) ||_{m,(1-K/n^2)sigma} <= (K/|n|) ||u-u~||_{r0,sigma}.
Meaning: on high frequencies, perturbing the shape function changes the
constraint almost exactly by the same perturbation -- the multiplier b_n is
1 plus a small error.
[sketch] Differentiating W_q in u (the interior mesh terms drop by
stationarity, the two endpoints cancel), the variation kernel on a cell is
explicit, with a leading piece that, after the change of Fourier variable,
integrates to the constant 1. The constant -q^2/(6pi) is chosen so that
this leading coefficient is exactly 1; the O(1/|n|) error is the sum of
q cells each contributing O(q^{-1})/O(q^{-2}) with a fixed integral
-(1/3) integral_0^1 ( (3/2) min(xi,2pi-xi)^2 - pi^2/2 ) e^{2pi i sgn(n) xi} d xi = 1.
This is a real computation; the statement is what matters for the Newton
step.
2.6 The collar makes every constraint vanish.
LEMMA. For the u coming from the ACTUAL table (via 2.3),
C_n(u) = 0 for all sufficiently large |n|.
Reason in words: the rational leaves of 2.2, of rotation 2*pi/q, crowd up
to the boundary as q grows; their x-coordinates form a positive stationary
mesh of q steps summing to 2*pi, and all steps shrink uniformly. A Taylor
argument with the same D_1 expansion shows consecutive steps s,t satisfy
t-s = O(s^4), hence all steps are uniformly comparable, hence each is
2*pi/q + O(q^{-3}) and each point is O(q^{-2}) from equally spaced -- which
is exactly the uniqueness ball of 2.4. So the geometric orbit IS the mesh
defining W_q. Moreover the orbit satisfies the Euler equation also at the
glued endpoint, so differentiating W_q the interior terms vanish and the
two endpoint derivatives cancel: W_q'(x) = 0. That forces all high modes
C_n(u) to vanish. (Only high |n| is used; low modes are irrelevant here.)
2.7 Inverting the linearization without losing width.
For a periodic function write the Wiener norms
|f|_sigma = sum_k |[f]_k| e^{sigma |k|},
|f|_{sigma,A} = sum_k (1+|k|)^A |[f]_k| e^{sigma |k|}.
Let Pi_{>=N} project onto modes |k|>=N and restrict the constraint map to
those outputs.
LEMMA (Wiener near-identity). With the bounds of 2.5 at m=2 and N large,
on the width-sigma Wiener space the coefficient formula for dC(u) defines
A(u) = Pi_{>=N} + R(u), ||R(u)|| <= C/N,
with ||A(u)-A(u~)|| <= C||u-u~||_{r0,sigma}. The high-mode restriction
A_H(u) has inverse of norm <= 2 for large N, and everything preserves real
data.
[sketch] Contour shift plus two integrations by parts gives
|[b_n(u)-1]_j| <= (C/|n|)(1+|j|)^{-2} e^{-(1-K/n^2)sigma|j|}. The point of
the mode-dependent width (1-K/n^2)sigma is that the weight "loses" only
2K/|n| in the matrix computation:
|n|-|k|-(1-K/n^2)|n-k| <= 2K/|n|, so column sums are O(1/N). A Neumann
series finishes. The operator is first built from smooth variations, then
extended by the bounded matrix; on every smaller strip it still equals the
genuine derivative, and it is consistent across strip widths by the real
mesh uniqueness of 2.4.
2.8 Newton iteration: smooth u becomes analytic. [deep -- this is the core]
Let u be the actual shape function. Fix m=2, r0=m+15, pick
M > 4e (1 + |u|_{0,r0}),
and choose N so 2.6, 2.7 hold with ||R||<=1/2, all fixed before P below.
Truncate u to |k|<=P (P>N) calling it u_0, and let sigma_0 = 1/P. Then
|u_0|_{sigma_0,a} <= e |u|_{0,a}
and the initial residual is tiny in every power:
|C(u_0)|_{sigma_0} = O(P^{-A}) for each fixed A.
[sketch] Split modes N<=|n|<=P (weight <=e; integrate dC along the real
segment from u to u_0, using C(u)=0 and the width-zero Wiener estimate) and
the tail |n|>P (use C(0)=0 and the high-frequency proposition with a fixed
larger m; the weighted convolution is summable). Each requested power costs
only one larger fixed smoothness order.
Now hold all modes |k|<N fixed and solve the high-mode equations by Newton.
Set sigma_j = (1+2^{-j})/(2P) (so sigma_j decreases but sigma_inf =
1/(2P) > 0), and eps_j = |C(u_j)|_{sigma_j}. The step is
eta_j = - A_H(u_j)^{-1} C(u_j), u_{j+1} = u_j + eta_j,
which has only high modes and satisfies |eta_j|_{sigma_j} <= 2 eps_j. Because
widening a strip costs a factor, for a = sigma_j - sigma_{j+1} = 2^{-j-2}/P,
||eta_j||_{r0,sigma_{j+1}} <= C P^{r0} 2^{r0 j} eps_j
(the sup_{xi>=0} xi^{r0} e^{-a xi} <= C a^{-r0} inequality). Integrating
the difference of derivatives along the correcting segment (the linear term
cancels exactly by construction, and the same coefficient-defined inverse
works on the smaller strip by 2.7) gives
eps_{j+1} <= C P^{r0} 2^{r0 j} eps_j^2.
Choosing P large so eps_0 <= P^{-2r0} and C P^{-r0} 2^{r0+1} < 1, induction
gives the geometric decay
eps_j <= P^{-2r0} 2^{-(r0+1) j}.
The corrections themselves satisfy sum_j ||eta_j||_{r0,sigma_{j+1}} <=
C P^{-r0} sum_j 2^{-j} < M/4 for large P; combined with |u_0|<M/4 this
keeps the whole iteration inside the bounded-input ball M. So the
bounded-input hypothesis is proved, not assumed.
[deep] This is the load-bearing technical construction of the chapter:
Newton's method run on a nested family of shrinking-but-nonzero strips,
with the contraction rate and the width loss both controlled by q/|n|. It
is where the paper spends most of its analytic effort. The reader should
take away the STRUCTURE: quadratic error decay + a positive surviving strip
= an analytic limit.
The corrections converge in C^{r0}(A_{sigma_inf}) to a real-symmetric
holomorphic u_* on the strip |Im| < 1/(2P). Its low modes match u's. By
continuity of the real mesh, each fixed constraint is 0 at u_*. The real
segment from u to u_* stays in the input ball. Let v = u_* - u; it has only
high modes, and at width zero
0 = C(u_*) - C(u) = v + integral_0^1 R(u+lambda v) v dlambda.
The integral operator has norm <= 1/2, so |v|_0 <= |v|_0/2, hence v = 0.
Therefore u = u_* is real analytic on a band.
Finally, trace back: u analytic => phi' = e^u > 0 analytic => phi and its
inverse analytic => r(phi(x)) = c e^{-3u} analytic => the support function h
solves the analytic ODE h'' + h = r, so h is analytic, and the boundary
gamma = h n + h' e is an analytic regular curve.
THEOREM (analytic regularity). Under the hypotheses of chapter 1, the
support function of the table and its boundary are real-analytic.
(We needed only the rational leaves of the collar for this; the full collar
returns in chapter 3.)
2.9 What we carry forward.
COROLLARY. For the actual table the normalized action D of 2.3 is
holomorphic in a neighborhood of the real diagonal, diagonally periodic,
and antisymmetric, with
D(X,X+T) = -T^3/24 + O(T^5), D_1(X,X+T) = T^2/8 + O(T^4)
locally uniformly with all fixed derivatives; and every sufficiently small
positive rational l/q has its unique periodic leaf from 2.2.
This (analytic action + unique rational leaves) is the raw material for
chapter 3, where we exploit the fact that the collar gives not just the
existence of these leaves but continuous dependence and parameter
derivatives across them.
------------------------------------------------------------------------
3. THE COLLAR AS AN ANALYTIC PARAMETRIZATION
------------------------------------------------------------------------
Chapter 2 gave us an analytic action D and unique rational leaves. That
still does not say the leaves depend ANALYTICALLY on a transverse
parameter -- only that each rational leaf exists. This chapter constructs
a parameterization of the whole one-sided collar in which the transverse
parameter is the (angular) step t, and proves it is jointly holomorphic
in (phase, parameter).
The engine is Newton's method on a residual E(U) that asks a periodic
curve U(.,t) of a shape function to be stationary for the sum of cell
actions t*D. The collar hypothesis supplies the residual-compatibility
data at every retained rational resonance t_0 = 2*pi*l/q. That data is
what lets us DIVIDE the two difference operators at their zeros; the
arbitrary-input machinery of chapter 2 alone would not suffice.
3.1 The normal form, residual, and its linearization.
Write T_z v(y)=v(y+z) for the shift, and the central-difference operator
partial v(y) = ( T_{t/2} v - T_{-t/2} v ) / t
(t-dependence understood; prime is d/dy). For a candidate U define
R(U) = D_1( U, T_t U ) + D_2( T_{-t} U, U ), (residual)
E(U) = t^{-3} U' R(U), (normalized)
a(U) = t^{-1} D_{12}( T_{-t/2} U, T_{t/2} U )
(T_{-t/2} U') (T_{t/2} U'). (coefficient)
The symbol a here is the linearized-equation coefficient; the complex
unit i is a separate symbol in later chapters.
LEMMA (regularity and automatic factorization). E and a extend
holomorphically through t=0, with
E(U)(y,0) = (U')^2 U'' / 4, a(U)(y,0) = (U')^3 / 4,
are even in t when U is even in t, and (E) = 0 (horizontal mean).
For a variation of the form deltaU = U' v (i.e. an infinitesimal change
of phase), the linearization is
E_U( U' v ) = partial( a * partial v ) + (E v)'.
At an EXACT stationary solution at a NONZERO real t_0, assign an arbitrary
parameter jet dot U = U' z; then the total physical-parameter derivative
of E at (U,t_0) is
dot E = partial( a * partial z + a / t_0 ).
Meaning: the linearized operator is a one-dimensional divergence operator
with coefficient a, a potential (E v)', and an extra forcing a/t_0 coming
from the parameter jet.
Proof idea. Antisymmetry gives R = D_1(U,T_tU) - D_1(U,T_{-t}U). Writing
v_+ = t A_+, v_-= -t A_- with A_\pm = integral_0^1 U'(y +/- lambda t,t) dlambda,
the leading part of t^{-3}R is (1/8)(A_++A_-)(A_+-A_-)/t, and the last
quotient is holomorphic (integral of U''). The O(t^4) terms of the D_1
expansion remain; evenness kills the linear part. For the linearized
formula, subtract from the off-diagonal Hessian terms the pieces with v
evaluated at y; what survives is exactly partial(a * partial v), and the
factored terms (including the variation of the prefactor U') are (E v)'.
The parameter-jet statement at an exact solution comes from differentiating
only the explicit shifts in R; the derivative of the prefactor t^{-3} dies
at an exact solution.
SLANTED DOMAINS. Fix s>d and work on
D^nu(s,d) = { (y,t) : |t|<d , |Im y| < s - nu |Im t| }, nu in {0,1/2,1}.
Shifts T_{\pm t/2} cost nu += 1/2. Smaller domains D_j^nu = D^nu(s-j delta, d-j delta),
Y_j = {|Im y| < s-j delta}. Cauchy across one step costs delta^{-1}. Every
constant below may increase between estimates; none depends on N, delta,
or on any formal initialisation order.
3.2 The Newton step, stated carefully. [deep]
PROPOSITION (joint Newton step). There are C, B_0 > 0 depending only on
the action domain and the fixed smallness bounds, with: for any
real-symmetric even-in-t candidate U with U-y periodic and
||E(U)||_{D^1_0} <= eps < 1,
for any integer N>=1 with e^{-N delta} <= eps, and with
Q := 1 + N + delta^{-1}, if C Q^{B_0} eps is small then there is a
real-symmetric, even, y-periodic correction deltaU with
max_{0<=j<=3} || partial_y^j deltaU ||_{D^0_{20}} <= C Q^{B_0} eps,
|| E(U + deltaU) ||_{D^1_{20}} <= C Q^{B_0} eps^2 .
The exponent B_0 is FIXED; it does not depend on N, on which rationals
we retained, or on how well the initialiser was built.
3.3 Why the collar matters: compatibility at rational resonances. [deep]
Let t_0 = 2*pi*l/q (reduced, q<=N, |t_0|<d-2delta). The mesh of q points
X~_j = U_j + U'_j xi_j, U_j(y)=U(y+jt_0,t_0), xi_0=xi_q=0
is what makes the collar directly usable: the endpoints differ by qt_0=2*pi*l,
so on real y these are genuine billiard trajectories in the given collar.
Their linearization is a cyclic Dirichlet stencil
(A_0 xi)_j = ( a_{j+1/2}(xi_{j+1}-xi_j) - a_{j-1/2}(xi_j-xi_{j-1}) ) / t_0^2
with a_{j+1/2}=a(y+(j+1/2)t_0,t_0); add a potential V_j = T_{jt_0}(E' -
(U''/U')E). Since a \approx 1/4, a Dirichlet flux computation gives
||A_0^{-1}|| <= C q^2 t_0^2, hence ||A_0^{-1}|| <= C Q^2.
The nonlinear residual (three adjacent nodes fixed derivatives of D, times
t_0^{-3}, plus the lower bound |t_0|^{-1}<N/(2*pi)) has second derivatives
bounded by C Q^3 in xi. So the fixed-point map preserves a radius-CQ^2*eps
ball and contracts when C Q^7 eps is small. Result: on Y_1, ||xi||<=C Q^2 eps.
Now U_*(y):=U(.,t_0) corrected by a phase reparameterization matches the
actual q-periodic leaf, so
U_*(y) = U(G^{-1}(y), t_0)
with G a periodic near-identity map satisfying G(T_t_0 y)=G(y)+t_0. On
real y, the leaf's periodicity forces xi_q=0 as chosen. On complex y,
U_*-U(.,t_0)=O(P eps) with P=C Q^{fixed}; the "P" always denotes some
C Q^{fixed exponent}; we may raise that exponent finitely many times.
Parameter jet. Start with dot U_* = U_t(.,t_0), residual jet O(P eps).
Let Pi be the orbit-average projection Pi v = (1/q) sum_j T_{j t_0} v. Then
Pi dot E_* = 0 (uses only partial = T_{-t_0/2}(T_{t_0}-1)/t_0 and
Pi T_{t_0}=Pi; the half-shift drops out). The cyclic inverse on ker Pi is
partial^{-1} = t_0 R_q T_{t_0/2}, R_q = (1/q) sum_{j=0}^{q-1} j T^j,
and since T^q=1, (T-1)R_q = 1 - Pi, giving the bound
|| partial^{-1} ||_{ker Pi} <= C q |t_0|.
The operator (1-Pi)a_* on ker Pi is invertible because a_* is close to 1/4.
So
|| (partial * a_* * partial)^{-1} ||_{ker Pi} <= C q^2 t_0^2.
Correct dot U_* by U_*'z for suitable z in ker Pi to make dot E_* = 0, at
cost O(P eps). Then w := (U_*-U)/U' and ẇ := (dot from this) satisfy
E + partial(a partial w) = O(P eps^2),
and the total t-derivative at t_0 of the left side is also O(P eps^2),
with w, ẇ = O(P eps) on Y_{10}. Those are the two rational compatibility
conditions. [The whole ledger of Q-powers here is what forces B_0 to be
large but FIXED; every denominator t_0 is bounded by t_0^{-1}<N/(2*pi).]
3.4 The buffered Fourier division at resonance.
For a periodic function write v = sum_n v_n(t) e^{i n y} and
d_n(t) = (2 i / t) sin(n t / 2),
so (partial v)_n = d_n v_n with d_n(0) = i n. The nonzero zeros of d_n
are simple, at t_j = 2*pi j / n, j nonzero.
LEMMA (buffered Fourier division). Fix nu in {1/2,1}; a mean-zero periodic
F holomorphic on D^nu_{j_0} with ||F||<=M. Suppose that for 1<=|n|<=N and
divisor zeros 0<|t_j|<d-b delta (b>=j_0+1),
|F_n(t_j)| <= eta e^{-|n|(s-a delta)}, a>=j_0.
Set, for retained modes,
R_n(t) = sum_{0<|t_j|<d-b delta} F_n(t_j) * sinc( n(t-t_j)/2 ),
H_n(t) = (F_n(t) - R_n(t)) / d_n(t),
and H_0=0, H_n=0 for |n|>N. Then every retained H_n is holomorphic on
|t|<d-(b+1)delta and, for any integer J>a, j_0, b+1,
||H||_{D_J^{nu-1/2}} <= P (M + eta),
||partial H - F||_{D_J^nu} <= P ( eta + M e^{-N delta} ),
with P=C Q^{fixed}. Real symmetry, evenness in t, and fixed-derivative
bounds are preserved on further strips.
Why: the sinc interpolant vanishes at all OTHER divisor zeros and equals
F_n at its own; so the apparent zeros of d_n in the quotient are removable.
On the "safe region" outside the small discs about the divisor zeros,
1/d_n is polynomially bounded; inside each disc, maximum modulus plus the
compactness of n delta gives a constant. The half-strip nu=1 input gives
nu-1/2 output because the omitted tail gains e^{-N delta} and the retained
interpolation pays no imaginary loss.
3.5 Assembling the Newton step. [deep]
Take f=-E (mean-zero). At every retained zero t_j with |t_j|<d-3delta,
the first compatibility inequality gives |f_n(t_j)|<=P eps^2 e^{-|n|(s-10delta)}.
Apply the division lemma (nu=1, j_0=0, a=10, b=3, J=11). Output h:
h=O(P eps) on D^{1/2}_{11}, partial h + E = O(P eps^2) on D^1_{11}.
Set b := a^{-1}( h + lambda(t) ), lambda(t) := - <a^{-1} h> / <a^{-1}>,
so b has zero mean and O(P eps) norm on D^{1/2}_{12}. The denominator is
bounded from zero because a^{-1}\approx 4.
Now we need NEW compatibility values before dividing b. Fix a retained
rational t_0 with |t_0|<d-14delta, let g = a * partial w from 3.3. For
retained nonzero modes the exact first-division identity is
d_n (h_n - g_n) = -(E + partial g)_n - R_n .
The two rational compatibility conditions bound the value AND the first
t-jet of both sides at t_0 by P eps^2 e^{-|n|(s-10delta)}. If d_n(t_0)≠0
the reduction l/q, |sin(pi n l/q)|>=c/q gives |1/d_n(t_0)|<=C q|t_0|; if
d_n(t_0)=0 one differentiates the product and uses |d_n'(t_0)|=|n|/|t_0|,
producing the same bound on h_n-g_n -- crucially this does NOT need the
resonant modes of g to vanish. For the omitted high modes h_n=0 and the
analytic tail of g on Y_{13} is P eps^2; summing retained + omitted:
h - (g - g_0) = O(P eps^2) on Y_{13}.
Mean choice gives lambda = g_0 + O(P eps^2), hence
b - partial w = O(P eps^2) on Y_{13}.
At a retained-nonzero divisor zero, (partial w)_n=0, so this supplies the
values for a second division (nu=1/2, j_0=12, a=13, b_DIV=15, J=17). Call
its output v. The remaining half-strip loss is now canceled:
v = O(P eps) on D^0_{17}, partial v - b = O(P eps^2) on D^{1/2}_{17}.
Both divisions, multiplication by a^{-1}, and the mean adjustment preserve
real symmetry and evenness.
Since a b = h + lambda(t) with partial lambda=0, the two divisions give
E + partial(a partial v) = O(P eps^2) on D^1_{18}.
Take deltaU = U' v. Cauchy on the surviving strips bounds deltaU through
three y-derivatives. Then E_U (deltaU) = partial(a partial v)+(Ev)' has
(E v)'=O(P eps^2), so the second-order remainder of E about U is O(P eps^2)
uniformly at t=0 as well. Enlarge B_0 to absorb every Q-power and every
smallness condition. All Q-powers used are FIXED; none depends on N, the
retained rationals, or the initialiser order.
[deep] The two subtleties a reader should trust rather than re-derive: the
rational-jet ledger (all exponents fixed but chosen large; every t_0^-k
dominated via t_0 > 2*pi/N), and the two divisions at the exact resonance,
where the interpolation must take values, and (when the interpolation
HAPPENS to miss one derivative, or several rationals collide) first
parameters as well.
3.6 Initialisation: arbitrarily accurate formal stationary data.
LEMMA. For every fixed integer m>=1 there are real-analytic periodic
u_1,...,u_m with U^(m)(y,t) := y + sum_{j=1}^m u_j(y) t^{2j} satisfying
E( U^(m) ) = O( t^{2m+2} )
with coefficients holomorphic on a common smaller strip; and for d_0 small
U^(m) satisfies the smallness conditions with slack and
||E(U^(m))|| <= C_m d_0^{2m+2}.
Why: by the t=0 formula E(U)(y,0) = (U')^2 U''/4 = u_1''/4 + ..., the only
term at order t^{2j} that involves the NEW u_j is u_j''/4; earlier orders
determine everything else. Their mean is zero because <E>=0 identically
for every candidate. Solve the periodic equation u_j'' = 4(forcing with
mean zero); division of nonzero Fourier modes by -n^2 keeps holomorphy on
any smaller strip. Evenness is built in.
3.7 Newton iteration -> the limit U.
Let d_0 be small, s_0>2d_0, and start from U^(m) with initial error with
slack. At step i, use loss delta_i = 2^{-i} d_0 / 200 and shrink BOTH
current radii by 20 delta_i. Limiting parameter radius >= (4/5) d_0,
limiting y half-width >= s_0 - d_0/5. Set N_i = ceil( delta_i^{-1} log(1/eps_i) )
if eps_i>0, else stop. Enlarge a fixed integer B if needed. Since every
power of (1 + log(1/eps_i)) is O(eps_i^{-1/4}) from above, the step gives
eps_{i+1} <= C d_0^{-B} 2^{B i} eps_i^{7/4},
|| deltaU_i ||_{C^3_y} <= C d_0^{-B} 2^{B i} eps_i^{3/4}
(the second on the next domains). Choose first m with 2m+2 > 4B+4, then d_0
small enough that the initial error <= d_0^{4B+4}. By induction
eps_i <= d_0^{4B+4} 2^{-(4B+4) i},
because the future-right-side / next-bound ratio is
C d_0^{2B+3} 2^{4B+4-(2B+3)i} < 1
for small d_0. Corrections are dominated by C d_0^{2B+3} 2^{-(2B+3)i},
summable, so they keep the smallness with slack. Limit U is holomorphic,
real symmetric, even, periodic modulo y, and solves the stationary
equation. At t=0, (U')^2 U''/4 = 0 and U'≠0 force U''=0; periodicity gives
U(y,0) = y + c_0; replace y by y-c_0 to kill c_0. Shrink to a product
domain with enough y-margin for both shifts.
THEOREM (joint glancing parametrization).
There are s>r_1>0 and a real-symmetric holomorphic U(y,t) on
{ |Im y| < 2s , |t| < r_1 }
with U-y well-defined 2*pi-periodic and U even in t, U(y,0)=y, satisfying
D_1(U(y,t), U(y+t,t)) + D_2(U(y-t,t), U(y,t)) = 0
for |Im y|<s, |t|<r_1. Then
Theta(y,t) := phi(U(y,t)), p(y,t) := -S_1(Theta(y,t), Theta(y+t,t))
are holomorphic, real symmetric, even in t, and for small real t>0 the
billiard sends the line (Theta(y,t),p(y,t)) to (Theta(y+t,t),p(y+t,t)).
The map (y,t^2) -> line gives analytic coordinates on a full one-sided
phase collar, and t^2 is a nondegenerate analytic first integral there.
3.8 The area form on the collar.
dp ^ dTheta = A(t) dy ^ dt, A(t) = (c/4) t + O(t^3),
with A holomorphic and real on the real axis.
Why: change from S to D was a telescoping endpoint subtraction + coordinate
change, so the stationary equation is equivalent to the billiard one. The
angle Theta is even; the support-point formula plus antisymmetry of S
gives p(y,-t)=p(y,t). Taylor on the support-point formula:
p(y,t) = h(Theta(y,t)) - (1/8) r(Theta(y,t)) phi'(y)^2 t^2 + O(t^4)
(write beta=Theta(y+t)-Theta so p=h(Theta)-r beta^2/8+O(beta^3), then
beta=phi'(y)t+O(t^2); evenness removes the cubic). Since Theta_y>0 at
t=0 and the inward momentum coefficient in t^2 is strictly positive
(namely (1/4) r(phi(y)) phi'(y)^3 t + O(t^3) = (c/4)t + O(t^3)),
the analytic inverse function theorem gives local collar coordinates;
modulo phase they match globally after shrinking, because Theta_y>0
determines y from angle at fixed t^2 and the t^2-derivative of the
inward momentum displacement is strictly positive.
Write the pulled-back area as B_0(y,t) dy^dt. In collar coordinates the
map is (y,t) -> (y+t,t), so invariance of dp^dTheta gives
B_0(y+t,t) = B_0(y,t).
For every small real t with t/2*pi irrational, the translation orbit is
dense, forcing B_0(.,t) constant; its nonconstant Fourier coefficients
vanish on a set of t's with accumulation at 0 and hence identically by
holomorphy. So B_0(y,t)=A(t). The coefficient in front of dy^dt is
determined by the p(y,t) t^2 coefficient: A(t)=(c/4)t + O(t^3).
What we carry forward: an analytic one-parameter family of stationary
meshes, the area coefficient A(t), the coefficient a(y,t) of the
linearized operator, evenness in t, and the mean-zero identity <E>=0.
Chapter 4 uses a completely different kind of determination -- algebraic
over endpoint data for FINITE reflected chains -- to rule out a first
continuation wall.
------------------------------------------------------------------------
4. NO PATCH AT A FIRST CONTINUATION WALL
------------------------------------------------------------------------
Chapter 3 gave us a hole-free analytic parametrization near the real
collar. This chapter continues it in the phase variable y, into the
complex strip |Im y| < H_F(t), and shows that the first obstruction
is a wall with a rigid alternative: no meromorphic patch can cross a
point of the wall, not even one. That rigidity drives the rest of the
paper -- any local analytic progress on a parameter disk then forces a
GLOBAL statement about the real width.
4.1 Joint geometry and widths.
Forget the orientation of the line with coordinates (Theta(y,t),p(y,t))
from chapter 3 and let L(y,t) denote it as a line in the complex
projective plane. Put
X(y,t) = L(y-t/2, t) ∩ L(y+t/2, t).
For small real t≠0 this is the boundary impact; the apparent singularity
at t=0 removes, and X(y,0)=Gamma(y). Both L and X are even in t,
2*pi-periodic in y, and real symmetric. The incidence identity
X(y,t) ∈ L(y-t/2,t) ∩ L(y+t/2,t), rank dX <= 1,
and the reflection identity at the tangent X_y, hold initially on the
real collar and continue as differential-algebraic identities. Treat
projective maps as meromorphic data (coordinates in any generic chart);
reflection in a nonisotropic tangent is rational in that tangent.
Fix a narrow basic strip {|Im y|<s_*} and a parameter disk B around 0.
DEFINITION. For F=L or F=X, let h_F(t) be the supremum of heights b>s_*
for which the primary branch of F has a single-valued meromorphic
extension to {-s* < Im y < b} x V for some parameter neighborhood V
of t. Value +infinity allowed. These h_F are lower semicontinuous
(extend -> open parameter neighborhood) and unique on the basic strip.
LEMMA (width relation). After shrinking B if needed,
h_L(t) = h_X(t) + |Im t|/2.
Why: intersection gives h_X >= h_L - |Im t|/2. For the opposite
inequality suppose Im t>=0; the line through X(y-3t/2,t) and
X(y-t/2,t) is the preceding billiard line, and reflecting it at the
second point gives L(y,t). Both points and the tangent exist whenever
Im y < h_X(t) + Im t/2, and the resulting rational formula agrees with
the primary line on the basic strip. For Im t<0 use the succeeding two
impacts and backward reflection.
4.2 Finite-web continuation lemma. [deep but standard]
Let (a,delta) range over a real rectangle I x (0,delta_*); suppose
psi_i, w_i (nonzero), and B are analytic there, that dpsi_i have
pairwise DISTINCT directions, and that formal one-variable germs f_i
at one interior basepoint satisfy sum_i w_i(a,delta) f_i(psi_i(a,delta))
= B(a,delta). Then the f_i converge and continue along the rectangle.
Moreover if psi_i(a,delta) = a+O(delta) and d_a psi_i = 1+O(delta)
analytically up to delta=0, they extend to that edge near every smaller
interior a-interval.
Why: differentiate by d_a^{m-r} d_delta^r for r=0..m-1. The coefficient
matrix in the m-th derivatives f_i^{(m)} is a Vandermonde
(w_i A_i^m rho_i^r) with A_i = d_a psi_i and rho_i = B_i/A_i = distinct,
hence invertible. Every f_i^{(m)} is thus analytic affine-linear in the
finite vector V = (f_i^{(l)} : l<m) with analytic coefficients, and the
chain rules turn this into a first-order linear PDE system. Solving
first along a horizontal section and then along vertical sections, with
analytic dependence, gives global convergence; the extra edge
hypothesis makes the ODE coefficients bounded enough. Uniqueness of the
solution of the two selected ODE problems identifies its Taylor series
with the given formal one, so the full web relation and its
differential identities hold near the basepoint.
4.3 Line-endpoint determination. [deep, four-part]
THEOREM (thm:lens-lines). Fix k>=2 and set L_i = L(y+it,t). Every
intermediate L_i (0<i<k) is algebraic over the differential field of
L_0,L_k. Finitely many endpoint derivatives suffice. All derivatives of
intermediate L_i are algebraic over the corresponding endpoint jets.
Sketch.
(a) Endpoint parameters. Use the initial angle a and total angular
increment delta as coordinates: alpha_i = a+i*delta/k+O(delta^2),
psi_i = (alpha_{i-1}+alpha_i)/2, d_i = (alpha_i-alpha_{i-1})/2. The
critical value of sum 2h(psi_i) sin d_i has endpoint-derivatives
-p_0, p_k, so ALL its positive-order derivatives are known from endpoint
jets. Near glancing the intermediate (alpha_i,psi_i) all COALESCE
(psii = a+(i-1/2)delta/k+O(delta^2)), so their gradient directions are
distinct; the stationary Hessian is invertible.
(b) Finite algebraic core. Let the k occurrences of h be independent
germs h_i. Core: interior half-angle exponentials and jets up to order
m_0=max(2,k) at sampled angles. Base stationarity + every positive
critical-value derivative equal to endpoint data. If m>m_0 the new
jets h_i^{(m)} enter the m-th derivative tensor with coefficients
2 sin d_i * (grad psi_i)^{tensor m}. The direct-substitution contribution
dominates (the stationary-point movement is order m-1, contributing to
at least 2m-2 > m). Tensor components with 0..k-1 delta-derivatives
are Vandermonde-in-(d_delta psi_i)/(d_a psi_i), invertible. So every
higher h_i jet is RATIONAL in the core + endpoint data.
(c) Non-isolated => contradiction by coalescing limit. Suppose the
actual core admits a nonconstant analytic arc, giving formal
deformations h_i^eps = h + sum_{j>=p} eps^j f_{i,j} with some f_{i,p}
nonzero. Coefficientwise (no convergence of eps-series needed) first
variation yields a web sum_i 2 sin d_i f_{i,j}(psi_i) = K_j - B_j.
4.2 applies to give analytic continuation to the coalescing edge. Then
substitute eps = e*delta^{2/p}: the scaled critical value coefficientwise
tends to
G(e^p) = crit_{sum z_i=1} sum_i [-r(a)/24 z_i^3 + e^p f_{i,p}(a) z_i],
z_i(0)=1/k.
Boundedness forces all positive x=e^p coefficients of G to vanish, so G
is constant. Differentiating the critical value in x and comparing with
the Lagrange multiplier Lambda = -r/(8k^2) gives
-r/8 z_i^2 + x f_i = Lambda = 3G = -r/(8k^2),
hence sum_i sqrt(1 + (8k^2/r) x f_i) = k. Every positive-degree
coefficient in the square-root series is nonzero, so sum_i f_i^m = 0
for every m>=1. Newton's identities make every elementary symmetric
function of the f_i vanish, so every f_i = 0. Contradicts f_{i,p}
nonzero.
(d) Specialization. An isolated point of a complex algebraic set
defined over a field K has coordinates algebraic over K. Apply this
over a countable field containing the endpoint coordinate functions,
their jets, the actual radicals, and the actual core. Choose a real
basepoint outside the exceptional zero/pole set (measure-zero), so
evaluation is injective; the polynomial identity lifts back to germs.
Eliminating radical extensions gives algebraicity over the original
unoriented endpoint differential field. In characteristic zero the
minimal polynomial gives algebraic expressions for jets by
differentiation. Translation of (y,t) gives the other index intervals.
4.4 Point-endpoint determination. [deep, similar spine]
One additional scalar datum is needed: the area coefficient A(t).
THEOREM (thm:lens-points). Fix k>=2, P_i = X(y+it,t). Every
intermediate point and edge of the chain is algebraic over the
differential field of P_0,P_k and A(t). Same finiteness, translation
and analytic-continuation clauses.
Sketch.
(a) Length functional. Near glancing choose orthogonal coords Y=F(x),
F''!=0; use endpoint abscissas a=x(P_0), b=x(P_k)=a+delta. Interior
x_i = a+ i*delta/k + O(delta^2). Allow interior graph occurrences to be
independent functions F_i (endpoints fixed). The critical length
functional A_point(a,b). Its endpoint derivatives are
(A_point)_a = -P_{0,a} . e_{theta_0}; and by the momentum identity
(A_point)_{ab} da^db = dp_0 ^ dtheta_0 = A(t) dy^dt. The two
derivatives plus the Jacobian recover every mixed length derivative.
(b) High graph jets. Subtract the telescoping graph arc; the leading
chord-arc defect is -F''(a)^2/(24(1+F'(a)^2)^{3/2}) * (x_i-x_{i-1})^3.
m>m_0 gives new jets entering with weights
w_i = (incoming unit velocity - outgoing unit velocity)_vertical
(nonzero on a nonvertical nondegenerate edge). Mixed tensor components
with 1..k-1 b-derivatives and remaining a-derivatives give a
Vandermonde after nonzero column scalings; x_{i,a} = 1-i/k+O(delta),
x_{i,b} = i/k+O(delta). Rational elimination leaves an algebraic set
of possible cores.
(c) Isolation by coalescing. Non-isolated => formal arcs
F_i^eps = F + sum_{j>=p} eps^j f_{i,j}, endpoints zero, positive-order
critical-value change = sum_j eps^j(U_j(a)+V_j(b)). First variation
gives a web sum_{i=1}^{k-1} w_i f_{i,j}(x_i) - U_j(a) - V_j(b) = B_j.
4.2 applies (all foliations x_1..x_{k-1},a,b distinct). Then substitute
eps = e*delta^{2/p}; scaled critical length coefficientwise tends to
G(x) = crit_{sum z_i=1} sum_{i=1}^k [
-r^2 z_i^3 / 24
- r x (f_i+f_{i-1}) z_i / 2
+ x^2 (f_i-f_{i-1})^2 / (2 z_i) ],
z_i(0) = 1/k,
with r = F''(a), x = e^p. The multiplier identity lambda = 3G - 2x G'
= 3 G_0 + g_1 x for G(x) = G_0 + g_1 x gives roots
(r z_i /2 + x(f_i-f_{i-1})/z_i)^2 = -6 G_0 - 2 x (g_1 + r f_{i-1}),
(r z_i /2 - x(f_i-f_{i-1})/z_i)^2 = -6 G_0 - 2 x (g_1 + r f_i).
Take each sqrt with value r/(2k) at x=0, sum both roots for every edge
and then over all edges; sum z_i=1 gives r = r, so
sqrt(1-beta x g_1) + 2 sum_{i=1}^{k-1} sqrt(1-beta x(g_1+r f_i))
+ sqrt(1-beta x g_1) = 2k, beta = 8k^2/r^2.
Newton's identities on the multiset {g_1+r f_0, g_1+r f_1 (twice), ..,
g_1+r f_{k-1} (twice), g_1+r f_k} kill every g_i; fixed endpoints
force g_1=0 and then every f_i=0. Contradiction.
(d) Specialization: as in 4.3, lift the polynomial to germs over the
endpoint field including derivatives of A(t); the intermediate graph
values give all points, joins give all edges, and differentiation of
separable polynomials handles jets.
4.5 Product continuation. [deep but standard]
LEMMA (lem:meromorphic-hartogs). Let D_w be a connected plane disk and
0<rho<R. If F(w,zeta) is meromorphic on D_w x {rho<|zeta|<R} and
extends meromorphically to D_0 x {|zeta|<R} for some nonempty open
D_0 subset D_w, then it extends to the whole product D_w x {|zeta|<R};
a holomorphic original and a holomorphic seed yield a holomorphic
extension.
Why: at w_0 choose a circle |zeta|=r (rho<r<R) on which
(w-w_0)^m F(w,zeta) is jointly holomorphic; clear w-w_0 powers from
denominators. Negative Laurent coefficients c_{-j}(w) are holomorphic.
The Hankel matrix (c_{-(i+j+1)}(w)) has finite rank on generic reached
slices; the vanishing of all minors of some size r_0+1 is an analytic
condition, contains an open reached set, and therefore holds
identically. Finite Hankel rank means a fixed dependence among the
first finitely many columns propagates by row shift, giving a
polynomial-in-zeta clearing of the negative coefficients. Clear
coefficient denominators locally, and Cauchy's integral finishes. The
set of reached w_0 is open and relatively closed in D_w; it is
nonempty, so all of D_w. In the holomorphic case no clearing polynomial
is needed.
COROLLARY (cont:disc-sweep). Continuation propagates along a
continuous family of analytic disks; the holomorphic version preserves
uniform boundary modulus bounds and bounds on the real or imaginary
part of a scalar holomorphic component.
4.6 Width superharmonicity.
PROPOSITION (prop:width-superharmonic). Both h_L and h_X are positive
lower semicontinuous superharmonic functions on B (allowing the
identically infinite case); that is, they satisfy superharmonic
comparison on circles.
Why: fix t_0 + r*D inside B, and a nonnegative trigonometric q(ζ)
strictly below h_F(t_0+rζ) on |ζ|=1; let g be a holomorphic polynomial
with boundary imaginary part q, so Im g(0) = mean of q. Consider the
disks (t,y) = (t_0+rζ, x_0 + lambda g(ζ) + w) with real x_0,
0<=lambda<=1, |w| small. On |ζ|=1 these lie strictly below the width;
compactness + lower semicontinuity gives a common annular collar and a
common w-disk on which the pullbacks are jointly meromorphic. At
lambda=0 the whole disks are inside the basic strip. Increase lambda in
fixed small increments; the full-disk data at the old lambda give a
full-disk seed at the new lambda. 4.5 fills the original w-disk.
Repeat. At ζ=0 the resulting strip centers on Im y = lambda Im g(0)
and reaches height Im g(0) at t_0; so
h_F(t_0) >= (1/2*pi) integral_0^{2*pi} q(e^{i theta}) d theta.
Approximate lower-semicontinuous positive boundary values from below
by trig polynomials with strict margin, giving the supermean
inequality. With lower semicontinuity this is superharmonic comparison.
4.7 No patch at a first wall.
A JOINT PATCH means a meromorphic continuation on a neighborhood in
both y and t, agreeing as joint germs with the primary branch below the
wall.
THEOREM (thm:no-patch). Let F=L or F=X and let t_0 in B be real. If
h_F(t_0) = H_1 < infinity, no joint meromorphic patch of the primary
branch passes through any point of Im y = H_1. The lower wall follows by
real symmetry.
Sketch for F=L. Suppose a patch exists. Fix a narrow band around H_1
where joint data are known; the patch matches the lower primary tube
across its bottom. Choose a narrower concentric band and eta>0 so
H_1+eta is strictly inside it. Shrink the parameter neighborhood so a
common strip below H_1 is available jointly.
(a) Endpoint equations across the band. For nearby real irrational
t_1, forward and backward translates of every real phase meet a small
"good arc". On a small phase box choose integers m<0<n so L(y+mt,t)
and L(y+nt,t) come from the patch through the narrower band. Finite
boxes cover the band. Apply 4.3 on the common bottom strip to get
nonzero polynomial equations for L(y,t) with coefficients meromorphic
across the band. Squarefree, with nonzero leading coefficients and
discriminants; these stay nonzero as germs (proper analytic curves
meet t=0 divisors discretely) and the primary branch satisfies them
below any first wall.
(b) A first obstruction has nontrivial monodromy. Exceptional sets are
coefficient poles and zeros of leading coeffs and discriminants. On
each of countably many analytic boxes, all but countably many
parameters t_2 (real irrational) miss the exceptional curves' images.
Suppose h_L(t_2) <= H_1+eta, so the first wall is strictly inside the
band. Away from exceptional points, simple roots pass through. At an
exceptional curve y = z(t), a small loop about z(t_2) generates the
fundamental group of the punctured bidisk. Trivial monodromy on that
loop makes the branch single-valued on the punctured-product; after
clearing denominators its leading behavior is a unit times a power of
y-z(t), so a polynomial root bound gives |L_coord| <= C |y-z(t)|^{-N}
and hence joint meromorphy. If every wall point had trivial monodromy,
these extensions glue on overlaps below the wall and a finite cover of
the periodic wall would continue the tube to a strictly greater height
on a parameter neighborhood, contradicting the definition of h_L(t_2).
So SOME small loop acts nontrivially.
(c) A defect affects only finitely many shifted indices. Put
L_j = L(y+jt,t). At a fixed point, a small loop changes L_j only if
translation by j*t_2 aligns its center with an exceptional point;
finitely many such j modulo 2*pi (since j -> jt_2 mod 2*pi is injective
on a finite set of indices at fixed t_2). Choose the loop small enough
for the finite set of aligned indices; further shrink preserves its
monodromy there. Among aligned indices with nontrivial monodromy take
the smallest; translate the center along the orbit so this index is 1.
Choose K bigger than every aligned |j| and b>K with b>=2, and a further
finite set of far indices including both signs. Shrink the loop to fix
all unaligned members and the unaligned members of {-1,0}. Then the
loop produces
M_1 != L_1
fixing L_{-1}, L_0, L_b, and all selected far jets.
(d) Descent to glancing. Below the wall at real t_2 all selected
primary fields are single-valued jointly near every strict subheight.
Descend there, then move along real phase and parameter to the original
collar; the small loop avoids every shifted exceptional point relevant
to the finite selection. On return M_1 is algebraic over L_0,L_b and
their jets (transport 4.3 around the loop). Continue the alternative
root down along paths avoiding its squarefree exceptional curves. Near
a generic point (y_0,0), only t=0 can be an exceptional divisor for the
polynomial; a power cover t=tau^d kills the finite monodromy, so M_1
has a joint meromorphic PUISEUX germ at glancing, its inequality from
L_1 persisting.
(e) The alternate germ is even. Put Z = L_0 ∩ M_1. Continued
incidence, rank, and reflection show Z has rank <= 1 and gives an
involutive relation. Z != L_{-1} ∩ L_0 (different as germs, else
continuing back would identify preceding and following intersections).
On the Puiseux cover there is a projective limit Z^0(y) which is
nonconstant because it lies on L(y,0) and tangents of the strictly
convex arc have no common projective point. If L(y,0) were generically
tangent to the fixed arc traced by Z^0, differentiated incidence and
tangency identify Z^0 with Gamma(y); near the simple quadratic tangency
of L_0(y,t), there are exactly two primary nearby intersections, so the
preceding is excluded and reflection at the following gives L_1,
contradicting M_1 != L_1. Thus the fixed arc is transverse to L(y,0).
Intersect it with the even line L_0(y,t) by the ordinary holomorphic
implicit function theorem, then reflect in its nonisotropic tangent.
The result is a joint holomorphic Puiseux germ M_1(y,t) with
M_1(y,-t) = M_1(y,t).
(f) The monodromy operation. The incoming leg can end at a regular
(y_0,0) itself; take the outgoing primary leg from the same point. Call
the resulting closed continuation path A. Let I be pullback under
t |-> -t. Do A, I, A^{-1}, I. The distinguished field follows
L_1 -> M_1 -> M_1 -> L_1 -> L_{-1}.
This fixes L_0 and every selected far field with its jets, provided both
signs of each far index have been selected (evenness of L sends L_j to
L_{-j}; the reversed path fixes that primary germ; the last pullback
restores L_j). So from the endpoints of any interval [B,B+b], the
immediately preceding line is algebraic over the same jets, as are all
interior lines (4.3) and, inductively, everything to the left.
(g) One polynomial, infinitely many roots. With A_1 = A, I, A^{-1}, I,
pick B>K+2 and b>=2. There is one fixed nonzero polynomial (over the
differential field of primary germs) for a generic direction coordinate
of L_1. For each n>=0, shear (y,t)->(y+2nt,t), apply the monodromy
operation with far indices including +/- (B+2n) and +/- (B+b+2n), shear
back. It sends L_{1-2n} -> L_{-1-2n} and fixes the coefficient jets. So
the same polynomial has direction coordinates of
L_1, L_{-1}, L_{-3}, L_{-5}, ...
as roots in the field of primary germs. They are DISTINCT: since
Theta(y,t) = phi(y) + O(t^2), the first t-derivative of a generic slope
coordinate of L_j is j phi'(y) times a nonzero chart factor. A nonzero
polynomial over a field cannot have infinitely many distinct roots.
Contradiction. Thus there is NO first wall at or below H_1+eta for
generic real irrational parameters in the neighborhood.
(h) Transfer to original real parameter. For every nearby irrational
t_1 we get a parameter neighborhood where h_L(t) >= H_1+eta for real t
outside a countable exceptional set (and H_1+eta is determined by the
patch). A countable subcover covers all nearby irrational real
parameters. Discard countable exceptions and countably many rational
shifts; we get the bound almost everywhere on an interval about t_0.
Put M = H_1+eta. Superharmonic comparison (4.6) on an upper half-disk
with diameter in that interval, approximating the lower-semicontinuous
positive lower bound from below by continuous functions with strict
margin, gives h_L(t) >= M omega_+(t) where omega_+ is the harmonic
measure of the diameter. The lower half-disk gives h_L(t) >= M omega_-.
Both harmonic measures tend to 1 near the interior diameter point t_0,
and the small-circle supermean inequality centered at t_0 (then
shrinking the circle) gives h_L(t_0) >= M. Contradicts h_L(t_0)=H_1.
For F=X patches: at real parameters, h_X=h_L by the width relation, and
forward/backward translates X(y+(j+1/2)t,t) of the patch supply
enclosing impacts for every edge. Choose a finite covering by phase
boxes and a common narrower vertical band. 4.4, with the analytic
coefficient A(t), supplies full-band joint polynomial equations for
the edge L(y,t), attached to the primary branch on the common bottom
strip. Those equations are the input to (g); no additional point
identity is needed there. Superharmonic transfer then contradicts
h_X(t_0)=H_1.
4.8 The boundary parametrization has the same natural wall.
COROLLARY (cor:shape-natural-boundary). Put H = h_X(0) = h_L(0). The
parametrization Gamma(y) = X(y,0) is meromorphic on the open strip
D_H = { y in C : |Im y| < H } .
If H<finite, Gamma has no meromorphic continuation patch through any
point of either boundary line Im y = +/- H. On its meromorphic domain,
det(Gamma',Gamma'')^4 / (Gamma' . Gamma')^3 = c^2, c = c_{normalize}
from chapter 3.
Why (upper edge; lower by real symmetry).
(1) At each strict upper subheight X has a joint meromorphic extension
near t=0, and there is no polar component of X along the entire zero
slice (such a component would meet the regular collar). Restriction to
t=0 gives meromorphic Gamma, and overlaps glue. So Gamma is meromorphic
on D_H, and the intrinsic identity holds on the real arc by
Gamma' = (c/(phi')^2) e_phi, so it holds meromorphically everywhere.
(2) Suppose a patch exists at H<infinity. Pick a regular point
s_edge = x_* + iH of the patch with det(Gamma',Gamma'') and
Gamma'.Gamma' nonzero; local square roots give a local normal, then a
local normal angle phi, an inverse angle coordinate, and a curvature
radius. Intrinsic identity shows r(phi)(phi')^3 = c * nonzero constant.
The zero slice L(y,0) is the tangent.
(3) A regular starting segment. Choose a row Im y = H - (small) below
the edge so short slanted segments with normalized vertical component
>=1/3 stay inside the patch. Choose the exact height avoiding joint
poles on the segment (indeterminacies intersect t=0 discretely). Then
the local action D gives uniform Theta(y,t)=phi(y)+O(t^2), and its
coefficient constancy reduces stationarity of the local action to
d_1 D(V, V+s) = kappa s^2 + O(s^4), kappa nonzero constant
(no cubic term). Antisymmetry => forward and backward O(s^2) match;
the analytic square root gives V_{j+1} = 2 V_j - V_{j-1} + O((V_j -
V_{j-1})^3).
(4) Steep propagation. For |Im t|>=|t|/3 (replace t by -t so it is
upward), start at V_0 = y - Nt, V_1 = y-(N-1)t with N = O(1/|t|). If
step sizes stay <=2|t| the recurrence error is O(|t|^3), so summing
twice V_j - (y-(N-j)t) = O(|t|) closes the step bound. Locally constant
N -> holomorphic; different N/choices glue on known slabs by
nonrepeated-successor uniqueness. Attach to the primary data on a
fixed lower part; at any chosen strict subheight, shrink |t| so that
comparison with the known primary branch holds.
(5) Quadratic disks fill remainder. Let eta be small. Choose r_0 small.
Consider
t = r_0 ζ, y = x_* + i sigma - i eta ζ^2 + w,
H - 2 eta <= sigma <= H,
|ζ|=1, w small. On near-real arcs |sin theta|<1/2, Im y|_{w=0} = sigma -
eta cos 2 theta <= H - eta/2, so joint primary data exist for r_0 small.
On the remaining arcs |Im t|/|t| >= 1/2 so steep propagation applies
with margin. Compactness gives a common annular collar and w-disk. At
sigma=H-2eta the full disks are inside the joint domain; raise sigma in
small steps using 4.5 and 4.5's corollary to sweep; at sigma=H the
disk center ζ=0 gives a joint meromorphic patch through
(x_*+iH, 0), contradicting 4.7. Lower edge by real symmetry.
What we carry forward: at every real t on B, the primary branch of
Gamma has a natural wall at +/- H(t) = h_X(t); there is NO patch
through any point of that wall in any parameter neighborhood; and the
intrinsic identity det(Gamma',Gamma'')^4/(Gamma' . Gamma')^3 = c^2
holds on the meromorphic domain. Chapters 5-11 study the two complex
fields f = phi' and q = f''/f, and push growth at the wall to
geometric contradictions.
------------------------------------------------------------------------
5. THE SHAPE STRIP AND THE INTRINSIC COORDINATE
------------------------------------------------------------------------
Chapter 4 gave us a curve Gamma on an open strip D_H = {|Im s| < H},
H possibly infinity, with a natural wall when H is finite and no
continuation patch across any point of the wall. This chapter studies
the two complex "shape fields"
f = phi' (the derivative of the normal angle phi)
a = e^{i phi} (the exponential of the normal angle)
and their relation to the boundary geometry. There are exactly two
cases, and the whole rest of the paper splits along them:
(E) a has a zero or a pole inside D_H => boundary is an ELLIPSE.
(N) a is zero-free and pole-free inside D_H => there is a global
shape coordinate lifting every continued impact.
5.1 Isotropic coordinates.
Write z = x_1 + i x_2, w = x_1 - i x_2 (isotropic; note
zz' + ww' = 0 for adjacent coordinates, and z,w carry the two
"characteristic directions"). On the real boundary let a=e^{i phi},
f=phi', and c = r(phi) f^3 > 0 (the printed normalization constant).
Then
Gamma' = (c/f^2) e_phi, Z' = i c a / f^2, W' = -i c / (a f^2),
where Gamma=(X_1,X_2) and Z=X_1 + i X_2, W=X_1 - i X_2. In particular
a^2 = - Z'/W', f = (1/(2i)) (a^2)'/(a^2), a = f^2 Z'/(ic).
The first relation is read meromorphically, so it defines a^2 as a
meromorphic function on D_H. The second gives f, and the last fixes a
from f and Z'. These are also the identities reconstructing the
NORMAL from the curve coordinates. A zero or pole of a is called an
ISOTROPIC CENTER (the corresponding tangent to Gamma is isotropic).
The INTRINSIC NORMALIZATION (from chapter 4) is
det(Gamma',Gamma'')^4 / (Gamma' . Gamma')^3 = c^2,
and follows from the displayed formulas above by direct computation.
Under a reparametrization s = sigma(u) it is multiplied by sigma'(u)^6;
consequently, once Gamma is meromorphic on D_H, the identity holds
there through all its mermorphic continuation.
5.2 Isotropic centers: (E) the boundary is an ellipse.
LEMMA (shape:isotropic-profile). Suppose a has a zero of order n>=1
at p in D_H. Then n != 3. Put D_+ = n+3, D_- = 3-n. There is a germ
xi(w) = w+O(1) at infinity such that xi^{D_+} and xi^{D_-} are BOTH
rational, and satisfy the reflection-functional equation
[xi(w+1)^{D_+} - xi(w)^{D_+}] [xi(w-1)^{D_+} - xi(w)^{D_+}]
------------------------------------------------------------
[xi(w+1)^{D_-} - xi(w)^{D_-}] [xi(w-1)^{D_-} - xi(w)^{D_-}]
= (D_+/D_-)^2 xi(w)^{2(D_+ - D_-)}.
The pole case is identical with Z and W interchanged.
Why: f = n/(i z) + O(1) gives Z(s)-Z(p) ~ c_Z z^{n+3}, W(s)-W_0 ~ c_W z^{3-n}
with nonzero constants. If n=3, then W' ~ const/z has nonzero residue,
impossible for a meromorphic W. So n != 3. Locally X(y,t)=Gamma(v(y,t))
with v(y,t)=y+sum_{j>=1} t^{2j} v_j(y); even powers by evenness of X.
The pole-bound ord_p v_j >= 1-2j is what makes the leading limits exist
under z=tw; it is proved by dividing the arc-minus-chord action
D(s,s+h) = h^3/24 + O(h^5) by t^3 and grouping: at order t^{2j} the new
coefficient v_j enters with -v_j''/4 (same as E at t=0), the forcing has
order at least -1-2j, so v_j'' gains order >= -1-2j, giving the bound.
Then z^{-b} P(z,z lambda) is holomorphic in (z,lambda) for every
constant-coefficient affine combination P whose zero-slice restrictions
have order b, by Weierstrass preparation and division. Its restriction
to z=0 is rational in lambda, so xi(w) = w U(1/w) makes
t^{-b} P(tw,t) -> c_P xi(w)^b
analytically with derivatives for |w|>R_0. Apply to Z-Z(p), W-W_0.
The reflection law in isotropic coordinates says the product of the
two incident chord slopes ((Z difference)/(W difference)) equals the
square of the tangent slope; substituting at w, w-1, w+1, all constants
cancel, and we get the function equation above first near infinity
then as an identity of rational functions.
LEMMA (shape:simple-centers). Every zero and pole of a in D_H is
SIMPLE. At such a center p the curve has the local symmetry
Gamma(2p - s) = Gamma(s), a(2p - s) = -a(s).
Why: Riemann-Hurwitz on one rational map. Put
d = gcd(D_+, |D_-|), R = xi^d, alpha = D_+/d, beta = D_-/d,
rho(w) = R(w+1)/R(w). Bezout from rationality of xi^{D_+} and xi^{D_-}
makes R rational, and R(w) ~ w^d, so rho(w) = 1 + d/w + O(w^{-2}) is
a nonconstant rational map of some degree s>=1 with rho(infinity)=1.
Set F(x)=(x^alpha-1)/(x^beta-1). After cancellation,
F(rho(w+1)) = (alpha/beta)^2 / F(1/rho(w)).
Case n=2: alpha=5,beta=1, F has four simple zeros at nontrivial fifth
roots; each of those fibers has multiplicities divisible by 4,
contributing >=3s ramification, contradicting 2s-2. Case n>3: at each
maximal run of poles of rho, the pole-order recurrence is
alpha p_{j+1} = beta p_j, and at the preceding point not a pole, the
seam relation alpha p_0 = m sum_j p_j + beta p_l with m=6/d>=1.
If m>=2 these seam points contribute >=ms>=2s ramification,
contradicting 2s-2. If m=1,gamma>=2, we get analogous contradictions;
the only remaining possibility m=gamma=1, alpha=2 is also impossible
for one fiber of a degree-s map. This leaves n=1.
Case n=1: d=2,alpha=2,beta=1. The equation becomes
(1+rho(w+1))(1+1/rho(w))=4,
so 2/(rho-1) increases by one under translation and is therefore
CONSTANT (rational one-periodic), giving R(w)=(w+b_0)(w+b_0+1). In
particular xi is NOT rational (xi^2 has two simple zeros; Bezout with
rationality of xi^{D_+}=xi^4 and xi^{D_-}=xi^2 would force xi rational).
Choose a local coordinate zeta with W-W_0 = c_W zeta^2. If Z had a first
odd term of order j in zeta, subtracting the higher even terms in W-W_0
would give xi^j rational for ODD j, hence xi rational, contradiction.
So both Z and W are even in zeta; the involution zeta -> -zeta induces
a local symmetry sigma of the shape parameter with sigma'(p)=-1. By
the intrinsic normalization sigma'^6=1; the symmetry is constant,
sigma(s)=2p-s; the tangent ratio gives a(2p-s)^2 = a(s)^2; a simple
zero picks the sign a(2p-s) = -a(s).
THEOREM (thm:isotropic-classification). If a has a zero or pole in
D_H, the boundary is an ELLIPSE.
Why: a center is nonreal (since |a|=1 on the real axis). If H<infinity,
the symmetry at p locally extends Gamma past the reflected overlap
of D_H, contradicting the natural wall. Hence H=infinity. Real symmetry
adds the conjugate center, and composing the two involutions 2p-s and
2q-s gives a NONREAL PERIOD in addition to 2*pi. Hence a is doubly
periodic; let Lambda be its lattice. Composing reflections about any
two centers shows 2(their difference) in Lambda, so all centers lie in
4 classes mod Lambda and all simple; degree of a on C/Lambda is <=2.
Degree one would be a biholomorphism C/Lambda->P^1, impossible;
degree is two. Riemann-Hurwitz: exactly four simple ramification points,
branch values beta_1..beta_4 distinct. The quotient
a'^2 / [(a-beta_1)...(a-beta_4)]
is holomorphic and nonvanishing on the torus, hence a nonzero constant;
so a'^2 = P(a) with deg P = 4. Differentiating a(2p-s)=-a(s) gives
a'(2p-s)=a'(s), so P(-a)=P(a): P is EVEN. Thus on the real curve
f^2 = -a'^2/a^2 = C + B cos(2 phi) + D sin(2 phi) > 0
for real constants B,C,D, i.e. f^2 = n_phi^T N n_phi for a real
positive-definite symmetric 2x2 matrix N. Direct computation gives
(sqrt(nNn))'' + sqrt(nNn) = det(N) / (nNn)^{3/2}.
Therefore the support h_0(phi) := |c/det N| sqrt(n_phi^T N n_phi)
has curvature radius c/f^3 = r(phi), which makes it the support of an
ELLIPSE with shape matrix N. Finally h-h_0 solves
(h-h_0)'' + (h-h_0) = 0
(a support function and h_0 both solve the ODE h+h''=r for the SAME r),
so h-h_0 = A cos phi + B sin phi, i.e. a translation. The ellipse
is rigid.
So from now on we work strictly in the NONISOTROPIC case:
a, and hence f=phi', is zero-free and pole-free in D_H.
5.3 (N) Global shape lift. [deep]
Throughout the rest of this chapter assume a has neither zeros nor
poles in D_H. Then a = e^{i phi} has a holomorphic log by simple
connectivity: phi, f are holomorphic with phi(s+2pi)=phi(s)+2pi. The
affine curve Gamma can still have poles (at zeros of f). We need to
control the impact coordinate even at those poles.
THEOREM (thm:shape-lift). Unless the boundary is a CIRCLE, there is a
single-valued holomorphic s = s(y,t) on the joint tube
T = { (y,t) : t in parameter disk, -h_X(ybar) < Im y < h_X(t) },
satisfying
X(y,t) = Gamma( s(y,t) ), |Im s(y,t)| < H,
along with s(y,0)=y, s(y+2pi,t)=s(y,t)+2pi, s(y,-t)=s(y,t), and real
symmetry s(ybar,tbar) = conj s(y,t). For each real t, y -> s(y,t) is
an orientation-preserving analytic circle diffeomorphism (after
shrinking the parameter disk). The bound |Im s|<H is void when
H=infinity.
Why (key ideas):
(a) Near real glancing the regular parametrization of the curve plus
rank-one gives the local lift. Its differential satisfies
c^2 s_y^6 = det(X_y,X_yy)^4 / (X_y.X_y)^3,
s_t/s_y = (X_t.X_y)/(X_y.X_y).
Remove a proper analytic set (poles, zeros in denominators, zeros of
the sixth root) and stay on a connected complement containing the
initial germ. Local primitives continue along every path. The
composition identity persists. At H<infinity the continued primitive
cannot hit Im s = H: local product coordinates (s, z) and a generic
constant-z section would continue Gamma through its edge, contradicting
4.8.
(b) Loop monodromy. On returning along a loop, the primitive is
s -> omega s + b, omega^6=1, giving Gamma(omega s+b) = Gamma(s) on the
connected overlap of the two strips. For H<infinity a nonreal omega
(and Im b != 0 for omega=+-1) shifts the strip past an edge -> edge
patch -> impossible; only omega=+-1, b in R remain. For H=infinity,
similar reality arguments (conjugating the real translation 2pi, and
for omega=-1 with Im b!=0, composing with its conjugate to get nonreal
period b-bbar) give the same conclusion; here a nonreal period of
Gamma would be a nonreal period of a, forcing a constant, contradicting
its real winding. Real embeddedness rules out a real reflection and
forces a real translation to be an integer multiple of 2*pi. Hence
v := e^{i s} descends to Gamma on the regular complement as a
single-valued function.
(c) Removable singularities. In the finite-strip case v and 1/v are
bounded (e^{+/-H} bounds) so they extend holomorphically across the
removed set with the same strict bounds (nonconstant v, maximum
principle). In the infinite-strip case use meromorphic mergel: ds is
single-valued (loop monodromy is translation) and extends
meromorphically, and if a single-valued holomorphic off an analytic set
has meromorphic sixth power, multiply by a local denominator of that
power: the product is bounded (its sixth power holomorphic) hence
removable. Either way, ds becomes holomorphic. Write Gamma = G(e^{i s})
with G meromorphic on C*. On a generic transverse disk around a
possible polar divisor, v cannot have an essential singularity (else
every punctured neighborhood has a dense image in C* and G is constant
by matching X=G(v)); a finite-order zero or pole transcribes to a
meromorphic extension of G; reality extends G at the other end, so
G is RATIONAL. Then a=f^2 Z'/(ic) makes a rational in w. A rational
function with no zeros/poles on C* is Cw^k; its real winding is the
degree, so k=1, hence f constant and the boundary a circle -- the
excluded case. Therefore v extends with nonzero finite values on every
generic transverse disk, and ds has no polar divisor; removability
in codimension two makes ds holomorphic throughout T.
(d) Single-valuedness. T is simply connected (its vertical fibers are
y-intervals containing the real axis, so vertical contraction retracts
it to R x parameter disk). Finite H: take a log of v. Infinite H: take
a primitive of the holomorphic closed one-form ds. Normalize by the
original germ. All monotonicity and symmetry identities follow by
analytic continuation, and once y -> s(y,t) satisfies s_y = 1+O(t^2)
uniformly, shrinking the parameter disk makes it a positive-degree-one
circle diffeomorphism.
5.4 (N) Speed zeros and the first displacement.
LEMMA (lem:f-zeros). Every zero of f in D_H is SIMPLE, and f''=0 at the
zero. Consequently q := f''/f extends holomorphically throughout D_H.
Why. Say f vanishes to order m>=1 at s_0. In fixed tangent/normal
coordinates, phi(s)-phi(s_0) has order m+1, so the tangent and normal
components of Gamma' have leading orders -2m and 1-m with nonzero
constants. m=2 gives a nonzero residue for the normal coordinate,
impossible for a meromorphic curve; m>=3 gives normal and tangent
coordinate POLES of orders m-2 and 2m-1 in the normal coordinates.
Then use the curve chord: in the lift X(y,t)=Gamma(s(y,t)), the
equation s(y,t)=s_0 defines y=s_0+O(t^2); for small t!=0 the two
neighboring shape coordinates are s_0 +/- t+O(t^2) with finite
impacts. Along the central curve the chord to the previous impact has
normal-to-tangent slope of order at least m+1, so its normal angle
theta is holomorphic with d = theta - phi(s) = O((s-s_0)^{m+1}).
Differentiating the signed offset p_chord = n_theta . Gamma(s),
dp_chord ^ dtheta = c f(s)^{-2} sin d ds ^ dd = +/- A(t) dy ^ dt,
using the continued area identity, and A(t) ~ (c/4)t is nonzero for
small t!=0. Substitute z=s-s_0 and d=z^{m+1} b: the LHS has order
-2m + (m+1) + (m+1) = 2, contradiction. So m=1. At a simple zero,
f=Az+Bz^2+..., A nonzero; from a'=i f a, a=a(s_0)+O(z^2); the residue
of a/f^2 is -2 a(s_0) B/A^3. It vanishes because ic a/f^2 = Z' is the
derivative of a meromorphic Z. Hence B=0, which is f''(s_0)=0.
Now fix the harmless freedom to translate phase as a function of t.
Put b(t) := (1/(2pi)) integral_0^{2*pi} (s(y,t)-y) dy. b is analytic,
even, real, vanishes at 0. Precompose L, X, s with y -> y-b(t), shrinking
the parameter disk. Contour translation gives the MEAN NORMALIZATION
<s(y,t)-y> = 0.
All rotation, area, and symmetry identities survive; the widths h_L,h_X
acquire the same harmonic summand Im b(t), so the width relation and
superharmonicity/no-patch still hold. Averaging on a horizontal row in
T now gives <Im s> = Im y, and the shape constraint |Im s|<H yields
h_X(t) <= H.
PROPOSITION (shape:first-displacement). In the mean normalization,
s(y,t) = y + t^2 V(y) + O(t^4),
where V is holomorphic and periodic on D_H, has zero mean, and satisfies
V' = [ f^2 + 8q - <f^2 + 8q> ] / 120.
Why. Continue from real nonsingular data. First, in the short-chord
action D(s,s+h) = h^3/24 + d_5(s+h/2) h^5 + O(h^7), a direct
midpoint-integration calculation (writing g=f^{-2},
g_pm = g e^{+/- i phi}, and the midpoint index d_5) gives
d_5 = - (f^2 + 8q) / 1920,
where q=f''/f. Then at the middle impact s=s(y,t) the previous and
next shape increments are h_pm = t + t^3 V' +/- (1/2) t^4 V'' + O(t^5);
the cubic part of the stationary action contributes -t^5 V''/4, the
fifth-order terms contribute -5 t^5 d_5' + t^5 d_5', and stationarity
at order t^5 reads -V''/4 - 4 d_5' = 0. Substitute d_5 above, integrate
once, and kill the integration constant by <s-y>=0.
5.5 What we carry forward. Unless the boundary is an ellipse (excluded
in the general case), we now have:
H > 0 possibly infinite; a = e^{i phi} and f = phi' holomorphic
and zero-free on D_H; q = f''/f holomorphic on D_H;
a global shape coordinate s : T -> D_H with |Im s| < H, mean
normalization <s-y>=0, h_X(t) <= H;
the first-displacement identity V' = (f^2 + 8q - <f^2+8q>)/120.
The M(s) = max{|f(s)|, |q(s)|^{1/2}} quantity, its growth at the wall
Im s = H, and the resulting complex-analytic contradictions occupy
chapters 6 through 11.
------------------------------------------------------------------------
6. THE NORMALIZED CHORD EQUATION AND CONE PASSAGE
------------------------------------------------------------------------
Chapters 4-5 set up the shape strip and the global lift. This chapter is
where the billiard equation is rewritten so that it survives (remains
analytic) at the poles and zeros of the affine shape curve, and where
that rewritten equation is then used to CONTINUE the lift toward the
wall Im s = H --- as long as the shape fields f=phi' and q=f''/f are
not too large in a sector ("cone") at the wall. The corollary "quiet
profiles" from this machinery is what chapters 7-11 will repeatedly use.
Throughout: nonisotropic case (5.5). Shape fields f, q holomorphic on
D_H. Set M(s) := max{ |f(s)|, |q(s)|^{1/2} }. Phase normalization
<s-y>=0, hence h_X(t) <= H.
6.1 Kernel action and the exact recurrence.
Fix a shape point s, a small nonzero parameter t, and a normalized step
J (so the second endpoint is s+tJ). Introduce the fast variable u and
fast fields
F(u) = t f(s+tu), Q(u) = t^2 q(s+tu), F'' = QF,
and the three potentials
Q_0 = Q, Q_pm = Q -/+ (3i/2) F' - F^2/4.
For each potential let k_j(u) solve k_j'' = Q_j k_j with k_j(0)=0 and
k_j'(0)=1. The normalized "arc-minus-chord" action is
p(s,J;t) = ( k_0(J) - sqrt( k_+(J) k_-(J) ) ) / ( F(0) F(J) ),
with the square root on the glancing branch (k_0 at zero field).
Why it is a clean object. Substituting
g_0(u) = F(u),
g_pm(u) = F(u) exp( -/+ (i/2) integral_0^u F(v) dv ),
and using F''=QF shows g_j'' = Q_j g_j. Reduction of order gives
k_j(J) = g_j(0) g_j(J) integral_0^J g_j(u)^{-2} du,
extended meromorphically across zeros of g. On the shape curve, the
arc length element is (c/f^2) ds, the squared chord between the two
endpoints is (c integral f^{-2} e^{+i phi}) (c integral f^{-2} e^{-i
phi}), and after substituting reduction-order and changing ds = t du
this matches the displayed p. We obtain thereby a formula that only
involves F, Q, and the solutions k_j, i.e. one that is LOCAL on the
shape curve (no affine coordinates anywhere).
LEMMA (kernel division). [deep]
Let Q be holomorphic near the segment [0,J]. Regard Q, J, F(0), F'(0)
(for F''=QF) as variables. Near F(0)=F'(0)=0 and any (Q,J) with
k_0(J)!=0, the right side of the p-formula above, with the square root
chosen equal to k_0(J) at zero initial data, extends ANALYTICALLY
through F(0) F(J)=0, is EVEN in F, and has bounds and analytic
dependence, including derivatives on smaller neighborhoods, that are
uniform on compact admissible (Q,J). In particular, for J in a fixed
small neighborhood of 1 and small Q, F(0), F'(0),
p - J^3/24 = O( ||Q|| + |F(0)|^2 + |F'(0)|^2 ),
and the same estimate holds for any fixed number of J-derivatives on a
smaller neighborhood.
Idea: if F(0)=0 the identity g_pm = g_pm'(0) k_pm and reality of the
Wronskian give k_+ k_- = k_0^2; if F(J)=0 use backward kernels (whose
value at 0 is -k_j(J), Wronskian again) and the two signs cancel in
the product. Thus on both zero hyperplanes the chosen square root
matches k_0. The map (F(0),F'(0)) -> (F(0),F(J)) has Jacobian k_0(J)
!= 0, so the two coordinates are analytic, independent; an analytic
function vanishing on both coordinate hyperplanes is divisible by
their product. Cauchy in these two coordinates gives the uniform
bounds. Replacing F by -F interchanges k_+ and k_-, so the quotient is
even; linear-in-data terms die; analytic interpolation from Q=0 bounds
the rest by O(||Q||).
EXACT RECURRENCE. For consecutive endpoints s_-, s = s_- + t J_-, and
s + t J, stationarity of the sum of the two actions (the derivative of
ct^3 p at the middle endpoint acts once as final endpoint of the left
cell -- giving c t^2 p_J there -- and once as initial endpoint of the
right cell -- giving c t^3 p_s - c t^2 p_J) is exactly
p_J(s,J;t) - t p_s(s,J;t) = p_J(s_-, J_-; t). (exact recurrence)
Because the kernel division extends through F(0)F(J)=0, this recurrence
also makes sense at the exceptional affine points of the curve.
6.2 The two frozen actions.
The limits we will need are constant fast fields satisfying FQ = 0.
For F = lambda and Q = 0, put z = lambda J / 2. Then k_0(J) = J and
k_pm(J) = 2 sin z / lambda,
so
p_F(lambda,J) = J/lambda^2 - 2 sin(z)/lambda^3,
E_F(lambda,J) := J p_F' - p_F = 2(sin z - z cos z) / lambda^3.
For F = 0 and Q = lambda^2, the analytically divided action is
p_Q(lambda,J) = J/(2 lambda^2) - tanh(z)/lambda^3,
E_Q(lambda,J) = (tanh z - z sech^2 z) / lambda^3,
where z = lambda J / 2 throughout. Why the Q-type frozen action comes
out this way: solve the Wronskian system for F = eta e^{lambda u} + ...
Inserting F = eta e^{lambda u} into F'' = QF with Q=lambda^2 and
expanding in eta gives the arc and chord above; the eta^{-2} terms
cancel; the glancing square root leaves
(1/(2 lambda^2)) ( J - ( integral_0^J e^{-lambda u} du )^2 /
( integral_0^J e^{-2 lambda u} du ) ).
The kernel lemma identifies the analytic value at F=0 with this limit.
Small-z series (used in chapters 8-9):
p_F = J^3/24 - lambda^2 J^5/1920 + O(lambda^4 J^7),
E_F = J^3/12 - lambda^2 J^5/480 + O(lambda^4 J^7),
p_Q = J^3/24 - lambda^2 J^5/240 + O(lambda^4 J^7),
E_Q = J^3/12 - lambda^2 J^5/60 + O(lambda^4 J^7).
(Note the Q_TYPE has the LARGER lambda^2 coefficient.)
LEMMA (analytic convergence to frozen action). Suppose analytic fast
fields with F''=QF converge locally uniformly to constant frozen states
(F=lambda,Q=0) or (F=0,Q=lambda^2) on complex neighborhoods of a compact
path, and that the branch of the glancing square root is analytically
continued along the perturbed path. Then their actions p converge
analytically, with all derivatives on smaller neighborhoods, to the
frozen action on that continued branch; the endpoint alone does not
select a branch after arbitrary loops.
Why: in the F-type both p_JJ = sin(z)/(2 lambda) and k_0 = J are nonzero
along the path avoiding degeneracies; in the Q-type
p_JJ = tanh(z) sech^2(z)/(2 lambda) and k_0=k_+=k_-=sinh(2z)/lambda, and
the only kernel zeros are incompatible with p_JJ != 0; so the ordinary
quotient formula for p applies continuously. Uniqueness of the square
root on overlaps tracks the branch. Cauchy gives convergence of
derivatives.
6.3 Cone passage. [load-bearing]
From now on suppose H < infinity for this subsection; "depth" always
means distance below the upper edge:
delta = H - Im y, delta_s = H - Im s.
The lower edge is by real symmetry.
THEOREM (thm:cone). Fix 1/2 < alpha < 1, a finite target aperture m_0,
and delta_b in (0,H/2). Then there is an input aperture L = L(alpha,m_0)
with: for any K<infinity, d->0, real centers x_0, and 0<r<=Kd, if
r M(s) <= eps (d/delta_s)^alpha
for d/10 <= delta_s <= delta_b and |Re s - x_0| <= L delta_s,
then for reasonably small eps>0 and then small d>0, the lift continues
JOINTLY HOLOMORPHICALLY along vertical paths from the fixed lower row
throughout
d < delta < delta_*, |Re y - x_0| < m_0 delta, |t| < r(delta/d)^alpha
(delta_*>0 fixed), and with
B := 1 + (eps/r) d^alpha
satisfies
|s(y,t) - y| <= C B^2 |t|^2 delta^{1-2alpha}.
The relative depth error stays small, so these continuations stay inside
D_H, and they agree with the original lift wherever the latter is
already defined below them.
Input aperture L is independent of K. Once it contains the fixed
geometric enlargements used in the proof, C and the eps-smallness are
independent of the target aperture m_0; they may depend on K, alpha and
the fixed lower-strip data.
Proof approach, in two parts.
(Part A) STEEP SECTORS via the exact recurrence. Fix a small gamma>0.
Work with |Im t| >= gamma |t| > 0 (whichever of t,-t moves upward, by
evenness). Start in a fixed slab about delta_s = delta_b/2 where the
lift gives s-y = O(t^2) and the difference quotient is 1+O(t^2), so the
largest parameters in the construction tend to zero. Follow the exact
recurrence; let x_j denote the sequence of unperturbed depths, dropped
by |Im t| per step, down to a depth x_j >= d/2, with |t| <= 10 r (x_j/d)^alpha.
Suppose inductively that position errors are small fractions of x_j
and normalized steps J_j are close to 1. The shape fields on every
required segment and a small thickening satisfy
|F(0)|+|F'(0)| <= C_K |t| B x_j^{-alpha},
||Q|| <= C_K |t|^2 B^2 x_j^{-2alpha},
by Cauchy on a disk of fixed fraction of x_j (which is well defined
because consecutive x_j's are comparable with constants depending on K).
Increasing the input aperture by the fixed cone slope accommodates the
horizontal displacement of these thickenings.
In this setting the kernel-division small parameter is bounded by
|t| B x_j^{-alpha} <= C (eps + r d^{-alpha}), so the kernel lemma
applies uniformly. Writing a(s,J;t) = p(s,J;t) - J^3/24,
|d_J^nu d_s^mu a| <= C_{K,mu,nu} |t|^2 B^2 x_j^{-2alpha-mu}, (nu,mu fixed)
again by Cauchy. Now test the exact recurrence with the trial J = J_-
(previous step). The cubic parts cancel; the difference of a_J between
adjacent basepoints together with t a_s is at most
C_K |t|^3 B^2 x_j^{-2alpha-1}; the derivative of the equation in the
new J is J/4+o(1) bounded away from zero. The analytic implicit
function theorem gives a unique nearby step with
|J - J_-| <= C_K |t|^3 B^2 x_j^{-2alpha-1}.
Sum over the depth mesh: for beta>1 the mesh sum satisfies
|t| sum_{x_j>=x} x_j^{-beta} <= C x^{1-beta}, and the last summand is
controlled since |t|/x is bounded. This gives
|J-1| <= C_K |t|^2 B^2 x^{-2alpha},
|s-y| <= C_K |t|^2 B^2 x^{1-2alpha} (using 2alpha > 1),
initial errors absorbed by B>=1, and closes the induction with
(|s-y|/x) + |J-1| <= C_K (eps + r d^{-alpha})^2.
Different integer numbers of steps back to the slab give the same
analytic germ by uniqueness of the near-identity recurrence, so these
germs define analytic continuation from below on every steep sector.
(Part B) FILLING THE REMAINING DIRECTIONS via analytic disks. Define
"reached" germs to be those obtained from the fixed lower row along the
vertical path with Re y and t fixed. Nearby parallel paths compare via
a finite chain of the same germs, so reached germs form a
single-valued holomorphic function of (y,t) already known on their
open domain, and they agree with the steep continuations above. Write
U(y,t)=(s(y,t)-y)/t^2 (regular at t=0 on the initial and reached data).
Pick D_0 large fixed with D_0^alpha > 10, then gamma small. For a
target (y_c,t_c) of depth delta, let R = r (delta/d)^alpha. Because
|t_c|<R, the parameter circle t = t_c + 3R zeta meets the non-steep
sector only on two uniformly short arcs. Choose a polynomial g with
g(0)=0 such that
|g| <= C_0 delta, Im g <= delta/4 on the whole circle,
Im g <= -D_0 delta on those arcs and a slightly larger collar.
This is done by smooth negative bumps on the arcs, then balancing the
mean, then approximating with a real trig polynomial with strict
margins, then taking its boundary-imaginary-part polynomial; the arc
widths are uniformly bounded below (the arcs are transverse), so the
bump widths and polynomial degree are also uniformly bounded.
Now choose an auxiliary target aperture m >= m_0 large enough that
m + C_0 < m D_0, and choose L that contains this aperture, the shifts
by C_0 delta, and the steep paths and thickenings, all with fixed slack.
We induct from a small fixed depth in successive bands of halved depth
threshold; in each band use the parameter circles t = t_c + 3R zeta,
y = y_c + g(zeta) + w with |zeta|<=1 and small complex w. On the
NON-steep boundary arcs the depth is at least D_0 delta, so those
arcs lie in the previous all-direction domain (the D_0^alpha > 10
inequality supplies the parameter margin; m + C_0 < m D_0 supplies the
aperture margin). On the STEEP arcs of the circle, part A applies at
depth at least delta/2 with the larger parameter range allowed there;
small collars of the arcs behave similarly. Start y_c lower by a fixed
multiple of delta so the whole disk and a neighborhood lie in the
reached domain; sweep sigma up in small steps using the product
continuation lemma of chapter 4, and the reached domain fills. At
sigma at the target depth the center of the disk gives the target
continuation. The constant C and the smallness of eps are fixed by the
recurrence estimates and are independent of the target aperture.
6.4 Quiet profiles, coarse bounds, subcritical exclusion.
At a real parameter t, ANY cone passage gives a global lower bound on
h_X(t): otherwise the local composition s(y,t)=Gamma^{-1}(X) with
|Im s|<H and joint analyticity in (y,t) would give a joint meromorphic
patch through the first wall, contradicting thm:no-patch.
COROLLARY (cor:quiet-cone). Let d_n -> 0, b_n>=1, x_n in R.
Suppose that on a sufficiently large fixed cone starting at depth d_n
(d_n/b_n) M(s) <= C (d_n/delta_s)^alpha, 1/2<alpha<1.
Then (i) on a strictly coarser cone starting at a fixed large multiple
of d_n, there is a passage with parameter radius c(d_n/b_n)(delta/d_n)^alpha
and error
|s-y| = O( b_n^2 |t|^2 d_n^{2alpha-2} delta^{1-2alpha} );
in rescaled variables Y=(y-x_n-iH)/d_n, S=(s-x_n-iH)/d_n, T=t b_n/d_n,
|S-Y| <= C|T|^2 (-Im Y)^{1-2alpha}.
(ii) If additionally (d_n/b_n) M(x_n + iH + d_n S) -> 0 uniformly on
every compact subset of {Im S<0}, then for every c_1,T_1>0 the lift
passages reach depth c_1 d_n for all |t| <= T_1 d_n / b_n eventually,
and the width deficit (H - h_X(T d_n/b_n))/d_n -> 0 uniformly for
|T|<=T_1.
(iii) The hypotheses in (ii) are IMPOSSIBLE if b_n stays bounded: a
vanishing weighted cone bound down to a sufficiently small fixed depth
ratio ALREADY forces a contradiction, via Part B of the cone passage
above (the disk argument reduces the quiet profile to a joint
meromorphic patch at a real parameter, contrary to thm:no-patch).
Sketch of (i): apply thm:cone with starting scale a fixed large multiple
of d_n and r a small fixed multiple of d_n/b_n; the resulting B is
bounded by const * b_n d_n^{alpha-1}, giving the displayed error;
increasing the fixed starting multiple permits any fixed bounded
interval of T. Sketch of (ii): pick beta with 1/2<beta<alpha; on a
rescaled cone put D=delta_s/d_n; the weighted norm to be made small is
D^beta (d_n/b_n) M, small on compacts of D by the quiet-profile
hypothesis and small on the tail D>=R by the coarse bound; take R large
then n large; then thm:cone with r a small multiple of c_1 d_n and
radius a large multiple of T_1 d_n/b_n gives a vanishing relative
error, and the real no-patch transfer gives h_X(t) >= H-c_1 d_n
uniformly. Since c_1 is arbitrary and h_X<=H, (ii) follows. For (iii),
bounded b_n turns a vanishing weighted cone bound into the same with
normalization d_n M; use thm:cone with starting scale d_n/20, radius
2d_n and target aperture at least 50, plus shifted impact data on the
disks t = d_n zeta, y = x_n + i sigma - (i/4) d_n zeta^2 + w with |zeta|
<= 1, sigma from H-4 d_n to H. On upper imaginary parameter arcs
recover L from the preceding two impacts (at y - t/2 and y - 3t/2) and
reflection at the nearer impact; on lower arcs use the two forward
impacts. The nearer impact has depth at least d_n/4 for zeta=e^{i theta}
on the circle (the formula is displayed in the source: the coefficient
of d_n is 1/4 - (1/2)sin^2(theta) + |sin(theta)|/2 which is >= 1/4),
the farther impact is deeper, and all horizontal shifts fit the fixed
aperture. Composition with Gamma gives a joint meromorphic patch of X
through (x_n+iH, 0) attached to the primary zero-parameter data; against
thm:no-patch.
PROPOSITION (subcritical exclusion). At a finite shape edge in the
nonisotropic alternative, a uniform bound
max_{Im s = H - delta} M(s) = O(delta^{-p_0})
with p_0 < 1 is IMPOSSIBLE.
Why: choose alpha with max{p_0,1/2}<alpha<1. On a cone with delta_s>=d,
d M(s) <= C d^{1-p_0} (d/delta_s)^alpha (delta_s/d)^{alpha-p_0}
and the weighted quantity d M(s)(delta_s/d)^alpha is bounded by
C d^{1-alpha} delta_s^{alpha-p_0}, which tends to 0 uniformly for
depths between any fixed multiple of d and the fixed starting row. So
cor:quiet-cone(iii) applies with b=1.
PROPOSITION (witness near every centre; cone:no-quiet-witness). If for
all small delta_s
delta_s max_{Im s = H-delta_s} M(s) <= C (1 + log(1/delta_s))^A
for fixed C,A, then there are constants L,C_1,C_2,c_*>0 with: for every
sufficiently large k=log(1/d) and every real center x_0, there is some
point s with
|Re s - x_0| <= L delta_s, k - C_1 log k <= log(1/delta_s) <= k + C_2,
and delta_s M(s) >= c_*.
Why: fix alpha and the cone aperture required in (iii). Choose C_2 so
e^{-C_2} d is below all the fixed ratios required there and then
C_1>A/(1-alpha). If no uniform c_* existed, there would be k_n->infinity
and centers such that sup delta_s M(s) -> 0 in their windows. On the
window,
d_n M(s)(delta_s/d_n)^alpha = delta_s M(s) (d_n/delta_s)^{1-alpha}
is uniformly small (last factor at most e^{(1-alpha)C_2}); BELOW the
window, delta_s/d_n >= k_n^{C_1}, the global polynomial bound gives
the same vanishing; the remaining fixed lower-strip portion has bounded
M and is smaller still. So the coarse quiet hypothesis of (iii) holds
with b_n=1, contradiction. Diagonalizing over a decreasing sequence of
eta>0 yields the witness window for every center.
6.5 What we carry forward. Cone passage, the coarse-bound corollary, the
subcritical exclusion, and the witness-near-every-center. Chapters 7-9
use them with b_n growing (the superlinear scales d_n = e^{-k_n},
m_n = e^{g(k_n)}, b_n = d_n m_n, b_n->infinity) to turn any superlinear
growth of M(s) at the wall into either a joint patch (impossible by
no-patch) or into a "pure limit" FQ=0 on the lower half-plane.
6bis. FINITE SUPERLINEAR GROWTH IS IMPOSSIBLE (supplement to ch6).
[This section is placed here, before the integration-of-growth-alternatives
section, to make the forward reference from 6.5 land on a stated proof.]
THEOREM (thm:finite-superlinear-exclusion). In the nonisotropic
non-circular case with H<infinity, if the growth order at the wall
g(p) := limsup_{k->infinity} (1/k) log max_x M(x+i(H-e^{-k}))
satisfies g(p) = p with 1 < p < infinity, this is impossible.
Idea, three cases of a compactness extraction. Choose deep levels
k_n -> infinity along which the row maximum is representative
(records, and slowly varying), and set
d_n := e^{-k_n}, m_n := max_{Im s = H-d_n} M, B_n := m_n,
b_n := d_n m_n = e^{(p-1)k_n+o(k_n)} -> infinity,
b_n / log m_n -> infinity.
Compactness of the rescaled fields
F_n(S) := (d_n / B_n) f(x_n + iH + d_n S),
Q_n(S) := (d_n / B_n)^2 q(x_n + iH + d_n S),
on the lower half-plane of scaled S, with |F_n|, |Q_n|^{1/2} at most 1
and one of them = 1 at 0 for suitable x_n, extracts a locally uniform
subsequence. THREE OUTCOMES.
(1) F-TYPE. F_n -> F not identically zero on a lower-half-plane
compact, with |F(S_0)| >= c_0 > 0 somewhere.
(a) All poles of f^{-2} in the shape strip are at simple zeros of f
(lem:f-zeros). At each such zero z, f''(z)=0, hence
Res_z(f^{-2}) = -f''(z) / f'(z)^3 = 0.
(b) Build a small closed contour C_n on the periodic cylinder, staying
within a fixed narrow band about depth d_n, whose points all avoid
the zero set of f and where |f| >= c_0 m_n (witness). A standard
detour construction (chapter 6 cone machinery + the zero-side
bound of witnesses) produces C_n of length O(1) with |f| bounded
below by c_0 m_n on it.
(c) Residue theorem with residues zero (all 0 by (a)) gives
integral_{C_n} ds / f(s)^2 = integral_0^{2 pi} ds / f(s)^2 > 0.
The right side is a FIXED POSITIVE number (since f > 0 on the real
arc). The left side has absolute value O(m_n^{-2}) -> 0. Contradiction.
(2) Q-TYPE. F_n -> 0, Q_n -> Q not identically zero on a lower-half-
plane compact. Use horizontal mean convexity of log|f| and evenness
(chapter 7, log-deficit stack) to choose centers x_n with
|f(x_n + i(H - d_n))| >= c_0 > 0 and pass to a subsequence with Q_n -> Q.
On a fixed short segment in a direction e along which Re(e^2 Q_n) >= c'
uniformly, the nonnegative function u(v) := |F_n(z_n + e v)|^2 satisfies
u'' = 2|e F_n'|^2 + 2 B_n^2 Re(e^2 Q_n) |F_n|^2 >= 2 c' B_n^2 u.
Comparison with the solution of v'' = 2 c' B_n^2 v having the same
endpoint bounds (subtract and use the maximum principle, a positive
interior max contradicts w''-2c'B_n^2 w>=0) gives u <= C e^{-c B_n} on
a shorter central segment. Maximum modulus on a disk about z_n
(constructed from the finite cover of the Q-nonvanishing region) gives
|F_n(z_n)| <= C e^{-c B_n}. But |F_n(z_n)| = (d_n/B_n)|f(z_n)| >=
(d_n/B_n) c_0, so c_0 d_n/B_n <= C e^{-c B_n}, contradicting
b_n / log m_n -> infinity in the form d_n e^{c' b_n} -> infinity.
(3) TRIVIAL PROFILE. F_n -> 0 AND Q_n -> 0 on lower-half-plane
compacts. This is exactly the configuration thm:no-trivial-profile
(chapter 8) rules out via the moment bootstrap and the Taylor + lower-
lift contradiction. Its proof does not use this result, so the argument
is not circular.
Since (1), (2), (3) are exhaustive, finite superlinear growth is
impossible. With chapter 6's subcritical exclusion (p_0<1) and
chapter 7's exclusion of infinite H and infinite upper order, the only
remaining case in the nonisotropic non-circular class is
H finite and growth order EXACTLY one.
------------------------------------------------------------------------
7. EXCLUSION OF INFINITE GROWTH AT THE WALL
------------------------------------------------------------------------
Chapter 6 gave the cone passage machinery. This chapter excludes the two
remaining noncritical cases, once H is finite: (a) H = infinity, (b)
H < infinity with M(s) growing faster than any power of inverse depth at
the wall. Then the only remaining case is the critical one (finite H,
growth order EXACTLY one), treated in chapters 8-9. Throughout:
nonisotropic, non-circular case (5.5). f = phi', q = f''/f holomorphic
on D_H, M(s) = max{|f(s)|, |q(s)|^{1/2}}. Impact lift s(y,t) with
<s-y>=0, s even in t.
LEMMA (log-deficit; lem:log-deficit). There is an absolute C such that:
for f holomorphic not identically zero on the unit disk, log|f| <= K,
and f''/f extending holomorphically through the zeros,
sup_{|z|<1/4} |f''(z)/f(z)|^{1/2} <= C (1 + integral_D (K - log|f|) dA).
On a disk of radius Delta, the left side is Delta sup|q|^{1/2} and the
integral is the disk average of the log deficit.
Why: U := K - log|f| >= 0, A := 1 + integral U. The distribution
-Delta U / (2pi) is the zero-counting measure; a cutoff test bounds
the zero count by C A. List zeros z_j with multiplicity; then
V := U + sum_j log|z-z_j| is harmonic on |z|<0.8, with L^1 norm O(A)
there; interior harmonic estimates give V', V'' bounded by C A on a
smaller disk. Off the zeros, differentiating the log of f gives
|f''/f|^{1/2} <= C A + C sum_j |z-z_j|^{-1},
and each summand is locally integrable in 2 real dimensions with a
uniform L^1 bound, at most C A of them. Submean for |q|^{1/2}
(subharmonic across zeros because q is holomorphic) gives the bound.
Periodic form (used below). The horizontal mean
L(h) := (1/(2 pi)) integral_0^{2 pi} log|f(x+i h)| dx
is convex by subharmonicity and even by real symmetry, hence
L(h) >= L(0) > -infinity. If K bounds log|f| at all heights of disks of
radius Delta, averaging their deficits over horizontal centers gives
|q|^{1/2}(s) <= C Delta^{-1} (1 + Delta^{-1}) (1 + K_+)
on a smaller concentric disk. (Enclose the disk in a rectangle of
height and width 2 Delta; each horizontal level is covered by
O(1+Delta) whole periods; integrate in height.)
7.1 Exponential-profile obstruction (statement). [load-bearing]
THEOREM (thm:exponential-obstruction). There is NO sequence of centers
s_n = x_n + i h_n in D_H, length scales ell_n > 0, and field scales
m_n > 0 such that, writing b_n = ell_n m_n and
F_n(S) = f(s_n + ell_n S)/m_n, Q_n(S) = q(s_n + ell_n S)/m_n^2,
Y = (y-s_n)/ell_n, T = m_n t,
the following all hold:
(a) b_n -> infinity;
(b) (D_H - s_n)/ell_n exhausts C;
(c) (F_n,Q_n) -> (c_0 e^{-iS}, 0) OR (F_n,Q_n) -> (0, c_0^2 e^{-2iS})
locally uniformly on C, c_0 != 0;
(d) either (i) H<infinity: h_n -> H, ell_n -> 0, and there is an
alpha_0 in (1/2,1) plus delta_b in (0,H/2) such that for every
fixed L<infinity, eventually
(d-i) M(s) <= C_L m_n (ell_n/delta)^{alpha_0} on
ell_n <= delta = h_n - Im s <= delta_b, |Re s - x_n| <= L delta;
OR (ii) H=infinity: h_n -> infinity, 0 < ell_n <= C, (d-i) holds
with a single constant for ALL horizontal positions on
ell_n <= delta <= h_n, and log(1+h_n/ell_n) = o(log m_n).
Key point: both profiles in (c) correspond to running scales of f or q
where the OTHER field vanishes to second order; the theorem rules out
both, because at that scale the deep model has a branch point, and the
branch point cannot coexist with a single-valued lift.
7.2 Deep passages (lemma 1).
LEMMA (profile:deep-data). Under hypotheses of 7.1 (without assuming
its conclusion), fix alpha in (1/2, alpha_0) and T_0 < infinity. There
are D_0, C independent of the target aperture A such that for every
fixed A<infinity and every sufficiently large n, there are joint
holomorphic vertical passages on compacts of
D := -Im Y > D_0, |Re Y| < A D, |T| < T_0,
satisfying |S_n(Y,T) - Y| <= C |T|^2 D^{1-2alpha}. They start in the
actual lift domain and stay inside D_H.
Proof approach (H<innity case, then H=infinity initializer). In the
finite-H case, apply thm:cone with reference height h_n in place of H
(no no-patch needed at that auxiliary height), with
d = ell_n D_0, r = 2 T_0 / m_n, eps = C' T_0 D_0^{-alpha}.
The cone hypothesis rM <= eps(d/delta)^alpha is exactly the coarser
cone (d-i); the auxiliary bound B = 1 + (eps/r) d^alpha <= C m_n
ell_n^alpha, and m_n ell_n^alpha -> infinity because b_n -> infinity
and ell_n -> 0. So the physical displacement estimate divided by ell_n
is the claimed |S_n - Y| bound. Increasing D_0 makes the required
smallness hold; the starting slab lies in a fixed strict substrip;
the relative error keeps the image in D_H. The D_0 and C do NOT depend
on A, using the weakening-trick: D^alpha M/m_n <= C_L D^{-(alpha_0 -
alpha)} beyond a finite depth threshold; on the remaining finite cone
exponential-pairs gives the bound eventually.
For H=infinity, the same construction is doctored: retain the same
d = ell_n D_0, r, B = C m_n ell_n^alpha; the log-scale condition
log(1+h_n/ell_n)=o(log m_n) gives
r(h_n/d)^alpha -> 0 and B h_n^{-alpha} -> infinity;
the known data near Im y=0 then has uniformly small parameters over
the entire range.
The remaining proof of the lemma is by Part A (steep sectors) as in
thm:cone, with the SAME kernel-division estimates: |J-1| <= C |t|^2 B^2
x^{-2alpha}, |s-y| <= C |t|^2 B^2 x^{1-2alpha}, the second because
2alpha>1; the accumulation argument is via the depth mesh sum, and the
ALL-DIRECTION extension is by the disk filling of Part B (chapter 6),
which enters only through the meromorphic-Hartogs setup. Then dyadic
halving from h_n/K_0 down to d; no factor accumulates across bands
because 1-2alpha<0 makes the deeper known arcs smaller.
7.3 The deep model and its uniqueness.
LEMMA (profile:energy-model). The deep passages of 7.2 converge,
locally uniformly with derivatives, to a unique holomorphic model
S_*(Y,T). With lambda = T c_0 e^{-i S_*} and J = d_Y S_*, it
satisfies E_F(lambda,J) = 1/12 (in the first profile) or
E_Q(lambda,J) = 1/12 (in the second profile), and as -Im Y -> infinity
(in any fixed cone) S_*-Y -> 0, J -> 1.
Proof core. Scaling recurrence: for a_n := T/b_n,
J_n(Y) = (S_n(Y+a_n,T) - S_n(Y,T))/a_n -> d_Y S_*(Y,T),
and the fast fields are T F_n(S+a_n u), T^2 Q_n(S+a_n u) -> (T c_0
e^{-iS}, 0) or (0, T^2 c_0^2 e^{-2iS}), by (c). The frozen actions
pf, pQ from chapter 6 apply on the corresponding branch. Divide the
exact recurrence
p_{n,J}(S,J) - a_n p_{n,S}(S,J) = p_{n,J}(S_-,J_-),
S = S_- + a_n J_-
by a_n; both sides are evaluations of p_{n,J} at neighboring states,
so analytic convergence gives (no rate of convergence vs a_n required)
d_Y p_J(S_*,J) = p_S(S_*,J).
Since S_{*,Y}=J, differentiating E := Jp_J - p along the solution
gives d_Y E = 0; the deep anchor fixes E = 1/12. Uniqueness: at
(lambda,J)=(0,1), dE/dJ = 1/4, so the frozen energy equation
lambda^3 = 24(sin z - z cos z) OR lambda^3 = 12(tanh z - z sech^2 z),
z = lambda J/2,
determines J analytically; w := T c_0 e^{-iY} satisfies
w d lambda/dw = lambda J = 2z, so lambda/w -> 1; integration
w = lambda exp( integral_0^lambda (1/J(v)-1)/v dv ),
S_*-Y = i log(lambda/w)
fixes the constant to a single value. Hence the deep model is unique;
a normal-family argument upgrades compact convergence to full
convergence.
7.4 A branch point of the model. [deep, load-bearing]
LEMMA (profile:branch-model). In the variable
w := T c_0 e^{-iY},
the deep model is analytic for |w|<R (some R>0), continues through the
boundary circle |w|=R EXCEPT possibly at w=R and w=-R, and a
sufficiently small loop around w=R from the inner side CHANGES the
value of S_*-Y. The loop and its two adjoining corridors can be chosen
so that all chord kernels and the twist p_JJ remain nonzero on the
changed branch.
Proof idea. In the F case (profile (F_n,Q_n)->(c_0 e^{-iS},0)), set
v=z and
g(v) = 3(sin v - v cos v)/v^3, P(v) = v g(v)^{2/3}/sin v, v_c=pi.
In the Q case set v=tanh z and
g(v) = (3/(2 v^3)) (v - (1-v^2) arctanh v), P(v) = g(v)^{2/3},
v_c=1.
Both have g(0)=P(0)=1. Differentiating the frozen equations gives
lambda = 2 v g(v)^{1/3}, w v'(w) = v P(v), v(w) ~ w/2.
Positivity and zero-freeness. In the first case every nonzero zero of
sin v - v cos v is real (via the quadratic form integration by parts);
the zeros r_k are simple with k pi < r_k < k pi + pi/2; Hadamard for
the even entire function of order one gives g(v) = product (1-v^2/r_k^2);
its log series has all strictly positive coefficients in |v|<pi. In
the second case the coefficient series a_j = 3/((2j+1)(2j+3)) with
increasing ratios; the reciprocal coefficient recursion has
0 <= b_n <= a_n; g zero-free on the unit disk and P has nonnegative
even coefficients. Then w(v) = 2v exp(integral_0^v (1/P(xi)-1)/xi d xi)
is increasing on (0,v_c), d log w/dv = 1/(vP(v)) -> finite positive R
as v->v_c. The power series v(w) reaches radius R (else it continues
through the closed circle by the regular ODE); at radius R its positive
sum is exactly v_c. At phases other than 0 and pi its modulus is STRICT-
LY less than v_c (equality in triangle inequality for the strictly
positive linear and cubic terms would force e^{2i theta}=1), so the ODE
continues through those phases.
For 0<|w|<R, v(w) never vanishes; lambda/w is holomorphic and nonzero
on the whole disk, so the impact model p lives there.
Change of value at R. In the F case, w(v) is analytic at v=pi with
simple quadratic critical point and w''(pi) = -4R/lambda_c^2 != 0,
lambda_c = (24 pi)^{1/3}; then v(w) has nonzero sqrt term and
lambda = lambda_c - 2 pi eta + (2 sqrt2 lambda_c / 3) eta^{3/2} +
O(eta^2), eta = 1 - w/R;
a turn around R changes eta^{3/2} -> -eta^{3/2}, so it changes lambda.
In the Q case put x=v-1, G=3/2, Lambda=2 G^{1/3};
g(v) = G[1 - x log(-x) + (log 2 - 2) x + O(x^2(1+|log|x||))] and
lambda = Lambda - (delta/R) log(-delta/a_0)
+ ((1+log 2)/R) delta + O(|delta|^2 (1+|log|delta||^2)),
a_0 = R/G^{2/3} != 0;
the -delta log(-delta) term changes the value under a turn (log jump).
Consequently S_*-Y = i log(lambda/w) changes. On a small bypass all
kernels and p_JJ stay nonzero because v stays in the open interval
(0, pi) or (0,1) away from endpoints, and lambda stays nonzero.
7.5 Tracking a regular corridor.
LEMMA (profile:tracking). Fix a compact corridor of a continued branch
of the model from 7.4 with nonzero kernels and twist, and let T range
over a compact set avoiding zero. Suppose two neighboring lift entries
are initialized in a joint patch where they converge to that model,
with convergence of their first divided difference. Then along segments
of O(1/|a_n|) increments a_n = T/b_n, the exact recurrence continues
these entries holomorphically and TRACKS the model with error o(1). The
same holds for increments -T/b_n. Uniform on compact thickenings.
Proof. Cover the compact model states by finitely many action charts
with kernels and p_JJ bounded away from zero. The scaled recurrence
has a unique nearby root for J (IFT), and Taylor expansion gives
J - J_- = a_n (p_S - J_- p_{SJ})/p_JJ (S_-,J_-) + o(|a_n|),
so the state map
(S_-,J_-) -> (S_-,J_-) + a_n (J_-, (p_S-J_- p_{SJ})/p_JJ) + o(|a_n|)
is the Euler step of the ODE d_Y p_J = p_S, which the model solves;
over O(1/|a_n|) steps with Lipschitz 1+C|a_n| the errors sum to
O(|a_n|)+ initial -> o(1). Holomorphic implicit function theorem handles
the complex neighborhoods and thickenings; even in lambda lets us use
-T as well.
Consequence: if |Im T| >= eta > 0, choose the parameter increment
a_n = +/- T/b_n; a target can be reached from an initialized lower slab
whenever its backward slanted segment remains in one regular corridor;
choose the number of steps locally constantly and use uniqueness of
every subsequent nearby root to make germs on overlaps agree.
7.6 Proof of thm:exponential-obstruction (assembling 7.2-7.5).
Set T_* := |c_0|^{-1}, r_0 := T_*/32; T_* is real and |T_* c_0|=1.
First reach, vertically with joint convergence, the inner regular
region
|T c_0 e^{-iY}| <= R e^{-nu}, |T - T_*| <= 2 r_0,
for each fixed nu>0 and every needed compact horizontal range. Fix
T_c in this parameter disk and use parameter circles
T' = T_c + 8 r_0 zeta,
Y' = Y_c - i log(T'/T_c) + g(zeta), g(0)=0.
(The parameter disk is bounded away from 0 so the log is single-valued;
note T' e^{-i(Y_c - i log(T'/T_c))} = T_c e^{-iY_c}.) Choose g with
boundary imaginary part equal to a sufficiently large negative constant
on the arcs |Im T'|<eta (plus a collar), at most nu/2 everywhere, and
mean zero; approximate the bumps by a trig polynomial with strict
margins; on the bad arcs the point lies in initialized data, on the
other arcs use the upward increment. The depth D_0 in thm:profile:deep-
data was independent of the aperture, so this initialization remains
available. Raise Im Y_c by discrete steps using meromorphic-Hartogs to
sweep the disks and the maximum principle on the difference (filled
lift vs analytic model) to promote convergence to the disk centers.
This gives the inner regular region.
Next reach two OUTER regular patches just beyond the model circle,
w = R e^{u + i theta} and w = R e^{u - i theta}, u <= nu,
T = T_*,
by doctoring the same construction with a polynomial whose Im is
large negative on |Im T'| < eta_1 (collar) and <= nu everywhere, with
|g| <= C_1 nu; raise centers from -C nu to +nu. At physical real
t = T_*/m_n, one of these passages reaches the physical height
h_n + ell_n (log R + nu),
still inside D_H (by deep data + compact exhaustion). Composing with
Gamma gives joint meromorphic germs pinned to the primary branch from
below, hence at real t gives a WIDTH LOWER BOUND:
h_X(T_*/m_n) > h_n + ell_n (log R + nu).
Now sweep the real parameter along the horizontal row Im Y = log R +
nu/2 from phase +theta to -theta, in increments a_n = T_*/b_n > 0, with
initialization in the first outer patch. Tracking follows the OUTSIDE
of w=R. At the endpoint the result differs from the INDEPENDENT radial
continuation to the other outer patch: close the comparison path
through the inner disk, and the resulting loop winds once around R,
avoiding -R. That's exactly the nontrivial continuation of 7.4. Both
must agree with the actual single-valued lift (by the tracking-vs-
actual-lift agreement from 7.5 extended by uniqueness of overlaps).
Contradiction: the same value cannot be two different things.
Hence NO sequence of scales and profiles as in thm:exponential-
obstruction exists. Whatever growth M(s) has near the wall, at every
scale it must violate one of (a)-(d). (The Q-type case is symmetric
once you replace F by Q in the above.)
7.7 Exclusion of entire and infinite-order growth.
THEOREM (thm:infinite-growth-exclusion). In the nonisotropic
non-circular case: (i) if H = infinity, the boundary is a CIRCLE;
(ii) if H < infinity,
limsup_{k -> infinity} log max_x M(x+i(H - e^{-k}))/k < infinity.
Why (i) and (ii) together. Exclude constant f (circle).
Suppose first that H<infinity and the displayed limsup is infinite.
The same is true with M replaced by |f| (otherwise log max|f|=O(k) on
those rows; disks of radius comparable to e^{-k} with fixed extra
depth margin have a global log upper bound O(k); periodic-q-bound
(7.7 above) then gives a finite power bound on |q|^{1/2}, contradicting
the assumed infinite order of M).
Now treat both (i) and (ii) via rho(h) := log max_x |f(x+ih)|. rho is
convex in height (three-lines) and even by symmetry. In the H<infty
case just identified it tends to +infinity and has infinite upper
order relative to log(1/(H-h)). In the H=infty case a nonconstant
real-symmetric periodic function has a growing Fourier mode, hence
rho(h) >= c h - C for some c>0. So in both cases rho -> +infinity
and is eventually strictly increasing with slope bounded below.
Let h(R) be the inverse at large values, N(R) its right slope there,
mu(R) := log N(R); h'(R) = e^{-mu(R)} a.e., and liminf_{R->infty}
mu(R)/R = 0 (otherwise inverse integration would give a finite
limiting height and H-h(R) <= C e^{-cR}, contradicting either H=infty
or infinite upper order). Pick R->infty along mu(R)=o(R); the positive
measure d mu on [R_b,R] has mass o(R); the weak maximal-interval
inequality gives a flat selection: eps_R->0 and a continuity point
v in [R/3,2R/3] with |mu(z)-mu(v)| <= eps_R |z-v| for R_b<=z<=R.
Set h_n=h(v), ell_n=1/N(v), m_n=e^v. Then uniform on each fixed
bounded real range rho(h_n + ell_n u) = v+u+o(1); from the max on
row h_n there is x_n with the limit shape (F_n(S) -> c_0 e^{-iS},
|c_0|=1), and the normalized q -> 0 on scaled compacts by
|q|^{1/2} <= C(ell_n^{-1}+ell_n^{-2})(1+v)=e^{o(v)}.
Finally, verify the coarser cone condition explicitly: for R_b<=v'<=v,
(h(v)-h(v'))/ell_n
= integral_{v'}^v e^{mu(v)-mu(z)} dz
<= (v-v') e^{eps_R(v-v')};
at height h(v') multiply |f|/m_n <= e^{-(v-v')} by the alpha_0 power of
the left side; the product is bounded uniformly. Hence
M(s)/m_n <= C (ell_n/delta)^{alpha_0}, ell_n<=delta<=h_n,
uniformly in horizontal position. In the entire case log(1+h_n/ell_n)
= o(v) supplies the (ii) log condition.
All hypotheses of thm:exponential-obstruction now hold with the FIRST
profile; contradiction. Hence H<infinity with finite upper order.
The growth alternatives now assemble: H finite and upper order finite;
upper order <1 is excluded by prop:subcritical-exclusion (chapter 6);
upper order >1 (finite superlinear) is excluded by
thm:finite-superlinear-exclusion (chapter 6, the superlinear residue
argument); so only the critical case (H finite, upper order EXACTLY one)
remains.
7.8 What we carry forward. Finite H > 0 and the growth condition
limsup_{k} log max_x M(x+i(H-e^{-k}))/k = 1.
Chapters 8-11 now work strictly in this critical case. They use moment
bounds and geometric arguments to rule it out, so the only remaining
possibility in the nonisotropic, non-circular, non-ellipse class is
no case at all. Circle was excluded by hypothesis; ellipse was excluded
because we are in nonisotropic; H=infinity excluded by 7.7(i); finite
supercritical excluded by 7.7(ii); the critical case is what remains.
...
8.1 Setup: means, tiles, envelopes.
Choose k_0 with d_{k_0}=e^{-k_0}<H/10; for k>=k_0 define
d_k := e^{-k}, mu_f(k) := d_k (E_x |f|^2)^{1/2},
mu(k) := d_k (E_x (|f|^2+|q|))^{1/2}, E_x = integral dx/(2 pi).
Then ||d_k M||_2 <= mu(k) <= sqrt(2) ||d_k M||_2, and critical growth
gives for every eps>0
mu(k) + d_k max_x M(x+i(H-d_k)) <= C_eps e^{eps k}.
Log-deficit applied on disks of radius c d_k, using the critical log
upper bound log|f| <= C(1+k), plus convexity of the horizontal mean of
log|f| (bounded below by its finite real-axis value), gives
d_k E_x |q(x+i(H-d_k))|^{1/2} <= C (1+k).
The first-displacement identity V' = (W - <W>)/120 with
W := f^2 + 8q gives P := -Im V: real harmonic, period, mean 0,
P=0 on the real row. Delta(t) := H - h_X(t) is nonnegative subharmonic
on the fixed disk, Delta(0)=0. The lower lift is
Im(y - s(y,t)) >= -Delta(t) for real t, 0 < Im y < h_X(t);
verified because the LHS is harmonic, periodic, and vanishes on the real
row, and Delta + eta bounds it below on the row at h_X(t)-eta.
ENVELOPES. N:[k_0,infinity)->[1,infinity) is an envelope when
N(j)/N(k) <= C_N max(e^{beta(k-j)}, e^{a(j-k)})
for fixed 0 <= beta < 1/2, a > 0. We impose the first-moment bound
E_x(d_k M) <= C_1 N(k). The bootstrap ratio sup_k mu(k)/N(k) is
assumed finite (it is for any envelope that grows at some positive
exponential rate, by subexponentiality).
TILES. For A>=1 and a base level k, tile T_A(k,x) is the set
e^{-A} d_k <= H - Im s <= e^A d_k, |Re s - x| <= A (H - Im s)
intersected with the upper-strip; the lower fixed region is handled by
the analytic bounds there. The elementary averaging M^p (subharmonic
because it is a maximum of |f|^p and |q|^{p/2}):
E sup_{s in T_A(k,x)} (d_k M(s))^p <= C_{A,p} sup_{|j-k|<=C_A} E (d_j M)^p.
The cone-cost
b(x,k) = 1 + sup_{c_0 d_k <= delta <= delta_b, |Re s - x| <= L delta}
[delta M(s) / (N(k) (delta/d_k)^{beta_1})]
satisfies
E_x b <= C, ||b(.,k)||_p <= C_p (1 + C_p^*),
splitting the cone into unit ranges of logarithmic depth and using the
summed bound ||d_j M||_p <= C_p^* N(j) on coarser levels.
8.2 One-sided real-width and Taylor estimates.
LEMMA (lem:one-sided). Under the envelope and first-moment bound, for
real t with z = |t| N(k)/d_k <= z_0,
Delta(t) <= C d_k z^gamma, gamma = 1/(1+a),
and, with m := 4(2-gamma)/(4-gamma) < 2,
d_k P^-(x+i(H-d_k)) / N(k)^2 <= C b(x,k)^m.
Idea: at a fixed center the cone theorem gives a vertical passage for
|t|<r_{x,k} = c d_k/(N(k) b(x,k)) with |s-y| <= C d_k. Passage attached
to the actual lift, evenness in t and Cauchy give Taylor error
|s - y - t^2 V(y)| <= C d_k (z b)^4 for |t| <= r_{x,k}/2.
Cone-moments provide a center with b(x,k) <= C, and the real
no-patch transfer gives Delta(t) <= d_k for |t| <= c d_k/N(k). Then for
j = k+h, envelope implies d_j/N(j) >= C d_k/N(k) e^{-(1+a)h}; choose
h so e^{-h} = C' z^{1/(1+a)} and the previous bound at level j gives
the first display. For the negative part, set z = c' b^{-4/(4-gamma)}
(fixed small c'). Both z b and Delta/d_k <= C z^gamma are small.
Combining the Taylor estimate with the lower lift gives
d_k P^-/N(k)^2 <= C (z^{gamma-2} + z^2 b^4) = C b^m.
8.3 Quadratic bootstrap.
LEMMA (disk-bound). On an inner tile with a fixed enlargement,
(d_k M)^2 <= C (1 + sup d_k^2 |W|).
Idea on a unit disk: g := d_k f, W~ := d_k^2 W, so
g'' = (1/8) g (W~ - g^2).
If sup_D |g| is not bounded by C (1+sup_{2D}|W~|)^{1/2}, take a
counterexample sequence maximizing |g(z)| times distance-to-boundary
on an intermediate disk; let m_n be that maximum and r_n its distance.
Failure forces r_n m_n -> infinity and sup|W~|/m_n^2 -> 0. Rescale on
the disk of radius r_n/2 (where |g|<=2 m_n by maximality) and normalize
to limit G with |G|<=2, |G(0)|=1, G''=-G^3/8. Liouville contradicts
|G(0)|=1. The identity 8q = W - f^2 bounds q on the same inner disk.
THEOREM (critical moment bootstrap; thm:critical-bootstrap). Let N be
an envelope with first-moment bound below and finite bootstrap ratio.
Then
mu(k) <= C N(k), E_x d_k|P|/N(k)^2 <= C,
||d_k M/N(k)||_{2+eta} <= C (some eta>0),
with suprema on fixed tiles as well; if N is nondecreasing,
sup_x d_k M <= C(1+k) N(k), sup_x d_k|P|/N(k)^2 <= C(1+k)^2.
Proof of the quadratic and higher moments (tight structural summary).
Let C_* = sup_j mu(j)/N(j). Lem:one-sided plus ||b||_2 <= C(1+C_*) gives
E d_k P^-/N(k)^2 <= C(1+C_*^m); horizontal mean zero makes |P| have
twice that integral. Interior harmonic estimates on tiles then bound
E sup_{T_A} d_k^2|W|/N(k)^2 by C_A (1+C_*^m). Disk bound gives
C_*^2 <= C(1+C_*^m). Since m<2, C_* <= C and ||d_k P^-/N(k)^2||_p <= C
for p=2/m>1. The concentration estimate is the heart of the moment
improvement. For a selected collection of cells of total fractional
length sigma (forcing sigma >= c d_k) with E the fixed-multiple
horizontal enlargement, lift the P-function via harmonic extension from
depth d' ~ d_k down to depth R d_k for fixed R, using the periodic strip
kernel
K(u) = 1/(2h') sin(pi l / h') / (cosh(pi u / h') - cos(pi l / h'))
with h' = H - d', l = R d_k - d'. On |u|<=R d_k it exceeds c/(R d_k);
K*P^+ = P_coarse + K*P^-, so
int_E P^+ dx/(2 pi) <= C( int|P_coarse| dx/(2 pi)
+ int_{E_R} K*P^- dx/(2 pi) )
with E_R the R d_k-enlargement of E. Coarse term is at most
C R^{2 beta - 1} (using N(k - log R) <= C R^beta N(k) and d_{k-log R}
= R d_k); convolution is an L^p contraction, so Holder and the negative
L^p bound yield C (R sigma)^vartheta with vartheta = 1 - 1/p. Directly
the negative mass on E has the same bound. So
int_E d_k|P|/N(k)^2 dx/(2 pi) <= C (R^{2 beta - 1} + (R sigma)^vartheta).
Interior harmonic estimates upgrade to
sum_{I in selected} |I| sup_{x in I} sup_{s in T_A(k,x)}
d_k^2 M(s)^2/N(k)^2
<= C_A (sigma + R^{2 beta - 1} + (R sigma)^vartheta).
Choose R = R_0 sigma^{-1/2} (allowed at large k since
sigma >= c d_k gives R d_k <= C sqrt(d_k)). Then RHS <= C_A sigma^zeta
for any 0 < zeta <= min((1-2 beta)/2, vartheta/2). Selecting cells
with sup M greater than b_0 >= 1 gives
P(A_k > b_0) <= C b_0^{-1/(1-zeta)}, 1/(1-zeta) > 1,
which gives higher-moment (including fixed-tile versions).
Proof of the global assertions when N is nondecreasing. Show
Delta(t) <= d_k/4 for |t| <= rho_k := c_2 d_k / ((1+k) N(k)).
Choose eps_0 > 0 with the real-deficit bound at |u|<=u_0 := eps_0 d_k/N(k)
giving Delta <= d_k/16. Harmonic upper comparison on a small parameter
half-disk (using the bounded subharmonic Delta) bounds Delta(t) by the
diameter contribution, O(|t|) from the curved arc, and
C|t| integral_{u_0}^{r_0} max(Delta(u),Delta(-u)) du/u^2,
the kernel coming from the upper half-plane Poisson kernel; lower half
symmetric. If N monotone, sum the geometric-like bound Delta(u)<=C d_j
on |u|<=c d_j/N(k) over j=k,k-1,... to get the integral <=
C_{eps_0}(1+k)N(k); the other terms likewise give << d_k. This gives
the claim; then analyticity of the lift on |t|<rho_k and the bound
Delta <= C d_k uniformly (a coefficient estimate for functions with
positive real part applied to C d_k + i(s-y)) gives
|V(y)| <= C d_k rho_k^{-2}, hence d_k|P|/N(k)^2 <= C(1+k)^2,
and Cauchy plus V'=(W-<W>)/120 gives |W| <= C rho_k^{-2}+C; disk bound
gives d_k M <= C d_k/rho_k + C.
8.4 Weighted version with an ancestor row.
Fix 0<d_a<<1; let K_{d_a} be the normalized kernel with density
comparable to d_a/(d_a^2+|x|_per^2). For probability omega on the
circle, nu := K_{d_a}*omega. Two facts: density of nu changes by at
most a fixed factor under translations of length <= A d_a; and for
d_j<d_a/2 the unnormalized upper-boundary harmonic-measure law from
x+i(H-d_a) to row at d_j is comparable to K_{d_a}, uniformly.
PROPOSITION (weighted bootstrap; prop:weighted-bootstrap). With the
bootstrap's hypotheses (fixed constants), small ancestor depth
d_a=e^{-k_a}, and uniform comparisons, if for every fixed tile
enlargement
||sup_{T_A(j,x)} d_j M||_{L^2(nu)} <= C_A N(j) k_0<=j<=k_a+3
and
int |P(x+i(H-d_a))| d omega(x) <= C N(k_a)^2/d_a,
then
||sup_{T_A(j,x)} d_j M||_{L^2(nu)} <= C'_A N(j)
on all finer levels. Constants independent of k_a, omega, bootstrap
ratio.
Idea: tile averaging on levels up to a fixed multiple of d_a; on the
coarse part use the assumed L^2(nu) bound; the ancestor contribution
via kappa_j (mixed harmonic measure) decays like e^{-(1-2 beta)(j-k_a)}
since 2beta<1; combine with negative-part bound and weighted harmonic
estimates.
8.5 Exclusion of vanishing profiles at individual centers.
THEOREM (thm:no-trivial-profile). Sequences k_n->infinity,
d_n=e^{-k_n}, B_n=mu(k_n)->infinity. Suppose envelopes N_n with the
bootstrap's constants, satisfying N_n(k_n) <= C B_n. Fix alpha in
(1/2,1) and large L. Then centers x_n with
sup_{c d_n <= delta <= delta_b, |Re s-x_n|<=L delta}
[d_n M(s) / B_n] (delta/d_n)^alpha -> 0
cannot exist (quiet cone). In particular if the scaled profiles
F_n(S) = (d_n/B_n) f(x_n+iH+d_n S),
Q_n(S) = (d_n/B_n)^2 q(x_n+iH+d_n S)
converge to (0,0) on compact lower-half-plane subsets, they cannot also
satisfy the bounded coarse cone
(d_n/B_n) M(s) <= C (d_n/delta)^{alpha_0} (d_n<=delta<=delta_b,
|Re s-x_n|<=L delta)
for any fixed alpha_0>1/2. Every compact holomorphic limit of the pair
(F_n,Q_n) is pure: FQ = 0.
Proof sketch. Apply the cone theorem at scale c d_n with parameter
radius T_0 d_n/B_n; the small parameter is supplied by the quiet-cone
smallness. Real no-patch transfer gives
Delta(T d_n/B_n) <= c d_n (|T|<=T_0)
for all large n, and c arbitrary gives
sup_{|T|<=T_0} Delta(T d_n/B_n)/d_n -> 0.
Then fix cutoff A>1, take t = T d_n/B_n for small nonzero T:
combining the lower lift, the Taylor error and the normalization gives
d_n P^-/B_n^2 <= C Delta(T d_n/B_n)/(T^2 d_n) + C T^2 A^4.
Choose A large (small complement probability), then |T| small (small
second term), then n->infinity: d_n P^-/B_n^2 -> 0 in probability.
Uniform integrability (from the 2/m-moment bound, 2/m>1) gives mean
convergence uniformly in fixed-level ranges. Horizontal mean zero
yields the same for d_n|P|/B_n^2. Interior harmonic estimates on tiles
give E sup d_n^2|W|/B_n^2 -> 0. Disk bound then yields
mu(k_n)^2/B_n^2 -> 0, contradicting mu(k_n)=B_n.
For the stated consequence, choose alpha in (1/2, min(1,alpha_0));
bounded coarse cone bounds the quiet-cone expression by C R^{alpha-alpha_0}
for delta/d_n >= R and assumption gives -> 0 on compacts c <=
delta/d_n <= R; R->infinity yields the quiet cone, which was excluded.
Finally, scaled identity d_S^2 F_n = B_n^2 Q_n F_n has bounded LHS by
compact convergence, so dividing by B_n^2 and passing to the limit
gives FQ = 0 on the connected lower half-plane, and one nonzero
analytic factor forces the other to vanish.
...
8.6 The sequence to be excluded.
DEFINITION (def:prohibited-sequence). A prohibited sequence consists of
data in the critical setting above:
(i) Levels k_n->infinity, amplitudes B_n=mu(k_n)->infinity, and
slow rates eps_n->0 with
mu(j) <= B_n e^{eps_n|j-k_n|} (j>=k_0).
(ii) Ancestors k_{a,n} = k_n - s_n with both ->infinity, and
probabilities omega_n, nu_n := K_{d_{a,n}}*omega_n where d_{a,n}=e^{-k_{a,n}}
and kernels are comparable to d_a/(d_a^2+|x|_per^2) with comparison
constants independent of n.
(iii) For every fixed 0<beta<1/2 and every fixed tile enlargement A
there is C_{beta,A} with, for all n large enough, simultaneously over
k_0<=j<=k_{a,n}+3,
||sup_{T_A(j,x)} d_j M||_{L^2(nu_n)} <= C_{beta,A} B_n e^{beta(k_n-j)};
and for every such beta there is C_beta with
int |P(x+i(H-d_{a,n}))| domega_n(x)
<= C_beta B_n^2 e^{2 beta(k_n-k_{a,n})}/d_{a,n}.
(iv) For every compact K subset of {Im S<0},
sup_{S in K} (d_n/B_n)^2 |q(x+iH+d_n S)| -> 0 in nu_n-probability.
Note: primary normality automatically controls unweighted moments by the
envelope N_n(j) = B_n exp(beta(k_n-j)_+ + a(j-k_n)_+) for any fixed
beta,a>0 with beta<1/2. So prop:weighted-bootstrap applies to this
envelope using (iii). This is why ancestors are included in the
definition.
We also use the elementary "propagation of a logarithmic deficit"
step: if U>=0 is superharmonic with two small disks of comparable radii
and nearby centers whose enlarged common disk lies in its domain, then
av_{D_2} U <= C av_{D_1} U.
Indeed for each point of D_1 a disk centered there of a fixed multiple
of the common radius contains D_2 and stays in the domain; positivity
plus the supermean inequality bounds av_{D_2} U by a constant times the
value at that point; integrate over D_1. In the upper strip, points
with logarithmic depth separation s can be joined by O(1+s) disks of
radii a fixed small multiple of their depths, all enlarged disks kept
in a fixed slightly wider range of logarithmic depths. The argument
applies to K - log|f| including its logarithmic poles at zeros; it
uses disk averages, not a pointwise Harnack.
8.7 Power saving.
PROPOSITION (prop:power-saving). If no prohibited sequence exists, then
for k>=k_0,
mu_f(k) <= C(1+k)^{1-eps_1}, some eps_1>0.
Idea. Suppose failure; produce a prohibited sequence. Select record
levels kappa_n->infinity with b_n=mu_f(kappa_n)->infinity,
b_n >= kappa_n^{1-o(1)}, mu_f(j)<=b_n (j<=kappa_n), and
mu_f(j)<=b_n e^{eta_n(j-kappa_n)} (j>=kappa_n, eta_n->0). For fixed
small eta>0, the nondecreasing envelope N_{n,eta}(j) =
(b_n+kappa_n) e^{eta(j-kappa_n)_+} controls first moment uniformly
in n (for |f|, trivially; for |q|^{1/2}, using
d_j E|q|^{1/2} <= C(1+j) <= C_eta (1+kappa_n) e^{eta(j-kappa_n)} on the
future side). Its finite ratio follows from exponential tail; so the
critical bootstrap gives mu(j)<= C_eta (b_n+kappa_n) e^{eta(j-kappa_n)_+}
and d_j max M <= C_eta (1+j)(b_n+kappa_n) e^{eta(j-kappa_n)_+}. Put
E_n := {x : d_{kappa_n}|f(x+i(H-d_{kappa_n}))| >= b_n/2}, p_n = Prob(E_n).
Holder's inequality with the higher-moment exponent p>2 gives
3/4 b_n^2 <= C (b_n+kappa_n)^2 p_n^{1-2/p},
so p_n >= c (b_n/(b_n+kappa_n))^{2p/(p-2)} = kappa_n^{-o(1)}.
Now select a normal level for the FULL moment. For each fixed small eps>0
maximize mu(j) e^{-eps|j-kappa_n|} over j; let k be a maximizer, B=mu(k).
Triangle gives mu(j)<=B e^{eps|j-k|}, B>=b_n. Using the previous
record-envelope with rate eps/2 also gives
|k-kappa_n| <= (2/eps) log(C_eps(b_n+kappa_n)/b_n) = o(log kappa_n)
for fixed eps by b_n >= kappa_n^{1-o(1)}. Diagonal: choose eps_n->0 and
maximizing levels k_n with B_n:=mu(k_n)>=b_n and
|k_n-kappa_n| = o(log kappa_n), and primary normality holds at eps=eps_n.
Let s_n->infinity slowly with s_n=o(log kappa_n) dominating both
|k_n-kappa_n| and log(1/p_n); possible since |k_n-kappa_n|=o(log kappa_n)
and log(1/p_n)=o(log kappa_n). Set k_{a,n}=k_n-s_n, omega_n = uniform
measure conditioned on E_n, nu_n=K_{d_{a,n}}*omega_n. Both omega_n and nu_n
have density at most p_n^{-1}=e^{o(s_n)} relative to uniform.
Ancestor bounds check. For fixed beta<1/2 and beta_0<beta and
enlargement A, primary normality plus tile averaging give the
unweighted bound ||sup_{T_A(j,x)} d_j M||_2 <= C_A B_n e^{beta_0(k_n-j)}
for j<=k_{a,n}+3. Passing to nu_n costs p_n^{-1/2}, absorbed by
e^{(beta-beta_0)(k_n-j)} uniformly on that range since k_n-j>=s_n-3.
For the anchor, apply the bootstrap P-estimate to envelope
B_n exp(beta_0(k_n-j)_+ + a_0(j-k_n)_+) for any fixed a_0>0; primary
normality supplies the hypotheses; conditioning costs p_n^{-1} absorbed
by e^{2(beta-beta_0)s_n}. This gives (iii).
It remains to prove (iv). Fix compact K subset {Im S<0} and A>1.
For a center x_0 in E_n, consider target disks covering x_0+iH+d_n K
for |x-x_0|_per<=A d_{a,n}. Connect the initial point at level kappa_n
to these disks by: up to the ancestor row, horizontally there, then
down to the target depth. The number of depth-sized disks needed is
O_A(1 + s_n + |k_n-kappa_n|) = o(log kappa_n). On a fixed enlargement
of this whole range, the record-envelope gives log|f| <= U_n^* :=
kappa_n + log b_n + C log kappa_n (with C fixed because all level
changes are o(log kappa_n) and log((b_n+kappa_n)/b_n)=o(log kappa_n)).
At the initial point in E_n, the nonnegative superharmonic deficit
U_n^* - log|f| is O(log kappa_n). The deficit-step chain bounds its
average on every target disk by kappa_n^{o(1)}, uniformly in the
centers. Then lem:log-deficit with fixed inner margins gives
sup_{S in K} d_n |q(x+iH+d_n S)|^{1/2} <= kappa_n^{o(1)}
for every center in the enlarged range. Since B_n >= b_n >=
kappa_n^{1-o(1)}, this is negligible relative to B_n. Under nu_n,
sample x_0 by omega_n and add the kernel; the probability the
displacement exceeds A d_{a,n} is <=C/A. Let n->infinity first, then
A->infinity. This proves (iv).
COROLLARY (cor:critical-polynomial-bounds). If no prohibited sequence
exists, then mu(k)<=C(1+k), d_k max M <= C(1+k)^2, d_k max|P| <= C(1+k)^4,
with fixed-tile versions.
Idea: by the power saving and d_k E|q|^{1/2} <= C(1+k), E d_j M <= C(1+j).
At a target level K, use the nondecreasing envelope N_K(j) = 1+j for
k_0<=j<=K and N_K(j) = (1+K)e^{j-K} for j>=K. Envelope and first-moment
constants are independent of K, and its exponential tail makes the
bootstrap ratio finite. Apply the bootstrap at K; N_K(K)=1+K gives all
three. Fixed-tile versions follow by the same row estimates on their
fixed depth ranges.
8.8 Growth charge and conditional angle budget.
LEMMA (lem:growth-charge). If no prohibited sequence exists, then for
every fixed 0<alpha<1/2 large K_0, c, C exist with: every kappa>=K_0
such that mu_f(kappa) >= kappa^alpha satisfies
G_f(kappa + C log kappa) - G_f(kappa - C log kappa)
>= c mu_f(kappa),
where G_f(j) := log max_x|f(x+i(H-e^{-j}))|.
Idea: G_f is nondecreasing (max principle between opposite rows plus
real symmetry). If not, take kappa_n->infinity and C_n->infinity with
C_n log kappa_n = o(kappa_n), b_n:=mu_f(kappa_n)>=kappa_n^alpha and
increment := o(b_n). Maximize mu(j) e^{-eps_n|j-kappa_n|} over j for
eps_n slowly small; call the maximizer k_n with B_n=mu(k_n)>=b_n; the
maximizing argument gives primary normality, and the polynomial bound
gives |k_n-kappa_n| = O(eps_n^{-1} log kappa_n). Arrange
|k_n-kappa_n|=o(C_n log kappa_n) and log kappa_n + |k_n-kappa_n| << s_n
<< C_n log kappa_n. Set k_{a,n}=k_n-s_n. Choose a horizontal maximizer
x_{a,n} for |f| on the ancestor row, take omega_n a point mass, and
nu_n=K_{d_{a,n}}*omega_n. For any fixed beta>0, (iii) follows directly
from the polynomial bounds: tile suprema on coarser levels are
O_A((1+k_n)^2), while B_n e^{beta(k_n-j)} >= kappa_n^alpha e^{beta(s_n-3)}
dominates; likewise the anchor O((1+k_{a,n})^4) is dominated by
B_n^2 e^{2 beta s_n}.
Fix compact K subset {Im S<0} and C' fixed with row k_n+C' above all
target disks and their fixed enlargements. On the periodic strip from
real row to that row put U_n(s) := G_f(k_n+C') - log|f(s)| >= 0. At the
ancestor point, monotonicity plus the failed charge hypothesis gives
U_n(x_{a,n}+i(H-d_{a,n})) = G_f(k_n+C') - G_f(k_{a,n}) = o(b_n) = o(B_n)
(since both levels lie in the failed interval). For any row in the
target range, positive superharmonic comparison to U_n from the
ancestor, using the real bottom (nonnegative contribution), gives
average of U_n over the target row <= C o(B_n) uniformly, since the
upper-boundary harmonic-measure law is comparable to nu_n on those
depths. Comparison first for min{U_n,R} and then monotone convergence
justifies the passage. Disk averages on the target compact set then
give, via lem:log-deficit,
int sup_{S in K} d_n |q(x+iH+d_n S)|^{1/2} dnu_n(x) <= C+o(B_n).
Since B_n->infinity, Markov gives (iv). So we have constructed a
prohibited sequence, contradiction.
THEOREM (conditional angle-budget reduction; thm:angle-budget-reduction).
If no prohibited sequence exists, then for some eps>0, C<infinity,
integral_{k_0}^K mu_f(k)^2 dk <= C(1+K)^{2-eps} (K>=k_0),
and also the polynomial bounds of cor:critical-polynomial-bounds.
Idea: fix 0<alpha<1/2 and split large levels into the high set
A:= {k: mu_f(k)>=k^alpha} and its complement. On A, growth-charge gives
int_{A cap [K_0,K]} mu_f(k) dk <= C int [G_f(k+C log k) - G_f(k-C log k)] dk.
The polynomial bound G_f(v) <= v + C log(2+v) bounds the integrand;
since G_f is nondecreasing, its positive Stieltjes measure counts each
point for a set of k's of length at most 2C log K, and its total mass on
[K_0, K+C log K] is O(K); so contribution to integral over A is
O(K log K). Multiply by the power saving mu_f <= C(1+k)^{1-eps_1}
gives int_A mu_f^2 <= C K^{2-eps_1} log K. On the complement
int mu_f^2 <= int_{K_0}^K k^{2 alpha} dk = O(K^{1+2alpha}). Both have a
positive power saving from K^2; decreasing that slightly absorbs the
logarithm and the fixed compact remainder. This gives the angle budget.
8.9 What we carry forward. Levels k, d_k=e^{-k}, B_n=mu(k_n), tiling
T_A(k,x), tiles and envelopes, thm:critical-bootstrap, thm:no-trivial-
profile, the polynomial bounds under "no prohibited sequence," and the
angle-budget reduction. Chapter 9 then constructs the model picture at
p = pure criticality (compact convergence to FQ=0), and chapter 10
excludes prohibited sequences (which by definition are NOT excluded by
chapters 6-8). Once excluded, the angle budget applies and chapters
10-11 close the remaining gap.