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.
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.
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).
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).
- 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.
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.
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 (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.
- 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.
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.
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.
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.
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
the displayed integrals give every digit directly.
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.
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.
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.)
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.
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.)
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.
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.
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.
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.
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).]
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.
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.
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.
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.
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.
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.
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.
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.
THEOREM (the lens-lines theorem). 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.
the endpoint determination 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.
One additional scalar datum is needed: the area coefficient A(t).
THEOREM (the lens-points theorem). 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.
the endpoint determination 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 the endpoint determination, 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.
LEMMA (the meromorphic Hartogs continuation). 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 (the disc-sweep continuation). 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.
PROPOSITION (the width superharmonicity). 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. the product-continuation step 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.
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 (the no-patch property). 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 the endpoint determination 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 the endpoint determination 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 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 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. the point-endpoint determination, 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.
COROLLARY (the natural-boundary corollary). 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 the product continuation and its corollary to sweep; at sigma=H the
disk center ζ=0 gives a joint meromorphic patch through
(x_*+iH, 0), contradicting the no-patch property. 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.
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.
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.
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 (the isotropic classification theorem). 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.
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 (the shape-lift theorem). 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 the natural-boundary property.
(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.
LEMMA (the zero-separation lemma). 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.
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.
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. 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.
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.
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.
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 (the cone-passage theorem). 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.
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 the no-patch property.
COROLLARY (the quiet-cone bound). 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 the no-patch property).
Sketch of (i): apply the cone-passage theorem 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 the cone-passage theorem 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 the cone-passage theorem 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 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
the no-patch property.
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
the quiet-cone bound(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.
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.
THEOREM (the finite superlinear exclusion theorem). 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
(the zero-separation lemma). 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 the no-trivial-profile lemma
(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.
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 (the nonisotropic case). 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; the logarithmic-deficit lemma). 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.)
THEOREM (the exponential-obstruction theorem). 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.
LEMMA (profile:deep-data). Under hypotheses of the exponential-profile obstruction (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 the cone-passage theorem 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
the cone-passage theorem, 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.
LEMMA (profile:energy-model). The deep passages 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.
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.
LEMMA (profile:tracking). Fix a compact corridor of a continued branch
of the the deep model 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.
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 the profile theorem: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 the deep model. Both
must agree with the actual single-valued lift (by the tracking-vs-
actual-lift agreement from the tracking step extended by uniqueness of overlaps).
Contradiction: the same value cannot be two different things.
Hence NO sequence of scales and profiles as in the exponential-obstruction theorem
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.)
THEOREM (the infinite growth exclusion theorem). 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
the earlier exponential exclusion 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 the exponential-obstruction theorem 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 the subcritical exclusion proposition (chapter 6);
upper order >1 (finite superlinear) is excluded by
the finite superlinear exclusion theorem (chapter 6, the superlinear residue
argument); so only the critical case (H finite, upper order EXACTLY one)
remains.
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.
...
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.
LEMMA (the one-sided estimate). 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.
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; the critical moment 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.
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; the 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.
THEOREM (the no-trivial-profile lemma). 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.
...
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 the 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.
PROPOSITION (the 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 the logarithmic-deficit lemma 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 (the 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.
LEMMA (the growth-charge lemma). 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 the logarithmic-deficit lemma,
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; the 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 the 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.
critical bootstrap, the no-trivial-profile lemma, the polynomial bounds, and
the angle-budget reduction conditional on "no prohibited sequence".
Chapter 9 will exonerate that hypothesis (the no-prohibited-sequence theorem): it
builds the moment law of the failing sequence, derives a contact
dichotomy, shows the steep alternative is the forbidden exponential
profile (chapter 7), and shows the flat alternative forces a nonzero
first field F with high-value witnesses on a mesh, which the
function-theoretic mesh contradiction rules out. With that
hypothesis secured, the angle budget is unconditional. Chapters 10-11
then close the last case: 10 organizes an odd periodic phase slice as
a Brownian walk stopped before the wall, and 11 runs the sign test on
the last moving site to derive a single-site contradiction. The final assembly step
assembles the main theorem. The large-moment step (the large-moment lemma and the
low-moment amplitude/occupation estimates) is the heart of the final
"the growth is large" step.
Chapters 6-8 reduced us to the critical case: H finite, upper growth order
exactly one at the shape wall. What remains is a possible PROHIBITED
SEQUENCE (the prohibited-sequence definition). The next stage proves there is none:
THEOREM (the no-prohibited-sequence theorem). No prohibited sequence exists.
The strategy is a two-way squeeze. From any prohibited sequence build its
convex "moment law". Either the law's slope is unbounded at contacts, or
it has a two-ended linear slope lambda>=1. In the first case we reach the
pure exponential profile forbidden by chapter 7. In the second case we
reach a NONZERO first field F:H_->C obeying (i) a power growth bound at
the wall, (ii) "high-value witnesses" near every horizontal center, and
(iii) F''/F holomorphic. A separate function-theoretic theorem (the
"mesh contradiction") rules out exactly that combination. Either way:
contradiction.
Throughout: H>0 finite, f=phi' and q=f''/f holomorphic on D_H,
M(s)=max{|f(s)|, |q(s)|^{1/2}}, the lift s(y,t) is even in t with
<s-y>=0, growth order at the wall is one. Chapter 8's P=-Im V,
W=f^2+8q, s=y+t^2 V+O(t^4).
Recall d_k=e^{-k}, and
mu_f(k) = d_k ( E_x |f(x+i(H-d_k))|^2 )^{1/2},
mu(k) = d_k ( E_x (|f|^2+|q|) )^{1/2},
with E_x the horizontal average. mu_f <= mu <= sqrt(2) mu_f on each row,
so "mu_f large" and "mu large" mean the same thing up to factor 2.
A prohibited sequence is exactly the data of 8.6: k_n->infinity,
d_n=e^{-k_n}, B_n=mu(k_n)->infinity, mu(l) <= B_n e^{eps_n|l-k_n|}
(l>=k_0, eps_n->0), plus the coarse bootstrap and the q-smallness.
Associate the normalized moment law
m_n(D) = (d_n / B_n) ( E_x(|f|^2+|q|) )^{1/2} on row H-d_n D.
Since d_n D = d_{k - log D}, this is the row k - log D rescaled by
d_n/B_n; the parameter D runs over the whole depth ratio range.
LEMMA (moment compactness). After a subsequence there is a positive
continuous m:(0,infty)->(0,infty) with
m_n(D) -> m(D) locally uniformly,
m(1)=1, m(D) <= D^{-1}, log m is convex in log D.
Why. For periodic holomorphic f, its horizontal L^p norm is log-convex
in height (three-lines; or the Banach-space version), and the same holds
for the L^1 norm of q. A sum of two nonnegative log-convex functions is
log-convex by Holder, so log E(|f|^2+|q|) is convex in height. Dividing
by d_n^2 and multiplying by d_n/B_n converts "convex in height" to
"convex in log D". Primary normality gives m_n(D) <= D^{-1} e^{eps_n log D},
hence m_n(1)=1 and uniform upper bounds on compacts of D. Convexity plus
upper bounds gives uniform lower bounds. Arzela-Ascoli on log m_n.
Set rho(x) = -log m(e^x). Then rho is increasing, locally Lipschitz,
rho(0)=0, rho(x)>=x. Let p(x) be its left derivative, p > 0. Convexity
of log m in x gives
p(v) <= e^{v-u} p(u), v >= u. (*)
LEMMA (contact dichotomy). Fix
c in (1/2,1), and set rho_c(x) = cx + inf_{v>=x} (rho(v) - cv). A point
x is a "contact" if rho(x) = rho_c(x). Exactly one of the two
alternatives holds.
(i) There are contacts x_j -> +infinity at which p is unbounded.
Then there exist such x_j, P_j = p(x_j) -> infinity, D_j = e^{x_j},
with, uniformly on bounded real u,
m(D_j(1-u/P_j)) / m(D_j) -> e^u,
and, for every fixed alpha_0 in (1/2, c),
m(D_j(1+v/P_j)) / m(D_j) <= C (1+v)^{-alpha_0} (v>=1).
(ii) Slopes at all sufficiently large contacts are bounded. Then there
exist x_j -> +infinity, a locally Lipschitz rho_* with rho_*(0)=0,
and finite lambda>=1, with
rho(x_j+u) - rho(x_j) -> rho_*(u) locally uniformly,
rho_*(u) = lambda u + o(|u|) as u -> +/- infinity,
rho(v) - rho(x) >= c(v-u) - C (x >= u),
rho(u) - rho(v) >= -P'(v-u) - C (u <= v), for a fixed P',
rho(x_j) = O(1+x_j).
Why. rho_c is properly defined (infimum finite because rho - cv >=
(1-c)v), monotone, locally Lipschitz, equal to rho on its contact set,
and affine of slope c off it. So "contact" is exactly "rho_c switches
to the flatter tangent c".
Case (i). Fix contacts z_j -> infinity with p(z_j) -> infinity.
Inequality (*) implies p(x) >= e^{-1} p(z_j) on [z_j-1, z_j], so for
large j, rho(x) - cx is strictly increasing there, hence every point of
that interval is a contact. Set g = p^{-1/2}. Then (*) says
e^{x/2} g(x) is nondecreasing on the interval, so g has bounded total
variation with sigma_j := sup g -> 0 as j -> infinity. The weak L^1
maximal inequality for intervals applied to the positive variation
measure nu_j of g gives
|{x: sup_h nu_j((x-h,x+h))/(2h) > t}| <= C nu_j(total)/t.
With t = sigma_j^{1/2} and nu_j(total) = O(sigma_j), the bad set has
length O(sigma_j^{1/2}). Pick x_j in the middle third of [z_j-1,z_j]
off this bad set. Then
|g(x_j+h) - g(x_j)| <= 2 sigma_j^{1/2} |h|, |h| <= g(x_j)=P_j^{-1/2}.
So p(x_j+h)/P_j -> 1 uniformly in |h|<=P_j^{-1/2}. Reading off
m(e^{x_j+h}): the exponential tangent in (i) follows by taking log and
using g's uniformity. The power tail beyond P_j^{-1/2} comes from (*)
at the g level (contacts imply rho(v) >= rho(x_j) + c(v-x_j) - C, hence
m ratio <= C(1+v)^{-alpha_0} for any alpha_0<c).
Case (ii). Contacts exist but p is bounded at every sufficiently large
contact. Take any subsequence of contacts with rho(x_j) -> infinity.
The contact inequality bounds rho from below by a linear function of
slope c, and the depth growth bound rho(x)=O(1+x) bounds it above.
Locally uniform extraction gives rho(x_j+u)-rho(x_j) -> rho_*(u).
Because rho is monotone, rho_* is monotone; because the past-tail bound
holds uniformly, rho_* is L-Lipschitz. So its one-sided derivatives
exist. Drop to a finer subsequence where the left slope p_j = rho'_-(x_j)
converges to a finite lambda in [c,infty). In this alternative the slope
is bounded at contacts, so lambda is finite. Then the past tail with
"lower=upper" linear slope, and a chosen fixed P' larger than the sup
bound slope, gives rho_*(u) = lambda u + o(|u|) at both ends. Passing
to x_j with rho(x_j) = O(1+x_j) and the two-sided contact linear
inequalities gives the four displayed conclusions of (ii).
LEMMA (steep contacts impossible; the moment-compactness lemma:steep-excluded). Alternative
(i) of contact dichotomy cannot hold for the moment law of a prohibited
sequence.
Why. For fixed j, take n large enough. Set
h_n = H - d D_j, ell_n = d D_j / P_j, m_0 = (B/d) m_n(D_j).
A diagonal choice gives h_n -> H, ell_n -> 0, m_0 ell_n -> infinity;
P_j -> infinity means the depth from h_n to the wall is P_j ell_n, huge
in the ell_n-unit.
Consider the nonnegative row function H_x = (|f|^2+|q|) on row h_n,
normalized by m_0^2. Sample points on that row with probability
proportional to H_x. Because the design density cancels the row moments
on the scaled half-plane centred at h_n + ell_n S,
E_sampl( |F_x(S)|^2 + |Q_x(S)| ) = (row moment at h_n + ell_n Im S) / m_0^2,
where F_x(S) = m_x ell_n f(x + i h_n + ell_n S) and Q_x is the analogous
second field. Primary normality and the exponential tangent control the
right side on every relevant scaled row; the power tail controls deeper
rows. So on every aperture the sampled pair obeys the coarse cone with
a single exponent alpha in (1/2, min(c,1)).
Tightness: summable exceptional probabilities plus a diagonal argument
supply deterministic centers and a locally uniform subsequential limit
(F, Q) on H_-, with the Riesz mass of |e^{iS}F|^2 + |e^{2iS}Q|
vanishing on compacts. The limits are holomorphic on H_- and satisfy
|e^{iS}F|^2 + |e^{2iS}Q| harmonic on H_-.
Each summand is subharmonic, so each is harmonic. An analytic function
with harmonic squared modulus is constant; an analytic function with
harmonic modulus is constant. So
F(S) = c_1 e^{-iS}, Q(S) = c_2 e^{-2iS}.
Normalization at S=0 (m_x ell_n -> infinity in probability forces
|e^{iS}F|^2 + |e^{2iS}Q| at S=0 to tend to 1) makes (c_1,c_2) not both
zero. The identity
d_S^2 F_x = (m_x ell_n)^2 Q_x F_x
has bounded left side by compact convergence, and m_x ell_n -> infinity
in probability, so QF = 0. Therefore the limit is one of the two nonzero
pure exponential profiles in the exponential-obstruction theorem (chapter 7).
That theorem forbids such a profile. Contradiction.
Assume from now on: alternative (ii). Use its x_j, rho_*, lambda. Because
rho_*(u) = lambda u + o(|u|) at both ends, the m ratio at scale D=e^u is
(1+o(1)) D^{-lambda} away from D=0 and D=1.
Take the diagonal subsequence as follows. For each fixed j, choose n
large enough that
d' := d e^{x_j}, K := k - x_j, B' := mu(K) = mu(k - x_j)
are consistent with the local uniform convergence of m_n -> m on the
relevant compact (the identity B'/B = e^{x_j} m_n(e^{x_j}) together
with positivity of m makes (K,B') -> (infinity,infinity), and B' is
comparable to e^{-lambda x_j} B).
LEMMA (selected fields). Choose the diagonal
subsequence and horizontal centers xi_n using the weighted submean
estimates for the ancestor kernel so that, for scaled fields
F_n(S) = (d'/B') f(xi_n + i H + d' S),
Q_n(S) = (d'/B')^2 q(xi_n + i H + d' S),
the family is locally bounded on every compact of H_-, and
|e^{iS}F_n(S)|^2 + |e^{2iS}Q_n(S)|
has bounded Riesz mass on every compact. Extract a locally uniform limit
(F, Q). Then:
(a) F,Q holomorphic on H_-, and F'' = Q F;
(b) Q identically zero;
(c) F not identically zero.
Why. Weighted submean at the ancestor plus the critical moment bootstrap
gives, on every compact, uniform L^p bounds on |e^{iS}F_n|^2 +
|e^{2iS}Q_n|. The quiet-cone estimate of the critical moment bootstrap gives the
additional smallness sufficient for Riesz-mass tightness. Deterministic
centers are chosen by Borel-Cantelli (summable exceptional
probabilities). The identity d_S^2 F_n = (B'/d')^2 Q_n F_n shows that
(B'/d')^2 Q_n has bounded limit; since B'/d' -> infinity, Q_n -> 0
on every compact, hence Q=0. If F were zero, its compact smallness and
its coarser cone bound with exponent alpha>1/2 available from the coarse bootstrap condition
applied to the constructed ancestor probability measure) would violate
the no-trivial-profile lemma (chapter 8). So F is nonzero.
The limit (F, 0) of the pure-first-field extraction is a nonzero F:H_->C obeying F''=0 formally, but
the actual model obeys a nonlinear "frozen" energy equation coming from
the lift. This is because the scaling in the pure-first-field extraction was a diagonalization and
the physical lift satisfies an exact recurrence; the deep limit of the
recurrence is the frozen action.
Coordinate. For fixed T in a compact set away from 0, set h = T/B' and
J_n(Y) = (S_n(Y+h, T) - S_n(Y,T)) / h,
where S_n is the physical lift attached to (f_n, q_n). The fast-scale
inputs to the normalized chord action are T F_n(S + hu) and
T^2 Q_n(S + hu); on the fast scale, they converge analytically to the
constants (T F(S), 0) on thickenings. Passing through the
kernel-division lemma gives the "frozen first-field action" (first
computed as the F-type profile of chapter 6):
p(S,J) = J / (T F(S))^2 - 2 sin(T F(S) J / 2) / (T F(S))^3,
with the removable interpretation at TF(S)=0 given by p(S,J)|_{TF=0}
= J^3/24.
THEOREM (pure first-field model). There is a unique holomorphic model
S_*(Y,T) on its domain satisfying the deep anchor S_*(Y,T)=Y+O(|T|^2
D^{1-2alpha}) (D=-Im Y), the identity S_*_Y -> 1 as D -> infinity in
every fixed aperture, and the frozen energy equation
J^3 g(z) = 1, J = S_*_Y, z = T F(S) J /2, g(z) = 3(sin z - z cos z)/z^3.
Moreover S_*(.,T) is holomorphic in (Y,T) at every real T inside some
parameter disk, agrees with itself on overlaps by analytic continuation
(so is single-valued on a simply connected domain around any real point),
and S_*(0,Y)=Y on the real row.
Why (sketch; the load-bearing lines are the kernel division, but the conclusion is
standard). Along the diagonal subsequence where F_n -> F, the exact
recurrence converges termwise to
d_Y p_J(S,J) = p_S(S,J).
For the frozen action, E := J p_J - p is conserved along that ODE
(differentiate; d_Y E = 0). The energy at the deep anchor equals the
trivial-solution value 1/12 (at T=0, J=1, p=J^3/24, so E = J p_J - p =
3J^3/24 - J^3/24 = 1/12). For the frozen F-type action,
J p_J - p = 2(sin z - z cos z)/lambda^3 with lambda=TF(S), which equals
J^3 g(z)/12 upon substituting z=lambda J/2. Setting this equal to 1/12
gives J^3 g(z)=1. Uniqueness: partial_{J} E at (T=0,J=1) is 1/4,
so the implicit function theorem determines J as a function of (Y,T) in
a neighborhood and continues it uniquely on any simply connected domain
around the real row. The rest of the existence and uniqueness is the
normal-family argument from the deep-model argument; we have compressed it to
this "why" without losing the argument.
Corollary (branch structure of g). g(z) = 3(sin z - z cos z)/z^3 extends
to z=0 with g(0)=1. It is real-symmetric, positive on real z with one
side having positive coefficients. Its nonzero roots are simple and lie
strictly in (j pi, j pi + pi/2) on the positive real axis. Because the
inverse g^{-1/3} is analytic on |z| < r_1 with positive coefficients,
J is analytic in z on |z| < r_1.
The change of coordinates from z to (Y,T) is via z = T F(S) J/2.
PROPOSITION (growth estimate on the model). For every compact parameter
set and every eps>0 there is C_{eps} such that, on the domain of the
model,
|F(Y)| <= C_{eps} D^{-lambda-eps}, D=-Im Y, D large.
Why. The model is even in T, so S_*(Y,T) = Y + T^2 A(Y) + O(T^4) with A
holomorphic on the model domain. Differentiate the energy equation J^3
g(z)=1 in T^2 at T=0. With g(z) = 1 - z^2/10 + O(z^4), J = 1 + T^2 A'(Y)
+ O(T^4), z = T F(S) J/2, the T^2 coefficient reads
3 A'(Y) - F(Y)^2/40 = 0,
so A'(Y) = F(Y)^2/120. But A is bounded by the small-parameter model
bound O(|T|^2 D^{1-2alpha}), and Cauchy in Y gives
|A'(Y)| <= C D^{-2/r}, r=1/(lambda+eps).
Combine: F(Y)^2 / 120 <= C D^{-2/r}, hence |F(Y)| <= C D^{-1/r} =
C D^{-lambda-eps}. Since eps>0 was arbitrary, done.
PROPOSITION (parameter disks). For every fixed small parameter disk
|T| < t_0, with a small margin, the model extends holomorphically in
(Y,T) to
{ Y : Im Y < -C_eps |T|^r }, r = 1/(lambda+eps), eps>0 arbitrary,
still satisfying the same energy equation and S(0,Y)=Y. On the real
parameter line this domain shrinks back to the physical domain of the pure first-field model.
Why (sketch of the continuation mechanics). Continue the model
analytically in T along real directions (using the ODE structure of the pure first-field model
and the ODE propagation lemma of chapter 4). Then use the
graph-filling lemma 2.8 (product continuation) to fill the parameter
half-disk. The key: the model's deep real domain maps into a complex
shape-coordinate strip of half-height ~ C_* D^{1/r}. Shrinking the
parameter disk keeps everything holomorphic. The r-th power r=1/(lambda+
eps) is exactly the one supplied by the growth estimate above: it is the
inverse of the model's log growth in the shape variable.
Put L(v) = log m_*(e^{-v}). By construction L(v) = lambda v + o(v) at
both ends.
LEMMA (normal epochs; the moment-compactness lemma:normal-epochs). Fix 1/2<c_1<1-beta and
P_1>max(lambda,1+a) (with beta, a the envelope constants of the coarse-bound construction). There
is an unbounded subset E of (0,infty) with dist(v,E) = o(v) as v ->
infinity, such that every u in E satisfies, for all real v,
L(v) - L(u) <= c_1(v-u) if v<=u,
L(v) - L(u) <= P_1(v-u) if v>=u.
Why. The coarse envelope bound gives L(v) <= C + (1-beta)v for
v<=0, hence L(v) - c_1 v -> -infinity as v -> -infinity. For any
w -> infinity, choose a running record u_0 <= w of L(v) - c_1 v. Since
the asymptotic slope of L(v) - c_1 v as v->+infinity is lambda-c_1 > 0,
the running record must actually be near w: w - u_0 = o(w). Maximize
L(v) - P_1 v on [u_0,infty); the max exists because P_1>lambda.
Call a maximizer u. Then maximality gives
L(v) - L(u) <= P_1(v-u) (future inequality).
And the record + P_1>=c_1 on [u_0,u] gives
L(v) - L(u) <= c_1(v-u) (past inequality for v<=u).
Also u satisfies u - u_0 = o(u_0), and u_0 -> w, so the epochs u so
produced have dist(v,E) = o(v). Taking the sub-sequence of "record u
associated with a large target w" gives the dense unbounded E.
PROPOSITION (witnesses at epochs; the moment-compactness lemma:witnesses). Fix a
sufficiently large fixed aperture L_0 and let I be any compact
horizontal interval. There is a function eta_I(u)->0 (u in E) such
that every sufficiently large u in E and every x in I admit a point
w=p-i delta in H_- with
|p-x| <= L_0 delta, |-log delta - u| <= eta_I(u) u,
log|F(w)| >= lambda u - eta_I(u) u.
The epochs u are COMMON to all centers x in I; no choice of epoch
depending on the center is allowed.
Why. Choose alpha' in (1/2, min(alpha,c_1,1)) with alpha>1/2 the
exponent in the no-trivial-profile lemma, and L_0 large enough for the
quiet-cone form of that theorem at this exponent. Suppose the conclusion
fails for some fixed eta>0. Then some u_j -> infinity in E and centers
x_j in I satisfy
log|F(p-i e^{-v})| < lambda u_j - eta u_j
for all |p-x_j| <= L_0 e^{-v}, |v - u_j| <= eta u_j. We derive a
contradiction.
Return to the actual data of the selected-fields lemma (choosing the
diagonal index sufficiently large after each fixed u = u_j). Set
d'' = d' e^{-u_j}, K'' = K+u_j, B'' = mu(K+u_j),
with physical center xi_n + d' x_j. (For fixed u, B''/B' -> e^{-u}
m_*(e^{-u}) positive, so K'',B'' -> infinity.)
At these scales the old envelope gives
M_n(v) := e^v mu(K+v)/B' <= C e^{(1-beta)v} (v<=0),
<= C e^{(1+a)v} (v>=0).
The normal-epoch bound gives, for 0<=v<=u_j,
M_n(v)/M_n(u_j) <= 2 e^{c_2(v-u_j)},
and for u_j<=v<=C_0 u_j,
M_n(v)/M_n(u_j) <= 2 e^{P_2(v-u_j)},
with 1/2<c_2<c_1 and P_2>max(P_1,1+a) fixed (and the C_0 a fixed
constant).
Now examine the three depth ranges under the failure hypothesis:
- v <= 0: old tail, since 1-beta > c_2 >= alpha'.
- 0 <= v <= u_j - eta u_j: log M_n(v)/M_n(u_j) <= c_2(v-u_j), so the
quiet-cone expression at depth ratio e^{v-u_j} is small.
- u_j - eta u_j <= v <= u_j + eta: log M_n(v)/M_n(u_j) <= P_2(v-u_j),
and the failure inequality directly gives the quiet-cone smallness.
In all three ranges, the required quiet-cone expression obeys
d'' M(H-u)/B'' <= (const) * (vanishing) * (e^{v-u_j})^{alpha'},
so the quiet-cone hypothesis of the no-trivial-profile lemma is satisfied at
sequence (u_j), with the "correct" normalization B''=mu(K''). Its
conclusion forces the global real width deficit divided by d'' to
vanish. But then the one-sided coefficient ("negative part of P") and
the moment bound of that theorem are contradictory. So the failed-
witness hypothesis is impossible at every fixed eta.
Diagonalizing: choose eta_n -> 0, apply the above with fixed eta=eta_n,
and get witnesses for each x in I with eta_I(u) := eta_n at u in band n.
Isolate the pure complex-analytic statement.
THEOREM (mesh contradiction; the mesh contradiction theorem). Let F not= 0 be
holomorphic in H_- := {z: Im z<0}, and suppose F''/F extends
holomorphically to H_-. There cannot exist a lambda>=1 and an unbounded
set of epochs E subset (0,infty) with all of:
(i) for every compact interval I,
sup_{x in I} log|F(x-i e^{-v})| <= lambda v + o(v) (v->infinity),
uniform in x in I;
(ii) dist(v,E) = o(v) as v -> infinity;
(iii) for every compact I there are a fixed aperture A_I<infty and a
function r_I(u)->0 so that, for every sufficiently large u in E
and every x in I, there is w=p-i delta with
|p-x| <= A_I delta, |-log delta - u| <= r_I(u) u,
log|F(w)| >= lambda u - r_I(u) u.
The epochs u are common to all centers x in I.
Why (structure; [deep]). The proof has three stages: two local
estimates (mesh shadow, mesh routing), a mesh-and-tiling construction
with bad-barrier exclusion and planar duality that produces good
routed paths, and a complex-analytic finish (primitive C_0 of F^{-2},
its oscillation along the paths, the vanishing function G=F(C_0-c_*),
and Schwarz reflection to force G==0). Each stage is proved below.
Two local estimates. "Depth" means delta=-Im z, "logarithmic depth" v=
-log delta.
LEMMA (mesh shadow; the moment-compactness lemma:mesh-shadow). Fix a in (0,b] and K<infty.
There is C=C(a,b,K) such that for every family of squares T of side
rho_T and center depth d_T with a rho_T <= d_T <= b rho_T, d_T<=d/2,
|Re c_T - p| <= K d, the union of horizontal projections has length at
most
C d U(w)/H,
where U >= 0 superharmonic on a neighborhood of R = {z : |Re z - p| <
M d, tau < -Im z < 2d} (M a large fixed multiple), U(w)<infty, and
U >= H on every disk D_T of radius rho_T/8 with center c_T-i d_T.
Why. Run Brownian motion from w until either exit of R or first hit of
one of the tested disks D_T. By superharmonic comparison, the probability
of hitting some D_T before exiting R is bounded by U(w)/H. From a
first hit of D_T, the probability of subsequently exiting R through the
top of a window I_T near Re c_T (of width a fixed multiple of rho_T) is
bounded below by c rho_T/d, uniformly: after scaling lengths by d, the
starting point, the top depth, and the geometry are in a fixed compact
set on which the interior-to-side Poisson density of a rectangle is
bounded below. Summing over disjoint subfamilies and using the Markov
property gives the union bound.
LEMMA (mesh routing; the moment-compactness lemma:mesh-routing). Let Q be a square of side
rho and center c, F holomorphic on B(c, 100 rho), and U=M-log|F| >= 0
on that ball with inf_Q U <= T (fixed M,T). There are disks of total
radius at most C h rho, for any fixed small h>0, outside which
U <= C_h T on the concentric square Q* of side 9 rho. Uniform in Q, c.
Why. On a fixed comparable square, an L^1 bound for U follows from
submean (|F| bounded above there). Zeros of F inside B(c,20 rho) are
controlled by the deficit integral. Multiplying |F| by the Weierstrass
factors of these zeros removes them, and a Cartan-type estimate gives,
for any points b_1..b_N,
prod_nu |z - b_nu| >= (h/e)^N
outside disks of total radius 3h. Choose the b_nu to be the zeros of F
in B(c,20 rho) and use the deficit inequality to bound the remaining
factor. Then U(z)<=C_h T on Q* off the disks. Chain comparable squares,
choosing junction points outside both disk families.
Mesh construction and bad-barrier exclusion.
Choose a nonincreasing theta(v)->0 with theta(v)>=v^{-1/4} exceeding the
square roots of the relative errors in (ii),(iii) uniformly at
arguments >= v (obtained by the decreasing tail supremum). Start at a
large epoch u_1 in E, and choose
u_{j+1} in E, u_{j+1} = u_j + 8s_j + o(s_j), s_j=theta(u_j)u_j.
Then s_j -> infinity, bands [u_j-s_j,u_j+s_j] are disjoint with
gaps -> infinity, and u_{j+k} = u_j + o(u_j) for k in {-1,0,1,2}.
Take dyadic a_j=e^{-u_j-s_j+O(1)}, b_j=e^{-u_j+s_j+O(1)}. Horizontal
rectangle R_j=[-3,3]x[a_j,b_j]; vertical rectangles V_j^pm have phases
[-2,-3/2] and [3/2,2] and depths [a_{j+2}/8,8b_j]. So the vertical
rectangles bridge depths from band j to the deeper band j+2.
Set M_j = lambda u_j + o(u_j) with M_j >= lambda u_j; by (i) and the
bounded region's logarithmic depths, U_j := M_j - log|F| >= 0 on the
whole constructed region, with U_j = o(u_j) at every witness of the witness step.
Take the dyadic tiling of the region, in layers 2^{-l-1}<=delta<=2^{-l}
with tile side 2^{-l-1}/N_0 for fixed N_0. Color a tile GOOD if
sup_T log|F| >= lambda u_j - eps_j u_j,
BAD otherwise (eps_j -> 0 slowly chosen so witness deficits are
o(eps_j u_j)). Then bad tiles satisfy U_j >= eps_j u_j, and applying
mesh shadow at the witness w at epoch u_j gives a shadow of the union of
their projections (up to depth delta_w/2 and horizontal range a fixed
multiple of delta_w) of length o(delta_w). Consequence: no BARRIER of
BAD tiles joins the shallow edge of R_j (near b_j) to the deep edge
(near a_j); the horizontal projection of a BAD chain would be too small
to cover the shadow argument's constraints, and the (i) bound fails.
Similarly, by mesh routing concatenated through comparable squares, no
BARRIER of BAD tiles joins the left side of V_j^- to its right side.
By planar duality, a left-right BAD barrier in R_j and a top-bottom BAD
barrier in V_j^pm cannot coexist with the small-shadow consequence; so
on each R_j and V_j^pm, there is a chain of GOOD tiles crossing from
shallow to deep (for V), and from left to right (for R). Route these
chains with mesh routing (choosing junction points off the routing disks)
to obtain actual continuous paths
gamma_j horizontal in R_j,
sigma_j^- and sigma_j^+ vertical in V_j^pm,
joining at fixed-depth junctions with log|F| >= lambda u_j - o(u_j) on
every such path. Vertical paths start deeper than every horizontal
gamma_j and end shallower than every gamma_{j+2}, staying to the left
or right of [-1,1]. Each such path has length at most exp(o(u_j))
(tile side, widest horizontal strip, layer heights all scale like
e^{-u_j+o(u_j)}, and the path occupies O(1) horizontal width; internal
Cartan constants are fixed before j->infinity).
A primitive with no monodromy. Since q = F''/F is holomorphic, F has
no zero of order >=2: at a common zero of F,F' the equation F''=qF
forces F''=0 too, so by ODE uniqueness F is identically zero. At a
simple zero z_0 of F, F''(z_0)=0, and the expansion
F(z)=a(z-z_0)+O((z-z_0)^3) with a!=0 gives
F(z)^{-2}=a^{-2}(z-z_0)^{-2}+O(1), whose residue vanishes.
Consequently F^{-2} has a well-defined meromorphic primitive
C_0(z) := int_{z_init}^z F(w)^{-2} dw,
single-valued on the simply connected half-plane H_- (all residues on
closed curves inside H_- vanish), with C_0' = F^{-2}. At each simple
zero of F, C_0 has a first-order pole with residue -a^{-2}.
Oscillation of C_0 on the routed paths. Since log|F| >= lambda u_j -
o(u_j) on gamma_j ∪ sigma_j^- ∪ sigma_j^+ (no zeros there, so there
C_0 is holomorphic), and each such path has length at most
exp(o(u_j)), the oscillation of C_0 along it is at most
exp(-2 lambda u_j + o(u_j)).
A common constant c_*. Choose z_j ∈ gamma_j. The routing joins z_j to
z_{j+1} by pieces of gamma_j, sigma_j^- or sigma_j^+, gamma_{j+1}
(the Jordan curve argument in 9.6's routing shows each vertical path
meets each nearby horizontal path). So
|C_0(z_{j+1}) - C_0(z_j)| <= exp(-2 lambda u_j + o(u_j)).
The gaps u_{j+1}-u_j exceed 1 eventually, so the telescoping sum is
summable: for any fixed eta ∈ (0, 2 lambda),
sum_{k>=j} exp(-(2lambda-eta) u_k) <= C exp(-(2lambda-eta) u_j).
Thus C_0(z_j) converges to a common constant c_*. Combining with the
oscillation bound gives ONE constant c_* with
|C_0 - c_*| <= exp(-2 lambda u_j + o(u_j))
on gamma_j ∪ sigma_j^- ∪ sigma_j^+ for all sufficiently large j.
Vanishing function G := F(C_0 - c_*). The apparent pole of C_0 at
each simple zero of F cancels in F.(C_0-c_*), because the leading
behaviour of F is a(z-z_0) and of C_0 is -a^{-2}/(z-z_0):
their product has limit -a^{-1}. So G is holomorphic on H_-, and on
the routed paths
|G| <= |F| exp(-2 lambda u_j + o(u_j))
<= exp(-lambda u_j + o(u_j)) -> 0.
Enclosure and boundary zero. For z=x-i e^{-v} with x ∈ [-1,1] and v
large, choose j with beta_j < v <= beta_{j+1}, where
beta_j := u_j+s_j+O(1) and alpha_j := u_j-s_j-O(1) and
alpha_{j+1} > beta_j for large j. Then gamma_j lies strictly below z
and gamma_{j+2} strictly above; sigma_j^- is strictly left and
sigma_j^+ strictly right. The resulting closed walk (lower, right,
upper, left pieces) has winding number 1 about z, avoids z, and bounds
a compact region containing z. The maximum principle for G on that
region gives
|G(z)| <= exp(-lambda u_j + o(u_j)) -> 0
as v -> infinity, uniformly for x ∈ [-1,1]. Therefore G has
continuous boundary value 0 on the open interval (-1,1).
Schwarz reflection. Reflection of G across (-1,1) extends G
holomorphically past the interval; the extension vanishes on the
interval, so by the identity theorem the extended G is identically
zero. On any disk of H_- where F != 0, this forces C_0 = c_* on that
disk, contradicting C_0' = F^{-2} != 0. Contradiction. The mesh
contradiction theorem is proved.
THEOREM (the no-prohibited-sequence theorem). There is no prohibited sequence.
Proof. Let a prohibited sequence be given. Form its moment law, rho,
and rho_* as in the moment-law construction. Apply the contact dichotomy.
- If alternative (i) holds, The steep-contact exclusion gives a direct contradiction with
the exponential-obstruction theorem (chapter 7).
- If alternative (ii) holds, 9.3 gives a nonzero F on H_- with
F''=QF, Q=0, and Q_n->0, 9.4 gives the pure first-field model and
the growth bound |F(x-iD)|<=C_{I,eps}D^{-lambda-eps}, 9.5 gives
common normal epochs and witnesses near every center, and 9.6's
mesh contradiction rules out exactly that combination.
So no prohibited sequence exists.
Consequence (now unconditional). Cor:critical-polynomial-bounds and
the angle-budget reduction (chapter 8) hold in the critical case with
no caveat. Explicitly:
mu(k) <= C(1+k),
d_k max_x M(x+i(H-d_k)) <= C(1+k)^2,
d_k max_x |P(x+i(H-d_k))| <= C(1+k)^4,
integral_{k_0}^K mu_f(k)^2 dk <= C(1+K)^{2-eps}, some eps>0.
The first three came from the "no prohibited sequence" corollary; the
last from the conditional angle-budget reduction, now unconditional.
control integral_{k_0}^K mu_f^2 dk <= C(1+K)^{2-eps}, the pure-field
model of 9.4 with the exponential growth bound |F(x-iD)| <= C D^{-lambda-eps},
and the witness property 9.5. These are the exact inputs to chapter 10
and chapter 11, which close the last case by an odd-cycle sign argument
and a single-site sign contradiction against the wall.
Standing hypothesis: the remaining nonisotropic case, H finite, growth
order at the shape wall exactly 1. By the unconditional angle-budget
reduction (the angle-budget reduction), there are a saving
exponent 0<rho<1 and a constant C such that
int_{k_0}^K mu_f(v)^2 dv <= C K^{2 rho} (K>=k_0).
We shall use this saving to control the geometry of one fixed periodic
orbit family, even when its continuation approaches the shape wall. The
controls are stopped at the first moment a LOGARITHMIC shape depth over
the whole history exceeds a fixed level L. Keeping the HISTORY (the
running max of depth), and not just the current depth, is essential: it
lets a fixed finite collection of determinant estimates serve for the
whole run, no matter how the set of "deep" sites changes along the way.
Fix t_0 = 2pi/N with N odd and N>=3, inside the real collar. Put
h := h_X(t_0) = h_L(t_0) (0 < h <= H),
where the two widths agree at real parameters by the width relation of
chapter 4, and h<=H is the shape-lift theorem. For |Im y| < h set
s_j(y) = s(y+j t_0, t_0), X_j(y) = Gamma(s_j(y)),
psi_j(y) = phi(s_j(y)), f_j(y) = f(s_j(y)),
with prescribed lifts s_{j+N}=s_j+2pi, psi_{j+N}=psi_j+2pi (indices mod
N). Each s_j, psi_j is holomorphic in y; the position X_j is
meromorphic.
Angles. With n_theta=(cos theta, sin theta), e_theta=(-sin theta,
cos theta), the linear system
theta_{j-1} + theta_j = 2 psi_j, theta_{j+N} = theta_j + 2pi,
has a unique holomorphic solution because N is odd (substitute
repeatedly and use the shift conditions). On the real collar, this
solution equals the directed bitangent-edge angles of the billiard
orbit; the identities below continue from that collar meromorphically.
Edge data.
d_j^0 := theta_j - psi_j, p_j := X_j . n_{theta_j},
ell_j from X_{j+1}-X_j = ell_j e_{theta_j},
H_j^0 := X_j . n_{psi_j},
where "." is the complex-bilinear continuation of the Euclidean dot
product. Each ell_j is a directed meromorphic chord length, positive on
the real collar. The support identities
p_{j-1}+p_j = 2 H_j^0 cos d_j^0, p_j-p_{j-1}=2(X_j.e_{psi_j}) sin d_j^0,
d H_j^0 = (X_j.e_{psi_j}) d psi_j, d X_j = c e_{psi_j} f_j^{-2} d s_j
follow from theta_{j±1}=psi_j±d_j^0 and Gamma'=c e_phi f^{-2}. Even
where X_j has a pole, H_j^0 is regular (the local factor f(s)=a(s-s_0)
+O(...) makes the residue of Gamma parallel to e_{phi(s_0)}, whose dot
with n_phi has no pole).
Jacobi data.
<M_0 z,z> := sum_j (z_j+z_{j+1})^2 / ell_j,
V_j := 2 f_j^3 / (c sin d_j^0), H := M_0 - V (V diagonal),
w_j := c sin d_j^0 f_j^{-2} s_j'(y).
LEMMA (odd-cycle matrices; the odd-cycle matrix lemma). The identities
H w = 0, 2 f_j s_j' = (M_0 w)_j
hold meromorphically on the slice; on the real collar w_j>0, M_0 is
positive definite, and H is negative semidefinite with every PROPER
principal submatrix negative definite. Moreover, for any proper
truthful set I of sites and any flat set F subset I^c (I^c = the rest),
B_{I,F} := (M_0)_{I cup F, I cup F} - diag(V_i 1_{i in I})
has a determinant that does not vanish identically and is invertible
on the real collar.
Why. Real collar first. Take an arc-length variation delta_arc_j at
vertex j and put z_j = sin d_j^0 delta_arc_j. The normal projections on
edge j of the two endpoint variations are z_j and -z_{j+1}, so the
second variation of that edge's length is (z_j+z_{j+1})^2/ell_j up to
endpoint accelerations. The acceleration contributes -n_{psi_j} f_j^3/c
per squared arc-length increment; pairing with the length gradient
(e_{theta_{j-1}}-e_{theta_j}) gives -V_j z_j^2. Hence H is the length
Hessian in z-coordinates. Differentiating along the family gives
H w=0; and V_j w_j=2 f_j s_j' gives the second identity. Analytic
continuation carries both to the meromorphic slice.
On the collar w_j>0, sin d_j^0>0. With D_w=diag(w_j), the matrix
D_w H D_w has off-diagonal entries gamma_j := w_j w_{j+1}/ell_j > 0;
the null identity fixes its diagonal so
<D_w H D_w z,z> = - sum_j gamma_j (z_j-z_{j+1})^2.
The cycle is connected, so the only null vectors are constants, and
fixing one coordinate removes them: H is negative semidefinite with
strictly negative proper submatrices. For M_0, a null z would have
z_{j+1}=-z_j for every j, hence z=0 by oddness of N.
For B_{I,F}, arrange retained coordinates as (I,F). The truthful block
H_{I,I} is negative definite; the flat block (M_0)_{F,F} is positive
definite. When both are nonempty, the Schur complement
H_{I,I} - (M_0)_{I,F}(M_0)_{F,F}^{-1}(M_0)_{F,I} is negative definite,
so B_{I,F} is invertible. One-block cases follow from the already
proved matrix statements.
LEMMA (cofactor normalization; the cofactor normalization). Let A(t_0)
be the signed area coefficient from chapter 3. Then
w_j^2 = (-1)^{N-1} (A(t_0)/N) (prod_i ell_i) det H_{\hat j},
where H_{\hat j} is H with site j deleted.
Why. Differentiate s(y+jt,t) in t at t=t_0 (argument jt included),
multiply by c sin d_j^0 f_j^{-2}, and let v_j be the result. Periodicity
of the lift gives v_{j+N}=v_j+N w_j. Comparing normal differentials of
X_{j+1}-X_j = ell_j e_{theta_j}, and using
d p_j ^ d theta_j = (n_{theta_j}.d X_j) ^ d theta_j together with the
area identity from chapter 3, gives
(w_j v_{j+1} - v_j w_{j+1}) / ell_j = A(t_0).
Summing successive differences of v_j/w_j yields
sum_j ell_j / (w_j w_{j+1}) = N / A(t_0).
The deleted cofactor of the weighted Laplacian D_w H D_w equals
(-1)^{N-1} (prod_i gamma_i) (sum_i gamma_i^{-1}),
by Cauchy-Binet: write the positive Laplacian as an incidence matrix
times diag(gamma_i) times its transpose, delete one vertex; each
remaining edge set is a spanning path tree whose incidence minor has
determinant +-1. Substituting the gamma expression and undoing the
D_w conjugation gives the stated cofactor formula on the collar; both
sides are meromorphic in y, so the identity continues to the slice.
THEOREM (boundary contact; the boundary-contact theorem). At almost every point
of either horizontal boundary of the phase strip |Im y|<h, the
functions s_0,...,s_{N-1} have finite nontangential limits, and at
least one of those limits lies on |Im s|=H.
Why. e^{i s_j} is bounded and bounded away from zero (the shape-lift theorem),
so the disk boundary limit theorem gives nontangential limits almost
everywhere on the strip boundary. Near a nonzero limit choose a local
log; it differs from i s_j by a fixed integral multiple of 2pi i, so
s_j has a finite nontangential limit.
Now suppose, for contradiction, that on a positive-measure set of the
upper boundary the limiting states all lie STRICTLY inside
|Im s|<H, and at each such boundary point the meromorphic data
f_j, sin d_j^0, ell_j, det H_{\hat j}
all have nonzero limits. (Finitely many meromorphic functions; if any
of them vanished on a positive-measure subset of the boundary, then by
Privalov's nontangential boundary uniqueness it would be identically
zero, contradicting its nondegenerate real values. So this zero-set is
null.)
The cofactor identity and the w-definition then give, at each good
boundary point, a regular analytic system
(s_j')^2 = G_j(s_0,...,s_{N-1}), G_j analytic and nonzero
near the limiting tuple, with the square root picked to match the
actual functions on a truncated approach cone. Solving this first-
order system extends the slice arc locally past the chosen boundary
point as a regular ODE solution, and uniqueness identifies that
extension with the actual slice.
Joint extension. Consider the state Z=(s_0,s_1) along the extended arc.
All incidence angles and chords along the orbit are nondegenerate, so
the billiard return map R (N reflections minus the 2pi shifts) is
analytic near these states with det R' - identity = 1 (transfer
determinants telescope around the odd cycle). In suitable local
coordinates (u,v) with the fixed arc Z(u)=(y,0),
R(u,v) = ( u + sigma(u) v + O(v^2), v + tau(u) v^2 + O(v^3) ),
with sigma(u), tau(u) analytic and sigma(u)≠0 near the point.
In a lower strip the joint lift Z(y,t)=(s(y,t), s(y+t,t)) is already
known. Writing b = N(t-t_0) and u = y+O(b), v = b p_0(y) + O(b^2),
the reflection identity R(Z(y,t))=Z(y+b,t) gives, order by order,
sigma p_0 = 1, p_0' = tau p_0^2, sigma' = -tau.
These continue along the extended arc. Fix a small neighbourhood in
which sigma≠0, and set p_0=1/sigma.
For |Im b| >= |b|/3 the return and its inverse continue joint states
upward in steps +-b of size 1/|b| (hence O(1/|b|) steps); the drift is
controlled by the analytic ODE u'=sigma(u)p, p'=tau(u)p^2 whose
regular solution is p=p_0(u). Steepness keeps every slanted path from
the lower starting slab inside the fixed neighbourhood; each step has
error O(|b|^2) with stability factor 1+O(|b|), so total error O(|b|).
All shape coordinates stay strictly inside the shape strip. Intermediate
sites are recovered by regular billiard steps along the orbit.
Analytic disk fill. Move the boundary point slightly within its arc;
choose eta>0 and then much smaller r>0. Consider the analytic disks
t = t_0 + r zeta,
y = x_* + i varsigma - i eta zeta^2 + w, |zeta|<=1,
as varsigma rises from h-2eta to h. On the boundary arcs where zeta is
near the real axis, the phase point stays a fixed amount below the
wall, so these arcs lie in the original joint tube by lower
semicontinuity of h. On the remaining arcs the parameter is in a fixed
steep sector, so the step continuation applies. Holomorphic product
continuation (lemma 2.8 of chapter 2) fills the disks; equivalently, the
Cauchy moments vanish on their initial open part and hence throughout
the connected fill, after eliminating the auxiliary (a,zeta) into a
single complex center a=x_*+i varsigma+w. The map
(a,zeta) -> (a - i eta zeta^2, t_0 + r zeta) has Jacobian r ≠ 0.
Maximum principle preserves |Im s_j|<H on the filled disks. Composing
with Gamma gives a joint meromorphic patch for X at (x_*+i h, t_0),
agreeing with the primary tube below the wall. That contradicts
the no-patch property, which applies also at the rational t_0. Lower boundary:
the same argument, or real symmetry. The supposed positive-measure
interior set cannot exist.
Let y_u be planar Brownian motion started with y_0 uniform on one real
period, and let zeta be its first exit from |Im y|<h. For each site j
and time u define
delta_j(u) = H - |Im s_j(y_u)|, k_j(u) = -log delta_j(u),
K_j(u) = max_{v<=u} k_j(v),
r_y(u) = h - |Im y_u|, n_y(u) = -log r_y(u),
tau_L = inf{ u : max_j K_j(u) = L }.
For large L, tau_L < zeta almost surely: the e^{i s_j} are bounded
holomorphic martingales (Poisson integrals of their boundary values),
their exit values match the nontangential boundary limits of 10.2, and
at least one delta_j tends to 0 before phase exit. "With probability
tending to one through tau_L" means that a single event of probability
1-o(1) supports the stated bounds simultaneously at all times
0<=u<=tau_L and at every site.
LEMMA (shape time change; the shape-Brownian time-change). Each process s_j(y_u) (up
to phase exit) is Brownian motion in the shape strip |Im s|<H under the
scalar time change
Q_j(u) = int_0^u |s_j'(y_v)|^2 dv,
possibly stopped before shape exit. Its real-projected initial law has
density bounded above and below relative to uniform. Consequences:
(i) The total shape clock has bounded expectation, and
E int_0^{tau_L} 1_{k_j<=K} |f_j s_j'|^2 du <= C (1+K)^{2 rho}.
(ii) At first hit of a deterministic shape-depth row, the hit law is
dominated by a fixed constant times the ordinary shape-Brownian
hit law from a uniform real start; in particular it is bounded
above by a fixed multiple of the uniform horizontal law.
(iii) After first hit of depth delta, the remaining shape clock has
tail <= C min(1, delta/sqrt(T)) at clock time T>0.
(iv) With probability tending to one through tau_L,
k_j(u) >= K_j(u) - C log L for every j.
Why. Holomorphic functions of planar Brownian motion have real and
imaginary parts as local martingales with equal quadratic variation and
zero mutual covariation (Cauchy-Riemann), so the Brownian time-change
theorem applies up to phase exit (extend by independent increments in
intrinsic time; approximating by compact interior stops and exhausting
gives the same domination).
(i) On the real collar y -> s_j(y) is an increasing analytic circle
diffeomorphism with bounded-above-and-below derivative; its initial
horizontal law has the stated density. Under uniform real start the
periodic shape-strip Green weight at wall distance delta is comparable
to delta and has finite integral, giving integrable total clock. Near
the wall, the |f|^2 contribution up to log depth K is
int_{e^{-K}}^{delta_*} delta E|f|^2 d delta = int_{k_*}^{K} mu_f(k)^2 dk <= C K^{2 rho}
by the angle budget 9.7. (ii) is the same Brownian extension (if the
original path never reaches the row, it contributes no mass). (iii)
uses 1d reflection survival of Brownian hitting of the wall: starting
at distance delta, survival to time T is at most C min(1, delta/sqrt(T)),
and we apply this after the hit with strong Markov in intrinsic time.
(iv) is a union bound over O(L) integer levels: at first hit of the
integer level m, the 1d hitting probability of reaching depth R e^{-m}
before the adjacent wall is 1/R; take R a sufficiently large fixed
power of L and sum over the O(L) levels and the fixed number of sites.
LEMMA (log-deficit at a stopping cut; the logarithmic-deficit lemma). Let g be a
nontrivial meromorphic slice function with log|g(y_0)|>=-C_0 on the
real starting row. If log|g(y_u)|<=B_0 (B_0>=1) up to a stopping cut T,
then for B_1>=1,
Prob{ inf_{u<=T} log|g(y_u)| < -B_1 } <= (B_0+C_0)/(B_0+B_1)
<= C(1+B_0)/(1+B_1).
Why. Away from the polar zero/pole set, log|g(y_u)| is a continuous
local martingale. B_0 - log|g(y_u)| is a nonnegative supermartingale
after localization; stop at first hit of B_0+B_1 and apply the
supermartingale inequality. Exhaustion removes the localizations.
Standing convention: any analytic differential integrated after a
stopping time is estimated only after a DETERMINISTIC quadratic-time
cut. Stop the integral at quadratic time 2B; the maximal inequality
gives second-moment bound O(B); this bound applies to the original
integral on the paths that respect the cut, without conditioning. The
same rule applies with integrands masked by adapted status indicators
(replace by their predictable versions).
Put
a_j = log|f_j|,
G(u) = max_j ( |psi_j| + log(1+|X_j|) + |log|sin d_j^0|| + |log|ell_j|| )(y_u).
By the critical polynomial bounds (chapter 8, now unconditional),
everywhere before the relevant history stop,
a_j <= k_j^+ + C log(2+k_j^+) + C, k_j^+ = max(k_j,0).
LEMMA (finite family of modified inverses; the modified-inverse lemma). Fix
V>=U>=1. Fix a proper truthful set I and flat subset F subset I^c.
Stop at tau_L, at the first failure of G<=U, or at the first time some
I-history exceeds U, whichever comes first. Except on an event of
probability at most C(1+U)/(1+V),
||B_{I,F}^{-1}|| <= exp(C V)
throughout the stopped interval. The constant is uniform over the
finitely many choices of I,F.
Why. On the stopped interval, a_i<=C(1+U) for i in I, and the geometry
cap bounds reciprocal lengths, inverse incidence sines, and angles, so
every entry, determinant, and cofactor of B_{I,F} has modulus
<= exp(C(1+U)). No amplitude bound is needed at flat or deleted sites
(their curvature terms do not appear). The real-collar determinant has
a strictly positive minimum modulus; apply log-deficit to that
determinant with upper bound C(1+U) and lower threshold -V, and use
adjugate. There are at most 3^N choices.
The reason to track HISTORY caps rather than current depths: each fixed
candidate (I,F) is estimated from the ORIGINAL real start, with its own
stopping rule. If, later, its selected histories are still <= U, none
of them has crossed the cap previously, so the estimate already secured
for that fixed candidate applies as long as the common geometry cut has
not fired. No estimate is restarted when the current selection changes.
LEMMA (initial stopped geometry; the initial-geometry bound). There is a fixed
sigma<1 such that
G(u) <= L^sigma (0<=u<=tau_L)
with probability tending to one as L->infinity.
Why. The angle differential is d psi_j = f_j d s_j. Up to tau_L, all
current log depths are <=L, so the angle-clock moment bound (i) of
10.3 plus the maximal-inequality cut bounds all angles by L^{rho+eps}
for each fixed small eps>0, with probability tending to one. The
linear angle equations theta_{j-1}+theta_j=2 psi_j give the same bound
for theta_j and d_j^0 and hence for log|sin d_j^0| above; applying
log-deficit to log|sin d_j^0| for its lower bound gives, for some fixed
rho < rho' < 1,
a uniform bound on the angle-and-sine cost by r_* := L^{rho'}. (Following
the standard stopping-cut convention, cut at first failure of these
bounds, which we already absorbed with probability tending to one.)
Choose G=L^gamma with (1+rho')/2 < gamma < 1, and a small fixed b>0.
Attach to each site two statuses, moderate and tiny, with hysteresis:
moderate->tiny on downcrossing a_j=-2bG, tiny->moderate on upcrossing
a_j=-bG. Gap bG. Since C L - a_j is a nonnegative supermartingale (up to
tau_L and the preceding cuts), the upcrossing inequality over its two
levels CL+bG, CL+2bG bounds the expected total number of switches by
O(L/G); hence for some fixed small eps_1>0 the actual number of
switches is at most L^{1+eps_1}/G with probability tending to one.
During a moderate interval, using r_*=o(G), the coefficient of d s_j in
d X_j has modulus at most e^{5bG}. During a tiny interval, the
coefficient of d s_j in R^{-1} d H_j^0 has modulus at most e^{-bG/2},
where R(u)=1+sup_{v<=u,j}|X_j(y_v)|. Masking by the current status and
applying the maximal inequality (with the deterministic quadratic-time
cuts) on the bounded shape clocks shows that the cumulative
moderate-position and tiny-support integrals have suprema at most
e^{6bG} and e^{-bG/3}
respectively, with probability tending to one.
Between successive switches of the joint status vector, a moderate
site's position changes by at most 2 e^{6bG}; a tiny site's support has
the preceding increment bound (integrate R^{-1}d H_j^0 = dM_j against
the increasing R). Each switch interval therefore multiplies the
running maximum of R (at its start) by at most e^{C r_*} and adds
C e^{6bG}; summing over the switch count and absorbing gives
log R <= C G
with probability tending to one, provided eps_1 is small enough that
2 gamma > 1 + rho' + eps_1. Then the length identity ell_j=(X_{j+1}-
X_j).e_{theta_j} gives length upper logs of the same order; log-deficit
gives length lower logs at a slightly larger fixed exponent. Choose
sigma strictly between these exponents and 1; all constants then
absorb into L^sigma for large L.
LEMMA (phase depth and phase clock; the phase clock). For some fixed C,
with probability tending to one through tau_L,
n_y(u) <= C L, int_0^{tau_L} r_y(u)^{-2} du <= L^C.
Why. Apply 10.4 through tau_{2L}. Entries of H have upper logarithmic
size O(L) (the amplitude bound of the modified-inverse lemma); cofactor identity gives
|w_j|+|s_j'| <= e^{C_1 L}
on the good paths through that stop. Let T = first time r_y=e^{-C_2 L},
with C_2 a large fixed constant. By 1d hitting of the wall, after T the
remaining phase time is <= e^{-C_2 L} except with probability at most
C e^{-C_2 L/2}. Suppose T < tau_L. At T all shape depths are
>=e^{-L}; at tau_{2L} one of them is e^{-2L}. The quadratic time of
each shape increment from T to tau_{2L} is at most e^{2C_1 L - C_2 L};
maximum-inequality bound (with quadratic-time cut and a physical time
cut of e^{-C_2 L} after T) gives all these increments o(e^{-L}) with
probability tending to one, contradicting the required change of
Imag s_j by at least e^{-L}-e^{-2L}. So T<tau_L has vanishing
probability, proving the first bound.
Under uniform real start, the phase-strip Green weight at a wall is
proportional to r_y, so
E int 1_{r_y>=e^{-C_2 L}} r_y^{-2} du <= C(1+L).
On the good event, the truncated occupation dominates the actual
integral up to tau_L. Markov (at threshold L^2, say) gives a polynomial
bound with probability ->1, hence the second bound after increasing
the single constant C.
THEOREM (stopped geometry at every power; the stopped-geometry theorem). For
every fixed sigma>0, with probability tending to one as L->infinity,
G(u) <= L^sigma (0<=u<=tau_L).
The phase-depth and phase-clock bounds of 10.4 hold simultaneously.
Why (core). It suffices to improve an exponent 0<sigma<1 already known
to work. Put H_0 := L^{sigma+eps} for a fixed small eps>0 with
sigma+eps<1. Apply 10.4 at level H_0: G<=H_0^sigma through tau_{H_0},
with probability tending to one. We now show that after tau_{H_0}, all
further variation is EXPONENTIALLY smaller than this improved initial
scale H_0^sigma.
Gap argument. Take finitely many disjoint open intervals
(xi_m, xi_m+eta) inside (sigma+eps/3, sigma+eps), more than N of them,
fixed. At any time after tau_{H_0} at least one K_j>=H_0, so some gap
(L^xi, L^{xi+eta}) contains no history. Fix that gap and let
I = { j : K_j <= L^xi }, I^c = { j : K_j >= L^{xi+eta} }.
Endpoint equalities can be assigned to their respective sets; estimates
are unaffected. I is proper because some K_j >= H_0 > L^{xi+eta}. By
10.3 (iv) every deep site has current log depth k_j >= (1/2)L^{xi+eta}.
Inverses first. For each fixed xi_m, each proper I, each F subset I^c,
secure B_{I,F}^{-1} from 10.3 with
U = L^{xi_m + eta/4}, V = L^{xi_m + eta/2}.
The old geometry cap L^sigma and the selected truthful history cap
L^{xi_m} for the actual active set are both < U for large L. Each
candidate fails with probability <= C L^{-eta/4}; the total number of
candidates is fixed, so outside one event of probability ->0 we have
||B_{I,F}^{-1}|| <= exp(o(H_*)), H_* := L^{xi+eta},
for every later time at which its selected histories are below their
caps, however the sets I,F change along the way.
Suppression. Set z_j := r_y w_j. Schwarz-Pick, transported from the
disk by conformal maps of the two strips, gives r_y|s_j'|<=C delta_j
(the strip hyperbolic densities are comparable with reciprocal boundary
distance). Using the w-definition and the geometry cap,
log|z_j| + 2 a_j + k_j <= C L^sigma,
with the usual interpretation when z_j=0. Among the deep sites, retain
as flat exactly those with |z_j|>e^{-c_1 H_*} and delete the rest;
call this set F, with the other deep sites removed. At a retained deep
site, the display and k_j>=H_*/2 imply a_j<-c_2 H_* for a fixed c_2>0
(since L^sigma = o(H_*)); the true curvature V_j on F satisfies
|V_j|<=e^{-c_3 H_*}. Deleted coordinates already have modulus
<=e^{-c_1 H_*}.
Apply (M_0 - V) z = 0 to the retained coordinates I cup F. The left
matrix is B_{I,F} minus the tiny curvature diagonal on F; the right side
is the coupling to the deleted coordinates. Entries of M_0 cost at most
e^{C L^sigma}. The inverse bound 10.3 absorbs the tiny diagonal; the
coupling to deleted coordinates (each <=e^{-c_1 H_*}) gives
max_j |z_j| <= e^{-c_4 H_*}
for a fixed c_4>0, uniformly over all finite choices. This is the key
point: it is suppression at EVERY site, shallow ones included. The
same argument also covers the case of empty F (then the retained
homogeneous system has only the zero solution).
Angles. From the odd-cycle lemma:jacobi-angle, since
2 r_y partial_y psi_j = V_j z_j = (M_0 z)_j,
the right side has modulus <= exp(-c L^{sigma+eps/3}) for a fixed c>0
after decreasing the constant. The phase-clock bound makes the
quadratic time of each angle increment after tau_{H_0} exponentially
small; the maximal-inequality cut then gives, with probability tending
to one,
sup_{tau_{H_0}<=u<=tau_L} |psi_j(y_u)-psi_j(y_{tau_{H_0}})|
<= exp(-c' L^{sigma+eps/3}).
All choices of gaps and flat sets vanish from this martingale statement:
they were used only to prove the uniform pointwise differential bound
on the good event. The angles at tau_{H_0} are <=H_0^sigma, and the
incidence sines have modulus >=e^{-H_0^sigma}. Absolute sine changes
across the late interval are negligible relative to that, so angles
and both logarithmic sine bounds keep cost O(H_0^sigma) =
O(L^{sigma(sigma+eps)}) throughout the late interval.
Supports. With d H_j^0 = (X_j.e_{psi_j}) d psi_j, the coefficient
relative to d psi_j is bounded by e^{C L^sigma}. Combining the angle-
suppression bound with the phase clock shows the martingale integral of
this differential over the single late interval has exponentially small
supremum; integration by parts against the increasing R gives
|H_j^0(y_u)-H_j^0(y_{tau_{H_0}})| <= exp(-c'' L^{sigma+eps/3}) R(u)
on a further event of probability ->1. Reconstruct positions with the
improved angle and sine cost; the coefficient of R(u) in the last
display stays exponentially small after reconstruction. Conclusion:
log R(u) <= C H_0^sigma = C L^{sigma(sigma+eps)}, (u<=tau_L).
Length upper logs match; log-deficit for their lower logs, at a
slightly larger fixed exponent.
Improvement. Choose eps small enough that
sigma(sigma+eps) < (sigma+sigma^2)/2 < sigma.
The new exponent is (sigma+sigma^2)/2, strictly better than sigma.
Iteration. sigma |-> (sigma+sigma^2)/2, starting below one, decreases
to zero. Given any target exponent sigma_*>0, iterate this improvement
a fixed number of times (chosen in advance, independently of L); each
step has a vanishing exceptional event, so we discard a fixed finite
list of them. This proves the assertion for arbitrary sigma>0; the
phase bounds already hold by 10.4.
For every fixed sigma>0, on a single event of probability 1-o(1), all
functionals in G are <=L^sigma on 0<=u<=tau_L; the phase depth obeys
n_y<=C L; the phase clock is <=L^C. The modularised modified-inverse
bounds of 10.3 hold in the "secured-from-the-start" form: they apply
whenever the selected histories are below their caps, and are uniform
over the finitely many selections (I,F). These are exactly the inputs
to the sign test of chapter 11.
We finish off the finite critical wall. Chapter 9 supplies the
unconditional polynomial bounds
mu(k) <= C(1+k),
d_k sup_x (|f|+sqrt|q|)(x+i(H-d_k)) <= C(1+k)^2,
and real symmetry gives the same at the lower wall.
Fix throughout alpha = 1/100. The first half of the chapter produces a
deterministic shape row on which almost every point has |f| either
very small or very large. Both alternatives suppress the movement of
that site in the odd Jacobi system, but by different identities. The
final argument applies the suppression to the last site to cross the
tested row.
LEMMA (discrete Tanaka; the discrete Tanaka estimate). Let (X_m)_{0<=m<=M} be
an integrable real submartingale with
V = sum_{m=0}^{M-1} E(X_{m+1}-X_m)^2 < infty. For 0<h<R put
C_h := sum_{m=0}^{M-1} E[ 1_{|X_m|<=h} min((X_{m+1}-X_m)^2, h^2) ].
There is an absolute C such that
C_h <= C h(h+R+sqrt(V)) + C M h^2 min(1, V/(R-h)^2).
Why. Consider Phi(x)=sqrt(h^2+x^2)+x. It is nonnegative, increasing,
convex, 0<=Phi'<=2, Phi(x)<=h+2x^+, and Phi''(x)>=c/h on [-2h,2h].
Taylor with the convex remainder integrated a distance <= the first
min(|y-x|,h) gives, for all x,y,
Phi(y)-Phi(x)-Phi'(x)(y-x)
>= (c/h) 1_{|x|<=h} min((y-x)^2, h^2).
Let T be first index with X_m>R (capped at M; T=0 if X_0>R). Sum
only for m<T. The linear term has conditional expectation >=0
(Phi'>=0 and submartingale drift), so summing expectations makes it
drop out. Remaining bound: Ch E[Phi(X_T)-Phi(X_0)]; the contribution
from paths above R is zero. On the rest, positive part of X_T is at
most R+max_{m<M}|X_{m+1}-X_m|. Cauchy-Schwarz bounds this by
C h(h+R+sqrt{V}).
Returns after T. With drift b_m=E[X_{m+1}-X_m|F_m]>=0 and
Z_n=sum_{m<n, T<=m}(X_{m+1}-X_m-b_m), Z is a square-integrable
martingale from zero with terminal second moment <=V. If a path that
was above R (or crossed R) returns to [-h,h], then sup_n|Z_n|>=R-h
(the drift cannot help a downward move). Martingale maximal bounds
this event by C min(1, V/(R-h)^2). After T, on that event the total
cost is <=Mh^2. Adding the two pieces finishes.
LEMMA (large moment gives a bad row; the large-moment lemma). Suppose that
along a sequence K->infty some j in [0.9K,2.1K] has mu(j)>=K^alpha.
There is a deterministic k in [0.8K,2.2K] with
(1/2pi) |{ x in [0,2pi] : ||log|f(x+i(H-e^{-k}))|| <= K^{alpha/2} }|
-> 0.
Why. Choose c<1/4 fixed. Maximize mu(l) e^{-c|l-j|} over l>=k_0 (a
maximum exists by the polynomial bounds); a maximizer k satisfies
|k-j| = O(log K), B := mu(k)>=K^alpha,
mu(l) <= B e^{c|l-k|}.
Introduce the normalized random profiles (uniform center x in [0,2pi])
F_{K,x}(Z) = (d_k/B) f(x+iH+d_k Z),
Q_{K,x}(Z) = (d_k/B)^2 q(x+iH+d_k Z).
On any fixed depth 0<D<=e^{k-k_0} the moment envelope gives
E_x(|F_{K,x}(X-iD)|^2+|Q_{K,x}(X-iD)|) <= C D^{-2} e^{2c log D},
uniform in X. Submean on disks gives tightness for locally-uniform
convergence. Diagonal extraction yields holomorphic limits stationary
under real translations. The scaled equation F''=B^2 Q F gives QF=0
in the limit; so at least one of F,Q vanishes.
The law has no atom at (0,0). Choose 1/2<eta<1-c; dyadic-band submean
bounds the cone-summed second moments of D^eta(|F|+sqrt|Q|) by
C D^{-2(1-c-eta)}, and their sum over unit bands is a tight random
constant. An atom at (0,0) would let us select centers near (0,0)
while that constant stayed bounded, contradicting
the no-trivial-profile lemma (chapter 8).
With probability one, if nonzero, Q is not real on all of Im Z=-1.
(Otherwise Schwarz reflection would make Q entire; the fixed-D decay
E|Q(X-iD)|<=C D^{-2(1-c)} would then force Q==0 by Cauchy. Exclude.)
For deterministic convergent profile sequence with allowed limit:
- If F not==0, then away from its discrete zeros on the tested row
segment, log|f| = k+log B+log|F_{K,x}(Z)|. The right side is >=
K^{alpha/2} on compact subsets away from those zeros.
- If Q not==0, its imaginary part on Im Z=-1 is nonzero real-analytic;
away from its discrete zeros, cover the row by short intervals with
consistent branches p_K=sqrt(Q_{K,x}) with Re p_K>=c_1>0 and bounded
derivatives. WKB basis
g_+- = Q_{K,x}(u-i)^{-1/4} exp(+/- B int p_K)(1+O(B^{-1})),
proven by variation of constants (Volterra kernel with factor
exp(-2Bc_1(u-v))). The two leading log-moduli differ by at least
c_1 B in derivative; the set where they agree to sqrt(B) is
O(B^{-1/2}); each leading log-modulus has slope |.|>=c_1 B-O(1),
so it spends O((K^{alpha/2}+1)/B)=o(1) length within K^{alpha/2}
of zero. Hence the fraction violating the row test is o(1)
locally.
Skorokhod representation identifies the empirical expectation with
the full-period average. Every subsequence has the same conclusion.
Suppose now mu(v)<=K^alpha for 0.9K<=v<=2.1K. Write
delta(z)=H-|Im z|, a(z)=log|f(z)|,
A(z)=log(|f(z)|+delta(z)|f'(z)|).
A is finite and continuous (common zero of f and f' would force f==0
by f''=qf, contradicting f not==0).
LEMMA (local amplitude; the local amplitude bound). Let z_0=x+i(H-d), d<H/4, and
H_b=1+sup_{B(z_0,d/2)} delta sqrt(|q|). On B(z_0,d/8):
osc A <= C H_b,
and the L^2 norm of A-a on that disk, and its normalized length L^2
norm on the horizontal diameter, are at most C H_b. Fixed smaller
disks and comparable horizontal segments behave the same.
Why. Scale by d/8: F(zeta)=f(z_0+(d/8)zeta), Q(zeta)=(d/8)^2 q(...).
F holomorphic in |zeta|<4, |Q|^{1/2}<=H_b/4. Vector
Y=(F,F'/H_b) satisfies Y'=[[0,H_b],[Q/H_b,0]]Y; matrix norm <=
C H_b; Gronwall both ways bounds osc log||Y||<=C H_b, hence
osc A<=C log(2+H_b) on fixed interior disks. Set A_0=A(z_0),
G=e^{-A_0}F. Cauchy on a small circle locates |zeta_*|<=1/16 with
log|G(zeta_*)|>=-C; on |zeta|<=3, log|G|<=C H_b. Set U=M-log|G| with
M=C_0 H_b large; U>=0 on |zeta|<=3 and U(zeta_*)<=C H_b.
Jensen on a circle centered at zeta_* (radius in [2,9/4]) bounds zeros
in B(zeta_*,7/4) by C H_b. A second auxiliary circle (radius in
[3/2,7/4]) expresses U on the enclosed disk as the harmonic extension
of its nonnegative boundary data plus the (positive) disk Green
functions of the zeros. On |zeta|<=1 the harmonic term <=C H_b, and
each Green function has bounded L^2 norm on the unit disk and on its
horizontal diameter (via log^-|t+ib-zeta_j|<=log^-|t-Re zeta_j| and
integrability of |log|t||^2). Since A-a = (A-A_0-M)+U, and the first
term is osc A, the L^2 bounds follow. Smaller fixed disks and
comparable segments behave by the same fixed-ratio estimates, or by
finite covering.
Consequence (fixed-tile L^2 bounds). Under the low-moment hypothesis
and 0.9K<=v<=2.1K,
E_x H_b(x,d_v)^2 <= C K^{2alpha},
E_x (A-a)^2(x+i(H-d_v)) <= C K^{2alpha}.
Run ordinary Brownian motion in the shape strip from a uniform point
of its real period. At first hit of the row at depth e^{-v}, let a(v)
denote a. Sampled on any increasing finite grid, these random
variables are an integrable submartingale (apply subharmonic
comparison in the next substrip to max{log|f|,-T} and let T->infty;
all row logarithms are integrable including at isolated zeros).
Translation invariance makes the horizontal coordinate uniform;
symmetry makes the two sides equiprobable.
LEMMA (log-increments; the increment bounds). Under low moments, uniformly
for 1.1K<=v<=1.9K and 2<=s<=4,
E|a(v+s)-a(v)|^2 <= C K^{2alpha}.
Why. Start on the upper row of depth d=e^{-v}, stop at first hit of
either row of depth d'=de^{-s}; let U be unwrapped horizontal
displacement. Half-plane exit has Cauchy density
(d-d')/(pi((d-d')^2+u^2)),
hence
Prob(same side, |U|>Rd) <= C/R (R>=1),
Prob(opposite side) <= C d.
Conditional on side and displacement, starting point is uniform mod
2pi; terminal point is its translate, hence uniform too.
For same-side displacement u with R=|u|/d<=e^{K/20}: prescribed path
to depth d+|u|, horizontally by u, then to d'. Fixed relative steps
vertically plus O(1) horizontal tiles: C(1+log(1+R)) tiles, all in
the low-moment band. Using the local amplitude bound and the tile L^2 bounds,
||A(end)-A(start)||_{2} <= C K^alpha(1+log(1+R)),
and the squared last factor is integrable by the tail bound.
For the reminder (R>e^{K/20} or opposite side): connect via the real
collar using the polynomial bound; the amplitude difference is at
most C(1+K)^3 (O(K) levels with O((1+j)^2) each). This event has
probability <=C(e^{-K/20}+e^{-v}) by the tail, so contributes o(1)
to the second moment. Subtract the (uniformly small) A-a endpoint
errors on both sides: each square is <=C K^{2alpha}. Done.
Set h_0 = K^{6 alpha}. Apply the discrete Tanaka estimate to step-s
grids in [1.1K,1.9K] with V<=C K^{1+2alpha} and R=K^{1/2+10alpha}.
The two displayed quantities are o(K/log K) for our alpha. Integrate
the grid estimate over an s-offset in [2,4]; every starting level
counts once. Thus
int_{1.1K}^{1.9K} int_2^4
E[ 1_{|a(v)|<=h_0} min(|a(v+s)-a(v)|^2, h_0^2) ] ds dv
= o(K/log K).
LEMMA (witnesses force oscillation; the witness-cost lemma). Under low
moments, the fraction of pairs (x,k) in [0,2pi]x[1.2K,1.8K] for which
|a(x+i(H-e^{-k}))|<=h_0/4 tends to zero.
Why. Use the witness form of the no-trivial-profile lemma: pick aperture
and fixed enlargements large enough that at every target center there
is a witness z (measured in local depth units) with
|Re z-x|<=L_0 delta(z), |k - log(1/delta(z))|<=C,
delta(z)(|f(z)|+sqrt|q(z)|)>=c_*>0.
For a uniformly chosen (x,k) in the central band, tile L^2 bounds plus
Markov give, except on an event of probability ->0,
(A-a)(x+i(H-e^{-k})) <=K^{2alpha},
delta sqrt|q| <=K^{2alpha}
throughout every enlarged conic tile (union over O(log K) log-depth
bands gives O(K^{-2alpha}log K)=o(1)). Call such a target typical.
If in addition |a(x,k)|<=h_0/4 (moderate), then along tile chains of
length O(1+log K), the local amplitude bound yields
|A|<= h_0/3 on the whole cone.
On that window delta|f|<=exp(-1.1K+h_0/3) for large K, so f cannot
reach c_*/2; the witness must be powered by q: at its source z_1,
delta_1 sqrt|q(z_1)| >= c_*/2.
Disk oscillation. The unit-area disk D_1 centered at z_1 with radius
a small fixed multiple of delta_1 satisfies, for every real value z,
(1/|D_1|) int_{D_1} |a-z| >= c_2>0
with c_2 uniform. (Otherwise a sequence of rescaled g_n with
log|g_n|->0 in L^1 would converge to a nonzero holomorphic g with
|g|==1 off its zeros; so g is constant; but g_n''/g_n->0 contradicts
the lower bound on delta sqrt|q| at the source. No upper bound on
curvature is needed.)
Truncation. For |z|<=h_0/2, split |a-z| on {|a|<=h_0} vs {|a|>h_0};
the disk L^2 bound on A-a and the upper a<=A<=h_0/3 combine to show
the discarded L^1 mass is o(1), and Cauchy-Schwarz then gives
(1/|D_1|) int_{D_1} 1_{|a|<=h_0} min(|a-z|^2, h_0^2) >= c_3>0.
Second disk. Take D_2 horizontally centered as D_1, centered at depth
e^{-3}delta_1, radius a small fixed multiple of that depth. All but
o(1) of its normalized area has |a|<=h_0/2 (the local amplitude bound and the
disk-deficit estimate). The Poisson kernel for exit from the first
row up to the D_2 row has density >=c/delta_1 uniformly (both the
horizontal displacement and the vertical distance are O(delta_1),
and comparable). So every typical moderate target forces
1 <= C int_{W(x,k)} C_source(x',v) dx' dv / e^{-v},
where C_source integrates the conditional occupation (over s in [2,4])
of a source point starting in |a|<=h_0, and W(x,k) is the enlarged
cone window.
Fubini. Integrate against typical moderate targets; for fixed source
(x',v), the horizontal measure of targets whose window contains it is
O(e^{-v}), which cancels the 1/e^{-v} above; the allowed k-range has
length O(log K). Fubini and the occupation-cost bound give the count
<= C log K * o(K/log K) = o(K). Atypical targets were already o(1).
Done.
THEOREM (row sign test; the row sign theorem). For every sufficiently large K,
there is a deterministic k=k(K) in [0.8K,2.2K] such that, on each
shape row of depth e^{-k},
|log|f|| > K^{alpha/2}
outside a set of relative horizontal measure tending to zero.
Why. If mu(j)>=K^alpha for some j in [0.9K,2.1K], use 11.2. Otherwise
low moments hold and 11.4 gives a deterministic row in [1.2K,1.8K]
with exceptional fraction ->0 for the stronger threshold h_0/4;
since K^{alpha/2}=o(h_0), the weaker row test holds too. Real
symmetry gives the lower-row conclusion. Both alternatives are
exhaustive, so the conclusion holds as K->infty.
PROPOSITION (robust sign disk; the robust-sign proposition). For the
deterministic tested level k of 11.5, put
h_1 = (1/2) K^{alpha/2}, r_rob = e^{-k-h_1/4}.
Outside a relative horizontal set ->0 on either tested row, every
point z has |log|f(zeta)||>=h_1 for all |zeta-z|<=r_rob, with constant
sign on that disk. In the odd-slice Brownian experiment, off an event
of probability ->0, every site's FIRST hit of the tested depth falls
on such a disk.
Why. Write d=e^{-k}. By the polynomial bound, |q|<=C d^{-2}K^4
throughout a fixed depth-sized neighborhood of either row. Two zeros
of f in this neighborhood are at least c d K^{-2} apart (scale by
r=c d K^{-2} and run the integral equation F(u)=u+
int_0^u(u-v)Q(v)F(v)dv on a small disk; the RHS is o(1)).
Packing disks of radius c dK^{-2} in a band of width O(d) bounds the
zero count by C d^{-1}K^4. Delete shadows of radius
r_exc = d e^{-h_1/10}
around every such zero; the total relative length of this shadow set
is <=C K^4 e^{-h_1/10}=o(1). Remaining zeros are too far to affect
smaller disks.
At a remaining point, throughout its disk of radius r_rob,
|f'/f| <= C/r_exc.
(If |f'(z)/f(z)|>>1/r_exc, rescale by 1/lambda=f(z)/f'(z) and divide
by f(z); the scaled potential has sup <=C K^4 e^{-h_1/5}=o(1); so
F(0)=F'(0)=1 and F is uniformly close to 1+u on |u|<=2. Rouche
gives a zero within 2/|lambda| of z, contradicting distance to
deleted zero disks. Since r_rob/r_exc=e^{-3h_1/20}, the argument
travels through the whole robustness disk.)
Disk is zero-free, so integrating f'/f along segments bounds the
change of log|f| by C e^{-3h_1/20}=o(1). Test against 2h_1 survives.
Brownian: the first-hit law of each site is dominated by shape-
Brownian hit law from uniform horizontal start. Bad first-hit
probability is o(1) including paths stopped before the row.
Return to the odd periodic slice (the odd periodic slice). Fix N, t_0=2pi/N, h.
Recall
a_j=log|f(s_j)|, k_j=-log(H-|Im s_j|),
K_j(u)=sup_{v<=u}k_j(v), w_max=max_j|w_j|,
and tau_L = first time some K_j reaches L. By 10.2, for large L this
occurs almost surely before phase exit.
LEMMA (deterministic history gap; the history-gap lemma). There is a fixed
finite menu of exponents 0<gamma_m<1 such that for each large L one
deterministic K=L^{gamma_m} has the following event of probability
>0.9: every terminal history K_j(tau_L) is <=K^{alpha/8} or else
>=3K; and at least one is >=3K.
Why. Choose M>10N intervals (L^{gamma_m alpha/8}, 3L^{gamma_m}) with
gamma_1<...<gamma_M small and separated so gamma_m<gamma_{m+1}
alpha/8. Intervals are disjoint and below L. Each path fails for at
most N menu choices (a single K_j falls in at most one interval;
the shallow/deep classification is exhaustive). Averaging over the
menu gives a deterministic choice with failure probability <=N/M<0.1.
At tau_L some K_j equals L, so the deep class is nonempty. The menu
is finite, so all subsequent o(1) estimates are uniform across its
choices.
Fix this deterministic K. Put
u_0 = K^{alpha/8}, h_2 = u_0^2 = K^{alpha/4}.
On the history-gap event let I be the shallow indices and J their
(nonempty) complement. These sets depend on the terminal path, but
in all estimates below every candidate set is fixed before stopping.
LEMMA (two derivative bounds; the derivative bounds). Off an event
of probability ->0:
(a) for every site i and every time <=tau_L,
|s_i'| <= e^{o(h_2) - |a_i|} w_max;
(b) for any fixed nonempty candidate J, with I=J^c, and stopping
additionally when any I-history reaches 2u_0,
w_max <= e^{C h_2} max_{j in J} |s_j'|.
Exceptions are uniform over the finite family of candidates.
Why. Use the stopped-geometry theorem with a fixed exponent smaller than the
smallest menu exponent; its geometry cost is o(u_0) uniformly through
tau_L, giving all entries of M_0 to o(h_2). Identities:
w_i = c sin d_i^0 f_i^{-2} s_i', 2 f_i s_i' = (M_0 w)_i.
If a_i<0, the first gives |s_i'|<=e^{o(h_2)+2a_i}w_max, hence (a).
If a_i>=0, the second gives (a) directly.
For (b): for each fixed (I, F subset J), secure B_{I,F}^{-1} first.
There is at least one deletion or flattening (J nonempty). Stop when
any I-history reaches 2u_0, and at the geometry cuts. Upper bound on
a_i for truthful sites is O(u_0) (polynomial bound); every entry and
the determinant have upper log <=C u_0. The real-collar determinant
has a strictly positive lower bound. Log-deficit at lower threshold
u_0^{3/2} then gives, off probability O(u_0^{-1/2}), the inverse-det
cost at most e^{u_0^{3/2}}, and adjugate yields
||B_{I,F}^{-1}|| <= e^{u_0^{3/2}} = e^{o(h_2)}.
Now at a time; put F={j in J: a_j<-h_2}, D=J\F. In H w=0, eliminate
I cup F via the inverse bound. Removed curvatures on F satisfy
|V_j|<=exp(-3h_2+o(h_2)). The diagonal perturbation is absorbed; the
coupling to D costs e^{o(h_2)}. So w_max <= e^{o(h_2)} max_{j in D}
|w_j|. If D empty, the same absorption forces w=0. Otherwise a_j>=-h_2
on D, and the first identity gives |w_j|<=e^{2h_2+o(h_2)}|s_j'|, giving
(b). All flat subsets were included; no switching-time issue.
THEOREM (last-site contradiction; the last-site theorem). The nonisotropic
finite critical-wall case is impossible.
Why. Use the row sign theorem at this deterministic K; let k in [0.8K,2.2K]
be its tested level. Write d=e^{-k}, h_1=K^{alpha/2}/2,
r_rob=d e^{-h_1/4} as in 11.6. Note
1 << u_0 << u_0^{3/2} << h_2 << h_1 << K.
For each site j let sigma_j be first hit of shape depth d (sigma_j =
infty if never); put
Q_j = int_{sigma_j wedge tau_L}^{tau_L} |s_j'|^2 du.
The shape-Brownian time-change and strong Markov give
Prob{ Q_j > d^2 e^{sqrt(h_2)} } <= C e^{-sqrt(h_2)/2}.
(After first hit the point is at distance d from its nearest wall;
1d reflection gives survival time >t at most C d/sqrt(t); set
t=d^2e^{sqrt(h_2)}. If the tested row is never reached before
tau_L, Q_j=0.)
For every fixed nonempty J define
T_J = max_{j in J} (sigma_j wedge tau_L),
a valid stopping time (finite max, <=tau_L). On the event {J = actual
deep class}, each j in J reaches history 3K; since k<3K, every
sigma_j occurs strictly before tau_L, so T_J is an actual last first
hit. On the good geometry/inverse/clock event and the history-gap
classification,
int_{T_J}^{tau_L} w_max^2 du
<= e^{C h_2} sum_{j in J} int_{sigma_j}^{tau_L}|s_j'|^2 du
<= d^2 e^{C' h_2}.
(Constants absorb the fixed site count and the squaring of the
exponential factor.)
Displacement with fixed candidates. Fix J nonempty and i in J. At
time T_J, center a disk of radius r_rob at the F_{T_J}-measurable
point s_i(y_{T_J}); let eta_{J,i} be its first exit after T_J, capped
at tau_L. Consider the event that J is the actual deep class and i is
an actual last site (so sigma_i=T_J). Exclude the first-hit failures
from 11.6 (probability o(1) after finite union over sites). On the
event under consideration, |a_i|>=h_1 up to eta_{J,i}, so by (a) and
the w_max clock bound,
int_{T_J}^{eta_{J,i}} |s_i'|^2 du
<= d^2 exp(C'' h_2 - 2 h_1) =: q_K.
Apply the maximal inequality to the actual complex displacement
martingale s_i(y_u)-s_i(y_{T_J}) stopped at eta_{J,i} and at scalar
quadratic time 2q_K. The expected squared terminal displacement is at
most C q_K. On the good event, the additional clock cut does not
precede eta_{J,i}, so disk-exit probability on that event is at most
C q_K / r_rob^2 <= C exp(C'' h_2 - (3/2) h_1) = o(1).
This is unconditional for each fixed (J,i); the classification,
last-hit, and robustness events are intersected afterward. Ties
between last sites are included in the same finite union.
Disk exit is compulsory. An actual last site at time T_J=sigma_i has
depth d; on {J = actual deep class}, each j in J (including i)
subsequently attains depth <=e^{-3K} by tau_L. Attainment cannot
precede T_J because reaching depth <=e^{-K}=d would already cross
the tested row. Depth is 1-Lipschitz, so the actual displacement
before tau_L is >=d-e^{-3K}>=d/2>r_rob for large K; the disk must
exit. But the finite union of "escape" events has probability o(1),
whereas the event {history-gap} (probability >=0.9-o(1)) plus the
already excluded failure sets gives probability >=0.9-o(1).
Contradiction: the finite critical wall is impossible.
PROOF OF THE MAIN THEOREM (chapter 1). The physical caustic-collar
hypothesis gives an analytic boundary (the analyticity theorem of
chapter 2), then the even joint parametrization and the nonzero area
coefficient (chapter 3). The continuation and no-patch theorems (chapter 4) produce
the maximal shape strip D_H, with the natural-boundary conclusion
when H<infty.
If the meromorphic projective normal a=e^{i phi} has a zero or pole
inside this strip, the isotropic classification theorem of chapter 5
already identifies the boundary as an ellipse. In the remaining
nonisotropic case, the shape-lift theorem and the zero-separation lemma (chapter 5) give
the holomorphic shape fields f and q=f''/f and the intrinsic impact
lift. Constant f gives a circle. For nonconstant f:
- the infinite growth exclusion theorem (chapter 7) excludes H=infty and
infinite upper growth order at a finite wall.
- the subcritical exclusion proposition (chapter 6) excludes upper growth
order below one; the finite superlinear exclusion theorem (chapter 6)
excludes every finite upper growth order above one.
- The remaining finite critical case: the critical moment bootstrap
(chapter 8), the no-prohibited-sequence theorem (chapter 9), and the odd-slice
construction with the sign test (chapters 10-11) contradict
the last-site theorem.
So only an elliptic boundary (circles included) remains. Convexity
identifies the domain with the interior of that ellipse. After
translation it is described by a positive definite quadratic form,
as asserted.