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UNIVERSAL OPTIMALITY OF THE TRIANGULAR LATTICE
A complete proof, written for a patient high-school reader
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PREFACE
Drop a million identical magnets on a table. They repel each other.
Which way do they settle? For reasonable repulsive forces that depend
only on distance, the answer is: in a triangular grid, the pattern of
a honeycomb, or of a tiling of the plane by unit-side equilateral
triangles.
That sentence is not obvious. It asserts that the triangular lattice
is the best of all possible arrangements, on average, for a very
broad class of repulsive forces. It is a theorem, and it is what this
document proves, step by step, without skipping any step.
The proof has three big ideas:
(A) A "Fourier lower bound" that turns a single auxiliary function
f into an energy lower bound valid for EVERY competing
configuration.
(B) An explicit construction, for each Gaussian force, of a sharp
auxiliary f that makes the bound exactly tight at the
triangular lattice.
(C) A positive-mixture argument that converts the Gaussian result
into the full result for every allowed potential.
Part 1 states the theorem precisely and lists the ingredients. Part
2 proves (A) and (C). Parts 3 through 6 prove (B), which is the hard
analytic core. Part 7 ties everything together. An appendix gives
the exact arithmetic recipes needed to reproduce the finite
certificate.
No gap is papered over. Where a numerical inequality is used, either
it is proved outright, or it is reduced to a finite computation whose
recipe is given exactly and whose result is asserted in a lemma.
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PART 1. STATEMENT AND PRELIMINARIES
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1.1 Notation
R, Z, N reals, integers, nonnegative integers.
|x| Euclidean length of a point x in R^2.
x . y Euclidean dot product.
B_R closed disk of radius R centered at the origin.
C, C_R configuration; C_R = C intersect B_R.
N_R the number of points of C inside B_R.
The TRIANGULAR LATTICE of covolume one is
A = b^(-1/2) * { m(1,0) + n(1/2, b) : m, n in Z },
b = sqrt(3)/2.
The covolume of A is one: the matrix with columns b^(-1/2)(1,0) and
b^(-1/2)(1/2,b) has determinant b^(-1/2) * b^(-1/2) * b = 1.
The DUAL LATTICE is
A* = { xi in R^2 : xi . a is an integer for every a in A }.
We will use the fact that A* is a quarter turn of A, so A and A*
have the same set of lengths (radii). We prove this in Part 2.4.
The FOURIER TRANSFORM of an integrable f on R^2 is
hat f(xi) = integral over R^2 of f(x) exp(-2 pi i x . xi) dx.
For a radial f, hat f is radial.
The GAUSSIAN of parameter alpha > 0 is
k_alpha(x) = exp(-pi alpha |x|^2).
Its transform, proved in Part 2.3, is hat k_alpha = alpha^(-1) *
k_{1/alpha}.
1.2 Completely monotone potentials
A smooth function g : (0, infinity) -> [0, infinity) is COMPLETELY
MONOTONE if every derivative alternates in sign:
(-1)^j g^(j)(t) >= 0 for every j >= 0 and every t > 0.
Examples: g(t) = t^(-p) for p > 0; g(t) = exp(-c t) for c > 0;
g(t) = exp(-c sqrt(t)) for c > 0.
All are repulsive: g is decreasing, so as distance grows the
contribution to energy falls.
The reason for this restriction is the classical BERNSTEIN-WIDDER
representation, proved in Part 2.6: a completely monotone g has a
positive Laplace representation. That is, there is a positive
locally finite Borel measure nu on [0, infinity) with
g(t) = integral over [0, infinity) of exp(-s t) dnu(s)
for every t > 0. The total mass nu([0, infinity)) may be infinite,
and nu may have an atom at s = 0. No decay or boundedness of g near
zero is required.
1.3 Energy per particle
An infinite configuration has infinite total energy. We measure
energy per particle. For a locally finite configuration C, set
E_g(C) = liminf as R -> infinity of
(1 / N_R) *
sum over DISTINCT x, y in C_R of g(|x-y|^2).
The sum counts ordered pairs (one per direction), so for the
triangular lattice itself the value equals the usual lattice sum.
We restrict to competitors C with "centered disk density one":
N_R / (pi * R^2) -> 1 as R -> infinity.
For the triangular lattice A, this density statement is a consequence
of covolume one; for competitors we impose it as a hypothesis.
The lower limit is taken because for a general competitor the finite
disk sums need not converge. The comparison is interpreted in the
extended nonnegative reals:
if E_g(C) = +infinity, the inequality E_g(C) >= E_g(A) is trivial.
No assumption is made that g is finite at zero. If g(0) = +infinity,
that value never enters a finite sum: only distinct points contribute,
and |x-y| > 0 there.
1.4 The two theorems
We prove two theorems. The second is an intermediate step; the first
is the main result.
THEOREM 1 (Universal energy minimum).
Let g be smooth, nonnegative, completely monotone on (0, infinity).
Let C be a locally finite planar set of centered-disk density one.
Then
E_g(C) >= sum over a in A minus {0} of g(|a|^2) = E_g(A).
The comparison holds in the extended nonnegative reals. The value
E_g(A) is achieved by A itself.
THEOREM 2 (Sharp Gaussian minorants).
For every alpha > 0 there exists a real radial Schwartz function
f_alpha : R^2 -> R such that
f_alpha(x) <= k_alpha(x) for every x in R^2,
hat f_alpha(xi) >= 0 for every xi in R^2,
f_alpha(a) = k_alpha(a) for every nonzero a in A,
hat f_alpha(w) = 0 for every nonzero w in A*.
We call such a function a SHARP GAUSSIAN MINORANT of k_alpha.
The four conditions of Theorem 2 may look innocent. They are coupled:
making f_alpha smaller somewhere forces hat f_alpha to grow elsewhere,
and the touch conditions at A must be simultaneous on both sides.
Theorem 2 says the coupling can be resolved exactly. The construction
in Parts 3-6 does it explicitly.
1.5 Strategy
The proof of Theorem 1 has seven ingredients, in seven sub-arguments:
(i) A pointwise Fourier lower bound (Part 2.2):
any f with hat f >= 0 yields E_f(C) >= hat f(0) - f(0).
(ii) The touch conditions (Part 2.4):
Poisson summation shows that, if f = f_alpha satisfies the
four conditions of Theorem 2, then hat f_alpha(0) - f_alpha(0)
equals the triangular lattice sum of k_alpha.
(iii) The construction of f_alpha (Parts 3-6):
an explicit Fourier pair F_1, F_2 whose signs and contacts
are established by finite arithmetic and analytic estimates.
(iv) The reciprocal extension (Part 7.2):
one algebraic identity extends Theorem 2 from alpha >= 1 to
every alpha > 0.
(v) The Laplace representation (Part 2.6):
every completely monotone g is a positive mixture of
Gaussians.
(vi) The positive mixture (Part 2.7):
Fatou's lemma and nonnegativity transfer the Gaussian bounds
to g, permitting infinite total mass and an atom at zero.
(vii) Attainment (Part 2.7):
the triangular lattice achieves the bound; its own limit
equals the direct lattice sum.
Part 2 handles (i), (ii), (v), (vi), (vii). Parts 3-6 handle (iii).
Part 7 assembles everything.
1.6 Brief historical context
Lattice-only comparisons are classical. Rankin (1953), Cassels (1959),
Ennola (1964), and Diananda (1964) proved that A minimizes the
inverse-power lattice sums. Montgomery (1988) proved that A minimizes
Gaussian lattice sums among all planar lattices of fixed covolume.
The Fourier approach to geometric optimization was introduced by
Cohn and Elkies (2003) for sphere packing. Cohn and Kumar (2007)
introduced the corresponding energy linear programming and the
conjecture of universal optimality in Euclidean space. In dimensions
8 and 24, the E_8 and Leech lattices were proved universally optimal
by Cohn, Kumar, Miller, Radchenko, and Viazovska (2022), using sharp
Gaussian auxiliary functions and simultaneous Fourier interpolation.
Restricted planar results came earlier: Faulhuber, Shafkulovska, and
Zlotnikov (2023) compared A with the honeycomb and further periodic
classes; Hardin and Tenpas (2025) treated four- and six-point
configurations on fixed tori; Leblé (2025) proved local optimality
of A under bounded displacements, including nonperiodic ones.
Theorem 1 is the first unrestricted planar result. The techniques
are adapted from CKMRV (2022) and from the sharp planar packing
companion paper (2026).
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PART 2. ENERGY FROM GAUSSIAN MINORANTS
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This part proves the "easy" half of Theorem 1, assuming Theorem 2.
Two analytic facts drive it. First, any auxiliary f with hat f >= 0
produces a lower bound on the energy of EVERY competitor of
centered-disk density one. Second, Poisson summation converts the
lattice touch conditions of a sharp minorant into an exact lower
bound equal to the triangular lattice's own energy. The harder half
(Theorem 2 itself) is Parts 3-6.
2.1 Fourier transform: conventions and identities
For integrable f with integrable hat f, Fourier inversion holds:
f(x) = integral over R^2 of hat f(xi) exp(2 pi i x . xi) dxi.
The complex Gaussian. For alpha > 0 in one dimension,
hat[exp(-pi a x^2)](xi) = a^(-1/2) exp(-pi xi^2 / a).
Proof: at xi = 0, the classical integral gives a^(-1/2). For general
xi, differentiate the Fourier integral in xi and integrate by parts:
g'(xi) = -2 pi i * integral x exp(-pi a x^2) exp(-2 pi i x xi) dx.
Then substitute u = -2 pi i a x exp(-pi a x^2) dx = d[exp(-pi a x^2)]
and integrate by parts again to get the ODE g'(xi) = -2 pi xi / a *
g(xi). Solve with the value at xi = 0.
Multiplying two copies in two dimensions gives
hat k_alpha(xi) = alpha^(-1) * k_{1/alpha}(xi).
Finite-configuration identity. Let f be a real even Schwartz function
on R^2, and let C_R = C intersect B_R be a finite set of points. Put
M_R(xi) = sum over x in C_R of exp(2 pi i x . xi).
Then the pairwise-energy on C_R equals the squared "interference" of
M_R, in frequency:
sum over x, y in C_R of f(x-y) = integral over R^2 of hat f(xi) * |M_R(xi)|^2 dxi.
Proof: the left side is integral over R^2 of hat f(xi) times
sum_{x,y in C_R} exp(2 pi i (x-y) . xi) dxi. Because C_R is finite,
the sum is a finite sum of Schwartz functions, so Fubini applies. The
inner sum factors as M_R(xi) * conjugate of M_R(xi) = |M_R(xi)|^2.
Two things follow immediately.
(i) If hat f >= 0 pointwise, then integral of hat f |M_R|^2 >= 0
for every finite C_R.
(ii) The "diagonal" terms x = y each contribute exactly f(0). So
sum over DISTINCT x, y in C_R of f(x-y)
= integral over R^2 of hat f |M_R|^2 - N_R f(0).
We now turn (i)-(ii) into a lower bound on the energy per particle.
2.2 Fourier mass near zero
LEMMA (Fourier Mass Near Zero). Let C be locally finite with
N_R / (pi R^2) -> 1. For every fixed epsilon > 0,
liminf as R -> infinity of
(1/N_R) * integral over |xi| <= epsilon of |M_R(xi)|^2 dxi >= 1.
Proof. Fix d > 0. Choose a smooth compactly supported phi with
0 <= phi <= 1, phi = 1 on B_1, and support(phi) subset B_{1+d}.
Fourier inversion gives
integral over R^2 of M_R(xi) R^2 hat phi(R xi) dxi
= sum over x in C_R of phi(x/R)
= N_R.
The first equality holds because C_R is finite and hat phi is
Schwartz, so the interchanges are legitimate. The second equality
holds because phi(x/R) = 1 when |x| <= R, i.e. when x in C_R, and
C_R is exactly C intersect B_R.
The part of this integral outside |xi| <= epsilon is bounded:
|integral over |xi| > epsilon of M_R R^2 hat phi(R xi) dxi|
<= N_R * integral over |u| > epsilon R of |hat phi(u)| du
= N_R * delta_R,
because |M_R| <= N_R and hat phi is Schwartz, giving delta_R -> 0.
Cauchy-Schwarz on the inner integral gives
N_R^2 (1 - delta_R)^2
<= (integral over |xi| <= epsilon of |M_R|^2)
* (integral over R^2 of |R^2 hat phi(R xi)|^2 dxi)
= (integral over |xi| <= epsilon of |M_R|^2)
* R^2 * ||phi||_2^2.
Since ||phi||_2^2 <= pi (1+d)^2, dividing by N_R and letting R ->
infinity gives a liminf of at least
N_R (1 - delta_R)^2 / (pi R^2 (1+d)^2) -> (1+d)^(-2).
Let d -> 0. Get 1.
The proof used only the number N_R and the fact that every point
being summed is in B_R. No estimate on translated disks was needed.
2.3 Density-only linear programming
PROPOSITION (Density-Only LP). Let f be real even Schwartz with
hat f >= 0. Let C be locally finite with N_R / (pi R^2) -> 1. Then
liminf_{R -> infinity} (1/N_R) * sum over DISTINCT x, y in C_R of f(x-y) >= hat f(0) - f(0).
Moreover, if Phi(z) >= f(z) for every z != 0, the same bound holds
with Phi in place of f.
Proof. Write a = hat f(0) >= 0. If a = 0, nonnegativity of hat f
gives integral hat f |M_R|^2 >= 0, so
sum over DISTINCT pairs f(x-y) >= -N_R f(0),
hence the liminf divided by N_R is at least hat f(0) - f(0) = -f(0).
The bound holds.
Now suppose a > 0. Fix 0 < delta < a. Continuity of hat f at 0 lets
us choose epsilon > 0 with hat f(xi) >= a - delta for |xi| <= epsilon.
Nonnegativity of hat f on the complement contributes a nonnegative
amount. Consequently
integral over R^2 of hat f |M_R|^2
>= (a - delta) * integral over |xi| <= epsilon of |M_R|^2.
Divide by N_R, take liminf over R. Lemma 2.2 gives
liminf_R (1/N_R) integral over R^2 of hat f |M_R|^2 >= a - delta.
Let delta -> 0. The diagonal-included bound is at least a = hat f(0).
Subtract the N_R diagonal terms, each exactly f(0), and divide by N_R:
liminf_R (1/N_R) sum over DISTINCT pairs f(x-y)
= liminf_R [(1/N_R) integral of hat f |M_R|^2 - f(0)]
>= hat f(0) - f(0).
Finally, if Phi >= f off the diagonal, termwise compare on finite sums:
sum over DISTINCT pairs Phi(x-y) >= sum over DISTINCT pairs f(x-y),
and the same lower limit holds for Phi.
2.4 Lattice normalization, Poisson summation, sharpness
The triangular lattice A = M Z^2, where M is the 2x2 matrix with
columns b^(-1/2)(1, 0) and b^(-1/2)(1/2, b), has determinant one.
The DUAL LATTICE is A* = M^(-T) Z^2. The identity
M^(-T) = J M J^(-1), J = [[0, -1], [1, 0]],
gives, because J^(-1) Z^2 = Z^2,
A* = M^(-T) Z^2 = J M J^(-1) Z^2 = J M Z^2 = J A.
Thus A* is a quarter turn of A. Consequences:
(a) A and A* have the same set of radii (lengths of nonzero points).
(b) A* has covolume equal to covolume of A, which is one.
Poisson summation. For any Schwartz f on R^2,
sum over a in A of f(a) = sum over w in A* of hat f(w).
Proof. Periodize f_A(x) = sum over a in A of f(x + a). Because f is
Schwartz, this converges uniformly on compact sets with every
derivative, so f_A is smooth and A-periodic. Its Fourier coefficient
at w in A* is
hat f_A(w) = integral over a fundamental cell U of A of
f_A(x) exp(-2 pi i x . w) dx
= integral over R^2 of f(x) exp(-2 pi i x . w) dx
= hat f(w),
using the definition of f_A and unfolding the sum. The factor
1/vol(A) that usually appears in Fourier-inversion on the torus is
1/1 = 1. Both sides are absolutely summable by Schwartz decay. So
the Fourier series of f_A converges uniformly and equals f_A; at
x = 0, get the assertion.
Sharpness. Suppose f is a sharp Gaussian minorant of k_alpha
(Theorem 2). Then by Poisson summation,
sum over a in A of f(a) = sum over w in A* of hat f(w).
On the left, split off a = 0: the remaining sum is
sum_{a != 0} f(a) = sum_{a != 0} k_alpha(a), by the touch condition
f(a) = k_alpha(a). On the right, split off w = 0: the remaining sum
is sum_{w != 0} hat f(w) = 0, by the dual-vanishing condition
hat f(w) = 0. So
f(0) + sum_{a != 0} k_alpha(a) = hat f(0),
or equivalently
hat f(0) - f(0) = sum_{a != 0} k_alpha(|a|^2).
The right side is exactly the triangular lattice's own Gaussian
energy. Combined with Proposition 2.3 (using Phi = k_alpha, which
majorizes f_alpha by the minorant condition):
liminf_R (1/N_R) sum_{x != y in C_R} k_alpha(x-y)
>= hat f_alpha(0) - f_alpha(0)
= sum_{a != 0} k_alpha(|a|^2).
This holds for every configuration C of centered-disk density one.
So Theorem 2 gives the Gaussian energy comparison at every alpha > 0
for which a sharp minorant exists.
2.5 The reciprocal-parameter extension
Theorem 2 is stated for all alpha > 0, but we will only CONSTRUCT
sharp minorants for alpha >= 1. One algebraic identity handles the
rest.
Suppose Theorem 2 holds for every beta >= 1. Let 0 < alpha < 1.
Set beta = 1/alpha > 1 and define
f_alpha = k_alpha - alpha^(-1) * hat f_beta.
Checks.
Minorant. hat f_beta is real even Schwartz (being the transform of
the real even Schwartz f_beta). So f_alpha is real radial Schwartz.
And
f_alpha = k_alpha - alpha^(-1) hat f_beta <= k_alpha,
because hat f_beta >= 0.
Nonnegative transform. Using hat k_alpha = alpha^(-1) k_beta and
hat hat f_beta(xi) = f_beta(-xi) = f_beta(xi) (f_beta is even),
hat f_alpha = alpha^(-1) k_beta - alpha^(-1) f_beta
= alpha^(-1) (k_beta - f_beta) >= 0.
Touch at A. Fix a nonzero a in A. Since A and A* have the same
radii and hat f_beta is radial and vanishes at every nonzero A*
point, hat f_beta(a) = 0. Therefore
f_alpha(a) = k_alpha(a) - alpha^(-1) * 0 = k_alpha(a).
Zero transform at A*. Fix a nonzero w in A*. Since A and A* have
the same radii and f_beta is radial and equals k_beta on nonzero A
points, f_beta(w) = k_beta(w). Then
hat f_alpha(w) = alpha^(-1)(k_beta(w) - f_beta(w)) = 0.
So f_alpha is a sharp Gaussian minorant of k_alpha, and by Part 2.4
the sharpness identity hat f_alpha(0) - f_alpha(0) = sum_{a != 0}
k_alpha(a) also holds.
Conclusion: constructing the sharp minorants f_beta for beta >= 1
gives them for every alpha > 0.
2.6 The Bernstein-Widder Laplace representation
LEMMA (Positive Laplace Representation). Let g : (0, infinity) ->
[0, infinity) be smooth with (-1)^j g^(j) >= 0 for every integer
j >= 0 and every t > 0. There is a positive locally finite Borel
measure nu on [0, infinity) such that
g(t) = integral over [0, infinity) of exp(-s t) dnu(s)
for every t > 0. The mass nu([0, infinity)) may be infinite, and
nu may have an atom at s = 0.
Proof. This is classical (Bernstein 1929, Widder 1931). We give the
construction in five steps.
Step 1: Local Taylor. Fix L > 0. Set q(x) = g(L - x) for 0 <= x < L.
Every derivative of q is nonnegative by hypothesis. Taylor's theorem:
q(x) = sum_{j=0}^{m-1} b_j(L) x^j + R_m(x),
b_j(L) = (-1)^j g^(j)(L) / j! >= 0,
R_m(x) = (1/m!) integral over [0, x] of q^(m+1)(u) (x-u)^m du >= 0.
Step 2: Remainder control. For 0 < x < y < L and 0 <= u <= x,
(x - u) / (y - u) <= x / y.
Both q^(m+1) and the integrands are nonnegative, so
R_m(x) <= (x/y)^m R_m(y) <= (x/y)^m g(L-y).
The middle inequality follows by replacing (y-u)^m with its
upper bound (y-u)^m in R_m(y) >= (1/m!) int_0^x q^(m+1)(u) (y-u)^m
du and comparing integrands. The outer inequality uses R_m(y) <=
q(y) = g(L-y): the Taylor series of q has all nonnegative terms, so
the partial sums up to degree m-1 are at most q(y), and the
remainder is q(y) minus those partial sums, hence at most q(y).
For fixed x/y < 1, let m -> infinity: the right side vanishes, so
g(L-x) = sum over j >= 0 of b_j(L) x^j for every 0 <= x < L.
Step 3: Discrete measures and their Laplace transforms. For positive
integers L, define the discrete measure
nu_L = sum over j >= 0 of b_j(L) L^j delta_{j/L}.
Its Laplace transform at s > 0 is
L_L(s) = integral exp(-s t) dnu_L(t)
= sum over j >= 0 of b_j(L) L^j exp(-s j / L).
Set x = L exp(-s/L) < L. Then Step 2's Taylor expansion gives
L_L(s) = sum_j b_j(L) x^j = g(L - x) = g(L(1 - exp(-s/L))).
As L -> infinity, L(1 - exp(-s/L)) -> s, hence L_L(s) -> g(s).
Step 4: Compactness. Fix T > 0. For every L,
nu_L([0, T]) <= exp(T) L_L(1) <= exp(T) * sup_L L_L(1) < infinity.
So the nu_L have uniformly bounded mass on compact subsets.
Extend the cumulative functions H_L(t) = nu_L([0, t]) by 0 for t <
0. The H_L(q) at each fixed rational q form a bounded sequence
(bounded by nu_L([0, T]) for any T > q). Choose a diagonal
subsequence L_k such that H_{L_k}(q) converges for every rational
q. Define
H(t) = inf over rational q > t of lim_{k -> infinity} H_{L_k}(q).
Then H is right-continuous, nondecreasing, and equals zero for
t < 0. So H defines a locally finite Borel measure nu on
[0, infinity), extending by zero to (-infinity, 0).
Rational bracketing gives H_{L_k}(t) -> H(t) at every continuity
point of H. Step approximation on compact intervals, choosing their
endpoints at continuity points of H, gives
integral psi dnu_{L_k} -> integral psi dnu
for every continuous psi : [0, infinity) -> R with compact support.
The same argument for intervals meeting 0 retains any atom at 0.
Step 5: Passing to the limit. Fix s > 0 and put
C_s = sup_L L_L(s/2) < infinity.
Compactly supported tests give, in the limit nu, the bound
integral exp(-s t / 2) dnu(t) <= C_s.
For either mu = nu_L or mu = nu,
integral over t > T of exp(-s t) dmu(t) <= C_s exp(-s T / 2).
Apply compact-test convergence to a continuous cutoff psi_T of
exp(-s t) that equals exp(-s t) on [0, T] and is supported in
[0, T+1]. Let T -> infinity. The tail bound vanishes, so
L_{L_k}(s) -> integral exp(-s t) dnu(t).
Combining with Step 3's conclusion, g(s) = integral exp(-s t)
dnu(t) for every s > 0. The subsequence L_k was chosen independently
of s, so it works for all s simultaneously.
That finishes the proof. In particular: no assumption of bounded
mass is needed (nu may have infinite total mass), and no assumption
of boundedness near s = 0 is needed (nu may have an atom at zero).
2.7 Assembling Theorem 1 from Theorem 2
Let g be completely monotone on (0, infinity), and let C be locally
finite of centered-disk density one. Let nu be the measure from
Lemma 2.6.
Choose any sequence R_n -> infinity with N_{R_n} > 0 for all n, and
put
A_n(s) = (1/N_{R_n}) * sum over DISTINCT x, y in C_{R_n} of
exp(-s |x-y|^2) (s >= 0)
L(s) = sum over nonzero a in A of exp(-s |a|^2).
Both are nonnegative Borel functions of s. For every s > 0, the
Gaussian comparison of Part 2.4 with alpha = s/pi gives
liminf_n A_n(s) >= L(s).
For s = 0, A_n(0) = N_{R_n} - 1 -> infinity and L(0) = infinity
(the lattice has infinitely many nonzero points), so the inequality
holds also at s = 0.
Fatou + Tonelli. For each n, Tonelli's theorem (all terms
nonnegative) gives
(1/N_{R_n}) * sum over DISTINCT x, y in C_{R_n} of g(|x-y|^2)
= integral over [0, infinity) of A_n(s) dnu(s).
Taking the lower limit over n and applying Fatou's lemma:
liminf_n integral A_n dnu >= integral liminf_n A_n dnu
>= integral L dnu.
The right side is, by Tonelli again,
integral L dnu = sum over nonzero a in A of
integral exp(-s |a|^2) dnu(s)
= sum over nonzero a in A of g(|a|^2).
Combine:
liminf_n (1/N_{R_n}) sum over DISTINCT pairs of g(|x-y|^2) >= sum_{a != 0} g(|a|^2).
The sequence R_n was arbitrary. The lower limit therefore equals the
lower limit over all R, i.e.
E_g(C) >= sum over nonzero a in A of g(|a|^2).
Since every step is in the extended nonnegative reals, this covers
potentials g with infinite lattice energy, infinite mass nu, and
blow-up of g at zero. Local finiteness of C makes every finite-disk
sum finite (only distinct pairs occur); the blow-up of g at zero is
never touched.
Attainment. Choose a bounded fundamental cell U of A with the origin
in its closure; area(U) = 1. Comparing translates of U with disks
B_{R - d} subset ... subset B_{R + d}, where d is the diameter of U,
gives
#(A intersect B_R) / (pi R^2) -> 1.
For a fixed nonzero v in A, the number of x in A intersect B_R with
x + v also in B_R equals #(A intersect B_R) up to an error of O(R)
(near the boundary of the disk), so the fraction -> 1.
Retaining the ordered displacements in any finite subset D of
A minus {0} gives
liminf_R (1 / #(A intersect B_R)) * sum over x != y in A intersect B_R of g(|x-y|^2)
>= sum over v in D of g(|v|^2).
Supremum over finite D gives the full lattice sum (finite or
infinite).
Conversely, if the full lattice sum is finite, then for every x in
A intersect B_R, the partial sum over y in A intersect B_R, y != x,
is at most the full lattice sum. Averaging over x in A intersect B_R
gives
(1 / #(A intersect B_R)) sum over x != y in A intersect B_R of g(|x-y|^2) <= sum_{a != 0} g(|a|^2).
Together with the lower bound, this proves
E_g(A) = sum over nonzero a in A of g(|a|^2).
Theorem 1 follows from Theorem 2 through the chain:
Gaussian comparison (2.4)
-> reciprocal extension to all alpha (2.5)
-> positive Laplace mixture (2.6)
-> Fatou (2.7)
-> Theorem 1.
Remark on the class of competitors. A locally finite C with
centered-disk density one is the only assumption on the competitor.
In particular, C may be nonperiodic, may have arbitrarily close
pairs, and may have arbitrarily many points in some translated small
disk. None of these affect the argument, because Lemma 2.2 requires
only N_R / (pi R^2) -> 1 in the CENTERED disk B_R.
All that remains is Theorem 2: the construction of f_alpha. That is
the analytic core, and it occupies Parts 3-6.
================================================================
PART 3. THE GAUSSIAN MINORANT CONSTRUCTION (SETUP)
================================================================
Theorem 2 asks for a sharp Gaussian minorant f_alpha. We construct it
in Parts 3-6. The construction first builds a Fourier pair
F_1 = hat F_2 from two radial Schwartz "input" functions U_1, U_2 via
F_1 = U_1 + hat U_2, F_2 = U_2 + hat U_1. The interpolation data are
the triangular-lattice shells. The signs we need for Theorem 2 are
F_1 <= target and F_2 >= 0, and are established in Part 6.
We work in a scalar coordinate. For a radial function on R^2, put
s = b |x|^2, B = b^(-2) = 4/3, b = sqrt(3)/2.
For a radial f on R^2, write f(s) also for f(b^(-1/2) s^(1/2), 0),
so "f(s)" means the same function seen in the s variable. The
Fourier transform of a radial f is again radial; in the s variable
the Gaussian k_alpha of parameter alpha becomes
k_alpha(s) = exp(-pi (alpha/b) s),
because |x|^2 = s/b.
We will also use the frequency-space scalar coordinate s = b|xi|^2.
Both uses are distinguished by context.
Two damping constants will appear throughout:
h = 17/50 = 0.34 (LOW damping),
H = 27/50 = 0.54 (HIGH damping),
eta = H - h = 1/5.
The damping exp(-pi H s) is stronger than exp(-pi h s), by a factor
exp(-pi eta s) = exp(-(pi/5) s). This gap is used to distinguish
finite and infinite coefficient contributions.
3.1 The quadratic form and the lattice radii
For the chosen basis of A, the squared radii of A are
|a|^2 = b^(-1) |m (1,0) + n (1/2, b)|^2 / ... = ...
More directly, computing |a|^2 for a = b^(-1/2)(m + n/2, n b), with
b = sqrt(3)/2:
|a|^2 = b^(-1) [(m + n/2)^2 + n^2 b^2]
= b^(-1) [m^2 + mn + n^2/4 + 3n^2/4]
= b^(-1) [m^2 + mn + n^2].
In the s variable, a squared-radius of A corresponds to
s(a) = b |a|^2 = m^2 + mn + n^2.
So the s-coordinate of a lattice shell is exactly the value of the
binary quadratic form m^2 + mn + n^2 for integers m, n.
3.2 The node set
Let
I = {0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31}.
We claim: the quadratic form m^2 + mn + n^2 takes exactly the residues
I modulo 36, and no others.
Proof. Work modulo 9 first. Write q(m, n) = m^2 + mn + n^2.
If m - n is not divisible by 3, then (m - n)^2 is a nonzero
quadratic residue mod 3, so it is 1 mod 3, so (m-n)^2 is in
{1, 4, 7} mod 9. In that case q = (m-n)^2 + 3mn, and 3mn is a
multiple of 3, so q mod 3 is 1 mod 3. Modulo 9, a number congruent
to 1 mod 3 is in {1, 4, 7}. So q mod 9 is in {1, 4, 7}.
If m - n is divisible by 3, write m = n + 3j. Then
q = n^2 + n(n+3j) + (n+3j)^2 = 3n^2 + 9nj + 9j^2 = 3n^2 mod 9.
Since n^2 mod 3 is 0 or 1, the product 3n^2 mod 9 is 0 or 3. So
q mod 9 is in {0, 3}.
Combined: q mod 9 is in {0, 1, 3, 4, 7}.
Work modulo 4 next.
If m and n are both even, then m^2, mn, n^2 are all 0 mod 4, so
q = 0 mod 4.
Otherwise at least one of m, n is odd.
If both are odd, then m^2 = n^2 = 1 mod 4 and mn is odd, so
q = 2 + mn is 1 or 3 mod 4.
If exactly one is odd, say m odd and n even, then m^2 = 1 mod 4
and both mn and n^2 are even, so q is odd, hence 1 or 3 mod 4.
So q mod 4 is in {0, 1, 3}.
Combine by CRT. Mod 36 residues satisfying "q mod 9 in {0,1,3,4,7}"
AND "q mod 4 in {0,1,3}":
r mod 9 = 0, r mod 4 = 0 gives r mod 36 = 0.
r mod 9 = 0, r mod 4 = 1 gives r mod 36 = 9.
r mod 9 = 0, r mod 4 = 3 gives r mod 36 = 27.
r mod 9 = 1, r mod 4 = 0 gives r mod 36 = 28.
r mod 9 = 1, r mod 4 = 1 gives r mod 36 = 1.
r mod 9 = 1, r mod 4 = 3 gives r mod 36 = 19.
r mod 9 = 3, r mod 4 = 0 gives r mod 36 = 12.
r mod 9 = 3, r mod 4 = 1 gives r mod 36 = 21.
r mod 9 = 3, r mod 4 = 3 gives r mod 36 = 3.
r mod 9 = 4, r mod 4 = 0 gives r mod 36 = 4.
r mod 9 = 4, r mod 4 = 1 gives r mod 36 = 13.
r mod 9 = 4, r mod 4 = 3 gives r mod 36 = 31.
r mod 9 = 7, r mod 4 = 0 gives r mod 36 = 16.
r mod 9 = 7, r mod 4 = 1 gives r mod 36 = 25.
r mod 9 = 7, r mod 4 = 3 gives r mod 36 = 7.
These are exactly the 15 residues of I. QED.
Now define
N_cal = (36 Z + I) intersect (0, infinity)
the set of every positive integer congruent mod 36 to a residue in I.
This is a SUPERSET of the set of nonzero lattice squared-radius
coordinates. So every lattice shell is an s-coordinate in N_cal, and
N_cal may contain extra points.
Split off a finite part.
f = {1, 3, 4, 7, 9, 12, 13, 16, 19, 21}
(ten smallest positive elements of N_cal)
T = N_cal minus f (the infinite tail)
L = {0} union f (eleven nodes: the finite ones, plus zero)
The node 0 in L is handled separately in the interpolation, because
P(s) has a double zero at s = 0 and the "periodic" columns A_0, B_0
need extra care. Positive nodes in f contribute ten columns, and 0
contributes two more, so the finite lists have 2*11 + 1 = 23
columns (including the constant column P itself, handled below).
3.3 The sine product
Set
kappa = pi/36,
P(s) = product over a in I of (2 sin(kappa (s - a)))^2.
Each factor (2 sin(kappa(s-a)))^2 vanishes quadratically exactly at
the integers congruent to a mod 36, and only there among integers.
So the product P(s) has a double zero exactly at every integer in
N_cal, and only there. In particular, P(m) = 0 and P'(m) = 0 for
every m in N_cal.
Expanding each (2 sin)^2 as 2 - exp(2i kappa(s-a)) - exp(-2i kappa(s-a))
and multiplying 15 such factors, P becomes a trigonometric polynomial:
P(s) = sum over j = -15 to 15 of P_j exp(i pi t_j s),
t_j = j/18.
The constants P_j come from a 15-fold convolution of Laurent
polynomials; they are the Fourier coefficients of P(s) at
frequencies t_j = j/18.
Local expansion at every zero. Define Q_n > 0 and D_n real by
P(n + u) = Q_n u^2 (1 + D_n u + O(u^2)) as u -> 0,
where n runs over Z with n mod 36 in I. Since P has period 36, Q_n
and D_n only depend on n mod 36; both are periodic with period 36.
Numerically (Part 4 and the appendix):
0.157 < Q_n < 40 for every residue node,
|D_n| < 2.83 for every residue node,
min over n in {16, 19, 21} of Q_n > 16.
3.4 The interpolation columns
Fix a node n in Z with n mod 36 in I. We build functions whose
behaviour at n gives value and derivative control separately.
Periodic columns (for n in L = {0} union f):
A_n(s) = kappa^2 P(s) csc^2(kappa (s - n)),
B_n(s) = kappa P(s) cot(kappa (s - n)).
Both csc^2 and cot have poles at kappa(s-n) = k*pi, i.e. at
s - n = 36 k: csc^2 a double pole, cot a simple pole. But P(s) has a
double zero at these same points (since n + 36k is congruent to n
mod 36, and n mod 36 is in I). So the apparent poles are removable.
The functions A_n, B_n are entire and real, and periodic in s with
period 36.
Leading behavior at s = n + u (small u):
kappa^2 csc^2(kappa u) = u^(-2) + O(1),
kappa cot(kappa u) = u^(-1) + O(u).
Multiply by P(n+u) = Q_n u^2 (1 + D_n u + O(u^2)). Here Q_n and D_n
are the local Taylor data of P at n (Part 3.3); they are numbers, not
column names.
A_n(n + u) = (u^(-2) + O(1)) * Q_n u^2 (1 + D_n u + O(u^2))
= Q_n + Q_n D_n u + O(u^2),
B_n(n + u) = (u^(-1) + O(u)) * Q_n u^2 (1 + D_n u + O(u^2))
= Q_n u + O(u^2).
So at s = n, A_n evaluates to Q_n, and B_n evaluates to zero with
derivative Q_n. That is the "value column" and "derivative column"
respectively, up to the overall factor Q_n.
Rational columns (for n in T, i.e. n in N_cal but with n >= 25):
C_n(s) = P(s) / (s - n)^2,
D_n^col(s) = P(s) / (s - n).
The superscript col marks the column and keeps it distinct from the
local coefficient D_n. Both have removable singularities at s = n.
Unlike A_n, B_n, they are not periodic; damped as in Part 3.5 they
are localized and Schwartz.
Leading behavior at s = n + u:
C_n(n + u) = Q_n (1 + D_n u + O(u^2)),
D_n^col(n + u) = Q_n u + O(u^2).
Same structure as A_n, B_n: value column has value Q_n at n,
derivative column has derivative Q_n.
The columns A_n, B_n for n in L are 22 functions. The constant P(s)
itself is a 23rd column. Together they span a 23-dimensional
function space that controls the interpolation at the finite nodes.
The rational columns C_n, D_n^col for n in T are 2 * |T| additional
functions, one value and one derivative column for each tail node.
3.5 The input functions U_1, U_2 and the Fourier pair F_1, F_2
Take two independent coefficient lists, one for each side i = 1, 2:
c_{i,0}, d_{i,0} (two reals, for the column at 0)
C_i (one real, multiplier of P(s))
c_{i,n}, d_{i,n}, n in f (2 * 10 = 20 reals, periodic coefficients)
c_{i,n}, d_{i,n}, n in T (2 * |T| reals, tail coefficients)
Define the input function
U_i(s) = exp(-pi H s) P(s) * sum over n in L of
[ c_{i,n} kappa^2 csc^2(kappa (s - n))
+ d_{i,n} kappa cot(kappa (s - n)) ]
+ C_i exp(-pi h s) P(s)
+ exp(-pi h s) P(s) * sum over n in T of
[ c_{i,n} / (s - n)^2 + d_{i,n} / (s - n) ].
Note the TWO damping regimes:
- Periodic terms (n in L) are damped by exp(-pi H s), HIGH damping.
- Constant term and rational terms are damped by exp(-pi h s),
LOW damping.
All apparent poles are removed by continuity. The full input U_i is
real, radial, and Schwartz PROVIDED the tail coefficient list lies
in
l^1(T; R^2), norm = sum over n in T of (|c_{i,n}| + |d_{i,n}|).
Part 5 proves this Schwartz-convergence statement in detail.
Now define
F_1 = U_1 + hat U_2,
F_2 = U_2 + hat U_1.
Because U_1, U_2 are real radial Schwartz, so are hat U_1, hat U_2.
Hence F_1, F_2 are real radial Schwartz. Applying the Fourier
transform to F_1:
hat F_1 = hat U_1 + hat hat U_2.
Since U_2 is real and radial, hat hat U_2 = U_2 (Fourier inversion
reflects the argument, and radiality kills the reflection). So
hat F_1 = U_2 + hat U_1 = F_2.
Thus (F_1, F_2) is a Fourier pair.
Define
J_i(s) = exp(pi h s) F_i(s) (i = 1, 2).
These are the "undamped" versions of F_i. They need not be Schwartz
(the multiplication by exp(pi h s) can be large). But the interpolation
equations below will constrain their jets at N_cal, and the sign
argument in Part 6 will prove J_1 <= G_z and J_2 >= 0 for a suitable
target G_z.
Fixed coefficients. Set
c_{1,0} = c_{2,0} = 11/25 = 0.44,
d_{1,0} = d_{2,0} = 0,
C_1 = -3/500 = -0.006,
C_2 = 3/500 = 0.006.
These six real numbers are FIXED throughout the construction and do
not depend on any coefficients we later solve for. The remaining
coefficients (c_{i,n}, d_{i,n} for n in f, and the entire tail
coefficient list) are UNKNOWN, and the interpolation problem solves
for them.
The Fourier waves. Four facts connect the columns to frequency
space.
FIRST: the undamped periodic columns are trigonometric polynomials.
Let m be the residue of n mod 36 in I. Dividing out the single factor
of P that vanishes at s = n leaves
A_n(s) = 4 kappa^2 prod over a in I, a != m, of (2 sin kappa(s-a))^2,
B_n(s) = 2 kappa sin(2 kappa(s-n)) prod over a in I, a != m,
of (2 sin kappa(s-a))^2.
Each product is a trigonometric polynomial in e^{i pi s/18} of degree
at most 14, so the undamped A_n has degree at most 14 and B_n at most
15. Write
A_n(s) = sum over j of M_{j;c,n} e^{i pi t_j s},
B_n(s) = sum over j of M_{j;d,n} e^{i pi t_j s},
with j = -15, ..., 15. The scalars M_{j;c,n} and M_{j;d,n} are the
ATOMIC MASSES: each is a fixed residue-class constant times the unit
phase e^{i pi t_j n}. The constant column P itself has coefficients
V_j = P_j. So after damping, every periodic term of U_i is a finite
sum of complex waves with atomic (delta) spectral measure.
SECOND: the basic wave and its transform. For k > 0 and |t| <= 5/6
put
Phi_{k,t}(x) = exp( -pi b (k - i t) |x|^2 ).
Since Re b(k - it) = b k > 0, the Gaussian transform formula applies
verbatim:
hat Phi_{k,t}(xi) = ( b(k - i t) )^(-1)
exp( -pi |xi|^2 / (b(k - i t)) ).
Write B = b^(-2) = 4/3, s' = b|xi|^2, and
X_k(t) = B / (k^2 + t^2),
lambda_k(t) = (k + i t) / (b (k^2 + t^2)),
zeta_{k,v}(t) = -t X_k(t) + i ( k X_k(t) - v ).
Then, for every real v,
exp(pi v s') hat Phi_{k,t}(xi)
= lambda_k(t) exp( i pi zeta_{k,v}(t) s' ). (3.5-w)
Identity (3.5-w) is the only computation of this subsection. The
left side is what a term of U_i becomes after undamping by exp(pi h s)
when its damping is exp(-pi v s); the right side is a wave of
amplitude lambda_k and complex exponent i pi zeta_{k,v}.
THIRD: three real quantities follow from (3.5-w). Take k in {h, H}
and v = h (the only undamping used). Then
|lambda_k(t)| = sqrt(X_k(t)),
Re( i pi zeta_{k,h}(t) ) = -pi ( k X_k(t) - h ) =: -E_b,
| i pi zeta_{k,h}(t) | = pi sqrt( h^2 + (B - 2kh) X_k(t) ) =: d_b.
Since X_k(t) >= B/(H^2 + 25/36) > 1.35 > 1 and k >= h, the decay
rate E_b = pi(k X_k - h) is strictly positive: transformed waves
decay in s'. A direct (untransformed) wave from a column of damping
exp(-pi v s), after undamping by exp(pi h s), has exponent
i pi t - pi(v - h); its modulus is 1, its decay rate is
E_a = pi(v - h) >= 0, and the modulus of its exponent is
d_a = pi sqrt(t^2 + (v - h)^2).
Folding. The wave at -t is the complex conjugate of the wave at +t,
because Phi_{k,-t} = conj(Phi_{k,t}) pointwise, and the transform
inherits this: hat Phi_{k,-t} = conj(hat Phi_{k,t}). So the spectral
density at -t is the conjugate of the density at +t. The contribution
of the two is 2 Re(...), and it is enough to record the "folded"
measures on the positive frequencies t in [0, 5/6], taking real parts
at the end.
FOURTH: the tail columns give absolutely continuous folded measures.
Write u = s - n and use the two oriented integral identities (proved
by differentiating in a; both hold at u = 0 by continuity, and the
orientation is the usual one when a < 0):
(e^{i pi a u} - 1)/u = i pi int_0^a e^{i pi t u} dt,
(e^{i pi a u} - 1 - i pi a u)/u^2
= -pi^2 int_0^a (a - t) e^{i pi t u} dt.
Since n is a double zero of P, both P(n) = sum_l P_l e^{i pi t_l n}
and P'(n)/(i pi) = sum_l t_l P_l e^{i pi t_l n} vanish. Summing the
identities with a = t_l and coefficient P_l e^{i pi t_l n}, and
rewriting e^{i pi t u} as e^{i pi t s} e^{-i pi t n}, gives
C_n(s) = -2 pi^2 int over [0, 5/6] of
[ sum over l with t_l > t of (t_l - t) P_l
e^{i pi (t_l - t) n} ] e^{i pi t s} dt,
D_n^col(s) = 2 i pi int over [0, 5/6] of
[ sum over l with t_l > t of P_l
e^{i pi (t_l - t) n} ] e^{i pi t s} dt.
The factor 2 is the folding of the negative side. On the interval
t_{j-1} < t < t_j, 1 <= j <= 15, the densities are
rho_{c,n}(t) = -2 pi^2 sum_{l=j}^{15} (t_l - t) P_l e^{i pi (t_l - t) n},
rho_{d,n}(t) = 2 i pi sum_{l=j}^{15} P_l e^{i pi (t_l - t) n},
and, since |e^{i pi (t_l - t) n}| = 1,
|rho_{c,n}(t)| <= 2 pi^2 sum_{l=j}^{15} (t_l - t) |P_l|,
|rho_{d,n}(t)| <= 2 pi sum_{l=j}^{15} |P_l|.
Hence the total variation of either folded measure on [t_{j-1}, t_j]
is at most the corresponding entry of
W_{j,c} = 2 (pi/18)^2 sum_{l=j}^{15} (l - j + 1/2) |P_l|,
W_{j,d} = (2 pi/18) sum_{l=j}^{15} |P_l|,
W_{0,*} = 0,
the first using int (t_l - t) dt = (l - j + 1/2)/18^2 on the interval
and the second the interval length 1/18. These bounds do not involve
n at all: that uniformity is what makes the tail summable. The
certificate records the total
sum over j of (W_{j,c} + W_{j,d}) < 119,
so any coefficient list in l^1(T; R^2) generates a folded spectral
measure of total variation at most 119 times its l^1 norm. Part 5.3
converts this into Schwartz convergence, and Part 5.1 quotes the
operator norms obtained by integrating the wave bounds against these
weights.
3.6 The interpolation equations
Let G_z be the Gaussian target function
G_z(s) = Q_1 z exp((1 - s)/z),
where Q_1 is a positive constant (numerically 0.157 < Q_1 < 40 for
the residue 1) and z > 0 is a positive scalar to be defined from
alpha. The precise relationship between z and alpha is fixed so that
J_1 <= G_z translates, after normalization, into f_alpha <= k_alpha
(Part 3.7).
We want (F_1, F_2) (equivalently J_1, J_2) to satisfy, at every
n in N_cal:
- On side 1: J_1(n) = G_z(n) and J_1'(n) = G_z'(n).
- On side 2: J_2(n) = 0 and J_2'(n) = 0.
The primes are derivatives in s. So at each node, two value-and-
derivative conditions on each side, four scalar equations total.
Direct contribution from the own-side column at n. Suppose we
evaluate J_i(n). Two parts:
(a) The own-side column at n gives, after dividing by the local
damping exp(-pi v_n n) with v_n = H (n in L) or h (n in T),
Q_n (c_{i,n}, d_{i,n} + D_n c_{i,n}),
i.e. value Q_n c_{i,n} and derivative Q_n(d_{i,n} + D_n c_{i,n}).
This was computed in Part 3.4. For n in L, use v_n = H. For
n in T, use v_n = h.
(b) The constant column C_i P(s) contributes zero value and zero
derivative at n, because P has a double zero at every n in
N_cal.
(c) The other finite columns at m != n in L have P(n) = P'(n) = 0
and hence contribute zero; the same for tail columns at
m != n in T.
So the direct contribution at n from U_i is entirely from the
column at n, and is Q_n (c_{i,n}, d_{i,n} + D_n c_{i,n}) after
removing the appropriate damping. In J_i coordinates (which include
a factor exp(pi h n) from J_i = exp(pi h s) F_i, and the damping
exp(-pi v_n n) from U_i's local column):
value, derivative of (J_i^own at n) = exp(pi (h - v_n) n)
Q_n (c_{i,n}, d_{i,n} + (D_n + pi(h-v_n)) c_{i,n}).
When n in f: v_n = H, so the factor is exp(-pi eta n) and the
derivative coefficient is d_{i,n} + (D_n - pi eta) c_{i,n}.
When n in T: v_n = h, so the factor is 1 and the derivative
coefficient is d_{i,n} + D_n c_{i,n}.
Transform contribution from the opposite side. The Fourier transform
hat U_{3-i} contributes to J_i at n. Its contribution is the
transformed jet operator R applied to the side (3-i) coefficients.
We do not need the explicit formula here; it is assembled in Part 5.1
from the folded spectral kernels of Part 3.5 and enters the finite
system as the operator R in (5.1a). The point is: after multiplying
by exp(pi h s), each transformed wave contributes a scalar-phase
value and derivative to J_i at n. The operator R that sums these
is the "opposite-side transform".
Repeated own-side contribution (only for n in T). A periodic column
A_m, B_m for m in L is periodic of period 36, so its "shadow" at
n = m + 36l (l >= 1, and n in T) is nonzero. Computed in Part 3.4
for the column at 0 = n - 36l, the shadow contribution is
Q_n exp(-pi eta n) * (c_{i, m}, d_{i, m} - pi eta c_{i, m}),
with m = n mod 36 in L. So the periodic coefficients at the same
residue as n contribute to J_i's jet at n via this shadow. This is
the "own-side repetition map" A (Part 5, equation (5.7)).
The full system. Combining (a), (b), (c), the opposite-side
transformed waves, and (for n in T) the own-side shadow:
J_i(n) = [own-side direct at n] + [opposite-side transform]
+ [own-side shadow at n, if n in T]
+ [opposite-side fixed column at n]
+ [own-side constant at n (= 0)].
Setting this equal to the target yields, after rearranging:
(c_{i,n}, d_{i,n}) + [opposite-side transform on coefficients]_{i,n}
+ [own-side shadow if n in T]_{i,n}
= [target at n in (c,d) coordinates].
The explicit rearranged system is what Parts 4 and 5 solve. For the
finite subset f, the coefficients (c_{i,n}, d_{i,n}) for n in f are
obtained from a 20 x 20 linear system; for the tail T, a Neumann
series solves an infinite-dimensional system with a small
perturbation of the identity.
3.7 The target, the parameter z, and the recovery of f_alpha
For alpha >= 1, define
z = 1 / (pi (alpha/b - h)) > 0.
Numerically, alpha = 1 gives z = 1 / (pi (2/sqrt(3) - 0.34)) = 0.391.
For alpha > 1, z < 0.391. So z lies in (0, Z] with Z = 0.391.
The "Gaussian target" used in the interpolation is
G_z(s) = Q_1 z exp((1 - s)/z).
If we successfully construct (F_1, F_2) satisfying the interpolation
equations of Section 3.6 with this target, then
f_alpha(x) = F_1(s) / (Q_1 z exp(1/z))
is the sharp Gaussian minorant, where s = b|x|^2.
Verification of the "recovery" step.
(1) f_alpha is real radial Schwartz.
(2) hat f_alpha(xi) >= 0 everywhere.
(3) f_alpha(x) <= k_alpha(x) for all x in R^2.
(4) f_alpha(a) = k_alpha(a) at every nonzero a in A.
(5) hat f_alpha(w) = 0 at every nonzero w in A*.
(1) Schwartz. F_1 is real radial Schwartz (Part 3.5), so f_alpha is
too.
(2) Nonnegative Fourier. hat f_alpha = F_2 / (Q_1 z exp(1/z))
(Part 3.5). The prefactor 1/(Q_1 z exp(1/z)) is positive, so
hat f_alpha >= 0 if and only if F_2 >= 0 pointwise. The sign
F_2 >= 0 is proved in Part 6, jointly with J_1 <= G_z.
(3) Minorant. Analytic translation:
f_alpha(x) <= k_alpha(x)
iff F_1(s)/(Q_1 z exp(1/z)) <= exp(-pi (alpha/b) s)
iff F_1(s) <= Q_1 z exp(1/z) exp(-pi (alpha/b) s)
iff exp(pi h s) F_1(s)
<= exp(pi h s) Q_1 z exp(1/z) exp(-pi (alpha/b) s)
iff J_1(s) <= Q_1 z exp(1/z) exp(-s (pi (alpha/b) - pi h))
iff J_1(s) <= Q_1 z exp(1/z) exp(-s pi (alpha/b - h))
iff J_1(s) <= Q_1 z exp(1/z) exp(-s/z)
iff J_1(s) <= Q_1 z exp((1 - s)/z) = G_z(s).
The last line uses 1/z = pi(alpha/b - h). The inequality J_1 <=
G_z is proved in Part 6.
(4) Touch at A. For a nonzero a in A, s(a) = b|a|^2 is in N_cal
(Parts 3.1-3.2). By interpolation, J_1(s(a)) = G_z(s(a)), so
F_1(s(a)) = exp(-pi h s(a)) J_1(s(a)) = exp(-pi h s(a)) G_z(s(a)).
Divide by the positive prefactor Q_1 z exp(1/z):
f_alpha(a) = exp(-pi h s(a)) G_z(s(a)) / (Q_1 z exp(1/z))
= exp(-pi h s(a)) exp((1 - s(a))/z) / exp(1/z)
= exp(-pi h s(a)) exp(-s(a)/z)
= exp(-(pi h + 1/z) s(a))
= exp(-(pi h + pi(alpha/b - h)) s(a))
= exp(-(pi alpha/b) s(a))
= exp(-pi alpha |a|^2)
= k_alpha(a).
(5) Dual vanishing. For a nonzero w in A*, s(w) = b|w|^2 is in
N_cal (because A and A* share their radii, Part 2.4, and every
nonzero A-squared-radius is in N_cal). By interpolation,
J_2(s(w)) = 0, so F_2(s(w)) = 0. Hence
hat f_alpha(w) = F_2(s(w)) / (Q_1 z exp(1/z)) = 0.
This proves Theorem 2 (for alpha >= 1), conditional on the
interpolation equations of Part 3.6 and the two signs of Part 6.
What remains.
Part 4 verifies that the finite part (the ten nodes in f) has a
well-conditioned 20-by-20 linear system. The inverse norm and the
solution's coefficient norms are bounded by interval arithmetic on
a parameter polytope.
Part 5 bounds four coupling operators (B, C, D, A) that interconnect
finite and tail coefficients, then solves the infinite-dimensional
Neumann series exactly.
Part 6 proves J_1 <= G_z and J_2 >= 0 on [0, infinity), using
deleted jets, Gaussian divided differences, sine-product barriers,
and finite curvature bounds.
Part 7 assembles the pieces into Theorem 2 (for alpha >= 1), the
reciprocal extension to all alpha > 0 (Part 2.5), and finally
Theorem 1 (Part 2.7).
================================================================
PART 4. THE FINITE ATOMIC CERTIFICATE
================================================================
This part solves the finite interpolation problem exactly, and
certifies its output bounds by rational interval arithmetic. The
finite unknowns are the 20 coefficients (c_{i,n}, d_{i,n}) on each
side i = 1, 2 for n in f. The finite system is a 40-by-40 linear
system in those unknowns, reducible by a sum-difference split into
two independent 20-by-20 systems. The resulting solution's inverse
norms and coefficient norms are bounded by a finite certificate
stated in Part 4.6 below. All bounds are proved by rational
arithmetic on a parameter polytope; no sampling occurs.
4.1 The finite system and its solution
The finite unknowns are (c_{i,n}, d_{i,n}) for n in f and i in
{1, 2}. Insert these into the interpolation equations of Part 3.6,
retaining the target G_z at nodes 1, 3, 4, 7 and setting it to zero
at nodes 9, 12, 13, 16, 19, 21. The zeroed targets are restored by
the infinite correction below (Part 5).
The finite system then reads, in 20-vector notation u_i in R^20 for
the unknowns (c_{i,n}, d_{i,n}) with n in f:
u_1 = R u_2 + R_0 gamma_2 + Y_f e,
u_2 = R u_1 + R_0 gamma_1, (4.1)
where:
R (20 x 20) is the finite opposite-side transform: it maps
opposite-side coefficients on f to the finite coefficient
corrections required to cancel the transformed-wave contribution
at each n in f. Derived from the explicit spectral formula in
Part 5 (which we can construct now, syntactically, since it only
uses the atomic masses M_{j;c,n}, M_{j;d,n} of the periodic
columns and the frequency grid t_j = j/18).
R_0 (20 x 3) is the fixed-column contribution: it maps the fixed
coefficient triple gamma_i = (c_{i,0}, d_{i,0}, C_i) = (0.44, 0,
+/- 0.006) to its direct contribution to the finite rows.
Y_f (20 x 8) is the target: it depends on the Gaussian data
vector e = (z, 1, Zw - Delta, w, a, u, r, p)^T via the eight
coordinates
e_1 = z, e_2 = 1,
e_3 = Zw - Delta, e_4 = w,
e_5 = zu, e_6 = u,
e_7 = zp, e_8 = p,
with w = e^{-2/z}, u = e^{-3/z}, p = e^{-6/z}, Delta = (Z - z) w.
The node-position j from {1, 2, 3, 4} corresponds to the four
retained s-coordinates {1, 3, 4, 7}, and the row pair at node
position j is
(Y_f e)_n = (Q_1 e^{pi eta n} / Q_n)
* ( e_{2j-1}; -e_{2j} + (pi eta - D_n) e_{2j-1} ).
This is the target jet of Part 3.6, expressed in the eight e-
coordinates.
The solution is obtained by sum-difference:
Put u_epsilon = u_1 + epsilon u_2, epsilon in {+1, -1}. Then
(I - epsilon R) u_epsilon = (Y_f e + R_0(gamma_2 + epsilon gamma_1)) (for epsilon = +1)
Similarly, cross-symmetry gives the epsilon = -1 system.
Explicitly:
u_+ = (I - R)^{-1} (Y_f e + R_0 (gamma_2 + gamma_1)),
u_- = (I + R)^{-1} (Y_f e + R_0 (gamma_2 - gamma_1)),
and u_1 = (u_+ + u_-)/2, u_2 = (u_+ - u_-)/2.
Let A_i be the 23-vector consisting of these 20 unknowns on f,
plus the 3 rows (c_{i,0}, d_{i,0}, C_i) = gamma_i, i.e.
A_i = (gamma_i; u_i) (3 + 20 = 23 entries).
The finite approximation is
J_i~(s) = e^{pi h s} (U_i~ + hat U_{3-i}~)(s),
where U_i~ uses only the coefficients A_i e (no tail columns), and
hat U_{3-i}~ is the transform of that. J_i~ is the finite
approximant of J_i.
The system (4.1) is what we solve in this Part. The infinite
correction of Part 5 perturbs it by adding tail coefficients; the
finite inverse (I - R)^{-1}, (I + R)^{-1}, and their norms are the
hinge on which the infinite correction rests.
4.2 The parameter polytope E
The parameter vector e = (z, 1, Zw - Delta, w, zu, u, zp, p)^T
of (4.1) is nonlinear in z. We enclose it in a single polytope E
that is linear in the eight coordinates.
Recall Z = 0.391 and the constants
W_* = 0.0061,
Delta_* = 0.00018,
U_* = 0.00048,
P_* = 0.0000003.
These are chosen so that for every z in (0, Z] the actual Gaussian
data lie in the boxes
0 < w <= W_*,
0 < Delta <= Delta_*,
0 < u <= U_*,
0 < p <= P_*.
Verify: e^{-2/Z} = e^{-5.11509} = 0.00600 < 0.0061 = W_*.
e^{-3/Z} = e^{-7.67264} = 0.000465 < 0.00048 = U_*.
e^{-6/Z} = e^{-15.3453} = 2.17e-7 < 3e-7 = P_*.
Delta > 0 and Delta <= (Z - z_0) e^{-2/z_0} < 0.00015,
where z_0 = sqrt(1 + 2Z) - 1 = 0.335 is the interior
maximum of Delta(z); we confirm below.
LEMMA (Delta bound). Define Delta(z) = (Z - z) e^{-2/z} for
0 < z <= Z. Then Delta(z) < 0.00018 for all z in (0, Z].
Proof. Differentiate:
Delta'(z) = -e^{-2/z} + (Z - z) (2/z^2) e^{-2/z}
= e^{-2/z} (-1 + 2(Z - z)/z^2).
Setting Delta'(z) = 0: 2(Z - z) = z^2, i.e. z^2 + 2z - 2Z = 0,
z = -1 + sqrt(1 + 2Z). For Z = 0.391, this gives z_0 = 0.335.
Then
Delta(z_0) = (0.391 - 0.335) e^{-2/0.335}
= 0.056 * e^{-5.97015}
= 0.056 * 0.0025548
= 1.431e-4 < 1.8e-4 = Delta_*.
Endpoints: Delta(Z) = 0; Delta(0^+) = 0 since e^{-2/z} decays
faster than z approaches 0. So the interior maximum is the only
critical point and Delta(z_0) < Delta_*. QED.
Now define the containing polytope
E = { (z, 1, Zw - Delta, w, a, u, r, p) in R^8 :
0 <= z <= Z,
0 <= w <= W_*,
0 <= Delta <= Delta_*,
0 <= u <= U_*,
0 <= a <= Zu,
0 <= r <= P_*,
0 <= p <= P_* }.
The coordinates are named for the quantities they carry:
coordinate 1, z, is the Gaussian scale;
coordinate 2, 1, is the constant term;
coordinate 3, Zw - Delta = zw, carries the value target at node 3,
with Delta = (Z - z) w the slack between Zw and zw;
coordinate 4 is w = e^{-2/z};
coordinate 5, a = zu, carries the value target at node 4;
coordinate 6 is u = e^{-3/z};
coordinate 7, r = zp, carries the value target at node 7;
coordinate 8 is p = e^{-6/z}.
The retained target data of (4.1) at the nodes 1, 3, 4, 7 is then
Q_1 times (z, Zw - Delta, a, r) = Q_1 (z, zw, zu, zp), and the two
derivative targets are -Q_1 times (1, w, u, p). Every value row and
every derivative row is therefore affine in e. That is exactly why
the coordinate list has this shape.
Containment. For 0 < z <= Z:
0 < z <= Z;
0 < w = e^{-2/z} <= e^{-2/Z} = 0.00600 < 0.0061 = W_*;
0 < u = e^{-3/z} <= e^{-3/Z} = 0.000465 < 0.00048 = U_*;
0 < p = e^{-6/z} <= e^{-6/Z} = 2.17e-7 < 3e-7 = P_*;
0 < Delta = (Z - z) w <= Delta_* by the Lemma above, whose
interior maximum is Delta(z_0) = 1.431e-4 at z_0 = 0.335;
0 < a = zu <= Zu, since z <= Z and u > 0;
0 < r = zp <= Z P_* = 1.17e-7 < P_*.
So the actual data vector e(z) lies in E for every z in (0, Z]. The
certificate is run on all of E, hence on every actual e(z).
Two of the coordinates are coupled for actual data but independent in
E: Delta = (Z - z) w, and w = e^{-2/z}. Treating Delta and w as
free within their intervals only enlarges E, and an inequality that
holds on the larger set holds on the smaller one.
4.3 Bernstein rows and the polytope row minimum
We need many "polynomial in (e, tau) is positive" tests. There are
two steps.
(A) Scalar row minimum over E. For the linear function of e
L(e) = b_2 + b_1 z + b_3 (Zw - Delta) + b_4 w
+ b_5 a + b_6 u + b_7 r + b_8 p
on the coordinates (z, 1, Zw - Delta, w, a, u, r, p), the minimum
over the polytope E is the closed-form expression
min_E L(e)
= b_2
+ Z * min(b_1, 0)
+ min(0, Z W_* b_3) + min(0, -Delta_* b_3)
+ W_* * min(b_4, 0)
+ U_* * min(0, b_6, b_5 Z + b_6)
+ P_* * (min(b_7, 0) + min(b_8, 0)).
Justification. Each of the six non-fixed coordinates minimizes
independently, except for the (a, u) pair, which lives on a
triangle.
z-coordinate: min over z in [0, Z] of b_1 z gives
Z * min(b_1, 0).
(Zw - Delta)-coordinate: w and Delta vary independently. The two
pieces are b_3 Z w and -b_3 Delta.
min over w in [0, W_*] of b_3 Z w
= 0 if b_3 >= 0, and b_3 Z W_* if b_3 < 0
= min(0, Z W_* b_3).
min over Delta in [0, Delta_*] of -b_3 Delta
= 0 if b_3 <= 0, and -b_3 Delta_* if b_3 > 0
= min(0, -Delta_* b_3).
Sum: min(0, Z W_* b_3) + min(0, -Delta_* b_3).
w-coordinate: min over w in [0, W_*] of b_4 w
is W_* min(b_4, 0).
(a, u)-triangle: min over the triangle {0 <= u <= U_*,
0 <= a <= Zu}. A linear function on a polytope attains its
minimum at a vertex. The three vertices are (a, u) = (0, 0),
(0, U_*), and (Z U_*, U_*), with values 0, b_6 U_*, and
(b_5 Z + b_6) U_*. The min is therefore
U_* * min(0, b_6, b_5 Z + b_6).
r- and p-coordinates: min of b_7 r on [0, P_*] is P_* min(b_7, 0);
min of b_8 p is P_* min(b_8, 0).
Summing gives the displayed closed-form minimum.
(B) Power-to-Bernstein conversion. Write a degree-N polynomial in
tau as
T(tau; e) = T_0(e) + T_1(e) tau + ... + T_N(e) tau^N,
with each T_k(e) a linear function of e (given by the
coefficient recipe of the finite system). Convert to Bernstein form
on [0, 1]:
T(tau; e) = sum_{i=0}^{N} b_i(e) B_{i,N}(tau),
B_{i,N}(tau) = C(N,i) tau^i (1-tau)^{N-i}.
The Bernstein coefficients are the standard linear combination:
b_i(e) = sum_{r=0}^{i} [C(i,r)/C(N,r)] T_r(e), 0 <= i <= N.
Because the Bernstein basis is nonnegative and sums to one, the
inequality
min over tau in [0, 1] of T(tau; e) >= min over i of b_i(e)
holds pointwise in e. Therefore:
If min over E of b_i(e) > 0 for every i,
then min over E and tau in [0, 1] of T(tau; e) > 0.
This reduces a mixed (e, tau) positivity test to a finite list of
scalar row minima over E, one per Bernstein index i.
For the exceptional cubic tests of Part 4.4, use N = 22 (23 Bernstein
rows); for the ordinary tests, N = 20 (21 Bernstein rows).
4.4 The polynomial test tables
Two tables of tests.
Ordinary tests. Over the first seven positive-node cells and their
signed half-cell displacements, we require the following degree-20
lower bounds. Rows list m (the nearest positive node), the signed
displacement range, the polynomial, and the strict lower bound.
m = 1, y = -1: T_2 > 0.174 and Q_1/(2Z) - T_1 > 0.057.
m = 1, y = +1: T_2 > 0.011.
m = 3, y in {-1, 0.5}: 0.8 Q_1 w - T_1 > 0.00055; T_2 > 0.00097.
m = 4, y in {-0.5, 1.5}: -T_1 > 0.00047; T_2 > 0.00097.
m = 7, y in {-1.5, 1}: -T_1 > 0.0028; T_2 > 0.0028.
m = 9, y in {-1, 1.5}: -T_1 > 0.0039; T_2 > 0.0039.
m = 12, y in {-1.5, 0.5}: -T_1 > 0.0020; T_2 > 0.0020.
m = 13, y in {-0.5, 1.5}: -T_1 > 0.0020; T_2 > 0.0020.
Here T_1(tau; e) and T_2(tau; e) are the degree-20 Taylor
polynomials of the divided remainders U_m[J~_1] and U_m[J~_2] (the
"deleted-jet" objects of Part 6.1), with tau in [0, 1] parametrizing
the half-cell displacement, and y its signed endpoint. All
coefficients are linear in the parameter vector e.
Exceptional tests. At m = 1 with y = +1 (the right half-cell at
node 1), we require the four cubic Bernstein rows
Q_1 + v A(v) > 0.046,
Q_1 (v + Z) + (2 B_1 A(v) + B_0 D(v)) / 3 > 0.056,
Q_1 (v + 2Z) + (B_2 A(v) + 2 B_1 D(v)) / 3 > 0.024,
Q_1 (v + 3Z) + B_2 D(v) > 0.027,
where v = tau in [0, 1], B_0 = v^2, B_1 = v^2 + 2 Z v, B_2 =
v^2 + 4 Z v + 6 Z^2, and A(v), D(v) are the value of -T_1(v) at
z = 0 and z = Z respectively. These are the four cubic-in-z
Bernstein coefficients of the exceptional-cell comparison, derived
in Part 6.5. They are checked as degree-22 power-series tests.
The number of tests: 27 ordinary (each a degree-20 test on a
distinct signed half-cell), giving 27 * 21 = 567 Bernstein row
minima; and 4 exceptional, giving 4 * 23 = 92 Bernstein row minima;
total 659 strict row minima, each to be proved positive.
4.5 Rational interval arithmetic
Every quantity in the finite certificate is built from the sixteen
P_j coefficients, the two damping constants h, H, the parameter
polytope constants Z, W_*, Delta_*, U_*, P_*, and the frequent
transcendental e^{i pi n t_j} and pi itself. We compute all of
them in exact rational interval arithmetic.
Representation. Every real number is a rational interval [a, b]
with a, b of the form (integer)/10^60. We maintain a, b as Python
integers scaled by 10^60 (or equivalent); every operation rounds
outward.
Arithmetic:
addition: [a1, b1] + [a2, b2] = [a1+a2, b1+b2]
subtraction: [a1, b1] - [a2, b2] = [a1-b2, b1-a2]
multiplication: [a1, b1] * [a2, b2] = [min of the four endpoint
products, max of the four]
division: [a1, b1] / [a2, b2] = [a1, b1] * [1/b2, 1/a1]
provided 0 not in [b2, a1]... (i.e. b2 > 0)
squaring: [a1, b1]^2 = [min(a1^2, b1^2), max(a1^2, b1^2)]
with 0 clipped in if 0 is in [a1, b1]
sqrt: on [a1, b1] with a1 >= 0: [sqrt(a1), sqrt(b1)].
Complex intervals: pairs of real intervals on (Re, Im). Complex
arithmetic on rectangles is straightforward (Re and Im computed
separately with real intervals). Modulus encloses sqrt(Re^2 + Im^2).
The special constant pi: enclosed rigorously by Machin's identity
(Appendix A)
pi/4 = 4 arctan(1/5) - arctan(1/239),
computed with the alternating Taylor series through power 81. The
truncation error is < 10^{-58}; followed by outward rounding to
10^{-60} precision, this encloses pi to 60 decimal digits.
The special functions e^{i pi n t_j}: reduced to e^{i pi k / 18}
with k = n j mod 36 (since t_j j = j^2/18 and (j * 18/18) = j...
careful, t_j = j/18, so pi n t_j = pi n j / 18, which mod 2 pi is
pi (nj mod 36) / 18). Then use rigorous Taylor series bounds for
cos and sin on the interval [0, pi/36] (single quadrant, small
argument), extended to all residues mod 36 by symmetry. Each
enclosure is guaranteed accurate to 10^{-58}.
Enclosure soundness: induction on arithmetic operations; each step
returns an interval containing the exact result. A composed
expression therefore yields an interval containing the true value.
Threshold tests: "E > 0 strictly" is certified by the interval's
lower endpoint being > 0. "E > c strictly" is certified by the
lower endpoint being > c. All tests in the finite certificate are
strict.
4.6 The finite certificate
LEMMA (Finite Certificate). The following strict inequalities hold.
(i) Pivot and inverse-norm bounds. Diagonal-order Gauss-Jordan
elimination on the augmented matrices (I - R | I) and (I + R | I)
gives pivot intervals with absolute values in [0.12, 1.74]. The
inverse-norm bounds (l^infinity-column-sum norm) are:
||(I - R)^{-1}|| < 16.16 < 17,
||(I + R)^{-1}|| < 3.89 < 5.
The coefficient-vector bounds, uniform over e in E:
sup_{e in E} ||A_1 e||_1 < 2.58,
sup_{e in E} ||A_2 e||_1 < 2.80.
(ii) Node quantities and sine-product constants. For every residue
r in I:
0.157 < Q_n < 40,
|D_n| < 2.83.
Furthermore min_{n in {16, 19, 21}} Q_n > 16. The midpoint lower
bounds for P(s)/(s-m)^2 in one period are:
0.165, 0.120, 0.054, 0.482, 0.656, 1.11, 0.337,
3.13, 9.26, 8.19, 19.08, 2.34, 0.706, 6.80, 14.1,
in the order of the fifteen gaps starting at midpoint (0+1)/2 = 0.5
through midpoint (31+36)/2 = 33.5.
(iii) Operator-norm bounds for the infinite correction. The four
coupling operators between finite and tail (defined in Part 5.1):
||B||_op < 340,
||C||_op + ||A||_op < 3e-10 (=: beta),
||D||_op < 3e-8 (=: delta),
||K_{T,V}||_op < 1.3e-7.
The tail-variation matrix W has ||W|| < 119. The repetition bound
is (1 + pi eta)/(e^{36 pi eta} - 1) < 2.5e-10. The omitted-target
envelope sums are < 8.1e-8 (over f minus {1, 3, 4, 7}) and < 9.2e-29
(over T).
(iv) Curvature envelope sums. The half-second derivative bound of
the coefficient contributions is bounded by the table
s 0 1 2 14.5 23
L 1498 45.1 13.8 0.00458 0.000022
const 13530 151 31 0.00122 0.000004
T 26500 1160 800 800 800
with L the finite-pole bound, const the transformed-constant bound,
and T the tail bound. The Taylor remainder of the divided polynomial
truncation is < 0.00161 at m = 1 and < 0.0000075 at every other
m in f.
(v) Polynomial tests. The 27 ordinary power-to-Bernstein tests and
4 exceptional tests of Part 4.4 have the displayed strict lower
bounds, uniformly over e in E and tau in [0, 1].
Proof. Every quantity in (i)-(v) is a finite expression in the P_j
coefficients, the constants h, H, Z, W_*, Delta_*, U_*, P_*, and
pi. Section 4.5's interval arithmetic certifies each inequality. The
specific computations are:
(i) Form R from the folded spectral kernels of Part 3.5 at the
frequencies t_j, with P_j coefficients.
Augment by identity; run Gauss-Jordan with interval arithmetic.
Verify each pivot interval excludes 0 and lies in the displayed
bounds. The right half of the augmented matrix gives an enclosure
of the inverse. The largest column sum bounds the operator norm.
For ||A_i e||: compute A_i e as a vector, then apply the polytope
row minimum of Part 4.3(A) to each coordinate (a linear function of
e) to bound ||A_i e|| from above (using max = -min of negation).
(ii) Form Q_n, D_n from the Fourier coefficients P_j:
Q_n = Re sum_{j=0}^{15} P_j (i pi t_j)^2 e^{i pi t_j n}
(this folds the frequencies onto nonnegative j),
D_n = Re sum_{j=0}^{15} P_j (i pi t_j)^3 e^{i pi t_j n} / (3 Q_n).
The folding factor cancels (Part 3.3). Compute each Q_n, D_n by
interval arithmetic. The midpoint values of P(s)/(s-m)^2 at each
midpoint are P(s) itself evaluated from P_j at s = (a + a')/2, all
interval-enclosed.
(iii) Form the atomic masses M_{j;c,n}, M_{j;d,n} from (5.2), the
folded tail-variation matrix W from (5.9), and the sums defining
the four operators; each operator norm is a max column sum, hence a
scalar row over a finite set of explicit real numbers.
(iv) The curvature envelope for a wave e^{w s} at damping e^{-pi h s}
has half-second-derivative factor |w|^2 exp(-Re(w) s - pi h s)
summed over the spectral measure. Linearity over the finite set of
frequencies t_j and (for the tail) the integral over t in [0, 5/6]
gives the displayed table; each entry is a finite sum evaluated by
interval arithmetic.
(v) Each polynomial test of Part 4.4 is converted to Bernstein form
via the power-to-Bernstein matrices of Part 4.3(B), with N = 20 or
N = 22. Then each Bernstein row (a scalar linear function of e) is
minimized over E via Part 4.3(A). Comparing the minimum to the
stated threshold finishes.
All computations are rational and outward-rounded; no step
involves sampling or floating-point arithmetic. Appendix A gives
the explicit recipes with concrete precision and operations.
This certifies all the strict inequalities in (i)-(v) at once,
using a single deterministic algorithm. The 659 strict row minima
of part (v) all succeed with margin well above the 10^{-60}
arithmetic precision.
What does this certificate give us? In Part 5, we use (iii) to
control the infinite correction and prove exact interpolation at
all nodes and the individual-error bound (5.16) below. In Part 6,
we use (ii) and (iv) to control curvature and (v) to establish the
near-node signs. Part 7 then assembles these pieces into Theorems 1
and 2.
================================================================
PART 5. THE INFINITE CORRECTION
================================================================
The finite system of Part 4 solves ten interpolation equations on
the nodes in f, with the target G_z truncated to zero at nodes
9, 12, 13, 16, 19, 21 and omitted entirely on the tail T. That
solution therefore does NOT satisfy the interpolation equations in
full: it misses the target at the six "extra" nodes of f, and it
misses all tail nodes.
Part 5 corrects this. The correction lives on an infinite coefficient
list w_i in l^1(T; R^2) for each side i = 1, 2. A Neumann series
on the tail produces the correction exactly; the series converges
rapidly because the operators coupling the tail to itself and to the
finite part are extremely small. The output is Proposition 5.7
below: existence, uniqueness, and quantitative error bounds.
5.1 The four coupling operators
Let s_i in R^20 denote the finite coefficient vector on side i
(the twenty unknowns (c_{i,n}, d_{i,n}) for n in f). Let w_i in
l^1(T; R^2) denote the tail coefficient list on side i.
The interpolation system couples these two flavors through four
bounded linear operators and one algebraic map:
B : l^1(T; R^2) -> R^20
tail-to-finite. Takes an opposite-side tail list and produces
the finite coefficient correction needed to cancel the
transformed-wave contribution at the ten nodes in f.
C : R^20 -> l^1(T; R^2)
finite-to-tail. Takes an opposite-side finite coefficient
vector and produces the tail coefficient correction needed to
cancel the transformed-wave contribution at tail nodes.
D : l^1(T; R^2) -> l^1(T; R^2)
tail-to-tail. Takes an opposite-side tail list and produces
the own-side tail coefficient correction needed to cancel the
transformed-wave contribution at tail nodes.
A : R^20 -> l^1(T; R^2)
own-side repetition. Takes an own-side finite coefficient
vector and produces the direct repeated-node contribution at
tail nodes (each n in L feeds the equally-spaced later nodes
n + 36 l, l >= 1).
K_{T,V} : R -> l^1(T; R^2)
constant-to-tail. Takes the constant-column coefficient
C_{3-i} and produces its transformed-wave contribution at tail
nodes.
The construction is straightforward. Every operator is built from
two pieces: the atomic masses M_{j;c,n} and M_{j;d,n} of the
periodic columns (Part 3.4), and the folded spectral kernels of the
rational tail columns (Part 3.5). The frequency index j runs over
the 31 modes t_j = j/18, j = -15, ..., 15; the positive index alone
suffices after folding (Part 3.5). The node-dependent phase has
modulus one. Linearity is over the coefficient lists.
The operator norms were computed by the finite certificate
(Lemma 4.6(iii)):
||B|| < 340,
||C||+||A|| < beta := 3e-10,
||D|| < delta := 3e-8,
||K_{T,V}|| < 1.3e-7.
The asymmetry is intrinsic. B evaluates the tail at FINITE nodes
1, 3, ..., 21 where the damping e^{-pi h n} is not yet tiny. D
evaluates the tail at TAIL nodes n >= 25 where the same damping is
astronomically small. So ||B|| is much larger than ||D||.
5.2 The tail targets and the exact system
Let s^{(T)}_1 in l^1(T; R^2) denote the side-1 target G_z restricted
to the tail, and s^{(T)}_2 = 0.
The full interpolation equations, in coefficient coordinates, read
s_1 = R s_2 + R_0 gamma_2 + Y_f e + B w_2, (5.1a)
s_2 = R s_1 + R_0 gamma_1 + B w_1, (5.1b)
w_1 = C s_2 + D w_2 - A s_1 + s^{(T)}_1, (5.1c)
w_2 = C s_1 + D w_1 - A s_2 + s^{(T)}_2, (5.1d)
where R (20 x 20) and R_0 (20 x 3) are as in Part 4.1,
gamma_i = (c_{i,0}, d_{i,0}, C_i) = (0.44, 0, +/- 0.006), and
Y_f e is the finite target (Part 4.2).
We prove below that (5.1) has a unique solution
(s_1, s_2, w_1, w_2) in R^20 x R^20 x l^1(T; R^2) x l^1(T; R^2).
5.3 Schwartz convergence of the tail series
Before we solve the system, we check that any summable coefficient
list produces a real radial Schwartz function via the tail series.
Recall the tail-series functions
U_i^{(T)}(s) = e^{-pi h s} P(s) sum_{n in T}
[ c_{i,n}/(s-n)^2 + d_{i,n}/(s-n) ]
and their Fourier transforms. Each kernel
k_{c,n}(s) = e^{-pi h s} P(s) / (s-n)^2,
k_{d,n}(s) = e^{-pi h s} P(s) / (s-n)
extends smoothly at s = n (removable). The n-dependence is a scalar
phase e^{i pi t_j n} (unit modulus) times a kernel of fixed
frequencies.
Uniform Schwartz bound. Work in the spatial variable, with s = b|x|^2.
By the fourth fact of Part 3.5, the folded spectral measure of
e^{-pi h s} C_n(s) is the density rho_{c,n}(t) dt on 0 <= t <= 5/6,
so
k_{c,n}(x) = int over [0, 5/6] of rho_{c,n}(t) Phi_{h,t}(x) dt.
For each pair of multi-indices (beta, gamma) there is a constant
C_{beta,gamma,h} with
sup over x in R^2 of |x^beta d_x^gamma Phi_{h,t}(x)|
<= C_{beta,gamma,h} for every t in [0, 5/6],
uniformly in t: differentiating the Gaussian gives a polynomial
times e^{-pi b h |x|^2}, and the coefficients are bounded on the
compact frequency interval. Hence
sup_x |x^beta d_x^gamma k_{c,n}(x)|
<= C_{beta,gamma,h} * int over [0, 5/6] of |rho_{c,n}(t)| dt
<= C_{beta,gamma,h} * sum over j of W_{j,c}
< C_{beta,gamma,h} * 119,
by the weights of Part 3.5, with no dependence on n. The same
argument with rho_{d,n} and W_{j,d} bounds k_{d,n}.
Consequently, writing C_{a,b} for C_{a,b,h} * 119, for any summable
list (c_n, d_n) in l^1(T),
|| sum_{n in T} c_n k_{c,n} ||_{a,b}
<= C_{a,b} sum_{n in T} |c_n| < infinity.
So the series converges in every Schwartz seminorm. There is NO
moment condition such as sum n^r |c_n| < infinity. This is the
essential fact that makes the tail correction convergent.
Corollary. For every summable (c, d) in l^1(T; R^2), and every
continuous linear functional of Schwartz functions, the series
converges. In particular, U_i^{(T)} is a real radial Schwartz
function; its Fourier transform is defined pointwise via the folded
spectral measures; differentiation and summation commute.
5.4 Sum-difference decoupling
The two sides of (5.1) are coupled. Decouple by the standard
sum-difference change of variables:
s_eps = s_1 + eps s_2, w_eps = w_1 + eps w_2, eps = -1, +1.
Substituting into (5.1):
s_eps = eps R s_eps + R_0(gamma_2 + eps gamma_1) + Y_f e
+ eps B w_eps,
w_eps = eps D w_eps + (eps C - A) s_eps + s^{(T)}_eps,
where s^{(T)}_eps = s^{(T)}_1 + eps s^{(T)}_2 = s^{(T)}_1.
This decouples into two INDEPENDENT signed systems, one for eps = +1
and one for eps = -1. Each is a system of the form
(I - eps R) s_eps - eps B w_eps = r_eps, (5.3a)
(I - eps D) w_eps - (eps C - A) s_eps = s^{(T)}_eps, (5.3b)
where r_eps = R_0(gamma_2 + eps gamma_1) + Y_f e.
Solving the two signed systems and then taking half-sums and
half-differences recovers s_1, s_2, w_1, w_2. No other coupling
remains.
5.5 Schur elimination and the Neumann series
Eliminate s_eps from (5.3) using (5.3a):
s_eps = L_eps(eps B w_eps + r_eps),
L_eps = (I - eps R)^{-1}. (5.4)
Substitute into (5.3b):
w_eps = eps D w_eps
+ (eps C - A) L_eps (eps B w_eps + r_eps)
+ s^{(T)}_eps.
Rearrange as a fixed point for w_eps:
(I - K_eps) w_eps = s^{(T)}_eps + (eps C - A) L_eps r_eps, (5.5)
where K_eps = eps D + (eps C - A) L_eps (eps B).
Operator bound. Using the operator norms of Lemma 4.6(iii) and the
inverse bounds of Chapter 4.6(i), N_+ < 17 and N_- < 5:
||K_eps|| <= ||D|| + ||eps C - A|| ||L_eps|| ||B||
<= delta + (beta + delta) N_eps * 340
<= 3e-8 + (3.0000003e-10)(17)(340)
<= 3e-8 + 1.7340e-6
<= 1.7640e-6 < 1.
So (I - K_eps) is invertible with
||(I - K_eps)^{-1}|| <= (1 - 1.7640e-6)^{-1} <= 1.0000018.
The geometric series
(I - K_eps)^{-1} = I + K_eps + K_eps^2 + K_eps^3 + ...
converges in operator norm. This gives a unique solution
w_eps = (I - K_eps)^{-1} ( s^{(T)}_eps + (eps C - A) L_eps r_eps )
in l^1(T; R^2). Recover s_eps = L_eps(eps B w_eps + r_eps). Take
half-sums and half-differences to get s_1, s_2, w_1, w_2. This
completes the proof that (5.1) has a unique solution.
Convergence of the tail series was proved in Part 5.3, so the
resulting U_1, U_2 are real radial Schwartz; the Fourier transforms
hat U_1, hat U_2 are too. So F_1 = U_1 + hat U_2 and
F_2 = U_2 + hat U_1 are real radial Schwartz, and hat F_1 = F_2.
5.6 Error relative to the finite interpolant
Let (s_i, w_i) denote the exact solution. Let (A_i e, 0) denote the
finite interpolant from Part 4: it solves the finite equations
(5.1a)-(5.1b) with the truncated target (retaining G_z at nodes
1, 3, 4, 7 and zeroing it at nodes 9, 12, 13, 16, 19, 21) and with
w_i = 0.
The finite interpolant's coefficient norms are bounded by the
finite certificate:
||A_i e||_1 < 3 for both i.
The omitted-target residual has norm
d < 8.2e-8 on the six nodes f \ {1, 3, 4, 7}
1.0e-26 on the tail T.
Because the target appears only on side 1, the same bound d applies
to each of the sum and difference residuals (r_eps has the same
residual structure).
Residuals also come from the finite coefficient vectors' OWN
feedback into the tail. The sum of the finite coefficient norms is
less than 6, so their total tail effect is bounded by 6 beta = 1.8e-9.
The constant columns cancel in the sum because C_1 + C_2 = 0; in
the difference they contribute (C_1 - C_2) * K_{T,V} with absolute
coefficient 0.012 and operator norm < 1.3e-7, giving an extra
0.012 * 1.3e-7 = 1.56e-9.
So valid tail residual bounds are
d'_+ = 1.0e-26 + 6 beta ~ 1.800e-9,
d'_- = 1.0e-26 + 6 beta + 0.012 (1.3e-7) ~ 3.360e-9.
Apply the same Schur elimination to the DIFFERENCE between the
exact solution and this finite interpolant. From (5.5)-(5.6), its
sum and difference errors on the tail are bounded by
e'_eps = ( d'_eps + beta N_eps d ) / ( 1 - delta - 340 beta N_eps ),
and on the finite unknowns by
e_eps = N_eps ( d + 340 e'_eps ).
Every constant here is rational. Substituting N_+ = 17, N_- = 5,
beta = 3e-10, delta = 3e-8, d = 8.2e-8, d'_+ = 1.8e-9, d'_- = 3.36e-9:
eps = +1: e'_1 < 1.800004e-9,
e_1 < 1.179803e-5,
eps = -1: e'_-1 < 3.360002e-9,
e_-1 < 6.122004e-6.
Taking half the sum of the two bounds gives the individual
side bounds:
||w_i||_1 < 3.0e-9,
||s_i - (A_i e)||_1 < 1.0e-5, i = 1, 2.
It is NOT necessary that either sum/difference finite bound be
below 1e-5 separately; the individual side is controlled by their
average, which is well under the stated bounds. The same
observation applies to the difference tail bound.
5.7 Proposition (Exact Interpolation)
We can now state the interpolation result.
PROPOSITION (Exact Interpolation). Let alpha >= 1, let z and e be
its Gaussian parameter and data vector (Parts 3.6-3.7), and retain
the fixed coefficients c_{i,0}, d_{i,0}, C_i. Then there are unique
real coefficient lists
s_i in R^20 and w_i in l^1(T; R^2), i = 1, 2
for which the functions J_1 = e^{pi h s} F_1 and J_2 = e^{pi h s} F_2
satisfy, for EVERY n in N_cal:
J_1(n) = G_z(n), J_1'(n) = G_z'(n),
J_2(n) = 0, J_2'(n) = 0. (5.7)
Moreover, relative to the finite coefficient lists A_i e of Part 4:
||w_i||_1 < 3.0e-9,
||s_i - (A_i e)||_1 < 1.0e-5 (i = 1, 2). (5.8)
The resulting F_1, F_2 are real radial Schwartz functions with
hat F_1 = F_2. Uniqueness is over these coefficient lists (with the
prescribed fixed columns), not over all radial Schwartz functions
with the same interpolation data.
Proof. Construct (s_i, w_i) by the sum-difference Neumann procedure
of Parts 5.4-5.5. Then J_1, J_2 are Schwartz by Part 5.3. The
equations (5.7) hold at every node of N_cal because the construction
was explicitly arranged to satisfy the finite equations (5.1a)-(5.1b)
and the tail equations (5.1c)-(5.1d), which ARE the jet equations
at each node after local removal of the damping. The bounds (5.8)
follow from Part 5.6.
One point deserves emphasis. The word "interpolation" in Proposition
5.7 is EXACT: the equality (5.7) holds at every n in N_cal, without
quantification error. The quantitative smallness in (5.8) is only the
error of the finite interpolation approximation, used below in
Part 6 to control the signs BETWEEN the interpolation nodes (see
Part 6.2). It is not an error in (5.7).
================================================================
PART 6. SIGNS BETWEEN THE INTERPOLATION NODES
================================================================
Proposition 5.7 supplies exact value and first derivative of J_1, J_2
at every node of N_cal. What remains is to prove the two POINTWISE
inequalities on the whole half-line s >= 0:
J_1(s) <= G_z(s), (6.A)
J_2(s) >= 0. (6.B)
Both are equalities (0 = 0 and J_1 = G_z) at each node. The proof
removes the first-order gap at a nearby node and compares second-order
remainders; between nodes, whether the "nearby node" is a finite node,
a larger node, or a distant tail node, a different dominant term
controls the comparison.
6.1 Divided jets and the deleted-jet identity
For smooth phi and m >= 0, define the DIVIDED JET
U_m[phi](v) = (phi(m+v) - phi(m) - v phi'(m)) / v^2 for v != 0,
and extend by continuity at v = 0:
U_m[phi](0) = (1/2) phi''(m).
For phi of class C^2 near m, the integral representation holds:
U_m[phi](v) = integral from 0 to 1 of (1-t) phi''(m + tv) dt.
This is Taylor's theorem with integral remainder to first order,
divided by v^2.
At an interpolation node m in N_cal, Proposition 5.7 gives the jets
J_1(m) = G_z(m), J_1'(m) = G_z'(m),
J_2(m) = 0, J_2'(m) = 0.
Subtracting the linear term at m and dividing by v^2 = (s - m)^2
(valid for v != 0; s = m + v), we obtain:
G_z(m+v) - J_1(m+v) = v^2 ( U_m[G_z](v) - U_m[J_1](v) ), (6.1)
J_2(m+v) = v^2 U_m[J_2](v). (6.2)
Since v^2 > 0 off the node, (6.A)-(6.B) become, on the interval
around m where numbers are well defined and m + v >= 0:
J_1 <= G_z on that interval <=> U_m[J_1] <= U_m[G_z],
J_2 >= 0 on that interval <=> U_m[J_2] >= 0.
The comparison has been pushed from the values of the functions to
their second-order divided jets.
6.2 Accuracy of the divided Taylor polynomials
Let J_i^{fin} = e^{pi h s} F_i^{fin} be the finite interpolant of
Part 4 (no tail coefficients w_i), and let T_i(v; e) denote the
degree-20 divided Taylor polynomial in v of U_m[J_i^{fin}], truncated
at degree 20. Every coefficient of T_i is computed exactly by the
finite 20 x 20 system of Part 4. Write
U_m[J_i^{fin}](v) = T_i(v; e) + R_i(v; e)
with |R_i| a Taylor remainder. Lemma 4.6(iv) gives:
|R_i(v; e)| < 0.002 at m = 1,
0.00001 at m >= 3, uniformly in v in the cell.
(These are the certificate values 0.00161 and 0.0000075 rounded up.)
For the FULL function J_i, not the finite interpolant, replacing the
coefficients s_i by the finite lists A_i e and dropping the tail gives
an additional coefficient error. By Part 5.7,
||s_i - (A_i e)||_1 < 1e-5, ||w_i||_1 < 3.0e-9.
Linearity of U_m plus the curvature envelopes of Lemma 4.6(iv) give
|U_m[J_i](v) - T_i(v; e)|
<= 1e-5 * (L-entry at s_0) + 3e-9 * (T-entry at s_0) + Taylor,
where s_0 is a lower bound for s in the cell. Inserting the curvature
table from Lemma 4.6(iv) (with the certificate entries rounded up
for margin: L = 1510, 46, 14 and T = 26500, 1160, 800):
left half-cell at s = 1 (s_0 = 0; v in [-1, 0]):
1e-5 * 1510 + 3e-9 * 26500 + 0.002 = 0.0171795 < 0.018.
right half-cell at s = 1 (s_0 = 1; v in [0, 1]):
1e-5 * 46 + 3e-9 * 1160 + 0.002 = 0.00246348 < 0.003.
every later cell m >= 3 (s_0 = 2; s >= 2):
1e-5 * 14 + 3e-9 * 800 + 0.00001 = 0.0001524 < 0.00016.
So the UNIFORM ACCURACY STATEMENT used below is:
|U_m[J_i](v) - T_i(v; e)| < 0.018 on the left half-cell at 1,
< 0.003 on the right half-cell at 1,
< 0.00016 on every cell with m >= 3.
6.3 The Gaussian divided-difference lemma
LEMMA (Gaussian Divided Difference). Let G_z(s) = Q_1 z e^{(1-s)/z}
with z > 0. For every m >= 0 and every v >= -m,
U_m[G_z](v) >= Q_1 e^{(1-m)/z} * pi_z(v),
where
pi_z(v) = 1 / (2z) for v <= 0,
pi_z(v) = (3z + v) / (6z^2 + 4zv + v^2) for v >= 0.
Both formulas give 1/(2z) at v = 0.
Proof. Compute G_z''(s) = Q_1 z^{-1} e^{(1-s)/z}.
Case v <= 0. Because G_z'' is decreasing in s and m + tv <= m for
0 <= t <= 1,
G_z''(m + tv) >= G_z''(m).
Since m + tv >= 0 (s = m + v >= 0 forces 0 <= m + tv <= m), the
integral form applies:
U_m[G_z](v) = int_0^1 (1-t) G_z''(m + tv) dt
>= G_z''(m) int_0^1 (1-t) dt
= G_z''(m)/2
= Q_1 e^{(1-m)/z} / (2z).
Case v >= 0. Put x = v/z. The integral form gives
U_m[G_z](v) / [ Q_1 e^{(1-m)/z} ]
= (1/z) int_0^1 (1-t) e^{-t x} dt
= (1/z) (x + 1 - e^{-x}) / x^2.
We want (x + 1 - e^{-x}) / x^2 >= (3 + x)/(6 + 4x + x^2). Clear the
positive denominator:
(x + 1 - e^{-x}) (6 + 4x + x^2) - x^2 (3 + x) = q(x),
where q(x) = e^{-x}(6 + 4x + x^2) - 6 + 2x. Direct differentiation:
q(0) = 6 - 6 = 0,
q'(0) = -2 + 2 = 0,
q''(x) = x^2 e^{-x} >= 0.
Therefore q is nonnegative on [0, infinity), and the inequality above
holds. Multiplying by 1/z gives the claimed bound for v >= 0.
Together, the two cases prove the lemma.
6.4 Cells, the seven half-cell pairs, and the ordinary comparisons
The set f = {1, 3, 4, 7, 9, 12, 13, 16, 19, 21}. The seven cells
centered at 1, 3, 4, 7, 9, 12, 13 tile [0, 14.5]:
m = 1: cell [0, 2], left v in [-1, 0], right v in [0, 1];
m = 3: cell [2, 3.5], left v in [-1, 0], right v in [0, 0.5];
m = 4: cell [3.5, 5.5], left v in [-0.5, 0], right v in [0, 1.5];
m = 7: cell [5.5, 8], left v in [-1.5, 0], right v in [0, 1];
m = 9: cell [8, 10.5], left v in [-1, 0], right v in [0, 1.5];
m = 12: cell [10.5, 12.5], left v in [-1.5, 0], right v in [0, 0.5];
m = 13: cell [12.5, 14.5], left v in [-0.5, 0], right v in [0, 1.5].
On each half-cell, the sign question (6.1)-(6.2) is a comparison of
the divided polynomials T_1, T_2, up to the accuracy of Section 6.2.
The finite certificate (Lemma 4.6(v)) gives, uniformly over the
parameter polytope E and the half-cell v-range, the following strict
lower bounds:
m = 1, left: T_2 > 0.174, Q_1 / (2Z) - T_1 > 0.057;
m = 1, right: T_2 > 0.011; side 1 handled in 6.5;
m = 3, either: 0.8 Q_1 w - T_1 > 0.00055, T_2 > 0.00097;
m = 4, either: -T_1 > 0.00047, T_2 > 0.00097;
m = 7, either: -T_1 > 0.0028, T_2 > 0.0028;
m = 9, either: -T_1 > 0.0039, T_2 > 0.0039;
m = 12,13: -T_1 > 0.0020, T_2 > 0.0020.
Here w is the coordinate e_4 = exp(-2/z) of the parameter vector e;
Q_1 is the fixed local coefficient at node 1; Q_1/(2Z) is a constant.
Case m = 1, left half-cell (v in [-1, 0]). Lemma 6.3 with v <= 0
gives U_1[G_z](v) >= Q_1/(2z) >= Q_1/(2Z). Accuracy 0.018.
Side 2: U_1[J_2] = T_2 + error >= 0.174 - 0.018 = 0.156 > 0.
Side 1: U_1[J_1] = T_1 + error <= T_1 + 0.018, and
U_1[G_z] - U_1[J_1] >= Q_1/(2Z) - T_1 - 0.018
> 0.057 - 0.018 = 0.039 > 0.
Case m = 1, right half-cell (v in [0, 1]). Accuracy 0.003.
Side 2: U_1[J_2] = T_2 + error >= 0.011 - 0.003 = 0.008 > 0.
Side 1: exceptional; see 6.5.
Case m = 3, both half-cells (v in [-1, 0.5]). Lemma 6.3 with v <= 0
gives U_3[G_z](v) >= Q_1 w/(2z) >= 0.8 Q_1 w (since 1/(2z) >= 1/(2Z)
> 1.27 > 0.8). For v >= 0 with v <= 0.5, the function
r(z,v) = (3z+v)/(6z^2+4zv+v^2) decreases in each variable; both
partial derivatives have negative numerators -18z^2 - 12zv - v^2
and -6z^2 - 6zv - v^2. So
r(z,v) >= r(Z, 0.5) = 836500/974643 > 0.8.
Together with the prefactor e^{(1-3)/z} = w, get U_3[G_z] >= 0.8 Q_1 w.
Accuracy 0.00016.
Side 1: U_3[G_z] - U_3[J_1] >= 0.8 Q_1 w - T_1 - 0.00016
> 0.00055 - 0.00016 = 0.00039 > 0.
Side 2: U_3[J_2] = T_2 + error >= 0.00097 - 0.00016 = 0.00081 > 0.
Case m in {4, 7, 9, 12, 13}. Since G_z'' > 0 everywhere, the convexity
of G_z gives U_m[G_z](v) >= 0 for all v in the cell. Accuracy 0.00016.
Side 1: U_m[J_1] = T_1 + error <= -0.00047 + 0.00016 < 0, and
U_m[G_z] >= 0, so U_m[J_1] < U_m[G_z]. (For m >= 7 even
stronger: -0.0020 + 0.00016 < 0.)
Side 2: U_m[J_2] = T_2 + error >= 0.00097 - 0.00016 = 0.00081 > 0.
(For m >= 7, >= 0.0020 - 0.00016 > 0.)
These cases exhaust [0, 14.5] except the right half-cell at m = 1.
6.5 The exceptional right half-cell at s = 1
Here v in [0, 1] and the accuracy is 0.003. Lemma 6.3 gives
U_1[G_z](v) >= Q_1 (3z + v) / (6z^2 + 4zv + v^2).
Since U_1[J_1] <= T_1 + 0.003, it is enough to prove
T_1 + 0.003 <= Q_1 (3z + v) / (6z^2 + 4zv + v^2).
Multiply by B(z, v) = 6z^2 + 4zv + v^2, positive for z > 0 or v > 0:
Q_1 (v + 3z) + B(z, v) ( -T_1 - 0.003 ) >= 0. (6.3)
On the parameter polytope E, at fixed other coordinates, -T_1 is
affine in z (Section 4.3). Write
A(v) = -T_1(v; e) at z = 0, D(v) = -T_1(v; e) at z = Z.
Then
-T_1(v; e) = (1 - z/Z) A(v) + (z/Z) D(v).
Substituting into (6.3):
F(z) := Q_1(v + 3z) + B(z, v) [ (1 - z/Z)(A - 0.003)
+ (z/Z)(D - 0.003) ] >= 0.
For fixed v in [0, 1] and fixed remaining parameters, F is cubic in z
on [0, Z]. Converting the pre-perturbation expression
Q_1(v + 3z) + B(z, v)[ (1 - z/Z) A + (z/Z) D ] to the Bernstein basis
on [0, 1] (in z/Z) gives the four coefficients
c_0 = v (Q_1 + v A),
c_1 = Q_1(v + Z) + (2 B_1 A + B_0 D)/3,
c_2 = Q_1(v + 2Z) + (B_2 A + 2 B_1 D)/3,
c_3 = Q_1(v + 3Z) + B_2 D,
with B_0 = v^2, B_1 = v^2 + 2 Z v, B_2 = v^2 + 4 Z v + 6 Z^2.
Lemma 4.6(v) gives, uniformly over E and v in [0, 1]:
c_0 > 0.046, c_1 > 0.056, c_2 > 0.024, c_3 > 0.027.
The perturbation subtracts 0.003 B(z, v) from F. B(z, v) is at most
B(Z, 1) = 6 Z^2 + 4 Z + 1 < 3.5. So the perturbed coefficients
satisfy, for j = 1, 2, 3,
c_j - 0.003 * 3.5 >= 0.02 - 0.0105 = 0.0095 > 0.
The leading coefficient shifts by 0.003 B(0, v) = 0.003 v^2, so its
post-perturbation value is v(Q_1 + v A - 0.003 v) >= v(0.046 - 0.003)
> 0, and it is zero only when v = 0. In that boundary case, the other
three perturbed coefficients are each > 0.0095, and F > 0 as soon as
z > 0; the physical case z > 0 covers it.
Since the Bernstein basis is nonnegative on [0, 1], F(z) >= 0 for all
z in [0, Z], hence for every admissible parameter. That proves (6.3),
and therefore U_1[J_1] <= U_1[G_z] on the right half-cell at s = 1.
6.6 The sine-product barrier for s >= 14.5
There are no more finite nodes at distance <= 15/2 once s > 14.5, but
there are always nodes in N_cal (the tail nodes beyond 21, interleaved
with the isolated node 16, 19, 21, and 25, 27, ...). To control the
sign far out, we exploit the sine product P itself.
LEMMA (Sine-Product Lower Bound). If m is a nearest zero of P to s
and s != m, then
P(s) > 0.054 (s - m)^2.
If 14.5 <= s <= 23 and m is the nearest of {16, 19, 21} to s, then
P(s) > 3 (s - m)^2.
Proof. On each half-gap from m to an adjacent midpoint (of two
consecutive zeros of P), the quotient G(s) = P(s)/(s-m)^2 is positive
and log-concave. Indeed,
log G(s) = 2 log|2 sin(kappa(s-m))| - 2 log|s-m|
+ sum over a in I, a != m of 2 log|2 sin(kappa(s-a))|.
The second derivative of each factor (2 sin(kappa(s-a)))^2 contributes
-2 kappa^2 csc^2(kappa(s-a)); the factor whose zero is divided out
contributes 2((s-m)^{-2} - kappa^2 csc^2(kappa(s-m))), which is <= 0
because |sin x| <= |x|. So (log G)'' <= 0 on the half-gap.
A concave function attains its minimum on an interval at an endpoint.
The two endpoints of the half-gap are the node m and the midpoint.
At m, G(m) = Q_m > 0.157 (certificate). At the midpoint, the values
are, from the certificate in the order of the 15 gaps in one period:
0.165, 0.120, 0.054, 0.482, 0.656, 1.11, 0.337,
3.13, 9.26, 8.19, 19.08, 2.34, 0.706, 6.80, 14.1.
All are >= 0.054. So G(s) >= 0.054 on the whole period.
For the stronger bound on [14.5, 23]: the four midpoints inside this
window are 14.5, 17.5, 20, 23, with G-values 3.13, 9.26, 8.19, 19.08.
The nearest zero m is one of 16, 19, 21, each with Q_m > 16. All these
are >= 3, so concavity of log G on each half-gap gives G > 3 on the
whole window.
Now the sign of J_i far out. Let m be the nearest zero of P to s,
with v = s - m. Write J_1 = e^{pi h s} F_1. The constant column of
side 1 contributes a term
-0.006 * P(s) in e^{pi h s} U_1,
from the "+C_i e^{-pi h s} P(s)" part of U_i with C_1 = -0.006 (its
dependence in s is P(s), after the e^{-pi h s} factors cancel). In
the divided jet at m this contributes -0.006 P(s)/(s-m)^2 exactly,
because P(m) = P'(m) = 0 forces U_m[P](v) = P(m+v)/v^2 = P(s)/(s-m)^2.
All other contributions to the divided jet of J_1 - G_z have total
absolute value bounded by the certificate constant
< 0.0140109 for 14.5 <= s <= 23,
< 0.00008363 for s >= 23.
The constant column dominates:
0.006 * 3 = 0.018 > 0.0140109 (margin 0.0039891),
0.006 * 0.054 = 0.000324 > 0.00008363 (margin 0.00024037).
For J_1: the divided jet receives the negative constant-column term
(at most -0.018 on the first range, at most -0.000324 on the second)
plus the bounded remainder, giving total U_m[J_1](v) < 0 in both
ranges. Since U_m[G_z](v) >= 0 by Lemma 6.3 (convexity alone suffices),
get U_m[J_1] < U_m[G_z]. This is (6.A).
For J_2: the constant column is +0.006 P(s), so its divided jet is at
least +0.000324; subtracting the bounded contribution of remaining
terms, get U_m[J_2](v) > 0.00024037 - 0.00008363 > 0 on s >= 23, and
> 0.018 - 0.0140109 > 0 on [14.5, 23]. So J_2 > 0. This is (6.B).
6.7 Proposition (Global Signs)
PROPOSITION. J_1 <= G_z and J_2 >= 0 for every s >= 0.
Proof. Section 6.4 proves the two signs on [0, 14.5] except the right
half-cell at 1; Section 6.5 proves it there. Section 6.6 proves the
two signs on [14.5, infinity). The two ranges overlap at s = 14.5, and
the arguments agree at the boundary. So the signs hold on all s >= 0.
Together with Proposition 5.7 and Part 3.7, this completes the proof
of Theorem 2 for alpha >= 1. Part 2.5 extends to all alpha > 0.
What remains is a short synthesis in Part 7, and the explicit
arithmetic recipe in Appendix A.
================================================================
PART 7. SYNTHESIS
================================================================
Everything is in place. This part assembles the pieces in the order
the theorem needs, states what is and is not proved, and gives a
short table of where each numeric constant enters.
7.1 Theorem 2 for alpha >= 1
Fix alpha >= 1 and set
z = 1 / ( pi (alpha/b - h) ).
Since alpha/b - h >= 1/b - h > 2h - h = h > 0, we have z > 0. And
alpha/b - h >= 1/b - h = 2/sqrt(3) - 0.34 > 0.81, so
z <= Z := 0.391.
Let e = (z, 1, Zw - Delta, w, a, u, r, p) be the Gaussian data vector
of Part 4.2, with w = e^{-2/z}, u = e^{-3/z}, p = e^{-6/z}, and
Delta = (Z - z) w. Part 4.2 checked e in E.
Apply Proposition 5.7 with this data. Get real radial Schwartz
functions F_1, F_2 with hat F_1 = F_2, and J_i = e^{pi h s} F_i
satisfying the exact jets at every n in N_cal. Apply Proposition 6.7:
J_1 <= G_z and J_2 >= 0 on s >= 0. Define, for x in R^2 with
s = b|x|^2,
f_alpha(x) = F_1(b|x|^2) / ( Q_1 z e^{1/z} ).
Then:
(i) f_alpha is a real radial Schwartz function (F_1 is).
(ii) hat f_alpha(xi) = F_2(b|xi|^2)/ (Q_1 z e^{1/z}) >= 0 for every
xi, since F_2 = hat F_1 (radial even) and F_2 >= 0 because
J_2 = e^{pi h s} F_2 >= 0.
(iii) f_alpha(x) <= k_alpha(x) = e^{-pi alpha |x|^2}. This follows
from J_1 <= G_z and the algebraic identity from Part 3.7: for s >= 0,
F_1(s) <= Q_1 z e^{1/z} e^{-pi (alpha/b) s}
<=> e^{pi h s} F_1(s) <= Q_1 z e^{1/z} e^{-(pi (alpha/b) - pi h) s}
<=> J_1(s) <= Q_1 z e^{(1-s)/z}
<=> J_1(s) <= G_z(s).
(iv) f_alpha(a) = k_alpha(a) at every nonzero a in A. Every nonzero
squared radius of A lies in N_cal (Part 3.2), and J_1 = G_z at every
n in N_cal. Substituting s = s(a) in the definition gives the identity
f_alpha(a) = k_alpha(a) as computed in Part 3.7.
(v) hat f_alpha(w) = 0 at every nonzero w in A*. Every nonzero
squared radius of A* also lies in N_cal (because A and A* share their
radii, Part 2.4). At every such s, J_2 = 0, so F_2 = 0, so
hat f_alpha = 0.
Properties (i)-(v) are precisely Theorem 2 for this alpha.
7.2 All alpha > 0
Part 2.5 gives the extension to 0 < alpha < 1 from the case alpha >= 1
by the reciprocal-parameter construction:
f_alpha = k_alpha - alpha^{-1} hat f_{1/alpha}.
The four required properties (minorant, nonnegative transform,
touch at A, dual vanishing at A*) were checked in Part 2.5. So
Theorem 2 holds for all alpha > 0.
7.3 Theorem 1
Part 2.7 used three ingredients: the Gaussian comparison from
Theorem 2 and Poisson summation (Part 2.4); the Bernstein-Widder
Laplace representation of a completely monotone g (Part 2.6); and
Fatou's lemma plus Tonelli (Part 2.7). These give, for every locally
finite C of centered-disk density one,
E_g(C) >= sum over nonzero a in A of g(|a|^2) = E_g(A),
with both sides in the extended nonnegative reals. Attainment by A
was checked in Part 2.7. So Theorem 1 holds.
7.4 What is and is not proved
Proved:
- The minimum value of E_g among locally finite planar sets of
centered-disk density one equals sum_{a != 0} g(|a|^2).
- The triangular lattice A attains it.
- The Fourier bound is sharp: there is no Schwartz function Phi
with hat Phi >= 0 that would produce a strictly better bound of
the Cohn-Elkies type (the touch conditions force the bound to
equal E_g(A)).
- All quantities are in the extended nonnegative reals, so g may
have an infinite value at zero and infinite total mass in its
Laplace representation, and E_g(A) may be infinite.
Not proved (and not claimed):
- Uniqueness of the minimizer. For some soft potentials there may
be many approximately optimal configurations.
- A classification of all minimizers.
- A rate of convergence of E_g(C) to E_g(A) for general C.
- A statement about competitors without the density hypothesis,
or with translated-disk instead of centered-disk densities.
The hypotheses on the competitor are minimal: locally finite, and
N_R / (pi R^2) -> 1 in the CENTERED disk. In particular, C may be
nonperiodic, may have arbitrarily close pairs, and may have
arbitrarily many points in some translated small disk. None of
these affect the argument, because the only place the competitor
enters is Lemma 2.2, which uses only N_R / (pi R^2) -> 1.
Local optimality of A under bounded displacements (Leblé, 2025) and
optimality within restricted periodic classes (Faulhuber-Shafkulovska-
Zlotnikov 2023; Hardin-Tenpas 2025) are separate results. Neither is
used here.
7.5 Where each number enters
A compact map of the constants, for reference:
Z = 0.391 upper bound on the Gaussian parameter z
W_*, Delta_*, U_*, P_*
bounds on the secondary coordinates of e
0.157, 40, 2.83 bounds on Q_n and |D_n| for residues n in I
16 lower bound on Q_n for n in {16, 19, 21}
17, 5 inverse norms of I - R and I + R
340 norm of B (tail-to-finite)
3e-10 norm of C plus norm of A (finite-to-tail)
3e-8 norm of D (tail-to-tail)
1.3e-7 norm of K_{T,V} (constant-to-tail)
119 total tail-variation weight |W|
2.5e-10 own-side repetition bound
3e-9, 1e-5 tail and finite coefficient errors
0.018, 0.003, 0.00016
divided-jet accuracy per cell class
0.002, 0.00001 Taylor remainders at m = 1 and m >= 3
0.174, 0.057, 0.011, 0.00055, 0.00097, 0.00047, 0.0028, 0.0039, 0.0020
lower bounds for the ordinary polynomial tests
0.046, 0.056, 0.024, 0.027
lower bounds for the four exceptional tests
0.054, 3 sine-product barrier constants
0.006 magnitude of the constant-column coefficient
0.0140109, 0.00008363
curvature bound for non-constant columns
8.2e-8, 1e-26 omitted-target residuals on f and T
0.012 |C_1 - C_2|
h = 17/50, H = 27/50 low and high damping; eta = H - h = 1/5
kappa = pi/36 sine-product mesh
That completes the proof.
================================================================
APPENDIX A. THE ARITHMETIC RECIPE FOR THE FINITE CERTIFICATE
================================================================
This appendix gives the explicit recipes that Lemma 4.6 asserts the
inequalities of. Any reader with the recipes and exact rational
arithmetic can reproduce them; there is no floating point, no
sampling of parameters, and no frequency quadrature. The companion
\tt{verification/check\_certificate.py} implements exactly these
recipes with integers and assertions.
A.1 The interval type and precision
Represent each real by a pair of integers (a, b) with a <= b,
interpreted as the interval [a/10^60, b/10^60].
addition: [a1,b1] + [a2,b2] = [a1+a2, b1+b2];
subtraction: [a1,b1] - [a2,b2] = [a1-b2, b1-a2];
multiplication: min and max of the four endpoint products;
division: only if 0 is not in [a2,b2]; then multiply by
[1/b2, 1/a2] (endpoints swapped because reciprocal
reverses order);
square: [a1,b1]^2 with 0 clipped in if 0 is in [a1,b1];
square root: [sqrt(a1), sqrt(b1)] with a1 >= 0.
All rounding is outward (down for lower endpoints, up for upper
endpoints). Complex intervals are pairs of real intervals. The
modulus sqrt(Re^2 + Im^2) is enclosed by the composition of the
above operations. Enclosure soundness follows by induction on the
expression tree: each step returns an interval containing the exact
result, hence so does the composed expression.
A.2 The constant pi
Use Machin's identity pi/4 = 4 arctan(1/5) - arctan(1/239) (from the
tangent addition formula; both angles lie in (0, pi/2)), and the
alternating Taylor series for arctan with the usual alternating
remainder bound. Keeping terms through power 81 gives
|error| <= 16/(83 * 5^83) + 4/(83 * 239^83) < 10^{-58}.
After outward rounding to 10^{-60}:
pi in [3.14159265358979323846264338327950288419716939937510,
3.14159265358979323846264338327950288419716939937510 + 1e-50].
A.3 The exponential e^w
Given a complex interval w with |w/1024| <= 1, divide w by 1024,
sum the Taylor series of e^{w/1024} through degree 60, add
[-10^{-60}, 10^{-60}] to each real component used, and square ten
times. The omitted modulus error at each real endpoint is at most
2/61! < 10^{-60} before squaring, so the result encloses e^w.
A.4 The phases e^{i pi n t_j}
Since t_j = j/18, the phase equals e^{i pi (n j mod 36)/18}. Reduce
n j modulo 36 to one of 0, 1, ..., 35 by integer arithmetic. Use
rigorous cos and sin Taylor enclosures on [0, pi/36] (a small
argument in one quadrant), extended to the other residues by sign
symmetry.
A.5 The coefficients P_j, the node data Q_n, D_n, and the masses
Form the Laurent coefficients of
prod_{a in I} ( 2 - e^{-i pi a/18} X - e^{i pi a/18} X^{-1} )
by ordinary convolution over the integer exponents -15, ..., 15.
The entries at array indices 15, ..., 30 give P_0, ..., P_15 (so
P(s) = sum_{j=0}^{15} P_j e^{i pi t_j s} with the folded convention;
the negative-frequency coefficients are the complex conjugates of
their reflections).
For p_j = P_j e^{i pi t_j n},
Q_n = Re sum_{j=0}^{15} p_j (i pi t_j)^2,
D_n = ( Re sum_{j=0}^{15} p_j (i pi t_j)^3 ) / (3 Q_n).
The missing factor 2 from folding cancels the factors 2 and 6 in
P''(n) = 2 Q_n and P'''(n) = 6 Q_n D_n. Both Q_n and D_n are
periodic in n with period 36.
Atomic masses for the finite columns. Use the explicit reduction of
the undamped periodic columns to trigonometric polynomials. Let m be
the residue of n mod 36 in I, and denote by A_n and B_n the functions
A_n(s) = kappa^2 P(s) csc^2(kappa(s-n)),
B_n(s) = kappa P(s) cot(kappa(s-n)).
Dividing out the one factor of P that vanishes at s = n:
A_n(s) = 4 kappa^2 prod_{a in I, a != m} (2 sin kappa(s-a))^2,
B_n(s) = 2 kappa sin(2 kappa(s-n)) prod_{a in I, a != m} (2 sin kappa(s-a))^2.
Each prod is a trigonometric polynomial in e^{i pi s/18} of degree
at most 14, so A_n has degree <= 14 and B_n has degree <= 15. Write
A_n(s) = sum_j M_{j;c,n} e^{i pi t_j s},
B_n(s) = sum_j M_{j;d,n} e^{i pi t_j s},
with j running over -15, ..., 15. The scalars M_{j;c,n} and M_{j;d,n}
are the atomic masses. Each depends on n only through the phase
e^{i pi t_j n} times a fixed residue class constant.
The constant column P is already a trigonometric polynomial; its
coefficients are V_j = P_j.
At a gap midpoint s = (a + a')/2 (where a, a' are consecutive zeros
of P in one period), the sine-product ratio is
P(s)/(s - a)^2 = 4 Re sum_j V_j e^{i pi t_j s} / (a' - a)^2.
A.6 The polynomial power rows
For each frequency t_j and each (k, S) pair with k = H and S = M, or
k = h and S = V, define
X_k(t_j) = B/(k^2 + t_j^2), B = b^{-2} = 4/3,
lambda_k(t_j) = (k + i t_j) / (b (k^2 + t_j^2)),
zeta_{k,v}(t_j) = -t_j X_k(t_j) + i (k X_k(t_j) - v),
w_a = i pi (t_j + i (k - h)),
w_b = i pi zeta_{k,h}(t_j).
For a cell centered at m with displacement bound y (positive), the
divided-jet power rows are generated by
a_0 = w_a^2 e^{m w_a} / 2, b_0 = w_b^2 e^{m w_b} lambda_k / 2,
a_{r+1} = y w_a a_r / (r+3), b_{r+1} = y w_b b_r / (r+3).
If A_i^S denotes the block of coefficient rows belonging to S, then
the degree-r row acting on the parameter vector e in T_i is
sum over j and over the two S of Re( a_r (S A_i^S)_j + b_r (S A_{3-i}^S)_j ).
Constants in the test inequalities are added to column 2 of the row;
multiples of w (the coordinate e_4 = e^{-2/z}) are added to column 4.
Each T_i is a degree-20 power form in tau = v/y with coefficients
linear in e.
Multiplying the test by tau or tau^2 shifts the power rows; this is
how the side-two tests for the near-1 cells are obtained from the
same envelope.
A.7 Power to Bernstein
For a degree-N power form with coefficients t_0, ..., t_N, the
Bernstein coefficients are
b_i = sum_{r=0}^{i} [ C(i, r) / C(N, r) ] t_r, i = 0, ..., N,
so
T(tau; e) = sum_{i=0}^{N} b_i(e) C(N, i) tau^i (1-tau)^{N-i}.
Lower bound: min over tau in [0,1] of T(tau; e) >= min_i b_i(e).
Upper bound: max of T <= max_i b_i(e). Use N = 20 for the ordinary
tests and N = 22 for the exceptional tests.
A.8 Bernstein row minimum on E
Each Bernstein row b_i(e) is a linear function of e alone:
b_i(e) = b_2 + b_1 z + b_3 (Zw - Delta) + b_4 w
+ b_5 a + b_6 u + b_7 r + b_8 p.
Its minimum over the polytope E is the closed-form expression
derived in Section 4.3:
min_E b_i = b_2
+ Z min(b_1, 0)
+ min(0, Z W_* b_3) + min(0, -Delta_* b_3)
+ W_* min(b_4, 0)
+ U_* min(0, b_6, b_5 Z + b_6)
+ P_* ( min(b_7, 0) + min(b_8, 0) ).
Comparing this minimum with the test threshold certifies the test
strictly.
A.9 The exceptional substitution
For the right half-cell at m = 1, replace z by 0 or z by Z in the
negative side-one rows: add column 1 times the chosen endpoint to
column 2, then set column 1 to zero. This produces A(v) and D(v) as
degree-22 power forms in v (from the degree-20 forms by a v or v^2
shift as needed). Then form the four cubic-in-z Bernstein coefficients
c_0, ..., c_3 of Section 6.5 as linear combinations of A and D
against the weights B_0 = v^2, B_1 = v^2 + 2 Z v, B_2 = v^2 + 4 Z v + 6 Z^2.
Convert each c_j in v to Bernstein form with N = 22, minimize each
Bernstein row over E as in A.8, and compare with the threshold in
Lemma 4.6(v).
A.10 Number of tests
The ordinary table has 27 degree-20 tests on the 14 signed
half-cells (side two on every half-cell; side one on every half-cell
except the right half-cell at 1). Each test has 21 Bernstein rows.
The four exceptional tests are degree 22 and use 23 Bernstein rows
each. Total strict row minima:
27 * 21 + 4 * 23 = 567 + 92 = 659.
All 659 are certified by the closed-form A.8 criterion with exact
rational arithmetic at 10^{-60} precision. No sampling of the
parameter space, no frequency quadrature, and no floating-point
arithmetic occur anywhere in the certificate.
END OF DOCUMENT