# Approaches 6–10: arithmetic, compression, detuning, and contour geometry

Research target throughout: for every finite set `A` of distinct positive integers,

\[
f_A(x)=\sum_{a\in A}\cos(ax),\quad N=|A|,\quad K(A)=-\min_x f_A(x),
\]

prove `K(A)>=c*sqrt(N)` with one absolute `c>0`, or construct an admissible
unbounded family with `K(A)/sqrt(N)->0`. No result below settles that target.
All circle integrals are normalized unless a different measure is displayed.

## 6. Quadratic Gauss sums as a probability measure biased toward negative phases

**Mechanism tested.** Use the Legendre symbol to place nonnegative probability
weights on prime roots of unity. Quadratic Gauss sums have square-root size, so
one might hope to make the expectation of the full cosine sum negative by
order `sqrt(N)` and hence obtain a negative point.

Let `p=1 mod 4` be prime, let `chi` be the Legendre symbol with `chi(0)=0`, and
for `s in {-1,1}` put

\[
\mu_s(k)=\frac{1-s\chi(k)}p,\qquad 0\le k<p.
\]

These are nonnegative and sum to one. The quadratic Gauss-sum identity gives

\[
\sum_{k=0}^{p-1}\mu_s(k)\cos(2\pi ak/p)
=\begin{cases}
1,&p\mid a,\\
-s\chi(a)/\sqrt p,&p\nmid a.
\end{cases}
\]

Writing `D_p=|{a in A:p divides a}|` and `S_p=sum_{a in A,p not dividing a}chi(a)`,
and selecting `s` optimally, proves the **valid partial estimate**

\[
K(A)\ge\frac{|S_p|}{\sqrt p}-D_p.
\]

It is useful when the frequencies have a strong common quadratic-residue bias.
It does not give a uniform estimate: a balanced character sum contributes zero,
and frequencies divisible by `p` contribute positively. For `p=3 mod 4`, the
unshifted quadratic Gauss sum is purely imaginary. The same real Legendre bias
then contributes zero to every nontrivial cosine coefficient; its square-root
magnitude cannot be treated as a negative real number.

**Actual coverage counterexample, including CRT.** Fix any finite collection
of denominators and let `L` be a common multiple. For `N>=3`, set

\[
A_L=\{L,2L,\ldots,(N-1)L,L+1\}.
\]

This set has `N` distinct positive frequencies and `gcd(A_L)=1`. At every tested
rational angle `x=2*pi*k/q` with `q` dividing `L`,

\[
f_{A_L}(x)=N-1+\cos x\ge N-2>0.
\]

If the collection consists of small primes, all CRT combinations have
denominator dividing their product, so choosing `L` divisible by that product
defeats the combined grid too. This obstruction applies to **every probability
measure supported on those angles**, not just the displayed Legendre weights.
It survives removing a common gcd.

**Status.** The Gauss-sum estimate is proved. Universal coverage by a fixed or
`N`-bounded list of small prime grids is false without a further structural
step: `L` can contain every prime on that list. Adaptive use of primes that
depend on the actual frequency arithmetic remains possible, but a uniform
bound from that strategy was not found.

**Primary source.** Keith Conrad, *Gauss and Jacobi Sums on Finite Fields and
Z/mZ*, Corollary 2.3 and the final appendix's explicit prime evaluation:
https://kconrad.math.uconn.edu/blurbs/gradnumthy/Gauss-Jacobi-sums.pdf .
The displayed probability estimate and CRT adversary are derived here from
that identity.

## 7. Arithmetic Mahler measure and resultants of the nonnegative polynomial

**Mechanism tested.** Put `m=ceil(K(A))`, `D=max(A)`, and form the Laurent polynomial

\[
H(z)=2m+\sum_{a\in A}(z^a+z^{-a}),\qquad F(z)=z^D H(z).
\]

On the unit circle `H=2m+2f_A>=0`. The ordinary polynomial `F` is monic,
reciprocal, integral, has constant term one, and has degree `2D`. Its noncentral
nonzero coefficients are exactly one. Let `M` denote multiplicative Mahler
measure. Jensen's inequality has the following direction:

\[
M(F)=M(H)=\exp\int\log H\le\int H=2m.
\]

Boundary zeros cause integrable logarithmic singularities, not a reversal of
this inequality. Thus the proposed **sufficient lemma**

\[
M(H)\ge c\sqrt N
\]

would imply `K(A)>=c*sqrt(N)/2-1`, which is asymptotically sufficient for Chowla.
The lemma for this exact unit-coefficient class was neither proved nor refuted.

**Actual arithmetic exceptions checked.** Integrality alone only guarantees
`M(F)>=1`. It cannot supply growth in the degree or in the total number of
coefficients for a general nonnegative integral cosine polynomial. For every
integer `n>=2`,

\[
H_n(z)=\left|1+z+\cdots+z^{n-1}\right|^2
=n+\sum_{a=1}^{n-1}(n-a)(z^a+z^{-a})
\]

is nonnegative on the circle and has `M(H_n)=1`: the underlying geometric sum
has only roots of unity and is monic. Its support grows without bound. For even
`n` the constant term already has the required form `2m`, but its offzero
coefficients vary, so **this example refutes the weighted relaxation, not the
unproved unit-coefficient lemma**.

There are root-of-unity exceptions even in the exact class. For `A={D}`, `m=1`,

\[
F(z)=z^{2D}+2z^D+1=(z^D+1)^2,
\]

so `M(F)=1` at arbitrarily large degree. The resultant with `z^(2D)-1` vanishes.
Consequently an argument that silently assumes nonzero resultants against
roots of unity is invalid.

There is also a universal obstruction to applying a nonreciprocal-polynomial
resultant shortcut: **every** `F` constructed above is reciprocal, so its
reciprocal polynomial is `F` itself and their resultant is zero. Results using
the separation of a polynomial from its reciprocal do not apply.

**Status.** The Mahler lower-bound target is a precise unproved sufficient
lemma. The generic extensions from integrality, root count, or a presumed
nonzero reciprocal resultant are false. No reverse Jensen estimate is used.

**Primary source.** Chris Smyth, *The Mahler measure of algebraic numbers: a
survey*, arXiv:math/0701397, especially the definition/Jensen formula and
Section 5.1 on the distinction between reciprocal and nonreciprocal polynomials:
https://arxiv.org/pdf/math/0701397 . The algebraic examples above are explicit
checks, not quoted theorems about the unresolved unit-coefficient class.

## 8. Frequency compression: exact attainment and an effective approximation

**Mechanism tested.** Replace a possibly enormous integer frequency vector by
a bounded vector that preserves its relevant additive relations. Then the
general frequency problem for fixed `N` might become a finite calculation.

### 8.1. A direct Freiman shortcut fails

Ordinary Freiman isomorphisms preserve equal-length additive relations. Even
preserving these at **every order** need not preserve a cosine minimum. Every
two-element set is Freiman-isomorphic, at every order, to every other
two-element set: equality of sums of `s` terms just says that each of the two
elements occurs the same number of times on both sides. But

\[
\min_x(\cos x+\cos 2x)=-\frac98,
\qquad
\min_x(\cos x+\cos 3x)=-2.
\]

The first identity follows by minimizing `2u^2+u-1` for `u=cos(x)`; the second
is attained at `x=pi`. A relation involving zero, or an unequal number of
positive terms on its two sides, matters for a real part.

Adding a fixed zero anchor repairs this particular issue but not exact
preservation at a fixed order. Given any fixed `s`, choose an even integer
`M>s`. The anchored sets `{0,1,M}` and `{0,1,M+1}` have no nontrivial equal-length
relations of order at most `s`, so the natural map is a Freiman `s`-isomorphism.
Nevertheless `{1,M+1}` has minimum `-2`, whereas `{1,M}` has minimum strictly
greater than `-2`: equality would require `x=pi mod 2*pi` and simultaneously
`Mx=pi mod 2*pi`, impossible for even `M`.

Thus no fixed relation order independent of the actual frequency data
preserves these minima **exactly**, already at `N=2`.

### 8.2. Proved existence: the optimal value at each fixed N is attained

**Proposition.** For every positive integer `N`,

\[
\kappa_N=\inf_{0<a_1<\cdots<a_N\atop a_j\in\mathbb Z}
\left[-\min_t\sum_{j=1}^N\cos(a_jt)\right]
\]

is attained by at least one tuple of distinct positive integers. An attaining
tuple may be chosen with gcd one.

**Proof.** Take an admissible minimizing sequence `a^(j)` and the compact
connected subgroups of the `N`-torus

\[
H_j=\{(a^{(j)}_1t,\ldots,a^{(j)}_Nt)\pmod{2\pi}:t\in\mathbb R\}.
\]

By compactness of the hyperspace of nonempty closed subsets of a compact
metric space, pass to a subsequence converging in Hausdorff distance to a
nonempty closed set `H`. This limit is a subgroup: approximate any two of its
points by points in `H_j`, subtract them, and pass to the limit. It is also
connected, although connectedness of the limit is not needed below.

For `h in Z^N` let `chi_h(theta)=exp(i*h dot theta)`. The annihilator

\[
L(H)=\{h\in\mathbb Z^N:\chi_h\equiv1\text{ on }H\}
\]

is a finitely generated subgroup of `Z^N`; choose a finite integer basis
`h_1,...,h_r`. For each such `h_i`, uniform continuity and Hausdorff convergence
imply that `chi_{h_i}(H_j)` eventually lies in a small arc around `1`. But
`H_j` is connected, so this image, a compact connected subgroup of the circle,
is either `{1}` or the whole circle. It must therefore be `{1}`. Taking a
maximum over the finitely many indices proves that all `h_i` annihilate `H_j`
for every sufficiently large `j`.

A closed subgroup of a torus is the intersection of the kernels of its
annihilating integer characters. Hence `H_j` is contained in `H` eventually.
For the fixed continuous function

\[
\Phi(\theta)=-\sum_{i=1}^N\cos\theta_i,
\]

Hausdorff convergence gives

\[
\max_H\Phi=\lim_j\max_{H_j}\Phi=\kappa_N.
\]

Eventual inclusion gives `max_{H_j}Phi<=kappa_N`. Conversely every `H_j` comes
from an admissible positive distinct integer tuple, so the definition of the
infimum gives `max_{H_j}Phi>=kappa_N`. Equality follows for sufficiently large
`j`; any such tuple attains the infimum. Dividing by its gcd does not change
the range of the cosine sum. QED.

**Scope.** This is exact existence, not an effective maximum-frequency bound.
The Hausdorff subsequence and the stage at which all relations stabilize give
no numerical stopping criterion. The proof does not assume that an arbitrary
point of the limit torus is realized by one time of a fixed original sum.

### 8.3. Proved effective approximation with controlled relation order

**Proposition.** Let `N>=1` and `0<epsilon<=1`, and set

\[
M=\left\lceil\frac{4\pi N}{\varepsilon}\right\rceil,
\qquad
r=\left\lceil\frac{4N^2\log M}{\varepsilon}\right\rceil.
\]

For every sorted positive distinct integer tuple `a` there is a sorted positive
distinct integer tuple `b` with

\[
\max_j b_j\le(r+1)^N,
\qquad |K(a)-K(b)|\le\varepsilon.
\]

Thus the extremal value for fixed `N` can be approximated to any prescribed
positive error by an explicitly bounded finite search. This does not provide
an exact stopping criterion or an asymptotic Chowla estimate.

**Step A: a uniform measure near an extremum.** Put `g_a=N-f_a` and
`G_a=max g_a=N+K(a)`, so `0<=g_a<=2N`. Partition each coordinate circle into `M`
half-open intervals of length `2*pi/M`. The map

\[
t\mapsto(a_1t,\ldots,a_Nt)\pmod{2\pi}
\]

partitions the time circle into at most `M^N` measurable cells. Some cell `E`
has normalized measure at least `M^(-N)`. Choose a point `t_0` in `E`. Every
`u=t-t_0` with `t in E` has each phase `a_j*u` at circle distance at most
`2*pi/M` from zero. Since cosine is 1-Lipschitz in the circular distance,

\[
|g_a(t_*+u)-g_a(t_*)|\le\frac{2\pi N}{M}\le\frac\varepsilon2,
\]

where `t_*` is a maximizing time. Translation preserves the measure of `E`.
Therefore

\[
\operatorname{meas}\{t:g_a(t)\ge G_a-\varepsilon/2\}\ge M^{-N}.
\]

**Step B: a sufficiently high moment approximates that extremum.** Let
`T_a=(int g_a^r)^(1/r)`. Then

\[
T_a\ge(G_a-\varepsilon/2)\exp(-N\log M/r),\qquad T_a\le G_a.
\]

Using `1-exp(-u)<=u` and `G_a<=2N` gives

\[
0\le G_a-T_a
\le\frac\varepsilon2+\frac{2N^2\log M}{r}
\le\varepsilon.
\]

**Step C: identify exactly which additive relations the moment needs.**
Expanding

\[
g_a(t)=N-\frac12\sum_{j=1}^N(e^{ia_jt}+e^{-ia_jt})
\]

shows that its `r`th moment is determined by the truth values of

\[
h\cdot a=0\quad(h\in\mathbb Z^N,\ \|h\|_1\le r).
\]

These include unbalanced signed relations; equal-length Freiman relations
without a zero anchor would not suffice.

**Step D: a small integer tuple preserving this finite data.** For every such
nonzero `h`, impose the following constraint on a real vector `v`:

- `h dot v=0` when `h dot a=0`;
- `h dot v>=1` when `h dot a>0`;
- `(-h) dot v>=1` when `h dot a<0`.

This finite rational polyhedron is nonempty because `v=a` is feasible.
It includes `v_i>=1` and `v_{i+1}-v_i>=1`, since `r>=2`. In particular, it is
pointed and lies in the positive orthant. Minimizing `sum v_i` and choosing
a vertex of the bounded minimizing face produces a vertex `v` of the
polyhedron. At it, `N` independent active constraints have integer coefficient
rows of `l1` norm at most `r` and right-hand sides zero or one.

Let `D_0` be the nonzero determinant of this integer constraint matrix. By
Cramer's rule, `b=|D_0|v` has integer coordinates and remains positive and
strictly increasing. Every relevant zero relation and nonrelation is preserved.
Each Cramer numerator is a determinant whose row Euclidean norms are at most
`sqrt(r^2+1)<=r+1`; Hadamard's inequality gives

\[
0<b_i\le(r+1)^N.
\]

Thus `T_b=T_a`. Step B places both `G_a` and `G_b` in the same interval
`[T_a,T_a+epsilon]`; their difference is at most `epsilon`, proving the claim.
QED.

The bound has the size `exp(O(N log(N/epsilon)))`, with the explicit constants
above. It is enormous as an algorithm, and bounding or approximating each
fixed `kappa_N` does not prove a lower bound uniform in `N`.

An asymptotic consequence is nevertheless useful. Set `epsilon=1`. If a
counterexample family with `N->infinity` and `K(A)/sqrt(N)->0` exists, then
another such family exists with `max(A)<=exp(O(N log N))`: replace each set
by its compressed approximation, whose error divided by `sqrt(N)` tends to
zero. Conversely a uniform square-root lower bound for all sets obeying this
explicit frequency bound would extend to all frequencies, first with an
additive error one and then, by adjusting the absolute constant for small
`N`, in the original form. The elementary bound `K(A)>=1/2` suffices for that
last adjustment: integrating `(f_A+K(A))*(N-f_A)>=0` gives
`N*K(A)>=int f_A^2=N/2`. This is a reduction of the general asymptotic
problem to an explicit growing search domain, not a solution of that problem.

**Actual checks and source/novelty boundaries.** The counterexamples in 8.1
are exact. As an additional check, `{1,6}` and `{1,7}` have identical cosine
moments through order six; at order seven the former has moment `7/64` and
the latter zero. This demonstrates precisely the first unbalanced resonance.

Yufei Zhao's MIT notes define the balanced Freiman data and explain the scope
of the Ruzsa modeling lemma:
https://live.ocw.mit.edu/courses/18-225-graph-theory-and-additive-combinatorics-fall-2023/mit18_225_f23_lec22-25.pdf .
For topology of compact spaces of torus subgroups, a primary reference is
D. Chatterjee, H. Lekharu, R. Shah, arXiv:2511.04111:
https://arxiv.org/abs/2511.04111 . The displayed two propositions are proved
in full here, rather than inferred from the abstract of that reference.

The primary Chowla reference inspected was Idris Mercer, *Finite searches,
Chowla's cosine problem, and large Newman polynomials*, arXiv:1709.06612:
https://arxiv.org/pdf/1709.06612 . It poses the challenge of exact finite
computation for general fixed `N` and treats small lengths. The proofs above
do not establish the exact effective reduction sought there. No novelty or
priority claim is made; the brief source search is not a literature-completeness
certificate.

**Status.** Two valid general partial facts: exact attainment for each fixed
`N`, and explicit approximation by bounded integer frequencies. Exact
preservation by a fixed-order Freiman shortcut is false. No square-root lower
bound follows yet.

## 9. Infinitesimal irrational detuning and independent phases

**Mechanism tested.** Replace integers `a_j` by `a_j+epsilon*alpha_j`, choose
rationally independent perturbed frequencies, and use density on the torus to
make all phases almost `pi`. Their full-time infimum is `-N`. Could sending
`epsilon` to zero transfer this to the original integer frequencies?

**Exact test of the interchange of limits.** Take

\[
f_0(t)=\cos t+\cos 2t,\qquad
f_m(t)=\cos t+\cos((2+\sqrt2/m)t),\qquad m\ge1.
\]

The two perturbed frequencies are rationally independent, so their continuous
torus orbit is dense. In particular,

\[
\inf_{t\in\mathbb R}f_m(t)=-2\quad\text{for every }m,
\qquad
\inf_{t\in\mathbb R}f_0(t)=-\frac98.
\]

Thus

\[
\lim_{m\to\infty}\inf_t f_m(t)=-2
\ne-\frac98=\inf_t\lim_{m\to\infty}f_m(t).
\]

The value `-2` is an infimum rather than an attained minimum for irrational
frequency ratio. There is no contradiction with continuity: on every fixed
finite window `|t|<=T`,

\[
|f_m(t)-f_0(t)|\le\frac{\sqrt2 T}{m}\longrightarrow0.
\]

More quantitatively, to reach a value below `-3/2`, a perturbed trajectory
must have `|t|>3m/(8*sqrt2)`, because the unperturbed minimum is `-9/8`.
The witness times necessarily escape every fixed window.

One can also exhibit such escaping times directly. Put `eta=sqrt(2)/m` and
choose an odd integer `n_m` closest to `1/eta`, so
`|eta*n_m-1|<=eta`. At `t_m=n_m*pi`,

\[
f_m(t_m)=-1-\cos\bigl(\pi(\eta n_m-1)\bigr)
\le -2+\frac{\pi^2\eta^2}{2}
=-2+\frac{\pi^2}{m^2}.
\]

Thus `t_m` has size `pi*m/sqrt(2)` and already gives values tending to `-2`.
This is a quantitative check of the limit-exchange failure, independent of
any numerical search for an irrational orbit.

For a general fixed choice of the `alpha_j`, rational independence should not
be asserted for every nonzero `epsilon`: exceptional relations can occur.
The explicit sequence above avoids that issue completely.

**Status.** Independent phases after detuning are a valid fact. The limit
exchange required to solve the integer-frequency problem is false, with an
exact two-frequency counterexample. Torus characters also give a short density
proof: a character annihilating the perturbed orbit would require
`h_1+(2+sqrt(2)/m)h_2=0`, so both integers vanish. Standard torus duality then
identifies its closure with the whole two-torus.

## 10. Complex time, winding, and the signed area of the image curve

**Mechanism tested.** Regard `P_A(z)=sum_{a in A}z^a` as a complex curve
`gamma(x)=P_A(exp(ix))`. Use its winding, large signed area, or continuation
to imaginary time to force a leftward excursion of order `sqrt(N)`.

**Exact geometric calculation.** Fourier orthogonality gives

\[
\mathcal A_{\rm signed}
=\frac12\int_0^{2\pi}\operatorname{Im}
   (\overline{\gamma(x)}\gamma'(x))\,dx
=\pi\sum_{a\in A}a.
\]

This is a **signed area counted with winding multiplicity**, not necessarily
the ordinary area inside a simple boundary. The curve can self-intersect.

The dilation test is decisive against identifying those notions. Replacing
`A` by `qA` leaves the image curve, `N`, and `K(A)` unchanged, but multiplies
the displayed signed area by `q`. Already `A={q}` traverses the same unit
circle `q` times, has winding number `q` and signed area `pi*q`, yet its
leftward excursion is always exactly one. Consequently a direct inference
`K >= c*sqrt(signed area)` is false, even for an eligible one-frequency sum.
The obstruction is not removed by requiring gcd one: `A={1,D}` has gcd one,
`K(A)<=2`, and signed area `pi*(D+1)` tending to infinity.
Dividing by a frequency parameter removes that contradiction but supplies
no new `sqrt(N)` estimate by itself.

The derivative energy is likewise degree-dependent:

\[
\int_0^{2\pi}|\gamma'(x)|^2\,dx=2\pi\sum_{a\in A}a^2.
\]

Applying an isoperimetric inequality to a curve with multiplicity requires the
corresponding hypotheses; substituting the signed area as ordinary enclosed
area is not valid. Argument-principle information counts zeros/winding and
does not quantify how far left the curve travels. It can even be undefined
for winding about zero if the curve passes through zero, unless a suitable
regularization is specified.

**Imaginary-time test.** For `0<rho<1`, harmonic extension gives

\[
\operatorname{Re}P_A(\rho e^{ix})
=\sum_{a\in A}\rho^a\cos(ax)
=(\text{Poisson kernel at }\rho)*f_A(x)\ge-K(A).
\]

Thus an interior negative point, if found, would be a valid boundary lower
bound. However the continuation introduces unequal decaying weights and
does not itself produce such a point. For imaginary time of the opposite
sign, `rho>1`, there is no such boundary implication. With `A={D}` one has
`min_x Re P_A(rho exp(ix))=-rho^D`, arbitrarily negative outside the circle,
although the original `K(A)=1`.

**Status.** The signed-area and Poisson-extension identities are valid partial
facts. The area-to-left-width inference and outside-to-boundary transfer are
false. Replacing a quantitative contour claim by zero-freeness would merely
return to the already refuted weaker shortcut; it is not a new solution.

## Verification scope

Each approach above has an actual analytical test: a proved estimate,
counterexample, or a complete partial theorem with proof. The companion
`arithmetic_checks.py` and `arithmetic_checks.json` provide small arithmetic
and numerical sanity checks, including exact integer moment coefficients.
They are not used as proofs of asymptotic statements. No external publication,
upload, human peer review, or formal proof-assistant verification is claimed.