# Approaches 27–29: complementary codes, power-sum cancellation, and support flattening

Research record, 18 October 2026 UTC. This document records mathematical
formulations, proofs, exact checks, and limitations suitable for a later
research article. It does not claim a solution to Chowla's conjecture or
mathematical priority. It is a scientific account of the checked arguments,
not a transcript of private deliberation.

## Target and evidence standard

For a nonempty finite set of distinct positive integers A, put

\[
f_A(x)=\sum_{a\in A}\cos(ax),\qquad
K(A)=-\min_{x\in\mathbb R}f_A(x),\qquad N=|A|.
\]

The target is one absolute c>0 such that K(A)>=c sqrt(N) for every A.
A counterexample construction must supply an unbounded sequence with
K(A)/sqrt(N)->0. Good behavior at finitely many sample points does not
certify this. The original unit coefficients and positive integer
frequencies are retained in the actual constructed sets below.

**Outcome:** approaches 27 and 28 produce admissible infinite families,
but a continuous negative witness proves that K/sqrt(N) tends to infinity
in both families. Thus these particular proposed counterexample families
fail asymptotically, not just at the finitely tested sizes. Approach 29
has an exact two-frequency obstruction for every fractional exponent
0<=p<1. None of these statements proves the general conjecture or excludes
all constructions inspired by the same subjects.

The two code constructions have different design premises. In 27 the
premise is cancellation of aperiodic autocorrelations between a pair of
polynomials. In 28 it is equality of increasing numbers of frequency
power sums between two sets. Their conversion to positive masks exposes
a related Dirichlet-kernel obstruction; that shared obstruction is made
explicit rather than presented as two independent general discoveries.

## 27. Complementary codes: erase unwanted correlations before removing signs

### 27.1 Proposed mechanism

Golay complementary pairs have signed coefficients but cancel all their
nonconstant autocorrelations when squared moduli are added. The proposal
was to exploit that exact cancellation, then convert the signs to 0/1
frequency masks, in the hope of producing an admissible positive cosine
sum with very small negative excursions.

Use the following completely explicit recursion:

\[
P_0(z)=Q_0(z)=1,
\]
\[
P_{m+1}(z)=P_m(z)+z^{2^m}Q_m(z),\qquad
Q_{m+1}(z)=P_m(z)-z^{2^m}Q_m(z).
\]

Both polynomials have length L=2^m and coefficients in {-1,+1}.
On |z|=1, the elementary parallelogram identity gives

\[
|P_{m+1}|^2+|Q_{m+1}|^2
=2(|P_m|^2+|Q_m|^2).
\]

Consequently

\[
\boxed{|P_m(z)|^2+|Q_m(z)|^2=2L},\qquad
|P_m(z)|\le\sqrt{2L}.
\]

These identities are proved by the displayed recursion; no numerical
flatness assumption is used.

### 27.2 Direct conversion to an admissible set

Write P_m(z)=sum_{j=0}^{L-1}p_j z^j and
J_L(z)=sum_{j=0}^{L-1}z^j. Since p_0=1, the positive mask with zero
frequency removed is

\[
A_m=\{j:1\le j<L,\ p_j=1\},\qquad
\sum_{a\in A_m}z^a=\frac{J_L(z)+P_m(z)}2-1.
\]

All retained coefficients are exactly 1 and every frequency is positive.
At z=1 the recursion has matrix [[1,1],[1,-1]], whose square is 2I.
It follows exactly that

\[
P_m(1)=2^{\lceil m/2\rceil},\qquad
N_m=\frac{L+2^{\lceil m/2\rceil}}2-1.
\]

Now choose a point in the continuous circle, not a DFT-grid certificate:

\[
\theta_L=\frac{3\pi}{2L},\qquad
\alpha_L=\frac{3\pi}{4L}.
\]

The finite cosine-sum identity gives

\[
\Re J_L(e^{i\theta_L})
=\frac12+\frac{\sin((L-1/2)\theta_L)}{2\sin(\theta_L/2)}
=\frac{1-\cot\alpha_L}{2}.
\]

Therefore

\[
f_{A_m}(\theta_L)
\le -\frac{\cot\alpha_L}{4}+\sqrt{\frac L2}-\frac34,
\]

and hence

\[
\boxed{K(A_m)\ge
\frac14\cot\!\left(\frac{3\pi}{4L}\right)
-\sqrt{\frac L2}+\frac34.}
\]

In fact the actual witness itself, divided by L, converges to
-1/(3*pi), because |P_m|/L tends to zero. Since N_m/L tends to 1/2,

\[
\frac{-f_{A_m}(\theta_L)}{N_m}\longrightarrow\frac{2}{3\pi},
\qquad
\liminf_{m\to\infty}\frac{K(A_m)}{N_m}\ge\frac{2}{3\pi}.
\]

Thus K(A_m)/sqrt(N_m)->infinity. This rules out the specific positive-mask
family as a counterexample to Chowla. It does not assert that the displayed
theta is a global minimizer or that K(A_m)/N_m has the same limit.

The same argument also applies to the negative mask, with its appropriate
constant term, and to Q_m: the bound |P_m|,|Q_m|<=sqrt(2L) cannot cancel
the order-L negative lobe introduced by J_L.

### 27.3 Does adding four positive masks restore the complementary miracle?

One can instead retain all four 0/1 polynomials

\[
B_\pm=(J_L\pm P_m)/2,\qquad
C_\pm=(J_L\pm Q_m)/2.
\]

Their squared moduli obey the exact identity

\[
|B_+|^2+|B_-|^2+|C_+|^2+|C_-|^2
=|J_L|^2+L
=2L+2\sum_{d=1}^{L-1}(L-d)\cos(dx).
\]

Nonnegativity survives, but the nonconstant Fourier coefficients are
L-d, rather than 0 or 1. This specific sign-removal operation reinstates
triangular multiplicities. Deleting those multiplicities is an additional
operation and does not follow from complementarity.

### 27.4 Reproducible evidence

The accompanying script verifies every aperiodic autocorrelation and
every coefficient of the four-mask identity for m=0,...,8 using integers.
It constructs five larger masks, at m=6,8,10,12,16, and supplies rigorous
rational lower bounds at theta_L.

For L>=2, alpha_L<sqrt(2), so
cos(alpha)>=1-alpha^2/2 and sin(alpha)<=alpha yield

\[
\frac14\cot\alpha_L\ge\frac{L}{3\pi}-\frac{3\pi}{32L}
>\frac{7L}{66}-\frac{33}{112L},
\]

where the final step uses pi<22/7. Hence the entirely rational quantity

\[
b_{27}(L)=\frac{7L}{66}
-\frac{\lceil\sqrt{2L}\rceil}{2}
+\frac34-\frac{33}{112L}
\]

is a strict lower bound for K(A_m). At m=16,

\[
L=65536,\quad N=32895,\quad
K(A_m)>\frac{1639845723071}{242221056}>6770,
\]

and integer comparison of squares proves K(A_m)>37 sqrt(32895).
The floating-point witness is approximately -6894.6886; this approximation
is only a sanity check and is not used to justify the rational certificate.

### 27.5 Source context and limitation

The established complementary-pair terminology and its polynomial
identity are described in Pantelimon Stanica, *A Boolean Functions View on
the Golay-Rudin-Shapiro Sequence*, author-hosted paper, equation (1.2):
https://faculty.nps.edu/pstanica/research/2020JCNT_RudinShapiroSeq.pdf .
The recursion and the obstruction above are proved explicitly here.
No novelty is asserted for these identities or consequences.

## 28. Prouhet cancellation: match more and more frequency moments

### 28.1 Proposed mechanism

Try to make the first m frequency power sums agree between two sets,
so that their signed difference is invisible to Taylor expansion to
increasing order. This is a different condition from cancellation of
autocorrelations in approach 27. Frequency power sums sum a^k must also
not be confused with distribution moments integral f_A(x)^k dx.

For L=2^m let

\[
t_j=(-1)^{s_2(j)},\qquad
T_m(z)=\sum_{j=0}^{L-1}t_jz^j
=\prod_{r=0}^{m-1}(1-z^{2^r}),
\]

where s_2(j) is the number of 1s in the binary expansion. Equivalently,
the sign word is replaced by its concatenation with its own negative
at each iteration.

The two positive-frequency sets are

\[
A_m^+=\{j+1:t_j=1\},\qquad
A_m^-=\{j+1:t_j=-1\}.
\]

Both have exactly N=L/2 terms. Shifting j to j+1 removes zero frequency
while preserving the power-sum equalities by the binomial formula.

### 28.2 Exact increasing-order cancellation

Each factor 1-z^(2^r) has a simple zero at z=1. Thus T_m has a zero of
order exactly m there. Applying (z d/dz)^k and evaluating at 1 gives

\[
\sum_{a\in A_m^+}a^k=\sum_{a\in A_m^-}a^k
\quad(0\le k<m).
\]

The first nonzero difference is exactly

\[
\sum_{a\in A_m^+}a^m-\sum_{a\in A_m^-}a^m
=(-1)^m m!\,2^{m(m-1)/2}.
\]

Indeed the leading coefficient at z=1 is
(-1)^m 2^{m(m-1)/2}(z-1)^m; lower derivative terms vanish.

### 28.3 Continuous witness: cancellation is too effective in the wrong term

The positive-frequency generating polynomial is

\[
\sum_{a\in A_m^+}z^a=\frac{z(J_L(z)+T_m(z))}{2}.
\]

Use the same explicit theta_L=3*pi/(2L), alpha_L=3*pi/(4L).
Because |1-e^(iu)|<=|u|,

\[
|T_m(e^{i\theta_L})|
\le\prod_{r=0}^{m-1}2^r\theta_L
=\left(\frac{3\pi}{2}\right)^m2^{-m(m+1)/2}
=:E_m.
\]

The right side tends to zero faster than exponentially in m.
For the shifted Dirichlet sum the exact formula is

\[
\Re(e^{i\theta_L}J_L(e^{i\theta_L}))
=\sum_{j=1}^{L}\cos(j\theta_L)
=-\frac{1+\cot\alpha_L}{2}.
\]

It follows that

\[
\boxed{K(A_m^+)\ge
\frac14\cot\!\left(\frac{3\pi}{4L}\right)
+\frac14-\frac{E_m}{2}.}
\]

The same inequality holds for A_m^- because the sign of T_m changes,
but its modulus bound does not. In particular,

\[
\frac{-f_{A_m^\pm}(\theta_L)}{N}
\longrightarrow\frac{2}{3\pi},\qquad
\frac{K(A_m^\pm)}{\sqrt N}\longrightarrow\infty.
\]

The mechanism of failure is precise. The signed polynomial has excellent
cancellation near z=1, but passing to an all-positive mask adds one half
of a Dirichlet polynomial. At the chosen scale the signed correction is
tiny and cannot cancel the Dirichlet negative lobe. Increasing the number
of matched powers does not repair this conversion.

This is not the previously tested product of positive cosine digit
polynomials. Here a signed substitution sequence is used to enforce
unbounded-order equal-power identities, and the target set is its binary
mask. The verified obstruction concerns this exact conversion.

### 28.4 Reproducible evidence

The script verifies with arbitrary-precision integers all power equalities
for m=1,...,10, including the stated first nonzero difference. It separately
generates the coefficient product to verify the sign formula.

At m=4,6,8,10,12,16 it also computes the exact rational witness bound

\[
b_{28}(L,m)=\frac{7L}{66}+\frac14-\frac{33}{112L}
-\frac12\left(\frac{33}{7}\right)^m2^{-m(m+1)/2},
\]

again using only pi<22/7. For m=16, N=32768, this yields
K(A_m^+)>6951 and K(A_m^+)>38 sqrt(32768), verified by rational and integer
comparisons. The floating-point witness is approximately -6953.8356,
recorded only as a sanity check.

### 28.5 Source context and limitation

The standard Prouhet-Tarry-Escott and generalized Thue-Morse context is
given by Ethan D. Bolker, Carl Offner, Robert Richman and Catalin Zara,
*The Prouhet-Tarry-Escott Problem and Generalized Thue-Morse Sequences*:
https://arxiv.org/abs/1304.6756 .
The required binary product, equal-power identities and continuous
negative witness are proved above, so the check does not depend on
importing an unstated theorem from that paper. No claim is made about
other automatic sequences or more elaborate encodings of matched powers.

## 29. Fractional Fourier powers: tropical removal of multiplicities

### 29.1 Proposed mechanism

Start with a nonnegative trigonometric polynomial whose Fourier
coefficients are nonnegative integers. Repeated frequencies appear as
coefficients greater than 1. For 0<p<1, replace each positive Fourier
coefficient c by c^p. As p tends to zero, every positive coefficient tends
to 1: this extracts the support and would erase multiplicity without
choosing individual copies. Applying the same operation to the constant
coefficient would also reduce the cost of the positivity certificate.

The crucial test is whether this operation preserves nonnegativity.
The answer is no, with a single exact example covering every p<1.

### 29.2 Exact obstruction for every fractional exponent

Use the nonnegative square

\[
H(x)=(1+2\cos x)^2=3+4\cos x+2\cos 2x.
\]

Its complex Fourier coefficients are c_0=3, c_{+/-1}=2, c_{+/-2}=1.
After the proposed coefficientwise power transform, the polynomial is

\[
H_p(x)=3^p+2\,2^p\cos x+2\cos 2x.
\]

Set y=cos x. Completing the square gives

\[
H_p(x)=4(y+2^{p-2})^2+3^p-2-4^{p-1}.
\]

For 0<=p<=1 the minimizing value y=-2^(p-2) lies in [-1,1], so the exact
continuous minimum is

\[
m(p)=\min_x H_p(x)=3^p-2-4^{p-1}.
\]

At p=1 the minimum is zero. On the whole interval 0<=p<=1,

\[
m'(p)=4^{p-1}\bigl(4\log3\,(3/4)^p-\log4\bigr)
\ge4^{p-1}(3\log3-\log4)>0,
\]

because 27>4. Therefore

\[
\boxed{\min_xH_p(x)<0\quad\text{for every }0\le p<1.}
\]

At p=1/2 the exact value is sqrt(3)-5/2<-3/4. At p=0, interpreting
zero coefficients as remaining zero, the support polynomial is

\[
H_0(x)=1+2\cos x+2\cos2x,
\qquad\min H_0=-5/4.
\]

This agrees with K({1,2})=9/8 after translating the normalization.

### 29.3 What is and is not excluded

The failed implication is precisely:
"nonnegative polynomial with nonnegative Fourier coefficients remains
nonnegative when every coefficient, including c_0, is replaced by its
fractional power."

This example does not refute every operation that flattens only the
nonconstant coefficients while keeping the original constant term. For
this particular example, keeping c_0=3 gives minimum 1-4^(p-1)>=0.
It also does not refute a corrected transformation with a quantified
additional constant term. Such a correction would need to be bounded
on the sqrt(N) scale to contribute to the target; no such general bound
was obtained here.

### 29.4 Evidence

The continuous minimum and all-p sign statement have exact analytic
proofs above. The script checks exact rational comparisons for p=0,
p=1 and p=1/2 and records eight floating-point samples only as sanity
checks. The conclusion for every p<1 is not extrapolated from samples.

## Reproduction, audit scope and article wording

Run:

    python chowla_research/round2/constructions_27_29.py

The script writes constructions_27_29_results.json next to itself. All
assertions passed on the recorded execution. No exact global minimum was
claimed for the large masks; their certificates establish single
negative values, which suffice to lower-bound K.

Appropriate article-level conclusions are:

1. The specified positive masks of the complementary recursion have
   continuous negative excursions of order their cardinality. They
   therefore fail as a proposed asymptotic counterexample family.
2. The specified Prouhet masks have the same failure despite equality of
   increasingly many frequency power sums.
3. Fractional coefficient powers do not preserve positivity, even for a
   fixed degree-two cosine polynomial and even for exponents arbitrarily
   close to one from below.

These are verified limited statements, not a claim that complementary
codes, automatic sequences, or positivity-preserving transforms cannot
contribute to another valid approach. External peer review and formal
proof-assistant verification have not been performed.