# General Chowla research checkpoint

18 October 2026 UTC. Intermediate working record. No general proof or
counterexample has been obtained. No novelty claim or external verification of
a new theorem is made. The application has not been changed.

## Target and scope

For every finite nonempty set A of distinct positive integers, put
f_A(x)=sum_{a in A}cos(ax), N=|A|, K(A)=-min_x f_A(x).
The target is one absolute c>0 with K(A)>=c sqrt(N), uniformly in both N and
the largest frequency. A disproof needs a sequence with N tending to infinity
and K(A)/sqrt(N) tending to zero. All amplitudes are exactly one and all phases
zero. Finite cyclic sampling, fractional amplitudes, signed coefficients, and
finite exhaustive searches do not resolve this target.

## General routes inspected in this pass

1. **Full neighborhood spectral certificate.** Positivity of 2K+2f_A gives
   both the odd and even blocks of the distance-graph Gram matrix on
   {0} union A union (-A). A universal negative eigenvalue of order sqrt(N)
   for this particular matrix would suffice, but is unproved and may be
   stronger than Chowla. Compressing the even block against the all-ones
   vector gives exactly the known Roth estimate.
2. **Damped analytic product.** For Q_delta(z)=product_(a in A)(1-delta z^a),
   ||Q_delta||_2^2 <= (1-delta)^(-2K). A uniform positive gap of this norm
   above one at delta=c0/sqrt(N) would suffice. Such a gap is not proved.
   An exponential/Laplace version has the same unresolved issue.
3. **Probability convolution law.** A nonnegative probability density mu in
   L1(T) satisfying mu*mu=rho mu+(1-rho)1 has Fourier coefficients exactly
   0 or rho off zero. Riemann-Lebesgue makes their support finite, so this
   formulation is exactly reversible to the original cosine problem.
   Bounding its support size by C rho^(-2) is equivalent to Chowla, not
   an independently established lemma.

## Verified reasons several proposed shortcuts fail

- A base-power Sidon difference family gives the sine-block spectrum
  k-2 (multiplicity k-1) and -2 (multiplicity (k-1)(k-2)/2). Its positive
  and negative spectral edges have no universal bounded ratio.
- The Bernoulli-Fejer construction gives actual sets A with T(A) of order
  N^2 and K(A)=O(sqrt(N log N)). Thus neither T<=C K N nor the analogous
  sine-block Frobenius estimate holds for all A. Root and spectral agent
  independently checked the counting, concentration, and constants.
- For every fixed moment-test degree d, there are genuine large cosine
  sums whose tests E[(f+L_d)p(f)^2]>=0 all hold for a constant L_d, for
  every polynomial p of degree at most d. Their first 2d+1 moments match
  a probability measure with uniformly bounded negative support. This
  concerns fixed-degree scalar-moment methods only, not arbitrary
  frequency-sensitive certificates or degree increasing with N.
- Adding a largest-frequency factor need not increase the damped product
  norm. A=1,...,100, delta=1/10, added frequency 140 gives an exact ratio
  less than 49/50. Root reran the integer-only certificate successfully.
- Fractional factor weights can destroy the proposed norm gap. Direct
  rounding to zero/one coefficients has an unavoidable L2 error of
  order sqrt(N), but this does not rule out all possible one-sided
  encodings or disprove Chowla.
- Stable spectral denominators and even a full Hardy H2 norm bound for
  the associated Schur transform do not imply the required pointwise
  positivity. The exact interval example is recorded in the notes.
- Connectedness of the circle does not replace absolute continuity:
  (1-rho)m+rho delta_0 satisfies the same convolution law with infinitely
  many nonzero Fourier coefficients. Fejer smoothing loses their exact
  zero/rho values.

These are obstructions to methods. None is a counterexample to Chowla.

## Source checks

The September 2026 preprint by Abhishek Shankar gives a general n^(1/5)
bound, using two analytic estimates from Bedert and an exact boundary
identity. It explicitly leaves the square-root exponent unresolved:
https://arxiv.org/html/2609.05338v1

Bedert's underlying version 3:
https://arxiv.org/html/2509.05260v3

The October 2026 OpenAI math catalogue was searched for cosine, Chowla,
Littlewood, Newman, and related terms. Its two-point multiplicative Chowla
result is a different problem. Its real-sign Littlewood results provide no
established transition to the zero/one cosine target here:
https://raw.githubusercontent.com/openai/math/main/CONTENTS.md

A specific integer-versus-modular-correlation error in the short Singer
flatness proof arXiv:2210.15480v2 was checked with S={0,1,3}, q=7. This
invalidates that proof step, not every possible flatness result. Numerical
Singer experiments in the companion files are explicitly uncertified and
are not used as asymptotic evidence.

## Detailed working records

- general_spectral/SPECTRAL_ATTEMPT.md
- general_fejer_barrier.md
- general_moments/moment_obstructions.md
- general_moments/product_route_audit.md
- general_moments/probability_convolution_audit.md
- general_factorization/FACTORIZATION_NOTES.md
- general_factorization/DAMPED_PRODUCT_NOTES.md
- general_factorization/exact_product_monotonicity.py
- general_factorization/exact_product_monotonicity.json

The earlier finite-search report and package predate these general-method
notes. Neither those finite results nor these method obstructions should be
described as a solution of the original universal conjecture.