The Parity Barrier

Why the binary Goldbach conjecture remains unproved: where the walls stand, how they are built, and which detours have already been checked and closed.

the Goldbach conjecture remains openno complete proof is known

The central barrier

“Average, $L^2$ , almost-all, or any other averaged form of control is not sufficient to prove positivity of $r(N)$ for every fixed even $N$ .”

Here $r(N)$ denotes the number of representations of an even integer $N$ as a sum of two primes:

r(N) = \#\{(p, q) : p + q = N,\ p, q\ \text{prime}\}

The Goldbach conjecture asserts that $r(N) \geq 1$ for every even $N \geq 4$ . Analytic methods are good at estimating $r(N)$ “on average” — over all $N$ at once, in the $L^2$ norm, or outside a small exceptional set. But the conjecture is a statement about each individual $N$ , and between “almost all” and “all” there is a principled gap.

This obstruction is known as the parity barrier: classical sieve methods do not distinguish primes from products of two primes — the two have the same parity of the number of prime divisors. In the language of the circle method the same defect appears as a one-logarithm deficit on the minor arcs: the minor-arc contribution is controlled to within an accuracy that falls short of a pointwise conclusion by exactly one logarithmic factor.

Below: how this barrier appears in concrete formulations, which six hard nodes must be untied, and which routes have already been honestly checked and found to be dead ends. The problem itself is stated on the Problem page; the status of the positive results is on the Results page.

The Friedlander–Iwaniec dichotomy

The landscape of the problem splits into two fundamentally different cases, according to whether a Siegel zero exists — an exceptional real zero of a Dirichlet $L$ -function lying abnormally close to $s = 1$ . This fork determines what one can hope to prove by present-day methods at all.

Case 1: a Siegel zero exists

In this world the anomalous zero itself becomes a resource: it yields a result that is unattainable in the “typical” situation. This is where the conditional theorem BK+ lives; it is discussed in detail on the Results page.

conditional branch — theorem BK+

Case 2: no Siegel zeros

If exceptional zeros do not exist, the problem remains open. In difficulty this branch is comparable to questions at the level of the generalized Riemann hypothesis (GRH): progress here would require control over the zeros of $L$ -functions that is currently beyond reach.

open — GRH level

Six hard nodes

The parity barrier is not a single obstacle but a bundle of concrete open problems. Untie any one of these six nodes and the landscape changes; for now all of them stand.

Node	The obstruction	Status
$\mathrm{SM}^*_{\alpha}$ pointwise shifted Möbius / Chowla	Pointwise (rather than averaged) control of shifted sums with the Möbius function — a pointwise form of the two-point Chowla problem, without which the minor arcs do not close.	Open problem
$\mathrm{EH}_{\mu}$ Elliott–Halberstam for Möbius-weighted sums	An analogue of the Elliott–Halberstam distribution-level conjecture — but for sums weighted by the Möbius function, where sign oscillation breaks the standard arguments.	Open problem
Zero location $\beta < 3/4$	One needs control over the zeros of $L$ -functions in the strip with real part $\beta < 3/4$ — deeper than the known zero-free regions allow.	Open problem
BMAH Bilinear Minor-Arcs Hypothesis	A hypothesis on bilinear sums over the minor arcs: it is exactly what would close the logarithm deficit, but it is itself an unproved assumption.	Open problem
Strengthening of Pilatte-type bounds	Pilatte's logarithmically averaged bounds must be strengthened to pointwise ones — a transition that is precisely the content of the parity barrier in its expansion formulation.	Open problem
A Chen → Zhao bridge	No bridge has been built between the $(1 + a)$ world (Chen Jingrun-type theorems on “prime + almost prime”) and the exceptional-set world (Zhao-type results on the density of exceptions) — the two approaches live in different coordinates.	Open problem

Ten languages of the parity barrier

One and the same obstruction can be formulated in ten different mathematical languages — from sieve functions to automorphic spectra. This vocabulary is collected from draft 2 (see Publications); each formulation illuminates the barrier from its own side.

sketchThe equivalence of all ten formulations has the status of a sketch, not of a proved theorem.

The Rosser–Iwaniec lower sieve. The lower-bound sieve function vanishes on the critical segment: $f(s) = 0$ for $s \leq 2$ . The classical linear sieve in its optimal Rosser–Iwaniec form gives no positive lower bound exactly where one is needed for Goldbach.
Pretentious number theory. In the Granville–Soundararajan language the obstruction is the “non-pretentiousness of $\mu$ ”: the Möbius function does not pretend to be any Dirichlet character of small order, which blocks the pretentious arguments that succeed in neighbouring problems.
Type I/II bilinear sums. The standard Vaughan decomposition splits the problem into Type I and Type II sums. The known estimates here are optimal (the Ford–Maynard results): there is no slack left to squeeze out of the technique without a new idea.
Expansion: the logarithmic average. A logarithmically averaged version of the two-point Chowla conjecture is a theorem of Tao, with improved quantitative bounds by Pilatte (arXiv:2310.19357, preprint) — but only in averaged form. Passing from the logarithmic average to a pointwise statement is the barrier in its expansion formulation.
The Fourier $L^2$ Cauchy–Schwarz gap. In estimating the minor-arc contribution, the Cauchy–Schwarz inequality unavoidably loses a factor of order $\sqrt{\log N}$ . This gap is the quantitative face of the “one-logarithm deficit”.
Weight architecture. Every weight construction runs into the distribution-level limit of its input sums. The current record for the relevant class is level $5/8$ (Pascadi, arXiv:2505.00653), and even that is insufficient for a pointwise conclusion.
The automorphic spectral obstruction. In automorphic language the barrier appears as a spectral obstruction: the contribution of the spectral parameters cannot be separated and estimated with the required accuracy — the obstruction sits in the very spectral decomposability of the problem.
The DFI/Delta method. The Duke–Friedlander–Iwaniec delta method detects diagonal conditions, but in the Goldbach configuration its accuracy runs into the same logarithmic deficit: the method reaches the barrier but does not pass through it.
The delta-symbol invariant. Under all rewritings of the problem through the delta symbol (extracting the condition $p + q = N$ ), an unchanging defective invariant survives — the transformations change the appearance of the sum but do not remove the obstruction.
Phase–amplitude co-peaking. In the language of harmonic analysis: the peaks of the phase and of the amplitude of the summands coincide (co-peaking). This coincidence blocks effective averaging — the cancellation on which estimates of oscillating sums usually rely never gains strength.

The no-go gallery: honest negative results

A separate value of the project is its recorded dead ends. Each item below is not a “we tried and failed” but an established statement that a specific route is closed.

(a) The PPE chain is refuted

The edge “Lemma IA $\Rightarrow$ Goldbach” of the PPE chain has fallen: there is a counterexample in which Möbius cancellation is present only over composite numbers, and this does not suffice to derive the conjecture. The refutation did not remain on paper — it is proved in Lean (file PPEChain.lean; see the Formalization page).

refuted — proved in Lean

(b) The $3/5$ Möbius level does not transfer

The Grimmelt–Teräväinen result on distribution level $3/5$ for the Möbius function — proved there for triply well-factorable weights (arXiv:2207.08805, Theorem 1.3) does not transfer to the Huang–Li hypothesis: the transfer is blocked by a coefficient-geometry obstruction — the coefficient structure of the required problem is incompatible with the geometry of their argument.

theorem — obstruction established

(c) The parametric limit of the Li–Liu architecture

Within the Li–Liu architecture, with the same constants, the parameter is bounded below: $a \geq 1.8999516$ . One cannot tune the parameters down to $a = 1$ — the defining function diverges:

g_3(a) = 8\,\log\frac{1}{a - 1}\ \xrightarrow[a \to 1^{+}]{}\ +\infty

The architecture has a built-in limit, and no tuning inside it reaches the Goldbach case $a = 1$ .

theorem — limit of the architecture

(d) The parity extremal: an irremovable factor $1/2$

There exists an extremal construction in which the local densities and the separate first moments coincide with the “correct” ones, while the quantity of interest vanishes: any method relying only on these data is left with an irremovable factor $1/2$ . This is the exact mathematical form of the parity barrier.

theorem — boundary of the method

Why an honest map of dead ends is a result

A page of obstructions may look like an admission of weakness. It is, in fact, a working tool. Every recorded dead end saves years: the next researcher — perhaps the reader — will not spend them on a route that is already proved to lead nowhere.

A negative result proved with the rigour of a positive one is navigation: a map on which not only the roads but also the cliffs are marked. This is why the PPE chain is formally refuted in Lean, and why the parametric limits are written out to the seventh decimal place. More on this style of work is on the Methodology page.

If you see a way to untie one of the six nodes — or spot an inaccuracy in the map itself — please write. Discussion is welcome at sergey@zheleznov.com.