Constants we actually certified

Do not read this as an exponent improvement. Balog 1989 remains the published binary record.

Trivial covering (not a dent)

For every integer $n\ge 2$,

via $n=m^2+r$ or $n=(m-1)^2+(2m-1)$. Independently checked on $[2,2\cdot 10^5]$ and on a handful of $10^{12}$–$10^{18}$ spot checks (certs/trivial_cover.json). Balog already calls the $N^{1/2}$ statement almost trivial.

Exact pointwise value

Lower bound: $131486759\equiv 7\pmod 8$ and $(-m/q)=1$ for every odd prime $q\le 79$, so the Jacobi obstruction forbids two 79-smooth summands. Upper bound: $649+131486110$ with $649=11\cdot 59$ and $131486110=2\cdot 5\cdot 7^2\cdot 53\cdot 61\cdot 83$. There is no prime strictly between 79 and 83.

Certified $G(y)$

See certs/g_certified.json. Through $y=79$ this independently reproduces OEIS A062241. In particular

with $727473$ 73-smooth and $808372$ 79-smooth integers through that point.

Exact exception lists on a prefix (residue)

Integer tests, C sweep plus Python recomputation of every listed $F$:

These lists are not a proof for all large $n$. Balog–Sárközy already have $n^{2/5}$ for large $n$; Balog 1989 has $n^{0.2695+\varepsilon}$. A prefix with no further exception is search residue.

What we did not obtain

• No explicit $\varepsilon<1/2$ with an infinite checkable covering.
• No finite residue cover that lifts.
• No exponent below $4/(9\sqrt e)$.
• No effective $N_0$ for Balog 1989.

2026-08-27 closed-form search (still not a dent)

See q1/certs/. Square / triangular / cube / power-of-two / floor-divisor all have holes persisting through $n=20000$ at $9/20$, $2/5$, $1/3$, and $27/100$. Floor-divisor and power-of-two have explicit infinite failure families. Two-factor $u\le n^{1/5}$ matches the $F$ exception prefixes and does not lift.