q1 — closed-form templates and polynomial obstruction

Continuation of the 2026-08-17 search. The parent folder already has the square covering, $F(131486759)=83$ , $G(y)$ through $y=79$ , and a failed finite-shift cover. This folder looks at templates that could have lifted: a closed-form $a(n)$ , a floor-divisor rewrite of the short-interval idea, a two-factor Balog–Sárközy shape, and the whole polynomial-value family.

Nothing here is an exponent below Balog $4/(9\sqrt{e})$ , and nothing is an infinite covering at any $\varepsilon<1/2$ .

Certificates

file	what it checks
`certs/poly_obstruction.json`	degree $d\ge 2$ remainder windows have exponent $1-1/d\ge 1/2$
`certs/infinite_family.json`	explicit $n=Pu+1$ failures of floor-divisor; $n=2^k+q$ failures of largest-power-of-two
`certs/q1_search.json`	first holes for square / triangular / cube / power-of-two / floor-divisor at $9/20,2/5,1/3,27/100$ , plus two-factor $u\le n^{1/5}$

Replay:

./run_all.sh

A last hole on a prefix is not $N_0$ . The square template at exponent $1/2$ is the covering already recorded in the parent folder.