OBJECTIVE AND SOURCE REVIEW
Erdős #782; independent attempt, 2026-09-23. Do the squares contain quasi-progressions of unbounded length with one fixed bound on the spread of consecutive gaps? Do they contain Hilbert cubes of unbounded dimension? I read the entire supplied
782.tex, including its Pell construction, five-term examples, and conditional cube obstruction. Its displayed numerical Pell solution has the wrong sign.
WORK: CORRECTING AND CERTIFYING THE PELL FAMILY
In fact \[
1068^2-73\cdot125^2=-1,
\] not $+1$. Nevertheless this yields a completely explicit infinite width-one family. Define $x_0=1,y_0=0$ and \[
x_{r+1}=1068x_r+9125y_r,\qquad
y_{r+1}=125x_r+1068y_r.
\] Multiplication of $x_r+y_r\sqrt{73}$ by $1068+125\sqrt{73}$ proves inductively \[
x_r^2-73y_r^2=(-1)^r.
\] For $r\ge1$, the four positive squares \[
y_r^2,\quad25y_r^2,\quad49y_r^2,\quad x_r^2
\] have consecutive gaps $24y_r^2,24y_r^2,24y_r^2+(-1)^r$. All are positive, and their spread is exactly one. The recurrence strictly increases $y_r$, so the families are distinct. Squaring the initial unit gives the genuine positive-norm solution $(x_2,y_2)=(2281249,267000)$. Thus the sign error does not invalidate the intended infinite construction; it changes which of the two adjacent possible gaps occurs. As a separate elementary check, the five squares $1,9,16,25,36$ have gaps $8,7,9,11$, hence spread four. This improves the width-six example in the supplied text, but is only a finite witness and is not asserted to be new.
THE CUBE QUESTION AND REMAINING GAP
The checked April 2026 preprint of Bremner--Elsholtz--Ulas proves infinitely many dimension-three square cubes, including a quantitative lower bound. This updates the supplied background, but fixed dimension three does not imply unbounded dimension. Neither the recurrence above, which always gives four terms, nor the five-term witness supplies unbounded quasi-progression length for one fixed width. I have no verified advance on the required uniformity, nor on the cube dimension beyond the cited known result. No conditional obstruction is promoted to an unconditional proof.
SOURCES CHECKED AND VERIFICATION
The complete local
782.tex; primary preprint abstract
https://arxiv.org/abs/2604.05459. The Pell norm, recurrence, positive-norm solution, and five gaps are checked by exact integer arithmetic in
verification/batch_3_final_checks.py. The failed raw GitHub fetch for
782.lean is not counted as a reviewed source.
FINAL
UNRESOLVED. A numerical error is repaired and small constructions checked; both unboundedness questions remain.
COMPLETION ESTIMATE
Subjective progress toward the original two questions.
COMPLETION: 10%