Define the height by \[ H(x,y,z)=\max\{|x|,|y|,|z|\}. \] The target asks for an integer triple satisfying \[ x^3+y^3+z^3=114,\qquad H(x,y,z)\leq10^{20}. \] Permutations count as the same representation for search purposes, although the equation itself uses ordered variables.

Booker and Sutherland listed 114 among the seven cases below 1,000 that remained unresolved in 2021. Their September 2019 computation included 114 and covered every solution with \[ \min\{|x|,|y|,|z|\}\leq10^{17}. \] Their later computation reached a larger rectangle in the search parameters, rather than the full height box. Grantham and Walsh's arXiv preprint, submitted on 22 November 2022 as version 1, reports that their substantial effort on \(k=114\) remained unsuccessful. Thus the current source record supplies neither a solution of height at most \(10^{20}\) nor a complete exclusion certificate.

The fully covered prefix leaves a precise unresolved part of the requested box: \[ 10^{17}<\min\{|x|,|y|,|z|\} \leq H(x,y,z)\leq10^{20}. \] Some of this region lies inside the later parameter search recorded separately.

Booker and Sutherland's algorithm permutes a nonexceptional solution so that \[ |x|>|y|>|z| \] and sets \[ d=|x+y|. \] For \(|z|>\sqrt{k}\), every such solution has \[ 0<d<(\sqrt[3]{2}-1)|z|. \] The exceptional equal-absolute-value cases are handled separately. Consequently, using \(z_{\max}=B\) and \(d_{\max}=(\sqrt[3]{2}-1)B\) searches every solution whose smallest absolute coordinate is at most \(B\).

In September 2019 the authors ran this complete parameter choice for all eleven then-unresolved targets, including 114, with \[ z_{\max}=10^{17},\qquad d_{\max}=(\sqrt[3]{2}-1)10^{17}. \] They report solutions for 42, 165, and 906. No solution for 114 appeared. This covers every triple in the smaller height box \(H\leq10^{17}\), and it also covers every triple in the requested height box having at least one coordinate of absolute value at most \(10^{17}\).

The authors caution that their crowd-sourced searches may contain an undetected hardware or software error. This record therefore preserves the result as exhaustive published computational evidence rather than a formal impossibility proof.

During 2020, Charity Engine searched the eight targets left after the first run, including 114, with \[ z_{\max}=10^{19},\qquad d_{\max}=z_{\max}/54. \] Since \(d\) is integral, its largest allowed value is \[ \left\lfloor\frac{10^{19}}{54}\right\rfloor =185185185185185185. \] The run produced a representation of 579 in January 2021 and reported none for 114.

This rectangle is much deeper in \(|z|\) than the complete \(10^{17}\) run, yet its \(d\)-bound is too small to cover every solution with \(|z|\leq10^{19}\). Full coverage at that \(z\)-bound would require \(d_{\max}=(\sqrt[3]{2}-1)10^{19}\). The computation therefore supplies substantial evidence inside a named parameter rectangle. It does not exclude the rest of the height-\(10^{20}\) box. Remark 5.1 gives the same possible-error caveat for the crowd-sourced computation.

Cubes modulo 9 have residues \(0,1,8\). Since \[ 114\equiv6\equiv-3\pmod9, \] the three cube residues must all equal \(-1\). Hence every solution satisfies \[ x,y,z\in\{2,5,8\}\pmod9, \] and in particular \[ x\equiv y\equiv z\equiv2\pmod3. \] This is a necessary filter. It supplies no global obstruction because the permitted residue classes exist.

Save the source lines as `verify.py`. Running `python3 verify.py` performs deterministic self-tests. Running ``` python3 verify.py X Y Z ``` checks any proposed integer triple. The program accepts the triple only when its exact cube sum equals 114 and its maximum absolute coordinate is at most \(10^{20}\). Python integers are arbitrary precision, so the calculation has no overflow or rounding step.

The residue test is reported separately. A genuine solution automatically passes it, while acceptance depends only on the exact equation and height definition.