Counting \((0,0)\) as position 1, the first P-position with \(b-a\geq50\) is \[ P_{42762}=(83742,83792),\qquad 83792-83742=50. \] The preceding position has a smaller gap, and an exhaustive scan of positions 1 through 42,761 finds maximum gap 49. Thus this is the first threshold witness in the requested ordering.

Across all 100,000 positions, the maximum gap is 67. Its first occurrence is \[ P_{96868}=(190625,190692). \] Gap 67 occurs three times in the prefix. There are 8,421 positions with gap at least 50. The final pair is \[ P_{100000}=(196828,196850). \] The final ten pairs and every strict gap record are stored with the executable certificate. The 68th and last record is the gap-67 pair above, so no larger gap occurs in the finite scope.

Let \(P_n=(a_n,b_n)\), with \(a_n\leq b_n\). Start with \(P_0=(0,0)\). After constructing the earlier pairs, let \(a_n\) be the least nonnegative integer absent from all their coordinates. Choose \(b_n\geq a_n\) as the least unused integer such that \[ (a_n-s^2,b_n-s^2)\notin\{P_0,\ldots,P_{n-1}\} \] for every positive square \(s^2\leq a_n\).

No two constructed pairs share a coordinate, so a single-heap move cannot connect them. The displayed exclusion prevents a legal square-diagonal move between them.

It remains to check that every other position can move to a constructed pair. If its smaller coordinate already occurs as the larger coordinate of an earlier pair, a single-heap reduction reaches that pair. Otherwise its smaller coordinate equals some \(a_n\). For a canonical position \((a_n,y)\), a value \(y<b_n\) was either used as the larger coordinate of an earlier pair or rejected because a square-diagonal move reaches an earlier pair. A value \(y>b_n\) can be reduced to \(b_n\) in one heap. These cases cover every position outside the constructed set. Hence the construction satisfies both defining P-position properties and lists the P-positions by increasing first coordinate.

Duchene, Fraenkel, Gravier, and Nowakowski define \(\operatorname{Wyt}_K\) for any subset \(K\) of the positive integers. Its moves reduce one heap arbitrarily or reduce both heaps by the same \(k\in K\). The present game is exactly their construction with \[ K=\{1,4,9,16,\ldots\}. \] Their explicit P-position results cover singleton sets and selected infinite sets of powers of two. The paper supplies the established name and ruleset framework.

A focused search covered the exact initial pairs, square-restricted diagonal moves, Wyt_K, Nimhoff games, and OEIS sequence searches for both coordinate sequences. It found no source stating the gap-50 witness or the 100,000-position maximum. This search is evidence about the sources checked on 2026-07-24; it leaves novelty of the finite computation unverified.

The optimized generator observes that a diagonal move preserves \(b-a\). For each new mex coordinate \(a\), it checks every square \(s^2\leq a\), reads the gap of the earlier pair whose first coordinate is \(a-s^2\), and forbids that gap. The independent generator stores complete prior pairs and tests each candidate \((a,b)\) directly against every shifted pair \((a-s^2,b-s^2)\). Both also exclude previously used coordinates.

The generators return identical lists of 100,000 pairs. Serializing that list as compact JSON with no spaces or trailing newline gives 1,486,349 bytes and SHA-256 digest ``` f5cacfeedfe15d229189db324a2a8457c4b765007c9f575184549578be15d55b ``` The compact 68-row record-gap table has SHA-256 digest `6a8a5adb0eb6794be663251ab7babce82198d8f95eb81466c45ecb59d094101a`.