A rainbow interval is first an ordinary gap between consecutive squarefree integers. Published computations cover ordinary squarefree gaps through \(10^{18}\). Marmet's first-occurrence table places the first run of 14 consecutive nonsquarefree integers at \[ 1043460553364, \] which exceeds the present cutoff. The first runs of lengths 15 through 18 occur still higher, and the exhaustive computation reports no longer run through \(10^{18}\). The first run of 13 begins at \[ 82462576220. \] Therefore an ordinary gap with upper endpoint at most \(10^{12}\) contains at most 13 interior integers, and its endpoint distance is at most 14. The rainbow target consequently satisfies \[ 7\leq G(10^{12})\leq14. \] The lower bound comes from the explicit interval at 30,922. Resolving the target requires checking the distinct-prime condition after five million.

The endpoint factorizations are \[ 30922=2\cdot15461, \qquad 30929=157\cdot197, \] and every displayed factor is prime. Thus both endpoints are squarefree.

The interior factorizations are \[ \begin{aligned} 30923&=17^2\cdot107,\\ 30924&=2^2\cdot3^2\cdot859,\\ 30925&=5^2\cdot1237,\\ 30926&=2\cdot7\cdot47^2,\\ 30927&=3\cdot13^2\cdot61,\\ 30928&=2^4\cdot1933. \end{aligned} \] Assigning the primes \[ (17,3,5,47,13,2) \] in increasing order of the interior integers gives six distinct labels, each with its square dividing the corresponding integer.

For consecutive squarefree endpoints \(a<b\leq5{,}000{,}000\), the largest distance admitting distinct square-prime labels on every interior integer is \[ \boxed{b-a=7}. \] The first attaining pair is \[ (a,b)=(30922,30929). \]

The replay marks every multiple of \(p^2\) for every prime \(p\leq\sqrt{5{,}000{,}000}\). It then visits every consecutive squarefree pair. Once the displayed distance-7 witness is checked, only gaps of distance at least 7 need a matching test. There are 344 such ordinary squarefree gaps and 33 admit a full distinct-prime matching. None has distance above 7.

Marmet gives an Eratosthenes-style algorithm for first occurrences of runs of nonsquarefree integers and reports the first runs through length 18. Mossinghoff, Oliveira e Silva, and Trudgian independently study the empirical distribution of squarefree gaps through \(10^{18}\). Kumchev, McCormick, McNew, Park, Scherr, and Ziehr cite those computations when proving explicit universal gap bounds.

Focused searches combined `squarefree gap`, `consecutive nonsquarefree`, `distinct prime squares`, `system of distinct representatives`, `Hall matching`, and `rainbow` with the cited authors and sequence data. The sources found treat ordinary gap length or its distribution. None imposes a different square-prime divisor at each interior position. This audit supports the upper bound and leaves novelty of the matching restriction unverified.

The program first sieves all primes through \(\sqrt{5{,}000{,}000}\), then marks every integer divisible by each prime square. This classifies every integer in the range as squarefree or nonsquarefree. For every ordinary squarefree gap of endpoint distance at least 7, it reconstructs all square-prime divisors of the interior integers and runs exact bipartite matching.

The canonical report records 344 tested ordinary gaps, 33 successful rainbow gaps, and the first maximum witness. Its SHA-256 digest is `17402d48f264f6d7d21e35e8694a02ce0e88e17a386f2a0b2edef83fd5c87f3c`.