From what I understand, the RAT test will continue to give a positive result well after the infection ceases to become active — Aaron Marshall

Quite frankly, you are wrong.
Not sure who told you that, but you seem to be muddled with PCR and rat tests, an easy thing to do.

from earlier in the thread
https://www.monocent.com/wp-content/uploads/2020/10/Web-Covid-gradph.png
PCR picks up cases well after infection ceases to be active.
Rats pick up active infection, your body is busy making the spike protein for the virus and exuding the freshly made virus. The rat detects the spike protein in your nose.
A NZ explainer https://thespinoff.co.nz/science/09-03-2022/siouxsie-wiles-toby-morris-feeling-covid-positive-but-your-rat-is-negative-heres-why
On day 7 you still had an active infection regardless of symptoms. This is why the day 7 test matters as a household contact, if you test negative earlier.
If you had have tested on day 3, you may have had the covid confirmation sooner, and would have isolated until day 10. You will never know now.

This is not a technical situation at all,it is quite clear that the day 7 test is positive and the person is now a confirmed covid case (looking like asymptomatic and yet carrying spike protein and virus in their nose), and should be at home. — Jane

The problem is it is not clear at all - it could either be a case of:

• Failure of the Covid infection on the Friday to produce a sample with sufficient nasal viral load to produce a positive RAT result until the following Saturday (either actual viral load or failure to swab it well enough), or
• A new Covid infection that has now taken hold after a coincidental common cold / influenza infection causing the first set of symptoms at the same time as the partners confirmed Covid infection.

If I was betting on the balance of probabilities, my money would be on the first scenario.

Unfortunately this is a technicality as it is a case in how the rules are technically interpreted, and without any further clarification on restricting the scope how far back you should consider "symptoms" prior to a positive RAT as being related to the RAT legally (as per the Order) in this case he would be free to leave self-isolation.
In this case whether it is the right decision to leave isolation when you have just returned a positive RAT is a moral choice.

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I think, given the discussions we've had here, it highlights that the rules aren't clear.

For those concerned, I didn't go out and party the night my isolation technically ended, I was at home for a further four days after symptoms had subsided.

A recent study of RAT test reliability found those used in NZ have a 31% false negative rate early in the infection period and an 11% false negative rate late in the infection period.

https://www.evidencealerts.com/Social/Article/nWo8EtxY6Roo7LTvH4lcXg?Source=twitter#.YiUR6J7Op0k.twitter