The £52,000 number is not a forecast — it is arithmetic
Two parallel scenarios. Same £200/month. Same 25 years. Same household.
Scenario A — overpay the mortgage. £200 a month against a 4.5% mortgage compounds (as interest avoided) to roughly £110,000 over 25 years. That is the future-value-of-an-annuity calculation at 4.5%, which is the Guardian's headline number.
Scenario B — fund the ISA instead. £200 a month into a global tracker at 7% nominal long-run compounds to roughly £162,000 over 25 years. That is the same future-value calculation at 7%.
The £52,000 difference is what the Guardian's "risk-free 4.5%" actually costs you. It is not pulled from thin air — it is the mechanical consequence of the rate gap, compounded for 25 years.
The Guardian will protest that the 7% is not guaranteed. Correct. The point is that the 7% does not need to be 7% for the Challenger to win. At 6%, the ISA still ends at roughly £138,000 — £28,000 ahead. At 5.5%, the ISA finishes at £126,000 — £16,000 ahead. The breakeven where overpayment beats investing is the mortgage rate itself. The hurdle is 4.5%, not 7%.