If we are taking over financing midstream the allocation of principle and interest will be at a different point than if we began adding principle on day one of a 30 yr loan for instance. The exact amount is not important but the method of calculating a way point at any given point in time as we will intervene at a different points in the existing loan cycle every time. The calculation I need will illustrate my taking over that loan 8 yr 3 mos into its life and in this illustration, at that point adding additional principle of a a given amount. The scenario is where there is an existing loan in place and we are say 8 yrs 3 mos into the amortization schedule of a 30 yr loan. My use for the formula you came up with has one nuance I didn't consider. I just realized something that will likely change the approach to what I was attempting and I apologize for the detour in advance. The challenge to be answered by traditional spreadsheet methods is "does an excess of simplicity itself create impenetrable and error-prone solutions?" The end of the Einstein quote you mentioned was ". The idea is that such a formula is less prone to errors of consistency than a traditional formula copied across a range. All a Lambda function does is allow one to write a formula in terms of parameters passed to it as variables.
You found the use of Lambda functions off-putting. The method is far closer to the world of professional programmer than it is to that of a normal spreadsheet end-user.
The 'simplicity' I set out to achieve is to generate each table from a single formula rather than the original 2240 individual formulas. The calculation produces an array of balance figures (columns H and M) and the other columns form no part of the calculation they are derived for information only. I accept that the calculation I presented is a mathematical abstraction of the problem and does not capture practical considerations no business practice is going to work with millionths of a cent.