Madame DeFarge:
Nothing you are saying in terms of how children learn (different rates/ experiencing plateaus/ suddenly leeping ahead - bit of a roller coaster - is unfair).
HOWEVER my dear I do object to your portraying using a huge amount of national data of years of KS2 SATs results as 'a magic formula'. (by the way my link clearly didn't work previously - so try this: ofqual.gov.uk/documents/analysis-use-key-stage-2-data-gcse-predictions/
I am not saying IT ISN'T POSSIBLE for someone attaining NC L5 end of Y7 to go on to achieve A/ A* at GCSE - I'm saying that the trend in thousands of students data is that it's highly unlikely. Of course there are exceptions.
Now we don't have the raw data (there isn't national testing until end KS4 - end of KS3 testing was abandoned in 2008) - so all that can be done is to track performance end KS2 to attainment at GCSE end KS4.
I accept that comparison of NC LEVELS 3-5 to GCSE letter grades is difficult - but if you look at the table 2.1 (on page 11 - look at top right corner) in the link above will see the percentage of pupils (all English pupils) who achieve A/ A at GCSE after attaining NC L5 at end KS2 - it's ~54%. If you attained a high 4A at end KS2 - you only have roughly a 25% chance of attaining A/ A on present trends.
So again Madame DeFarge NC L5 end of Y7 DOES NOT GURANTEE in any way A/ A* at GCSE and it would be unwise for teachers to be indicating this to parents.
Whilst studiously avoiding any conversation about the OP's decision on punishing her child for what she deems a poor result, I do feel that the OP is not entirely wrong to be disappointed at slow progress in Year 7.
I personally think that some have posted helpfullly that there can be any number of reasons why someone may not be doing well in a new environment: bullying/ slow to settle in/ not making many friends in the new school/ not enjoying classes/ etc...
nonetheless - if OPs son was say a high level 5 end of KS2 and there was a long chat about how they narrowly missed a level 6 last year - and now a year on there's no improvement - that would be upsetting to a parent.
One thing that does occur CLL65 - is if there isn't a current 'issue' for poor performance at the new school - then I think you might want to reflect on how much 'hot housing' was going on at the old primary in the run up to SATs.
Certainly it is clear many teachers in secondadry are not content with the KS2 SATs levels awarded to pupils they receive. This may in fact be a case of preparing a child well for a test, but not in such a way (that old fashioned, slow and cumulative build up of knowledge/ skills) that they could retain that information/ consistently work at that level subsequently.
HTH