The government’s defeat in the House of Lords has triggered a U-Turn from George Osborne, who will announce measures to ameliorate the effects of cuts to tax credits in the Autumn Statement on 25 November.

But the Conservatives’ problems may only just be beginning. At the election, the party made a virtue of tying its hands – increases to any of income tax, value added tax and national insurance were not only ruled out but accompanied with a promise to make any rises illegal for the course of the parliament.

The fiscal charter – designed as a trap for Labour – commits the government to an overall surplus by 2020, and further limits the Conservatives’ options. (The easiest – and most economically sensible – route would be to commit to a slightly slower pace of deficit reduction, giving the government the wiggle room to avoid cuts)

As Jolyon Maugham has shown, you can find the revenue without breaking any of the government’s fiscal undertakings – but that involves unraveling two of the Conservatives’ signature policies, the threshold raise – an increasingly lucrative bung to dual-earner childless couples, the only working age demographic to do better under the Coalition than Labour and a crucial plank of the Cameron-Osborne electoral coalition – and their cuts to corporation tax.

And if the government goes ahead with its current timetable for reducing the deficit, it is difficult to see where they can get the money from without cutting tax credits, as Flip Chart Rick shows on his blog:

Those gold bits at the top are tax credits - almost everything else, from child benefit to pensioners, has been ringfenced by the government, which makes keeping their fiscal promises impossible without cuts to tax credits.

What could the government do? A further increase in the threshold raise would be incredibly expensive – as it benefits all taxpayers, not just the poorest or those in receipt of tax credits – further knocking the government’s fiscal timetable out of whack. In addition, previous increases in the income tax threshold have “significantly reduced the overlap between the populations of tax payers and tax credit recipients”, in the words of the Resolution Foundation’s Torsten Bell. Households will start to lose out due to cuts to tax credits if they earn above £3,850. To benefit from increased rises in the income tax threshold, you have to earn more than £11,000. So the government can’t get out of its hole that way.

National insurance kicks in a little earlier – people earning above £8,164 pay that – but the significant costs of the raise again, cost more than the savings from cutting tax credits and once again won’t adequately compensate the poorest earners.

Well, what about an increase in the minimum wage? As I’ve set out before, increases in the minimum wage are double-edged. Osborne’s current increase – which falls 70p short of where the real living wage is today – is probably too high for small and medium-sized businesses. Trade union organisers from Usdaw – the shopworkers union – and the GMB – which represents many parts of the hospitality industry – are already concerned that the net effect of the wage rise will be job losses and reduced hours.

To compensate for the cuts to tax credits, the national minimum wage would be far higher than the £7.20 wage proposed by Osborne – to do without in-work benefits, the minimum wage would have to be £12 an hour. That would result in large numbers of job losses – which, philosophically, may be perfectly just if you think the state shouldn’t bail out employers. Good luck getting elected after making that many people unemployed, though.

What about hitting the rich as well? That’s the solution proposed by Janan Ganesh in the FT this morning. Well, frankly, while the symbolism of tax credit cuts is not great for the government, the reason why they are so unpopular – and why Conservative backbenchers and ministers alike are privately worried – is not because it looks bad, but because the £1,200 that most households will lose out represents a significant reduction in takehome pay – effectively over a month’s salary for some tax credit recipients. No-one who is pushed into poverty by cuts in tax credits will take comfort from the knowledge that a mansion tax has been implemented elsewhere.

The big political – as well as the big policy problem – is that the government’s fiscal plans force it to take large amounts of money off some of Britain’s poorest workers. The easiest thing would simply be to extend the timetable for balancing the books – having done so and won re-election in 2015, the longterm political costs would likely be low for the Conservatives. Any other solution looks next to impossible.