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Even if your Luxembourg visa includes the 15-day "period of grace" contemplated by the Schengen Visa Code in Article 24, the visa will not be sufficient for an additional 1-month stay in France. It would, however, be within the rules for you to stay in France for a week or two. There is however no guarantee that they will include a grace period.
For a one-month additional stay, therefore, you will definitely require another visa.
There should be no problem with your change in plans from the original Luxembourg visa application. Problems that could arise in such a case include: