Budget negotiators have not found a way to avert a government default on federal debt obligations, but with their ideas to cut Medicare and Medicaid they have managed to provoke opposition from almost every major group that represents beneficiaries and healthcare providers. The latest provocation was a list of proposed savings presented at the White House this week by the House majority leader, Representative Eric Cantor, Republican of Virginia. Cantor said Tuesday that the ideas had all been seriously discussed, with varying levels of Democratic support, in seven weeks of negotiations led by Vice President Joseph R. Biden Jr. But with House Republicans adamantly opposed to new taxes, Democrats said they would not accept cuts in Medicare that reduced benefits. Cantor’s list included 27 proposals that he said would save up to $353 billion over 10 years, in the context of a budget deal that could save anywhere from $2 trillion to $4 trillion over the same period.