The Massachusetts School Building Authority has invited Carver back into consideration for state school construction funding. State Treasurer Steve Grossman, the authority’s chairman, said he was looking forward to working with local officials to get an elementary school building proposal ready for state backing during this year’s eligibility period, as long as the town can demonstrate that it is ready financially and has sufficient community backing. Despite parent support, proposals to build a new elementary school to replace the town’s two old elementary schools have failed repeatedly at the polls in recent years. Opponents said the proposals were too costly. After the last failure two years ago, the building authority removed Carver from funding consideration. Carver school advocates then sought to reopen its bid for state funding support after a public survey showed strong support for a new school and a divided Board of Selectmen also gave its backing.