The soundproofing materials must be installed as specified. If there is no specification for the materials, then the court would have to try to determine the objective intent of the parties, and where there is an ambiguity that cannot be resolved by the contract language or extrinsic evidence, then it is resolved against the person who drafts the contract provision.
The term, "sound proof" has a very different meaning than does wall insulation "R" thermal rating. Even a high R rating does not equate to "sound proof." The proper rating system is the STC system. Even the highest typical STC rating (66) is not truly soundproof. However, soundproofing methodology is far different from thermal insulation, and were I representing you in court, I would simply argue that insulation is not soundproofing, and the contract requires sound proof walls, which is at least STC 66.
If sound proofing is important to you, because of some fundamental acoustic issue in the environment (such as a shared wall with a neighbor), then I wouldn't pay. Otherwise, I might offer to pay some proportion of the bill, given that a complete tear out and replacement is probably unlikely.
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