= Solution
Use the usual <independent private values model>, <quasilinear utility>, and voluntary participation with zero outside utility. These assumptions matter: without <individual rationality>, arbitrary type-independent entry charges make revenue unbounded, and correlated types cannot in general be described by their marginal priors alone.
The <revelation principle> lets us optimize over <direct revelation mechanisms> satisfying <Bayesian incentive compatibility>. Write $x(v)\in[0,1]$ for the common project allocation, $X_i(t)=\mathbb E_{V_{-i}}x(t,V_{-i})$ for player $i$'s interim allocation, and $P_i(t)$ for its interim payment. The <interim payment identity> gives
$$
P_i(t)=tX_i(t)-\int_{\underline v_i}^tX_i(s)ds-U_i(\underline v_i).
$$
Since <interim individual rationality> requires $U_i(\underline v_i)\geq0$, the <virtual-surplus revenue identity> bounds expected revenue by
$$
\mathbb E\sum_iP_i(V_i)\leq\mathbb E\left[x(V)\sum_i\phi_i(V_i)\right],
\qquad\phi_i(t)=t-\frac{1-F_i(t)}{f_i(t)}.
$$
The best feasible common allocation at each valuation profile therefore provides the project when total <virtual surplus> is nonnegative:
$$
\boxed{x^*(v)=\mathbf1_{\{\sum_i\phi_i(v_i)\geq0\}}.}
$$
Because each <regular prior> has a nondecreasing <virtual valuation>, this allocation is a <nondecreasing function> of each player's report. Hold $v_{-i}$ fixed and charge the <critical-value payment>
$$
p_i^*(v)=v_i x^*(v)-\int_{\underline v_i}^{v_i}x^*(t,v_{-i})dt.
$$
This is the winning threshold when it lies in the support, the lowest allowed value if every type wins, and zero if the player loses. A truthful winner never pays more than its value; a losing type cannot profit by crossing the threshold. Thus the mechanism has <dominant-strategy incentive compatibility> and <ex post individual rationality>, with zero utility at every lowest type. It attains the revenue bound, proving optimality even among mechanisms requiring only <Bayesian incentive compatibility>. At a zero-virtual-surplus tie, choose any fixed rule that preserves monotonicity.
For independent <uniform distributions> on $[0,1]$, the <virtual valuations> are $\phi_i(v_i)=2v_i-1$. The <revenue-optimal public-project auction> becomes
$$
\boxed{\text{Provide the project exactly when }\sum_i v_i\geq\frac n2.}
$$
When it is provided, player $i$ pays
$$
\boxed{p_i^*(v)=\max\left\{0,\frac n2-\sum_{j\ne i}v_j\right\};}
$$
otherwise every payment is zero. If the displayed threshold exceeds one, player $i$ cannot induce provision within its allowed support; if it is negative, provision is independent of its own report and its payment is zero. For $n=1$, this specializes to a reserve value and payment of $1/2$.
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