= Solution
The <t-J model> approximates the low-energy sector of a repulsive <Hubbard model> with $U\gg t$ near one <electron> per site. Large on-site repulsion suppresses double occupancy. Virtual hopping out of and back into that low-energy sector produces antiferromagnetic <superexchange> with $J=4t^2/U$. One conventional effective Hamiltonian is
$$
H=-t\sum_{\langle ij\rangle,\sigma}
\left(\widetilde c_{i\sigma}^\dagger\widetilde c_{j\sigma}+\mathrm{h.c.}\right)
+J\sum_{\langle ij\rangle}\left(\mathbf S_i\cdot\mathbf S_j-\frac14n_in_j\right),
$$
with <projected electron operator> $\widetilde c_{i\sigma}=c_{i\sigma}(1-n_{i\bar\sigma})$. At half filling, nearest-neighbor hopping cannot move an <electron> without entering a forbidden doubly occupied state, leaving the leading spin-exchange model. Doped holes can move and couple the charge dynamics to these spin correlations. Additional terms, including three-site processes at order $t^2/U$, are omitted in this simplified Hamiltonian.
The exchange term energetically favors a nearest-neighbor spin singlet. With $b_{ij}^\dagger=(c_{i\uparrow}^\dagger c_{j\downarrow}^\dagger-c_{i\downarrow}^\dagger c_{j\uparrow}^\dagger)/\sqrt2$ in the no-double-occupancy subspace, $\mathbf S_i\cdot\mathbf S_j-n_in_j/4=-b_{ij}^\dagger b_{ij}$. A square-lattice pairing ansatz can therefore use bond singlets with opposite signs along the two axes. This <d-wave bond order> has
$$
\boxed{\Delta_{\mathbf k}\propto\cos(k_xa)-\cos(k_ya),\qquad\Delta_x=-\Delta_y.}
$$
It changes sign under a quarter-turn and vanishes along the zone diagonals. It is an even-parity, spin-singlet pairing pattern, not a statement of triplet pairing or necessarily broken time reversal. Phase-sensitive <interference> experiments distinguish this sign structure from a merely anisotropic nonnegative gap in <cuprate superconductors>.<Pairing symmetry and flux quantization in a tricrystal superconducting ring by C. C. Tsuei and others>[1994 phase-sensitive tricrystal test]{ref}
The model captures strong local repulsion, antiferromagnetism and mobile doped carriers. Its interacting phase diagram is not solved by writing down the Hamiltonian: a mean-field d-wave ansatz is a proposed approximation, not a proof that every parameter choice has that exact <ground state> or a complete microscopic explanation of every high-temperature superconductor.
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