```python
import torch
import torch.nn as nn
from torch.utils.cpp_extension import load_inline

_cuda_source = r"""
#include <torch/extension.h>
#include <ATen/cuda/CUDAContext.h>
#include <c10/cuda/CUDAGuard.h>
#include <c10/cuda/CUDAException.h>
#include <cuda_runtime.h>

constexpr int TILE = 64;
constexpr int TILE_K = 16;
constexpr int THREADS = 128;

__global__ __launch_bounds__(THREADS)
void triangular_matmul_kernel(
    const float* __restrict__ A,
    const float* __restrict__ B,
    float* __restrict__ C,
    const int N
) {
    const int tile_row = blockIdx.y * TILE;
    const int tile_col = blockIdx.x * TILE;
    const int tid = threadIdx.x;
    const int tx = tid & 15;
    const int ty = tid >> 4;

    if (blockIdx.x > blockIdx.y) {
        #pragma unroll
        for (int i = 0; i < 8; ++i) {
            const int row = tile_row + ty + i * 8;
            #pragma unroll
            for (int j = 0; j < 4; ++j) {
                const int col = tile_col + tx + j * 16;
                if (row < N && col < N) {
                    C[(int64_t)row * N + col] = 0.0f;
                }
            }
        }
        return;
    }

    __shared__ float As[TILE_K][TILE + 1];
    __shared__ float Bs[TILE_K][TILE];

    float accum[8][4] = {};

    // For lower triangular inputs, only this interval can contribute.
    const int k_end = min(tile_row + TILE, N);
    for (int k_base = tile_col; k_base < k_end; k_base += TILE_K) {
        #pragma unroll
        for (int p = 0; p < TILE * TILE_K / THREADS; ++p) {
            const int index = tid + p * THREADS;

            const int a_row = index / TILE_K;
            const int a_k = index % TILE_K;
            const int global_a_row = tile_row + a_row;
            const int global_a_k = k_base + a_k;

            As[a_k][a_row] =
                (global_a_row < N && global_a_k < N)
                ? A[(int64_t)global_a_row * N + global_a_k]
                : 0.0f;

            const int b_k = index / TILE;
            const int b_col = index % TILE;
            const int global_b_k = k_base + b_k;
            const int global_b_col = tile_col + b_col;

            Bs[b_k][b_col] =
                (global_b_k < N && global_b_col < N)
                ? B[(int64_t)global_b_k * N + global_b_col]
                : 0.0f;
        }
        __syncthreads();

        #pragma unroll
        for (int k = 0; k < TILE_K; ++k) {
            float a_values[8];
            float b_values[4];

            #pragma unroll
            for (int i = 0; i < 8; ++i) {
                a_values[i] = As[k][ty + i * 8];
            }

            #pragma unroll
            for (int j = 0; j < 4; ++j) {
                b_values[j] = Bs[k][tx + j * 16];
            }

            #pragma unroll
            for (int i = 0; i < 8; ++i) {
                #pragma unroll
                for (int j = 0; j < 4; ++j) {
                    accum[i][j] =
                        fmaf(a_values[i], b_values[j], accum[i][j]);
                }
            }
        }
        __syncthreads();
    }

    #pragma unroll
    for (int i = 0; i < 8; ++i) {
        const int row = tile_row + ty + i * 8;
        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const int col = tile_col + tx + j * 16;
            if (row < N && col < N) {
                C[(int64_t)row * N + col] =
                    row >= col ? accum[i][j] : 0.0f;
            }
        }
    }
}

torch::Tensor triangular_matmul_cuda(torch::Tensor A, torch::Tensor B) {
    TORCH_CHECK(A.is_cuda() && B.is_cuda(), "Inputs must be CUDA tensors");
    TORCH_CHECK(A.device() == B.device(), "Inputs must share a device");
    TORCH_CHECK(
        A.scalar_type() == at::kFloat && B.scalar_type() == at::kFloat,
        "Inputs must have dtype float32"
    );
    TORCH_CHECK(
        A.dim() == 2 && B.dim() == 2 &&
        A.size(0) == A.size(1) &&
        B.size(0) == A.size(0) && B.size(1) == A.size(1),
        "Inputs must be equally sized square matrices"
    );
    TORCH_CHECK(A.size(0) <= 2147483583LL, "Matrix dimension is too large");

    const c10::cuda::CUDAGuard guard(A.device());
    A = A.contiguous();
    B = B.contiguous();

    auto C = torch::empty(A.sizes(), A.options());
    const int N = static_cast<int>(A.size(0));
    if (N == 0) {
        return C;
    }

    const dim3 grid((N + TILE - 1) / TILE, (N + TILE - 1) / TILE);
    const auto stream = at::cuda::getCurrentCUDAStream();

    triangular_matmul_kernel<<<grid, THREADS, 0, stream>>>(
        A.data_ptr<float>(),
        B.data_ptr<float>(),
        C.data_ptr<float>(),
        N
    );
    C10_CUDA_KERNEL_LAUNCH_CHECK();
    return C;
}
"""

_extension = load_inline(
    name="triangular_matmul_t64_k16",
    cpp_sources=(
        "torch::Tensor triangular_matmul_cuda(torch::Tensor A, torch::Tensor B);"
    ),
    cuda_sources=_cuda_source,
    functions=["triangular_matmul_cuda"],
    extra_cflags=["-O3"],
    extra_cuda_cflags=["-O3"],
    verbose=False,
)


class ModelNew(nn.Module):
    def __init__(self):
        super().__init__()

    def forward(self, A, B):
        return _extension.triangular_matmul_cuda(A, B)
```