```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 BM = 128;
constexpr int BN = 64;
constexpr int BK = 16;

__global__ __launch_bounds__(256)
void triangular_matmul_kernel(
    const float* __restrict__ A,
    const float* __restrict__ B,
    float* __restrict__ C,
    int N
) {
    const int tid = threadIdx.x;
    const int tx = tid & 15;
    const int ty = tid >> 4;
    const int row_base = blockIdx.y * BM;
    const int col_base = blockIdx.x * BN;

    if (col_base >= row_base + BM) {
        for (int index = tid; index < BM * BN; index += 256) {
            const int row = row_base + index / BN;
            const int col = col_base + index % BN;
            if (row < N && col < N)
                C[(int64_t)row * N + col] = 0.0f;
        }
        return;
    }

    __shared__ float As[BM][BK];
    __shared__ float Bs[BK][BN];

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

    // For lower triangular inputs, only col <= k <= row contributes.
    const int k_end = min(row_base + BM, N);
    for (int k_base = col_base; k_base < k_end; k_base += BK) {
        #pragma unroll
        for (int offset = 0; offset < BM * BK; offset += 256) {
            const int index = tid + offset;
            const int local_row = index / BK;
            const int local_k = index % BK;
            const int row = row_base + local_row;
            const int k = k_base + local_k;
            As[local_row][local_k] =
                (row < N && k < N) ? A[(int64_t)row * N + k] : 0.0f;
        }

        #pragma unroll
        for (int offset = 0; offset < BK * BN; offset += 256) {
            const int index = tid + offset;
            const int local_k = index / BN;
            const int local_col = index % BN;
            const int k = k_base + local_k;
            const int col = col_base + local_col;
            Bs[local_k][local_col] =
                (k < N && col < N) ? B[(int64_t)k * N + col] : 0.0f;
        }

        __syncthreads();

        #pragma unroll
        for (int k = 0; k < BK; ++k) {
            float a[8];
            float b[4];

            #pragma unroll
            for (int i = 0; i < 8; ++i)
                a[i] = As[ty + i * 16][k];

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

            #pragma unroll
            for (int i = 0; i < 8; ++i) {
                #pragma unroll
                for (int j = 0; j < 4; ++j)
                    acc[i][j] = fmaf(a[i], b[j], acc[i][j]);
            }
        }

        __syncthreads();
    }

    #pragma unroll
    for (int i = 0; i < 8; ++i) {
        const int row = row_base + ty + i * 16;
        #pragma unroll
        for (int j = 0; j < 4; ++j) {
            const int col = col_base + tx + j * 16;
            if (row < N && col < N) {
                C[(int64_t)row * N + col] =
                    row >= col ? acc[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 be float32"
    );
    TORCH_CHECK(
        A.dim() == 2 && B.dim() == 2 &&
        A.size(0) == A.size(1) && B.sizes() == A.sizes(),
        "Inputs must be equally sized square matrices"
    );
    TORCH_CHECK(A.is_contiguous() && B.is_contiguous(),
                "Inputs must be contiguous");

    const c10::cuda::CUDAGuard device_guard(A.device());
    auto C = torch::empty_like(A);
    const int N = static_cast<int>(A.size(0));

    if (N > 0) {
        const dim3 grid((N + BN - 1) / BN, (N + BM - 1) / BM);
        triangular_matmul_kernel<<<
            grid, 256, 0, at::cuda::getCurrentCUDAStream()
        >>>(
            A.data_ptr<float>(),
            B.data_ptr<float>(),
            C.data_ptr<float>(),
            N
        );
        C10_CUDA_KERNEL_LAUNCH_CHECK();
    }
    return C;
}
"""

_extension = load_inline(
    name="lower_triangular_matmul_128x64",
    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):
        if (
            A.is_cuda
            and B.is_cuda
            and A.dtype == torch.float32
            and B.dtype == torch.float32
            and not (
                torch.is_grad_enabled() and (A.requires_grad or B.requires_grad)
            )
        ):
            return _extension.triangular_matmul_cuda(
                A.contiguous(), B.contiguous()
            )
        return torch.tril(torch.matmul(A, B))
```