"""
Batched Pentachoron CV — Fast Geometric Volume Measurement
============================================================
Replaces the sequential Python loop with fully batched operations.
One torch.linalg.det call on (n_samples, 6, 6) tensor.

Usage:
    from cv_batch import cv_metric, cv_loss

    # Non-differentiable monitoring (fast)
    cv_value = cv_metric(embeddings, n_samples=200)

    # Differentiable loss (fast, for training)
    loss = cv_loss(embeddings, target=0.22, n_samples=64)
"""

import torch
import torch.nn.functional as F
import math


def _batch_pentachoron_volumes(emb, n_samples=200, n_points=5):
    """Compute pentachoron volumes in one batched operation.

    Args:
        emb: (N, D) L2-normalized embeddings on S^(d-1)
        n_samples: number of random pentachora to sample
        n_points: points per simplex (5 = pentachoron)

    Returns:
        volumes: (n_valid,) tensor of simplex volumes (may be < n_samples if some degenerate)
    """
    N, D = emb.shape
    device = emb.device
    dtype = emb.dtype

    # Sample all pentachora indices at once: (n_samples, n_points)
    # Batched randperm via argsort on random values
    pool = min(N, 512)
    rand_keys = torch.rand(n_samples, pool, device=device)
    indices = rand_keys.argsort(dim=1)[:, :n_points]  # (n_samples, n_points)

    # Gather points: (n_samples, n_points, D)
    pts = emb[:pool][indices]  # advanced indexing

    # Gram matrices: (n_samples, n_points, n_points)
    gram = torch.bmm(pts, pts.transpose(1, 2))

    # Squared distance matrices: d2[i,j] = ||p_i - p_j||^2
    norms = torch.diagonal(gram, dim1=1, dim2=2)  # (n_samples, n_points)
    d2 = norms.unsqueeze(2) + norms.unsqueeze(1) - 2 * gram  # (n_samples, n_points, n_points)
    d2 = F.relu(d2)  # numerical safety

    # Build Cayley-Menger matrices: (n_samples, n_points+1, n_points+1)
    M = n_points + 1
    cm = torch.zeros(n_samples, M, M, device=device, dtype=dtype)
    cm[:, 0, 1:] = 1.0
    cm[:, 1:, 0] = 1.0
    cm[:, 1:, 1:] = d2

    # Prefactor for volume from CM determinant
    k = n_points - 1  # dimension of simplex
    pf = ((-1.0) ** (k + 1)) / ((2.0 ** k) * (math.factorial(k) ** 2))

    # Batched determinant — the one expensive call, fully parallel
    dets = pf * torch.linalg.det(cm.float())  # (n_samples,)

    # Filter valid (positive volume squared) and take sqrt
    valid_mask = dets > 1e-20
    volumes = dets[valid_mask].to(dtype).sqrt()

    return volumes


def cv_metric(emb, n_samples=200, n_points=5):
    """Non-differentiable CV for monitoring. Target band: 0.20–0.23.

    Args:
        emb: (N, D) embeddings (will be L2-normalized internally)
        n_samples: pentachora to sample (200 is robust, 100 is fast)
        n_points: points per simplex (5 = pentachoron)

    Returns:
        float: coefficient of variation of simplex volumes
    """
    with torch.no_grad():
        vols = _batch_pentachoron_volumes(emb, n_samples=n_samples, n_points=n_points)
        if vols.shape[0] < 10:
            return 0.0
        return (vols.std() / (vols.mean() + 1e-8)).item()


def cv_loss(emb, target=0.22, n_samples=64, n_points=5):
    """Differentiable CV loss for training. Weight: 0.01 or below.

    Args:
        emb: (N, D) L2-normalized embeddings
        target: CV target value
        n_samples: pentachora to sample (32-64 for training)
        n_points: points per simplex

    Returns:
        scalar tensor: (CV - target)^2, differentiable w.r.t. emb
    """
    vols = _batch_pentachoron_volumes(emb, n_samples=n_samples, n_points=n_points)
    if vols.shape[0] < 5:
        return torch.tensor(0.0, device=emb.device, requires_grad=True)
    cv = vols.std() / (vols.mean() + 1e-8)
    return (cv - target).pow(2)


def cv_multi_scale(emb, scales=(3, 4, 5, 6, 7, 8), n_samples=100):
    """CV at multiple simplex sizes. Returns dict: {n_points: cv_value}.

    Useful for diagnosing whether geometry is scale-invariant.
    Target: all scales in [0.18, 0.25] for healthy geometry.
    """
    results = {}
    with torch.no_grad():
        for n_pts in scales:
            vols = _batch_pentachoron_volumes(emb, n_samples=n_samples, n_points=n_pts)
            if vols.shape[0] >= 10:
                results[n_pts] = round((vols.std() / (vols.mean() + 1e-8)).item(), 4)
            else:
                results[n_pts] = None
    return results