submission 877576

#!POPCORN leaderboard eigh
#!POPCORN gpu B200

"""Batched symmetric eigendecomposition (eigh) — all-PTX embed build.

GENERATED FILE — do not hand-edit. Sources: kernels/{jacobi32,jacobi64,
bisect,invit}.cu + submission.tmpl.py, assembled by build_submission.py.
Kernels are cross-compiled locally to compute_100a PTX, .version patched to
8.7 (floor supported by both runner drivers), hex-encoded and loaded at first
use via the CUDA driver API — nothing compiles on-box.

  n <= 32 : jacobi32 — one CTA/matrix two-sided cyclic Jacobi in SMEM,
            rank-sort epilogue (validated path from v1).
  n == 512: one-stage blocked Householder tridiagonalization (latrd panels,
            fp16 symv with fp32 accumulation) -> split at negligible
            couplings -> block-local bisection (bisect kernel) -> block-
            masked inverse iteration (invit kernel) with a stein repair loop
            over cluster runs -> polar orthonormalization -> compact-WY
            back-transform -> Rayleigh-Ritz + first-order eigenvector
            refinement + adaptive dynamic-ordering cleanup (jacobi64) +
            per-matrix fallback for gate rejects.
  else    : torch.linalg.eigh fallback (cuSOLVER) — the iteration frontier.
"""

import ctypes
import torch

from task import input_t, output_t

_PTX = {
    b"jacobi32": "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}
_SMEM = {b"bisect": 8192,        # 2 * n * 4B (d + e^2 staged in SMEM)
         b"latrd512": 67584}     # fp16 mirror of the panel V,W (512 x 66
                                 # padded: 64-half rows are 128B = one bank)

N32_SWEEPS = 12     # jacobi32 sweeps (n <= 32 path)
INNER_SWEEPS = 4    # jacobi64 subsolver sweeps per dynamic-ordering round
NB = 32             # tridiagonalization panel width
EPS512 = 1.19209290e-7
SPLIT_C = 1000.0    # zero |e| <= SPLIT_C*eps*tnorm: decouples degenerate
                    # clusters into independent blocks; perturbs the spectrum
                    # by <= 1.2e-4 relative (checker allows 1.2e-2)
GTOL_F = 64.0       # cluster-run grouping gap for the stein repair loop
BISECT_ITERS = 45
SEP_F = 2.0         # tiny in-block shift separation (ulp scale). Large values
                    # clump sub-resolution clusters onto one eigendirection.
STEIN_REPS = 2      # {run-orth, re-solve} repair passes (the orthonormal-
                    # bucket skip makes pass 2 cheap when pass 1 sufficed;
                    # 1 rep let a mixed-batch matrix collapse through)
POLAR_ITERS = 5     # Newton-Schulz polar iterations on the invit basis
DYN_MAX = 6         # max dynamic-ordering cleanup rounds (~2/15 sweep each)
CLEANUP_MAX = 2     # max full tournament sweeps if dyn rounds aren't enough
POLISH = True       # fp32 Newton-Schulz on Q + Rayleigh eigenvalues at the end
REFINE = True       # one first-order eigenvector refinement before dyn rounds

try:
    _lib = ctypes.CDLL("libcuda.so.1")
except OSError:
    _lib = ctypes.CDLL("libcuda.so")

_lib.cuGetErrorString.argtypes = [ctypes.c_int, ctypes.POINTER(ctypes.c_char_p)]
_lib.cuInit.argtypes = [ctypes.c_uint]
_lib.cuModuleLoadData.argtypes = [ctypes.POINTER(ctypes.c_void_p), ctypes.c_char_p]
_lib.cuModuleGetFunction.argtypes = [ctypes.POINTER(ctypes.c_void_p),
                                     ctypes.c_void_p, ctypes.c_char_p]
_lib.cuFuncSetAttribute.argtypes = [ctypes.c_void_p, ctypes.c_int, ctypes.c_int]
_lib.cuLaunchKernel.argtypes = [
    ctypes.c_void_p,
    ctypes.c_uint, ctypes.c_uint, ctypes.c_uint,
    ctypes.c_uint, ctypes.c_uint, ctypes.c_uint,
    ctypes.c_uint, ctypes.c_void_p,
    ctypes.POINTER(ctypes.c_void_p), ctypes.POINTER(ctypes.c_void_p)]


def _ck(res, what):
    if res != 0:
        m = ctypes.c_char_p(); _lib.cuGetErrorString(res, ctypes.byref(m))
        raise RuntimeError("driver %s failed: rc=%d %s" % (what, res, m.value))


# torch's primary context must be current before any driver call.
torch.cuda.init(); torch.zeros(1, device="cuda"); _lib.cuInit(0)

_loaded = {}


def _get(kname):
    ent = _loaded.get(kname)
    if ent is None:
        hexstr = _PTX[kname]
        if not hexstr:
            raise RuntimeError("kernel %s not embedded in this build" % kname)
        mod = ctypes.c_void_p()
        _ck(_lib.cuModuleLoadData(ctypes.byref(mod), bytes.fromhex(hexstr)),
            "cuModuleLoadData")
        fn = ctypes.c_void_p()
        _ck(_lib.cuModuleGetFunction(ctypes.byref(fn), mod, kname),
            "cuModuleGetFunction")
        smem = _SMEM.get(kname, 0)
        if smem:
            _ck(_lib.cuFuncSetAttribute(fn, 8, smem), "cuFuncSetAttribute")
        _loaded[kname] = ent = (mod, fn, smem)
    return ent


def _launch(kname, grid, block, *args):
    _, fn, smem = _get(kname)
    holders = [ctypes.c_void_p(a.data_ptr()) if isinstance(a, torch.Tensor)
               else ctypes.c_int(a) for a in args]
    params = (ctypes.c_void_p * len(holders))(
        *[ctypes.cast(ctypes.byref(h), ctypes.c_void_p) for h in holders])
    _ck(_lib.cuLaunchKernel(fn, grid[0], grid[1], grid[2],
                            block[0], block[1], block[2],
                            smem, None, params, None), "cuLaunchKernel")


def _jac64(S, V):
    # S, V: (nb, 64, 64) contiguous. V <- rotations, columns sorted ascending.
    _launch(b"jacobi64", (S.shape[0], 1, 1), (32, 32, 1), S, V, INNER_SWEEPS)


def _row32(V, Arows, Out):
    # fp32 cleanup-sweep variant of _row (torch keeps matmul at full fp32).
    Out.copy_(V.mT @ Arows)


def _col32(Acols, V, Out):
    Out.copy_(Acols @ V)


def _eigh32(a):
    a = a.contiguous()
    batch, n, _ = a.shape
    q = torch.empty_like(a)
    vals = torch.empty((batch, n), device=a.device, dtype=a.dtype)
    _launch(b"jacobi32", (batch, 1, 1), (32, 16, 1), a, q, vals, n, N32_SWEEPS)
    return q, vals


def _roundrobin(m):
    # Circle-method tournament: 15 rounds x 8 disjoint pairs covering all 120.
    # Pairs oriented (I < J): jacobi64 sorts each pair's eigenvalues ascending,
    # so consistent orientation pushes small eigenvalues to lower blocks every
    # round; mixed orientation makes the outer iteration plateau.
    pl = list(range(m)); res = []
    for _ in range(m - 1):
        res.append([tuple(sorted((pl[i], pl[m - 1 - i]))) for i in range(m // 2)])
        pl = [pl[0]] + [pl[-1]] + pl[1:-1]
    return res


_perm_cache = {}


def _adjacent_rounds(m=16):
    # Two pairings that cover the 64-grid: the identity round diagonalizes the
    # eight diagonal 64-blocks; the offset round covers couplings straddling
    # the 64-block boundaries.
    ident = [(2 * i, 2 * i + 1) for i in range(m // 2)]
    offset = [(2 * i + 1, 2 * i + 2) for i in range(m // 2 - 1)] + [(m - 1, 0)]
    return [ident, offset]


def _perms512(dev, n, rounds=None):
    key = (str(dev), n,
           "rr" if rounds is None else tuple(tuple(r) for r in rounds))
    if key not in _perm_cache:
        b, m = 32, n // 32
        rlist = _roundrobin(m) if rounds is None else rounds
        Ps = []
        for pairs in rlist:
            p = []
            for (I, J) in pairs:
                p += list(range(I * b, I * b + b)) + list(range(J * b, J * b + b))
            Ps.append(torch.tensor(p, device=dev))
        inv = [torch.argsort(P) for P in Ps]
        # Composed transitions: T maps the previous round's frame straight to
        # this round's frame, so each round costs ONE gather instead of an
        # unpermute + repermute pair.
        trans = [Ps[0]] + [inv[r - 1][Ps[r]] for r in range(1, len(Ps))]
        wrap = inv[-1][Ps[0]]           # last round's frame -> round-0 frame
        _perm_cache[key] = (trans, wrap, inv[-1])
    return _perm_cache[key]


def _sweeps512(A, Q, nsweeps, rowf, colf, bufs, rounds=None):
    """Run pairing sweeps; A and Q enter and leave in the identity frame.

    rounds=None -> full round-robin tournament (all 120 pairs per sweep);
    otherwise a list of disjoint 32-block pairings (e.g. _adjacent_rounds()).
    A must be a fresh contiguous tensor (its storage is recycled); the tensor
    passed in is never written, because the first op of every round gathers
    into a new allocation.
    """
    if nsweeps <= 0:
        return A, Q
    B, n = A.shape[0], A.shape[1]
    trans, wrap, unperm = _perms512(A.device, n, rounds)
    nb = n // 64                                     # 8 pair-blocks per round
    A2, S, V, Pk, Pk2 = bufs
    for sw in range(nsweeps):
        for r in range(len(trans)):
            T = trans[r] if (sw == 0 or r > 0) else wrap
            A = A[:, T[:, None], T[None, :]]
            Q = Q.index_select(2, T)

            # 8 diagonal 64x64 pair-subproblems -> V (sorted ascending)
            S.view(B, nb, 64, 64).copy_(
                A.as_strided((B, nb, 64, 64), (n * n, 64 * n + 64, n, 1)))
            _jac64(S, V)

            # A <- R^T A R with R = blockdiag(V): row pass into A2, then the
            # column pass reads packed column bands and writes back into A.
            rowf(V, A.view(B * nb, 64, n), A2.view(B * nb, 64, n))
            Pk.view(B, nb, n, 64).copy_(A2.view(B, n, nb, 64).permute(0, 2, 1, 3))
            colf(Pk, V, Pk2)
            A.view(B, n, nb, 64).copy_(Pk2.view(B, nb, n, 64).permute(0, 2, 1, 3))

            # Q <- Q R
            Pk.view(B, nb, n, 64).copy_(Q.view(B, n, nb, 64).permute(0, 2, 1, 3))
            colf(Pk, V, Pk2)
            Q.view(B, n, nb, 64).copy_(Pk2.view(B, nb, n, 64).permute(0, 2, 1, 3))
    A = A[:, unperm[:, None], unperm[None, :]]
    Q = Q.index_select(2, unperm)
    return A, Q


def _dyn_round(Bp, Qc, rowf, colf, bufs):
    """One dynamic-ordering block-Jacobi round: per matrix, greedily pair the
    16 32-blocks by descending off-diagonal block energy (cuSOLVER-syevj
    style), permute so those pairs are the eight diagonal 64-blocks, subsolve
    + update, unpermute. Costs ~2 tournament rounds; targets long-range
    coupling that fixed pairings never touch."""
    B, n = Bp.shape[0], Bp.shape[1]
    m, b = n // 32, 32
    dev = Bp.device
    E = Bp.view(B, m, b, m, b).square().sum(dim=(2, 4))
    E = E + E.transpose(-1, -2)
    E[:, torch.arange(m), torch.arange(m)] = -1.0
    order = torch.empty(B, m, dtype=torch.long, device=dev)
    bidx = torch.arange(B, device=dev)
    for k in range(m // 2):
        flat = E.view(B, -1).argmax(dim=1)
        i, j = flat // m, flat % m
        order[:, 2 * k] = torch.minimum(i, j)
        order[:, 2 * k + 1] = torch.maximum(i, j)
        E[bidx, i, :] = -1.0; E[bidx, :, i] = -1.0
        E[bidx, j, :] = -1.0; E[bidx, :, j] = -1.0
    T = (order[:, :, None] * b + torch.arange(b, device=dev)).reshape(B, n)

    nb = n // 64
    A2, S, V, Pk, Pk2 = bufs
    Bp = Bp[bidx[:, None, None], T[:, :, None], T[:, None, :]]
    Qc = torch.gather(Qc, 2, T[:, None, :].expand(B, n, n))
    S.view(B, nb, 64, 64).copy_(
        Bp.as_strided((B, nb, 64, 64), (n * n, 64 * n + 64, n, 1)))
    _jac64(S, V)
    rowf(V, Bp.view(B * nb, 64, n), A2.view(B * nb, 64, n))
    Pk.view(B, nb, n, 64).copy_(A2.view(B, n, nb, 64).permute(0, 2, 1, 3))
    colf(Pk, V, Pk2)
    Bp.view(B, n, nb, 64).copy_(Pk2.view(B, nb, n, 64).permute(0, 2, 1, 3))
    Pk.view(B, nb, n, 64).copy_(Qc.view(B, n, nb, 64).permute(0, 2, 1, 3))
    colf(Pk, V, Pk2)
    Qc.view(B, n, nb, 64).copy_(Pk2.view(B, nb, n, 64).permute(0, 2, 1, 3))
    invT = torch.argsort(T, dim=1)
    Bp = Bp[bidx[:, None, None], invT[:, :, None], invT[:, None, :]]
    Qc = torch.gather(Qc, 2, invT[:, None, :].expand(B, n, n))
    return Bp, Qc


# ---------------------------------------------------------------------------
# n = 512: one-stage blocked Householder tridiagonalization (latrd panels).
# ---------------------------------------------------------------------------

class _Sytrd:
    """Blocked one-stage tridiagonalization (latrd, lower). Each panel is ONE
    latrd kernel launch (one CTA per matrix; correction, Householder, fp16
    symv with fp32 accumulation, and the W recurrence all in-kernel). The
    rank-2*nb trailing update stays in torch (two fat GEMMs per panel). Q's
    orthogonality is structural; only the tridiagonal picks up ~1e-3 noise,
    absorbed by the finisher's RR. (The two-stage SBR reduction was cheaper
    in the reduction itself but its all-eigenvectors Q2 back-transform cost
    ~70ms/solve -- 6x this path's WY -- so one-stage wins overall.)"""

    def __init__(self, B, n, dev="cuda"):
        self.B, self.n = B, n
        self.A = torch.zeros(B, n, n, device=dev)
        self.A16 = torch.zeros(B, n, n, device=dev, dtype=torch.float16)
        self.P = torch.zeros(B, n, 2 * NB, device=dev)
        self.d = torch.zeros(B, n, device=dev)
        self.e = torch.zeros(B, n, device=dev)      # (B, n): kernel stride
        self.taus = torch.zeros(B, n, device=dev)
        self.Vs = [torch.zeros(B, n - k, min(NB, n - 2 - k), device=dev)
                   for k in range(0, n - 2, NB)]

    def panel(self, k):
        n = self.n
        m = n - k
        nbk = min(NB, n - 2 - k)
        self.A16[:, :m, :m].copy_(self.A[:, k:, k:])
        _launch(b"latrd512", (self.B, 1, 1), (512, 1, 1),
                self.A, self.A16, self.P, self.d, self.e, self.taus,
                n, k, nbk)
        return nbk

    def _post_panel(self, idx, k, nbk):
        m = self.n - k
        self.Vs[idx].copy_(self.P[:, :m, 0:2 * nbk:2])
        V = self.Vs[idx][:, nbk:, :]
        W = self.P[:, nbk:m, 1:2 * nbk:2].contiguous()
        self.A[:, k + nbk:, k + nbk:] -= V @ W.mT + W @ V.mT

    def run(self, A):
        n = self.n
        self.A.copy_(A)
        for idx, k in enumerate(range(0, n - 2, NB)):
            nbk = self.panel(k)
            self._post_panel(idx, k, nbk)
        self.d[:, n - 2] = self.A[:, n - 2, n - 2]
        self.d[:, n - 1] = self.A[:, n - 1, n - 1]
        self.e[:, n - 2] = self.A[:, n - 1, n - 2]
        return self.d, self.e[:, :n - 1], self.Vs, self.taus


_sytrd_cache = {}


def _sytrd_get(B, n):
    key = (B, n)
    st = _sytrd_cache.get(key)
    if st is None:
        st = _Sytrd(B, n)
        _sytrd_cache[key] = st
    return st




def _split512(d, e):
    """Zero negligible couplings; per-position block bounds [s, t)."""
    B, n = d.shape
    dev = d.device
    tnorm = (d.abs().amax(-1) + 2 * e.abs().amax(-1)).clamp_min(1e-30)
    es = torch.where(e.abs() <= (SPLIT_C * EPS512 * tnorm).unsqueeze(-1),
                     torch.zeros_like(e), e)
    idx = torch.arange(n, device=dev).expand(B, n)
    bnd = torch.cat([torch.ones(B, 1, dtype=torch.bool, device=dev),
                     es == 0], dim=1)
    s = torch.cummax(torch.where(bnd, idx, torch.zeros_like(idx)), 1).values
    nxt = torch.cat([torch.where(bnd[:, 1:], idx[:, 1:],
                                 torch.full_like(idx[:, 1:], n)),
                     torch.full((B, 1), n, device=dev, dtype=idx.dtype)], 1)
    t = torch.flip(torch.cummin(torch.flip(nxt, [1]), 1).values, [1])
    return es, s.int().contiguous(), t.int().contiguous(), tnorm.contiguous()


def _bisect512(d, es, s, t, tnorm):
    B, n = d.shape
    e2 = torch.cat([torch.zeros(B, 1, device=d.device), es * es], 1).contiguous()
    lam = torch.empty(B, n, device=d.device)
    _launch(b"bisect", (B, 1, 1), (n, 1, 1),
            d.contiguous(), e2, s, t, tnorm, lam, n, BISECT_ITERS)
    return lam


class _Invit:
    """State for the invit kernel: factor arrays are step-major (n, S)."""

    def __init__(self, B, n):
        S = B * n
        self.B, self.n, self.S = B, n, S
        dev = "cuda"
        self.dd = torch.empty(n, S, device=dev)
        self.dl = torch.empty(n, S, device=dev)
        self.du = torch.empty(n, S, device=dev)
        self.du2 = torch.empty(n, S, device=dev)
        self.piv = torch.empty(n, S, device=dev, dtype=torch.uint8)
        self.rhs = torch.zeros(n, S, device=dev)

    def bind(self, d, es, s, t, tnorm, lam):
        B, n = self.B, self.n
        self.dc = d.contiguous()
        self.esp = torch.cat([es, torch.zeros(B, 1, device=d.device)], 1).contiguous()
        self.lam = lam.reshape(-1).contiguous()
        self.s = s.reshape(-1).contiguous()
        self.t = t.reshape(-1).contiguous()
        self.tn = tnorm
        self.rhs.zero_()

    def solve(self, init):
        grid = ((self.S + 255) // 256, 1, 1)
        _launch(b"invit", grid, (256, 1, 1),
                self.dc, self.esp, self.lam, self.s, self.t, self.tn,
                self.dd, self.dl, self.du, self.du2, self.piv, self.rhs,
                self.n, self.S, 1 if init else 0, 1234)

    def Z(self):
        return self.rhs.view(self.n, self.B, self.n).permute(1, 0, 2).contiguous()

    def setZ(self, Z):
        self.rhs.copy_(Z.permute(1, 0, 2).reshape(self.n, self.S))


_invit_cache = {}


def _invit_get(B, n):
    key = (B, n)
    if key not in _invit_cache:
        _invit_cache[key] = _Invit(B, n)
    return _invit_cache[key]


def _find_runs512(lam, s, tnorm):
    """Vectorized cluster-run detection: maximal chains of consecutive
    positions in the same block with adjacent gaps < gtol. Returns
    (batch_of_run, lo, len) 1-D tensors, or None if no runs."""
    B, n = lam.shape
    dev = lam.device
    gtol = (GTOL_F * EPS512) * tnorm
    link = ((s[:, 1:] == s[:, :-1])
            & ((lam[:, 1:] - lam[:, :-1]).abs() < gtol.unsqueeze(-1)))
    if not bool(link.any()):
        return None
    z = torch.zeros(B, 1, dtype=torch.bool, device=dev)
    left = torch.cat([z, link], 1)
    right = torch.cat([link, z], 1)
    start = right & ~left
    end = left & ~right
    pos = torch.arange(n, device=dev).expand(B, n)
    bidx = torch.arange(B, device=dev).unsqueeze(-1).expand(B, n)
    lo = pos[start]
    hi = pos[end] + 1
    bof = bidx[start]
    return bof, lo, hi - lo


def _runs_orth512(Z, runs, jitter=1e-5):
    """Within-run orthonormalization via padded batched jitter-CholQR,
    bucketed by run length to bound padding waste. Rank injection from the
    jitter is intentional: the following invit solve purifies the injected
    directions back into the cluster subspace (stein iteration)."""
    bof, lo, lens = runs
    B, n = Z.shape[0], Z.shape[1]
    dev = Z.device
    Zc = Z.permute(0, 2, 1).reshape(B * n, n)   # columns as rows (copy)
    for lo_cap, hi_cap in ((0, 8), (8, 24), (24, 64), (64, 160), (160, 1 << 30)):
        mset = (lens > lo_cap) & (lens <= hi_cap)
        if not bool(mset.any()):
            continue
        bo, l, ln = bof[mset], lo[mset], lens[mset]
        W = int(ln.max())
        ar = torch.arange(W, device=dev)
        cols = l.unsqueeze(-1) + ar
        valid = ar < ln.unsqueeze(-1)
        flat = (bo.unsqueeze(-1) * n + cols).clamp(max=B * n - 1)
        V = Zc[flat.reshape(-1)].reshape(-1, W, n) * valid.unsqueeze(-1)
        G = V @ V.mT
        G = G + torch.diag_embed((~valid).float() + jitter)
        # skip the factor/apply when this bucket is already orthonormal
        # (later stein passes mostly are)
        eye = torch.eye(W, device=dev)
        if float((G - eye).abs().amax()) < 3e-4 + jitter:
            continue
        L, info = torch.linalg.cholesky_ex(G)
        if bool((info > 0).any()):
            G = G + torch.diag_embed(torch.full_like(G[..., 0], 1e-3))
            L = torch.linalg.cholesky(G)
        V = torch.linalg.solve_triangular(L, V, upper=False)
        V = V / V.norm(dim=-1, keepdim=True).clamp_min(1e-30)
        Zc[flat[valid]] = V[valid]
    return Zc.reshape(B, n, n).permute(0, 2, 1).contiguous()


def _polar512(Q):
    """Newton-Schulz polar orthonormalization; TF32 GEMMs (the finisher's
    fp32 polish scrubs the TF32 noise floor afterwards)."""
    B, n = Q.shape[0], Q.shape[1]
    prev = torch.backends.cuda.matmul.allow_tf32
    torch.backends.cuda.matmul.allow_tf32 = True
    # M = Q^T Q only feeds a power-iteration spectral-norm ESTIMATE (with a
    # 1.05 safety margin), so TF32 is plenty and saves an fp32 512-GEMM.
    v = Q.new_ones(B, n, 1)
    M = Q.mT @ Q
    for _ in range(6):
        v = M @ v
        v = v / v.norm(dim=1, keepdim=True).clamp_min(1e-30)
    sn = (M @ v).norm(dim=1).squeeze(-1).sqrt() * 1.05
    X = Q / sn.clamp_min(1e-30).view(-1, 1, 1)
    for _ in range(POLAR_ITERS):
        X = 1.5 * X - 0.5 * (X @ (X.mT @ X))
    torch.backends.cuda.matmul.allow_tf32 = prev
    return X


def _wy_apply512(Vs, taus, Z):
    """Z <- Q Z with Q = P_0 P_1 ... P_L (panel WY products), applied
    right-to-left. T per panel from T^-1 = diag(1/tau) + strictUpper(V^T V).
    GEMMs run TF32: Householder V entries are bounded by 1, the ~1e-3 noise
    floor lands on Q and the finisher's fp32 Newton-Schulz scrubs it."""
    n = Z.shape[1]
    ks = list(range(0, n - 2, NB))
    prev = torch.backends.cuda.matmul.allow_tf32
    torch.backends.cuda.matmul.allow_tf32 = True
    try:
        for idx in range(len(ks) - 1, -1, -1):
            k = ks[idx]
            V = Vs[idx]
            nbk = V.shape[2]
            tk = taus[:, k:k + nbk]
            safe = torch.where(tk == 0, torch.ones_like(tk), tk)
            Tinv = torch.triu(V.mT @ V, diagonal=1) + torch.diag_embed(1.0 / safe)
            Y = V.mT @ Z[:, k:, :]
            Z[:, k:, :] -= V @ torch.linalg.solve_triangular(Tinv, Y, upper=True)
    finally:
        torch.backends.cuda.matmul.allow_tf32 = prev
    return Z


def _eigh512(data):
    B, n, _ = data.shape
    dev = data.device

    # per-matrix scale guard: keeps the fp16 GEMM inputs in range for the
    # *_high/low_magnitude spectra; eigenvalues are rescaled at the end.
    sc = data.abs().amax(dim=(-2, -1)).clamp_min(1e-30)
    As = (data / sc.view(B, 1, 1)).contiguous()

    d, e, Vs, taus = _sytrd_get(B, n).run(As)
    es, s, t, tnorm = _split512(d, e)
    lam = _bisect512(d, es, s, t, tnorm)

    iv = _invit_get(B, n)
    iv.bind(d, es, s, t, tnorm, lam)
    iv.solve(init=True)
    iv.solve(init=False)
    iv.solve(init=False)
    runs = _find_runs512(lam, s, tnorm)
    if runs is not None:
        for _ in range(STEIN_REPS):
            iv.setZ(_runs_orth512(iv.Z(), runs))
            iv.solve(init=False)
        Z = _runs_orth512(iv.Z(), runs)
    else:
        Z = iv.Z()

    _, order = torch.sort(lam, dim=-1)
    Z = torch.gather(Z, 2, order.unsqueeze(1).expand(B, n, n))
    Z = _polar512(Z)
    Q = _wy_apply512(Vs, taus, Z)

    # ---- finisher: Rayleigh-Ritz + adaptive dynamic-ordering cleanup ----
    nb = n // 64
    bufs = (torch.empty(B, n, n, device=dev),
            torch.empty(B * nb, 64, 64, device=dev),
            torch.empty(B * nb, 64, 64, device=dev),
            torch.empty(B * nb, n, 64, device=dev),
            torch.empty(B * nb, n, 64, device=dev))
    Q = 1.5 * Q - 0.5 * (Q @ (Q.mT @ Q))
    Bp = Q.mT @ (As @ Q)
    Bp = (0.5 * (Bp + Bp.mT)).contiguous()
    Qc = torch.eye(n, device=dev).expand(B, n, n).contiguous()
    gate = (0.5 * 200.0 * n * EPS512
            * As.abs().sum(dim=1).amax(dim=-1).clamp_min(1e-30))
    rootn = float(n) ** 0.5

    def _ok(M):
        off2 = (M.square().sum(dim=1)
                - torch.diagonal(M, dim1=-2, dim2=-1).square()).clamp_min(0.0)
        return rootn * off2.amax(dim=-1).sqrt() <= gate

    # first-order eigenvector refinement: the fp16 reduction noise lands
    # as small DISTRIBUTED offdiag(Bp) mass, which dynamic-ordering rounds
    # grind at (measured 6 rounds + a full sweep on mixed/rankdef). One
    # E-step kills well-separated pairs quadratically for ~4 GEMMs; the
    # gap guard leaves near-degenerate mass to the dyn rounds (their
    # specialty). E is skew to first order, the NS re-orth cleans the rest.
    if REFINE and not bool(_ok(Bp).all()):
        w0 = torch.diagonal(Bp, dim1=-2, dim2=-1)
        den = w0.unsqueeze(-2) - w0.unsqueeze(-1)   # den[j,i] = w_i - w_j
        E = torch.where(den.abs() > 10.0 * Bp.abs(), Bp / den,
                        torch.zeros_like(Bp))
        # only refine matrices that (a) fail the gate and (b) have a
        # moderate correction: E is exactly antisymmetric so (I+E) is
        # non-orthogonal by ||E||^2 — clustered spectra hit ||E||~0.8
        # where the linearization is meaningless (measured post-ratio 60
        # and a 640-matrix fallback storm). polar (prescaled 5-round NS)
        # restores orthogonality from en up to ~0.3; one NS round cannot.
        en = E.square().sum(dim=1).amax(dim=-1)
        E = E * ((~_ok(Bp)) & (en < 0.3)).view(B, 1, 1).to(E.dtype)
        Q = _polar512(Q + Q @ E)
        # polar's TF32 NS floor (~1e-3) sits ON the finisher's orth gate
        # (0.0015): one fp32 NS round squares it away (a 640-matrix
        # fallback storm without it, measured).
        Q = 1.5 * Q - 0.5 * (Q @ (Q.mT @ Q))
        Bp = Q.mT @ (As @ Q)
        Bp = (0.5 * (Bp + Bp.mT)).contiguous()

    dyn_ran = False
    for _ in range(DYN_MAX):
        if bool(_ok(Bp).all()):
            break
        Bp, Qc = _dyn_round(Bp, Qc, _row32, _col32, bufs)
        dyn_ran = True
    for _ in range(CLEANUP_MAX):
        if bool(_ok(Bp).all()):
            break
        Bp, Qc = _sweeps512(Bp, Qc, 1, _row32, _col32, bufs)
        dyn_ran = True
    bad = ~_ok(Bp)
    if dyn_ran:                       # Qc is still I when no round ran
        Q = Q @ Qc
    # Eigenvalues = the Rayleigh quotient = diag(B) with B = Q^T A Q,
    # tracked EXACTLY through the dynamic-ordering rotations (each round
    # applies the same Qc to both sides). No separate A@Q GEMM: the final
    # polish perturbs Q by ~1e-5, far under the 1.22% eigenvalue gate.
    w = torch.diagonal(Bp, dim1=-2, dim2=-1)
    w, order = torch.sort(w, dim=-1)
    Q = torch.gather(Q, 2, order.unsqueeze(1).expand(B, n, n))
    # The Bp gate is blind to a non-orthogonal Q (collapsed directions make
    # Q^T A Q look diagonal while A Q != Q diag(w)); gate orthogonality of
    # the final Q explicitly, reusing the polish's Gram. Checker allowance
    # is 100*n*eps (L1 columns); polish squares the pre-polish error, so
    # gating the PRE-polish Gram at the same threshold is conservative.
    G = Q.mT @ Q
    Gd = G - torch.diag_embed(torch.ones(B, n, device=dev))
    bad |= Gd.abs().sum(dim=1).amax(dim=-1) > 0.5 * 100.0 * n * EPS512
    if POLISH:
        Q = 1.5 * Q - 0.5 * (Q @ G)
    if bool(bad.any()):
        idx = torch.nonzero(bad, as_tuple=False).squeeze(-1)
        wf, qf = torch.linalg.eigh(As[idx])
        w = w.clone(); Q = Q.clone()
        w[idx] = wf
        Q[idx] = qf
    return Q, (w * sc.unsqueeze(-1)).contiguous()


def custom_kernel(data: input_t) -> output_t:
    n = data.shape[-1]
    if n <= 32:
        return _eigh32(data)
    # n=1024 measured SLOWER through this path (110-129ms vs eigh's 88-92:
    # B=60 underfills the 1-CTA-per-matrix kernels and the batched stages);
    # keep the cuSOLVER fallback there.
    if n == 512:
        return _eigh512(data)
    values, vectors = torch.linalg.eigh(data)
    return vectors, values