This directory contains dense linear-algebra front ends and operation descriptors. It is the policy layer that selects or describes dense operations before they lower to backend wrappers and kernels.

linalg.hpp: public include point for the dense linalg layer.
operation_tags.hpp: central catalogue of dispatchable operation values and their diagnostic names, including callable-carrying elementwise operation descriptors.
elementwise_functions.hpp: named backend-neutral function objects that may have explicit precompiled backend lowerings.
reduction_axes.hpp: normalized reduced/surviving axis descriptors shared by reduction front ends and backends.
contraction_axes.hpp: normalized contracted/surviving axis descriptors for fixed-rank pairwise contractions.
contraction_strides.hpp: jointly merged M/N/K stride metadata plus direct and one-residual-axis rank-two GEMM projections.
backend_selector.hpp: ordered backend selector values, Uni20-owned operation/storage defaults, user selector overrides, and the stateless host backend entries shared with tensor storage.
dispatch.hpp: operation-value backend-list dispatch helpers.
dispatch_diagnostics.hpp: disabled-by-default structured observation of ordered backend walks.
kernel_attempt.hpp, dispatch_error.hpp, dispatch_error_presentation.hpp: clean backend-decline results and terminal dispatch-failure diagnostics.
async.hpp: opt-in include point for scheduled Async<Tensor> copies, transforms, reductions, contractions, dense solves and factorizations.
`async/`: all-async Tensor wrappers over the synchronous operation layer.
`blas/`: mdspan-to-BLAS-compatible descriptor and direct wrapper helpers.
`cublas/`: provider-ready cuBLAS operations over staged CUDA mdspans and synchronized buffer access.
`cpu/`: accessor-respecting reference GEMM and GEMV leaves over resolved host mdspans.
`ops/`: Tensor-facing dense operation wrappers.
`backends/`: operation-tag backend implementations.
`backends/direct_gemm/`: direct tensor-contraction lowering through a retained GEMM selector.
`backends/looped_gemm/`: residual-M/N tensor-contraction loops through a retained GEMM selector.
`backends/blas/`: operation-tag BLAS backend adapters.
`backends/cublas/`: provider-ready cuBLAS backend adapters.
`backends/cpu/`: generic CPU operation-tag kernels, dense matrix norms and solves, and the dense matrix exponential implementation.
`backends/lapack/`: LAPACK-backed linalg entry points.
`backends/cusolver/`: cuSOLVER-backed linalg entry points.

Notes

Keep dense linalg APIs separate from matrix-free Krylov APIs in krylov/.
Backend-specific code should stay in backends/ and call provider wrappers under backend/ where appropriate.
Backend implementations include operation_tags.hpp; do not redeclare operation identities locally.
Bare-mdspan GEMM, GEMV, and copy calls use dispatch_kernel with their operation tag and an explicit selector. Tensor-view operands use the fixed-output gemm or gemv front end so it can derive their common storage selector, or they may supply an explicit selector override.
Async Tensor operations live in the opt-in async/ layer. They resolve the static storage selector before scheduling and await Tensor values. Fixed operands then lower to mdspecs for coroutine-aware operation dispatch; value-oriented wrappers may call the corresponding synchronous Tensor front end. Ordinary backends do not depend on the async runtime.
Async reductions validate axes where applicable, defer output preparation and dispatch until inputs are readable, and return either a storage-preserving async Tensor, an explicit output epoch, or a nonblocking host scalar result.
copy_op is the semantic element-copy operation used by Tensor copy and make_tensor. Its CPU backend respects accessors. CudaReferenceBackend handles canonical contiguous host/device and device/device transfers plus registered same-device positive-strided accessor lowerings. Future BLAS matrix-copy extensions may accept representable rank-two layouts and conjugating views; strict BLAS/LAPACK compute wrappers still do not materialize implicitly.
transform_op<F> and transform_inplace_op<F> carry a const-invoked callable through dispatch. The CPU reference backend supports arbitrary rank and input arity. The CUDA reference backend explicitly registers same-element-type unary and binary arithmetic function objects plus stateful linalg::scale<Factor>; other optimized callable/layout combinations belong in specialized backends.
sum_reduction_op<R, N> carries normalized reduced and surviving axes. Tensor front ends remove the selected axes, preserve canonical result layout, and use the generic CPU reference executor when no earlier backend accepts.
matrix_norm_op computes maximum-entry, induced one/infinity, or Frobenius norms. Complex norms use the mathematical complex magnitude; LAPACK handles complex Frobenius norms but declines its incompatible component-sum forms.
linear_solve_op is the destructive general-system workspace operation. solve_inplace exposes that contract directly, while solve preserves its inputs by materializing column-major host work matrices. The recoverable solve_inplace_with_info frontend returns the kernel's SolveInfo output; numerical failure does not mean dispatch decline. See square solves for pivot policy and precision.
qr_op and lq_op are destructive reduced real factorization operations. Their preserving value APIs materialize column-major host work and return owning reduced factors.
Dense linalg operations use operation values, kernel_accepts_types, and try_kernel; the former backend-tag selector hierarchy has been removed.
Default selector resolution gives a user backend_selector_override complete replacement priority over Uni20's backend_selector_default; absent either, it uses the storage policy's general selector.
kernel_type_candidates(...) filters a selector to its ordered yes and maybe type candidates while preserving backend values, allowing shared conformance tests to exercise every compatible backend independently.
Scalar-generic code should use Uni20 scalar traits and numeric limits from core/.

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Notes

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