New in 0.3.0 Core Engine (sft.core) W-computation with 5 backends: Dense (einsum), Sparse, Edge-Laplacian, Coordinate-Diagonal, Repeated-Identity — each with optimised kernel computation, automatically selected by _resolve_basis_backend. OperatorFamily with fluent API: .at(k) / .shift(dk) / .toward(target) / .flow(target) / .solve(target) chaining. InverseResult as a tuple subclass backward-compatible with (k, err, converged) while exposing .trajectory, .residual_history, .eigh_count, .n_refresh, .time_ms. Adaptive inverse design: damped Newton with stagnation detection, periodic W-refresh (every N steps), support for linear / hessian / trust / homotopy methods. Non-Hermitian support: biorthogonal kernel ⟨Lᵢ|Bⱼ|Rᵢ⟩, _eig_biorthogonal, BiorthogonalState. Caching: build() result cached on _last_build_key; invalidate_cache() for manual invalidation. Complexity measure: fam.complexity = rank(W)/N, fam.condition_number(), fam.isospectral_dimension() = M - rank(W). Isospectral flow: fam.isospectral_flow(direction) walks along ker(W) and monitors spectral drift. Operator Calculus (sft.operator_algebra + sft.algebra) OperatorSpec: is_symmetric, is_hermitian, is_sparse, bandwidth, stability score, basis_kind inference from matrix structure. OperatorCost: memory_estimate, eigh_cost, predict_cost, inverse_cost with human-readable formatting. OperatorLaw: spectral_law with type-checking for ⊕, ⊗, ∘, ○, ↻ operations; LawSet.verify(). OperatorInvariant: expectation(model) — monte carlo gating. CostModel: predict(problem_size, M) via cached benchmarks. Direct sum ⊕: block-diagonal composition, preserves OperatorFamily interface. Tensor sum ⊗: A⊗I + I⊗B, eigenvalues add. Linear composition ∘: W_new = W_outer @ C (chain rule). Jordan fusion: jordan_fuse / multi_jordan_fuse / jordan_fuse_chain — construct exceptional-point operators with controlled nilpotent index (EP 2/4/8/16). AlgebraTransform: deferred operations via .then(transform) fluent API. Spectral Topology (sft.topology) Monodromy: eigenvalue tracking along closed loops via Hungarian matching on eigenvector overlaps; detects braiding/swaps; parallelisable via joblib. Berry holonomy: Z₂ ±1 sign flip detection after 2π loops. Eigenvalue winding: gap winding number around zero for non-Hermitian families. Complex monodromy: full summary with tracked eigenvalues + swaps + pairwise windings. Exceptional point atlas: grid+refinement scan of eigenvalue gaps with eigenvector coalescence scores, candidate reporting. Spectral flow: wrapper around monodromy returning tracked eigenvalues. Second-Order Analysis (sft.hessian) hessian (finite-difference): (N, M, M) tensor from central 2nd-order FD with eigh memoisation (4M² → ~2M calls). hessian_analytic: fully vectorised analytic formula 2 Σ_{p≠i} (vᵢᵀBⱼvₚ)(vₚᵀBₗvᵢ)/(λᵢ−λₚ) — O(N²·M²). hessian_sparsity: fraction of near-zero entries. spectral_curvature: d²λ/dα² along a given direction. Pre-Built Families (sft.families) random(N, M, seed, sparsity) — random symmetric family. graph_laplacian(adjacency) — edge-weight Laplacian family; sparse/dense dispatch. toeplitz(N, diagonals) — banded Toeplitz/filter family. diagonal(N) — repeated-identity family (ORDER regime, rank(W)=1). avoided_crossing_2x2(Delta) — classic avoided crossing. Physics Operators (sft.physics) exceptional_point_2x2() — standard [[0,1],[z,0]] family. pt_symmetric_2x2(gamma) — PT-symmetric 2x2. schrodinger_1d(x, potential, ...) — 1D Schrödinger with parametric potential. laplacian_grid(shape, bc) — grid Laplacian with Dirichlet/Neumann BC. 12 Domain Adapters (sft.adapters) | Adapter | Signal → OperatorFamily | |---------|------------------------| | AudioAdapter | Audio signal → Hankel/autocorrelation operator | | ImageAdapter | Image patches → covariance operator | | GraphAdapter | Adjacency matrix → graph Laplacian family | | TextAdapter | Text corpus → co-occurrence/embedding operator | | TimeseriesAdapter | Timeseries → delay-embedding operator | | VideoAdapter | Video frames → spatiotemporal covariance | | VoxelAdapter | 3D voxel grid → volumetric Laplacian | | PointCloudAdapter | Point cloud → affinity/Laplacian | | MolecularAdapter | Molecular structure → Coulomb/molecular operator | | FinancialAdapter | Financial returns → correlation/risk operator | | TabularAdapter | Tabular data → cross-covariance operator | | MeshAdapter | Triangular mesh → FEM/cotangent Laplacian | Each adapter provides .to_family(n_kernels) returning an OperatorFamily ready for spectral analysis. Graph Analysis (sft.graphop, sft.embed) GraphOperator: bridge detection (Tarjan), articulation points, k-cores (Batagelj-Zaversnik), graph density. GraphEmbedder: deterministic spectral embeddings with heat/diffusion/Wavelet kernels. LogicalGraphEmbedder: AND/NOT/IMPLY logical projections + ternary (0/1/2) embeddings. Streaming & Order Statistics (sft.order, sft.streaming) UniversalRankOperator: CDF-based ranking, quantiles, zeta distribution fitting. DefectPrecomputedCDF: precomputed queryable CDF. StreamingCDF: online/streaming CDF with Welford-style quantile updates. StreamingOrderOnline: streaming order statistics with quantile estimation. carleman_cdf: GF(2)/GF(3) + Complex Hermitian CDF. Task Routing (sft.tasks) OperatorGenus: 14 canonical problem genera (SORT, CLUSTER, FILTER, CLASSIFY, etc.). classify_task(nl_text): NLP → OperatorGenus mapping. cdf_rank_sort: spectral-rank via CDF. dct_matrix, filter_via_dct: DCT-based filtering. route_and_solve: NL pipeline → family → solve. task(), pipe(): decorators for composing operators. Inverse Design Suite (sft.inversion) bottleneck_inverse — projection onto low-rank subspace. fixed_point_inverse — Newton on a linear subspace. monodromy_inverse — drift-based inversion via loop tracking. Global Invariants (sft.invariants) svd_kurtosis(W) — detects sparse-rank vs ORDER regimes (≈1.8 → ORDER, >3 → fold/sparse). hessian_sparsity(family) — structured curvature measure. poisson_preimage(W) — KS statistic for parameter-space separability. w_coherence(W) — mean column cross-correlation of W. zeta_fingerprint(W) — log₁₀ quantile curve of singular values. all_invariants(family) — all 5 at once. Numerical Methods Arnoldi iteration (sft.arnoldi): Arnoldi Krylov basis, Rayleigh-Ritz eigenpair estimation, krylov_solve. Basis (sft.basis): Toeplitz embedding, Gaussian affinity, covariance basis construction. Homotopy (sft.homotopy): regularised pseudoinverse, trust-region corrector, track_homotopy with adaptive steps. Carleman (sft.carleman): GF(2)/GF(3) + complex Hermitian structured operators. Optimal Transport (sft.transport): W2 distance between spectral distributions. Compression (sft.compress): DCT codec for spectral data. Verification Suite (sft.verify) 8 independent verification gates covering: Core HF kernel correctness (batch vs sequential) Prediction error scaling (O(dk²)) Inverse convergence (Newton step count) Invariant consistency (kurtosis vs complexity) Graph response kernel (edge formula) Domain adapter round-trips Streaming CDF exactness for sorted input Task classification accuracy