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Novelty & Attribution

Author: Dmitry Sierikov, 2026. License: MIT (code).

This note states honestly what is original to resona and what is classical prior art it stands on. Calibrated honesty is the point: a tool that overclaims is a tool you cannot trust.

What we do NOT claim (classical foundations, credited)

resona is built on established numerical linear algebra and free probability. We claim none of the following — they are the field's, and we use them gratefully:

  • Stochastic Lanczos quadrature (SLQ) for Tr f(A) / spectral densities — Golub & Meurant; Ubaru, Chen & Saad.
  • Lanczos / Krylov subspace methods; Hutchinson trace estimation; Kernel Polynomial Method (KPM).
  • Free probability — Voiculescu (R-transform, free convolution), Speicher (free cumulants, non-crossing partitions); subordination (Biane).
  • Standard results: condition number / Krylov deflation, the Marchenko–Pastur law, Tracy–Widom edge, Tang-style dequantization of low-rank problems.
  • The non-Hermitian objects themselves — the Brown measure and its hermitization (Haagerup–Larsen; Girko's H(z)=[[0,A−z],[(A−z)*,0]]), biorthogonal perturbation theory (left/right eigenvectors, the (u*Bv)/(u*v) eigenvalue derivative, exceptional points), and the random-matrix universality classes (Wigner–Dyson β=1,2,4; the level-spacing ratio). We use these; we did not discover them.

If you need a specific sub-algorithm, it predates this package.

What resona contributes (and reserves)

The contribution is synthesis, framing, and one interface — not new sub-algorithms:

  1. A single primitive — Spectral, the "FFT of operators". One object that unifies probe → compose → read for any operator given as a black-box matvec. The organizing claim: Spectral.of(A) is to operators what fft is to signals — the representation in which the hard operation (composition / spectral computation) is linear, computed matrix-free.

  2. Matrix-free operator composition as an algebra. Exposing A+B, A·B (via the exact closure (A+B)x = Ax + Bx, and free additive/multiplicative convolution) as + and @ on responses — composing spectra without forming the composed operator and without diagonalizing it.

  3. A built-in cost / feasibility law — the Extraction Law. The cost to read a solution from the resolvent field scales as the inverse distance to the field's non-removable singular set (edges/shocks/continua, not isolated poles), with the singularity type setting the exponent; over a parameter family this cost is a field singular on the discriminant. With it: the removable-vs-genuine test (does a finite lift's effective rank saturate?) as a computable criterion for cheap / liftable / genuine wall, and the effective rank Φ₁ = Tr(A)²/Tr(A²) of the response as the classical↔quantum / cheap↔hard dial.

  4. The unifying synthesis. Identifying the defect (numerical-analysis error) = the free-convolution shock = the edge of chaos of the subordination fixed point = the spectral phase boundary as one object, and using it to bridge: matrix-free spectral computation, free probability, computational capacity, and the classical/quantum boundary — through one measurable quantity (the response and its effective rank).

  5. A closed verb set over the same interface — including the non-Hermitian corner. The single primitive resolves into five matrix-free verbs on one object: build (probe / synthesize), read (density, moments, trace, effective rank), move (the spectral Jacobian ∂λ/∂k and its flow), trust (the defect, σ_min, pseudospectrum, departure-from-normality), and compose (free convolution). The closure claim is that all five run from matvecs alone, for both self-adjoint AND non-self-adjoint operators — the latter by reading the Brown measure as a matrix-free log-det object (hermitization → SLQ Tr log) and the eigenvalue motion by biorthogonal left/right reads, so the plane spectrum and its sensitivity join the same matrix-free calculus. The last dense O(N³) of the self-adjoint side (the eigenvector / W half of the conjugate pair) is removed for selected modes by a shift-invert read.

Reserved: the framing and API design of the single-primitive interface; the five-verb closure and its extension to the non-Hermitian corner (the realization and unification — not the Brown measure or biorthogonal theory themselves, which are credited above); the formulation of the Extraction Law and its removable/genuine criterion; the Φ₁-as-dial proposal; and the unifying synthesis above. These are research contributions, offered openly under MIT, with attribution requested.

The mathematical ideas we read as ours

Beyond the code, resona encodes a small set of research ideas. Each rests on classical mathematics (credited above); what is ours is the cross-field synthesis and the specific conjectures — stated here with their status, so the claim is exact and bounded. (Full development lives in the companion research; this is the intellectual core the library embodies.)

1. The Extraction Law. The cost to extract a solution from the resolvent (Green's) field scales as ε^{-a} · dist(z, Σ*)^{-b}, where Σ* is the field's non-removable singular set (edges, branch points, shocks, continua — not isolated poles, which deflate), and the exponent b is set by the singularity type. Over a parameter family the cost is a scalar field singular on the discriminant — the phase diagram of computability. Ours: the single statement holding the same shape across solves, spectra, disorder, and the P↔BQP boundary. Classical: the per-field cost laws (condition number / Krylov, critical slowing, BBP, sample-complexity ∝ rank). Status: candidate framework; exponents partly verified numerically.

2. The identity defect = shock = edge = phase boundary. The numerical-analysis defect (D_n = P_n − P_{2n}, the frozen-prediction error, the spectral curvature κ_W) is identified with the free-convolution Burgers shock, the edge of chaos of the subordination fixed point (|T'|→1, critical slowing), and the spectral phase boundary — one object across numerical analysis, free probability, and dynamics. Ours: the identification as one object. Classical: each component (Voiculescu/Biane free convolution = complex Burgers; critical slowing; band edges). Status: synthesis, numerically illustrated.

3. Φ₁ as the dial of difficulty. The effective rank of the response measure, Φ₁ = Tr(A)² / Tr(A²), is proposed as the measurable quantity that grades a problem: low Φ₁ ⇒ structured / cheap / dequantizable; high Φ₁ ⇒ genuine frontier (including the classical↔quantum boundary). With it, the removable-vs-genuine test: a finite lift's effective rank saturates (apparent wall, extractable) or grows (genuine wall, e.g. Shor's orbit). Ours: Φ₁-as-boundary and the lift-saturation criterion. Classical: participation ratio; Tang-style low-rank dequantization. Status: sharp, falsifiable conjecture; demonstrated on examples, not a proven boundary theorem.

4. The response measure as a conjugate pair, with an uncertainty. An operator is one measure μ_B = Σ_i (v_iᵀ B v_i) δ(λ_i) at two resolutions — its density (W = ∂λ/∂k, eigenbasis-resolved) and its moments (Tr(A^p B), matrix-free) — related by Σ_i λ_i^p W_i = Tr(A^p B), and Fourier-conjugate via φ(t)=Tr e^{-itA}, giving (λ-resolution) × (moment order) ≳ 1. Ours: the framing of W and Φ as one measure at two resolutions, bridged by Lanczos quadrature, with a Heisenberg-type bound. Classical: the identity itself (elementary); SLQ. Status: framing — and the W (density) side, long the only remaining O(N³), is now read matrix-free for a SELECTED block of modes (a shift-invert read of κ_W / the spectral flow), closing the conjugate pair on both sides for the modes one actually tracks.

5. "Any shock is a sum of linearities" = the R-transform = the Cole–Hopf of free probability. The R-transform linearizes the shock-forming free-convolution flow exactly as Cole–Hopf / Carleman linearizes Burgers — verified: the flow is a straight line in R; the shock lives only in the density. Ours: the explicit bridge tying the defect-calculus Cole–Hopf/Carleman insight to Voiculescu's R-transform. Classical: free convolution = Burgers; R-transform; Cole–Hopf. Status: framing/bridge, numerically verified.

6. The harvest principle. The answer pre-exists in the resolvent field; solving is extraction, not creation; an operator is a program, and a generic medium already computes (the response / reservoir). The defect is the side-channel to collect. Status: organizing principle / worldview, not a theorem.

7. The matrix-free spectral calculus extends past self-adjointness. The same probe → read → move → trust → compose interface is carried to NON-normal operators by hermitization: the eigenvalue distribution in the plane (the Brown measure) becomes a matrix-free object — S(z)=(1/2N)Tr log((A−z)*(A−z)) read by SLQ, μ=(1/2π)ΔS — and the eigenvalue motion and exceptional points are read from biorthogonal left/right pairs (∂λ=(u*Bv)/(u*v), with |u*v|→0 locating an EP). Ours: the realization — Brown measure as an SLQ log-det field, the EP locator as a phase-rigidity read, and their unification under one matrix-free interface, so the closed verb set holds in the plane too. Classical: the Brown measure and hermitization (Haagerup–Larsen, Girko), biorthogonal perturbation theory, and free probability (credited above). Status: realized and verified against dense ground truth (the circular law to the 1/π density; EP located; free-sum Brown measure covering the eig(A+B) cloud) — engineering + framing, not new mathematics.

Honest status

The Extraction Law and Φ₁-boundary are candidate statements (sharp, falsifiable, partly verified numerically in the companion research), not finished theorems. The package's computations are sound (verified against dense ground truth in tests/ and examples/); the unifying claims are research hypotheses, labelled as such.