Quantum memory and numerical span contracts

Read side by side

The deep audit uses independent trajectories, finite integral references and changes of generators to check behavior beyond the existing verification suite.

Schrödinger memory convention

QuantumFeshbachReduction.self_energy(z) retains the Hamiltonian convention

memory_kernel() returns the kernel for the derivative equation obtained from psi' = -i H psi. With zero hidden initial state,

Both coupling maps in the returned MemoryKernel carry -i. The kernel's RationalMemoryRealization therefore reconstructs the full Schrödinger trajectory after the retained and hidden coordinates are ordered consistently. In the open upper half plane, the convention is kernel.laplace(-1j*z) = -1j*self_energy(z).

For H=[[0,1],[1,0]], the retained amplitude from initial state [1,0] is cos(t). A positive derivative kernel would instead produce cosh(t). The regression checks the direct matrix exponential and the analytic cosine.

Infinite and finite Laplace transforms

MemoryKernel.laplace(z) requires the sufficient half plane

This domain checks the full hidden realization. It is conservative when unstable hidden modes cancel from B exp(tD) C. The eigenvalue and residual checks use floating point arithmetic and retain numerical certificate status.

rational_transform(z) separately returns B solve(zI-D,C) whenever the resolvent exists. It makes no claim that the infinite Laplace integral converges. For the kernel exp(t), its value at zero is -1 as an algebraic continuation, while laplace(0) rejects the divergent integral and certificate(0) is invalid.

finite_laplace(z,horizon) evaluates the finite integral by the upper right block of an augmented matrix exponential. It does not invert zI-D, so it also works at a resolvent pole. For K(t)=exp(t), z=1 and horizon 2, the integral is 2. For the corresponding size two Jordan kernel K(t)=t exp(t), it is 2 as well. The finite integral remains a numerical result, with no rigorous floating point enclosure implied.

Constraint span closure

constraint_closure preserves its coefficients, residual matrix and maximum residual in the declared generator coordinates. Its validity decision uses the spectral norm of the quotient commutator map in a Hilbert Schmidt orthonormal basis of the resolved span. A unitary change of orthonormal basis acts unitarily on the input tensor coordinates, so this norm is invariant.

Multiplying the two Pauli generators X,Z by 1e-6 does not close their span. Its raw commutators shrink quadratically with the generator scale, while the resolved span is unchanged. The span obstruction rejects this example over the tested uniform scales from 1e-100 through 1e100.

The SVD rank threshold and singular spectrum are certificate diagnostics. Extremely ill conditioned generating families remain a finite precision rank boundary. Numerical span resolution does not certify an exact symbolic rank.

Spectral dimension

spectral_dimension cancels the common exponential factor associated with the smallest eigenvalue before computing the heat trace ratio. It therefore returns 2000 for the scalar identity at time 1000, even though the heat trace itself underflows.

For a finite nonnegative operator the limit at time zero is zero. The large time limit is zero when a zero eigenvalue is present. A strictly positive operator instead has asymptotic spectral dimension 2*t*lambda_min.

Mutable inputs and finite validation

Lindblad generators and memory kernels snapshot their constructor matrices. Caller mutation of the original arrays no longer changes the represented model. The snapshots are read only. Lindblad operations also recheck the declared form and reject reassigned nonfinite Hamiltonians, jumps or computed outputs before they can produce a positive certificate.

Shared quantum Hermitian, unitary, projector and density validation rejects nonfinite or negative tolerances. These are numerical tolerance decisions, without an exact theorem claim about unrounded real inputs.