Metcore-suite

What is live in this deployment?

Is my kernel rational (MET applies) or power-law?

Memory modes and embedding order K

All in one call, plain-language verdict

Asymptotic Markovian rate gamma_M without simulation

Synchronization threshold K_c(M) for retarded Kuramoto

BoltZ-Kernel pipeline: the bath spectral density J(omega) is supplied, the correlation C(tau) is computed by FFT, the MaxEnt kernel K*(tau)=exp[-e(tau)/T_eff]/Z is built, the master equation is solved, and the Lindblad gap G_int is produced.

The Metcore suite applies the same MET engine across four scientific domains. The central core is the Markov Embedding Theorem (MET).

Coherence rho_01(t): Lindblad (grey dashed) vs MaxEnt non-Markovian kernel (gold curve). The MaxEnt kernel reveals population revivals and slower coherence decay than Lindblad predicts for a structured bath.

Shape of the MaxEnt kernel K*(tau) for three bath types: Ohmic (fast decay), Lorentzian / structured cavity (oscillatory), and photonic / band-edge (power-law tail). Each shape corresponds to a distinct physical regime.

ILLUSTRATIVE MODEL from the suite's documented behaviour. Not a live solver.

Quantum physics · Scientific tools · Metcore suite

Metcore-suite: one theorem, four domains, three interfaces

BoltZ-Kernel (MaxEnt-Kernel) + the Metcore suite: measure whether your system's memory matters, before deciding to simulate beyond Lindblad

Every quantum physics laboratory that measures the spectral density J(omega) of its environment uses the Lindblad-GKSL equation by default to predict decoherence. It is convenient, but often wrong. The problem is not with the form of the equation - it is that it implicitly assumes the bath instantly forgets all correlations with the system. For structured baths (resonant cavity, photonic crystal, waveguide), this assumption is structurally incorrect.

The Metcore suite is built around a practical question: is your system's memory significant enough to change predictions? The BoltZ-Kernel tool answers this question without requiring expertise in advanced non-Markovian theory. You just need to supply J(omega). The answer is quantified: a numerical Lindblad gap G_int, a regime verdict (Markovian / weakly NM / strongly NM), and the memory kernel shape explaining the observed behaviour.

2. The problem: Lindblad's memoryless assumption

The Markov approximation and its limits

The Lindblad-GKSL master equation writes drho/dt = -i[H, rho] + mathcalL[rho], where mathcalL is a time-local superoperator. This form assumes the bath relaxation rate is instantaneous - that the bath correlation C(tau) decays so fast that the memory kernel K(t,s) reduces to a Dirac delta. This is true for high-temperature ohmic baths. It is not true for cavities, photonic crystals, or any environment with a structured spectral density.

The practical consequences are significant. Lindblad predicts exponential coherence decay - smooth and monotone. Real systems with structured baths exhibit population revivals, slower-than-predicted decay, and power-law tails instead of exponentials. Lindblad cannot predict these effects because it has erased bath memory before the calculation begins.

The BoltZ-Kernel pipeline follows five steps. (1) You supply the spectral density J(omega) - measured, fitted, or theoretical. (2) The bath correlation C(tau) = g^2 integral_0^inf J(omega) e^-i(omega - omega_0)tau domega is computed by fast FFT (165x-326x faster than adaptive quadrature for typical parameters). (3) The MaxEnt memory kernel K^*(t,s) = exp[-e(t,s)/T_eff] / Z(t) is built, where e(t,s) = integral_s^t | C(tau - s)|^2 dtau is the accumulated correlation energy. This is the least-biased kernel consistent with the bath correlations - the derivation follows Jaynes 1957. (4) The full integro-differential equation is solved. (5) The Lindblad equation is solved in parallel, and the gap G_int is computed.

faster than adaptive quadrature for typical parameters (FFT vs adaptive integration)

K* is the least-biased kernel: maximises Shannon entropy under the constraint of bath correlations.

The originality of this kernel is that it is derived by a variational principle (Jaynes Maximum Entropy) rather than a perturbative approximation. We do not assume a particular form for K - we derive the most neutral form consistent with what the bath correlations impose. This neutrality is both the strength and the limit of the approach: the MaxEnt kernel is not the true kernel of the system, it is the reference kernel against which to compare.

4. Lindblad gap G_int and non-Markovian dynamics

G_int = integral | rho_NM(t) - rho_Lindblad(t)|_1 dt is the cumulative error of the Markov approximation over the full temporal trajectory. Small G_int ( 0.1) means Lindblad is wrong, and the solver shows by how much and how: population revivals, slower coherence decay, power-law tails.

The shape of K^*(tau) encodes the physical regime. An ohmic bath generates a fast-decaying kernel close to a Dirac delta - Lindblad is a good approximation. A Lorentzian cavity generates an oscillatory kernel with a slowly decaying envelope - G_int becomes significant. A photonic crystal with band-edge generates a power-law kernel that never falls completely to zero - G_int is large, Lindblad is systematically wrong.

Geometric non-Markovianity N_G and CP-divisibility

The suite includes a geometric measure N_G of non-Markovianity, computed via the obliquity-ng component. N_G quantifies the information backflow from the bath to the system - N_G = 0 is strictly Markovian. CP-divisibility is checked via the zero-frequency Choi matrix: C(omega = 0) = integral_0^inf C_K(tau) dtau ge 0 is the necessary and sufficient condition. Both measures are exposed via the MCP tools nonmarkovianity_ng and cptp_certify.

5. The Metcore suite: one theorem, four domains

The central engine of the suite is the Markov Embedding Theorem (MET): any dynamics with a rational memory kernel is exactly equivalent to Markovian dynamics in an enlarged space of dimension equal to the Hankel rank of the kernel. This theorem is not specific to quantum physics - it applies to any system whose memory can be represented by a Prony decomposition (sum of exponentials).

This generality is what allows the suite to address four distinct domains with the same central code (the memkern package). (1) Quantum: J(omega) -> BoltZ-Kernel -> decoherence diagnosis. (2) Classical Kuramoto: retarded kernels of the generalised Kuramoto model -> critical coupling threshold K_c(M) by closed analytical form, saving a simulation campaign. (3) Viscoelastic rheology: Maxwell-Wiechert relaxation spectrum from measured G(t), with entropy-based (MaxEnt) order selection. (4) Neural synaptic kernels: dendritic or synaptic retarded kernels -> reaction-time scaling RT = tau_l + tau_d ln(theta/I_0).

Four closed forms replace hours of simulation. (1) Asymptotic Lindblad rate: gamma_M = Re(sum_k alpha_k / beta_k) from Prony parameters - computed in < 1 ms. (2) Kuramoto critical coupling: K_c(M) = 2gamma_g / M(0) from the zero-frequency kernel transform. (3) RF line shape: resonance shift as a function of perturbation frequency - analytical Lorentzian. (4) Behavioural reaction-time scaling: RT = tau_l + tau_d ln(theta/I_0).

6. How to use the suite

Desktop app (Windows / macOS / Linux): five specialty tabs (Quantum, Classical, Rheology, Neural, Diagnostics), live update of figures while moving parameters, academic export formats (PNG 300/600, PDF, SVG, EPS, TIFF), batch processing by CSV. No Python required.

MCP server (metcore-mcp): 17 tools exposed to MCP-compatible LLM agents (Claude Desktop, Claude Code, Cursor, VS Code, Windsurf, Cline). pip install metcore-mcp then add one line to the client config.

Python library (metcore / memkern): complete API for integration into pipelines, notebooks, or custom analyses. from memkern import full_diagnosis, zoo.

Source code and setup on github.com/hopenmind

Group 1 - Diagnosis (9 tools): answer the question "does memory matter here?"

Group 2 - Analytical shortcuts (4 tools): replace hours of simulation with closed forms.

Group 3 - Export and branding (4 tools): export_formats, export_curve (7 image formats + csv/xlsx/xml), branding_get, branding_set. Same files, same visual identity as the desktop app.

BoltZ-Kernel is valid in the weak-coupling regime (g ll omega_0). At strong coupling (g > 1), the mean-field factorisation rho_SE ~ rho_S (x) rho_E breaks and the solver is no longer reliable - this is a stated domain of validity, not a bug. The MaxEnt kernel itself cannot be falsified as an inference principle (Jaynes 1957); what is falsifiable is whether the MaxEnt kernel for a given J(omega) matches the experimental data of the real system. A discordance is a publishable result.

The suite is tested with 26 tests for BoltZ-Kernel alone (all passing) + MCP integration tests. The fft and prony backends are cross-validated: agreement of both backends on a given J(omega) strongly constrains the result against numerical artefacts. A divergence between backends signals a regime where physical assumptions are being strained.

Both deposits are dual-licensed Apache-2.0 (academic / non-commercial use, DOI citation sufficient) or proprietary commercial licence (contact@hopenmind.com). All data, source code and tests are public on GitHub. No experimental data is included: BoltZ-Kernel is a theoretical diagnostic tool, and comparing its predictions with laboratory measurements is what closes the scientific loop.