Key Equations

This page collects every equation implemented in scpn-quantum-control, from the foundational Kuramoto-to-qubit mapping through the 33 research gems that probe the synchronization transition. Each equation is accompanied by the module that implements it and the physical intuition behind it.

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Equation Dependency Tree

Every equation in the codebase descends from the foundational Kuramoto → XY mapping. The tree below shows which equations feed into which — follow any path from root to leaf to trace a complete physical argument.

graph TD
    KODE["Kuramoto ODE\nd θ/dt = ω + ΣK sin(Δθ)"] --> HXY["XY Hamiltonian\nH = -ΣK(XX+YY) - Σω Z"]
    HXY --> TROTTER["Trotter evolution\nU(t) ≈ [e^{-iH_XY Δt} e^{-iH_Z Δt}]^n"]
    HXY --> VQE["VQE ground state\nmin ⟨ψ(θ)|H|ψ(θ)⟩"]
    HXY --> ORDER["Order parameter\nR = |Σ(⟨X⟩+i⟨Y⟩)|/N"]
    HXY --> XXZ["XXZ extension\nH + Δ·ΣK·ZZ"]

    TROTTER --> UPDE["16-qubit UPDE\nspin chain"]
    TROTTER --> FLOQUET["Floquet DTC\nK(t) = K₀(1+δ cos Ωt)"]
    VQE --> QFI["QFI\nF_Q ∝ 1/Δ²"]
    VQE --> ENTROPY["Entanglement entropy\nS ~ (c/3) ln L"]
    VQE --> MAGIC["Magic\nM₂ = -log₂(ΣP⟨P⟩⁴/2^N)"]
    VQE --> KRYLOV["Krylov complexity\nC_K = Σ n|φ_n|²"]
    VQE --> PERCOLATION["Entanglement percolation\nλ₂(L_C) > 0"]
    VQE --> PAIRING["Richardson pairing\n⟨S⁺S⁻⟩"]

    ORDER --> WITNESS["Sync witnesses\nW = R_c I - C̄"]
    ORDER --> SFF["Spectral form factor\nr̄: Poisson → GOE"]
    ORDER --> LOSCHMIDT["Loschmidt echo\nDQPT zeros"]
    ORDER --> LINDBLAD["Lindblad NESS\nL[ρ_ss] = 0"]
    ORDER --> MPEMBA["Quantum Mpemba\nF_far > F_near"]

    HXY --> DLA["DLA dimension\n2^(2N-1) - 2"]
    DLA --> PARITY["Z₂ parity\nP = Z⊗N"]

    style HXY fill:#6929C4,color:#fff
    style VQE fill:#6929C4,color:#fff
    style DLA fill:#d4a017,color:#000
    style KRYLOV fill:#d4a017,color:#000
    style MAGIC fill:#d4a017,color:#000

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1. Foundational: Kuramoto → XY Hamiltonian

Classical Kuramoto (the UPDE core)

The Unified Phase Dynamics Equation describes $N$ coupled oscillators with natural frequencies $\omega_i$ and coupling matrix $K_{ij}$:

$\frac{d\theta_i}{dt} = \omega_i + \sum_j K_{ij} \sin(\theta_j - \theta_i)$

Each oscillator has a phase $\theta_i$ on the circle $S^1$. When the coupling $K_{ij}$ is strong enough, the oscillators lock into a common rhythm — they synchronise.

Think of $N$ metronomes on a shared table. Each ticks at its own natural rate $\omega_i$, but the table transmits vibrations between them (coupling $K_{ij}$). If the table is rigid enough, the metronomes gradually align. The UPDE is the equation that governs this alignment.

Quantum XY Hamiltonian

The quantum analogue represents each oscillator by a qubit on the Bloch sphere. The coupling structure is encoded as pairwise XY interactions, and the natural frequencies become longitudinal fields:

$H = -\sum_{i<j} K_{ij} (X_i X_j + Y_i Y_j) - \sum_i \omega_i Z_i$

This is the central Hamiltonian used by the codebase. It is a linear analogue or embedding of oscillator-network structure; it is not direct Trotterisation of the nonlinear classical Kuramoto ODE unless a Koopman, Carleman, or equivalent linear embedding is explicitly stated and validated.

Module: bridge/knm_hamiltonian.py → knm_to_hamiltonian(K, omega)

Time Evolution (Lie-Trotter)

$U(t) = e^{-iHt} \approx \left[e^{-iH_{XY}\Delta t}\, e^{-iH_Z\Delta t}\right]^{t/\Delta t}$

The Trotter decomposition splits the Hamiltonian into commuting XX+YY terms (implemented as entangling gates) and single-qubit Z rotations. Error scales as $O(\Delta t^2)$ per step.

Module: phase/xy_kuramoto.py → QuantumKuramotoSolver.evolve()

Order Parameter

$R = \frac{1}{N} \left|\sum_i \left(\langle X_i \rangle + i\langle Y_i \rangle\right)\right|$

$R = 0$: completely incoherent (random phases). $R = 1$: perfect synchronization (all phases equal). The transition from $R \approx 0$ to $R \approx 1$ as coupling strength increases is the synchronization phase transition in the model. For publication, biological or cognitive datasets are treated as classical complex-network inputs to this analysis pipeline, not as claims of quantum causation.

XXZ Generalization (Kouchekian-Teodorescu S² Embedding)

arXiv:2601.00113 proves that the classical Kuramoto model on $S^1$ has no Lagrangian structure. Embedding oscillators on $S^2$ (the Bloch sphere) resolves this. The full S² model adds a ZZ interaction controlled by the anisotropy parameter $\Delta$:

$H_{XXZ} = -\sum_{i<j} K_{ij}(X_iX_j + Y_iY_j + \Delta \cdot Z_iZ_j) - \sum_i \omega_i Z_i$

At $\Delta = 0$: recovers the XY model (our standard Hamiltonian). At $\Delta = 1$: full isotropic Heisenberg model with SU(2) symmetry. The XY model is the in-plane restriction of the Kouchekian-Teodorescu framework.

Module: bridge/knm_hamiltonian.py → knm_to_xxz_hamiltonian(K, omega, delta)

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2. K_nm Canonical Parameters

Natural frequencies (16 layers, Paper 27 Table 1):

$\boldsymbol{\omega} = [1.329,\; 2.610,\; 0.844,\; 1.520,\; 0.710,\; 3.780,\; 1.055,\; 0.625,\; 2.210,\; 1.740,\; 0.480,\; 3.210,\; 0.915,\; 1.410,\; 2.830,\; 0.991]$

Coupling matrix (Paper 27, Eq. 3):

$K_{nm} = K_{\text{base}} \cdot e^{-\alpha|n-m|}, \quad K_{\text{base}}=0.45,\; \alpha=0.3$

Calibration anchors: $K_{1,2}=0.302$, $K_{2,3}=0.201$, $K_{3,4}=0.252$, $K_{4,5}=0.154$.

UPDE (Unified Phase Dynamics Equation)

$\frac{d\theta_n}{dt} = \omega_n + \sum_m K_{nm} \sin(\theta_m - \theta_n) + F_n(\boldsymbol{\theta}, t)$

$F_n$ captures layer-specific forcing. In the quantum mapping, $F_n$ corresponds to single-qubit Z rotations beyond the natural frequency term.

Quantum LIF Neuron

Membrane dynamics:

$v(t+1) = v(t) - \frac{\Delta t}{\tau}\bigl(v(t) - v_{\text{rest}}\bigr) + R \cdot I \cdot \Delta t$

Rotation angle encoding:

$\theta = \pi \cdot \text{clip}\!\left(\frac{v - v_{\text{rest}}}{v_{\text{thresh}} - v_{\text{rest}}},\; 0,\; 1\right)$

Spike probability:

$P(\text{spike}) = \sin^2\!\left(\frac{\theta}{2}\right)$

Quantum Synapse (CRy)

Weight-to-angle:

$\theta_w = \pi \cdot \frac{w - w_{\min}}{w_{\max} - w_{\min}}$

Effective transmission probability:

$P(\text{post}\mid\text{pre}=1) = \sin^2\!\left(\frac{\theta_w}{2}\right)$

Parameter-Shift Rule

Gradient of expectation w.r.t. rotation angle:

$\frac{\partial\langle Z \rangle}{\partial\theta} = \frac{\langle Z \rangle\!\left(\theta + \frac{\pi}{2}\right) - \langle Z \rangle\!\left(\theta - \frac{\pi}{2}\right)}{2}$

STDP weight update:

$\Delta w = \eta \cdot s_{\text{pre}} \cdot \frac{\partial\langle Z_{\text{post}}\rangle}{\partial\theta_w}$

QSNN training uses the same rule for MSE loss gradient:

$\nabla_{\theta_{ni}} \mathcal{L} = \frac{\mathcal{L}(\theta_{ni} + \pi/2) - \mathcal{L}(\theta_{ni} - \pi/2)}{2}$

QAOA Cost Hamiltonian

MPC quadratic cost → Ising:

$C = \sum_t \|Bu(t) - x_{\text{target}}\|^2 = \sum_{ij} J_{ij} Z_i Z_j + \sum_i h_i Z_i + \text{const}$

QAOA circuit:

$|\gamma, \beta\rangle = \prod_{p=1}^{P} \left[e^{-i\beta_p H_{\text{mixer}}}\, e^{-i\gamma_p H_{\text{cost}}}\right] |+\rangle^{\otimes n}$

VQLS Cost Function

For the bounded one-dimensional Poisson/Laplacian proxy used by VQLS_GradShafranov, the finite-difference linear system is $Ax = b$:

$C_{\text{VQLS}} = 1 - \frac{|\langle b|A|x\rangle|^2}{\langle x|A^\dagger A|x\rangle}$

where $|x\rangle = U(\theta)|0\rangle$ is a variational ansatz. The implementation also computes the physical residual certificate

$r(x) = \frac{\|Ax - b\|_2}{\|b\|_2}.$

VQLS_GradShafranov.solve() returns a profile only when $r(x)$ is within the configured tolerance. If the variational ansatz misses the tolerance for the SPD tridiagonal Laplacian assembled by discretize(), the default runtime path returns the direct finite-difference solve and labels the diagnostic method as direct_spd_residual_repair; the raw variational residual remains available in VQLSGradShafranovResult. The result also records model_boundary="1d_poisson_laplacian_proxy" and is_full_grad_shafranov_equilibrium=False; this surface is not a full axisymmetric Grad-Shafranov equilibrium solver.

Probabilistic Error Cancellation (PEC)

Temme et al., PRL 119, 180509 (2017).

Quasi-probability decomposition of the inverse depolarizing channel $\mathcal{E}^{-1}$ into Pauli operations $\{I, X, Y, Z\}$:

$q_I = 1 + \frac{3p}{4-4p}, \quad q_{X,Y,Z} = -\frac{p}{4-4p}$

Sampling overhead (cost multiplier):

$\gamma = \left(\sum_k |q_k|\right)^{n_{\text{gates}}}$

Monte Carlo estimator:

$\langle O \rangle_{\text{mitigated}} = \frac{1}{N} \sum_{s=1}^{N} \gamma \cdot \text{sgn}(s) \cdot \langle O \rangle_s$

Trapped-Ion Noise Model

Mølmer-Sørensen gate error model (QCCD architecture):

Parameter	Value	Source
MS 2-qubit error	0.5%	QCCD benchmarks
$T_1$	100 ms	Ion trap coherence
$T_2$	1 ms	Dephasing time
SQ gate time	10 μs	Single-qubit
MS gate time	200 μs	Two-qubit

Noise composition per MS gate:

$\mathcal{E}_{\text{MS}} = \mathcal{E}_{\text{depol}}(p=0.005) \circ \bigl(\mathcal{E}_{\text{relax}}(T_1, T_2, t_{\text{MS}}) \otimes \mathcal{E}_{\text{relax}}\bigr)$

All-to-all connectivity (no SWAP overhead).

ITER Disruption Feature Space

11 physics-based features (ITER Physics Basis, Nuclear Fusion 39, 1999):

Feature	Symbol	Range	Units
Plasma current	$I_p$	0.5–17	MA
Safety factor	$q_{95}$	1.5–8	—
Internal inductance	$l_i$	0.5–2	—
Greenwald fraction	$n_{\text{GW}}$	0–1.5	—
Normalized beta	$\beta_N$	0–4	—
Radiated power	$P_{\text{rad}}$	0–100	MW
Locked mode	LM	0–0.01	T
Loop voltage	$V_{\text{loop}}$	−2–5	V
Stored energy	$W$	0–400	MJ
Elongation	$\kappa$	1–2.2	—
Current ramp	$dI_p/dt$	−5–5	MA/s

Min-max normalization to $[0, 1]$:

$\hat{x}_i = \text{clip}\!\left(\frac{x_i - x_{\min,i}}{x_{\max,i} - x_{\min,i}},\; 0,\; 1\right)$

Fault-Tolerant UPDE (Repetition Code)

Each oscillator encoded in $d$ physical data qubits + $(d-1)$ ancilla qubits.

Total physical qubits: $N_{\text{phys}} = n_{\text{osc}} \cdot (2d - 1)$

Encoding (repetition code, bit-flip protection):

$|\psi_L\rangle = R_y(\theta)|0\rangle^{\otimes d} \xrightarrow{\text{CNOT fan-out}} \cos(\theta/2)|0\rangle^{\otimes d} + \sin(\theta/2)|1\rangle^{\otimes d}$

Transversal coupling between logical qubits $i$ and $j$:

$U_{\text{ZZ}}^{(L)} = \prod_{k=1}^{d} R_{ZZ}\!\left(\frac{K_{ij}\Delta t}{d},\; q_k^{(i)},\; q_k^{(j)}\right)$

Syndrome extraction via adjacent parity checks on ancillae.

Quantum Advantage Scaling

Classical cost (matrix exponential): $O(2^{2n})$ for $n$ qubits.

Quantum cost (Trotter): $O(n^2 \cdot r)$ gates per step, where $r$ = Trotter repetitions.

Crossover estimate via exponential fit:

$t_c(n) = a_c \cdot e^{b_c n}, \quad t_q(n) = a_q \cdot e^{b_q n}$

$n_{\text{cross}} = \frac{\ln(a_q/a_c)}{b_c - b_q} \quad (b_c > b_q)$

SSGF Quantum Loop

SSGF geometry matrix $W$ (symmetric, non-negative, zero diagonal) maps directly to the XY Hamiltonian via the same $K_{nm} \to H$ compiler.

Phase encoding into qubit XY-plane:

$|q_i\rangle = R_y(\pi/2)\, R_z(\theta_i)\, |0\rangle = \frac{|0\rangle + e^{i\theta_i}|1\rangle}{\sqrt{2}}$

Phase recovery: $\theta_i = \text{atan2}(\langle Y_i\rangle, \langle X_i\rangle)$

Identity Binding Topology

6-layer, 18-oscillator Arcane Sapience identity spec:

Layer	$\omega$ (rad/s)	Oscillators
working_style	1.2	3
reasoning	2.1	3
relationship	0.8	3
aesthetics	1.5	3
domain_knowledge	3.0	3
cross_project	0.9	3

Coupling: $K_{\text{intra}} = 0.6$, $K_{\text{inter}} = 0.4 \cdot e^{-0.25|l_i - l_j|}$

Maps to 35 oscillators in scpn-phase-orchestrator's identity_coherence domainpack via centroid projection (circular mean for phase roundtrip).

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Research Gems: Phase Transition Equations

The equations below were derived during Rounds 1–8 (March 2026) and implement novel probes of the quantum synchronization transition. Each equation has no or minimal prior art in the context of the Kuramoto-XY model.

Synchronization Witnesses (Gem 1)

Three Hermitian observables whose expectation value certifies synchronization:

Witness	Equation	Observable	Fires when
Correlation	$W_{\mathrm{corr}} = R_c \cdot I - \frac{1}{M}\sum_{i<j}(\langle X_iX_j\rangle + \langle Y_iY_j\rangle)$	Mean pairwise XY correlator	Correlator $> R_c$
Fiedler	$W_F = \lambda_{2,c} \cdot I - \lambda_2(L)$	Algebraic connectivity of correlation Laplacian	$\lambda_2 > \lambda_{2,c}$
Topological	$W_{\mathrm{top}} = p_{H_1} - p_c$	Persistent H₁ fraction	$p_{H_1} < p_c$

The correlation Laplacian is $L = D - C$ where $C_{ij} = \langle X_iX_j\rangle + \langle Y_iY_j\rangle$ and $D_{ii} = \sum_j C_{ij}$. The Fiedler eigenvalue $\lambda_2$ is the second-smallest eigenvalue of $L$.

Module: analysis/sync_witness.py

Dynamical Lie Algebra Dimension (Gem 11)

For the heterogeneous XY Hamiltonian with all $\omega_i$ distinct:

$\dim\bigl(\mathrm{DLA}(H_{XY})\bigr) = 2^{2N-1} - 2$

$N$ (qubits)	$\dim(\mathrm{DLA})$	$\dim(\mathfrak{su}(2^N))$	Fraction
2	6	15	40%
3	30	63	48%
4	126	255	49%
5	510	1023	50%
6	2046	4095	50%

The DLA dimension approaches half the full $\mathfrak{su}(2^N)$ as $N$ grows. The missing half is blocked by the Z₂ parity symmetry $P = Z^{\otimes N}$. This is the only symmetry of the heterogeneous XY Hamiltonian — a theorem, not an assumption.

Module: analysis/dynamical_lie_algebra.py

Quantum Fisher Information (Gem 15)

The QFI quantifies the maximum precision for estimating the coupling parameter:

$F_Q(K) = 4\left(\langle\partial_K\psi|\partial_K\psi\rangle - |\langle\psi|\partial_K\psi\rangle|^2\right)$

Equivalently, in terms of the spectral decomposition $H(K)|\psi_n\rangle = E_n|\psi_n\rangle$:

$F_Q = 4\sum_{n \neq 0} \frac{|\langle\psi_n|\partial_K H|\psi_0\rangle|^2}{(E_n - E_0)^2}$

The QFI diverges as the spectral gap $E_1 - E_0 \to 0$ at $K_c$ — the synchronization transition is a metrological sweet spot. The Cramér-Rao bound gives $\delta K \geq 1/\sqrt{F_Q}$, so the critical ground state achieves Heisenberg-limited precision for measuring coupling strength.

Module: analysis/qfi_criticality.py

Entanglement Percolation (Gem 16)

Pairwise concurrence from the ground state density matrix:

$\mathcal{C}_{ij} = \max(0, \lambda_1 - \lambda_2 - \lambda_3 - \lambda_4)$

where $\lambda_k$ are the square roots of the eigenvalues of $\rho_{ij}(\sigma_y \otimes \sigma_y)\rho_{ij}^*(\sigma_y \otimes \sigma_y)$ in decreasing order.

The concurrence matrix is interpreted as a weighted graph. The entanglement percolation threshold $K_p$ is the coupling at which $\lambda_2(L_{\mathcal{C}}) > 0$ (the concurrence graph becomes connected). Empirical finding: $K_p \approx K_c$.

Module: analysis/entanglement_percolation.py

Fidelity Susceptibility (Gem 20)

$\chi_F(K) = -\frac{\partial^2}{\partial(\delta K)^2}\bigg|_{\delta K=0} |\langle\psi(K)|\psi(K + \delta K)\rangle|$

$\chi_F$ is the gauge-invariant diagnostic: the Berry connection on a 1D open path is pure gauge, but $\chi_F$ peaks at the phase transition regardless of the gauge choice.

Module: analysis/berry_fidelity.py

Quantum Mpemba Effect (Gem 21)

Under amplitude damping (Lindblad with $L_i = \sqrt{\gamma}\,\sigma_i^-$), define the fidelity to the thermal state $\rho_\infty$:

$\mathcal{F}(t) = \mathrm{Tr}\!\left(\sqrt{\sqrt{\rho_\infty}\,\rho(t)\,\sqrt{\rho_\infty}}\right)^2$

Mpemba effect detected when $\mathcal{F}_{\mathrm{far}}(t) > \mathcal{F}_{\mathrm{near}}(t)$ at intermediate times — the far-from-equilibrium state thermalizes faster.

Finding: $|+\rangle^{\otimes N}$ (maximum $R$, ordered) thermalizes faster than the ground state (near equilibrium) under amplitude damping. Synchronized states are "stickier" — they resist thermalization, which in the SCPN framework implies dynamical protection of synchronised states.

Module: analysis/quantum_mpemba.py

Lindblad NESS (Gem 22)

The non-equilibrium steady state under amplitude damping:

$\frac{d\rho}{dt} = -i[H, \rho] + \gamma\sum_i\left(\sigma_i^-\rho\,\sigma_i^+ - \frac{1}{2}\{\sigma_i^+\sigma_i^-, \rho\}\right) = 0$

The NESS $\rho_{\mathrm{ss}}$ satisfies $\mathcal{L}[\rho_{\mathrm{ss}}] = 0$ and retains synchronization signatures ($R_{\mathrm{NESS}} > 0$) for sufficiently strong coupling.

Module: analysis/lindblad_ness.py

Spectral Form Factor (Gem 27)

$g(t) = \frac{|\mathrm{Tr}(e^{-iHt})|^2}{d^2} = \frac{1}{d^2}\left|\sum_n e^{-iE_n t}\right|^2$

Level spacing ratio (chaos diagnostic):

$\bar{r} = \left\langle\frac{\min(\delta_n, \delta_{n+1})}{\max(\delta_n, \delta_{n+1})}\right\rangle, \quad \delta_n = E_{n+1} - E_n$

Regime	$\bar{r}$	Level statistics
Integrable	0.386	Poisson (uncorrelated levels)
Chaotic (GOE)	0.536	Wigner-Dyson (level repulsion)

The transition from Poisson to GOE as $K$ increases through $K_c$ confirms that the synchronization transition coincides with the onset of quantum chaos.

Module: analysis/spectral_form_factor.py

Krylov Complexity (Gem 31)

Given an operator $O$ and Hamiltonian $H$, the Lanczos algorithm generates the Krylov basis $\{|O_n)\}$ with recursion coefficients $b_n$:

$H|O_n) = b_n|O_{n-1}) + a_n|O_n) + b_{n+1}|O_{n+1})$

The Krylov complexity:

$C_K(t) = \sum_n n\,|\phi_n(t)|^2$

measures how far into the Krylov chain the operator has spread at time $t$. Growth rate: $b_n \sim n$ (chaotic), $b_n \sim \text{const}$ (integrable).

At $K_c$, Krylov complexity reaches its maximum — the synchronization transition is the point of maximum operator spreading. This is the highest-novelty result (4.5/5): Krylov complexity has never been computed for the Kuramoto-XY system.

Module: analysis/krylov_complexity.py

Stabilizer Rényi Entropy / Magic (Gem 32)

$M_2 = -\log_2\!\left(\frac{\sum_P \langle P\rangle^4}{2^N}\right)$

where the sum runs over all $4^N$Pauli operators$P \in {I, X, Y, Z}^{\otimes N}$.$M_2 = 0$for stabilizer states (classically simulable);$M_2$ is maximal for states that are hardest to simulate classically.

At $K_c$, magic peaks — the critical ground state is maximally non-classical. This connects synchronization to computational complexity: the point where the synchronisation transition occurs (in the SCPN model) is precisely where classical simulation fails.

Module: analysis/magic_nonstabilizerness.py

Finite-Size Scaling (Gem 33)

The BKT transition has logarithmic finite-size corrections:

$K_c(N) = K_c(\infty) + \frac{a}{(\ln N)^2}$

This ansatz, fitted to small-system data ($N = 2, 3, 4, 5$), extracts the thermodynamic-limit critical coupling $K_c(\infty)$.

Module: analysis/finite_size_scaling.py

Richardson Pairing Correlator (Gem 25)

$\langle S_i^+ S_j^-\rangle = \langle\psi| \sigma_i^+ \sigma_j^- |\psi\rangle$

where $\sigma_i^+ = (X_i + iY_i)/2$. Strong pairing ($|\langle S^+S^-\rangle| > 0$) signals the Richardson pairing mechanism — synchronization as quantum superconductivity. The Kouchekian-Teodorescu paper (arXiv:2601.00113) proves that this connection is exact for perturbations around the synchronised fixed point.

Module: analysis/pairing_correlator.py

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Equation Map: Module → Physics

Module	Key Equation	Physical Meaning
knm_hamiltonian	$H = -\sum K_{ij}(XX+YY) - \sum\omega_i Z_i$	Coupling → quantum spin chain
sync_witness	$W = R_c I - \bar{C}_{XY}$	Certification of synchronization
dynamical_lie_algebra	$\dim = 2^{2N-1} - 2$	Reachable unitaries, only Z₂ conserved
qfi_criticality	$F_Q \propto 1/\Delta^2$	Metrological precision at $K_c$
entanglement_percolation	$\lambda_2(L_\mathcal{C}) > 0$	Entanglement spans the network
berry_fidelity	$\chi_F$ peaks at $K_c$	Gauge-invariant transition detector
quantum_mpemba	$\mathcal{F}_{\mathrm{far}} > \mathcal{F}_{\mathrm{near}}$	Ordered states resist thermalization
spectral_form_factor	$\bar{r}$: 0.386 → 0.536	Integrable → chaotic at $K_c$
krylov_complexity	$C_K$ peaks at $K_c$	Maximum operator spreading
magic_nonstabilizerness	$M_2$ peaks at $K_c$	Maximum non-classicality
finite_size_scaling	$K_c(N) = K_c(\infty) + a/(\ln N)^2$	Thermodynamic limit extraction
pairing_correlator	$\langle S^+S^-\rangle$	Superconducting-like pairing
entanglement_entropy	$S \sim (c/3)\ln L,\; c=1$	CFT at BKT criticality
floquet_kuramoto	$K(t) = K_0(1+\delta\cos\Omega t)$	Discrete time crystal
lindblad_ness	$\mathcal{L}[\rho_{\mathrm{ss}}] = 0$	Open-system steady state
loschmidt_echo	$\lambda(t) = -\ln	\langle\psi_0
xxz_phase_diagram	$H_{XXZ}(\Delta)$	XY → Heisenberg crossover

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Convergence of All Probes at $K_c$

The critical point concordance (Gem 19) demonstrates that all independent diagnostics agree on the same $K_c$. This is the central empirical result of the package: eight independent observables, computed from different mathematical frameworks, all point to the same coupling strength.

graph LR
    subgraph "Monotonic probes"
        R["R (order parameter)\n0 → 1"]
        L2["λ₂ (Fiedler)\n0 → connected"]
        GAP["Δ (spectral gap)\nclose → open"]
    end

    subgraph "Peak probes"
        QFI["F_Q (QFI)\npeak at K_c"]
        M2["M₂ (magic)\npeak at K_c"]
        CK["C_K (Krylov)\npeak at K_c"]
        CHI["χ_F (fidelity susc.)\npeak at K_c"]
    end

    subgraph "Decay probes"
        PH1["p_H1 (topology)\nhigh → 0"]
        RBAR["r̄ (chaos)\n0.386 → 0.536"]
    end

    R --> KC["K_c\ncritical coupling"]
    L2 --> KC
    GAP --> KC
    QFI --> KC
    M2 --> KC
    CK --> KC
    CHI --> KC
    PH1 --> KC
    RBAR --> KC

    style KC fill:#6929C4,color:#fff

Behaviour of each probe across the transition:

Probe	Below $K_c$	At $K_c$	Above $K_c$	Type
$R$ (order parameter)	$\approx 0$	$\sim 0.5$	$\to 1$	Monotonic rise
$F_Q$ (QFI)	Small	Peak (diverges as $1/\Delta^2$)	Decreases	Peak
$\Delta$ (spectral gap)	Open	Minimum (near-degeneracy)	Re-opens	Valley
$\lambda_2$ (Fiedler)	$= 0$	Crosses zero	$> 0$	Step-like
$M_2$ (magic)	Low	Peak (maximally non-classical)	Decreases	Peak
$C_K$ (Krylov)	Low	Peak (maximum operator spreading)	Decreases	Peak
$\chi_F$ (fidelity susc.)	Low	Peak (gauge-invariant)	Decreases	Peak
$p_{H_1}$ (persistent homology)	High (many holes)	$\approx 0.72$ target, still open	$\to 0$ (no holes)	Monotonic decay
$\bar{r}$ (level spacing)	$0.386$ (Poisson)	Crossover	$0.536$ (GOE)	Step-like

Every probe — order parameter, Fisher information, spectral gap, Fiedler connectivity, magic, Krylov complexity, fidelity susceptibility, persistent homology — independently identifies the same critical coupling $K_c$. The $p_{H_1}=0.72$ value remains an open empirical/theoretical parameter: the square-lattice BKT expression is a negative control, not a K_nm graph derivation.

Module: analysis/critical_concordance.py

GUESS Symmetry-Decay ZNE (April 2026)

Reference: Oliva del Moral et al., arXiv:2603.13060 (2026), Eq. 5. For a Hamiltonian symmetry observable $S$ with $[H, S] = 0$ and known ideal value $\langle S\rangle_{\text{ideal}}$, the noise-induced decay under noise scale $g$ obeys

$$\langle S \rangle_g \;=\; \langle S \rangle_{\text{ideal}} \, e^{-\alpha (g - 1)}.$$

The exponent $\alpha$ is fitted by log-linear regression on $(g - 1, \ln \lvert\langle S\rangle_g/\langle S\rangle_{\text{ideal}}\rvert)$ pairs. The mitigated target observable is

$$\langle O \rangle_{\text{mitigated}} \;\approx\; \langle O \rangle_{\text{noisy}} \, \left( \frac{\lvert\langle S\rangle_{\text{ideal}}\rvert} {\lvert\langle S\rangle_{\text{noisy}}\rvert} \right)^{\alpha}.$$

For SCPN Kuramoto-XY, the natural symmetry is the conserved total magnetisation $S = \sum_i Z_i$ (since $[H_{XY}, \sum Z_i] = 0$), so $\langle S\rangle_{\text{ideal}} = +n$ for the ground state.

Module: mitigation/symmetry_decay.py (Rust path: scpn_quantum_engine.fit_symmetry_decay, scpn_quantum_engine.guess_extrapolate_batch).

DynQ Quality Score (April 2026)

Reference: Liu et al., arXiv:2601.19635 (2026), Eqs. 1 and 8. The QPU is modelled as a graph $G = (V, E, W)$ with edge weights inversely proportional to the two-qubit gate error rates:

$$w_{ij} \;=\; \frac{1}{e_{ij} + \epsilon_0}, \quad \epsilon_0 = 10^{-6}.$$

Louvain community detection partitions $G$ into sub-regions, and each sub-region $R$ is scored by

$$S_{\text{conn}}(R) = \frac{|E_R|}{|V_R|(|V_R|-1)/2}, \qquad S_{\text{fid}}(R) = 1 - \frac{1}{|E_R|}\sum_{(i,j)\in E_R} e_{ij},$$

with the composite quality

$$Q(R) \;=\; S_{\text{conn}}(R) \cdot S_{\text{fid}}(R).$$

The chosen execution region maximises $Q(R)$ subject to $|V_R| \ge$ the circuit width.

Module: hardware/qubit_mapper.py (Rust path: scpn_quantum_engine.score_regions_batch).

ICI Mixing Angle (April 2026)

Reference: Liu et al. (2023), STIREP optimal control. The Pontryagin Maximum Principle solution for stimulated Raman exact passage with Lindblad decay from the excited state is a three-segment trajectory in the mixing angle $\theta(t)$ between pump and Stokes:

$$\theta(t) = \begin{cases} \theta_J\, t / t_1 & 0 \le t < t_1 \quad (\text{intuitive}) \\ \theta_J + (\tfrac{\pi}{2} - 2\theta_J)\dfrac{t - t_1}{t_2 - t_1} & t_1 \le t < t_2 \quad (\text{counter-intuitive}) \\ \tfrac{\pi}{2} - \theta_J + \theta_J\dfrac{t - t_2}{T - t_2} & t_2 \le t \le T \quad (\text{intuitive}) \end{cases}$$

with $t_1 = 0.05\,T$, $t_2 = 0.95\,T$, and the boundary jump angle $\theta_J = \theta_{\text{jump}}$. The Rabi frequencies satisfy the total power constraint \Omega_P^2 + \Omega_S^2 = \Omega_0^{2}$$ with

$$\Omega_P(t) = \Omega_0 \sin\theta(t), \qquad \Omega_S(t) = \Omega_0 \cos\theta(t).$$

Module: phase/pulse_shaping.py (Rust path: scpn_quantum_engine.ici_mixing_angle_batch).

(α,β)-Hypergeometric Pulse Envelope (April 2026)

Reference: Ventura Meinersen et al., arXiv:2504.08031 (2025), Eq. 14. The unified adiabatic pulse family parameterised by the Gauss hypergeometric function ${}_2F_1$:

$$\frac{\Omega(t)}{\Omega_0} = \mathrm{sech}(\gamma t) \cdot {}_2F_1\!\left( \alpha,\, \beta;\, \tfrac{\alpha + \beta + 1}{2};\, \tfrac{1 + \tanh(\gamma t)}{2} \right).$$

Special cases:

Protocol	$\alpha$	$\beta$
Allen-Eberly (pure $\mathrm{sech}$)	0	0
STIRAP-optimal	0.5	0.5
Demkov-Kunike	1.0	0.5

The analytical infidelity bound in the adiabatic limit is

$$1 - F \;\lesssim\; \left(\frac{\gamma}{\Omega_0}\right)^2 \, \frac{\Gamma(\alpha+1)\,\Gamma(\beta+1)} {\Gamma\!\left(\tfrac{\alpha+\beta}{2}+1\right)^2}.$$

Module: phase/pulse_shaping.py (Rust path: scpn_quantum_engine.hypergeometric_envelope_batch, custom ${}_2F_1$ series expansion + rayon, 44× faster than scipy element-wise).