C15_sec Ethical Cost Term¶

1. Mathematical Formalism¶

The Ethical Lagrangian¶

The SCPN framework augments the standard SSGF cost with an ethical constraint term from Layer 15 (the ethical governance layer):

\[\mathcal{L}_{\text{ethical}} = U_{\text{total}} + w_{c15} \cdot C_{15,\text{sec}}\]

where \(C_{15,\text{sec}}\) is the Social Ethical Cost — a scalar measuring the ethical compliance of the synchronisation dynamics.

C15_sec Decomposition¶

\[C_{15,\text{sec}} = (1 - J_{\text{sec}}) + \kappa \cdot \Phi_{\text{ethics}}\]

The cost has two terms: 1. (1 - J_sec): How far from ethically optimal the system is 2. κ · Φ_ethics: Penalty for violating hard safety constraints

\[J_{\text{sec}} = \alpha \cdot R + \beta \cdot K_{\text{norm}} + \gamma \cdot Q - \nu \cdot S_{\text{dev}}\]

where:

Symbol	Definition	Meaning
\(R\)	\(\frac{1}{N}\\|\sum_j e^{i\theta_j}\\|\)	Kuramoto order parameter (coherence)
\(K_{\text{norm}}\)	\(\lambda_2(L) / N\)	Normalised algebraic connectivity
\(Q\)	\(\frac{\text{nnz}(W)}{N(N-1)}\)	Coupling density (quality)
\(S_{\text{dev}}\)	\(\text{std}(\theta) / \pi\)	Phase deviation from uniformity
\(\alpha, \beta, \gamma, \nu\)	weights	Default: 0.4, 0.3, 0.2, 0.1

\(J_{\text{sec}} \in [0, 1]\) approximately. Higher = more ethically compliant. The four terms balance: - Coherence (\(\alpha R\)): System should be synchronised (collective benefit) - Connectivity (\(\beta K_{\text{norm}}\)): Network should be globally integrated (Wiener cybernetic ethics — no isolated subgroups) - Quality (\(\gamma Q\)): Coupling should be dense enough to function - Equity (\(-\nu S_{\text{dev}}\)): Phase deviation penalises systems where some oscillators are left behind

CBF Constraint Penalties (Control Barrier Functions)¶

\[\Phi_{\text{ethics}} = \sum_{k=1}^{3} \max(0, g_k)^2\]

where the constraint functions \(g_k\) encode hard safety boundaries:

Constraint	\(g_k\)	Meaning
Non-harm	\(R_{\min} - R\)	Minimum coherence (system must function)
Connectivity	\(\lambda_{2,\min} - \lambda_2\)	Network must not fragment
Coupling limit	\(\max(K_{ij}) - K_{\max}\)	Coupling must not exceed safety bound

The quadratic penalty \(\max(0, g_k)^2\) is zero when the constraint is satisfied and grows quadratically with the violation magnitude. This is a relaxed CBF (Control Barrier Function) formulation.

Fiedler Value (Algebraic Connectivity)¶

The algebraic connectivity \(\lambda_2\) is the second-smallest eigenvalue of the graph Laplacian:

\[L = D - |W|\]

where \(D = \text{diag}(\sum_j |W_{ij}|)\) is the degree matrix.

\(\lambda_2 > 0\) if and only if the graph is connected. A larger \(\lambda_2\) indicates stronger global integration — harder to partition the network into disconnected components.

Jacobi Eigenvalue Algorithm (Rust)¶

The Rust implementation computes eigenvalues of the symmetric Laplacian using the classical Jacobi rotation method:

Find the largest off-diagonal element \(|L_{pq}|\)
Apply a Givens rotation to zero out \(L_{pq}\)
Repeat until all off-diagonal elements are below \(\varepsilon = 10^{-12}\)
Diagonal entries are the eigenvalues

The algorithm converges quadratically for well-separated eigenvalues and has complexity \(O(N^2)\) per rotation with \(O(N^2)\) rotations in the worst case, giving \(O(N^4)\) total for dense matrices.

2. Theoretical Context¶

Ethical Governance in Autonomous Systems¶

The C15_sec term operationalises ethical principles as mathematical constraints within the synchronisation dynamics. This approach draws from four traditions:

Harsanyi Aggregation (1955)¶

The SEC functional \(J_{\text{sec}}\) is a weighted sum of individual metrics, analogous to Harsanyi's utilitarian aggregation theorem: the social welfare function is a weighted sum of individual utilities under certain rationality axioms. The weights \((\alpha, \beta, \gamma, \nu)\) encode the relative importance of different ethical principles.

MacAskill's Effective Altruism (2022)¶

The equity term \(-\nu S_{\text{dev}}\) reflects the principle that a system which benefits some oscillators at the expense of others is less ethically compliant. This operationalises the concern for "leaving no one behind" in resource allocation.

Lyapunov/CBF Safety (Ames et al. 2017)¶

Control Barrier Functions (CBFs) provide forward invariance guarantees: if the system starts in a safe set, it remains there. The constraint functions \(g_k\) define the safe set boundary. The quadratic penalty is a relaxation — it does not guarantee hard invariance but provides a continuous optimisation signal.

Wiener Cybernetic Ethics (1950)¶

Norbert Wiener's The Human Use of Human Beings argued that communication systems should maintain connectivity and resist fragmentation. The \(\beta K_{\text{norm}}\) term directly implements this: penalising network topologies that allow subgroups to become disconnected.

Historical Context¶

Harsanyi, J. C. (1955): "Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility." Weighted utilitarian aggregation theorem.
Wiener, N. (1950): The Human Use of Human Beings: Cybernetics and Society. Ethical implications of communication and control.
Ames, A. D. et al. (2017): "Control barrier function based quadratic programs for safety critical systems." CBF framework.
MacAskill, W. (2022): What We Owe the Future. Long-termist ethical framework for AI systems.
Floridi, L. (2013): The Ethics of Information. Information ethics as a foundation for AI governance.

Layer 15 in the SCPN Stack¶

In the 15+1 layer SCPN architecture, Layer 15 is the Ethical Governance Layer. It receives telemetry from all lower layers and applies ethical constraints before the Layer 16 Director makes high-level decisions. The C15_sec cost is the quantitative output of Layer 15.

3. Pipeline Position¶

 UPDEEngine.step() ──→ phases
 CouplingBuilder/SSGF ──→ W (knm)
                    │        │
                    ↓        ↓
 ┌── compute_ethical_cost(phases, knm, ...) ──────┐
 │                                                 │
 │  Step 1: Compute SEC inputs                    │
 │    R = order parameter                          │
 │    λ₂ = fiedler_value(L(W))                    │
 │    Q = coupling density                         │
 │    S_dev = phase deviation                      │
 │                                                 │
 │  Step 2: Compute J_sec (weighted sum)          │
 │                                                 │
 │  Step 3: Compute CBF penalties (Φ_ethics)      │
 │    g₁: R_min - R                                │
 │    g₂: connectivity_min - λ₂                   │
 │    g₃: max(K) - max_coupling                    │
 │                                                 │
 │  Step 4: Assemble C15_sec = (1-J) + κΦ         │
 │                                                 │
 │  Output: EthicalCost(J_sec, Φ, C15, n_violated)│
 └─────────────────────────────────────────────────┘
                    │
                    ↓
         SSGFCosts adds w_c15 · C15_sec to U_total
                    │
                    ↓
         GeometryCarrier minimises L_ethical

Input Contracts¶

Parameter	Type	Default	Range	Meaning
`phases`	`NDArray[float64]`	—	\([0, 2\pi)\)	Current phases
`knm`	`NDArray[float64]`	—	\(\geq 0\)	Coupling matrix
`alpha_R`	`float`	0.4	\([0, 1]\)	Coherence weight
`beta_K`	`float`	0.3	\([0, 1]\)	Connectivity weight
`gamma_Q`	`float`	0.2	\([0, 1]\)	Quality weight
`nu_S`	`float`	0.1	\([0, 1]\)	Equity weight
`kappa`	`float`	1.0	\(> 0\)	CBF penalty multiplier
`R_min`	`float`	0.2	\([0, 1]\)	Min. coherence (CBF)
`connectivity_min`	`float`	0.1	\(\geq 0\)	Min. \(\lambda_2\) (CBF)
`max_coupling`	`float`	5.0	\(> 0\)	Max. coupling (CBF)

Output Contract¶

@dataclass
class EthicalCost:
    J_sec: float             # SEC functional, ∈ [0, 1] approx.
    phi_ethics: float        # CBF penalty, ≥ 0
    c15_sec: float           # Total ethical cost
    constraints_violated: int # Number of violated constraints (0-3)

4. Features¶

Weighted SEC functional — multi-objective ethical compliance (coherence, connectivity, quality, equity)
CBF constraint penalties — hard safety boundaries with quadratic relaxation
Three safety constraints — non-harm (minimum R), connectivity (minimum \(\lambda_2\)), coupling limit
Algebraic connectivity — Fiedler value via Jacobi eigenvalues (Rust) or NumPy eigvalsh (Python)
Violation counting — reports how many constraints are violated
Decomposable cost — \(C_{15}\) = deficit + penalties, separable for diagnosis
Rust FFI acceleration — 5.7x faster for small N (N ≤ 8)
Configurable weights — all 4 SEC weights and 3 CBF thresholds adjustable
Pipeline composable — feeds into SSGFCosts as an additive term

5. Usage Examples¶

Basic: Compute Ethical Cost¶

import numpy as np
from scpn_phase_orchestrator.ssgf.ethical import compute_ethical_cost

N = 8
rng = np.random.default_rng(42)
phases = rng.uniform(0, 2 * np.pi, N)
knm = np.full((N, N), 0.5)
np.fill_diagonal(knm, 0.0)

result = compute_ethical_cost(phases, knm)
print(f"J_sec = {result.J_sec:.4f}")
print(f"Φ_ethics = {result.phi_ethics:.4f}")
print(f"C15_sec = {result.c15_sec:.4f}")
print(f"Violations: {result.constraints_violated}")

Synchronised vs Desynchronised¶

import numpy as np
from scpn_phase_orchestrator.ssgf.ethical import compute_ethical_cost

N = 8
knm = np.full((N, N), 0.5); np.fill_diagonal(knm, 0.0)

# Synchronised: all same phase
sync_phases = np.ones(N) * 1.0
sync_cost = compute_ethical_cost(sync_phases, knm)

# Desynchronised: uniform spread
desync_phases = np.linspace(0, 2 * np.pi, N, endpoint=False)
desync_cost = compute_ethical_cost(desync_phases, knm)

print(f"Sync: J={sync_cost.J_sec:.4f}, C15={sync_cost.c15_sec:.4f}")
print(f"Desync: J={desync_cost.J_sec:.4f}, C15={desync_cost.c15_sec:.4f}")
# Sync should have lower C15 (more ethical)

Custom Safety Thresholds¶

import numpy as np
from scpn_phase_orchestrator.ssgf.ethical import compute_ethical_cost

N = 8
phases = np.ones(N) * 0.5  # synchronised
knm = np.full((N, N), 0.3); np.fill_diagonal(knm, 0.0)

# Strict safety: high R_min, high connectivity_min
strict = compute_ethical_cost(
    phases, knm,
    R_min=0.8,
    connectivity_min=0.5,
    max_coupling=1.0,
)

# Relaxed safety: low thresholds
relaxed = compute_ethical_cost(
    phases, knm,
    R_min=0.1,
    connectivity_min=0.01,
    max_coupling=10.0,
)

print(f"Strict: C15={strict.c15_sec:.4f}, violations={strict.constraints_violated}")
print(f"Relaxed: C15={relaxed.c15_sec:.4f}, violations={relaxed.constraints_violated}")

Integration with SSGF Loop¶

import numpy as np
from scpn_phase_orchestrator.ssgf.carrier import GeometryCarrier
from scpn_phase_orchestrator.ssgf.ethical import compute_ethical_cost
from scpn_phase_orchestrator.upde.engine import UPDEEngine

N = 8
gc = GeometryCarrier(N, z_dim=8, lr=0.005, seed=42)
eng = UPDEEngine(N, dt=0.01)
rng = np.random.default_rng(42)
phases = rng.uniform(0, 2 * np.pi, N)
omegas = np.ones(N)
alpha = np.zeros((N, N))

for outer in range(10):
    W = gc.decode()
    for _ in range(100):
        phases = eng.step(phases, omegas, W, 0.0, 0.0, alpha)

    ethical = compute_ethical_cost(phases, W)

    def cost_fn(W_flat):
        W_tmp = W_flat.reshape(N, N)
        for _ in range(50):
            p = eng.step(phases.copy(), omegas, W_tmp, 0.0, 0.0, alpha)
        return compute_ethical_cost(p, W_tmp).c15_sec

    state = gc.update(ethical.c15_sec, cost_fn=cost_fn)
    print(f"Step {outer}: C15={ethical.c15_sec:.4f}, violations={ethical.constraints_violated}")

6. Technical Reference¶

Function: compute_ethical_cost¶

ethical ¶

Ethical-cost diagnostic term for SEC and CBF-style constraint penalties.

The module computes the C15_sec term from coherence, spectral connectivity, coupling density, phase dispersion, and squared control-barrier violations. It is a numeric diagnostic used by the SSGF cost surface, not an autonomous policy authority or clinical decision surface. Rust and Python paths expose the same result fields so callers can audit SEC score, ethics penalty, total term, and violation count independently.

Classes¶

EthicalCost `dataclass` ¶

EthicalCost(
    J_sec: float,
    phi_ethics: float,
    c15_sec: float,
    constraints_violated: int,
)

C15_sec ethical cost: SEC functional, CBF penalties, violations.

Functions:¶

compute_ethical_cost ¶

compute_ethical_cost(
    phases: FloatArray,
    knm: FloatArray,
    *,
    alpha_R: float = 0.4,
    beta_K: float = 0.3,
    gamma_Q: float = 0.2,
    nu_S: float = 0.1,
    kappa: float = 1.0,
    R_min: float = 0.2,
    connectivity_min: float = 0.1,
    max_coupling: float = 5.0,
) -> EthicalCost

Compute C15_sec ethical cost term.

J_sec = α·R + β·K_norm + γ·Q - ν·S_dev where: R = Kuramoto order parameter (coherence) K_norm = λ₂(L) / max(λ₂) (normalized connectivity, Wiener) Q = 1 - sparsity (coupling quality) S_dev = std(phases) / π (phase deviation from uniform)

Φ_ethics = Σ max(0, g_k)² where g_k are CBF constraint violations: g_1: R_min - R (non-harm: minimum coherence) g_2: connectivity_min - λ₂ (Wiener: maintain connectivity) g_3: max(K_ij) - max_coupling (boundary: coupling limits)

Parameters¶

phases : FloatArray Oscillator phases in radians, shape (N,). knm : FloatArray Coupling matrix K_nm, shape (N, N). alpha_R : float Order-parameter cost weight. beta_K : float Coupling-cost weight. gamma_Q : float Quality-cost weight. nu_S : float Symbolic-cost weight. kappa : float Coupling/curvature parameter. R_min : float Minimum order-parameter target. connectivity_min : float Minimum algebraic connectivity. max_coupling : float Maximum allowed coupling value.

Returns¶

EthicalCost C15_sec ethical cost term.

Source code in src/scpn_phase_orchestrator/ssgf/ethical.py

def compute_ethical_cost(
    phases: FloatArray,
    knm: FloatArray,
    *,
    alpha_R: float = 0.4,
    beta_K: float = 0.3,
    gamma_Q: float = 0.2,
    nu_S: float = 0.1,
    kappa: float = 1.0,
    R_min: float = 0.2,
    connectivity_min: float = 0.1,
    max_coupling: float = 5.0,
) -> EthicalCost:
    """Compute C15_sec ethical cost term.

    J_sec = α·R + β·K_norm + γ·Q - ν·S_dev
    where:
      R = Kuramoto order parameter (coherence)
      K_norm = λ₂(L) / max(λ₂) (normalized connectivity, Wiener)
      Q = 1 - sparsity (coupling quality)
      S_dev = std(phases) / π (phase deviation from uniform)

    Φ_ethics = Σ max(0, g_k)² where g_k are CBF constraint violations:
      g_1: R_min - R                   (non-harm: minimum coherence)
      g_2: connectivity_min - λ₂       (Wiener: maintain connectivity)
      g_3: max(K_ij) - max_coupling    (boundary: coupling limits)

    Parameters
    ----------
    phases : FloatArray
        Oscillator phases in radians, shape ``(N,)``.
    knm : FloatArray
        Coupling matrix ``K_nm``, shape ``(N, N)``.
    alpha_R : float
        Order-parameter cost weight.
    beta_K : float
        Coupling-cost weight.
    gamma_Q : float
        Quality-cost weight.
    nu_S : float
        Symbolic-cost weight.
    kappa : float
        Coupling/curvature parameter.
    R_min : float
        Minimum order-parameter target.
    connectivity_min : float
        Minimum algebraic connectivity.
    max_coupling : float
        Maximum allowed coupling value.

    Returns
    -------
    EthicalCost
        C15_sec ethical cost term.
    """
    n = len(phases)
    if n == 0:
        return EthicalCost(
            J_sec=0.0, phi_ethics=0.0, c15_sec=1.0, constraints_violated=0
        )

    if _HAS_RUST:
        p: FloatArray = np.ascontiguousarray(phases, dtype=np.float64)
        k: FloatArray = np.ascontiguousarray(knm.ravel(), dtype=np.float64)
        j, phi, c15, nv = _rust_ethical_cost(
            p,
            k,
            n,
            alpha_R,
            beta_K,
            gamma_Q,
            nu_S,
            kappa,
            R_min,
            connectivity_min,
            max_coupling,
        )
        return EthicalCost(
            J_sec=j,
            phi_ethics=phi,
            c15_sec=c15,
            constraints_violated=nv,
        )

    R, _ = compute_order_parameter(phases)
    lam2 = fiedler_value(knm)
    lam2_max = float(n)  # max λ₂ for complete graph with unit weights
    K_norm = lam2 / lam2_max if lam2_max > 0 else 0.0

    n_nonzero = np.count_nonzero(knm)
    n_possible = n * (n - 1)
    Q = n_nonzero / n_possible if n_possible > 0 else 0.0

    S_dev = float(np.std(phases)) / np.pi

    J_sec = alpha_R * R + beta_K * K_norm + gamma_Q * Q - nu_S * S_dev

    # CBF constraint violations
    g = [
        R_min - R,
        connectivity_min - lam2,
        float(np.max(knm)) - max_coupling if np.any(knm > 0) else 0.0,
    ]
    violations = [max(0.0, gi) ** 2 for gi in g]
    phi_ethics = kappa * sum(violations)
    n_violated = sum(1 for gi in g if gi > 0)

    c15_sec = (1.0 - J_sec) + phi_ethics

    return EthicalCost(
        J_sec=J_sec,
        phi_ethics=phi_ethics,
        c15_sec=c15_sec,
        constraints_violated=n_violated,
    )

Dataclass: EthicalCost¶

@dataclass
class EthicalCost:
    J_sec: float             # SEC functional value
    phi_ethics: float        # CBF penalty sum
    c15_sec: float           # Total: (1 - J_sec) + κ·Φ
    constraints_violated: int # Count of g_k > 0

Rust Engine Functions¶

pub fn compute_ethical_cost(
    phases: &[f64], knm: &[f64], n: usize,
    alpha_r: f64, beta_k: f64, gamma_q: f64, nu_s: f64, kappa: f64,
    r_min: f64, connectivity_min: f64, max_coupling: f64,
) -> (f64, f64, f64, usize)  // (j_sec, phi_ethics, c15_sec, n_violated)

Internal helpers: - compute_sec_inputs — R, λ₂, Q, S_dev - compute_cbf_penalties — penalty sum and violation count - fiedler_value_inline — Laplacian construction + Jacobi eigenvalues - jacobi_eigenvalues — iterative eigenvalue solver - find_max_offdiag — largest off-diagonal element - jacobi_rotate — single Givens rotation

Auto-Select Logic¶

try:
    from spo_kernel import compute_ethical_cost_rust as _rust_ethical_cost
    _HAS_RUST = True
except ImportError:
    _HAS_RUST = False

Python Eigenvalue Path¶

The Python path uses fiedler_value() from coupling.spectral, which calls numpy.linalg.eigvalsh — a LAPACK wrapper using the symmetric divide-and-conquer algorithm (\(O(N^3)\) but with very small constant factor due to BLAS optimisation).

7. Performance Benchmarks¶

Measured on Intel Core i5-11600K @ 3.90 GHz, 32 GB DDR4-2400. Random phases and coupling, median of 50-100 iterations.

N	Python (µs)	Rust (µs)	Speedup
8	82.6	14.5	5.7x
16	124.4	94.8	1.3x
32	735.2	1711.5	0.4x

Why Does Rust Slow Down at Large N?¶

The bottleneck is the Fiedler value computation: - Python: numpy.linalg.eigvalsh → LAPACK dsyevd, \(O(N^3)\) with highly optimised BLAS (SIMD, cache-aware) - Rust: Jacobi rotation, \(O(N^4)\) worst case with naive loops

For N=8, the Jacobi overhead is small and Rust wins via reduced Python overhead. For N=32, the \(O(N^4)\) Jacobi dominates.

Recommendation: Use the Rust path for \(N \leq 16\) and the Python path for \(N > 16\). Future work: integrate LAPACK bindings (ndarray-linalg) in Rust for \(O(N^3)\) eigenvalue computation.

Cost Breakdown (N=16)¶

Component	Python (µs)	Rust (µs)
Order parameter R	~5	~2
Fiedler value λ₂	~100	~80
Density Q	~1	~0.5
Phase deviation S_dev	~2	~1
CBF penalties	~1	~0.5
Total	~124	~95

Memory Usage¶

Laplacian: \(N^2\) floats (temporary, 8 KB for N=32)
Working copy for Jacobi: \(N^2\) floats
Output: 4 scalars

Test Coverage¶

Rust tests: 7 (ethical module in spo-engine)
Empty phases, synchronised high coupling, no coupling violation, high coupling violation, C15 decomposition, Fiedler complete graph, Fiedler disconnected
Python tests: 13 (tests/test_closure_ethical.py)
Shape/type, synchronised lower cost, constraints detected, decomposition identity, coupling limit, kappa scaling, weight sensitivity, pipeline wiring, edge cases
Source lines: 277 (Rust) + 125 (Python) = 402 total

8. Citations¶

Harsanyi, J. C. (1955). "Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility." Journal of Political Economy 63(4):309-321. DOI: 10.1086/257678
Wiener, N. (1950). The Human Use of Human Beings: Cybernetics and Society. Houghton Mifflin. ISBN: 978-0-306-80320-8.
Ames, A. D., Xu, X., Grizzle, J. W., & Tabuada, P. (2017). "Control barrier function based quadratic programs for safety critical systems." IEEE Transactions on Automatic Control 62(8):3861-3876. DOI: 10.1109/TAC.2016.2638961
MacAskill, W. (2022). What We Owe the Future. Basic Books. ISBN: 978-1-5416-1862-6.
Floridi, L. (2013). The Ethics of Information. Oxford University Press. ISBN: 978-0-19-964132-1.
Fiedler, M. (1973). "Algebraic connectivity of graphs." Czechoslovak Mathematical Journal 23(98):298-305.
Jacobi, C. G. J. (1846). "Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen." Journal für die reine und angewandte Mathematik 30:51-94.
Russell, S. (2019). Human Compatible: Artificial Intelligence and the Problem of Control. Viking. ISBN: 978-0-525-55861-3.

Edge Cases and Limitations¶

Empty Phases (N = 0)¶

Returns \(J_{\text{sec}} = 0\), \(\Phi = 0\), \(C_{15} = 1.0\) (maximum cost), 0 violations.

Zero Coupling Matrix¶

When \(W = 0\): - \(\lambda_2 = 0\) (disconnected graph) - \(Q = 0\) (no connections) - Connectivity constraint is violated (\(g_2 = \lambda_{2,\min} > 0\)) - \(J_{\text{sec}}\) is low (only \(\alpha R\) contributes)

Perfect Synchronisation (R = 1)¶

With \(R = 1\), all phases are identical: - \(S_{\text{dev}} = 0\) (no deviation) - \(J_{\text{sec}} = \alpha + \beta K_{\text{norm}} + \gamma Q\) (maximum without equity penalty) - Non-harm constraint satisfied (R ≥ R_min)

All Weights Zero¶

If \(\alpha = \beta = \gamma = \nu = 0\): - \(J_{\text{sec}} = 0\) - \(C_{15} = 1 + \kappa \Phi\) (only CBF penalties remain)

Troubleshooting¶

Issue: C15_sec is Negative¶

Diagnosis: \(J_{\text{sec}} > 1\) is possible when R, K_norm, and Q are all high. Then \(1 - J_{\text{sec}} < 0\) and \(C_{15}\) can be negative if \(\Phi = 0\).

Solution: This indicates excellent ethical compliance — all metrics are high and no constraints violated. Negative \(C_{15}\) is valid.

Issue: Constraints Always Violated¶

Diagnosis: The thresholds \(R_{\min}\), \(\lambda_{2,\min}\), or \(K_{\max}\) may be set too strictly for the current dynamical regime.

Solution: Reduce thresholds or increase coupling strength.

Issue: Fiedler Value Very Slow for Large N¶

Diagnosis: The Rust Jacobi algorithm is \(O(N^4)\).

Solution: For \(N > 16\), ensure the Python path is used (_HAS_RUST = False for this module). The Python path uses LAPACK eigvalsh at \(O(N^3)\).

Integration with Other SPO Modules¶

With SSGFCosts¶

The ethical cost integrates into the SSGF cost functional as an additive term:

\[U_{\text{total}} = w_1 C_1 + w_2 C_2 + w_3 C_3 + w_4 C_4 + w_{c15} C_{15,\text{sec}}\]

The weight \(w_{c15}\) controls the relative importance of ethical compliance versus synchronisation efficiency.

With GeometryCarrier¶

The geometry carrier can minimise the ethical cost directly:

def ethical_cost_fn(W_flat):
    W = W_flat.reshape(N, N)
    for _ in range(100):
        phases = eng.step(phases, omegas, W, 0.0, 0.0, alpha)
    return compute_ethical_cost(phases, W).c15_sec

state = carrier.update(current_cost, cost_fn=ethical_cost_fn)

With RegimeManager¶

The constraints_violated count provides a direct signal for regime transitions: - 0 violations: NOMINAL regime - 1 violation: DEGRADED — supervisor should act - 2-3 violations: CRITICAL — immediate intervention required

With ActiveInferenceAgent¶

The ethical cost serves as the variational free energy for the active inference agent's ethical beliefs. The agent's policy selection minimises expected \(C_{15,\text{sec}}\) over future states.