Structural Stability and Dynamics

This document collects the stability analysis, phase transition theory, lifecycle dynamics, and integrity monitoring that emerge from the nodal equation $\partial\mathrm{EPI}/\partial t = \nu_f \cdot \Delta\mathrm{NFR}(t)$. Each section corresponds to a verified implementation in the codebase.

Status: CANONICAL — All results derived from the nodal equation and validated computationally.

1. Lyapunov Stability Analysis

1.1 Energy Functional

The structural energy functional serves as a Lyapunov candidate:

For grammar-compliant evolution: $dE/dt \le 0$ (Lyapunov stability).

1.2 Per-Operator Lyapunov Role

The structural energy functional $E$ above is emergent: it is built entirely from the tetrad fields $(\Phi_s, |\nabla\phi|, K_\phi, J_\phi, J_{\Delta\mathrm{NFR}})$ and contains no $\mathrm{EPI}$ or $\nu_f$ term. Measured directly: scaling the form $\mathrm{EPI}$ or the capacity $\nu_f$ on every node leaves $E$ unchanged ($\Delta E = 0$), while the phase $\theta$ (through $|\nabla\phi|, K_\phi, J_\phi$) and the structural pressure $\Delta\mathrm{NFR}$ (through $\Phi_s, J_{\Delta\mathrm{NFR}}$) do enter it. The coherence $C(t) = 1/(1 + \mathrm{mean}|\Delta\mathrm{NFR}| + \mathrm{mean}|\mathrm{dEPI}|)$ likewise responds to the pressure channel. Both Lyapunov candidates share one structural-pressure channel, $|\Delta\mathrm{NFR}|$.

Consequently each operator's Lyapunov role is its canonical grammar U2 role, derived from config.physics_derivation (the single source of truth, identical to the U2 stabiliser/destabiliser classification) — not a separate energy algebra. An operator contracts the Lyapunov functional iff it provides negative feedback on $|\Delta\mathrm{NFR}|$, and expands it iff it raises $|\Delta\mathrm{NFR}|$. The contraction/expansion rate is the operator's own structural-pressure factor.

Stabilisers ($\Delta E \le 0$ — reduce $|\Delta\mathrm{NFR}|$)

Destabilisers ($\Delta E \le \kappa \cdot E$ — raise $|\Delta\mathrm{NFR}|$)

Neutral ($\Delta E \approx 0$ by grammatical role)

Dual-lever vs Lyapunov role: The dual-lever structure (STRUCTURAL_OPERATORS.md §17.1) classifies operators by which right-hand-side factor of the nodal equation they modulate — the capacity lever $\nu_f$ (UM, SHA, VAL), the pressure lever $\Delta\mathrm{NFR}$ (IL, OZ, THOL, ZHIR, NAV), both (NUL), or neither/the form on the LHS (AL, EN, RA, REMESH). The Lyapunov role above is the grammar U2 role (stabiliser/destabiliser), which depends on the sign of the $|\Delta\mathrm{NFR}|$ feedback: VAL engages the capacity lever yet is a U2 destabiliser, while NAV engages the pressure lever yet is U2-neutral because its trajectory is controlled. The two classifications are related but distinct. See example 39 and src/tnfr/physics/lyapunov.py.

1.3 Grammar U2 Lyapunov Theorem

Grammar rule U2 (CONVERGENCE & BOUNDEDNESS) requires that every destabiliser be accompanied by a stabiliser. The formal proof shows that the net energy change across a grammar-compliant sequence is non-positive:

This confirms Lyapunov stability for the full 13-operator algebra.

Refinement: The formal bound $\sum \Delta E_{\text{op}} \le 0$ is sufficient but not necessary for energy descent. Experimental observation (example 38) shows grammar-compliant sequences with cumulative Lyapunov product $\Pi \approx 1.288$ (formally non-contractive) that still achieve net energy decrease ($\Delta E = -9.59$). The multiplicative bound is conservative because operator interactions on the shared graph state are nonlinear.

1.4 Spectral Gap Characterisation

The algebraic connectivity $\lambda_1$ of the graph Laplacian controls the relaxation time:

Implementation: src/tnfr/physics/lyapunov.py — OperatorStabilityClass, OperatorEnergyBound, LyapunovPerOperator, analyze_spectral_gap().

2. Phase Transitions

2.1 Order Parameter

The symmetry breaking field $\mathcal{S}$ serves as the order parameter for phase transitions:

2.2 Phase Classification

The phase is decided by the sampling-noise z-score of the symmetry breaking, $z = |\langle\mathcal{S}\rangle| / \mathrm{SE}$ with $\mathrm{SE} = \sqrt{\mathrm{Var}(\mathcal{S})/N}$ — the statistical significance measured from the system itself. The only cut is $z = 1$ (one sampling sigma); likewise $z_\chi$ for the chirality field.

2.3 Critical Exponent (measured observable)

Near the critical point the order parameter follows a power law:

The exponent $\beta$ is an observable to be measured (fit_critical_exponent), not a derived universal constant. A measurement across sweep protocols gives protocol-dependent values, so there is no universal closed-form exponent. The constant $\gamma/\pi \approx 0.1837$ is retained only as a calibrated reference / noise-floor scale (TIER-2), not a prediction of the nodal equation.

2.4 Constants

2.5 Susceptibility

The structural susceptibility diverges at the critical point:

2.6 Critical Exponent Fitting

For systems near the transition, the critical exponent can be fit from the scaling law $|\langle\mathcal{S}\rangle| \sim |p - p_c|^{\gamma_{\text{fit}}}$. The theoretical prediction $\gamma_{\text{fit}} \to \gamma/\pi$ serves as validation.

Implementation: src/tnfr/physics/phase_transition.py — Phase enum, PhaseTransitionTelemetry, PhaseSnapshot, compute_order_parameter(), classify_phase(), detect_phase_transition(), fit_critical_exponent().

3. Self-Sustaining Dynamics and Autopoiesis

3.1 Autopoietic Coefficient

The autopoietic coefficient measures a system's capacity for self-generation relative to external driving:

where the self-generation function follows logistic growth:

3.2 Self-Sustaining Threshold

When $A > 1$, the system generates more structural change through self-organisation than through external forcing — the defining property of autopoietic systems in the sense of Maturana & Varela.

3.3 Auxiliary Indices

3.4 Self-Sustaining Threshold Detection

The threshold time $t_{\text{self}}$ is found by interpolation at the $A(t) = 1.0$ crossing. The LifeTelemetry dataclass records the complete trajectory $(V_i(t), A(t), S(t), M(t))$.

Implementation: src/tnfr/physics/life.py — detect_life_emergence(), LifeTelemetry dataclass.

4. Node Lifecycle

4.1 Lifecycle States

Each TNFR node passes through a sequence of canonical states determined by its structural attributes:

Priority order for classification: mutation > propagation > stabilization > activation > dormant. Collapse is checked first.

4.2 Collapse Conditions

Four canonical collapse reasons, checked in priority order:

4.3 Default Thresholds

Implementation: src/tnfr/operators/lifecycle.py — LifecycleState, CollapseReason, get_lifecycle_state(), check_collapse_conditions().

5. Internal Hamiltonian Construction

5.1 Definition

The internal Hamiltonian governs structural evolution:

5.2 Components

Coherence potential (attractive interaction):

where $w_{ij}$ is the coherence weight from structural similarity and $C_0 = -1.0$ (attractive).

Frequency operator (diagonal):

Each node's $\nu_f$ becomes its diagonal energy.

Coupling Hamiltonian (topology):

All components are $N \times N$ Hermitian matrices ($N$ = number of nodes).

5.3 Time Evolution

The unitary time evolution operator:

Propagates states: $|\psi(t)\rangle = U(t)|\psi(0)\rangle$.

5.4 Energy Spectrum

The eigenvalue equation:

gives stationary states $|\phi_n\rangle$ with energies $E_n$ (maximally stable configurations).

5.5 ΔNFR from Hamiltonian

The $\Delta\mathrm{NFR}$ operator follows from the Hamiltonian commutator:

Per-node: $\Delta\mathrm{NFR}_n = (i/\hbar_{\text{str}})\langle n|[\hat{H}_{\text{int}}, \rho_n]|n\rangle$ where $\rho_n = |n\rangle\langle n|$.

Implementation: src/tnfr/operators/hamiltonian.py — InternalHamiltonian class with get_spectrum(), time_evolution_operator(), compute_delta_nfr_operator().

6. Structural Integrity Monitor

6.1 Purpose

The integrity monitor verifies postconditions of all 13 canonical operators after each application. This closes the loop between theoretical contracts and runtime behaviour.

6.2 Postconditions (13/13 Operators)

6.3 Monitor Modes

6.4 Corrective Suggestions

When a violation is detected in OBSERVE or ENFORCE mode, the monitor provides corrective suggestions (e.g. "apply IL after OZ to restore convergence").

Implementation: src/tnfr/physics/integrity.py — IntegrityReport, IntegritySummary, MonitorMode, StructuralIntegrityViolation, POSTCONDITIONS registry.

Tests: tests/test_integrity.py

Implementation Reference

Implementation & Examples

SDK Entry Points

from tnfr.sdk import TNFR

net = TNFR.create(20).ring().evolve(5)
report = net.integrity_check()    # IntegrityReport (13/13 operators)

Executable Demonstrations

Key Source Modules

• src/tnfr/physics/integrity.py — Structural integrity monitor (13/13 operator postconditions)
• src/tnfr/physics/conservation.py — Energy functional (Lyapunov candidate)
• src/tnfr/physics/phase_transition.py — Phase transition detection
• src/tnfr/operators/lifecycle.py — Node lifecycle management

Cross-References

• Lyapunov energy in conservation: STRUCTURAL_CONSERVATION_THEOREM.md §8
• Grammar U2 (convergence): UNIFIED_GRAMMAR_RULES.md
• Hamiltonian/Lagrangian formulation: TNFR_VARIATIONAL_PRINCIPLE.md
• Order parameter $\mathcal{S}$: EXTENDED_FIELDS_AND_DERIVED_QUANTITIES.md §3.2
• Dissipative extensions: DISSIPATIVE_AND_OPEN_SYSTEMS.md
• Gauge structure: GAUGE_SYMMETRY_AND_UNIFICATION.md