Extended Fields and Derived Quantities

Beyond the core structural field tetrad $(\Phi_s, |\nabla\phi|, K_\phi, \xi_C)$ , the nodal equation $\partial\mathrm{EPI}/\partial t = \nu_f \cdot \Delta\mathrm{NFR}(t)$ gives rise to transport fields and derived invariants that complete the multi-scale characterisation of TNFR network state.

Status: CANONICAL — All fields promoted November 2025 after multi-topology validation.

1. Extended Canonical Fields

Two flux fields complement the core tetrad by adding directed transport dynamics.

1.1 Phase Current ( $J_\phi$ )

J_\phi(i) = \frac{1}{|\mathcal{N}(i)|}\sum_{j \in \mathcal{N}(i)} \sin(\phi_j - \phi_i)

Property	Value
Physics	Geometric phase confinement drives directed transport
Sign convention	Positive = net inward flow; negative = net outward flow; zero = equilibrium
Promotion evidence	48 samples across WS, BA, Grid topologies; anticorrelation $r(J_\phi, K_\phi) \approx -0.854$ to $-0.997$ (see §2.2); 100% sign consistency
Canonical status	Promoted November 12, 2025

1.2 $\Delta$ NFR Flux ( $J_{\Delta\mathrm{NFR}}$ )

J_{\Delta\mathrm{NFR}}(i) = \frac{1}{|\mathcal{N}(i)|}\sum_{j \in \mathcal{N}(i)} \left(\Delta\mathrm{NFR}_j - \Delta\mathrm{NFR}_i\right)

Property	Value
Physics	Potential-driven reorganisation transport
Sign convention	Positive = net inward reorganisation pressure; negative = net outward
Canonical status	Promoted November 12, 2025

1.3 Research-Level Fields

Additional fields under investigation (not yet canonical):

Field	Definition	Status
Phase strain	Second-order phase gradient tensor	Research
Phase vorticity	Curl analogue of phase field	Research
Reorganisation strain	Second-order $\Delta$ NFR gradient	Research

Implementation: src/tnfr/physics/extended.py — compute_phase_current(), compute_dnfr_flux().

2. Complex Geometric Field

2.1 Definition

The complex geometric field unifies curvature and transport into a single quantity:

\Psi = K_\phi + i \cdot J_\phi

Property	Expression
Magnitude	$
Phase	$\arg(\Psi) = \arctan(J_\phi / K_\phi)$

2.2 Anticorrelation Evidence

Systematic validation reveals near-perfect anticorrelation between the real and imaginary components:

r(K_\phi, J_\phi) \approx -0.854 \text{ to } -0.997

This anticorrelation is a structural consequence: increasing phase curvature (confinement) suppresses phase current (transport) and vice versa.

2.3 Physical Interpretation

$\Psi$ unifies two complementary aspects of the geometric sector:

$\operatorname{Re}(\Psi) = K_\phi$ : Static geometric confinement (how much the phase field bends)
$\operatorname{Im}(\Psi) = J_\phi$ : Dynamic transport (how much phase flows through the node)

The unified field enables gauge structure (§ GAUGE_SYMMETRY_AND_UNIFICATION.md).

Implementation: src/tnfr/physics/unified.py — compute_complex_geometric_field().

3. Derived Invariants

Seven derived fields emerge from the 6D field space $(\Phi_s, |\nabla\phi|, K_\phi, J_\phi, J_{\Delta\mathrm{NFR}}, \xi_C)$ .

3.1 Chirality ( $\chi$ )

\chi(i) = |\nabla\phi|(i) \cdot K_\phi(i) - J_\phi(i) \cdot J_{\Delta\mathrm{NFR}}(i)

Detects structural handedness. Nonzero chirality signals broken parity in the network — the field configuration has a preferred rotational direction.

3.2 Symmetry Breaking ( $\mathcal{S}$ )

\mathcal{S}(i) = \left(|\nabla\phi|^2 - K_\phi^2\right) + \left(J_\phi^2 - J_{\Delta\mathrm{NFR}}^2\right)

Order parameter for phase transitions. $\mathcal{S} \approx 0$ indicates balanced sectors; large $|\mathcal{S}|$ signals structural asymmetry between geometric and transport fields.

3.3 Coherence Coupling ( $\mathcal{C}$ )

\mathcal{C}(i) = \Phi_s(i) \cdot |\Psi(i)|

Multi-scale connector that measures how strongly the global potential field $\Phi_s$ couples to the local geometric-transport intensity $|\Psi|$ .

3.4 Energy Density ( $\mathcal{E}$ )

\mathcal{E}(i) = \Phi_s^2 + |\nabla\phi|^2 + K_\phi^2 + J_\phi^2 + J_{\Delta\mathrm{NFR}}^2

Total structural energy per node. This is a gauge-invariant scalar (see GAUGE_SYMMETRY_AND_UNIFICATION.md §1.2).

3.5 Action Density ( $\mathcal{A}$ )

\mathcal{A}(i) = \Phi_s \cdot |\nabla\phi| + K_\phi \cdot J_\phi + |\nabla\phi| \cdot J_{\Delta\mathrm{NFR}}

Mixed-sector coupling density. Connects potential, geometric, and transport contributions through cross-terms.

3.6 Topological Charge ( $\mathcal{Q}$ )

\mathcal{Q}(i) = |\nabla\phi|(i) \cdot J_\phi(i) - K_\phi(i) \cdot J_{\Delta\mathrm{NFR}}(i)

Conserved topological quantity under grammar-compliant evolution. Forms a U(1) doublet $(\mathcal{Q}, \tilde{\mathcal{Q}})$ whose norm $|\mathcal{T}|^2 = \mathcal{Q}^2 + \tilde{\mathcal{Q}}^2$ is gauge-invariant.

3.7 Dual Charges

\tilde{\mathcal{Q}}(i) = K_\phi \cdot |\nabla\phi| + J_\phi \cdot J_{\Delta\mathrm{NFR}}

\tilde{\chi}(i) = |\nabla\phi| \cdot J_\phi + K_\phi \cdot J_{\Delta\mathrm{NFR}}

Implementation: src/tnfr/physics/unified.py — compute_chirality(), compute_symmetry_breaking(), compute_coherence_coupling(), compute_energy_density(), compute_action_density(), compute_topological_charge().

4. Energy Decomposition

The Lyapunov energy functional decomposes into kinetic and potential sectors:

E = \frac{1}{2}\sum_i \mathcal{E}(i) = T + V

where:

T = \frac{1}{2}\sum_i \left(J_\phi(i)^2 + J_{\Delta\mathrm{NFR}}(i)^2\right) \qquad\text{(kinetic — transport sector)}

V = \frac{1}{2}\sum_i \left(\Phi_s(i)^2 + |\nabla\phi|(i)^2 + K_\phi(i)^2\right) \qquad\text{(potential — geometric sector)}

Per-node Hamiltonian density: $H(i) = \frac{1}{2}\mathcal{E}(i)$ .

4.1 Two-Sector Structure

Sector	Fields	Conjugate pair	Physical role
Geometric	$K_\phi$ , $J_\phi$	$(K_\phi, J_\phi)$ via $\Psi$	Confinement vs transport
Potential	$\Phi_s$ , $J_{\Delta\mathrm{NFR}}$	$(Φ_{s}, J_{Δ N}$

The phase gradient $|\nabla\phi|$ mediates between sectors, appearing in the potential energy but driving transport through its gradient.

5. Structural Signatures

Element-specific structural signatures are constructed from the field tetrad and extended fields:

Threshold	Value	Derivation
$	\nabla\phi	$ bound
$K_\phi$ canonical	$0.9\pi \approx 2.827$	90% of wrap-angle maximum
Permissive curvature (Au)	$0.95\pi \approx 2.985$	95% of the π phase-wrap (operational margin)

Implementation: src/tnfr/physics/signatures.py — compute_element_signature(), spectral coherence metrics.

6. Field Hierarchy Summary

text

Core Tetrad (CANONICAL)          Extended Transport (CANONICAL)
├── Φ_s  (structural potential)  ├── J_φ       (phase current)
├── |∇φ| (phase gradient)       └── J_ΔNFR    (ΔNFR flux)
├── K_φ  (phase curvature)
└── ξ_C  (coherence length)
                     ↓
            Complex Unification
            Ψ = K_φ + i·J_φ
                     ↓
         Derived Invariants (CANONICAL)
         ├── χ  (chirality)
         ├── S  (symmetry breaking)
         ├── C  (coherence coupling)
         ├── E  (energy density)
         ├── A  (action density)
         └── Q  (topological charge)

Causal chain: The hierarchy above is diagnostic, not dynamical. The experimentally confirmed causal chain is: Operator → (ν_f, ΔNFR) → dEPI/dt → Tetrad → (ℰ, Q). All fields and invariants are uniquely determined by the nodal equation state; they do not feed back into the dynamics. See STRUCTURAL_OPERATORS.md §17.5 and example 39.

Implementation Reference

Module	Content
`src/tnfr/physics/extended.py`	$J_\phi$ , $J_{\Delta\mathrm{NFR}}$ computation
`src/tnfr/physics/unified.py`	$\Psi$ , $\chi$ , $\mathcal{S}$ , $\mathcal{C}$ , $\mathcal{E}$ , $\mathcal{A}$ , $\mathcal{Q}$
`src/tnfr/physics/signatures.py`	Element signatures, spectral coherence
`src/tnfr/physics/telemetry.py`	Optimised computation pipeline

Tests: tests/test_extended_fields.py, tests/test_unified_fields.py

Implementation & Examples

SDK Entry Points

python

from tnfr.sdk import TNFR

net = TNFR.create(20).ring().evolve(5)
invariants = net.tensor_invariants()    # energy_density, topological_charge
emergent = net.emergent_fields()        # chirality, symmetry_breaking, coherence_coupling

Executable Demonstrations

Example	Concept from this document

| 33_complex_field_unification.py | Ψ = K_φ + i·J_φ anticorrelation, emergent fields χ/𝒮/𝒞, energy decomposition | | unified_fields_showcase.py | Ψ = K_φ + i·J_φ, emergent fields χ/𝒮/𝒰, tensor invariants |

Key Source Modules

src/tnfr/physics/fields.py — compute_tensor_invariants(), compute_emergent_fields(), compute_complex_geometric_field_arrays()

Cross-References

Core tetrad: FUNDAMENTAL_THEORY.md §2
Gauge symmetry of $\Psi$ : GAUGE_SYMMETRY_AND_UNIFICATION.md
Conservation laws: STRUCTURAL_CONSERVATION_THEOREM.md
Variational principle: TNFR_VARIATIONAL_PRINCIPLE.md
Lyapunov stability: STRUCTURAL_STABILITY_AND_DYNAMICS.md

Extended Fields and Derived Quantities

Status: CANONICAL — All fields promoted November 2025 after multi-topology validation.

1. Extended Canonical Fields

Two flux fields complement the core tetrad by adding directed transport dynamics.

1.1 Phase Current ( $J_\phi$ )

J_\phi(i) = \frac{1}{|\mathcal{N}(i)|}\sum_{j \in \mathcal{N}(i)} \sin(\phi_j - \phi_i)

Property	Value
Physics	Geometric phase confinement drives directed transport
Sign convention	Positive = net inward flow; negative = net outward flow; zero = equilibrium
Promotion evidence	48 samples across WS, BA, Grid topologies; anticorrelation $r(J_\phi, K_\phi) \approx -0.854$ to $-0.997$ (see §2.2); 100% sign consistency
Canonical status	Promoted November 12, 2025

1.2 $\Delta$ NFR Flux ( $J_{\Delta\mathrm{NFR}}$ )

J_{\Delta\mathrm{NFR}}(i) = \frac{1}{|\mathcal{N}(i)|}\sum_{j \in \mathcal{N}(i)} \left(\Delta\mathrm{NFR}_j - \Delta\mathrm{NFR}_i\right)

Property	Value
Physics	Potential-driven reorganisation transport
Sign convention	Positive = net inward reorganisation pressure; negative = net outward
Canonical status	Promoted November 12, 2025

1.3 Research-Level Fields

Additional fields under investigation (not yet canonical):

Field	Definition	Status
Phase strain	Second-order phase gradient tensor	Research
Phase vorticity	Curl analogue of phase field	Research
Reorganisation strain	Second-order $\Delta$ NFR gradient	Research

Implementation: src/tnfr/physics/extended.py — compute_phase_current(), compute_dnfr_flux().

2. Complex Geometric Field

2.1 Definition

The complex geometric field unifies curvature and transport into a single quantity:

\Psi = K_\phi + i \cdot J_\phi

Property	Expression
Magnitude	$
Phase	$\arg(\Psi) = \arctan(J_\phi / K_\phi)$

2.2 Anticorrelation Evidence

Systematic validation reveals near-perfect anticorrelation between the real and imaginary components:

r(K_\phi, J_\phi) \approx -0.854 \text{ to } -0.997

This anticorrelation is a structural consequence: increasing phase curvature (confinement) suppresses phase current (transport) and vice versa.

2.3 Physical Interpretation

$\Psi$ unifies two complementary aspects of the geometric sector:

$\operatorname{Re}(\Psi) = K_\phi$ : Static geometric confinement (how much the phase field bends)
$\operatorname{Im}(\Psi) = J_\phi$ : Dynamic transport (how much phase flows through the node)

The unified field enables gauge structure (§ GAUGE_SYMMETRY_AND_UNIFICATION.md).

Implementation: src/tnfr/physics/unified.py — compute_complex_geometric_field().

3. Derived Invariants

Seven derived fields emerge from the 6D field space $(\Phi_s, |\nabla\phi|, K_\phi, J_\phi, J_{\Delta\mathrm{NFR}}, \xi_C)$ .

3.1 Chirality ( $\chi$ )

\chi(i) = |\nabla\phi|(i) \cdot K_\phi(i) - J_\phi(i) \cdot J_{\Delta\mathrm{NFR}}(i)

Detects structural handedness. Nonzero chirality signals broken parity in the network — the field configuration has a preferred rotational direction.

3.2 Symmetry Breaking ( $\mathcal{S}$ )

\mathcal{S}(i) = \left(|\nabla\phi|^2 - K_\phi^2\right) + \left(J_\phi^2 - J_{\Delta\mathrm{NFR}}^2\right)

Order parameter for phase transitions. $\mathcal{S} \approx 0$ indicates balanced sectors; large $|\mathcal{S}|$ signals structural asymmetry between geometric and transport fields.

3.3 Coherence Coupling ( $\mathcal{C}$ )

\mathcal{C}(i) = \Phi_s(i) \cdot |\Psi(i)|

Multi-scale connector that measures how strongly the global potential field $\Phi_s$ couples to the local geometric-transport intensity $|\Psi|$ .

3.4 Energy Density ( $\mathcal{E}$ )

\mathcal{E}(i) = \Phi_s^2 + |\nabla\phi|^2 + K_\phi^2 + J_\phi^2 + J_{\Delta\mathrm{NFR}}^2

Total structural energy per node. This is a gauge-invariant scalar (see GAUGE_SYMMETRY_AND_UNIFICATION.md §1.2).

3.5 Action Density ( $\mathcal{A}$ )

\mathcal{A}(i) = \Phi_s \cdot |\nabla\phi| + K_\phi \cdot J_\phi + |\nabla\phi| \cdot J_{\Delta\mathrm{NFR}}

Mixed-sector coupling density. Connects potential, geometric, and transport contributions through cross-terms.

3.6 Topological Charge ( $\mathcal{Q}$ )

\mathcal{Q}(i) = |\nabla\phi|(i) \cdot J_\phi(i) - K_\phi(i) \cdot J_{\Delta\mathrm{NFR}}(i)

3.7 Dual Charges

\tilde{\mathcal{Q}}(i) = K_\phi \cdot |\nabla\phi| + J_\phi \cdot J_{\Delta\mathrm{NFR}}

\tilde{\chi}(i) = |\nabla\phi| \cdot J_\phi + K_\phi \cdot J_{\Delta\mathrm{NFR}}

4. Energy Decomposition

The Lyapunov energy functional decomposes into kinetic and potential sectors:

E = \frac{1}{2}\sum_i \mathcal{E}(i) = T + V

where:

T = \frac{1}{2}\sum_i \left(J_\phi(i)^2 + J_{\Delta\mathrm{NFR}}(i)^2\right) \qquad\text{(kinetic — transport sector)}

V = \frac{1}{2}\sum_i \left(\Phi_s(i)^2 + |\nabla\phi|(i)^2 + K_\phi(i)^2\right) \qquad\text{(potential — geometric sector)}

Per-node Hamiltonian density: $H(i) = \frac{1}{2}\mathcal{E}(i)$ .

4.1 Two-Sector Structure

Sector	Fields	Conjugate pair	Physical role
Geometric	$K_\phi$ , $J_\phi$	$(K_\phi, J_\phi)$ via $\Psi$	Confinement vs transport
Potential	$\Phi_s$ , $J_{\Delta\mathrm{NFR}}$	$(Φ_{s}, J_{Δ N}$

The phase gradient $|\nabla\phi|$ mediates between sectors, appearing in the potential energy but driving transport through its gradient.

5. Structural Signatures

Element-specific structural signatures are constructed from the field tetrad and extended fields:

Threshold	Value	Derivation
$	\nabla\phi	$ bound
$K_\phi$ canonical	$0.9\pi \approx 2.827$	90% of wrap-angle maximum
Permissive curvature (Au)	$0.95\pi \approx 2.985$	95% of the π phase-wrap (operational margin)

Implementation: src/tnfr/physics/signatures.py — compute_element_signature(), spectral coherence metrics.

6. Field Hierarchy Summary

text

Core Tetrad (CANONICAL)          Extended Transport (CANONICAL)
├── Φ_s  (structural potential)  ├── J_φ       (phase current)
├── |∇φ| (phase gradient)       └── J_ΔNFR    (ΔNFR flux)
├── K_φ  (phase curvature)
└── ξ_C  (coherence length)
                     ↓
            Complex Unification
            Ψ = K_φ + i·J_φ
                     ↓
         Derived Invariants (CANONICAL)
         ├── χ  (chirality)
         ├── S  (symmetry breaking)
         ├── C  (coherence coupling)
         ├── E  (energy density)
         ├── A  (action density)
         └── Q  (topological charge)

Implementation Reference

Module	Content
`src/tnfr/physics/extended.py`	$J_\phi$ , $J_{\Delta\mathrm{NFR}}$ computation
`src/tnfr/physics/unified.py`	$\Psi$ , $\chi$ , $\mathcal{S}$ , $\mathcal{C}$ , $\mathcal{E}$ , $\mathcal{A}$ , $\mathcal{Q}$
`src/tnfr/physics/signatures.py`	Element signatures, spectral coherence
`src/tnfr/physics/telemetry.py`	Optimised computation pipeline

Tests: tests/test_extended_fields.py, tests/test_unified_fields.py

Implementation & Examples

SDK Entry Points

python

from tnfr.sdk import TNFR

net = TNFR.create(20).ring().evolve(5)
invariants = net.tensor_invariants()    # energy_density, topological_charge
emergent = net.emergent_fields()        # chirality, symmetry_breaking, coherence_coupling

Executable Demonstrations

Example	Concept from this document

Key Source Modules

src/tnfr/physics/fields.py — compute_tensor_invariants(), compute_emergent_fields(), compute_complex_geometric_field_arrays()

Cross-References

Core tetrad: FUNDAMENTAL_THEORY.md §2
Gauge symmetry of $\Psi$ : GAUGE_SYMMETRY_AND_UNIFICATION.md
Conservation laws: STRUCTURAL_CONSERVATION_THEOREM.md
Variational principle: TNFR_VARIATIONAL_PRINCIPLE.md
Lyapunov stability: STRUCTURAL_STABILITY_AND_DYNAMICS.md

EXTENDED_FIELDS_AND_DERIVED_QUANTITIES.md

Extended Fields and Derived Quantities

1. Extended Canonical Fields

1.1 Phase Current (JϕJ_\phiJϕ​)

1.2 Δ\DeltaΔNFR Flux (JΔNFRJ_{\Delta\mathrm{NFR}}JΔNFR​)

1.3 Research-Level Fields

2. Complex Geometric Field

2.1 Definition

2.2 Anticorrelation Evidence

2.3 Physical Interpretation

3. Derived Invariants

3.1 Chirality (χ\chiχ)

3.2 Symmetry Breaking (S\mathcal{S}S)

3.3 Coherence Coupling (C\mathcal{C}C)

3.4 Energy Density (E\mathcal{E}E)

3.5 Action Density (A\mathcal{A}A)

3.6 Topological Charge (Q\mathcal{Q}Q)

3.7 Dual Charges

4. Energy Decomposition

4.1 Two-Sector Structure

5. Structural Signatures

6. Field Hierarchy Summary

Implementation Reference

Implementation & Examples

SDK Entry Points

Executable Demonstrations

Key Source Modules

Cross-References

EXTENDED_FIELDS_AND_DERIVED_QUANTITIES.md

Extended Fields and Derived Quantities

1. Extended Canonical Fields

1.1 Phase Current (JϕJ_\phiJϕ​)

1.2 Δ\DeltaΔNFR Flux (JΔNFRJ_{\Delta\mathrm{NFR}}JΔNFR​)

1.3 Research-Level Fields

2. Complex Geometric Field

2.1 Definition

2.2 Anticorrelation Evidence

2.3 Physical Interpretation

3. Derived Invariants

3.1 Chirality (χ\chiχ)

3.2 Symmetry Breaking (S\mathcal{S}S)

3.3 Coherence Coupling (C\mathcal{C}C)

3.4 Energy Density (E\mathcal{E}E)

3.5 Action Density (A\mathcal{A}A)

3.6 Topological Charge (Q\mathcal{Q}Q)

3.7 Dual Charges

4. Energy Decomposition

4.1 Two-Sector Structure

5. Structural Signatures

6. Field Hierarchy Summary

Implementation Reference

Implementation & Examples

SDK Entry Points

Executable Demonstrations

Key Source Modules

Cross-References

1.1 Phase Current ( $J_\phi$ )

1.2 $\Delta$ NFR Flux ( $J_{\Delta\mathrm{NFR}}$ )

3.1 Chirality ( $\chi$ )

3.2 Symmetry Breaking ( $\mathcal{S}$ )

3.3 Coherence Coupling ( $\mathcal{C}$ )

3.4 Energy Density ( $\mathcal{E}$ )

3.5 Action Density ( $\mathcal{A}$ )

3.6 Topological Charge ( $\mathcal{Q}$ )

1.1 Phase Current ( $J_\phi$ )

1.2 $\Delta$ NFR Flux ( $J_{\Delta\mathrm{NFR}}$ )

3.1 Chirality ( $\chi$ )

3.2 Symmetry Breaking ( $\mathcal{S}$ )

3.3 Coherence Coupling ( $\mathcal{C}$ )

3.4 Energy Density ( $\mathcal{E}$ )

3.5 Action Density ( $\mathcal{A}$ )

3.6 Topological Charge ( $\mathcal{Q}$ )