Extended Fields and Derived Quantities¶

Beyond the core structural field tetrad $(\Phi_s, |\nabla\phi|, K_\phi, \xi_C)$, the nodal equation $\partial\mathrm{EPI}/\partial t = \nu_f \cdot \Delta\mathrm{NFR}(t)$ supports transport fields and derived diagnostic contractions. They add directional and interaction read-outs to a declared graph state; they do not make the tetrad an injective or complete representation of arbitrary TNFR states.

Status: CANONICAL DIAGNOSTIC DEFINITIONS — finite multi-topology validation supports the implemented definitions; completeness and unrestricted conservation claims remain open.

1. Extended transport diagnostics¶

Two flux fields complement the core tetrad with directed neighbor statistics; their read-outs do not select or measure an evolution law by themselves.

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
Definition scope	Signed mean sine over unique support neighbors; a directional statistic, not an observed transport rate
Sign convention	Sign of the summed neighbor sine displacement; zero permits cancellation or antipodal phases and does not establish nodal equilibrium
Finite evidence	48 historical samples across WS, BA and grid topologies reported anticorrelation $r(J_\phi, K_\phi) \approx -0.854$ to $-0.997$ under that protocol (see §2.2)
Engine status	Canonical diagnostic definition; no universal correlation theorem

On a regular exact-real chart with nonzero neighbor resultant, J_phi(i)=-|S_i| sin(K_phi(i))/degree_i. Nonzero current and curvature have opposite signs on this chart; that identity does not prescribe the magnitude of a sample correlation. The numerical readers retain their separate trigonometric and angular rounding.

The implementation counts parallel neighbors once and self-loops once, uses successors on directed graphs, and includes zero-conductance edges. Isolates use an explicit zero convention. For reciprocal support, the pair-cost and state-dependent metric identity relates this statistic to canonical phase pressure. A phase evolution law remains a separate premise; none is selected by this read-out.

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
Definition scope	Signed pressure contrast over unique support neighbors; no physical transport-rate factor is supplied
Sign convention	Positive means the neighbor-average pressure exceeds the node pressure; negative means the reverse
Engine status	Canonical diagnostic definition

Like the phase current, this contrast uses unique outgoing support neighbors, including zero-conductance edges. It reads stored pressure without refreshing it. Scalar and array paths reuse the stable neighbor-difference reducer: a uniform field has zero contrast even near the binary64 range limit, and signed cancellation is retained. Unrepresentable final values raise. Cached maps are detached from their public returns and bind admitted state and neighbor order. These arithmetic and cache contracts add no pressure law.

1.3 Research-Level Fields¶

Additional fields under investigation (not yet canonical):

Field	Definition	Status
Phase strain	Population variance of neighboring phase-gradient magnitudes	Static research statistic; only scale 1 is implemented
Phase vorticity	Historical name for the mean inverse-length weighted wrapped neighbor displacement	Not a curl or cycle winding; nonzero even on a tree
Reorganisation strain	Population standard deviation of neighboring pressure	Static research statistic, not a time derivative or force balance

The legacy vorticity reader uses the shared distance convention: explicit length, then weight fallback, then unit length; parallel lengths combine by minimum. Every consumed inverse distance must be positive and finite. Zero conductance with an explicit positive length remains admissible. Actual topological winding requires an oriented cycle and branch evidence from certify_phase_winding; it cannot be recovered by interpreting this local mean as circulation. Missing pressure uses the common zero default and invalid authoritative pressure is rejected. These definitions introduce no extra nodal evolution law.

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	$\vert \Psi\vert = \sqrt{K_\phi^2 + J_\phi^2}$
Phase	$\arg(\Psi) = \operatorname{atan2}(J_\phi, K_\phi)$

2.2 Anticorrelation Evidence¶

The cited finite protocol measured strong anticorrelation between the real and imaginary components:

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

This is a protocol-specific relationship, not a universal consequence of the definitions. Graph topology, phase distribution and weighting can change it.

2.3 Physical Interpretation¶

$\Psi$ unifies two complementary aspects of the geometric sector: - $\operatorname{Re}(\Psi) = K_\phi$: signed circular mismatch with the neighbor mean; - $\operatorname{Im}(\Psi) = J_\phi$: signed neighbor sine average.

Both are instantaneous read-outs of primitive phase. Neither supplies a phase velocity or a confinement law. The harmonic coordinate interpretation and its graph-realizability obstructions are owned by the variational note.

The pair can be embedded in the auxiliary rotation/gauge comparison described in GAUGE_SYMMETRY_AND_UNIFICATION.md; forming a complex number alone does not derive a gauge law.

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

3. Derived diagnostic contractions¶

The named read-outs below are computed from the field tuple $(\Phi_s, |\nabla\phi|, K_\phi, J_\phi, J_{\Delta\mathrm{NFR}}, \xi_C)$. They are algebraic contractions; names such as “charge” or “action” do not by themselves establish conservation, topology or a variational principle.

3.1 Chirality ($\chi$)¶

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

Signed handedness diagnostic in the declared field convention. A nonzero value does not by itself prove graph-level parity breaking.

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

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

Candidate order parameter used by the finite transition diagnostics. $\mathcal{S} \approx 0$ means that these squared terms balance; a phase transition requires an independently specified finite-size protocol.

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

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

Signed product of the source aggregation and local geometric magnitude. It does not measure a dynamical coupling coefficient or demonstrate energy transfer.

3.4 Energy Density ($\mathcal{E}$)¶

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

Nonnegative quadratic energy diagnostic per node. It is invariant under the specified auxiliary orthogonal rotations; this scoped algebraic fact is not a general gauge-conservation theorem.

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 ($\mathcal{Q}$)¶

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

The public API retains the historical name “topological charge”. It is a bilinear diagnostic. Grammar compliance alone does not conserve it, and no topological quantization theorem is claimed. Together with $\tilde{\mathcal{Q}}$, its Euclidean norm is invariant only under the declared auxiliary U(1) rotation.

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 quadratic energy diagnostic, used as a Lyapunov candidate, decomposes by definition into transport and static-field 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	$K_\phi$, $J_\phi$	Auxiliary pair $(K_\phi, J_\phi)$ via $\Psi$	Curvature vs transport
Potential	$\Phi_s$, $J_{\Delta\mathrm{NFR}}$	Auxiliary pair $(\Phi_s, J_{\Delta\mathrm{NFR}})$	Source aggregation vs reorganisation transport

The phase-gradient magnitude appears in the quadratic diagnostic and several cross-products. This algebraic participation does not derive transport from its gradient. The shared normalization boundary also applies before interpreting these sums and products as physical energies.

5. Graph-field probe signatures¶

compute_element_signature() is a compatibility-named diagnostic over the supplied graph. It reports the $\xi_C$ estimator, phase-gradient and phase-curvature summaries, and the response of $\Phi_s$ to an optional declared probe. The default probe runs the valid word [Emission, Coherence, Silence] once on a detached graph copy and defines

\[ \Delta_{\mathrm{probe}}\Phi_s = \max_i |\Phi_s^{\mathrm{after}}(i)-\Phi_s^{\mathrm{before}}(i)|. \]

The input graph is not evolved. The returned synthetic_probe_word and synthetic_step_applied fields identify the intervention. If the probe is disabled, its zero drift is a compatibility sentinel for “not measured,” not evidence of stability: potential_drift_assessed=False and potential_drift_ok=None preserve that unavailability. A resulting signature_class="stable" then reflects only the phase gates; it does not report a passed pressure-drift check. If a performed probe fails that check, the classifier returns "unstable" regardless of the other gates.

Diagnostic policy	Value	Status
Mean $\vert \nabla\phi\vert $ gate	$\pi/16 \approx 0.196$	Selected early-warning threshold; the exact kinematic bound is $\pi$
Maximum $\vert K_\phi\vert $ gate	$0.9\pi \approx 2.827$	Selected margin within the exact wrapped bound $\pi$
Probe $\Phi_s$ drift gate	$\pi/2$	Selected U6 policy, applied only when the probe runs
Fitted $\xi_C$ category	relative to $\sqrt{\vert V\vert }$	Heuristic: localized below $0.3\sqrt{\vert V\vert }$, extended above $1.2\sqrt{\vert V\vert }$
Legacy Au-like curvature gate	$0.95\pi \approx 2.985$	Compatibility policy

These outputs are a response signature for a specified graph and intervention. The coherence estimator can return a product-fit length, a dimensionless spectral fallback or an unavailable value. Legacy size-relative categories must not silently identify these branches; the tetrad guide owns their units and provenance requirements. They are not spectral-coherence metrics, physical-element signatures, chemical stability tests, or evidence of autonomous restoring dynamics. The historical compute_au_like_signature() name adds a compatibility boolean from declared thresholds; it does not establish an Au or metallic correspondence.

Implementation: src/tnfr/physics/signatures.py — compute_element_signature(), compute_au_like_signature().

6. Field Hierarchy Summary¶

Core Tetrad (DIAGNOSTIC)         Extended Transport (DIAGNOSTIC)
├── 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 Diagnostic Contractions
         ├── χ  (chirality)
         ├── S  (symmetry breaking)
         ├── C  (coherence coupling)
         ├── E  (energy density)
         ├── A  (action density)
         └── Q  (topological charge)

Dependency chain: The hierarchy above is diagnostic, not a general causal identification theorem: Operator → updated nodal/graph state → structural read-outs → derived contractions. For a fully specified state and graph these functions are deterministic. The read-outs alone need not reconstruct that state or predict its future. See STRUCTURAL_OPERATORS.md §17.5 and example 39.

6.1 Emergent-property audit and useful derived margins¶

Adding a function of already observed fields does not restore information lost by those fields. Derived quantities are useful when they expose a specific invariant, response or domain boundary; naming another combination does not create an independent state coordinate or an evolution law.

Quantity or property	What is available	What it does not establish
Complex fields, bilinears and energy totals	One shared algebraic suite over the captured five base maps	New independent degrees of freedom, dynamics, or conservation
Cycle winding and branch/U3 margins	Existing declared-cycle certificate and period geometry; cycle edges and orientation are required	Creation of the cycle, unrestricted preservation or an integer value on an open path
Local normalized resultant	On the admitted acute cycle, the existing cosine margin bounds it away from zero	A nonzero global resultant or a universal synchrony clock
Conditional restoring stiffness	The same margin bounds the supplied sine law's Hessian and its existing decay rate	Stiffness of every pressure or phase law; the Arg-pressure Jacobian is different
Effective capacity and memory	Existing reduction/quotient and hidden-coordinate owners under their closure hypotheses	Identification with primitive capacity or elimination of unresolved history by renaming it
THOL record count, record rate and emergence index	Retrospective heuristics with an explicit observation/window convention	Autonomous generation, mathematical bifurcation, causal efficiency or a persistence certificate
Wave spectrum and nodal pulse	Auxiliary modal frequencies versus stored capacity/phase summaries	Measured oscillations, mode excitation or equality of those two clocks

The positive margin derivation is centralized in the acute-cycle theorem. It reuses phase_radius_upper, cosine_lower_bound and the existing spectral rate bounds; no new adjustable coefficient is introduced. The full configured pressure mixture is admissible there because its capacity/topology gradients vanish under the theorem's common-capacity, fixed-simple-cycle premises.

Snapshot conservation boundary. The tensor facade formerly differentiated charge density along sorted node labels. That operation is neither a graph divergence nor a time derivative: relabeling the same P4 changed the resulting score. conservation_quality is now None, with conservation_sample_available=False and conservation_scope="single_snapshot_no_temporal_balance". Temporal balance still belongs to the existing conservation observer on declared snapshots and times. A zero unavailable score would incorrectly claim a measurement. Array facades retain an explicit nodes order instead of sorting arbitrary node identifiers; composite telemetry reuses one base-field collection.

THOL heuristics retain signed scalar EPI admission and exact intermediate ratios before their approximate cube root, avoiding false zero/infinity from overflow or underflow. An explicit operator counter cannot be advanced merely to accommodate a future-dated record. Their EPI baseline must still match the retained THOL window: eviction does not preserve a lifetime denominator. These arithmetic and provenance repairs do not promote the heuristics into dynamical mechanisms. See emergence controls.

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`	Graph-field summaries and declared-probe response signatures
`src/tnfr/physics/telemetry.py`	Optimised computation pipeline

Tests: tests/physics/test_field_readout_consistency.py, tests/physics/test_gauge.py, tests/physics/test_readout_reuse.py

Implementation & Examples¶

SDK Entry Points¶

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	Seeded supplied-state snapshots, shared complex/derived fields and explicit unavailable temporal conservation; no operator evolution

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

Diagnostic policy	Value	Status
Mean $\vert \nabla\phi\vert $ gate	\(\pi/16 \approx 0.196\)	Selected early-warning threshold; the exact kinematic bound is \(\pi\)
Maximum $\vert K_\phi\vert $ gate	\(0.9\pi \approx 2.827\)	Selected margin within the exact wrapped bound \(\pi\)
Probe \(\Phi_s\) drift gate	\(\pi/2\)	Selected U6 policy, applied only when the probe runs
Fitted \(\xi_C\) category	relative to \(\sqrt{\vert V\vert }\)	Heuristic: localized below \(0.3\sqrt{\vert V\vert }\), extended above \(1.2\sqrt{\vert V\vert }\)
Legacy Au-like curvature gate	\(0.95\pi \approx 2.985\)	Compatibility policy