TNFR Glossary

Purpose: Operational quick reference for the Resonant Fractal Nature Theory (TNFR)
Status: Canonical reference, aligned with the current engine and TNFR.pdf
Version: 0.0.3.5 (June 2026)
Authority: AGENTS.md is the single source of truth; this glossary mirrors it API-first

Scope: API-focused definitions for developers implementing TNFR networks — the nodal equation, the structural triad and the fractal-resonant node (NFR), the structural-field tetrad, the 13 operators and the unified grammar (U1–U6). For full derivations see AGENTS.md, FUNDAMENTAL_THEORY.md and UNIFIED_GRAMMAR_RULES.md.

Core Variables

Fractal-Resonant Node (NFR)

What: Nodo Fractal Resonante — a region of structural coherence coupled to a network (TNFR.pdf §1.4.1), the fundamental entity of TNFR. The structural triad (EPI, νf, φ) defines it; it is read out as a whole by Network.nfr().
Properties: multiscalar (an NFR can nest other NFRs — operational fractality), autopoietic (emerges by local reorganization, no external support), relational (exists only by coupling) and temporal (persists while it reorganizes its coherence).
Nodal topology: radial (one central nucleus), annular (passive center, peripheral ring) or multinodal (several centers), classified from the structural-potential geometry by classify_nodal_topology(G).
Equilibrium: ΔNFR = 0 is not an NFR but its resonant-coherence attractor (C → 1); under canonical relaxation the EPI channel diffuses to a uniform field, so a fully relaxed network is one uniform NFR. Scale-relative: a single node is a micro-NFR, a coherent region a macro-NFR.
API: tnfr.physics.fields.classify_nodal_topology, tnfr.sdk.simple.Network.nfr, tnfr.structural.create_nfr
Theory: AGENTS.md §2 (The fractal-resonant node), TNFR.pdf §1.4.1

Primary Information Structure (EPI)

Code: G.nodes[n]['EPI'], ALIAS_EPI
Symbol: (\text{EPI}) or (E)
What: Coherent structural form of a node
Space: (B_{\text{EPI}}) (Banach space)
Rules: Modified only via structural operators, never directly
API: tnfr.structural operators
Math: FUNDAMENTAL_THEORY.md §2.2 (Structural Triad — Banach space B_EPI)

Structural Frequency (νf)

Code: G.nodes[n]['vf'], ALIAS_VF
Symbol: (\nu_f)
Units: Hz_str (structural hertz)
Range: (\mathbb{R}^+) (positive reals; node collapse when (\nu_f \to 0))
What: Rate of structural reorganization
API: adapt_vf_by_coherence(), operators
Math: FUNDAMENTAL_THEORY.md §2 (Governing Dynamics)

Internal Reorganization Operator (ΔNFR)

Code: G.nodes[n]['dnfr'], ALIAS_DNFR
Symbol: (\Delta\text{NFR})
What: Structural reorganization pressure — the gradient driving evolution.
Four gradient channels: (\Delta\text{NFR} = w_\phi,\partial\phi + w_E,\partial\text{EPI} + w_{\nu},\partial\nu_f + w_\tau,\partial\text{topo}) (phase desync, EPI gradient, νf gradient, topology). Operational default weights (tunable, free parameters) DNFR_WEIGHTS = {phase ≈ 0.737, epi ≈ 0.155, vf ≈ 0.090, topo = 0.0} in config/defaults_core.py; normalized and applied by _configure_dnfr_weights in dynamics/dnfr.py.
EPI channel = graph diffusion (KEY): the EPI channel is exactly the random-walk graph Laplacian, (\Delta\text{NFR}\text{epi}(i) = \overline{\text{EPI}}{\mathcal{N}(i)} - \text{EPI}(i) = -(L_\text{rw},\text{EPI})(i)), so (\partial\text{EPI}/\partial t = -\nu_f L_\text{rw},\text{EPI}) is the discrete diffusion (heat) equation with diffusivity νf — eigenmode decay (e^{-\nu_f \lambda_k t}), conserved degree-weighted total, equilibrium ⟺ uniform field.
Sign: positive = expansion, negative = contraction.
Compute: default_compute_delta_nfr hook, automatic in step().
Math: FUNDAMENTAL_THEORY.md §2.1, AGENTS.md §2 (Transport content), src/tnfr/physics/structural_diffusion.py

Phase (φ, θ)

Code: G.nodes[n]['theta'], collect_theta_attr()
Symbol: (\theta) or (\phi)
Range: ([0, 2\pi)) or ([-\pi, \pi)) radians
What: Network synchrony parameter (relative timing)
Phase difference: (\Delta\theta = \theta_i - \theta_j)
API: Phase adaptation in dynamics
Math: FUNDAMENTAL_THEORY.md §2.2 (Structural Triad — phase)

Total Coherence (C(t))

Code: compute_coherence(G) → float ∈ [0,1]; per-node kernel structural_coherence(dnfr, depi)
Symbol: (C(t))
Formula: (C(t) = 1/(1 + \overline{|\Delta\text{NFR}|} + \overline{|d\text{EPI}|})) (canonical; derived from the nodal equation — equilibrium (\Delta\text{NFR}\to 0 \wedge d\text{EPI}\to 0 \Rightarrow C\to 1))
Range: ([0, 1]) where 1 = perfect coherence, 0 = total fragmentation
What: Global stability measure (recorded in history['C_steps']). Dual status: beyond a telemetry read-out, its per-node kernel structural_coherence (= (1/(1+|\Delta\text{NFR}|+|d\text{EPI}|))) is the single constitutive coherence map every domain reads (graph, arithmetic, chemical) — an NFR is a region of structural coherence, so (C) measures the coherence that defines NFR-hood. compute_coherence delegates to this kernel.
Thresholds: strong (C > \pi/(\pi+1) \approx 0.7585); fragmentation risk (C < 1/(\pi+1) \approx 0.2415) (the coherence band; π the sole structural scale).
Code: src/tnfr/metrics/common.py (compute_coherence, structural_coherence)
Math: FUNDAMENTAL_THEORY.md §5.1, AGENTS.md §7

Structural Equilibrium (ΔNFR = 0)

Code: is_structural_equilibrium(dnfr, depi=0, *, eps_dnfr, eps_depi) → bool
What: The canonical fixed-point predicate of the nodal equation: a node sits at the resonant-coherence attractor when (|\Delta\text{NFR}| \le) eps_dnfr and (|d\text{EPI}| \le) eps_depi (default EPS_DNFR_STABLE = 1e-3). This is the ONE deep invariant recurring fractally across TNFR: a relaxed graph node, a structural prime ((\Delta\text{NFR}\text{arith} = 0)) and a noble-gas element ((\Delta\text{NFR}\text{chem} = 0)) are the same fixed point under a domain-specific ΔNFR. The tolerance is a per-domain numerical scale (1e-12 for exact integer arithmetic), not a different logic.
Code: src/tnfr/metrics/common.py (is_structural_equilibrium)
Theory: AGENTS.md §2, §7

Coherence Operator (Ĉ)

Code: coherence_matrix(G) → (nodes, W)
Symbol: (\hat{C})
Matrix element: (w_{ij} \approx \langle i | \hat{C} | j \rangle)
Properties: Hermitian ((\hat{C}^\dagger = \hat{C})), positive semi-definite
What: Operator measuring structural stability between nodes
Math: src/tnfr/metrics/coherence.py (coherence_matrix)

Sense Index (Si)

Code: G.nodes[n]['Si'], ALIAS_SI, compute_Si_node()
Symbol: (\text{Si}) (global) or (S_i) (node i)
Formula: (\text{Si} = \alpha \cdot \nu_{f,\text{norm}} + \beta \cdot (1 - \text{disp}\theta) + \gamma \cdot (1 - |\Delta\text{NFR}|{\text{norm}}))
Range: ([0, 1^+]) typically, higher = more stable reorganization
What: Reorganization-capacity predictor. Unlike C(t), Si is a heuristic composite (weighted νf, phase sync, (|\Delta\text{NFR}|)) — predictive/diagnostic, not constitutive of NFR-hood. Si > 0.8 excellent; Si < 0.4 bifurcation-prone.
Weights: operational defaults (\alpha \approx 0.737), (\beta \approx 0.155), (\gamma_w \approx 0.114) (SI_WEIGHTS in config/defaults_core.py; free parameters, sum (\approx 1))
Math: Mathematical Foundations - Metrics

Phase Gradient (|∇φ|) - CANONICAL

Code: compute_phase_gradient(G) → Dict[NodeId, float]
Symbol: (|\nabla\phi|(i))
Formula: (|\nabla\phi|(i) = \text{mean}_{j \in N(i)} |\theta_i - \theta_j|) (circular mean)
What: Local phase desynchronization / stress proxy field
Status: CANONICAL (Nov 2025)
Physics: Captures dynamics C(t) misses due to scaling invariance
Threshold: Kinematic bound |∇φ| ≤ π (phase wrap — same as K_φ); ≈ 0.18 is only a heuristic early-warning level, NOT a derived bound (measured sync-onset ≈ 0.29, σ-dependent)
API: tnfr.physics.fields.compute_phase_gradient()
Usage: Stress detection, local instability prediction
Documentation: docs/STRUCTURAL_FIELDS_TETRAD.md

Phase Curvature (K_φ) - CANONICAL

Code: compute_phase_curvature(G) → Dict[NodeId, float]
Symbol: (K_\phi(i))
Formula: (K_\phi = \text{wrap_angle}(\phi_i - \text{circular_mean}(\text{neighbors})))
What: Phase torsion and geometric confinement field
Status: CANONICAL (Nov 2025)
Physics: Flags mutation-prone loci via geometric constraints
Threshold: Classical bound |K_φ| < 2.8274 (90% of π theoretical maximum)
API: tnfr.physics.fields.compute_phase_curvature()
Usage: Geometric confinement monitoring, bifurcation prediction
Documentation: docs/STRUCTURAL_FIELDS_TETRAD.md

Coherence Length (ξ_C) - CANONICAL

Code: estimate_coherence_length(G) → float
Symbol: (\xi_C)
Formula: Spatial correlation function (C(r) = A \exp(-r/\xi_C))
What: Spatial correlation scale of local coherence
Status: CANONICAL (Nov 2025)
Physics: Critical phenomena and finite-size scaling analysis
Thresholds:

• Critical: ξ_C > 1.0 × diameter (finite-size scaling dominates)
• Watch: ξ_C > π ≈ 3.14 × mean_distance (RG scaling)
• Stable: ξ_C < mean_distance (bulk behavior)
  API: tnfr.physics.fields.estimate_coherence_length()
  Usage: Critical point detection, correlation analysis
  Documentation: docs/STRUCTURAL_FIELDS_TETRAD.md

Structural Potential (Φ_s) - CANONICAL

Code: compute_structural_potential(G, alpha=2.0) → Dict[NodeId, float]
Symbol: (\Phi_s(i))
Formula: (\Phi_s(i) = \sum_{j \neq i} \frac{\Delta\text{NFR}_j}{d(i,j)^\alpha}) where (\alpha = 2)
What: Global structural potential field from ΔNFR distribution
Status: CANONICAL (Nov 2025)
Validation: 1,599 tests across 5 topologies
Physics: Passive equilibrium confinement landscape
Grammar: U6 STRUCTURAL POTENTIAL CONFINEMENT (Δ Φ_s < π/2 ≈ 1.571 confinement bound, half phase-wrap; ceiling 2.0 binary escape)
API: tnfr.physics.fields.compute_structural_potential()
Threshold: Per-node bound |Φ_s| < π/4 ≈ 0.785 (quarter phase-wrap; π-derived, tied to the one genuine structural scale π)
Documentation: docs/STRUCTURAL_FIELDS_TETRAD.md

• src/tnfr/physics/fields.py - Implementation

Interpretation:

• Φ_s minima = passive equilibrium states
• Δ Φ_s < π/2 ≈ 1.571 = confinement (safe regime, half phase-wrap)
• Δ Φ_s ≥ 2.0 = binary escape threshold (fragmentation risk)
• Valid sequences: Δ Φ_s ≈ 0.6 (≈ 38% of the π/2 bound)
• Violations: Δ Φ_s ≈ 3.9 (≈ 248% of the π/2 bound)

Mechanism: Grammar U1-U5 acts as passive confinement (NOT active attractor). Reduces escape drift by 85%.

The Nodal Equation

The fundamental equation of TNFR governs structural evolution:

[ \frac{\partial \text{EPI}}{\partial t} = \nu_f \cdot \Delta\text{NFR}(t) ]

Where:

• (\frac{\partial \text{EPI}}{\partial t}): Rate of change of structure
• (\nu_f): Structural frequency (reorganization rate) in Hz_str
• (\Delta\text{NFR}(t)): Reorganization gradient (driving pressure)

Interpretation:

• Structure changes only when both (\nu_f > 0) (capacity) and (\Delta\text{NFR} \neq 0) (pressure) exist
• Rate of change is proportional to both frequency and gradient
• When (\nu_f \to 0), evolution freezes (node collapse)
• When (\Delta\text{NFR} = 0), structure reaches equilibrium

Implementation: See src/tnfr/dynamics/ for numerical integration
Theory: Nodal equation §2

Structural Diffusion & Spectral Parameters

The EPI channel of ΔNFR makes the nodal equation a graph diffusion, so the dynamics are governed by the spectrum of the graph Laplacian. These are the factors that set how fast a network relaxes and toward what pattern.

• The conservative pulse. The substrate vibrates at ω_k = √λ_k; the ΔNFR = 0 equilibria are the beats. The grammar temporal windows are derived from this relaxation: with q = 1 − νf·dt·ρ (ρ = trace(L_rw)/N = 1, exact), the U4b / repeat window (= 3) is the relaxation time to the band 1/(π+1) and the U2 debt (= 2) is the relaxation absorption capacity ⌊1/(1−q)⌋ — a time and a capacity from the same q. No e (the canonical relaxation is the discrete geometric decay qⁿ, not the continuous exponential).
• λ₁ = 0 is the conserved uniform mode; the degree-weighted total Σ deg·EPI is invariant (degree_weighted_total). This EPI-channel conservation is distinct from the tetrad Noether charge Q = Σ(Φ_s + K_φ) (compute_noether_charge) — TNFR carries two conservation laws (EPI field vs tetrad fields).
• A fully relaxed network is one uniform NFR; differentiated nodal topology (radial/annular/multinodal) lives off-equilibrium.
• API: src/tnfr/physics/structural_diffusion.py, src/tnfr/physics/conservation.py.

Key Parameters & Factors

The quantities that govern TNFR dynamics, with their canonical status.

Only π is a genuine structural constant (the phase-wrap bound). φ, γ, e are not structural scales and no longer appear in the engine; every parameter other than π is derived from the nodal dynamics / spectral gap (e.g. ξ_C ∝ 1/√λ₂, the π-derived ΔΦ_s bound π/2) or is a free operational parameter (e.g. the ≈ 0.18 |∇φ| early-warning level).

Structural Operators

The 13 canonical operators are the only way to modify nodes in TNFR. They're not arbitrary functions—they're resonant transformations with rigorous physics.

For complete specifications with physics derivations, contracts, and usage examples, see AGENTS.md § The 13 Canonical Operators.

Quick Reference

Operator Composition

Operators combine into sequences that implement complex behaviors:

• Bootstrap = [Emission, Coupling, Coherence]
• Stabilize = [Coherence, Silence]
• Explore = [Dissonance, Mutation, Coherence]
• Propagate = [Resonance, Coupling]

Critical: All sequences must satisfy unified grammar (U1-U6).

API:

• tnfr.structural.<OperatorName>() - Individual operators
• run_sequence(G, node, ops) - Execute operator sequences
• validate_sequence(ops) - Check grammar compliance

Grammar: See UNIFIED_GRAMMAR_RULES.md for complete rules
Detailed Specs: See AGENTS.md § The 13 Canonical Operators
Math: Mathematical Foundations

Canonical Invariants (Optimized Set)

From AGENTS.md - Optimized from 10 to 6 invariants based on mathematical derivation:

1. Nodal Equation Integrity: EPI evolution only via ∂EPI/∂t = νf · ΔNFR(t)
2. Phase-Coherent Coupling: |φᵢ - φⱼ| ≤ Δφ_max required for resonant operations
3. Multi-Scale Fractality: Operational fractality and nested EPIs maintained
4. Grammar Compliance: All operator sequences must satisfy U1-U6 validation
5. Structural Metrology: Units consistency (νf in Hz_str) and telemetry exposure
6. Reproducible Dynamics: Deterministic evolution with seed-based control

Quick Reference Tables

Variable Summary

Common API Patterns

# Access node attributes
epi = G.nodes[node_id]['EPI']
vf = G.nodes[node_id]['vf']
theta = G.nodes[node_id]['theta']

# Compute metrics
C_t = compute_coherence(G)
nodes, W = coherence_matrix(G)
Si = compute_Si_node(G, node_id)

# Apply operators
from tnfr.structural import Emission, Coherence, Resonance
run_sequence(G, node_id, [Emission(), Coherence(), Resonance()])

# Evolution step
from tnfr.dynamics import step
step(G, use_Si=True, apply_glyphs=True)

# Canonical fixed point (per-node kernel + equilibrium predicate)
from tnfr.metrics.common import structural_coherence, is_structural_equilibrium
C_node = structural_coherence(G.nodes[node_id]['dnfr'])
at_equilibrium = is_structural_equilibrium(G.nodes[node_id]['dnfr'])

# Whole-NFR read-out (region) + nodal topology (radial/annular/multinodal)
from tnfr.sdk import TNFR
net = TNFR.create(20).ring().evolve(5)
nfr = net.nfr()  # topology, centers, coherence, equilibrium_fraction, coherence_length

Telemetry & Traces

Expose in telemetry:

• C(t) - Total coherence
• νf per node - Structural frequency
• phase per node - Synchrony state
• Si per node/network - Sense index
• ΔNFR per node - Reorganization gradient
• Operator history - Applied transformations
• Events - Birth, bifurcation, collapse

API: tnfr.utils.callback_manager, history tracking in G.graph['_hist']

Domain Neutrality & the Two-Layer Ontology

TNFR is domain-neutral: the structural operators apply to any graph-coupled network, with no built-in domain assumptions. The genuinely canonical invariant — derivable directly from the nodal equation — is the fixed point ΔNFR = 0 (structural equilibrium / resonant-coherence attractor); every domain realizes its own ΔNFR but shares this one fixed point, read through the single kernel structural_coherence and the predicate is_structural_equilibrium. Around that invariant the read-outs span a spectrum of emergence, contrasted as two layers:

• Physical layer — particles (direct emergence): the integer winding W ∈ ℤ is a topological invariant of the phase field — its integrality emerges (any single-valued phase field on a loop has integer winding; nothing imposes it). The class (boson/fermion/composite) is an output of measuring W, not an imposed label (tnfr.physics.emergent_particles).
• Symbolic / informational layer — numbers, chemistry (projection of the same fixed point): a structural prime is ΔNFR_arith = 0 and a noble gas is ΔNFR_chem = 0. The per-node arithmetic/chemical ΔNFR consumes its domain data (divisibility τ/σ/ω; the aufbau order) — the informational shadow of the structural grammar, not a direct topological emergence.

Refinement — emergence is a spectrum, not a clean binary. The symbolic layer is not uniformly "consuming". Per TNFR_NUMBER_THEORY.md §9.5, primality has three sectors: A (arithmetic ΔNFR = 0, what the SDK primes()/primality() expose) is an exact but circular re-expression that consumes Ω/τ/σ; B (spectral — the Paley/residue Fiedler gap, input only x² mod n) is genuinely emergent (primes-OUT, non-circular); C (representation-theoretic irreducibility) is refuted. So numbers do carry a non-circular emergent core, but it is spectral, partial (the real spectrum reaches only n ≡ 1 (mod 4)), and never lives in the per-node substrate — the residual is the same Fix(G)^⊥ obstruction as the paused TNFR-Riemann program. The two-layer split is the SDK-level contrast; the trichotomy is the precise account.

Only the equilibrium criterion (is_structural_equilibrium) and the coherence kernel (structural_coherence) are shared; each domain realizes its own ΔNFR.

Guideline: avoid domain-specific hard-coding in the core engine; be honest about the emergence spectrum (direct topological / spectral-emergent / divisibility-consuming) per read-out.

Reproducibility

All simulations must be:

1. Seeded: Explicit RNG seeds
2. Traceable: Log operators, parameters, states
3. Deterministic: Same seed → same trajectory

Tools: RNG scaffolding, structural history, telemetry caches

Unified Grammar Terms

Unified Grammar

The consolidated TNFR grammar system (U1-U6) that replaces the old C1-C3 and RC1-RC4 systems.

Source of Truth: UNIFIED_GRAMMAR_RULES.md
Quick Reference: AGENTS.md § Unified Grammar (U1-U6)
Implementation: src/tnfr/operators/grammar.py

Grammar Completeness: The canonical TNFR grammar consists of exactly six rules (U1-U6) and is COMPLETE. No additional rules (U7, U8, etc.) are required or planned. Extended dynamics (flux fields) add telemetry, not prescriptive constraints.

Six Canonical Constraints:

Canonicity Levels:

• ABSOLUTE: Mathematical necessity (direct consequence of nodal equation)
• STRONG: Strong empirical/theoretical support (1,599 tests for U6)

Recent Updates:

• U5 added 2025-11-10 (hierarchical REMESH stabilization)
• U6 promoted to canonical 2025-11-11 (Φ_s field validation complete)
  • Replaces experimental "Temporal Ordering" research proposal
  • Validated across 5 topologies: ring, scale_free, small-world, tree, grid
  • Correlation: corr(Δ Φ_s, ΔC) = -0.822 (R² ≈ 0.68)
• 2025-11-15: Grammar declared COMPLETE (U1-U6) - no U7/U8 required

Not Part of Grammar (telemetry/dynamics, NOT rules):

• Structural Field Hexad: Tetrad (Φ_s, |∇φ|, K_φ, ξ_C) + Flux Pair (J_φ, ∇·J_ΔNFR)
• "Proposed U7": Historical research direction (Temporal Ordering) - NOT canonical, NOT implemented

See Also:

• UNIFIED_GRAMMAR_RULES.md - Complete derivations from physics
• AGENTS.md § Unified Grammar - Quick reference
• docs/grammar/U6_STRUCTURAL_POTENTIAL_CONFINEMENT.md - U6 complete specification
• docs/grammar/U6_STRUCTURAL_FIELD_TETRAD.md - Why no U7/U8
• STRUCTURAL_FIELDS_TETRAD.md - U6 validation details
• src/tnfr/physics/fields.py - Φ_s implementation

Generator Operator

Operator that can create EPI from null/dormant states.

Set: GENERATORS = {emission, transition, recursivity}

Physics: Only these operators can initialize when EPI=0

Grammar Rule: U1a (STRUCTURAL INITIATION)

See: UNIFIED_GRAMMAR_RULES.md § U1a

Closure Operator

Operator that leaves system in coherent attractor state.

Set: CLOSURES = {silence, transition, recursivity, dissonance}

Physics: Terminal states preserving coherence

Grammar Rule: U1b (STRUCTURAL CLOSURE)

See: UNIFIED_GRAMMAR_RULES.md § U1b

Stabilizer Operator

Operator that provides negative feedback for convergence.

Set: STABILIZERS = {coherence, self_organization}

Physics: Ensures ∫νf·ΔNFR dt converges (bounded evolution)

Grammar Rule: U2 (CONVERGENCE & BOUNDEDNESS)

See: UNIFIED_GRAMMAR_RULES.md § U2

Destabilizer Operator

Operator that increases |ΔNFR| through positive feedback.

Set: DESTABILIZERS = {dissonance, mutation, expansion}

Physics: Without stabilizers, leads to divergence

Grammar Rule: U2 (CONVERGENCE & BOUNDEDNESS)

See: UNIFIED_GRAMMAR_RULES.md § U2

Coupling/Resonance Operator

Operators that require phase verification for valid coupling.

Set: COUPLING_RESONANCE = {coupling, resonance}

Physics: Resonance requires |φᵢ - φⱼ| ≤ Δφ_max

Grammar Rule: U3 (RESONANT COUPLING)

See: UNIFIED_GRAMMAR_RULES.md § U3

The Structural-Field Tetrad

Theory: The four structural fields are the minimal derivative tower (DERIVED). Only π is a genuine structural scale (the phase-wrap bound shared by |∇φ| and K_φ).

The one structural scale

Only π is a genuine structural scale — it bounds the phase sector (|∇φ| ≤ π and |K_φ| ≤ π). φ, γ, e are not structural scales and no longer appear in the engine; the coherence length is set by the spectral gap (ξ_C ∝ 1/√λ₂) and the Φ_s confinement bound is π-derived. Every other parameter is derived from the nodal dynamics or is a free operational parameter.

Structural Fields and their bounds

1. Φ_s (0th order): π-derived confinement Δ Φ_s < π/2 ≈ 1.571 (half phase-wrap; per-node |Φ_s| < π/4 ≈ 0.785)
2. |∇φ| (1st order): bound |∇φ| ≤ π (phase wrap); γ/π ≈ 0.184 is a heuristic early-warning only
3. K_φ (2nd order): bound |K_φ| < 0.9×π ≈ 2.827 (phase wrap — GENUINE); K_φ = L_rw·φ
4. ξ_C (correlation): scale set by the spectral gap, ξ_C ∝ 1/√λ₂ (not base e)

Documentation: Structural-field tetrad

Flux Fields & the Emergent Symplectic Substrate

The tetrad has two conjugate flux fields that complete it into a symplectic structure (the field hexad). They are the currents paired with the static fields.

• Phase current J_φ — geometric, phase-driven transport; conjugate to curvature K_φ. Compute: compute_phase_current(G).
• ΔNFR flux J_ΔNFR — potential-driven reorganization transport; conjugate to the potential Φ_s. Compute: compute_dnfr_flux(G).
• API: tnfr.physics.extended (compute_phase_current, compute_dnfr_flux).

Emergent symplectic substrate. The dynamics generate a symplectic phase space \(\mathbb{R}^{4N}\) with two canonical conjugate pairs per node — geometric \((K_\phi, J_\phi)\) and potential \((\Phi_s, J_{\Delta\text{NFR}})\) — with brackets \(\{K_\phi, J_\phi\} = \{\Phi_s, J_{\Delta\text{NFR}}\} = 1\) and Hamiltonian \(H_\text{sub} = \tfrac{1}{2}\sum(K_\phi^2 + J_\phi^2 + \Phi_s^2 + J_{\Delta\text{NFR}}^2)\). The flow is a symplectomorphism (Liouville: phase volume preserved), so the 13 operators are canonical volume-preserving transforms. The complex coordinate \(\Psi = K_\phi + i\,J_\phi\) carries a U(1) gauge symmetry; the substrate further carries a U(2) polarization symmetry (per-node Poincaré sphere, classical Stokes texture — not a quantum state). The nodal equation is the overdamped projection of this Hamiltonian flow.

API: tnfr.physics.symplectic_substrate, Network.symplectic_substrate().
Documentation: AGENTS.md §4 (Emergent geometry), src/tnfr/physics/symplectic_substrate.py

Bifurcation Trigger

Operators that may trigger phase transitions.

Set: BIFURCATION_TRIGGERS = {dissonance, mutation}

Physics: ZHIR (Mutation) is the canonical bifurcation operator — it transforms θ when the structural change rate crosses the mutation threshold, dEPI/dt > ξ.

Grammar Rule: U4a (requires handlers)

See: UNIFIED_GRAMMAR_RULES.md § U4a

Bifurcation Handler

Operators that manage structural reorganization during bifurcations.

Set: BIFURCATION_HANDLERS = {self_organization, coherence}

Physics: Provide stability during phase transitions

Grammar Rule: U4a (BIFURCATION DYNAMICS)

See: UNIFIED_GRAMMAR_RULES.md § U4a

Transformer Operator

Operators that perform threshold-crossing structural phase transitions.

Set: TRANSFORMERS = {mutation, self_organization}

Physics: Require recent destabilizer for threshold energy

Grammar Rule: U4b (requires context + prior IL for ZHIR)

See: UNIFIED_GRAMMAR_RULES.md § U4b

Related Documentation

Core References (Essential)

• AGENTS.md ⭐ - Single source of truth for TNFR agent guidance, invariants, and philosophy
• UNIFIED_GRAMMAR_RULES.md ⭐ - Grammar single source of truth (U1-U6 complete derivations)
• Mathematical Foundations ⭐ - SINGLE SOURCE FOR ALL MATH (formalization, proofs, spectral theory)

Theory & Physics

• TNFR.pdf - Original theoretical companion (paradigm, nodal equation, foundational physics)
• docs/grammar/U6_STRUCTURAL_POTENTIAL_CONFINEMENT.md - U6 complete specification
• STRUCTURAL_FIELDS_TETRAD.md - Structural fields validation (Φ_s, phase gradients)

Implementation & API

• ARCHITECTURE.md - System design and architecture patterns
• Foundations - Runtime/API guide
• API Overview - Package architecture
• Structural Operators - Operator implementation details
• Examples - Runnable scenarios across domains

Grammar & Migration

• docs/grammar/ - Grammar documentation directory (U6, fundamental concepts, etc.)

Testing & Development

• TESTING.md - Test conventions and invariant verification
• CONTRIBUTING.md - Detailed contribution guidelines
• REPRODUCIBILITY.md - Determinism requirements

Cross-References and Documentation Hub

Primary Sources:

• AGENTS.md - Single source of truth for TNFR theory
• UNIFIED_GRAMMAR_RULES.md - Complete U1-U6 grammar derivations
• Structural Fields and the Tetrad - Mathematical foundations

Implementation References:

• src/tnfr/physics/fields.py - Unified Structural Field Tetrad (Canonical)
• src/tnfr/dynamics/self_optimizing_engine.py - Self-optimization & auto-optimization
• docs/STRUCTURAL_FIELDS_TETRAD.md - Technical field specifications
• docs/grammar/PHYSICS_VERIFICATION.md - Grammar physics verification

Development Resources:

• src/tnfr/sdk/ - Simplified & Fluent API
• examples/ - Complete tutorial suite
• ARCHITECTURE.md - System design patterns

Molecular Chemistry from TNFR

Technical approach: chemistry on atomic-scale graph networks — the symbolic-layer read-out of the nodal fixed point. A closed shell is ΔNFR_chem(Z) = 0 (is_structural_equilibrium), the chemical mirror of the primality criterion; the magic numbers (noble-gas Z) combine genuinely-emergent subshell capacities 2l+1 (manifold eigenmode degeneracies) with an assumed (n+l) aufbau order. See src/tnfr/physics/emergent_chemistry.py (classify_element, emergent_magic_numbers) and the SDK TNFR.element(Z) / TNFR.magic_numbers().

Element Signatures

Code: tnfr.physics.signatures
What: Structural field-based classification of coherent patterns
Metrics: ξ_C, |∇φ|, |K_φ|, ΔΦ_s drift, stability classification
API: compute_element_signature(G), compute_au_like_signature(G)
Physics: Elements as coherent attractors in structural space

Au-like Patterns

Symbol: Au (from Latin 'aurum')
What: Complex coherent patterns exhibiting metallic properties
Criteria: Extended ξ_C, phase synchrony (|∇φ| < 2.0), evolution stability
Detection: compute_au_like_signature()["is_au_like"]
Physics: Optimal multi-scale coordination under nodal dynamics

Chemical Bonds (TNFR Structural Analogy)

Traditional: Force between atoms
TNFR: Phase synchronization with U3 verification: |φᵢ - φⱼ| ≤ Δφ_max
API: Coupling operators with phase compatibility check
Strength: Determined by phase coherence and coupling stability

Chemical Reactions (TNFR Structural Analogy)

Traditional: Collision/transition state theory
TNFR: Operator sequences: [Dissonance→Mutation→Coupling→Coherence]
Grammar: Must satisfy U1-U6 constraints
API: Sequence validation via grammar.py
Example: Bond formation = [OZ, ZHIR, UM, IL] sequence

Molecular Geometry (TNFR Structural Analogy)

Traditional: VSEPR, orbital hybridization
TNFR: ΔNFR minimization in coupled network topology
Prediction: Stable configurations minimize reorganization pressure
API: Network topology analysis after coupling sequences

Implementation: src/tnfr/physics/emergent_chemistry.py, src/tnfr/physics/signatures.py

Self-Optimizing Engine

Self-Optimization: The TNFR engine includes self-optimization capabilities using unified field telemetry.

Core Components

TNFRSelfOptimizingEngine: src/tnfr/dynamics/self_optimizing_engine.py
Purpose: Closes feedback loop via unified field monitoring
Monitors: Complex Geometric Field (Ψ), Chirality (χ), Symmetry Breaking (𝒮), Coherence Coupling (𝒞)
Detects: Inefficiencies via tensor invariants (Energy Density ℰ, Topological Charge 𝒬)
Usage: engine = TNFRSelfOptimizingEngine(G); success, metrics = engine.step(node_id)

Auto-Optimization API

Fluent Integration: TNFRNetwork(G).focus(node).auto_optimize().execute()
Field Analysis: analyze_optimization_potential(G) - Mathematical structure analysis
Strategy Recommendations: recommend_field_optimization_strategy(G) - Optimization strategies
Automatic Execution: auto_optimize_field_computation(G) - Self-optimizing computation

Unified Field Framework (Nov 2025)

Mathematical Unification: Discovery of complex field relationships and conservation laws.

Complex Geometric Field (Ψ)

Definition: Ψ = K_φ + i·J_φ (unifies geometry + transport)
Evidence: r(K_φ, J_φ) = -0.854 to -0.997 (near-perfect anticorrelation)
API: compute_complex_geometric_field(G)
Usage: Unified geometry-transport analysis

Emergent Fields

Chirality (χ): χ = |∇φ|·K_φ - J_φ·J_ΔNFR - Handedness detection
Symmetry Breaking (𝒮): Phase transition indicator
Coherence Coupling (𝒞): Multi-scale connector field
API: compute_emergent_fields(G)

Tensor Invariants

Energy Density (ℰ): ℰ = Φ_s² + |∇φ|² + K_φ² + J_φ² + J_ΔNFR²
Topological Charge (𝒬): 𝒬 = |∇φ|·J_φ - K_φ·J_ΔNFR
Conservation Law: ∂ρ/∂t + ∇·𝐉 = S_grammar where S_grammar → 0 under U1-U6
API: compute_tensor_invariants(G)

Unified Telemetry: compute_unified_telemetry(G) - Complete dual-face field suite: the dissipative read-out (canonical tetrad + coherence, relaxes to the ΔNFR=0 attractor) plus the conservative pulse (the resonant rhythm ω_k=√λ_k, dominant beat, vibration energy; does not saturate)

Operator-Tetrad Synergies (Experimental, March 2026)

Six experimentally validated results connecting canonical operators to the structural field tetrad. Reference: STRUCTURAL_OPERATORS.md §17, examples 37-39.

Dual-Lever Structure (channel partition)

What: Each operator's primary effect lands on exactly one nodal channel — the partition is simultaneously the dual-lever (capacity νf vs pressure ΔNFR), the tetrad driver and the number-theory grading (AGENTS.md §5):

• νf (capacity): Silence (SHA), Expansion (VAL), Contraction (NUL).
• ΔNFR (pressure): Coherence (IL), Dissonance (OZ), Self-organization (THOL), Transition (NAV).
• θ (phase): Coupling (UM), Mutation (ZHIR).
• EPI (written directly): Emission (AL), Reception (EN), Resonance (RA), Recursivity (REMESH).

Source of truth: src/tnfr/operators/operator_contracts.py. Evidence: examples/02_physics_regimes/39_nodal_equation_decomposition.py.

Operator-Tetrad Fingerprint Matrix

What: Each operator produces a unique signature across the four tetrad fields. The fingerprint matrix tabulates relative changes (dPhi_s, d|grad_phi|, dK_phi, dxi_C) per operator. Example: UM modifies all four fields (strongest Phi_s at -73.7%); SHA is tetrad-neutral. Evidence: examples/02_physics_regimes/37_operator_tetrad_synergy.py.

IL-OZ Tetrad Symmetry

What: Coherence (IL) and Dissonance (OZ) produce identical tetrad perturbation magnitudes and identical energy changes (dE = -0.011) despite opposite physics. They share the same |d(DNFR)| = 0.0096, differing only in sign. Interpretation: IL and OZ are structural mirrors on the energy manifold, explaining U2 balancing.

Linear Response of Phi_s

What: Phi_s responds linearly to DNFR perturbations with |r| = 1.000 (Pearson correlation). This confirms its 0th-order position in the operator-derivative tower. Contrast: xi_C transitions to strongly nonlinear behaviour above DNFR ~ 0.3. Evidence: examples/02_physics_regimes/39_nodal_equation_decomposition.py.

Complete Causal Chain

What: The information flow is strictly unidirectional: Operator -> (vf, DNFR) -> dEPI/dt -> Tetrad -> (E, Q). Tetrad fields are diagnostic outputs, not independent dynamical variables. They are fully determined by the nodal equation state.

Grammar-Energy Landscape

What: Lyapunov contractivity (cumulative multiplier Pi < 1) is sufficient but not necessary for energy descent. Experimentally: lambda = 1.288 (non-contractive) yet net dE = -9.59 (energy descent). Interpretation: The Lyapunov bound is conservative; actual grammar-compliant sequences may descend more steeply than the bound predicts. Evidence: examples/02_physics_regimes/38_grammar_energy_landscape.py.

Structural Conservation Theorem

Main Result: Grammar symmetry (U1-U6) implies an approximate Noether-like structural conservation law.

Charge Density: ρ = Φ_s + K_φ (potential + geometric sectors)
Current: 𝐉 = (J_φ, J_ΔNFR) (transport channels)
Conservation: ∂ρ/∂t + ∇·𝐉 = S_grammar where S_grammar → 0 under U1-U6
Two Sectors: Potential (Φ_s ↔ J_ΔNFR) and Geometric (K_φ ↔ J_φ), coupled through Ψ = K_φ + i·J_φ
Lyapunov: E = ½Σ(Φ_s² + |∇φ|² + K_φ² + J_φ² + J_ΔNFR²) ≥ 0 with dE/dt ≤ 0 observed under grammar (proof sketch; complete proof open)
Validation: 62 tests, charge drift < 0.03% across tested topologies and seeds
API: tnfr.physics.conservation — Noether charge Q, energy functional E, Ward identities, spectral decomposition
Documentation: theory/STRUCTURAL_CONSERVATION_THEOREM.md

Integrity Monitor

What: Closed-loop postcondition verification ensuring all 13 canonical operators satisfy their structural contracts after execution.
API: tnfr.physics.integrity
Coverage: 13/13 operators verified — monotonicity (IL), ΔNFR increase (OZ), phase preservation (SHA), EPI creation (AL), coupling validity (UM/RA), etc.
Usage: Automatic contract verification in apply_glyph_with_grammar() pipeline.
Documentation: src/tnfr/physics/integrity.py

Grammar-Aware Dynamics

What: Incremental U1-U6 enforcement during step-by-step operator selection, bridging grammar validation with the dynamic operator selection layer.
API: tnfr.operators.grammar_dynamics.GrammarAwareDynamics
Checks: U1a initiation, U2 destabilizer/stabilizer debt tracking, U3 phase compatibility for UM/RA, U4a/U4b bifurcation context
Physics: Proactive enforcement prevents grammar violations before they corrupt graph state.
Documentation: src/tnfr/operators/grammar_dynamics.py

Grammar Application

What: Pre-validated, grammar-enforced operator application at runtime.
API: tnfr.operators.grammar_application.apply_glyph_with_grammar()
Pipeline: Grammar check → operator application → postcondition verification (integrity monitor)
Physics: Ensures every structural mutation passes U1-U6 before modifying graph state.
Documentation: src/tnfr/operators/grammar_application.py

Per-Node Φ_s Threshold

Value: PHI_S_VON_KOCH_THRESHOLD = π/4 ≈ 0.785
What: Per-node safety threshold for structural potential |Φ_s|.
Derivation: π-derived — a quarter phase-wrap, tying the bound to the one genuine structural scale (π). The constant name retains "VON_KOCH" for code-compatibility only; there is no golden-ratio or von-Koch content (the earlier empirical 0.7711 / Γ(4/3)/Γ(1/3) framing is superseded).
Usage: |Φ_s(i)| < π/4 ≈ 0.785 indicates safe per-node structural potential.
API: tnfr.constants.canonical.PHI_S_VON_KOCH_THRESHOLD
Relation to U6: Part of three-tier Φ_s monitoring: π/4 ≈ 0.785 (per-node) → π/2 ≈ 1.571 (drift confinement, half phase-wrap) → 2.0 (escape ceiling).

Regime Correspondences

Theory: The single nodal dynamics produces two empirically-anchored regimes. The external labels "classical"/"quantum-like" are comparisons only, not TNFR primitives.

Smooth-Trajectory / Overdamped-Drift Regime (High Coherence)

Condition: C(t) → 1, |∇φ| → 0
Correspondence: first order in time, q̇ = νf·F — drift velocity ∝ force, so νf is mobility (Stokes/Einstein), not inverse mass; F = ΔNFR (force ↔ structural pressure). The inertial (second-order) regime lives in the conservative symplectic substrate, not here.
API: tnfr.physics.structural_diffusion, tnfr.physics.classical_mechanics

Discrete-Mode Regime (High Dissonance)

Condition: |∇φ| ~ π, near phase singularities
Emergent: on a bounded graph the diffusion operator has a discrete spectrum of orthonormal standing-wave eigenmodes (vibrating-string / Chladni analogue), with nodal-domain ordering (Courant); uncertainty (Fourier ΔEPI·Δνf ≥ K), superposition.
API: tnfr.physics.structural_diffusion, tnfr.physics.quantum_mechanics

TNFR-Riemann Program

What: A structural attack surface relating discrete TNFR operators to the Riemann zeta function. Not a proof of RH — the bridge is the open conjecture T-HP (gap G4).
Core Operator: H^(k)(σ) = L_k + V_σ where L_k = graph Laplacian, V_σ = diagonal potential
Numerical result: the critical parameter σ_c^(k) → 1/2 as k → ∞, verified across topologies (σ_c^(k) = 1/2 + O(log⁻¹ k)).
Honest scope: TNFR-internal structural results and numerical evidence; no classical open problem is closed.
Implementation: src/tnfr/riemann/
Documentation: theory/TNFR_RIEMANN_RESEARCH_NOTES.md

Key Canonical Thresholds

Quick reference for canonical threshold values from src/tnfr/constants/canonical.py:

Contributing Guidelines

When adding new functionality:

1. Verify theoretical foundation: Align with AGENTS.md physics
2. Preserve canonical invariants: Follow optimized 6-invariant set
3. Use established terminology: Reference this glossary for consistency
4. Map to canonical operators: All functions must correspond to 13 canonical operators
5. Validate grammar compliance: Ensure U1-U6 satisfaction
6. Maintain English-only policy: All documentation in English for canonical terminology
7. Write comprehensive tests: Cover invariants and operator contracts

Development Workflow:

1. Read AGENTS.md completely - SINGLE SOURCE OF TRUTH
2. Study UNIFIED_GRAMMAR_RULES.md for physics foundations
3. Follow CONTRIBUTING.md for detailed guidelines
4. Test with TESTING.md requirements

Version: 0.0.3.5 (June 2026)
Status: Canonical operational reference, aligned with the current engine, AGENTS.md and TNFR.pdf
Language: English only (canonical documentation policy)