The Structural-Field Tetrad as the Minimal Basis

Status: Canonical — the tetrad of fields is the DERIVED minimal basis; only π is a genuine structural scale Foundation: the nodal equation ∂EPI/∂t = νf·ΔNFR(t) Prerequisites: MINIMAL_STRUCTURAL_DEGREES.md, FUNDAMENTAL_THEORY.md §4

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1. Statement

The state of any coherent system on a graph is read through four structural fields — the structural-field tetrad. These four fields are the four orders of the discrete derivative tower over the graph, and they are the minimal complete structural read-out of a scalar phase field coupled to a scalar source. The minimality is derived (see MINIMAL_STRUCTURAL_DEGREES.md).

Field	Symbol	Tower order	Genuine structural scale
Structural potential	Φ_s	0th — global aggregation	π-derived confinement (π/4 per-node, π/2 drift)
Phase gradient	|∇φ|	1st — local derivative	π (phase-wrap bound)
Phase curvature	K_φ	2nd — discrete Laplacian	π (phase-wrap bound); K_φ = L_rw·φ
Coherence length	ξ_C	non-local — correlation	Spectral gap, ξ_C ∝ 1/√λ₂

The one genuine structural constant is π: both phase derivatives are wrapped angles, so |∇φ| ≤ π and |K_φ| ≤ π for any configuration. The Φ_s confinement bound is π-derived (per-node π/4 ≈ 0.785, drift π/2 ≈ 1.571 — quarter / half phase-wrap), and the coherence length is set by the spectral gap λ₂ (ξ_C ∝ 1/√λ₂). φ, γ, e are not structural scales and no longer appear in the engine; everything other than π is derived from the nodal dynamics or is a free operational parameter.

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2. The one structural scale: π

Within TNFR only π carries a genuine structural role — it is the phase-wrap bound of the phase sector. Both |∇φ| and K_φ are means of wrapped angles, so each is ≤ π; π is the period of e^{ix} (angular closure), and that geometric closure is exactly what bounds the two phase derivatives.

φ, γ, e are not structural scales of the tetrad fields and no longer appear in the engine: every parameter other than π is either derived from the nodal dynamics / spectral gap or is a free operational parameter. In particular the coherence length is set by the spectral gap (ξ_C ∝ 1/√λ₂) and the Φ_s confinement bound is π-derived (quarter / half phase-wrap).

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3. The field scales

• π — genuine (geometric, exact). |∇φ| ≤ π and |K_φ| ≤ π for any configuration; |K_φ| < 0.9·π ≈ 2.827 is the operational safety bound. π is the one constant that scales the whole phase sector.
• ξ_C — set by the spectral gap. The correlation length is set by the Fiedler value: ξ_C ∝ 1/√λ₂. The structural content is the spectral gap.
• Φ_s — π-derived. The confinement bounds are tied to the π phase scale: per-node |Φ_s| < π/4 ≈ 0.785 (quarter phase-wrap) and drift ΔΦ_s < π/2 ≈ 1.571 (half phase-wrap).
• |∇φ| onset — heuristic. The synchronization onset is a measured ≈ 0.29 and σ-dependent; a fixed ≈ 0.18 level is retained only as a heuristic early-warning level, not a derived bound.

The field computations themselves (compute_structural_potential, compute_phase_gradient, compute_phase_curvature, estimate_coherence_length) read these scales directly from the graph and the nodal equation.

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4. References

• Minimality of the tetrad: MINIMAL_STRUCTURAL_DEGREES.md
• Field definitions and scales: FUNDAMENTAL_THEORY.md §4, STRUCTURAL_FIELDS_TETRAD.md
• Implementation: src/tnfr/physics/fields.py, src/tnfr/physics/canonical.py, src/tnfr/constants/canonical.py