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Structural Tetrad: Diagnostic Degrees and Reconstruction Scope

Status: Four canonical diagnostic channels. Universal minimality and complete state reconstruction are open questions with explicit hypotheses required. Definitions are centralized in the field guide; proofs and counterexamples are centralized in the mathematical scope note.

1. Statement

The structural tetrad (Φ_s, |∇φ|, K_φ, ξ_C) is the engine's selected readout of pressure accumulation, local phase stress, local circular curvature, and spatial correlation. All four fields retain canonical status.

The selection is a diagnostic design. It does not establish that every scalar observable is a function of four reported quantities, or that the tetrad determines a full graph state and its evolution under

∂EPI/∂t = ν_f · ΔNFR.

A completeness claim must specify a state space, admissible observables, gauge identifications, and what information is retained by the readout. A minimality claim must additionally prove that no selected channel is redundant for that specified task. These stronger claims are not established by naming four structural questions or by computing the fields on a collection of graphs.

2. Four diagnostic questions

Question Field Construction
How much pressure accumulates from other nodes? Φ_s Distance-weighted aggregation of ΔNFR
How misaligned is a node with its neighbors? |∇φ| Mean magnitude of wrapped phase differences
How does its phase differ from the neighborhood mean? K_φ Wrapped deviation from the circular mean
At what spatial scale does coherence correlate? ξ_C Correlation-decay fit with spectral fallback

This organizes local phase information and non-local aggregation/correlation. Φ_s is a per-node value with global dependence; it is not a purely local quantity. ξ_C compresses a spatial correlation profile into a length. The table does not exhaust every possible graph observable.

3. The operator-derivative tower

3.1 Construction

ΔNFR → distance-kernel aggregation → Φ_s
phase → wrapped neighbor differences → |∇φ|
phase → circular-neighborhood deviation → K_φ
local coherence → spatial correlation → ξ_C

The first two phase operations are related to first and second derivatives. For an undirected graph, an oriented incidence matrix gives an edge gradient, and composing it with its weighted adjoint gives the graph Laplacian. For small phase spread within a consistent branch and matching neighbor weights, circular curvature approaches L_rw φ. The equality is a linearized description; the exact implemented field uses circular means and wrapping.

3.2 Algebraic generation does not terminate information at second order

Higher Laplacian powers are compositions of the same operator. This fact does not mean their actions are linearly dependent or recoverable from lossy pointwise summaries. If L has at least four distinct eigenvalues, then I, L, L², and L³ are linearly independent. Otherwise a polynomial of degree at most three would vanish at four distinct eigenvalues.

The finite-dimensional cutoff is controlled by L's minimal polynomial. It depends on the graph and is not universally second order. In addition, |∇φ| discards signs and ξ_C discards most of the correlation profile. A generating set of operators and a sufficient set of state statistics are different mathematical objects.

3.3 Dynamical non-reconstruction

At fixed graph, phase, EPI, and pure-EPI pressure, a uniform rescaling of ν_f leaves the tetrad unchanged while scaling ∂EPI/∂t = ν_f ΔNFR. Consequently the tetrad alone does not determine the full dynamical state. The structural triad, frequency, pressure law, and graph remain part of the model.

3.4 Conditional reconstruction in the pure EPI channel

The negative result above identifies capacity as a necessary predictive model parameter. A graph-specific positive result is also available when the complete nodal structural-potential field is retained.

For fixed connected reciprocal nonnegative conductance with positive nodal strengths and fixed positive capacities, let K be the canonical inverse-square shortest-path kernel on its declared distance domain. With unit pure-EPI channel coefficient, pressure gives

Phi_s = -K L_rw x,    x = EPI.

Write O=-K L_rw. Since L_rw 1=0, uniform EPI shifts are always invisible to Phi_s. If rank(O)=N-1, this is the only invisible EPI direction. Appending the fixed-capacity conserved mean

m(x) = sum_i (d_i/nu_i)x_i / sum_i(d_i/nu_i)

makes [O; m] injective and hence reconstructs absolute EPI in exact arithmetic. Together with the capacity vector and known graph, this determines the future pure-diffusion trajectory. Thus, for this restricted algebraic task, the full potential field plus one zero-mode scalar is sufficient for form reconstruction, while capacity remains necessary to determine the clock/generator. Numerical promotion additionally requires scale-aware rank and residual checks.

epi_diffusion_reconstruction_certificate computes the potential and augmented ranks with separate scale-aware SVD thresholds, then requires a small relative reconstruction residual before it reports numerical success. The exhaustive NetworkX graph atlas through six nodes gives rank(O)=N-1 on all 142 connected simple graphs in that finite domain. This is a measured finite-domain result, not a proof for every graph, weight or structural-potential kernel.

Algebraic rank is also weaker than robust reconstruction. On a four-node weighted star with edge weights [10^-3, 10^3, 10^3], the rank remains N-1 but the nonzero singular-value condition number exceeds 10^10. Thus an exact observer can be practically unstable under extreme geometry. Uniformly scaling all conductances to 10^-8 or 10^8 also preserves the structural nullity but can make the unscaled augmented observer numerically singular. The certificate reports its thresholds, condition number and relative residual so downstream inverse methods can distinguish algebraic identifiability from numerical resolution.

Global tetrad summaries remain insufficient because they discard the nodal potential field. Changing capacity ratios also removes the fixed conserved mean, as shown by the time-varying result in TNFR_DIFFUSION_STABILITY_THEOREM.md.

4. Field scales and selected thresholds

The exact phase-wrap maximum π must be distinguished from policy values expressed as fractions of π. The numerical values below remain unchanged in canonical.py.

Field Exact or conditional mathematical scale Canonical telemetry policy
Φ_s ‖Φ_s‖∞ ≤ ‖B_G‖∞ ‖ΔNFR‖∞ for a fixed distance kernel Per-node π/4; U6 drift π/2
|∇φ| At most π by wrapped-angle definition Warning level π/16
K_φ Absolute value at most π by wrapping Warning level 0.9π
ξ_C Spectral reference 1/√λ_gap when a positive graph gap is used Configured diameter and mean-distance comparisons

4.1 Structural potential

The canonical exponent is α = 2 in B_ij = d(i,j)⁻². The resulting field is linear in ΔNFR and unbounded under pressure rescaling. Even at zero phase, unit pressure on K₄ gives Φ_s = 3. No phase-wrap argument can enforce π/4 or π/2 for this linear field without an additional pressure/geometry constraint.

The chain sums Σ d⁻² = ζ(2) and Σ d⁻⁴ = ζ(4) are correct particular values. They do not uniquely select exponent 2: absolute chain sums converge for all α > 1 and independent unit-variance pressure sums for all α > 1/2. Graph families with growing distance shells need their own summability assumptions. The inverse-square kernel is the canonical modeling choice, not a uniqueness theorem from those series.

4.2 Phase gradient

The mean of absolute wrapped neighbor differences is bounded by π. The warning value π/16 is selected policy; a measured synchronization onset depends on the experiment's frequency spread, topology, and coupling protocol.

Canonical aggregate coherence is

C(t) = 1 / (1 + mean|ΔNFR| + mean|dEPI|).

It does not identify the spatial location of phase stress. It also is not scale invariant: with dEPI = 0 and uniform |ΔNFR| = 1, C(t) = 1/2; doubling pressure gives 1/3. Only the separate normalized dispersion statistic C_disp = 1 − σ_ΔNFR/ΔNFR_max is invariant under proportional positive pressure scaling when its denominator is nonzero.

4.3 Circular curvature

Where the circular mean is available, K_φ = wrap(φ_i − circular_mean(neighbors)) gives |K_φ| ≤ π by definition. The 0.9π threshold is an operational margin inside that exact bound. The shared reader accepts every nonzero represented phasor resultant without an arbitrary small-magnitude cutoff; exact represented cancellation on nonempty support makes curvature unavailable. Isolates retain the documented zero convention. These arithmetic availability conditions do not prove a trigonometric singularity or a singularity of a continuous manifold.

4.4 Correlation length

The current estimator first fits decay of a pressure-derived coherence correlation against graph distance. If that fit is unsuitable, it uses a spectral reference from the graph: the inverse square root of its smallest normalized-Laplacian eigenvalue above 1e-9. On an admitted connected undirected graph this is 1/√λ₂ only when that gap clears the numerical cutoff. A fitted correlation length need not equal that reference for every state; exponential spatial decay is itself a fitting assumption.

On disconnected graphs, a second-smallest eigenvalue and a smallest positive eigenvalue need not coincide. Reports must identify the estimator/fallback and graph regime. A failed fit or large estimate alone does not prove a phase transition.

5. Diagnostic removal studies

Each field addresses a distinct diagnostic task: accumulated pressure, local mismatch magnitude, local curvature, or correlation range. This motivates retaining all four channels. A formal non-redundancy study should construct explicit pairs of states with identical retained readouts and a differing omitted readout, or specify a detection problem and prove the omitted channel changes the decision.

Descriptions such as “a blind spot appears” require the actual fixture, readout resolution, and detection threshold. Passing many implementation tests does not prove universal irreducibility. Nor does usefulness of each channel prove that no additional observable can carry independent information.

The strong claim of a minimal complete representation therefore remains open until its target state space and equivalence relation are specified and the appropriate injectivity and non-redundancy results are proved.

A positive example with those choices fixed is the P5 reflection quotient. Its three even coordinates and constrained quadratic hidden-mode matrix reconstruct form up to reflection under the declared pure-EPI law. This is not the four-field tetrad, a universal state basis, or a reduction to three dimensions near the reflection-fixed stratum. The linked derivation owns the injectivity, rate, decoder and field-covariance results.

6. Energy, currents, and conservation

The tetrad can participate in energy-like diagnostics with phase and pressure currents:

V = ½ Σ_i (Φ_s(i)² + |∇φ|(i)² + K_φ(i)²),
T = ½ Σ_i (J_φ(i)² + J_ΔNFR(i)²).

These are nonnegative functionals of measured fields. Their existence does not establish state reconstruction, does not make ΔNFR equal to −∂V/∂EPI, and does not turn all implemented operators into Hamiltonian transformations. Conjugate coordinates belong to a separately specified substrate model.

Likewise a continuity residual or measured energy descent is evidence about the tested trajectory. Universal Noether conservation and asymptotic stability require the stated symmetry, flow, and Lyapunov hypotheses. The mathematical scope is developed in the variational note and the conservation note; the tetrad's canonical diagnostic status does not supply those proofs.

7. Symmetry-restricted research

Symmetry can restrict which sectors a graph readout detects. A statement about the tetrad on a particular prime-ladder graph with uniform parameters must name that graph, its action, and the parameter restrictions. Invariance under that action constrains reach; it does not imply that the tetrad spans every vector in the invariant sector.

The REMESH-∞ and Riemann programs use separate projection and observability constructions. Their particular graph certificates do not establish a universal complete “smooth structural sector” on arbitrary networks. See REMESH_INFINITY_DERIVATION.md and TNFR_RIEMANN_RESEARCH_NOTES.md for the precise program assumptions and open conjectures.

8. Verification scope

Claim Supported scope
Four canonical field APIs exist Definitions and implementation contracts
Phase magnitudes are at most π Exact wrapped-angle bounds
Potential is linear in pressure Exact fixed-kernel identity
Potential has a uniform π-fraction maximum False without additional pressure/geometry assumptions
Three operations bound every diffusion relaxation time False; the 21-node path is a counterexample
Laplacian compositions prove tetrad completeness False inference; operator generation is not reconstruction
Tetrad is a universal minimal sufficient state statistic Not established; task-specific quotients require explicit injectivity and dynamics proofs
Full Φ_s plus one zero-mode scalar reconstructs EPI Conditional on rank(-K L_rw)=N-1, fixed graph and pure EPI pressure
Rank condition holds on every connected graph Open; measured on all 142 connected simple graph-atlas cases through six nodes
Full-rank potential reconstruction is uniformly well-conditioned False across unrestricted weights; extreme weighted stars are arbitrarily ill-conditioned candidates

Operator gains, discretization steps, numerical clamps, and safety thresholds remain configured policies unless an explicit derivation with its assumptions is supplied. A formula using π alone is not sufficient evidence of a physical necessity.

References