TNFR Directed and Non-Normal Dynamics (R9)¶

Status: exact finite-dimensional results are available for the fixed linear pure-EPI channel; sampled gains and benchmark correlations remain finite-family evidence. The canonical directed U2 metric, nonlinear operator extension, changing-topology theory, and universal U6 trajectory bound remain open.

Modules: directed_diffusion.py, transient_u2.py, nonnormal_prediction.py, heterogeneous_vf.py, and structural_diffusion.py.

1. Declared transport model¶

For finite nonnegative outgoing conductance, the EPI-only pressure is

\[p=\Delta\mathrm{NFR}_{\mathrm{epi}}=-L_{\mathrm{rw}}x, \qquad L_{\mathrm{rw}}=I-P,\]

where positive rows of $P$ are normalized by outgoing strength and a zero-strength row is absorbing. The fixed scalar-capacity evolution is

\[\dot x=-\nu_f L_{\mathrm{rw}}x.\]

This is one channel of the canonical pressure, on a fixed graph. It does not cover phase, frequency, topology, nonlinear operator gains, or graph mutation. Even for symmetric conductance, the raw random-walk matrix is generally only similar to a symmetric normalized Laplacian; Euclidean normality is a separate property.

Sections 2–5 use normalized capacity time s = nu_f t for a fixed common positive capacity, so dx/ds = -L_rw x. The matrix readers implement this unit-capacity generator; their window and rate parameters refer to s unless an explicit clock conversion is supplied. At zero common capacity the unforced state and its pressure are static. Section 6 distinguishes a scalar clock change from heterogeneous-capacity dynamics.

2. Normality, spectrum, and metric¶

A non-normal generator can have a nonpositive spectral abscissa while its Euclidean semigroup norm grows transiently. Consequently:

eigh is used only for symmetric/Hermitian representations;
right eigenvectors are not converted into projectors as $QQ^*$;
Schur/Riesz methods or direct semigroup and resolvent diagnostics are used;
the Kreiss resolvent quantity is a lower bound on peak semigroup gain, not an equality or an infinite-window measurement.

Directed circulants are normal and form a benign special case. General directed graphs need not be normal.

For a strongly connected row-stochastic $P$ with stationary distribution $\pi$, Jensen's inequality gives

\[\|Pf\|_{2,\pi}\leq\|f\|_{2,\pi},\qquad \|f\|_{2,\pi}^2=\sum_i\pi_i f_i^2.\]

Because $e^{-sL}=e^{-s}\sum_{k\geq0}s^kP^k/k!$, the diffusion semigroup is a contraction in $L^2(\pi)$. This exact weighted result does not make the raw Euclidean norm canonical for U2. The APIs expose both metrics.

3. Exact pressure-transient criterion¶

For fixed $L=L_{\mathrm{rw}}$,

\[p=-Lx\quad\Longrightarrow\quad \frac{dp}{ds}=-Lp, \qquad \frac{dp}{dt}=-\nu_f Lp.\]

When the graph has one consensus mode, $\operatorname{range}(L)$ is the non-consensus pressure space $\{p:\pi^Tp=0\}$. Let $V$ be an orthonormal basis of that space and $L_{\mathrm{sub}}=V^TLV$. Every vector in this restricted space is a reachable pressure. In the declared Euclidean metric, the logarithmic norm

\[\mu_2(-L_{\mathrm{sub}})= \lambda_{\max}\!\left(\frac{-L_{\mathrm{sub}}-L_{\mathrm{sub}}^T}{2}\right)\]

has an exact qualitative meaning:

$\mu_2\leq0$ implies contraction for every time;
$\mu_2>0$ implies that some reachable pressure direction grows immediately.

A negative spectral abscissa gives asymptotic decay and cannot exclude the second case. A fixed weighted counterexample in the test suite disproves the older conjecture that every random-walk digraph contracts in unweighted Euclidean pressure energy.

nonnormal_prediction.py applies this theorem to a predeclared deterministic family. Its peak gains, rank correlations, and calibration/holdout accuracies are measurements of that finite family. The sign theorem itself is an exact matrix result and uses no fitted threshold.

4. U2 readings¶

The implementation distinguishes two finite-window quantities:

\[\left\|\int_0^S\frac{dx}{ds}\,ds\right\| =\|x(S)-x(0)\|, \qquad \int_0^S\left\|\frac{dx}{ds}\right\|\,ds.\]

The first allows cancellation; the second is total variation, and the triangle inequality gives net <= total. Neither a sampled finite window nor a stable full-state spectrum alone proves an infinite-horizon U2 theorem. On the non-consensus subspace, a declared exponential envelope

\[\|e^{-sL}Q\|\leq M e^{-\omega s}\]

yields the sufficient bound

\[J\leq \frac{M\|LQ\|\|x_0\|}{\omega}.\]

The bound is restricted to the fixed linear EPI channel and its stated norm. Choosing the canonical directed U2 metric and extending the result to the full four-channel dynamics remain open.

5. U6 scope¶

Canonical U6 compares structural-potential fields from two declared graph states using the centralized $\pi/2$ drift policy. The sampled quantity

\[\max_{s,i}|\Phi_s(s)_i|\]

is an absolute magnitude, not a U6 drift. TransientU2Certificate therefore reports it as peak_structural_potential_magnitude, exposes its operator-norm bound, and sets u6_drift_assessed=False. The old peak_structural_potential, structural_potential_bound, and u6_confined names remain compatibility aliases with explicit legacy semantics.

finite_schedule_readout can evaluate mean potential drift from the initial state at declared segment endpoints. It still does not cover values between endpoints or an unobserved tail.

6. Structural time and heterogeneous capacity¶

For a common scalar schedule $\nu_f(t)\geq0$ on a fixed graph,

\[x(t)=e^{-s(t)L}x_0,\qquad s(t)=\int_0^t\nu_f(\tau)\,d\tau.\]

This is an exact clock change because all generators are scalar multiples of one fixed $L$. Total variation expressed in structural time is invariant under that reparameterization, subject to the stated integral and decay hypotheses.

For heterogeneous capacity $D_{\nu_f}(t)$, generators $D_{\nu_f}(t)L$ generally do not commute, so one scalar clock does not describe the trajectory. Several useful results survive:

A frozen nonnegative generator $-D_{\nu_f}L$ is Metzler with zero row sums, so its semigroup preserves the scalar convex hull.
For fixed positive capacity on a strongly connected graph, the invariant measure is proportional to $\pi_i/\nu_{f,i}$.
On fixed symmetric conductance $B=D-W$, the Dirichlet energy $E_D=x^TBx/2$ obeys $$\dot E_D=-(Bx)^TM(t)(Bx)\leq0, \qquad M_{ii}(t)=\nu_{f,i}(t)/d_i,$$ for finite nonnegative time-varying capacity. Uniformly positive mobility plus a divergent mobility clock gives a sufficient consensus bound.
For switching among symmetric fixed-node regimes, the exact common-metric theorem applies only when their normalized diagonal metrics agree up to a positive scalar. Without that condition the provided certificate reports that the theorem is unavailable; it does not infer instability.

These statements do not create a single conserved mean for arbitrary changing capacity ratios, and they do not cover changing node sets.

7. Claim ledger¶

Claim	Status
Directed circulants are normal	Exact algebraic property; numerically checked
General non-normal stable generators may amplify in Euclidean norm	Exact possibility; finite witnesses measured
Diffusion contracts in stationary $L^2(\pi)$	Derived by Jensen
$p=-Lx$ implies $dp/ds=-Lp$	Exact for the fixed unit-capacity generator; in original time, $dp/dt=-\nu_f Lp$
Sign of $\mu_2(-L_{\mathrm{sub}})$ classifies possible Euclidean pressure gain	Exact finite-dimensional theorem
Spectral abscissa alone classifies transient pressure gain	False; counterexamples retained
Every random-walk digraph contracts in restricted Euclidean pressure energy	False; weighted counterexample retained
Scalar $\nu_f(t)$ is a clock change	Exact on a fixed graph
General heterogeneous $\nu_f(t)$ is one scalar clock	False when generators do not commute
Fixed-symmetric Dirichlet energy decreases under nonnegative heterogeneous capacity	Exact instantaneous identity
Absolute $\lvert\Phi_s\rvert$ magnitude is U6 drift	False; compatibility aliases are marked
Canonical directed U2 metric and full nonlinear bound	Open
Universal U6 interval/tail safety from finite samples	Open

No result here changes U1-U6 or resolves an external open mathematical problem.