Variational diagnostics and conditional constitutive models¶

Status: Exact restricted EPI balances, conditional forced-potential family and reciprocal closure constraints; decoupled product and fixed-P2 harmonic realizability obstruction; broader coupled tetrad/nodal bridge unresolved Implementation: physics/variational.py owns field-action readouts; the pressure, diffusion, phase-response and constitutive owners linked below supply the separate dynamical calculations. Tests: tests/physics/test_diffusion_energy_balance.py, tests/physics/test_variational.py, tests/physics/test_symplectic_substrate.py, tests/physics/test_metriplectic_product.py, tests/physics/test_symplectic_graph_realizability.py, tests/physics/test_constitutive_variational_scope.py Updated: 2026-10-07

1. Main Result¶

The TNFR nodal equation

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

is the unforced nodal row; a complete model must also supply pressure and the other consumed evolution laws. This note distinguishes three responsibilities:

Question	Section and mathematical scope
What do the field-action diagnostics compute?	Sections 2–12 define the readouts and the separate harmonic-coordinate model. Its brackets and flow are supplied premises.
Which diffusion and harmonic bridges actually hold?	Sections 3.4–3.8 retain exact restricted balances, a decoupled product and graph-realizability obstructions.
What does a proposed joint variational law have to satisfy?	Section 13 derives conditional source, metric, work and reciprocity constraints; it does not uniquely select a constitutive law.

A derivation of the full nodal law from the diagnostic field potential is not established. The conditional pressure-compatible potentials in section 13 are different objects. Neither a field identity nor an auxiliary harmonic motion proves that an engine trajectory obeys the corresponding action principle. The current native relational law and its competing storage premises are owned by relational exchange admission.

2. The TNFR Lagrangian¶

2.1 Lagrangian Density¶

On a graph $G=(V,E)$ the implemented Lagrangian diagnostic at node $i$ is

\[\mathcal{L}(i) \;=\; T(i) \;-\; V(i)\]

where the historical kinetic and potential labels denote the following quadratic field groups:

\[T(i) = \tfrac{1}{2}\bigl[J_\varphi(i)^{\,2} + J_{\Delta\mathrm{NFR}}(i)^{\,2}\bigr]\]

\[V(i) = \tfrac{1}{2}\bigl[\Phi_s(i)^{\,2} + |\nabla\varphi|(i)^{\,2} + K_\varphi(i)^{\,2}\bigr]\]

These graph read-outs are not independently supplied positions and velocities. The harmonic model in section 3 adds that coordinate interpretation explicitly. Physical energy units require field scales or a metric; raw pressure, angle and path-length quantities cannot be added dimensionally without them. The normalization boundary owns that distinction for all field-energy and charge diagnostics.

2.2 Action Functional¶

\[S_{\mathrm{TNFR}} = \int \mathrm{d}t \;\sum_{i\in V} \mathcal{L}(i,t)\]

In discrete form (time steps of width $\Delta t$):

\[S_{\mathrm{TNFR}} = \Delta t \;\sum_n \sum_i \mathcal{L}(i,\,t_n)\]

2.3 Hamiltonian¶

Treating the two conjugate sectors as independent harmonic-model coordinates gives the substrate Hamiltonian plus held-fixed background:

\[\mathcal{H}(i) = T(i) + V(i) = \tfrac{1}{2}\bigl[\Phi_s^2 + |\nabla\varphi|^2 + K_\varphi^2 + J_\varphi^2 + J_{\Delta\mathrm{NFR}}^2\bigr]\]

This is exactly $\tfrac{1}{2}\,\mathcal{E}(i)$ where $\mathcal{E}$ is the energy density from unified.compute_energy_density(), and the total Hamiltonian $H = \sum_i \mathcal{H}(i)$ equals conservation.compute_energy_functional().

3. Harmonic Model and the Unresolved Nodal Correspondence¶

3.1 Canonical Conjugate Pairs¶

The substrate model assigns two canonical coordinate pairs $(q,\,p)$ to the extracted field sectors:

Sector	Coordinate $q$	Momentum $p$	Graph-field interpretation
Geometric	$K_\varphi$ (curvature)	$J_\varphi$ (phase current)	Two instantaneous phase statistics; no continuity law follows from pairing them
Potential	$\Phi_s$ (potential)	$J_{\Delta\mathrm{NFR}}$ (ΔNFR flux)	Two pressure statistics; their graph balance has a measured residual

The ambient canonical brackets are a model premise. Conservation diagnostics do not establish those brackets or a small continuity residual. Sections 3.7–3.8 test whether the resulting auxiliary velocity is realizable by the graph fields.

3.2 Hamilton's Equations¶

\[\frac{\partial q_i}{\partial t} = \frac{\partial \mathcal{H}}{\partial p_i} = p_i \qquad\text{(dynamics)}\]

\[\frac{\partial p_i}{\partial t} = -\frac{\partial \mathcal{H}}{\partial q_i} = -q_i \qquad\text{(force law)}\]

3.3 Scope of Hamilton's Equations¶

The equations above define isotropic oscillators, q''=-q, on the specified ambient coordinate space. Extracting coordinates from a graph does not prove that all ambient points are realizable by graph fields or that an evolved ambient point is the image of an engine trajectory.

3.4 Overdamped Graph Diffusion and the Missing Bridge¶

For a separate graph-wave model,

q'' + gamma*q' + L_rw*q = 0,

the overdamped slow rates approach lambda_k/gamma, producing the EPI channel's diffusion with mobility nu_f=1/gamma. This requires an explicit stiffness operator and damping law. It does not identify the graph wave with the isotropic substrate oscillator, whose stiffness is the identity.

The previously stated identity DeltaNFR=-dV/dEPI fails for the implemented potential even on one edge. With pure EPI-channel pressure, equal phases, and EPI=[1,0], one obtains DeltaNFR=[-1,1], Phi_s=[1,-1], and V=(EPI_0-EPI_1)^2. Hence -gradient(V)=[-2,2], not DeltaNFR.

The EPI-only diffusion does have a degree-metric gradient formulation using its Dirichlet energy. A bridge to the different tetrad potential and to the full pressure still requires a specified coordinate map, metric and damping mechanism. The nodal equation makes nu_f a mobility; it does not derive an inverse inertial mass.

3.5 Exact Restricted Dirichlet Balance¶

Hold a finite symmetric nonnegative adjacency W fixed. Let d_i=Σ_j W_ij, B=D−W, and x=EPI. For the isolated EPI channel, define

E_D(x) = ½ xᵀ B x = ¼ Σ_ij W_ij (x_i−x_j)²,
∇E_D = Bx,
M_ii = νf_i/d_i if d_i>0, otherwise 0.

Symmetry gives ∇E_D=Bx. Multiplying by the nonnegative mobility yields −M∇E_D=−diag(νf)L_rw x, exactly the canonical pure EPI pressure times capacity. The chain rule therefore proves

dE_D/dt = −(∇E_D)ᵀ M (∇E_D) = −Σ_i M_ii (∇E_D_i)² ≤ 0.

For positive strengths and capacities, the metric is diag(d_i/νf_i). Zero capacity instead gives degenerate mobility: nonzero pressure may coexist with zero velocity and constant positive energy. Isolates have zero flow; self-loops add row strength but no Dirichlet energy. Parallel-edge conductance is summed. Disconnected components need not share a uniform equilibrium value.

The read-only implementation compute_diffusion_energy reuses the diffusion module's validated adjacency, capacity and EPI readers. Its edge difference evaluation avoids cancellation under large common EPI offsets and requires only the core NumPy dependency. The generic spectral smoothness helper evaluates xᵀLx for a supplied matrix and imports optional SciPy; it neither supplies this mobility nor validates this transport scope.

Tests compare the gradient and directional energy derivative against independent edge-energy finite differences, then compare velocity with the canonical pressure callback for weighted graphs, parallel edges, loops and heterogeneous capacities. This establishes the restricted identity, not a theorem about the different tetrad potential, changing graph weights, all four pressure channels, arbitrary discrete time steps, or a lift to the isotropic substrate.

3.6 Exact Decoupled Metriplectic Product¶

The harmonic substrate and the restricted EPI gradient flow can coexist in one block system X'=J grad(H)-G grad(E_D). With zero cross blocks, J grad(E_D)=0 and G grad(H)=0; therefore the substrate Hamiltonian is conserved while Dirichlet energy decreases. The implementation verify_metriplectic_product verifies both degeneracies and both component vector fields.

Its numerical Dirichlet energy, gradient, mobility and rate now come from the shared diffusion balance, aligned explicitly to the substrate's node order. Recomputing these quantities locally previously admitted false zero motion after underflow: on P2 with conductance 1e200, capacity 1e-200 and EPI (1,0), a computed zero nu/d erased representable nodal rates of magnitude 1e-200. The exact Dirichlet derivative is -2*W*nu, approximately -2 for those binary64 inputs. The shared owner rejects this unrepresentable positive mobility, as well as lost nonzero energy/rate terms, instead of certifying their zero substitutes. Normal-range and reordered weighted-loop controls independently compare the edge energy and dissipation. This numerical admission does not extend the mathematical model.

This is an exact direct product, not the missing coupled derivation. Since the cross blocks vanish, it supplies no pulse-relaxation feedback. A full bridge would require a TNFR-derived interaction and preservation of the realizable graph-field image, not just nonzero cross tensors with those degeneracy laws. See TNFR_SCALE_GEOMETRY_AND_BRIDGE.md.

3.7 P2 Read-Out Realizability: Obstruction and Derived Flow¶

An ambient harmonic solution need not be a trajectory of the graph fields used to initialize it. For a graph state X, actual vector field F and read-out h, a proposed smooth autonomous field flow f requires Dh(X) F(X) = f(h(X)). In particular, f must be tangent to the realizable image of h. An instantaneous coordinate pairing does not establish this.

Exact geometric obstruction. On fixed mutual-singleton P2 support, choose an open wrap branch and set k=wrap(theta_i-theta_j) in (-pi,pi). The production definitions of curvature and phase current give

K_phi(i) = k,                 J_phi(i) = -sin(k),
c(k,j) = j + sin(k) = 0.

The singleton phasor resultant has unit magnitude, so no ambiguous circular mean is used. The auxiliary harmonic velocity (k_dot,j_dot)=(j,-k) has constraint derivative

c_dot = -k + cos(k)*j = -k - sin(k)*cos(k).
k + sin(k)*cos(k) = integral_0^k 2*cos(u)^2 du.

The integral has the strict sign of k, hence tangency holds only at k=0. No nonzero continuous harmonic geometric trajectory can therefore be produced by any differentiable phase evolution with this fixed support and these same read-outs. Changing capacity or EPI does not change that constraint. A nonvanishing scalar rescaling of time cannot remove the obstruction. Synchronized phases may rotate together while the geometric read-out stays zero; the theorem does not freeze all possible phase evolution.

This is an exact-real, fixed-support, open-branch result. It is not a claim about larger graphs, different derived observables, or branch-crossing events. It also does not exclude isolated endpoint coincidences: a harmonic half-turn maps (k,-sin(k)) to (-k,sin(k)), again on the geometric image, although the intermediate harmonic path leaves that image. No graph event is certified by that isolated coincidence.

A field flow that does derive from the nodal equation. Now restrict to unit conductance and unit metric length on P2, positive fixed capacities nu_0,nu_1, scalar EPI x, fixed phase, pure-EPI pressure refreshed from the current state, and no additional forcing. Write d=x_1-x_0. The nodal law and the same production field definitions give

DeltaNFR = (d,-d),            d_dot = -(nu_0+nu_1)*d,
Phi_s = (-d,d),               J_DeltaNFR = (-2*d,2*d) = 2*Phi_s,
K_phi_dot = J_phi_dot = 0,
Phi_s_dot = -(nu_0+nu_1)*Phi_s,
J_DeltaNFR_dot = -(nu_0+nu_1)*J_DeltaNFR.

Thus the extracted potential/flux sector has a closed dissipative flow with rate determined by the nodal capacities, without a fitted coefficient or an imported interaction. The lost common EPI offset does not prevent this read-out closure. This does not establish complete state reconstruction. On the realized potential line p=2q, the auxiliary harmonic velocity instead has (p-2q)_dot=-q-2p=-5q, nonzero for q!=0. Consequently even synchronized phases do not rescue that harmonic comparison when EPI disagreement remains. The synchronized uniform-EPI state is a common zero-field fixed point.

Executable boundary. The fixture x=(0,1), nu=(1,1) and theta=(0,pi/3) gives K_phi(0)=-pi/3, J_phi(0)=sqrt(3)/2 and geometric tangency defect pi/3+sqrt(3)/4. Tests use the production pressure callback, field extractors and shared nodal integrator for three explicitly refreshed Euler segments of length 1/8, with no active clipping or added forcing. The exact ideal segment disagreement factor is 1-(nu_0+nu_1)*dt; this finite check is not an asymptotic binary64 solver theorem. Additional fixtures check heterogeneous capacities, both signs of the geometric defect, the zero control and the half-turn boundary. Numerical tolerances verify represented readings; the nonzero-for-all-k claim follows from the analytic integral above.

Regression owner: test_symplectic_graph_realizability.py. Field owners: canonical.py, extended.py, symplectic_substrate.py. No new physical primitive or general TNFR-to-Hamiltonian equivalence is assumed.

3.8 Prepared prism harmonic geometric velocity has no phase lift¶

The current unit-prism preparation supplies a separate exact realizability test. At repeated phases (0,pi/3,pi/6), the actual geometric fields are K=(-pi/6,pi/6,0) and J_phi=((1+sqrt(3))/6,-(1+sqrt(3))/6,0), repeated in both triangles. Let R be the shared phasor mean-response matrix. Then DK=I-R has rank five and kernel equal to common phase rotation.

The auxiliary harmonic direction requires K_dot=J_phi and J_phi_dot=-K. Since DK*J_phi=J_phi, every possible primitive phase velocity satisfying the first row is theta_dot=J_phi+q*1. The actual mean-sine derivative annihilates 1, and necessarily gives

\[ DJ_\phi\,\dot\theta= \left(-\frac{5+3\sqrt3}{36},\frac{5+3\sqrt3}{36},0\right)_{\rm repeated}, \]

which differs from -K. The stacked derivative has rank five, while adjoining the proposed harmonic velocity raises the rank to six. Therefore no primitive phase velocity, even a nonrepeated one, lifts this auxiliary geometric direction at this state. This is a prepared-state obstruction, not a classification of every prism state or every coupled Hamiltonian. It reuses the existing field extraction and harmonic-vector-field owners; no new flow is executed.

Reciprocal exchange also does not automatically replenish loss. In the conditional linear family z_dot=-e*z-k*zeta, zeta_dot=a*z+b*zeta, with a,e,k>0, b<=0, the positive storage S=a*|z|^2+k*|zeta|^2 satisfies S_dot=-2*a*e*|z|^2+2*k*b*|zeta|^2<=0. After scaling coordinates by sqrt(a),sqrt(k), the generator is an antisymmetric exchange plus diagonal loss. At b=0, zero instantaneous loss need not imply zero velocity: a state with z=0,zeta!=0 immediately leaves that set. The largest invariant zero-loss set is only the origin; the characteristic polynomial in the joint-response classification gives asymptotic decay throughout this passive class, including damped oscillations where its discriminant is negative. This storage is a comparison quadratic, not a derived canonical joint energy.

Controls: prepared prism lift and passive exchange.

4. Grammar-Labelled Heuristics¶

analyze_grammar_stationarity() reports energy and interaction statistics labelled by related grammar rules. Each result now explicitly carries verification_scope="heuristic". Its legacy is_satisfied flag evaluates that statistic only. In particular:

Nonzero Lagrangian density does not establish a generator history.
Potential dominance does not prove an attractor or valid closure.
A finite sampled action does not prove infinite-horizon convergence.
Bounded bilinear interactions do not verify the U3 phase gate.
T/V does not determine a Hessian, bifurcation, or handler history.
Energy dispersion does not establish nested identities or per-level stabilizers.
Absolute potential magnitude is not U6 drift from a reference state.

Canonical sequence validation and confinement telemetry remain responsible for the actual grammar requirements.

5. Operator Energy Heuristics and Local Symplectic Checks¶

Older code attached sign labels to operators. Those labels are priors for a snapshot statistic, not consequences of U2 or of the glyph factors. The five-field Hamiltonian has no explicit EPI or capacity term, while phase and pressure changes propagate through graph-dependent fields. Every realized sign therefore remains state-dependent and must be measured.

Operators	U2 bookkeeping role	Historical sign prior	Valid conclusion
IL, THOL	Stabilizer	decrease	No universal $\Delta H$ sign
OZ, ZHIR, VAL	Destabilizer	increase	No universal $\Delta H$ sign
AL, EN, UM, RA, SHA, NUL, NAV, REMESH	Neutral	unchanged	U2 neutrality does not imply $\Delta H=0$

The policy-multiplier layer in physics.lyapunov now exposes this scope through is_energy_bound=False and is_lyapunov_certificate=False. A canonical transformation can also change a particular Hamiltonian; energy conservation and symplecticity are separate properties.

check_symplectic_preservation(before, after, ...) preserves its positional arguments and legacy ratio fields, but snapshots alone now return is_canonical=None, classification="inconclusive". The old ratio category is available as heuristic_classification; sum(abs(q*p)) is not a volume.

For a supplied local derivative, use:

result = check_symplectic_preservation(
    before, after, "map name", jacobian=M, jacobian_tolerance=1e-9
)

The matrix must be finite, real, and (4N,4N), with coordinates interleaved (K_phi,J_phi,Phi_s,J_DNFR) per node. The shared substrate helper evaluates max(abs(M.T @ omega @ M - omega)). A passing result concerns this supplied Jacobian only; establishing that it is an engine operator's derivative and proving preservation over its domain requires additional evidence.

The functions historically named compute_phase_space_volume and compute_poisson_bracket_estimate retain their numeric statistics for compatibility. They no longer claim those statistics are geometric certificates.

6. Telemetry Thresholds Are Regular Points of the Quadratic Potential¶

For each field coordinate, V(x)=x^2/2, V'(x)=x, and V''(x)=1. The only critical point is zero. Nonzero configured thresholds do not create extrema or saddles of this potential, and the restoring force does not reverse sign when a threshold is crossed.

analyze_potential_critical_points() retains its API and configured comparison values. It reports the correct derivatives and critical_type="regular" at nonzero thresholds. The added near_threshold_count preserves the observation of nodes near a threshold without mislabelling them as stationary points. A nonlinear constrained effective potential would require an explicit formula and a separate derivative calculation.

7. Interaction Lagrangian¶

The bilinear cross-sector coupling is:

\[\mathcal{A}(i) = \Phi_s \cdot |\nabla\varphi| + K_\varphi \cdot J_\varphi + |\nabla\varphi| \cdot J_{\Delta\mathrm{NFR}}\]

This equals the snapshot contraction unified.compute_action_density(). One can formally write a candidate interaction functional:

\[\mathcal{L}_{\mathrm{full}} = T - V - \mathcal{A}\]

This algebra alone does not derive a coupled equation of motion for the graph or its extracted fields. Such a claim requires a declared state space, admissible variations, metric/mobilities and a realizability check against the nodal dynamics. The specified harmonic substrate supplies its own independent sector flow; the snapshot bilinear term does not establish the missing phase/capacity feedback or the restoring contribution in section 13.5.

8. Energy Partition¶

For the harmonic conjugate core, a complete-period average satisfies <T>=<V_core>. This does not imply instantaneous equipartition, and the full potential also contains the held-fixed |grad_phi| background. The instantaneous T/V ratio is an energy-partition statistic; it does not by itself certify an attractor or bifurcation.

9. Consistency Relations¶

The following algebraic readout identities use shared field definitions:

Relation	Verification
$\mathcal{H}(i) = \tfrac{1}{2}\,\mathcal{E}(i)$	`compute_hamiltonian_density()` vs `unified.compute_energy_density()`
$\sum_i \mathcal{H}(i) = E$	Total Hamiltonian vs `conservation.compute_energy_functional()`
$\rho = \Phi_s + K_\varphi$	Charge density from conjugate pair coordinates
$\mathcal{A} = $ action density	`compute_interaction_density()` vs `unified.compute_action_density()`
Recorded $dH/dt$	Finite-difference diagnostic; no universal sign is certified here

Tests cover these readout identities across deterministic graph fixtures. Passing identity tests does not prove the unresolved model correspondence in section 3.

10. Single Source of Truth Architecture¶

The five fields in the energy expression admit three readout groupings. This algebraic regrouping does not assert a six-dimensional independent phase space. The implementation ensures a single computational path with no duplicated formulae:

unified.py                    (Layer 3 — RAW quadratic forms)
  ├── compute_energy_density   ℰ(i) = Φ_s² + |∇φ|² + K_φ² + J_φ² + J_ΔNFR²
  └── compute_action_density   𝒜(i) = bilinear coupling term
       │
       ├──► variational.py      H(i) = ½·ℰ(i),  T, V, ℒ = T−V
       │    (delegates via _raw_energy_density / _action_density)
       │
       └──► conservation.py     E = ½·Σℰ(i),  ρ, div(J), Q
            (delegates via _raw_energy_density)

10.1 Sector Decompositions¶

Decomposition	Split	Grouping criterion
Variational (T / V)	$T = \tfrac{1}{2}[J_\varphi^2 + J_{\Delta NFR}^2]$, $V = \tfrac{1}{2}[\Phi_s^2 + \vert \nabla\varphi\vert ^2 + K_\varphi^2]$	Temporal role (kinetic / configuration)
Conservation ($\rho$ / J)	$\rho = \Phi_s + K_\varphi$, $\mathbf{J} = (J_\varphi, J_{\Delta NFR})$	Diagnostic charge/current grouping; conservation requires a separate law
Unified ($\Psi$)	$\Psi = K_\varphi + i\,J_\varphi$	Dual structure (geometry-transport)

All three satisfy the consistency identity $T(i) + V(i) = \tfrac{1}{2}\,\mathcal{E}(i)$ at every node, verified by translate_sectors() with residual < $10^{-12}$.

10.2 translate_sectors() API¶

from tnfr.physics.variational import translate_sectors

result = translate_sectors(G)
# result['variational']        -> {'T': {...}, 'V': {...}}
# result['conservation']       -> {'rho': {...}, 'J_phi': {...}, 'J_dnfr': {...}}
# result['unified_psi']        -> {node: K_φ + i·J_φ}
# result['energy_density']     -> raw ℰ from unified.py
# result['consistency_check']  -> max |T+V − ½ℰ|  (should be ~0)

11. API Summary¶

from tnfr.physics.variational import (
    # Core densities
    compute_kinetic_density,       # T(i) = ½[J_φ² + J_ΔNFR²]
    compute_potential_density,     # V(i) = ½[Φ_s² + |∇φ|² + K_φ²]
    compute_lagrangian_density,    # ℒ(i) = T(i) − V(i)
    compute_hamiltonian_density,   # H(i) = ½·ℰ(i) — delegates to unified.py
    compute_interaction_density,   # 𝒜(i) — delegates to unified.compute_action_density

    # Sector translation
    translate_sectors,             # All three decompositions + consistency check

    # Phase space
    identify_conjugate_pairs,      # (K_φ,J_φ), (Φ_s,J_ΔNFR)
    compute_phase_space_volume,    # Legacy sum |q*p| statistic
    compute_poisson_bracket_estimate,

    # Snapshot & tracking
    capture_lagrangian_snapshot,    # Complete instant analysis
    compute_euler_lagrange_residual,
    compute_action_functional,     # S = ∫ℒ dt
    VariationalTracker,            # Time-series accumulation

    # Canonical checks
    check_symplectic_preservation,
    classify_operator_canonical,

    # Grammar stationarity
    analyze_grammar_stationarity,  # Explicitly heuristic grammar-labelled readouts
    analyze_potential_critical_points,

    # Comprehensive
    compute_variational_suite,     # Full analysis in one call
)

12. Symplectic Reduction Scope¶

For J=H_sub=|z|^2/2>0, the diagonal U(1) action is free and the quotient has dimension 4N-2. Its global space is complex projective space CP^(2N-1), not a flat linear space. The certificate restricts the symplectic form to the actual horizontal tangent space orthogonal to z and omega*z. Relative phase checks use only nonzero complex coordinates and a defined reference.

At J=0, the level set and quotient are a point. The certificate reports reduced_dimension=0, reduction_status="singular_zero_level", and is_valid_reduction=False, meaning the regular-level test is inapplicable. The legacy constant reduced_symplectic_form_matrix(n_nodes) remains a local non-orthonormal action-angle basis matrix; it does not certify a supplied state.

13. Forced-Potential Family and Reciprocal Closure Constraints¶

The full canonical pressure constrains a possible variational completion more strongly than the EPI-only energy does. These are exact conditional identities, not an added engine law. Fix a connected symmetric conductance W, positive strengths d, unweighted support U, and constant channel coefficients. Write D=diag(d), B=D-W, L_W=D^-1 B, and L_U for the support random-walk Laplacian. On a regular phase chart with nonzero neighbor resultants,

p = -e L_W x + F(theta,nu,U),
F = w_phi g(theta) - v L_U nu - t L_U k,
Dg = (R-I)/pi,

where k is support degree and R is the exact neighbor-mean derivative from phase_response. Positive capacity gives the already established EPI mobility M_x=diag(nu_i/d_i). Suppose a C2 scalar potential supplies the EPI component specifically as x'=-M_x gradient_x E, with no cross-mobility contribution to that component. The nodal equation then requires

gradient_x E = e Bx - DF,
E(x,theta,nu,W) = (e/2) x^T Bx - x^T DF + Psi(theta,nu,W).

Integration in x proves both necessity and sufficiency on a connected EPI coordinate domain. The arbitrary x-independent function Psi is genuine remaining freedom; setting it to zero is an additional choice. The expression is not identified with the different field energy in section 2.

13.1 Mixed derivatives require reciprocal coupling¶

For this entire potential family, equality of mixed partial derivatives fixes the following dependence on EPI:

gradient_theta E = -(w_phi/pi)(R-I)^T D x + gradient_theta Psi,
gradient_nu E    = v L_U^T D x + gradient_nu Psi.

Thus a proposed joint gradient completion with positive definite block mobilities M_theta(theta,nu,W) and M_nu(theta,nu,W), independent of x, would necessarily have these terms in theta'=-M_theta gradient_theta E and nu'=-M_nu gradient_nu E. An x-independent phase relaxation cannot satisfy that completion on an open full-state domain when w_phi>0 and R-I!=0. An x-independent capacity relaxation likewise cannot satisfy it when v>0 and L_U!=0. An x-independent Psi cannot cancel an x-linear term for all x. The obstruction does not cover degenerate mobilities, cross blocks, other metrics, constrained submanifolds or non-gradient dynamics. In particular it does not turn the engine's discrete phase/capacity policies into smooth laws or assume that their complete state dependence is absent.

Even if a gradient completion is selected, the EPI equation determines neither Psi nor the two auxiliary mobilities and their time scale. For smoothly varying positive off-diagonal edge conductances on fixed support, it also forces

partial E / partial W_ij = (e/2)(x_i-x_j)^2 - x_i F_i - x_j F_j
                          + partial Psi / partial W_ij,

where each undirected edge i<j is one coordinate and F is independent of its weight on this fixed-support chart. This supplies a necessary energy derivative, not an edge evolution law. Edge birth/deletion changes U and the pressure realization; it is a separate hybrid closure obligation.

13.2 Lower bounds require the actual source compatibility¶

At held theta,nu,W, shifting x by c*1 changes every compatible potential by -c*d^T F. Therefore no member of this family is bounded below over all real EPI fields if d^T F!=0. An arbitrary Psi cannot repair this tilt. Conversely, for e>0 and connected conductance, d^T F=0 yields a solution e Bx_*=DF, and completing the square gives

E = (e/2)(x-x_*)^T B(x-x_*) - (e/2)x_*^T Bx_* + Psi.

This is bounded below in x, with the common-offset freedom retained for the held source. Section 13.9 tests whether that freedom extends jointly to phase variations. This held-source result reuses the compatibility and profile in forced_support; it supplies no new stationary phase or capacity maintenance law. A weighted three-node path with edge weights (1,2), capacities (1/2,1,3/2), phase consensus and channel weights (e,v,t)=(1/2,1/4,1/4) has F=(3/8,-1/4,1/8) and d^T F=-1/8. This is an exact canonical-source example of the unbounded tilt, despite symmetric conductance.

13.3 Canonical phase pressure is not generally a fixed-metric gradient¶

It is also necessary to check integrability before treating g itself as a phase gradient flow. On the unit star K1,3, index the center by zero and choose phases (0,0,0,alpha) with 0<alpha<pi/2. Every edge satisfies strict U3, all resultants are nonzero and the wrap chart is regular. Put c=cos(alpha). The shared circular-mean derivative gives

R_01=R_02=(2+c)/(5+4c),  R_03=(1+2c)/(5+4c),
R_j0=1 for j=1,2,3; all other entries are zero.

A constant positive diagonal metric H could represent g as -H^-1 gradient V only if H(R-I) were symmetric throughout the domain. At consensus this forces H proportional to diag(3,1,1,1). At every nonzero arbitrarily small alpha its center/first-leaf antisymmetry is

3 R_01 - R_10 = (1-c)/(5+4c) > 0.

Consequently no such constant diagonal metric works on any open neighborhood of consensus. For the exact planar cosine Gram with c=3/5, the center row is (0,13/37,13/37,11/37), its squared resultant is 37/5, and the metric antisymmetries are 2/37,2/37,-4/37. This does not exclude state-dependent metrics, non-diagonal metrics, different phase laws or a variational completion of the joint pressure family above. Strict U3 and symmetric graph support alone do not establish the missing gradient identity. Section 13.6 derives an exact state-dependent diagonal representation on the regular phase domain. It preserves this constant-metric obstruction and does not by itself select a joint evolution law.

13.4 Relation to the full tetrad¶

Neither this conditional potential nor its mixed derivatives close the four read-outs Phi_s, phase-gradient magnitude, K_phi and xi_C. Their evolution still requires actual phase/capacity/support laws and the derivative of their common graph observation map, with branch and estimator boundaries handled separately. The five-field energy in section 2 has no explicit xi_C term; its algebraic identity cannot establish a correlation-length evolution. The complete closure obligations remain in DIAGNOSTIC_AND_GRAMMAR_SCOPE section 13.

Exact finite regression controls reuse the pressure decomposition, support-transport and phase-response owners in test_constitutive_variational_scope.py. They check source signs, the required mixed coupling and the fixed-metric obstruction without trajectories, new feedback coefficients or fitted laws.

13.5 Necessary restoring contribution on the compatible P2 preparation¶

The finite joint compatibility result does not select a maintenance mechanism. The same two-node preparation gives an exact test of what the conditional potential family in section 13 would require. Keep its EPI mobility premise and fixed unit P2 support. Write xi=x_0-x_1, eta=nu_0-nu_1, and hold phase and both common coordinates fixed only as a perturbation subspace. This does not assert that the subspace is invariant under an unspecified joint flow.

For regular singleton phase means, set a(theta)=w_phi*g_0(theta), so g_1=-g_0. The topology channel vanishes on this support. Restricting the already-required potential to the plane gives

\[ E(\xi,\eta)=\frac e2\xi^2+v\xi\eta-a\xi+\Psi(\eta), \qquad e>0,\quad v>0. \]

Here Psi is the restriction of the still-undetermined, EPI-independent potential contribution; it is not the complex geometric field with the same letter elsewhere. The source owners determine the mixed coefficient v. They do not determine Psi. Assume it is twice continuously differentiable near the preparation, with all held coordinates understood as parameters.

Stationarity and restoring curvature are distinct requirements. A local minimum in the interior requires

\[ e\xi_*+v\eta_*-a=0,\qquad \Psi'(\eta_*)=-v\xi_*. \]

The first equation is the zero-pressure condition; it does not imply the second. At the retained consensus-phase preparation

x=(1/4,0), nu=(1,3/2), theta=(0,0),
e=1/2, v=w_phi=1/4, xi*=1/4, eta*=-1/2, a=0,

the additional requirement is Psi'(eta*)=-1/16. Setting Psi=0 therefore does not even make this point stationary on the plane. Calling that noncritical point a saddle would be incorrect.

Let k=Psi''(eta*). The restricted Hessian and its quadratic form are

\[ H=\begin{pmatrix}e&v\\v&k\end{pmatrix},\qquad (u,s)H(u,s)^\top =e\left(u+\frac ve s\right)^2 +\left(k-\frac{v^2}{e}\right)s^2. \]

Consequently a local energy minimum necessarily requires

\[ k\ge\frac{v^2}{e};\qquad \text{at this preparation: }k\ge\frac18. \]

Together with stationarity, strict inequality suffices for a strict local minimum on this two-coordinate plane. It does not establish a minimum in the full phase/capacity/common-coordinate space or dynamical stability. With [xi]=X, [eta]=T^-1 and unit conductance, [E]=X^2, [v]=XT and [k]=X^2*T^2; 1/8 belongs to this normalized fixture, not a universal dimensionless structural threshold.

Exact insufficient-curvature obstruction. If stationarity is supplied but k<v^2/e, take delta_eta=s and delta_xi=-(v/e)*s. Taylor's theorem gives

\[ E(\xi_*-(v/e)s,\eta_*+s)-E(\xi_*,\eta_*) =\frac12\left(k-\frac{v^2}{e}\right)s^2+o(s^2)<0 \]

for sufficiently small nonzero s. The pure xi direction has positive quadratic curvature, so a stationary point in this case is a saddle on the plane. At the retained fixture, the perturbation is

\[ x(s)=(1/4-s/4,\ s/4),\qquad \nu(s)=(1+s/2,\ 3/2-s/2),\qquad\theta=0. \]

For 0<s<1/2, both capacities stay strictly inside the earlier closed bands [3/4,7/4], and EPI remains nonnegative. Thus the descending direction is not an artifact of negative capacity or a forbidden phase separation. Each point has exactly zero canonical pressure: the capacity source compensates the new EPI contrast. The curve is not a nodal trajectory, since EPI changes along it while the nodal equation would give xdot=0 at each point.

Equality is genuinely undecided by the Hessian. Put s=eta-eta* and r=xi-xi*+s/2. At this fixture the exact identity is

\[ \Delta E=\frac{r^2}{4} +\Psi(\eta_*+s)-\Psi(\eta_*)+\frac{s}{16}-\frac{s^2}{16}. \]

The three logical completions

\[ \Psi(\eta_*+s)-\Psi(\eta_*) =-\frac{s}{16}+\frac{s^2}{16}+\{0,\ +s^4,\ -s^4\} \]

share the required slope and k=1/8. They yield a flat valley, a strict restricted minimum, and a descending null direction, respectively. These polynomials show why second-order evidence is insufficient at equality; none is installed or proposed as a selected TNFR law.

Tetrad and physical scope. Along the displayed curve, zero pressure gives Phi_s=0; consensus phase gives zero phase gradient and K_phi=0. The graph-distance metric and static pressure-coherence inputs to xi_C are unchanged; its actual fit/fallback provenance is retained. All four current diagnostics can therefore agree while the conditional joint potential changes. This is a failure of observation sufficiency, not an executed relaxation or a new diagnostic feedback law. Auxiliary field-energy positivity cannot substitute for the missing capacity derivatives.

The obstruction concerns a local energy minimum in the stated passive gradient class. It is not a no-go theorem for TNFR patterns, non-gradient motion, oscillatory identity, hybrid events or constrained/degenerate mobilities. Those possibilities need their own independently justified laws. Even a strict minimum on the plane supplies neither an activation mechanism nor a route by which a preparation forms spontaneously.

Required ingredient	Provenance and remaining obligation
EPI force and mixed coefficient `v`	Fixed by the declared pressure realization and the section 13 EPI mobility premise; checked through the shared pressure and support owners.
`Psi'(eta)` and lower bound on `Psi''(eta)`	Necessary consequences for a restricted stationary minimum, not a derivation or selection of `Psi`.
Phase, capacity and support evolution	Not supplied by the nodal product or the local Hessian; auxiliary mobilities and any event occurrence remain unjustified unless separately derived.
Existing capacity adaptation	adaptation.py implements configured averaging and pressure/Si gates; its gains, timing and activation policy do not establish this missing potential.
Reduction and memory	epi_memory.py derives its closure/kernel from supplied dynamics with held capacity/forcing; operator quotients check supplied maps. Neither derives a law for those held inputs.
Physical emergence	Requires a justified complete mechanism, spontaneous formation and independently testable consequences; the present conditional obstruction establishes none of these.

The exact regression extension in test_constitutive_variational_scope.py reuses the existing forced-potential helper and exact semidefinite primitive. No new public certificate, runtime potential or phase/capacity update is needed for this bounded result. The single execution plan records the constitutive decision required before another origin experiment.

13.6 Exact state-dependent metric for canonical phase pressure¶

The nonlinear pressure admits a geometric representation that the constant-metric test in section 13.3 could not decide. Fix reciprocal, unweighted unique neighbor support, including any zero-conductance support edges. This is the phase channel's support, not the weighted EPI conductance. At a node with a nonzero neighbor resultant, put

\[ S_i=\sum_{j\in N(i)}e^{i\theta_j},\quad r_i=|S_i|>0,\quad \delta_i=\operatorname{wrap}(\arg S_i-\theta_i),\quad g_i=\delta_i/\pi. \]

Work on the open branch |delta_i|<pi at every node. Let

\[ V_\phi(\theta)=\sum_{\{i,j\}\in U}\bigl(1-\cos(\theta_j-\theta_i)\bigr) =\frac12\sum_{\{i,j\}\in U}|e^{i\theta_j}-e^{i\theta_i}|^2. \]

Each undirected distinct-neighbor edge is counted once. A self-loop has zero cost and zero derivative, although its own phasor still contributes to S_i and changes the metric below. Parallel multiplicity and conductance weights do not multiply this cost. It is an alignment cost derived from the specified phasor geometry, not a new definition of tetrad energy or physical energy. Choosing a scale for it as part of a joint potential would require the corresponding units and a constitutive premise.

Reciprocity gives the exact derivative

\[ \partial_i V_\phi=-\sum_{j\in N(i)}\sin(\theta_j-\theta_i) =-r_i\sin\delta_i. \]

Writing sinc(delta)=sin(delta)/delta, continuously extended by one at zero, therefore gives

\[ H_\phi(\theta)=\operatorname{diag} \left(\pi r_i\operatorname{sinc}\delta_i\right)>0,\qquad g=-H_\phi^{-1}\nabla V_\phi. \]

The diagonal mobility is M_i=delta_i/(pi*r_i*sin(delta_i)), with the regular limit M_i=1/(pi*r_i) at zero displacement. These factors are fixed by the displayed pressure/cost representation; they are not primitive capacity or an angular clock. A positive constant rescaling of the cost rescales this metric by the same factor without changing g, so no absolute energy scale is inferred.

Equivalently, set z_i=exp(-i*theta_i)*S_i=C_i+i*A_i, using A_i here only for the imaginary resultant component. The full regular domain is z_i outside the nonpositive real axis. It is wider than either strict acute edges or positive real relative resultants. On that domain

\[ H_i=\pi A_i/\operatorname{Arg}z_i \quad(\operatorname{Arg}z_i\ne0),\qquad H_i=\pi C_i\quad(z_i=C_i>0). \]

The imaginary component and principal argument have the same sign, including when C_i<0; the metric therefore stays positive. The second formula is its removable limit, not a value assigned at a negative-real branch or zero resultant. No new state variable or phase law is introduced by this wider domain. The relational execution owner separately certifies sufficient point and straight-proposal margins using rational enclosures; it retains the existing acute and positive-resultant options. This representation does not establish phase dynamics from the nodal form equation alone or turn every regular state into a stable pattern.

The previously established constant-metric obstruction remains correct. Here differentiating H_phi*g=-gradient V_phi includes (dH_phi)*g as well as H_phi*Dg. Testing only the latter for symmetry would incorrectly reject a variable metric. The star counterexample in section 13.3 is a useful control for this distinction.

Complete response identity. Write J=Dg, H=H_phi, and let DH denote the matrix with entries partial_j H_i for the diagonal coefficients. On this regular domain, differentiating every component gives

\[ H J+\operatorname{diag}(g)DH=-\operatorname{Hess}V_\phi. \]

Thus the antisymmetric part of HJ is exactly the negative of the antisymmetric part of diag(g)DH. This accounts for the entire matrix, not just one off-diagonal entry. It does not make J reversible or remove the retained response's cycle ratio 4/5. At g=0, the correction vanishes and HJ=-Hess(V_phi) is symmetric. For direct evaluation, with T_ij=1[j in N_i]*sin(Arg S_i-theta_j),

\[ \partial_j H_i=\pi\operatorname{sinc}(\delta_i)T_{ij} +\pi r_i\operatorname{sinc}'(\delta_i)(R_{ij}-\delta_{ij}), \]

where sinc(0)=1, sinc'(0)=0. All 36 entries are checked at the retained nonrepeated preparation and its repeated comparator, including removable zero displacements. The underlying covector identity is stronger:

\[ g^TH\,d\theta=-dV_\phi, \qquad \oint g^TH\,d\theta=0 \]

for a closed regular phase loop. There is no independent circulating work in this metric-weighted phase covector. The unweighted source response and the joint EPI source work are different objects; neither vanishes by this argument. In particular it specifies no phase velocity. The maintenance implication is the joint budget in section 13.18, not an invented phase propagator R. Controls: complete metric and phase-work identities.

Current and curvature share this geometry. With n_i=|N(i)|, the existing phase current reads J_phi(i)=r_i*sin(delta_i)/n_i. The regular canonical curvature has K_phi(i)=-delta_i; hence

\[ \nabla_i V_\phi=-n_iJ_\phi(i),\qquad J_\phi(i)=-\frac{r_i}{n_i}\sin K_\phi(i). \]

Nonzero current and curvature have opposite signs on this domain. Current and phase pressure have the same sign. The former documentation's same-sign/current-curvature claim does not match these definitions and has been removed. None of these instantaneous directional statistics measures an actual phase velocity without a phase evolution law.

The domain exclusions matter:

At delta=0 with r>0, the apparent 0/0 is removable.
At an exact half-turn, the sine torque can vanish while |g|=1; no finite positive diagonal mobility represents that component. The metric degenerates on approach to that branch boundary.
At a zero neighbor resultant the circular mean has no direction. Engine fallback conventions or represented residuals do not supply this theorem's missing ideal direction. Isolates likewise require their separate zero convention rather than division by zero.
Strict edge U3 (|wrap(theta_j-theta_i)|<pi/2) implies a positive real component of every nonempty rotated neighbor sum and therefore r>0, |delta|<pi/2. It is a sufficient current-state domain condition, not a proof that a future trajectory remains there.
Directed neighbor support does not generally yield this gradient from the undirected cost; reciprocal support is part of the claim.

Exact trigonometric identities and represented observations remain distinct. For example binary64 sin(pi) is not ideal zero. The shared forcing capture and current/curvature owners verify numerical readings with their explicit residual or tolerance; exact symbolic controls establish the limiting and branch statements. No trigonometric reimplementation or energy API is needed.

Classification of edge-additive storage aligned with native pressure¶

The cosine cost is also selected within a specific additional storage class, not merely an available representation. Restrict here to all finite connected simple unit graphs with at least two vertices, and the full regular phase domain above. Let the same even, 2*pi-periodic, continuously differentiable edge cost u define

\[ U_\phi=\sum_{\{i,j\}\in E}u(\theta_i-\theta_j),\qquad u(0)=0. \]

Assume that, on every admitted state and graph, a finite positive diagonal metric satisfies grad(U_phi)=-H_U*g, with the native resultant pressure g unchanged. Edge additivity and this alignment are explicit premises; neither follows from the nodal product. Then, for one positive constant kappa,

\[ \boxed{u(\delta)=\kappa(1-\cos\delta),\qquad U_\phi=\kappa V_\phi.} \]

C1 regularity is sufficient; no assertion that it is minimal is needed. Metric continuity is not required to classify the cost, but has a separate role at zero pressure below.

Proof. Write f=u', which is odd with f(0)=0. Reuse the phase-balanced star construction. For any finite nonantipodal gaps delta_1,...,delta_m with zero sine sum, append m+1 aligned leaves. The center has positive real resultant at least one and g_center=0, and every leaf is regular. Finite-metric alignment therefore requires

\[ 0=\partial_{\theta_{center}}U_\phi =-\sum_{r=1}^m f(\delta_r). \]

Aligned leaves contribute f(0)=0. The balanced pair delta,-arcsin(sin(delta)) shows that f factors through the sine, including obtuse gaps. Define b(s)=f(arcsin(s)) on [-1,1]. Balanced triples give b(s+t)=b(s)+b(t) whenever all three arguments are in that interval. The same continuous additive-function argument as in the linked proof gives b(s)=kappa*s. Thus u'(delta)=kappa*sin(delta) on nonantipodal gaps; continuity and periodicity include the endpoints. Integrating and using u(0)=0 yields the displayed cost. On P2 at a nonzero regular gap, alignment requires H_U=kappa*pi*sinc(delta)>0, so kappa>0. Conversely, the existing representation supplies H_U=kappa*H_phi, proving sufficiency.

Metric at zero pressure. Where g_i!=0, the product fixes (H_U)_i=kappa*(H_phi)_i. Where g_i=0, both sides vanish, and any finite positive value at that point satisfies the product. Continuity on the full regular phase domain selects the same removable limit: nonzero g_i states are locally dense, since varying the node phase with neighbors held gives partial_i g_i=-1/pi. Without that continuity condition, changing only the zero-pressure metric entries preserves alignment but not the regular metric field. A numerical tolerance is not an exact zero set or a continuation rule.

An acute obstruction to the alternative edge cost. Section 13.8's V_phi+Q_phi, with Q_phi=sum_edges(1-cos(delta))^2, retains its stated symmetries and consensus Hessian, but fails this stronger alignment condition even if a different finite diagonal metric is allowed. Take a three-leaf star with center phase zero and leaf phases (alpha,alpha,-pi/6), where alpha=arcsin(1/4). Every edge is strictly acute. The center resultant is (sqrt(15)+sqrt(3))/2>0, its sine sum is zero, and therefore

\[ g_{center}=\partial_{center}V_\phi=0,\qquad \partial_{center}Q_\phi=\frac{\sqrt{15}-2\sqrt3}{4}>0. \]

No finite diagonal metric can multiply zero pressure into that nonzero gradient. This is a static obstruction, not a failed trajectory. It does not invalidate section 13.8's auxiliary-potential nonselection, which did not impose the present native-pressure alignment.

Limits of selection. If alignment is required only on strictly acute edge states, the argument fixes the cost only for |wrap(delta)|<pi/2. A smooth even periodic addition epsilon*b(cos(delta)), with b(t)=exp(-1/t^2) for t<0 and zero otherwise, vanishes throughout that acute interval but changes obtuse costs. Global classification therefore uses the stated full regular graph family; a restricted execution chamber cannot establish that global classification on its own. Nor does alignment derive edge additivity: U=V_phi+V_phi^2 has the positive metric (1+2*V_phi)*H_phi on the regular domain. That different construction generally consumes global storage and is outside the shared edge-cost class; its induced global metric factor is not an admitted primitive-local phase completion under the separate locality theorem.

Scale and joint-law integration. With fixed Dirichlet form storage, E_D+beta_U*U_phi=E_D+beta_eff*V_phi, where beta_eff=kappa*beta_U. The existing relational engine fixes the edge-cost normalization to one and already consumes this effective value as RelationalExchangeModel.storage_scale. No second cost-scale parameter is needed. Holding beta_U fixed while changing kappa changes the phase rates; it is not a harmless relabeling. The relative invariant remains chi=beta_eff*(e/w)^2, whose value is not selected here. The condition u(0)=0 also fixes an otherwise free constant per edge; such a constant drops out of fixed-support flow but can change support-event storage jumps. A convention for the former is not a derived event budget for the latter.

The alignment controls retain the exact obstruction, metric continuation, nonadditive alternative and effective-scale accounting, alongside the existing balanced-star controls. This result classifies a cost under explicit premises. It does not derive Dirichlet form storage, joint separability, an absolute energy scale, physical energy or primitive support formation. The separate joint-cost theorem does derive the quadratic/cosine split from a prescribed exact loss and uniform primitive-local completion on the full regular graph family. It admits mixed terms initially and needs neither the alignment premise used here nor a preselected phase row. Its stronger premises retain their own proof and scope.

13.7 Phase alignment representation does not close the joint law¶

Representing g as a gradient does not establish theta_dot=g. To examine that tempting completion, retain the section 13 premises: the EPI block uses M_x=diag(nu/d), there are no cross-mobility blocks, the phase block uses the newly derived H_phi^-1, and Psi is independent of EPI. With J_g=(R-I)/pi, the required phase velocity is

\[ \dot\theta=w_\phi H_\phi^{-1}J_g^\top D x -H_\phi^{-1}\nabla_\theta\Psi. \]

Making this equal g=-H_phi^-1*gradient V_phi would require gradient_theta Psi=gradient V_phi+w_phi*J_g^T*D*x. For w_phi>0 and J_g!=0, an EPI-independent Psi cannot satisfy that identity throughout an open EPI domain. The metric representation thus exposes the missing reciprocal interaction rather than removing it. Other block metrics, cross mobilities, constraints and non-gradient laws remain separate cases; the nodal identity still selects none of them automatically.

The retained unit prism gives a concrete same-phase comparison. Write q=(1,-1,0,1,-1,0), r_0=1+sqrt(3) and keep the prepared phases (0,pi/3,pi/6) in each triangle. Then Rq=R^Tq=0, D=3I and H_phi*q=3*r_0*q. Two EPI states differing by epsilon*q therefore require phase velocities differing by

\[ -\frac{w_\phi\epsilon}{\pi(1+\sqrt3)}q. \]

Every Psi term cancels in this comparison. For w_phi=1/4 and epsilon=1/16, the difference is -q/(64*pi*(1+sqrt(3))), whereas the isolated pressure vector g is identical in the two preparations. This is an exact conditional model obstruction, not a phase update or a runtime derivative certificate.

There is also a useful conditional budget, without installing that completion. Suppose a supplied exact-real model does use theta_dot=g and stays on the regular unit-prism domain for all time, with H_phi>=h_min*I, h_min>0. Assume fixed unit capacity and conductance, fixed EPI coefficient e>0, fixed phase coefficient w>=0, no other nonzero forcing, events or pressure residuals. Then

\[ \dot V_\phi=-\sum_i H_{\phi,i}g_i^2, \qquad \int_0^\infty\|g\|^2dt\le V_\phi(0)/h_{\min}. \]

Combining this with the existing internal-form budget and Young's inequality gives, for S=|z_0|^2+|z_1|^2,

\[ \dot S\le-eS+\frac{w^2}{e}\|g\|^2,\qquad \frac{d}{dt}\left(S+\frac{w^2}{e h_{\min}}V_\phi\right)\le-eS. \]

Consequently integral S dt<=S(0)/e+w^2*V_phi(0)/(e^2*h_min). The scalar comparison bound is an exponentially decaying initial term plus the convolution of exp(-e*t) with an integrable nonnegative input; both tend to zero, so S(t)->0. Thus this separately supplied phase-only relaxation cannot maintain nonzero internal amplitude under the stated uniform regularity hypotheses, even if it initially supports growth. It may retain phase structure with zero source; the result is not a theorem of phase consensus or a rejection of other TNFR completions. If the metric approaches zero or a branch/resultant boundary, this uniform bound does not apply. No future-domain or engine-trajectory claim is made here.

Portable controls reuse the existing current, curvature, forcing, phase response and regional-balance owners: phase alignment metric and joint reciprocity and cost budget. No second field-energy implementation or phase solver is introduced.

The new result is the intrinsic metric/cost representation and its conditional implications. A law generating a joint source, its reciprocal response, and an appropriate phase/capacity time scale remain open. The single execution plan retains those dependencies before any maintenance experiment.

13.8 Structural restrictions do not select the auxiliary potential¶

Retain the section 13.7 potential family and phase metric. Even common-phase rotation invariance, individual 2*pi periodicity, graph relabeling covariance, undirected edge locality, nonnegativity and the same consensus Hessian do not select Psi. In the already declared nondimensional comparison, put

\[ Q_\phi=\sum_{\{i,j\}\in U}(1-\cos(\theta_j-\theta_i))^2. \]

The two logical alternatives Psi=V_phi and Psi=V_phi+Q_phi have all those properties. Neither is installed as a constitutive law. Since each edge cost a=1-cos(delta) lies in [0,2], 0<=Q_phi<=2*V_phi; the alternatives have identical zeros. The added term has zero gradient and Hessian at consensus, so their quadratic normalization and linearization agree. On strict U3 edges, the respective edge curvatures are

\[ \cos\delta>0,\qquad \cos\delta+2(1-\cos\delta)(1+2\cos\delta)>0. \]

Thus even local convexity on that phase domain does not remove the freedom. Dimensional analysis can require a cost scale; it cannot select between these two dimensionless edge functions after that scale is fixed.

At the prepared prism phases, with q=(1,-1,0,1,-1,0) and r_0=1+sqrt(3), exact evaluation gives

\[ Q_\phi=\frac{15}{2}-4\sqrt3,\qquad \nabla Q_\phi=-q. \]

For the same state, coefficients and phase block H_phi^-1, replacing the first alternative by the second leaves the instantaneous EPI rate unchanged but changes the conditional rates by

\[ \Delta\dot\theta=\frac{q}{3r_0},\qquad \Delta\dot p=-\frac{q}{12\pi r_0},\quad w_\phi=\frac14. \]

These are ideal derivatives of two supplied joint models, not derivatives of binary64 trigonometric execution. The captured current-pressure residual is retained separately. Nor does V_phi+Q_phi represent the original isolated g under the same metric: this comparison concerns the still-free auxiliary part of the joint potential. No phase clock or full-pattern stability follows. The edge-storage classification gives the separate acute zero-pressure obstruction to representing V_phi+Q_phi with any finite diagonal metric while retaining native g. Its added alignment premise does not change the present auxiliary-potential result.

13.9 Joint source compatibility and the common-mean saddle¶

Fix support, conductance and capacity, and write C(theta)=d^T F(theta). The letter C here denotes source compatibility, not the coherence metric. The conditional potential family forced by the EPI block is

\[ \mathcal E(x,\theta)=\frac e2x^TBx-x^TDF(\theta)+\Psi(\theta), \qquad B\mathbf1=0. \]

Other held arguments of Psi and F are suppressed. Under a common EPI shift,

\[ \mathcal E(x+c\mathbf1,\theta)-\mathcal E(x,\theta)=-cC(\theta), \qquad \Delta\dot\theta=cH_\phi^{-1}\nabla C. \]

Pointwise source compatibility C=0 does not establish phase-rate covariance, whose condition here is gradient C=0. These two pointwise conditions are logically independent. Indeed phase-rotation invariance gives 1^T*gradient C=0. A positive definite phase mobility cannot map a nonzero gradient C to a pure common rotation, since that would contradict its positive quadratic form. This tests inheritance of the frozen nodal flow's shift symmetry; it does not declare absolute EPI unphysical or remove the Emission/vacuum contracts.

Centering changes the law unless justified separately. For fixed weights rho^T*1=1, replacing the cross term by -(Px)^T D F, P=I-1*rho^T, changes the pressure to

\[ p_{\rm centered}=p-C D^{-1}\rho. \]

Its phase derivative differs by -D^-1*rho*(gradient C)^T, even at a point where C=0. Degree-weighted centering subtracts uniform pressure C/sum(d); centering with weights proportional to d/nu subtracts a uniform nodal drift. An invertible chart retaining the mean retains the omitted coupling term. A fixed-mean constraint or an invariant C=0 manifold therefore needs its own derivation; it is not supplied by recentering a diagnostic.

Stationary-pattern obstruction, independent of Psi. Assume a C^2 potential on an open EPI/phase chart. At a point with C=0 and gradient C!=0, choose a phase direction v with b=gradient C dot v!=0. Restrict its second variation to x=x_*+a*1, theta=theta_*+t*v. Its Hessian on this plane is

\[ \begin{pmatrix}0&-b\\-b&k\end{pmatrix},\qquad \det=-b^2<0, \]

where k includes every possible C^2, EPI-independent Psi. The zero entry follows from B*1=0, and the mixed entry from -a*C(theta). Consequently any full critical point there is a saddle; if it is not critical, it already cannot be an interior local minimum. Smooth symmetric positive definite gradient mobility makes such a critical point linearly unstable: -M*Hessian(E) is similar to the symmetric matrix -sqrt(M)*Hessian(E)*sqrt(M), which has a positive eigenvalue. This last statement also applies to cross mobility for this same potential family; allowing cross mobility when deriving the EPI law in the first place can change the family and is not excluded by this argument.

The retained source has exactly this obstruction. For the prepared phases, unit capacity/conductance and e=1/2, w_phi=1/4, put

\[ q=(1,-1,0,1,-1,0),\quad s=(1,1,-2,1,1,-2),\qquad x_*=\tfrac12\mathbf1+\tfrac1{12}q. \]

This is an ideal exact zero-pressure reference with source F=q/24; it is distinct from the earlier growing preparation with amplitude 1/16. Although C=0,

\[ \nabla C=-\frac{w_\phi\gamma}{\pi}s\ne0,\qquad \gamma=\frac{3(2-\sqrt3)}{2(1+\sqrt3)}>0. \]

The reference EPI lies strictly between zero and one and the largest edge phase gap is pi/3<pi/2, so sufficiently small two-sided variations are admissible. No binary64 zero-pressure equilibrium is asserted. As an additional control, the hypothetical completion Psi=V_phi at this reference has C_dot=(1/2)*gradient C^T*H_phi^-1*gradient C>0: initial compatibility does not even persist under that particular trial law.

Repeated-triple family corollary. This is not just an isolated phase choice. On the unit prism, let each triangle carry the same arbitrary three phases, with common nonzero neighbor resultant r*exp(i*beta) and a regular phase-source chart. Every row contains one copy of each of those phases. Column sums of R are 3*cos(theta_j-beta)/r. Hence J_g^T*d=0 forces each of the three cosines to equal r/3>0. Their sines have equal magnitude and sum to zero by definition of beta. Three signed copies of a nonzero magnitude cannot sum to zero, so every sine vanishes. The positive cosine then forces phase consensus modulo 2*pi. No semicircle premise needs to be added; derivative compatibility itself implies it.

For w_phi>0 and uniform capacity on this unit support, every nonconsensus regular repeated triple therefore has either C!=0, excluding a critical point in the common-EPI direction, or C=0 and gradient C!=0, giving the saddle above. This excludes an interior stationary minimum supported by such phases throughout this family, for every admissible Psi. It does not classify arbitrary six-phase configurations, resultant singularities, other supports or moving patterns.

The obstruction does not require the two-dimensional test plane to be dynamically invariant. Allowing capacity to vary cannot restore an interior minimum while this negative direction remains admissible. Conversely a derived fixed-mean constraint, a boundary point, degenerate mobility, non-gradient motion, moving identity or a different justified joint law lies outside the conclusion. This rejects stabilization of this preparation by choosing Psi alone in the stated gradient class, not autonomous TNFR patterns generally.

Controls reuse the existing forced-support solver and the shared prism geometry/phase-metric fixture: auxiliary freedom and compatibility and stability. The single execution plan owns the next admission gate; no new potential, dynamics API or trajectory is introduced here.

13.10 Persistent motion is a separate dynamical target¶

A maintained NFR need not be a stationary point. The interior-minimum obstructions above concern one conditional gradient class; they do not define persistence for the whole TNFR programme. A periodic state, a relative equilibrium under a stated symmetry, a response sustained by its environment, and a transient oscillatory observation are different claims. Specify the full state, the identity observation and any allowed symmetry before testing one. Constant amplitude alone does not establish identity or robustness.

There is nevertheless a useful stronger exclusion within pure gradient dynamics. Let the complete continuous state obey

\[ \dot z=-M(z)\nabla\mathcal E(z),\qquad M=M^T\succeq0, \]

with a single-valued, time-independent differentiable energy and sufficiently regular autonomous flow. Then

\[ \dot{\mathcal E}=-\nabla\mathcal E^T M\nabla\mathcal E\le0. \]

A periodic return, recurrence to the initial state, or motion entirely along an energy-invariant symmetry orbit forces the energy to be constant. For a positive semidefinite symmetric matrix, zero quadratic form implies M*gradient E=0, hence z_dot=0. Thus even degeneracy cannot generate a nonconstant recurrent orbit in this pure gradient model. Positive definiteness additionally forces gradient E=0. This does not exclude a transient oscillatory projection, a nonrecurrent drifting state or recurrence of a lossy observation while hidden energy is consumed.

A common phase rotation preserves canonical phase differences and regular g; it changes neither the phase source nor its work by itself. If the joint energy is rotation invariant, that rotation is also excluded as nonzero motion of a pure gradient law. A supplied angular clock can generate it in a different law, but the phase symmetry does not derive that clock.

Possible recurrent completions must therefore justify something absent from this restricted class: for example a circulation term, an explicit sustaining boundary, a time-dependent input, or hybrid energy transfers with their actual occurrence law. The algebra z_dot=(J-M)*gradient E, J^T=-J, shows why an antisymmetric term does no energy work. It does not derive J, a Poisson structure or a TNFR phase velocity. The existing auxiliary substrate and decoupled product in section 3 retain their realizability/closure obligations.

13.11 Cycle budgets reuse the nodal transport owners¶

Pointwise pressure equilibrium is too strong a test for a moving pattern. For fixed symmetric connected conductance, positive fixed capacity and a declared source F(t), the actual EPI model is

\[ \dot x=\operatorname{diag}(\nu)(-eL_{\rm rw}x+F(t)),\quad H=\operatorname{diag}(d/\nu),\quad m_H=\frac{\mathbf1^THx}{\mathbf1^TH\mathbf1}. \]

Writing C(t)=d^T F(t) gives

\[ \dot m_H=\frac{C(t)}{\sum_i d_i/\nu_i}. \]

A periodic EPI field requires integral_0^T C(t) dt=0, not C(t)=0 at every instant. A relative pattern translating in common EPI may have a nonzero mean drift; it is not a bounded recurrent full EPI state. The held case is already implemented by derive_forced_support_balance; no second Poisson solver or mean-centering law is needed.

For the retained unit prism with unit capacity, let y be the within-triangle centered EPI, z_a=sqrt(2)*u_a+i*sqrt(6)*v_a, S=|z_0|^2+|z_1|^2=||y||^2 and D_z=|z_0-z_1|^2. Summing the existing exact regional balances gives

\[ \dot S=-2eS-\frac{2e}{3}D_z+2\langle y,F\rangle. \]

Whenever S(T)=S(0),

\[ \int_0^T\langle y,F\rangle\,dt =e\int_0^T S\,dt+\frac e3\int_0^T D_z\,dt. \]

For positive e and continuous nonzero internal form this work is positive. A zero-average source can still supply positive work through its correlation with the moving form. Returning S is only an accounting condition, not a return of the full pattern. Stored-pressure defects add their own work; changing capacity/geometry, boundaries, Euler steps and events require the existing derivative/finite/reset owners. None may be silently omitted.

An exact driven control with canonical phase pressure. Set q=(1,-1,0,1,-1,0) and prescribe identical phase triples (-a(t),a(t),0) in the two triangles. For a(t)=A*cos(omega*t), 0<A<pi/4, omega>0, every neighbor resultant is the positive real number 1+2*cos(a) and the largest edge gap is below pi/2. The canonical ideal pressure is therefore g=(a/pi)*q. Use nondimensional structural time with common capacity normalized to one. With F=w*g, L_rw*q=q and e,w>0,

\[ x=\mu\mathbf1+u(t)q,\qquad u(t)=\frac{wA}{\pi(e^2+\omega^2)} (e\cos\omega t+\omega\sin\omega t) \]

solves u_dot=-e*u+w*a/pi exactly. In an unscaled time with common capacity nu, replace the decay coefficient by nu*e and forcing coefficient by nu*w; the denominator becomes (nu*e)^2+omega^2. Here S=4u^2 and the cycle source work equals dissipation. If a scalar EPI band is required, choose a declared mean and amplitude whose entire analytic range lies in that band. The general scalar solution differs by constant*exp(-e*t) under this supplied source. No phase evolution is derived, no graph is integrated and no autonomous periodic orbit of the full state is certified.

The response amplitude is w*A/(pi*sqrt(e^2+omega^2)), strictly decreasing for omega>0. It has no nonzero-frequency resonance peak in this particular first-order channel. This does not rule out resonance in another justified joint law; it prevents mistaking a periodically driven relaxation for such a derivation. Neither this frequency nor the auxiliary graph-wave sqrt(lambda) is selected by the nodal product alone.

The exact controls are in test_resonant_pattern_scope.py. They reuse the shared prism and regional budgets, and separate the analytic source from represented runtime pressure. The resonant-persistence audit maps the existing implementations; the single execution plan retains the constitutive gate for a real autonomous mechanism.

13.12 Oriented source/form work without a selected clock¶

The preceding cycle balance concerns the squared internal amplitude. A different identity tests the oriented joint loop without first choosing its traversal speed. Fix symmetric conductance with positive row strengths, positive fixed capacities and coefficient e>0. Set B=D-W, H=diag(d/nu) and retain the actual EPI law

\[ H\dot x=-eBx+DF. \]

Taking its scalar product with dx=dot(x)*dt and integrating an actual trajectory segment yields

\[ \int F^TD\,dx =\frac e2[x^TBx]_{t_0}^{t_1} +\int_{t_0}^{t_1}\dot x^TH\dot x\,dt. \]

When x(t_1)=x(t_0), this becomes

\[ \boxed{\ \int_{t_0}^{t_1} F^TD\,dx=\int_{t_0}^{t_1}\|\dot x\|_H^2dt>0\ } \]

whenever its continuously differentiable EPI path is nonconstant. This is work against form displacement, distinct from the amplitude-budget integral integral <y,F> dt in section 13.11. The left side is an oriented line integral along the traced joint (x,F) path. If the source also returns, this is a closed joint loop and may be written with oint. Closing EPI alone does not guarantee that stronger return. The integral is unchanged by an orientation-preserving reparameterization, negated by reversing the path, and zero for a constant held source on a closed EPI path. A nonconstant periodic EPI path under a held source is therefore excluded in this fixed model without choosing a global time unit.

The right side uses the actual time/capacity of an admitted solution. Arbitrarily changing its speed while keeping the same fixed capacities and pressure law does not generally produce another solution. Under a consistent change d tau/dt=h>0, dx/dtau=dot(x)/h and the effective EPI metric is H_tau=hH, so its kinetic integral is preserved. This is a change of coordinate for the complete supplied vector field, not a constitutive invariance theorem with unchanged coefficients. The multichannel pressure's capacity dependence must also be transformed consistently. The parameter foundations derive the separate constant-time-unit special case; they do not establish arbitrary state-dependent clock covariance of the configured engine. The line integral is not a license to rescale one channel in isolation.

Reciprocal phase geometry. If all nonphase source components are held and F=w_phi*g(theta)+F_0, a closed joint source/form loop satisfies integration by parts:

\[ \oint F^TD\,dx =-w_\phi\oint x^TD J_g(\theta)\,d\theta. \]

The regular source Jacobian J_g=(R-I)/pi is the existing response owner. Common rotation has J_g*dtheta=0 and cannot supply this loop work. More generally a finite path preserving the entire source gives zero work when EPI closes. Relative source/form variation with the proper orientation is necessary here; a high synchronization score alone says nothing about it. Positive varying capacity alone leaves the first work identity valid with the instantaneous H(t)=diag(d/nu(t)): no metric derivative is taken in this velocity-action identity. If capacity variation also changes the source, integration by parts must include that part of dF; the phase-only expression then does not suffice. Changing support, conductance or e requires the corresponding Dirichlet derivative and event terms. These belong to the existing derivative and event budgets, not an unchanged fixed-coefficient formula.

For the supplied prism control of section 13.11, in normalized structural time, exact evaluation gives

\[ \oint F^TD\,dx =\frac{12w^2A^2\omega}{\pi(e^2+\omega^2)}. \]

This equals the positive nodal-rate integral and the reciprocal phase line integral. Reversing the same geometric loop makes the left side negative, so that orientation cannot solve the same positive-capacity nodal law. Changing omega in the displayed control changes the EPI amplitude and lag, hence changes the loop itself; it is not just a relabeling of one loop.

The positive value is a necessary admission result, not a derivation of the loop, its clock, attraction or autonomous phase feedback. The existing common-mean/phase saddle and gradient recurrence exclusions are not bypassed: any proposed reciprocal completion still needs its full energy/work balance. The relational-clock tangent test supplies another necessary condition before a curve may be called a trajectory. Exact controls reuse the shared conductance and phase geometry in test_phase_form_cycle_work_scope.py. No new force, integration path or physical-time calibration is introduced.

13.13 Work pulled back to a proposed geometric phase relation¶

A phase observation proposed as an endogenous source must pass a geometric test as well as the earlier full-vector-field test. Let z be coordinates on a regular open chart, and declare x=chi(z), theta=Theta(z). Hold conductance, the EPI coefficient, positive capacities and all nonphase source components fixed. Put F=w*g(Theta)+F_0, with fixed w>0. The source work one-form on this chart is

\[ \alpha=F^TD\,D\chi\,dz. \]

For twice differentiable chi and a differentiable regular phase map, define

\[ K=w(D\chi)^TDJ_gD\Theta. \]

The mixed derivatives of chi cancel in the exterior derivative of alpha; the constant source contributes the exact differential d(F_0^TD chi). Consequently d alpha=0 exactly when K=K^T. This criterion must hold throughout the chart, not merely at one sampled state. On a simply connected chart it makes alpha exact, hence every closed embedded loop has zero work. Section 13.12 then excludes a closed embedded-state cycle with nonconstant EPI under the stated law. EPI return alone suffices only if chi is injective on the chart or otherwise guarantees return of the latent state; a lossy EPI map can hide different endpoints of the work primitive. On a domain with holes, closedness alone does not establish global exactness. A nonzero antisymmetric part only permits local circulation; it proves neither nodal tangency nor an autonomous orbit.

This reuses the canonical source Jacobian J_g, the actual conductance metric D, and the proposed observation map. Testing symmetry of the raw phase Jacobian, or only counting available coordinates, would answer a different question. No additional field, energy API or phase solver is needed.

Exact two-coordinate prism control. Use the retained repeated-triple chart, with P=(1,-1,0)^2, Q=(1,1,-2)^2, x=mu*1+u*P+v*Q, fixed mu, and the declared phase relation Theta=beta*1-pi*kappa*(x-mu*1), kappa>0. Choose a simply connected neighborhood of (u,v)=(0,0) with a fixed nonzero-resultant branch and strict U3. Each node has one neighbor of each triple type, so its neighbor resultant is common and

\[ g=\kappa(uP+vQ)+c(u,v)\mathbf1. \]

The common component drops out of the fixed-mean work because 1^T P=1^T Q=0, D=3I, P^T P=4, Q^T Q=12 and P^T Q=0:

\[ \alpha=w\kappa(12u\,du+36v\,dv) =d\{w\kappa(6u^2+18v^2)\}. \]

This phase-from-form relation may support instantaneous amplitude growth but cannot supply positive net work around a closed embedded loop. It is a declared geometric candidate, not a derived phase law. Its fixed-mean chart also need not be invariant: at u=0, v=1/12, kappa=1, the common resultant has imaginary part 1/2-2*sin(pi/12)!=0, so the common phase source generally moves the EPI mean. Dropping that source would alter the model.

Allowing independent source and form coordinates can instead give nonzero circulation. For x=mu*1+uP, theta=-aP, the work is alpha=(12w/pi)*a du; its exterior derivative is nonzero. This is the geometry already used by the supplied loop in section 13.12, not proof that a has its own justified equation. A freely drawn loop can fail nodal tangency at an EPI turning point despite positive total work. Controls: pullback scope.

13.14 Existing optional feedback: pressure consistency before recurrence¶

The optional extended integrator contains a phase response depending on pressure, so it is a concrete candidate to audit rather than inventing a feedback term. Its full pressure block, however, evolves stored pressure independently and fails the fresh-pressure chain rule on the retained prism. The constitutive audit owns that exact positive-squared-Laplacian obstruction. It is not a canonical phase/form closure or a source of admitted resonant motion.

A separate conditional comparison isolates what the existing phase formula would do if paired with freshly realized pressure. This substitution is not the implemented optional model and is not installed as a new one. Retain unit capacity, the repeated antisymmetric prism family x=mu*1+uP, theta=-aP, no other source, and let c=w_phi/pi>0, e=w_epi>0. Then

\[ p=-\eta P,\quad \eta=eu-ca,\qquad J_\phi=j(a)P,\quad j(a)=\frac{\sin(2a)+\sin a}{3}. \]

Write the existing configured phase coefficients as A=0.5, B=0.15, G=0.135, and hold the prism's common configured coupling kappa>0 fixed. Set f(eta)=A*sin(pi*eta)+B*eta, k=G*kappa>0. Its phase formula and the fresh EPI product give

\[ \dot u=-\eta,\qquad \dot a=f(\eta)-k j(a),\qquad \dot\eta=-e\eta-c f(\eta)+ck j(a). \]

No coefficient or clock has been inferred by writing these supplied laws. On |eta|<1, |a|<pi/4, consider the derived comparison function

\[ V=\int_0^\eta f(s)ds+ck\int_0^a j(s)ds =\frac A\pi(1-\cos\pi\eta)+\frac B2\eta^2 +ck\left(\frac{1-\cos2a}{6}+\frac{1-\cos a}{3}\right). \]

Direct substitution yields

\[ \dot V=-e\eta f(\eta)-c\{f(\eta)-k j(a)\}^2. \]

Here eta*f(eta)>0 for nonzero eta and j(a)=0 only at a=0. Thus V_dot<0 away from (eta,a)=(0,0) on this chart. A nonconstant periodic orbit contained there is excluded. This neither proves forward invariance of the chart nor classifies its exterior or the actual optional runtime. It shows why pressure-dependent feedback alone is insufficient evidence for sustained resonance: the fresh-pressure comparison is dissipative here, while the implemented independent-pressure completion has a different and incompatible pressure slope. Exact and production-reader controls share one test module.

The phase/form exchange owner extends these work tests to radial/tangential source projections, transported internal frames and a supplied circular phase-contrast response with a moving mean. Its primitive source and clock remain prescribed. The full work includes the mean even when the time integral of the mean source is zero.

The phase-origin admission study complements this work test with actual prism reflection constraints and the relative oriented-area budget of independent phase/form state. Its joint linear classification places the same hypothetical fresh-pressure feedback in a nonoscillatory stable contrast class near uniform state. The nonlinear one-direction proof above and that local two-direction result have distinct scopes; neither certifies the implemented independent-pressure extension.

13.15 Full repeated-triple feedback and the mean-work term¶

The one-direction result in section 13.14 extends to the two internal directions only when the common phase pressure is retained. This remains a hypothetical fresh-pressure comparison of the configured optional phase formula, not the implemented independent-pressure model or a selected TNFR completion. Fix the repeated unit prism, unit capacities, positive weights e,w, k=w/pi, and held common g>0. Write

\[ p=h+q\mathbf1,\quad h=-ey-k\eta,\quad q=w c(\eta),\quad v=\dot\eta=\Pi f(p)+gJ(\eta),\quad \dot h=-eh-kv, \]

where sum h=sum eta=0, Pi=I-11^T/3, and f(s)=A*sin(pi*s)+B*s, A,B>0. The common EPI rate is mu_dot=q, and the common phase rate is beta_dot=mean f(p). Let

\[ F(s)=\frac A\pi(1-\cos\pi s)+\frac B2s^2,\qquad V(\eta)=\sum_{i<j}[1-\cos(\eta_i-\eta_j)]. \]

The actual three-neighbor mean-sine current is J=-grad V/3. Define the comparison storage, counted once per repeated triple,

\[ \mathcal E=\sum_i[F(h_i+q)-F(q)]+\frac{kg}{3}V(\eta). \]

If all pressures and their mean lie in a convex interval where f'>=m>0, Jensen's inequality gives E>=m*||h||^2/2+(kg/3)V>=0. Exact differentiation using both nodal rows yields

\[ \boxed{\quad \dot{\mathcal E}=-e\sum_i h_i f(p_i)-k\|v\|^2 +\underbrace{\left[\sum_i f(p_i)-3f(q)\right] w\,Dc(\eta)[v]}_{\mathcal W_{\rm mean}}. \quad} \]

The last term is a nonlinear mean-pressure contribution. It vanishes for affine f, but can have either sign for the sine response, even at a state where c=0: zero current value does not imply zero derivative. Omitting it would turn the one-direction dissipation proof into a false unrestricted two-direction proof. It is a balance term, not evidence of an independently generated energy supply.

A sufficient complete-mean dissipation criterion. Suppose on a declared domain that f'>=m>0, |f''|<=M, ||h||<=H and the derivative of c restricted to centered phase directions has norm at most C. Taylor's theorem and sum h=0 give

\[ \left|\sum_i f(p_i)-3f(q)\right|\le\tfrac M2\|h\|^2, \qquad \sum_i h_i f(p_i)\ge m\|h\|^2. \]

Thus, with gamma=w*M*C*H/2,

\[ \dot{\mathcal E}\le-em\|h\|^2-k\|v\|^2+ \gamma\|h\|\|v\|. \]

The right side is strictly negative away from h=v=0 whenever gamma^2<4*e*m*k. This is a declared-domain test; its bounds are not new evolution coefficients. Within the strict phase chart, h=v=0 also implies eta=0: then v=gJ, and eta^T J=-(1/3)sum_pairs (eta_i-eta_j)*sin(eta_i-eta_j)<0 for nonzero centered eta. The complete stationary contrast has q=0.

An explicit finite-pressure class. On any centered triple whose phase spread is strictly less than pi/2, write a=Arg(sum exp(i eta_i)) and s_i=cos(eta_i-a)/|sum exp(i eta_i)|. Each s_i is positive and sum s_i=1, so

\[ \|Dc|_{\mathbf1^\perp}\|^2 =\frac{\sum_i s_i^2-1/3}{\pi^2}\le\frac{2}{3\pi^2}. \]

For the declared exact-real coefficients e=1/2, w=1/4, A=1/2, B=3/20, take the natural monotonic-sine pressure interval |p_i|<=1/2. Then one may use m=3/20, M=pi^2/2, H=sqrt(3)/2, C=sqrt(2/3)/pi. Consequently

\[ \gamma=\frac{\pi\sqrt2}{32},\qquad \frac{\gamma^2}{4emk}=\frac{5\pi^3}{192}<\frac{64}{75}<1, \]

where pi<16/5 suffices for the strict rational bound. This proves strict contrast dissipation on the stated class despite the potentially positive mean-work term. It excludes a nonconstant periodic orbit wholly contained in that class. It does not prove forward invariance, convergence of all initial states, behavior beyond this pressure interval, binary64 runtime stability or the independent-pressure extension's consistency. The existing local coupling estimator depends only on degree, so it is common and constant on this fixed prism. A changing coefficient would add (k*g_dot/3)*V to the displayed balance and would require a new bound; it cannot silently inherit the result.

The finite-amplitude orientation witness lies in this pressure class and creates oriented area while this storage decreases. Angular generation and sustained identity are therefore distinct even in one explicitly specified nonlinear comparison. The single G3 queue must retain both tests. No phase formula, coefficient or pressure-refresh policy is changed by this derivation.

Exact controls: complete mean-work balance.

13.16 Full-prism feedback: boundary exchange does not remove all dissipation¶

The repeated-triple restriction can be removed on the same unit prism. Keep fixed unit capacities, fixed e,w,g>0, k=w/pi, and the conditional fresh-pressure comparison of sections 13.14-13.15. In a regular phase chart, write the actual six-node equations as

\[ p=-eLx+w g_\phi(\theta),\qquad \dot x=p,\qquad v=\dot\theta=f(p)+gJ(\theta),\qquad \dot p=-eLp+k(R-I)v. \]

Here L is the unit prism's symmetric random-walk Laplacian, R is the actual neighbor-phasor mean derivative, R1=1, and J is the mean-sine current. This is not the optional runtime's independent pressure derivative. No repeated phase, form, or triangle mean is assumed. Set q=mean p, h=p-q1, u=Pi v, Pi=I-11^T/6. With F'=f as above and V=sum_edges(1-cos(theta_i-theta_j)), each undirected edge counted once, grad V=-3J and sum J=0. The full-graph comparison storage is

\[ \mathcal E_6=\sum_{i=1}^6 F(p_i)-6F(q)+\frac{kg}{3}V, \]

with the exact balance

\[ \boxed{\quad \dot{\mathcal E}_6 =-e f(p)^TLp-k\|u\|^2 +k\,[f(p)-f(q)\mathbf1]^T R u.\quad} \]

The last term contains the full nonlinear neighbor response, including the common pressure rate. To verify cancellation of the common phase velocity, use v=u+mean(f(p))*1, R1=1, and sum J=0; do not set that velocity to zero. On repeated triples the formula reduces to twice section 13.15's balance. Its last term is not generally zero and cannot be discarded as internal exchange. Thus the previous proof is reused through its storage and exact chain rule, not extended by assuming the same reduced equations.

Suppose all six lifted phases have spread less than pi/2, and all |p_i|<=1/4. The regular phasor geometry makes R nonnegative with row sums one, giving ||R||_2<=sqrt(6). For A=1/2, B=3/20,

\[ m=B+A\pi/\sqrt2\le f'(p_i)\le M=B+A\pi. \]

These bounds hold on the whole pressure interval, including its mean. The graph's gap is lambda_2=2/3. Pairwise monotonicity and the gap give f(p)^T Lp>=m lambda_2 ||h||^2; componentwise Lipschitz continuity gives ||f(p)-f(q)1||<=M||h||. Therefore

\[ \dot{\mathcal E}_6\le -em\lambda_2\|h\|^2-k\|u\|^2+kM\sqrt6\|h\|\|u\|. \]

For e=1/2, w=1/4, strict negativity away from h=u=0 follows from 6kM^2<4em lambda_2. The constants need no numerical parameter scan: 3<pi<16/5 and sqrt2<3/2 imply

\[ 6kM^2<\frac{49}{32}<\frac{23}{15}<4em\lambda_2, \qquad 735<736. \]

Storage is nonnegative, with E_6>=m||h||^2/2+(kg/3)V. In this strict phase chart, h=u=0 forces J=0, hence uniform phase by connected attractive sine coupling. Then g_phi=0, q=0, and Lp=0 with p=-eLx forces uniform form. Consequently no nonconstant periodic full-state orbit, or recurrent nonuniform quotient orbit, can lie wholly within this domain. Inter-triangle exchange does not evade this sufficient dissipation test.

This six-node theorem uses a narrower pressure band than the repeated theorem (1/4 versus 1/2) and controls additional directions. Neither declared band is proved forward invariant. This is not global convergence, a theorem for larger graphs or variable capacities, a binary64 solver result, or admission of the comparison as a physically derived law. Controls: full nonlinear balance.

13.17 All spatial modes and local nonlinear attraction¶

Near uniform form and phase, the same conditional model has an exact local classification on the whole prism. At uniform phase, both implemented support derivatives share L: Dg_phi=-L/pi, DJ=-L. Write sigma=f'(0)=A*pi+B>0. Every Laplacian mode lambda carries the block

\[ \frac{d}{dt}\begin{pmatrix}\delta x_\lambda\\\delta\theta_\lambda\end{pmatrix} =\lambda\begin{pmatrix}-e&-k\\-\sigma e&-\sigma k-g\end{pmatrix} \begin{pmatrix}\delta x_\lambda\\\delta\theta_\lambda\end{pmatrix}. \]

The full spatial spectrum is 0,2/3,1,1,5/3,5/3. For every lambda>0, the temporal characteristic polynomial is

\[ r^2+\lambda(e+\sigma k+g)r+\lambda^2eg. \]

Its discriminant divided by lambda^2 is (e-g)^2+2*sigma*k*(e+g)+(sigma*k)^2>0. Thus both roots are real and strictly negative for any positive e,k,sigma,g. The inter-triangle mean mode and the opposite internal modes add no growing or oscillatory linear direction. The two neutral directions are the independent uniform shifts of EPI and primitive phase; their neutrality is not a pulsation.

There is also a local nonlinear consequence, distinct from an instantaneous linear diagnostic. The regular continuous comparison is smooth and invariant under both uniform shifts. Its ten-dimensional centered quotient is therefore autonomous, with a Hurwitz Jacobian at the origin. Local linearization stability gives a sufficiently small exponentially attracting quotient neighborhood. The absolute means can drift during that decay; their smooth rate functions vanish with the contrast and are integrable, so both approach finite constants. For example, along the nonrepeated small phase preparation theta=t*(0,0,0,1,1,-2), the mean phase pressure is t^3/(18*pi)+O(t^5). Mean stationarity cannot be imposed even this close to consensus. Here t is a preparation amplitude, not trajectory time.

This proves existence of a local basin for the exact conditional ODE, not that every state in section 13.16's displayed pressure/phase band belongs to it. No basin radius, global attraction, recurrence outside these domains, event law, or implemented-runtime stability is established. It closes the small-amplitude, missing-spatial-mode loophole without a trajectory campaign. Controls: full local mode and mean tests.

13.18 Source sensitivity, joint work and the passive-transfer limit¶

This closes the source-asymmetry check before returning to the maintenance mechanism. Fix the reciprocal unit prism, unit capacities, fixed e,w>0, regular phase geometry and freshly constituted pressure p=-eLx+w*g(theta). Let Pi center the complete six-node field, y=Pi*x, mu=mean(x), omega=theta_dot and J=Dg. Then

\[ \dot\mu=w\,\overline g,\qquad \frac{d}{dt}\frac{\|y\|^2}{2} =-e\,y^TLy+w\,y^Tg, \qquad \dot p=-eLp+wJ\omega. \]

The instantaneous contribution to form contrast is w*y^T*g. The matrix J enters its change, through the still-unspecified omega; matrix cycle imbalance is not an additional term in the nodal equation. In particular,

\[ \ddot\mu=w\,\overline{J\omega},\qquad \frac{d^2}{dt^2}\|y\|^2 =2\|\Pi p\|^2-2e\,y^TLp+2w\,y^TJ\omega. \]

Unchanged-preparation controls. Keep the shared EPI x=(3/4,1/4,1/2) repeated, e=1/2, w=1/4. At the retained phase theta=(-a,a,0,-a,a,a), a=pi/6, put gamma=atan(1/(3*sqrt(3))). The phase reflection has exactly the same R and cycle ratio, but opposite g. Its contribution to d||y||^2/dt is respectively +1/12 and -1/12; total rates are -1/6 and -1/3. Both lose contrast. The mean rates are opposite as well: mu_dot=+/- (3*gamma-pi/6)/(24*pi). Detached production pressure readings verify those exact-model expressions with their floating-point scope.

At the same unreflected state, the three tangent controls omega=0,+u,-u, u=(-1,1,0,0,0,0), give phase contributions to d^2||y||^2/dt^2 of zero and +/- (40-7*sqrt(3))/(112*pi), and mean accelerations zero and +/-1/(168*pi). These are local derivatives, not chosen trajectories. Common phase rotation contributes zero. Thus neither the sign of work nor its subsequent change is determined by the cycle defect alone. Controls: source work, sensitivity and mean.

A conditional storage criterion for maintenance. For a fixed declared comparison coefficient beta>0, with the units needed to compare the two terms, define

\[ \mathcal E=\frac12\|y\|^2+\beta V_\phi\ge0,\qquad \mathcal R=w\,y^Tg-\beta g^TH\omega. \]

This is a mathematical storage and its signed transfer residual, not a new TNFR parameter, physical energy law, external power meter or selector. No beta is fitted or installed in the dynamics. The exact identity is

\[ \dot{\mathcal E}=-e\,y^TLy+\mathcal R. \]

It retains both work exchanged with form and the change of phase storage. In the class R<=0, phase/form transfer cannot maintain nonzero contrast indefinitely. For example the supplied comparison family omega=(w/beta)*H^-1*y+lambda*g+c*1, lambda>=0, gives R=-beta*lambda*g^THg<=0. State-dependent skew exchange preserves power; this does not prove a Poisson structure or derive that comparison law. Common rotation cancels because g^TH*1=-1^T gradient V_phi=0.

More generally, let B(T) be an independently justified upper bound on integral_0^T R(t) dt for the specified complete trajectory. Since this prism has gap lambda_2=2/3,

\[ e\lambda_2\int_0^T\|y(t)\|^2dt \le \mathcal E(0)+B(T). \]

If ||y||>=rho>0 throughout that interval, its duration obeys e*lambda_2*rho^2*T<=E(0)+B(T). With a uniform finite cumulative bound, indefinite uniformly nonzero form and nonzero periodic centered form are excluded while the hypotheses persist. Gauge motion at zero contrast is not excluded. Integral control alone proves neither pointwise convergence nor domain invariance; no binary64 or changing-support theorem is implied.

Conversely, sustained contrast with bounded storage requires the cumulative residual to pay its integrated loss. Sufficient positive cumulative or average residual is a necessary accounting condition, not pointwise positivity or a generation law: held phase support can satisfy it without explaining why the phases are held. Nor does this diagnostic residual establish a literal physical external supply. A proposed complete law must determine its value independently and retain any changing capacity, support, boundary, memory or event contributions. Those terms cannot be inserted retrospectively to make the balance close. Controls: passive joint storage and duration bounds.

The relational passive-loss comparison reuses this work method with Dirichlet form storage and heterogeneous held capacities. Its equilibrium, response and local-recovery statements belong to that complete law; the centered-storage prism budget here is not transferred by replacing symbols or setting a damping coefficient in the production engine.

The geometric asymmetry branch is therefore resolved in scope. Its metric correction and actual work have separate owners, and no further matrix or amplitude sweep is needed. The remaining task is the law supporting the joint transfer, not another reinterpretation of the response matrix.

Validation: 418 tests across 32 relevant modules, including 14 new metric, work, storage and configured-coordination controls, pass. The shared symbolic fixture replaces duplicate retained preparation/response algebra in tests. No production dynamics changes. Source-bound record: artifacts/research/phase_transfer_maintenance_validation_2026_09_19.json.

13.19 Structural closure tests: exchange, Jacobi and the remaining potential¶

This tests whether stronger structural conditions determine the missing phase row. Keep the same fixed unit prism, canonical regular phase source, positive e,w, complete EPI field and declared comparison beta from section 13.18. No new dynamical coefficient or trajectory is installed. Write a=H*g=-gradient V_phi, M=H^-1 and y=Pi*x.

Power balance leaves four nongauge phase directions. Prescribing a storage residual R_* imposes only

\[ a^T\omega=(w y^Tg-R_*)/\beta. \]

At a!=0 the complete solution is

\[ \omega=\frac w\beta My- \frac{R_*}{\beta}\frac a{a^Ta}+v,\qquad a^Tv=0. \]

There are five free instantaneous directions on six nodes. Because a^T 1=0, common phase rotation is one of them; four remain after taking that quotient. These are directions of the existing phase state, not four new coordinates or attributes. At a=0, positive H implies g=0; only R_*=0 is compatible and the balance constrains no phase velocity. Prescribing the residual is itself a premise, not a measured law inferred from the desired outcome.

Locality and graph symmetries do not eliminate all this freedom. With W_U the unit reciprocal support adjacency, form the derived matrix

\[ B(a)=\operatorname{diag}(W_U(a\odot a)) -\operatorname{diag}(a)W_U\operatorname{diag}(a),\qquad r=B(a)Lx. \]

It has B(a)a=0 and z^T B(a)z=sum_{edges ij}(a_j z_i-a_i z_j)^2>=0. Thus a^T r=0. This construction uses only the existing nodal fields and support, respects relabeling, common phase rotation and common EPI shifts, and has radius-two dependence in the primitive fields. It is a witness against uniqueness under finite locality, not a new operator or an assertion of strict one-hop dependence.

At the unchanged preparation x=(3/4,1/4,1/2) repeated and theta=(-pi/6,pi/6,0,-pi/6,pi/6,pi/6),

\[ \|\Pi r\|^2=\frac{131}{96}+\frac{283\sqrt3}{768}>0. \]

The two comparison rows omega_0=(w/beta)My and omega_1=omega_0+(w/beta)r have the same zero residual and the same instantaneous full EPI rate. They have different subsequent responses:

\[ \Delta\ddot\mu=\frac{w^2}{\beta} \frac{155\sqrt3-267}{2688\pi},\qquad \Delta\frac{d^2}{dt^2}\|y\|^2=\frac{w^2}{\beta} \frac{13\sqrt3-333}{448\pi}. \]

The displayed zero-residual baseline uses the global mean through y; only its null addition r has the asserted radius-two dependence. There is also a fully local pair, omega_bar_0=(w/beta)M Lx and omega_bar_1=omega_bar_0+(w/beta)r. They share the independently specified residual function R_*=w*g^T*(y-Lx) and preserve the stated symmetries. This residual is not generally zero. At the retained form Lx=y, they coincide with the preceding pair, including the instantaneous response differences. The residual is part of these mathematical countercontrols, not a rule inferred retrospectively or installed in the engine.

Neither pair is selected as a TNFR law. The zero-residual family retains the previous dissipation budget, so its freedom does not bypass that maintenance obstruction; the local pair has its own nonzero-residual budget away from the retained point. Controls: intrinsic exchange freedom.

A power-canceling skew tensor need not be Poisson. The lossless comparison above can be written, before adding (-eLx,0), as

\[ \binom{\dot x}{\dot\theta}= P(\theta)\nabla_{x,\theta}\mathcal E,\qquad P=c\begin{pmatrix}0&-M\\M&0\end{pmatrix},\quad c=w/\beta. \]

Since gradient_x E=y, this gives the full source row xdot=w*g and theta_dot=c*M*y, including mean(xdot)=w*mean(g). No centering of the source row is implicit. Antisymmetry cancels exchange power, but the Jacobi identity for this particular full tensor additionally requires

\[ \{x_i,\{x_j,\theta_k\}\}+\text{cyclic} =c^2(\delta_{jk}m_i\partial_i m_j -\delta_{ik}m_j\partial_j m_i)=0, \qquad m_i=1/H_i. \]

Because every m_i>0, this is equivalent to partial_i m_j=0 for every i!=j. Together with the canonical metric's common-phase-shift invariance, it would require a locally constant metric. The retained nonrepeated preparation violates it: H_0=H_1=3(1+sqrt(3)), partial_1 H_0=3-9sqrt(3)/pi, and the (x_1,x_0,theta_0) Jacobiator is

\[ \frac{c^2(3\sqrt3-\pi)}{9\pi(1+\sqrt3)^3}>0. \]

This rejects promotion of that skew tensor to a Hamiltonian structure. It does not reject other cross blocks, reduced brackets or the existing closed wave pullback. Adding a dissipative term does not repair this tensor's Jacobi identity. The substrate's constant bracket and supplied tangent-map tests are different owners and cannot certify it. Controls: Jacobi condition and exact obstruction.

Even a selected genuine symplectic structure leaves a potential to derive. Reuse the existing local pressure coordinates, (y,theta modulo rotation)->(y,v=Pi*p), on global phase spread below pi/2. In orthonormal centered coordinates the selected canonical form is 3 sum(dy_alpha wedge dv_alpha). For an autonomous Hamiltonian on this ten-dimensional chart, requiring the nodal first row ydot=v forces, on connected velocity fibers,

\[ \mathcal H(y,v)=\frac32\|v\|^2+U(y),\qquad \dot v=-\frac13\nabla U(y). \]

The function U remains undetermined. The existing wave chooses U_L=3*y^T L y/2; the distinct U_L2=3*y^T L^2 y/2 shares positivity, graph relabeling covariance and common-offset invariance. Both have the same genuine coordinate pullback and local nodal first-row lift. A separate one-hop acceleration condition could exclude L^2, but is not implied by symplecticity or a one-hop assumption about a different state variable. At the original EPI contrast, an eigenvector of L with eigenvalue one, these two accelerations coincide. Their derivative in the existing inter-triangle mean direction differs because its eigenvalue is 2/3. This distinguishes full local laws without changing the retained preparation or constructing a favorable new trajectory. The executable phase-lift controls reuse the existing repeated phase comparator from section 21.4; they do not claim to have executed a lift at the nonrepeated phase point of the preceding two tests. The acceleration equality at L y=y is independent of that choice of regular phase.

For every such comparison the complete mean still obeys mu_dot=w*mean(g) and mu_ddot=w*mean(Dg*omega); the centered Hamiltonian does not justify setting either to zero. Common phase rotation is also outside its ten-dimensional form. No hypothesis here establishes global chart invariance, autonomous law selection, runtime realization or physical validation. Controls: Hamiltonian family and retained-state derivatives.

The constitutive gap is now sharper: a specified storage balance, finite locality and symmetries do not determine phase evolution; imposing a genuine Hamiltonian structure in the known pressure chart still leaves U and the choice of that structure as premises. Further calculation inside those same assumptions cannot establish uniqueness. The single G3 plan must identify an independent structural relation that constrains this remaining freedom, or report it as an unresolved premise. It must not rename a chosen potential, tensor or diagnostic as a derivation.

The integration also corrects the decoupled product observer's validation order: all nodal scalar channels pass its existing finite-real contract before substrate fields are extracted. Invalid phase/pressure booleans no longer reach a different downstream exception after field/cache work. Validated values retain the substrate's canonical node order. This repairs the observer boundary, without changing its valid-input product dynamics.

Validation: 193 tests in 19 modules pass, including 12 new exact controls. The common metric differential and centered pressure-coordinate fixture replace duplicate test construction. Source-bound checkpoint: artifacts/research/joint_constitutive_conditions_validation_2026_09_19.json.

13.20 Locality and exact diffusion compatibility¶

The next structural restrictions can be tested without a new trajectory. Retain the same unit-conductance triangular prism, degree three, unit capacity, e,w>0 and complete nodal equation xdot=-e*L*x+w*g(theta). Here L=L_rw, Pi=I-11^T/6, y=Pi*x and v=Pi*xdot. The selected centered symplectic chart from section 13.19 has H=3||v||^2/2+U(y). These are conditional restrictions on that comparison family; they do not establish its physical origin.

One-hop acceleration and symmetry leave two edge coefficients. For a quadratic U=y^T Q y/2, require a constant symmetric Hessian, Q*1=0, zero off-support entries and invariance under all twelve prism automorphisms. The graph is vertex-transitive but has two edge orbits: six triangle edges and three edges joining the triangles. Exactly

\[ Q=aB_{\triangle}+bB_{\mathrm{matching}},\qquad \operatorname{spec}(Q)=\{0,2b,3a,3a,3a+2b,3a+2b\}, \]

where the B matrices are the combinatorial edge-orbit Laplacians and B_triangle+B_matching=3L. The nine edge coefficients obey seven independent symmetry constraints. Positivity on centered forms requires and is implied by a,b>0; it does not fix their ratio. The original form only probes the triangle coefficient. Its existing inter-triangle tangent probes the other coefficient, without changing the preparation.

Requiring acceleration -Q*y/3=-sigma*L*y on every centered form forces a=b=sigma. This transport-matching premise is stronger than graph symmetry and leaves the scalar unspecified. Matching only the consensus Hessian is weaker still: the unit edge quadratic and that quadratic plus sum_edges (x_i-x_j)^4/4 have the same Hessian there, all graph symmetries and one-hop gradients, but different responses at the retained form. The quartic is a dimensionless mathematical countercontrol; a physical use would additionally need its scale and justification. Controls: local potential classification.

Exact embedded diffusion differs from a leading overdamped balance. For the supplied auxiliary model

\[ \dot y=v,\qquad \dot v=-Ky-\gamma v, \]

the complete diffusion graph v=-eL*y is invariant if and only if K*Pi=gamma*e*L-e^2*L^2. With K*1=0 this gives

\[ K=\gamma eL-e^2L^2,\qquad s^2+\gamma s+\gamma e\lambda-e^2\lambda^2 =(s+e\lambda)(s+\gamma-e\lambda). \]

Damping remains free and the initial velocity must lie on the specified graph. Positive centered stiffness requires gamma>e*lambda_max, with lambda_max=5/3; the diffusion root is the strictly slower root in every nonuniform mode only when gamma>2e*lambda_max. The leading choice K=gamma*e*L instead has exact tangency residual -e^2*L^2*y. Discarding inertia in tau*vdot+v=-eL*y gives a leading balance, not an exact finite-tau embedding or a proof of convergence of that limit.

On ordered nodes ((0,0),(0,1),(0,2),(1,0),(1,1),(1,2)), nodes 0 and 4 are nonadjacent, yet (L^2)[0,4]=2/9. Thus the exact embedding requires K[0,4]=-2e^2/9, independently of damping: strict one-hop acceleration and this exact embedding cannot both hold on the retained support. At gamma=0 the embedding requires negative centered stiffness K=-e^2*L^2, not a positive conservative wave. When gamma>e*lambda_max, the positive centered energy satisfies

\[ \mathcal H=\frac32(\|v\|^2+y^TKy),\qquad \dot{\mathcal H}=-3\gamma\|v\|^2. \]

A periodic solution then has v=0 throughout and y=0; this family does not supply nonzero periodic maintenance. These centered auxiliary identities are not primitive phase or full runtime certificates. Controls: exact diffusion embedding.

Primitive phase locality is a separate restriction. At the existing repeated phase comparator theta=(0,pi/3,pi/6,0,pi/3,pi/6), let J=Dg, B=pi*J and T=(Pi*B+11^T/6)^(-1)*Pi, the shared exact pressure-coordinate inverse. Consider the selected Hamiltonian acceleration vdot=-kappa*L*y, kappa>=0. Differentiating the nodal lift at fixed phase gives

\[ w\Pi J D_x\omega=-(\kappa L+e^2L^2),\qquad D_x\omega=-\frac\pi w T(\kappa L+e^2L^2)+\mathbf1 c^T. \]

The row difference eliminates every common-phase gauge derivative:

\[ \partial_{x_0}\omega_4-\partial_{x_0}\omega_5 =\frac\pi w\left[ \frac{193-111\sqrt3}{1716}\kappa +\frac{37-19\sqrt3}{396}e^2\right]>0. \]

Both 4 and 5 are nonneighbors of 0. A continuously differentiable phase law depending only on each node's and its neighbors' primitive EPI, phase and fixed capacity would require both derivatives to vanish. Such a law cannot realize this selected acceleration on an open EPI neighborhood at this phase. Relabeling and choosing another common phase velocity do not remove the obstruction. The complete form mean is retained: its acceleration derivative in the existing repeated Q_MODE=(1,1,-2) direction is (kappa+e^2)*(5-3sqrt(3))/2, not zero.

At phase consensus a sparse lift exists, so this is a state-dependent obstruction rather than an artifact of centering. It excludes the specified kappa*L family under this strict locality requirement, not all local stiffnesses or every TNFR law. Access to freshly computed neighboring pressures can already have radius-two primitive dependence; it is a different restriction. Neither the nodal equation nor the current global coordinator establishes strict primitive one-hop phase locality. Controls: primitive phase locality and mean.

The same study repairs a separate numerical diagnostic error: subtracting near-equal terms lost the slow damped-wave root at large damping. All graph-wave observers now share a root kernel that recovers the small root from the root product. On P2 with gamma=1e10, the nonuniform rate is approximately -2e-10, rather than zero. Decimal comparison, zero/critical/ complex-root controls and existing coordinate regressions cover this fix. It does not change native evolution, prove a wave limit, remove all floating-range limitations or repair the distinct critical-trajectory fallback. Controls: root precision.

These restrictions do not identify a unique potential or phase law. Locality must name its primitive variables, and diffusion compatibility must distinguish a limiting approximation from an exact invariant graph. The single G3 plan retains the remaining independent closure premise; no coefficient, clock or maintaining trajectory is selected here.

Validation: 144 tests in 11 modules pass, including 12 new exact mathematical controls and five numerical regression cases. Shared owners supply the support, symmetries and pressure-coordinate inverse. Earlier research artifacts are preserved byte-for-byte. Source-bound checkpoint: artifacts/research/locality_transport_closure_validation_2026_09_19.json.

13.21 Cotangent phase exchange: a conditional constitutive candidate¶

The failed Jacobi tensor in section 13.19 can be repaired by an actual coordinate construction. This supplies a new conditional model, not a derivation of the engine's unchanged pressure. Work in declared nondimensional coordinates, on finite simple loopless fixed regular reciprocal support with unit conductances and unique-neighbor degree d>0, with held unit capacity, positive phase weight w, and EPI weight e>=0. Let H=H_phi(theta)>0, M=H^-1 and g=-M*gradient V_phi have exactly their section 13.6 meanings on the regular phase chart.

The independent premise is that

\[ \pi_i=H_i(\theta)x_i \]

is the canonical momentum conjugate to theta_i, with Hamiltonian E=||x||^2/2+w*V_phi. This chooses a symplectic structure, an uncentered form storage and a time normalization. The existing pressure metric alone does not force that identification or its units. Momentum adds no primitive state on this chart: x_i=pi_i/H_i is invertible. It is distinct from the centered (y,v) wave construction in sections 13.19-13.20.

The correction is forced by this premise. Pushing the canonical bracket through x=H^-1*pi, in coordinate order (x,theta), gives

\[ \mathcal J=\begin{pmatrix}K&-M\\M&0\end{pmatrix},\qquad K_{ij}=\frac{x_j\partial_i H_j-x_i\partial_j H_i}{H_iH_j}. \]

Here partial_i differentiates theta_i. This tensor satisfies Jacobi because it is a genuine coordinate pullback; its additional K=-K^T block is fixed by the chosen momentum map. Dropping it generally recovers the non-Poisson tensor already rejected in section 13.19. The Hamiltonian rows, with optional existing form diffusion added separately, are

\[ \boxed{\dot x=-eLx+wg+Kx,\qquad \dot\theta=Mx,\qquad\dot\nu=0.} \]

Thus the proposed pressure is p_new=-eLx+wg+Kx; p_old=-eLx+wg remains the configured two-channel engine pressure. A different phase row alone cannot turn the latter into this model. The correction is quadratic in form and cannot generally be absorbed into a phase-only potential Psi. On the retained nonrepeated prism, taking x=(1,0,0,0,0,0) gives

\[ (Kx)_1=\frac{3-9\sqrt3/\pi}{9(1+\sqrt3)^2}<0. \]

This is an exact, regular-state discriminator, not a numerical trajectory. The phase row now has real form feedback, D_x dot(theta)=M; two states with identical phase and different form have different phase rates. At P2 consensus, changing form from zero to (1,-1)/4 changes phase velocity by (1,-1)/(4*pi) and phase acceleration by -e*(1,-1)/(2*pi).

H_i reads the node and its support neighbors; Kx generally reaches their neighbors and therefore has radius-two primitive dependence. Absolute form enters the phase row. A common EPI translation is not automatically a redundancy of this joint model, and centering its storage would change it. No operator selector, tetrad threshold, gain fitted to persistence, capacity adaptation or external phase clock has been added.

Complete storage and momentum balances. Regularity of the support makes L symmetric positive semidefinite. Antisymmetry of K and Hg=-grad V_phi give the exact continuous balance

\[ \dot E=-e\,x^TLx\le0. \]

The correction transfers form between components with x^T Kx=0; it does not inject storage. The complete form mean nevertheless obeys mean(x)'=w*mean(g)+mean(Kx), which need not vanish. Common phase-rotation invariance supplies a different conserved quantity for the reversible row, the canonical momentum P=sum_i H_i*x_i. With diffusion included,

\[ \dot P=-e\sum_i H_i(Lx)_i. \]

It need not remain conserved on a nonuniform phase state. Neither quantity is the engine's coherence telemetry or an experimentally identified energy.

There is a useful forward-domain guarantee on any finite connected regular support. If initially E_0<w, every edge obeys 1-cos(gap)<=E_0/w<1 for all future times. Put a=acos(1-E_0/w)<pi/2. All edge gaps and local mean displacements stay within [-a,a], all resultants have magnitude at least d*cos(a), and H_i>=pi*d*cos(a)*sinc(a)>0. Bounded form and the compact phase torus then keep the smooth vector field away from its branch/resultant singularities, giving global existence within this conditional model. The bound establishes admission, not nontrivial persistence or full-state convergence.

A geometry-dependent linear prediction. At zero form and phase consensus, H=pi*d*I, DH=0 and Dg=-L/pi. A Laplacian mode with eigenvalue ell>0 has characteristic polynomial

\[ s^2+e\ell s+\frac{w\ell}{\pi^2d}=0. \]

This is a damped oscillatory mode exactly when e^2*ell<4*w/(pi^2*d). With the same retained coefficients e=w=1/2, unit cycles have d=2: the lowest C6 mode ell=1/2 fails this test, whereas the lowest C8 mode ell=1-sqrt(2)/2 satisfies it. The elementary bounds 9<pi^2<10 separate both cases. Geometry supplies this difference without coefficient adjustment. This is a linear prediction of the new conditional model, unrelated to the earlier C6 full-runtime stability campaign. It does not certify a nonlinear waveform or maintained NFR.

A nonlinear sign-reversal discriminator without fitting a trajectory. On that C8 comparison let v_i=cos(2*pi*i/8), so ||v||^2=4, and prepare x(0)=epsilon*v, theta(0)=0. Observe A(t;epsilon)=v^T*x(t)/4, not a refitted pattern amplitude. Write alpha=e*ell/2 and omega=sqrt(w*ell/(pi^2*d)-alpha^2)>0. The linear initial-value problem predicts

\[ A_1(t)=e^{-\alpha t}\left(\cos(\omega t) -\frac{\alpha}{\omega}\sin(\omega t)\right),\qquad A_1(T)=-\frac{\alpha}{\omega}e^{-\alpha T}<0, \quad T=\frac{\pi}{2\omega}. \]

This implies a genuine nonlinear existence result. The metric is even under phase reflection, its derivative and g are odd, and K(-x,-theta)=K(x,theta). The full vector field is therefore odd under the joint reflection and smooth near consensus. Smooth dependence on the signed preparation amplitude gives, at fixed finite T, A(T;epsilon)=epsilon*A_1(T)+O(epsilon^3). Consequently there exists epsilon_0>0 such that every sufficiently small positive preparation reverses this form mode's sign by T. Its initial storage is 2*epsilon^2; abs(epsilon)<1/2 guarantees the regular domain but is not a computed error bound or value of epsilon_0.

The old held-capacity, two-channel continuous comparison with its supplied gated-sine phase row instead keeps phase consensus up to common rotation and has A_old(t)=epsilon*exp(-e*ell*t)>0. This distinguishes the two complete laws beyond merely renaming a phase variable. It does not compare the native operator runtime or prove formation. This existence argument alone does not select a finite amplitude. Section 13.22 supplies the quantitative nonlinear remainder, separate ideal-integration budget and frozen executable comparison.

A complete nonlinear P2 sector. On P2 write delta=theta_1-theta_0 in (-pi,pi). Then H=h(delta)*I, h=pi*sin(delta)/delta, with h(0)=pi. The zero-momentum sector x=(a,-a) is invariant, Kx=0 there, and

\[ \dot a=-2ea+w\delta/\pi,\qquad \dot\delta=-2a/h(\delta),\qquad E=a^2+w(1-\cos\delta),\qquad \dot E=-4ea^2. \]

For E_0<2w its compact sublevel stays strictly inside the regular chart; E_0<w additionally preserves strict U3. At e=0, every nonzero energy level below 2w is a regular closed curve with a nonvanishing vector field, so it gives periodic reversible form/phase exchange. At e>0, the largest invariant subset of dot(E)=0 requires both a=0 and delta=0; LaSalle's argument proves convergence to that point in this sector. The common phase is constant there. Positive diffusion therefore does not give indefinite nontrivial maintenance. In particular e=w=1/2 is overdamped near P2 consensus: the model is not made oscillatory by describing it as resonant.

The constructive result is a closed, genuinely Poisson exchange law with a forced geometric pressure correction, an admitted global small-energy domain, and a discriminating graph-dependent response. Its cotangent premise remains open to independent justification or rejection. It is not installed in the motor, physically validated, uniquely selected by NFR identity, or shown to generate support/capacity. Exact pullback, Jacobi, production-source boundary, balance and mode controls are in cotangent exchange scope. The single plan owns subsequent work; no trajectory was used to select this law.

13.22 A finite C8 prediction fixed before numerical evaluation¶

Keep every constitutive choice of section 13.21: the unit C8, unit held capacity, e=w=1/2, no forcing or operator events, and the cotangent pressure including Kx. Fix epsilon=1/1024, x_i(0)=epsilon*cos(2*pi*i/8) and theta(0)=0. The observable remains A=v^T*x/4. The amplitude, observable and response time are chosen from the preceding analytic prediction, before inspecting a trajectory. This comparison does not modify engine defaults.

A quantitative continuous bound. On this degree-two support, with continuous phase lifts from the initial consensus, put

\[ a_i=(\theta_{i+1}+\theta_{i-1})/2-\theta_i=-(L\theta)_i, \qquad b_i=(\theta_{i+1}-\theta_{i-1})/2. \]

While edges are acute, the exact metric and pressure source simplify to H_i=2*pi*cos(b_i)*sinc(a_i) and g_i=a_i/pi; the latter has no nonlinear remainder. For any 0<epsilon<=1/16, the previously proved energy balance gives ||x||<=2*epsilon and V_phi<=4*epsilon^2. Each edge separation is less than 1/2, since its potential cost is below 1-cos(1/2). The bound 1-cos(delta)>=(47/96)*delta^2 on this interval then gives ||D theta||^2<9*epsilon^2, where D is the oriented cycle incidence map. Consequently abs(a_i),abs(b_i)<=3*epsilon and

\[ H_i\ge2\pi(1-6\epsilon^2),\qquad \|M\|\le1/6. \]

For B_ij=partial_j log(H_i), the cycle difference operators have norms ||D_theta a||<=2 and ||D_theta b||<=1. Using abs(tan b)<=abs(b)/(1-b^2/2) and abs(sinc'(a)/sinc(a))<=abs(a)/(3*(1-a^2/6)) therefore yields

\[ \|B\|\le\left(\frac3{1-9\epsilon^2/2} +\frac2{1-3\epsilon^2/2}\right)\epsilon \le\frac{21}{4}\epsilon. \]

In particular, the exact expression Kx=M*B^T*(x*x)-diag(x)*B*M*x, with componentwise x*x, satisfies ||Kx||<=7*epsilon^3. Also ||(M-I/(2*pi))*x||<=2*epsilon^3.

Use the phase-rotation quotient and its energy norm

\[ N(x,\theta)^2=\|x\|^2+\theta^TL\theta. \]

The linearized generator mathcal A is dissipative in this norm; its semigroup is contractive. The complete nonlinear remainder R has N(R)<=sqrt(57)*epsilon^3<8*epsilon^3 along the exact trajectory, because ||L||=2. Duhamel's formula and ||v||/4=1/2 now give the finite bound

\[ \boxed{|A(t;\epsilon)-\epsilon A_1(t)|\le4t\epsilon^3.} \]

For the fixed response time of section 13.21, T=pi/(2*omega)=34.630474785..., the linear response is epsilon*A_1(T)=-1.2485970787...e-4, whereas the continuous nonlinear allowance is less than 1.291e-7. Thus this specific finite preparation, not merely an unspecified sufficiently small one, has A(T)<0 in the exact continuous model. The displayed decimals summarize the explicit formulas; the sign has a wide margin under their rounding.

A separate exact-arithmetic Euler bound. Freeze h=1/2048, n=ceil(T/h)=70924 and tau=n*h=34.630859375, and evaluate the same observable at tau. A coarse, independently checkable error allowance suffices; no step-size search against an observed response is needed.

For this purpose put q^2=theta^T*L*theta. At any state with N<=0.1, abs(a_i),abs(b_i)<=q, and m=1-2*q^2/3>=149/150. The identities above give H_i in [2*pi*m,2*pi], abs(H_a)<=2*pi*q/3, abs(H_b)<=2*pi*q, and both the maximum row-sum and column-sum norms of D H at most 10*pi*q/3. Hence

\[ \begin{split} \|Kx\|&\le\frac5{3\pi m^2}q\|x\|^2,\\ \|L^{1/2}(M-I/(2\pi))x\| &\le\frac{\sqrt2}{3\pi m}q^2\|x\|. \end{split} \]

The Euclidean norm of these two coefficients is below 0.6, so the convenient bound N(R)<=N^3 holds throughout this domain. In energy coordinates each linear modal block is [[-ell/2,-sqrt(ell)/(2*pi)],[sqrt(ell)/(2*pi),0]]. For 0<=ell<=2, its squared operator norm is at most c^2=10/9. Linear dissipativity therefore bounds one Euler step by ||I+h*mathcal A||<=exp(5*h^2/9).

Let B0=4*epsilon. The bootstrap

\[ N(u_k)\le2\epsilon\exp[t_k(5h/9+B_0^2)]<2.021\epsilon<B_0 \qquad(0\le k\le n) \]

holds inductively, because tau<=35 and the exponential is below 2. Thus the numerical iterates also remain inside the domain used for the remainder estimate. Separating full Euler from linear Euler, and linear Euler from the exact linear flow, gives the observable bounds

\[ \begin{split} |A_{\rm Euler}(\tau)-A_{{\rm linear\ Euler}}(\tau)| &\le\frac\tau2 e^{5h\tau/9}B_0^3,\\ |A_{{\rm linear\ Euler}}(\tau)-\epsilon A_1(\tau)| &\le\frac{\tau h}{2}\frac{10}{9}\epsilon e^{5h\tau/9}. \end{split} \]

Their combined allowance is less than 1.044e-5, using only tau<=35 and exp(5*h*tau/9)<1.01. The declared linear target is epsilon*A_1(tau)=-1.2485754095...e-4; even this coarse allowance leaves the exact-arithmetic Euler response strictly negative. The old supplied phase law instead predicts the positive epsilon*exp(-ell*tau/2)=6.1255393889...e-6.

These are two separate bounds: the continuous nonlinear deviation and the full Euler deviation from its linear target. Adding them is not needed to establish either statement. The Euler bound concerns exact arithmetic; it does not certify binary64 rounding or assign an unproved floating-point error allowance. A subsequent executable comparison must retain the frozen law, initial state, step, horizon and observable, and report floating-point execution as finite numerical evidence. Neither sign reversal nor its prediction establishes maintained identity, substrate generation or physical identification. The cotangent constitutive premise remains conditional.

Frozen finite execution, 2026-09-26. The bounded instrument executed one preparation with the amplitude and clock above. Every step reuses the shared fused production pressure and dynamics._euler_kernel.euler_update for both rows from the same preceding state; Kx is added explicitly. At steps 0,17731,35462,53193,70924, the ordinary graph pressure owner independently checks the old-pressure component. Detached SDK diagnostics then observe the candidate pressure and its declared model rate. This is a research execution path, not the default phase coordinator or an operator study.

Quantity	Frozen prediction or admission	Observed binary64 result
Final mode amplitude	Linear target `-1.2485754095182383e-4`; negative	`-1.2485839574377552e-4`
Absolute deviation from linear target	At most `1.044e-5`	`8.5479195169e-10`
Quotient energy norm, all endpoints	At most `4*epsilon=0.00390625`	Maximum `0.001953125`
Old pressure residual	Each checkpoint and all-step formula residual at most `1e-12`	`0` and `5.4211e-20`, respectively
Instantaneous storage balance residual	At most `1e-15`	Maximum `3.1764e-22`
Geometric form-work residual	At most `1e-15`	Maximum `3.9444e-30`

The old continuous triangular comparison predicts +6.1255393889e-6 at the same endpoint. No amplitude, coefficient, step, observation or acceptance cut was changed after the reserved execution. Maximum adjacent separation was 0.000521212 radians (rounded upward), with minimum metric 6.2831844537. The tetrad was available at all five checkpoints; xi_C=1.8477590650 came from the spectral fallback at every checkpoint, not an observed growing correlation length. No field selected or corrected the dynamics.

The discrete energy fell overall from 1.9073486328e-6 to 7.4318907298e-8, but explicit Euler had a maximum positive single-step increment of 5.6298e-16. Its cumulative energy-balance defect reached 1.1397e-10. These observations must not be called exact discrete dissipation or a validated floating-point enclosure. The analytic continuous and ideal Euler claims above remain separate from this finite check.

Reproduce the fixed run with:

python benchmarks/cotangent_phase_exchange.py --amplitude 0.0009765625 --timestep 0.00048828125 --steps 70924 --checkpoint 17731 --checkpoint 35462 --checkpoint 53193 --output artifacts/research/cotangent_c8_execution_2026_09_26.json

The retained local records are artifacts/research/cotangent_c8_protocol_2026_09_26.json, artifacts/research/cotangent_c8_execution_2026_09_26.json and artifacts/research/cotangent_c8_verdict_2026_09_26.json. The protocol SHA-256 is a1ee858cece6371f177827658626adcea349aaeab5649ec44957ff0cbae14775; the execution SHA-256 is 20d2aa43311dec4b13c49a41d3b7ef8ac9d56f3be0b0820d83470cf74c3e65f5. Execution used Python 3.13.6, NumPy 2.5.3, Windows 11, binary64, no randomness, and base revision f27413a0a29dddc132e5aafea081b0caee0b0135 with scoped working-source digest sha256:af0c6fe2dd2428df0da278be4960aa90e271f375d0322538d40f341fd2b97940. The source scope is this instrument and src/tnfr; the report retains full state at checkpoints and all-step extrema. The local artifacts are generated evidence, not a substitute for the reproducible recipe and scope above. Small execution controls check independent derivatives, pressure, balances, simultaneous integration and diagnostic provenance without regenerating the full trajectory.

What this comparison cannot select. Suppressing Kx while retaining theta_dot=Mx has the same linearization at consensus; its storage balance also holds because x^T Kx=0. The sign reversal therefore discriminates reciprocal feedback from the old triangular law, not the cotangent premise from every alternative reciprocal law. Moreover, this preparation is antiperiodic under a four-node shift: x_{i+4}=-x_i and theta_{i+4}=-theta_i. Since the paired metrics agree, its exact total momentum is zero by symmetry. Near-zero observed momentum is not an independent test of the general momentum balance. The existing nonrepeated state and Jacobi witnesses in section 13.21 already distinguish the geometric correction algebraically; no second C8 sweep is needed to repeat that result.

13.23 Nonzero momentum and relative identity on P2¶

The same cotangent law admits an exact reduction beyond the zero-momentum sector in section 13.21. This changes the prepared state, not the closure. On P2, take the regular chart delta=theta_1-theta_0 in (-pi,pi), and set

\[ h(\delta)=\pi\operatorname{sinc}\delta,\quad \psi=(\theta_0+\theta_1)/2,\quad a=(x_0-x_1)/2,\quad P=h(\delta)(x_0+x_1). \]

Here H=h*I, so the general momentum budget reduces to P_dot=0, even when e>0: h*sum(Lx)=0. This P2 simplification is not momentum conservation on every graph. Substitution into the complete law gives

\[ \begin{gathered} x_{0,1}=\frac{P}{2h}\mathbin{\pm}a,\qquad U_P(\delta)=\frac{P^2}{4h(\delta)^2}+w(1-\cos\delta),\\ \boxed{\dot\delta=-\frac{2a}{h},\qquad \dot a=-2ea+\frac{U_P'(\delta)}h,\qquad \dot\psi=\frac{P}{2h^2}.} \end{gathered} \]

In particular U_P'/h=w*delta/pi-P^2*h'/(2*h^4). The momentum-dependent term comes from Kx; it is not a new potential inserted to maintain a pattern. The reduced storage and balance are exactly E=a^2+U_P(delta) and E_dot=-4e*a^2.

Regularity and the possible identity. On 0<delta<pi, h>0 and h'<0: sin(delta)-delta*cos(delta) has derivative delta*sin(delta)>0 and vanishes at zero. Thus U_P increases strictly with abs(delta). For P!=0 it diverges at the antipodal boundary. Every finite-energy sublevel consequently stays compactly inside the regular chart, without requiring the earlier sufficient bound E_0<w. Strict acute admission additionally follows from E_0<U_P(pi/2)=P^2/16+w; regularity alone must not be reported as U3 admission.

For e>0, the only invariant part of E_dot=0 in a fixed-P sublevel requires a=0 and U_P'=0, hence delta=0. The reduced LaSalle argument therefore gives

\[ \delta,a\longrightarrow0,\qquad x_0,x_1\longrightarrow\frac{P}{2\pi},\qquad \dot\psi\longrightarrow\frac{P}{2\pi^2}. \]

The resulting family is a uniform rotating relative equilibrium: its relative state is fixed while its common phase advances. It is labelled by prepared momentum. It does not retain a nonzero internal form or phase contrast, produce momentum from nothing, or identify a physical particle. At e=0, nonminimal fixed-energy curves give periodic relative exchange; the accumulated common phase need not make the full state periodic.

Geometric contribution and a local discriminator. Since h=pi*(1-delta^2/6+O(delta^4)), the characteristic polynomial of relative perturbations about this rotating equilibrium is

\[ \lambda^2+2e\lambda+\frac{2w}{\pi^2} +\frac{P^2}{3\pi^4}=0. \]

Thus prepared momentum changes relative restoring stiffness with no change of coefficients. Omitting Kx while keeping theta_dot=Mx preserves the same storage balance but generally gives P_dot=h'*(x_1^2-x_0^2)/h, rather than zero. The zero-momentum preparation hides that defect. Storage dissipation, a plausible oscillation and a genuine cotangent structure are therefore distinct obligations. The exact scope controls derive the reduced rows and momentum boundary from the full pullback law. On larger supports, use the actual momentum balance and test relative identity and its transverse stability; this P2 result does not supply them.

13.24 Integration audit: inherited structure and missing selection¶

Sections 13.21-13.23 establish consequences of a conditional completion. Before another pattern study, distinguish its integration with the nodal framework from selection of a fundamental law. No new production default, controller, fitted parameter or physical identification follows here.

Ingredient	Status in this comparison
Signed scalar form, circular phase, support and nodal product	Existing state/flow contracts; the comparison restricts capacity to one
`g=-H^-1 grad(V_phi)`	Derived from the configured phasor pressure and declared cosine cost; neither its absolute cost scale nor a phase clock follows
Canonical momentum `H*x`, uncentered quadratic form storage	Additional constitutive premises, not consequences of the NFR definition
`Kx`, reciprocal phase row, Jacobi and balance	Forced after those premises; independently checked and conditionally predictive
Support, initialization and held capacity	Supplied; no birth, adaptation or autonomous partition-selection law is derived
Tetrad and grammar	Existing observers and event-admission contracts; neither selects this continuous law

The existing principles admit more than one law. The cost normalization freedom already identified in section 13.6 becomes dynamically consequential after the momentum premise. For any constant eta>0, take

\[ \pi_\eta=\eta Hx,\qquad E_\eta=\tfrac12\|x\|^2+\eta wV_\phi,\qquad \mathcal J_\eta=\mathcal J/\eta. \]

The genuine coordinate pullback gives

\[ \dot x=-eLx+wg+\eta^{-1}Kx,\qquad \dot\theta=\eta^{-1}Mx,\qquad \dot E_\eta=-e\,x^TLx. \]

The configured form/phase pressure part, circular symmetry, finite primitive locality, genuine Poisson structure and storage dissipation all survive. Nevertheless the consensus-mode polynomial becomes s^2+e*ell*s+w*ell/(eta*pi^2*d). At fixed nonzero e,w, state chart and clock, changing eta preserves the trace while changing the determinant; it is not a common rescaling of time. The earlier C6/C8 prediction remains valid for its declared eta=1, not a parameter-free consequence of the phase-pressure identity.

There is an exact finite-state discriminator without another trajectory. On P2 with x=(a,-a) and phase consensus, every member gives a_dot(0)=-2e*a, but

\[ \ddot a(0)=4e^2a-\frac{2wa}{\eta\pi^2}. \]

This is exact because the zero-momentum P2 sector has Kx=0, not merely a linearized formula. The family is an underdetermination witness, not a new gain to tune until a pattern persists. It establishes that the listed conditions do not select eta=1; it does not establish that no stronger structural principle can do so.

The difference is not removable by a state-dependent common clock either. At the same consensus preparation with a,e>0, the joint contrast tangent is F_eta=(-2*e*a,-2*a/(eta*pi)) in coordinates (a,delta). Consequently

\[ \frac{d\delta}{da}=\frac1{\eta\pi e},\qquad \det[F_1,F_\eta]=\frac{4ea^2}{\pi}(\eta^{-1}-1). \]

A positive clock change divides both components by the same local value; it cannot change this slope. Only eta=1 aligns the two tangents in the fixed state chart. This is an exact orbit discriminator, separate from a comparison of periods or a fitted time unit. The clock control derives it from the full nonlinear P2 field without another trajectory.

Genuine unit/chart conversion remains distinct: for x_new=A*x and t_new=B*t, with positive constants, the same family has e_new=e/B, w_new=A*w/B, eta_new=A*B*eta. Thus eta*e^2/w is invariant under these conversions. Changing only eta changes that invariant. Likewise a fixed-formula EPI shift adds beta/(eta*H_i) to the phase rate; when H is nonuniform, relative phase rates change. The same physical law can be pushed into a shifted chart, but its storage and momentum origin must transform too. Pure-EPI offset covariance cannot silently be imposed on this joint law while retaining its unshifted formulas.

An exact hierarchy result supplies part of the scale information. Reuse the counted-support reduction. Start with a regular simple base graph. Replace every node by m replicas, place no edges within a block, and replace every base edge by all m^2 cross-block edges with unit conductance. Keep fine capacity one, eta=1, and identical form/phase within each block. Within-block permutation symmetry makes this manifold invariant under the complete cotangent vector field. This is an imposed exact preparation, not formation of coherent blocks.

On that manifold, with base variables X,Theta,

\[ H_{\rm fine}=mH,\qquad g_{\rm fine}=g,\qquad L_{{\rm rw,fine}}X=L_{{\rm rw,base}}X,\qquad (K_{\rm fine}x_{\rm fine})_{i\alpha}=(KX)_i/m. \]

For distinct blocks, each fine K entry is the base entry divided by m^2; summing its m equal neighbors gives the last identity. Off-diagonal entries inside a block vanish. Hence the exact inherited law is

\[ \dot X=-eLX+wg+KX/m,\qquad \dot\Theta=MX/m. \]

Its effective exchange scale is eta_eff=m, although every fine capacity is still one and the old pressure is unchanged. There is no fitted rate or new primitive state. Fine storage divided by m equals ||X||^2/2+m*w*V_phi; the pulled-back canonical one-form divided by m is m*sum_i(H_i*X_i*dTheta_i), reproducing the same effective momentum. For a common fine scale eta, the inherited scale is m*eta; repeated equal-replica reductions multiply the counts consistently.

This is a constructive link to nested effective NFR descriptions and a counterexample to discarding multiplicity and resetting eta=1 at every scale. It does not select the microscopic exchange premise, make the chosen partition emerge, or establish transverse stability of its synchronized blocks. Outside this invariant manifold, existing closure and memory tools must retain hidden state or bound its effect; they cannot derive an unknown fine law from a chosen projection.

If one additionally requires a degree-only scale eta(d) to retain the same bare-graph recipe under every such reduction, with unchanged form, clock and capacity, consistency imposes eta(d)=m*eta(m*d). For all positive integer degrees this is equivalent to eta(d)=C/d: use d=1 to obtain the necessary formula, which also satisfies every replication identity. Global C>0 remains free. The extra requirement treats this particular replication as constitutive equivalence; physical replication is not automatically an observational redundancy. It would change the previous cross-degree eta=1 predictions, and is not adopted here as a default or a chosen sustaining mechanism.

Capacity and support cannot be restored by substitution. Common prescribed capacity kappa(t)>=0 can multiply both rows, giving the same orbits under s(t)=integral(kappa dt) and storage rate -kappa*e*x^T*L*x. Infinite-structural-time results require s(t) to diverge; finite accumulated activity can stop earlier. Pausing both rows at zero is a chosen joint law: the nodal identity alone pauses only form. Multiplying by a positive state-dependent clock still reparameterizes orbits, but in general does not preserve the Poisson identity in dimension at least four. Time reparameterization and a Hamiltonian structure are separate claims.

For held heterogeneous N=diag(nu_i)>0, naively multiplying both rows by N gives a form bracket NK, which is not generally skew. A genuine pullback through pi_i=H_i*x_i/nu_i instead requires

\[ (K_\nu)_{ij} =\frac{\nu_i x_j\partial_iH_j-\nu_jx_i\partial_jH_i}{H_iH_j}. \]

Even this corrected pullback, with the same storage and diffusion -eNLx, has E_dot=-e*x^T*N*L*x, not an established nonpositive expression. At phase consensus on P2, N=diag(1,4) and x=(2,1) give x^T*N*L*x=-2. For e=1/2 the storage grows at rate 1, despite valid stable diffusion in its own capacity-weighted metric. This refutes a proposed extension of the storage theorem, not the existing diffusion theorem.

The map H/nu also excludes zero capacity. On P2 at delta=pi/3, x=(1,1) and nu_1=1, its form rate as nu_0 tends to zero is 2*(3*sqrt(3)-pi)/(9*pi)>0 at node zero. It therefore cannot extend continuously as a finite-pressure unforced nodal row there. Changing to inverse-capacity storage or another pressure can repair selected balances, but defines another model and does not resolve that boundary automatically. Evolving capacity additionally introduces moving-coordinate/storage terms.

There is an independent graph-domain obstruction. On unit P3 with phase consensus and x=(1,3/2,1), x^T*L_rw*x=-1/4; the unchanged Euclidean storage would grow at rate e/4. A degree-weighted Hamiltonian E_D=x^T*D*x/2+w*V_phi repairs diffusion on this reciprocal graph, but with momentum Hx its Hamiltonian rows are K*D*x+wg and M*D*x. Both change. Naming a weighted energy while retaining the old joint rows would not repair the proof. The full pressure also has configured capacity and topology source channels; none is included by these two-channel results.

Engine, events and formation. The default runtime interleaves pressure, glyphs, held-pressure form integration, clamps, phase coordination and capacity adaptation. Disabling glyphs or supplying a pressure callback does not implement the joint cotangent flow. The bounded producer correctly reuses pressure, Euler arithmetic and detached SDK diagnostics through a separate declared path. A future opt-in joint model must preserve that ownership and admit its scalar, support, capacity, clock and phase-chart domains explicitly.

Operator/grammar admission does not imply preservation of this storage, momentum or regular chart. Support changes also change H and V_phi; THOL birth changes the state dimension. Their event balances and occurrence laws remain independent obligations. On fixed cycles in the strict acute domain, continuous phase motion preserves integer winding. Thus the small-energy theorem cannot also establish creation of nonzero winding from zero within that domain. This does not exclude non-topological patterns or finite lifetimes. Neither indefinite persistence nor a Hamiltonian form is required by the definition of every NFR.

The audit verdict is conditional model with a derived hierarchy law and explicit integration gaps, not a complete emergent theory and not a failed nodal identity. Known principles permit several completions; no reviewed law yet derives formation, identity, interaction, capacity and support together from those principles. The single plan places constitutive/scale admission before further C8 identity experiments. The scope tests retain the exact alternative-family, inherited-scale and production-pressure counterexamples without another trajectory or a public tuning parameter.

13.25 Unequal coherent blocks inherit a joint law, not a bare graph¶

Keep the section 13.21 fine law with eta=1, unit capacity, fixed unit C4 support and no inputs or events. Prepare blocks A={0}, B={1,3}, C={2} with equal form and phase inside each block. The automorphism exchanging nodes 1 and 3 leaves the full vector field equivariant on its regular chart. Its fixed-point manifold is therefore invariant for as long as the solution stays in that chart. This is an exact preparation, not attraction or formation.

The existing counted-support owner retains the neighbor counts and block mass

\[ N=\begin{pmatrix}0&2&0\\1&0&1\\0&2&0\end{pmatrix},\qquad S=\operatorname{diag}(1,2,1),\qquad \bar L=I-N/2. \]

Choose a consistent real lift with acute edge differences, and put u=Theta_B-Theta_A, v=Theta_B-Theta_C, a=-(u+v)/2, b=(u-v)/2. In particular, abs(u),abs(v)<pi/2; the midpoint formulas below are not asserted across a resultant branch change. The inherited per-node phase metric, pressure source and phase cost are

\[ H=\operatorname{diag}(2\pi\operatorname{sinc}u, 2\pi\cos b\operatorname{sinc}a,2\pi\operatorname{sinc}v),\qquad g=\pi^{-1}(u,a,v)^T, \]

\[ \bar V=2(1-\cos u)+2(1-\cos v),\qquad SHg=-\nabla_\Theta\bar V. \]

Here sinc(0)=1. The factors of two count actual fine edges and neighbors; they are not fitted coefficients. Restricting the canonical one-form and storage gives

\[ \bar\pi=SHX,\qquad \bar E=\tfrac12X^TSX+w\bar V. \]

For block sizes n=(1,2,1) and diagonal entries h_i of H, the pullback connection is

\[ \bar K_{ij}= \frac{n_jX_j\partial_i h_j-n_iX_i\partial_jh_i} {n_in_jh_ih_j}. \]

It is skew, but acts on grad_X E=S*X. The exact restricted rows are

\[ \boxed{\dot X=-e\bar L X+wg+\bar K SX,\qquad \dot\Theta=H^{-1}X.} \]

They follow either by restricting the fine field or by using the pulled-back canonical coordinates; no new primitive capacity is introduced. The balances are

\[ \dot{\bar E}=-eX^TS\bar LX =-e[(X_A-X_B)^2+(X_C-X_B)^2],\qquad \dot P=-e\,\mathbf1^TSH\bar LX, \quad P=\mathbf1^TSHX. \]

The first uses S*Kbar*S skew and the phase-gradient identity. The second uses common phase-rotation symmetry for the Hamiltonian part and retains the diffusion contribution. It does not assert conserved P in general.

Recomputing a bare unit P3 keeps Lbar and g, but gives metric diag(1/2,1,1/2)*H. With the bare unweighted storage, the endpoint phase rates double and the geometric correction changes. No common exchange factor repairs this: nonzero endpoint forms require eta=2, whereas a nonzero center form requires eta=1. Thus preserving old pressure is insufficient to preserve the joint law. Inherited mass and multiplicity carry information needed for composition of effective regions.

The exact controls reuse production joint-quotient metadata, independently differentiate the macro trigonometric expressions, and compare fine and restricted fields at Theta=(0,pi/6,pi/2) for symbolic arbitrary X. There, H=diag(6,6*(sqrt(3)-1),3*sqrt(3)). These finite controls accompany the invariant-manifold argument; they do not test every phase preparation or establish an all-state quotient. Hidden differences inside block B, changing support, evolving capacity and the microscopic momentum premise remain outside this result.

13.26 Identical snapshot fields can have different futures under one law¶

Keep the same P2 cotangent law, unit capacity and clock, with e=w=1/2. Compare x=(a,-a) and x=(m+a,m-a) at phase consensus. Both have Kx=0, initial pressure (-a,a) and the same phase fields. Their stored nodal rates and coherence agree. This comparison fixes the law; it does not exploit uncertainty about which phase equation to choose.

The full P2 reduction in section 13.23 gives, at that preparation,

\[ \dot a=-2ea,\qquad \ddot a=4e^2a-\frac{2wa}{\pi^2}-\frac{4m^2a}{3\pi^2}. \]

Indeed P=2*pi*m there, and differentiating U_P gives U_P''(0)=w+P^2/(6*pi^2). The resulting acceleration difference is in relative form, so it cannot be removed by common phase rotation. The uncentered form/storage premise makes the hidden mean dynamically relevant.

For a=1/8, m=0 versus m=1/4, the actual shared SDK diagnostic reports identical tetrads: Phi_s=(1/8,-1/8), zero phase gradient and curvature, and xi_C=1/sqrt(2) with the same spectral-fallback provenance. Yet

\[ \Delta\ddot a=-\frac1{96\pi^2},\qquad \Delta\dot\Phi_s= \frac1{\pi^2}\left(\frac1{64},-\frac1{192}\right). \]

For the last identity, unit-length P2 has Phi_s=(p_1,p_0) and the declared unit-capacity field has p=xdot. Pressure is independently given by the law, not reconstructed from an observed derivative. Its derivatives are computed from that law's full field before imposing consensus.

Thus the snapshot tetrad does not close even its own evolution in this conditional model. This does not exclude observability from a time series or require a new fundamental variable: the omitted EPI mean was already part of the full state. The single witness checks both the exact field derivative and the actual SDK readout. At this initial state the default two-channel pressure agrees because Kx=0; the subsequent law is the declared cotangent comparison, not default runtime. No trajectory, synthetic measurement history or physical identification is claimed.

A passive chart translation does preserve the law. This state comparison does not privilege a physical absolute EPI origin. For y=x+beta*1, with fixed coordinate offset beta, transform momentum and storage to

\[ \widetilde\pi=H(y-\beta\mathbf1),\qquad \widetilde E=\tfrac12\|y-\beta\mathbf1\|^2+wV_\phi. \]

Writing K_theta(x) to expose the connection's form dependence, the same law in the new chart is

\[ \dot y=-eLy+wg+K_\theta(y-\beta\mathbf1)(y-\beta\mathbf1),\qquad \dot\theta=M(y-\beta\mathbf1). \]

This uses L*1=0, not a new force. Resetting momentum to Hy and uncentered storage to ||y||^2/2 is a different premise. A positive control in the same test owner independently pushes the canonical bracket through y=pi/h+beta on regular P2, retaining both absolute phase rows, momentum and the storage balance. It complements the state-shift counterexample; it does not select the origin or microscopic law through mathematical consistency alone.

13.27 Storage admission beyond the cotangent choice¶

The relational exchange owner compares the existing Dirichlet form storage with the uncentered cotangent storage. A declared nodewise work-cancellation premise supplies a different joint phase/form law while keeping the native two-channel pressure intact. It retains origin covariance, heterogeneous held capacity and inherited coherent replication; its modal restoring coefficient scales as the square of the diffusion eigenvalue, rather than the cotangent first power. The same owner gives a P2 Noether obstruction independent of the chosen smooth Hamiltonian storage: conserving both the native form mean and the cotangent momentum precludes relative phase motion on a full open state domain. These are conditional admission results, not a unique law selected by the nodal equation. Proofs and controls remain in that single owner.

Implementation & Examples¶

Executable Demonstrations¶

Example	Concept from this document
27_variational_principle_demo.py	Finite field-action readouts, auxiliary coordinate pairs, harmonic momentum residual and grammar-labelled heuristics; snapshot-only symplecticity is inconclusive

Key Source Modules¶

variational.py — field-action diagnostics and supplied harmonic-coordinate checks
conservation.py — field-energy diagnostic (a Lyapunov candidate, not a general certificate)
forced_support.py and phase_response.py — source compatibility and phase derivatives consumed by section 13
classical_mechanics.py — auxiliary mechanical readouts with their own scope

References¶

Nodal equation and model premises: Fundamental theory, parameter foundations
Conservation theorem: STRUCTURAL_CONSERVATION_THEOREM.md
Grammar rules: UNIFIED_GRAMMAR_RULES.md
Classical mechanics analogy: src/tnfr/physics/classical_mechanics.py
Unified fields (canonical source): src/tnfr/physics/unified.py
Energy functional: src/tnfr/physics/conservation.py

Sector	Coordinate \(q\)	Momentum \(p\)	Graph-field interpretation
Geometric	\(K_\varphi\) (curvature)	\(J_\varphi\) (phase current)	Two instantaneous phase statistics; no continuity law follows from pairing them
Potential	\(\Phi_s\) (potential)	\(J_{\Delta\mathrm{NFR}}\) (ΔNFR flux)	Two pressure statistics; their graph balance has a measured residual

Relation	Verification
\(\mathcal{H}(i) = \tfrac{1}{2}\,\mathcal{E}(i)\)	`compute_hamiltonian_density()` vs `unified.compute_energy_density()`
\(\sum_i \mathcal{H}(i) = E\)	Total Hamiltonian vs `conservation.compute_energy_functional()`
\(\rho = \Phi_s + K_\varphi\)	Charge density from conjugate pair coordinates
$\mathcal{A} = $ action density	`compute_interaction_density()` vs `unified.compute_action_density()`
Recorded \(dH/dt\)	Finite-difference diagnostic; no universal sign is certified here

Decomposition	Split	Grouping criterion
Variational (T / V)	\(T = \tfrac{1}{2}[J_\varphi^2 + J_{\Delta NFR}^2]\), \(V = \tfrac{1}{2}[\Phi_s^2 + \vert \nabla\varphi\vert ^2 + K_\varphi^2]\)	Temporal role (kinetic / configuration)
Conservation (\(\rho\) / J)	\(\rho = \Phi_s + K_\varphi\), \(\mathbf{J} = (J_\varphi, J_{\Delta NFR})\)	Diagnostic charge/current grouping; conservation requires a separate law
Unified (\(\Psi\))	\(\Psi = K_\varphi + i\,J_\varphi\)	Dual structure (geometry-transport)