Relations, support and the zero-interaction boundary

This owner asks what a connection means in a declared TNFR model before asking how one forms. It distinguishes implemented graph data, causal properties of a complete law and additional support-evolution premises. It does not install an edge law or identify a graph relation with a physical bond. The parameter ledger owns the general variable definitions; the research plan owns the next task.

1. A relation state card

Write the nodal coordinates as X=(x,theta,nu). The graph can carry distinct relational data; only the selected model's consumed coordinates belong to its dynamical state.

U is discrete and unit-free; W is a nonnegative matrix with a declared conductance scale, while its normalized row ratios are dimensionless. Directionality, reciprocity and allowed loops follow the selected model. ell has an independently declared metric unit. A proposed continuous amplitude a>=0 also needs a scale: in W=a*W_0, dimensionless W_0 gives [a_dot]=[W]/[t]; with the conductance scale retained in W_0, a is instead dimensionless and [a_dot]=1/[t]. The nodal, channel and storage units remain with the joint-law unit owner. Numerically normalized coefficients do not erase those chart/unit premises. Present support, weights and lengths are supplied or held inputs where consumed; derived field geometry and effective interactions are outputs. A candidate equation for a would add a dynamical law, not merely rename an existing input.

Object	Operational meaning	What it does not establish
Bare support U	Current directed or undirected neighborhood incidences consumed by the selected law	A nonzero current, an event time or an independently derived origin
Conductance W	Nonnegative coefficients of weighted form transport on that support	Absolute interaction strength in a row-normalized law; W_ij=0 does not remove a bare neighbor
Structural length ell	The metric/path input consumed by specified field diagnostics	Physical distance or a law deriving support from geometry
Candidate relation	Which pairs a support-changing action is permitted to inspect	An existing transport edge or a justified occurrence rule
Direct dynamical dependence	Changing a donor coordinate can change a receiver's row in the complete vector field	A nonzero derivative at every state or a persistent material bond
Effective relation	A specified donor intervention changes a retained receiver through the full dynamics, possibly via hidden nodes and memory	A new primitive edge between the observed regions

The current mixed pressure uses weighted EPI means, but unique bare neighbors for phase, capacity and degree contrast. Directed rows read outgoing neighbors; the relevant input/output convention must therefore accompany any causal claim. Parallel conductances and diagnostic lengths also have different reductions. These contracts remain centralized in the pressure owner and edge semantics. The fallback from missing length to weight is a compatibility convention, not a physical identification or an equation for emergent distance.

Three tests of absence must not be conflated. A row can depend on another node although its instantaneous rate vanishes at equilibrium. An instantaneous Jacobian block can vanish although a higher derivative or a mediated response does not. A causal-null claim instead specifies the donor intervention, receiver observation, hidden initial state and complete evolution. In the held-capacity relational law, zero receiver capacity freezes both receiver rows, but that node can still supply a fixed boundary to active neighbors. The effective-link theorem and hidden-memory owner retain these distinctions.

2. What currently changes support

The continuous relational executor selects a simple connected unit support, held capacities and an admitted phase domain. Its equations do not contain a support derivative. This is a declared model boundary, not a proof that support must be fundamental or immutable in every TNFR realization.

UM can create actual edges through the shared coupling proposal. It first needs an existing compatible neighbor at its target, then searches all nodes or a supplied sample for candidates. Its phase/form/Si score, threshold, sampling and invocation are configured. Topological REMESH similarly constructs support from declared candidate access and settings; ordinary delayed REMESH mixes form on existing support. Neither is an unrecorded autonomous connection theorem. The all-operator mechanism map owns their complete scope.

Candidate access is logically distinct from realized coupling. Supplying all unordered pairs is a possible kinematic domain; it does not assert that every pair already transmits. But a rule that reads across otherwise disconnected components adds a cross-component dependency and must declare it. A strictly component-local, fixed-support law cannot select and execute a cross-component change using information it never consumes.

Joint nodal reorganization can pay a positive attachment cost in the same declared action. The actual UM reset witness already proves this conditional possibility. Conversely, symmetry, clock covariance, passivity and recovery do not select a unique event or its time. The question below is different: whether a continuously varying relation coordinate has a consistent zero limit.

3. Native support and zero conductance have different limits

First retain the implemented distinction between bare phase support and weighted form transport. Start with edges (0,1) and (2,3), unit weights, uniform form and phases (0,0,delta,delta), 0<delta<pi/2. Without the bridge (1,2), both components have zero phase pressure. With that bridge present at any positive conductance a, the native phase sources at its ports are

\[ g_1=\delta/(2\pi),\qquad g_2=-\delta/(2\pi), \]

independently of a. They are the wrapped displacements to the unweighted two-neighbor means. A supported zero-weight bridge has these same sources; deleting it gives zero. Its unweighted phase-storage contribution beta*(1-cos(delta)) likewise does not shrink with a.

Even coincidence of endpoint phases and forms does not generally make a support event dynamically invisible. With phases zero and forms (0,1,1,1), adding (1,2) costs zero storage. At port 1, its form gradient remains one, but the phase metric changes from pi to 2*pi. The unit-support relational phase rate therefore changes from w*nu_1/(beta*pi) to half that value. This compares the separate component field and the joined unit-support field, not a weighted relational execution. The attachment owner already retains the necessary degree and resultant information.

Thus making only transport conductance small does not regularize native bare-support birth. A continuous relation model must either revise that phase/support identification or keep the change as a hybrid event with its actual reset and budget.

4. Weighting every channel does not resolve an empty row

To separate the second obstruction, consider an explicitly different, conditional mathematical extension, not an installed engine model. Take symmetric nonnegative weights on a finite potential-pair domain and set

\[ \begin{gathered} d_i=\sum_j W_{ij}>0,\quad D=\operatorname{diag}(d),\quad B=D-W,\quad q=Bx,\\ Z_i=\sum_jW_{ij}e^{i(\theta_j-\theta_i)},\quad g_i=\arg(Z_i)/\pi,\quad H_i=\pi|Z_i|\operatorname{sinc}(\arg Z_i),\\ V_W=\sum_{i<j}W_{ij}[1-\cos(\theta_j-\theta_i)]. \end{gathered} \]

Use an admitted regular phase chart with H_i>0; strict acute gaps suffice. Then H*g=-grad_theta(V_W). With held positive capacities N, fixed e,w,beta>0, the rows

\[ \dot x=N(-eD^{-1}q+wg),\qquad \dot\theta=(w/\beta)H^{-1}Nq \]

recover the existing relational model at unit weights. This extension weights phase storage and resultants too; it differs from the fixed-bare-phase reinforcement calculation. Their common unit specialization does not select between their other premises.

Let W=a*W_0, a>0, for any fixed admitted positive-strength geometry. Then

\[ D=aD_0,\quad B=aB_0,\quad H=aH_0,\quad g=g_0,\qquad F_{aW_0}(X)=F_{W_0}(X). \]

Every factor a cancels from the complete nodal field at fixed capacity. For generic state it therefore has no continuous extension to a zero-weight configuration whose nodes have no cross-node interaction. Different directions W_0 can also give different limiting responses. Weighting phase consistently removes the earlier bare-neighbor jump on a regular positive background, but does not remove this empty-strength normalization obstruction.

The implemented EPI channel already gives a minimal exact witness. On two nodes with one edge of weight a>0,

\[ \dot x_i=e\nu_i(a)(x_j-x_i), \]

whereas its zero-strength convention makes that channel inactive. At fixed nonzero contrast and e>0, continuity of this rate at zero requires nu_i(a)->0. A locally Lipschitz extension, or a finite derivative there, further requires nu_i(a)=O(a). Mere continuity allows other vanishing rates; none is selected by this argument. Pressure itself retains its nonzero directional limit even when its product with capacity vanishes.

For the complete weighted field, vanishing capacities are sufficient along a fixed regular geometry, or uniformly on a phase cone where H_i/d_i stays bounded below and forms stay bounded. They are not sufficient without such phase regularity: a vanishing resultant can make q_i/H_i unbounded. A relation between capacity and strength, an absolute mobility factor or a revised unnormalized pressure would each be an additional model premise. An evolving capacity must also retain its actual law; it is not the held-capacity model.

5. Passive onset has different branches

For the fully weighted extension define

\[ S=\tfrac12x^TBx+\beta V_W,\qquad \dot S=-e\sum_i\nu_iq_i^2/d_i +\sum_{i<j}\dot W_{ij} \left[\tfrac12(x_i-x_j)^2+\beta(1-\cos(\theta_j-\theta_i))\right]. \]

This follows by differentiation and cancellation of the form/phase exchange terms. Support work uses the same weights in both storage terms. Requiring S_dot<=0 is an additional passivity premise, not a consequence of the nodal identity and not an occurrence law.

For uniform onset W=a*W_0, write S=a*S_0(X) and Diss_0=e*sum_i nu_i*q_0i^2/d_0i. Wherever S_0>0, passivity requires

\[ \dot a\le a\,\mathrm{Diss}_0/S_0\le4e\nu_{\max}a. \]

The last inequality is Cauchy's inequality applied to each weighted gradient: sum_i q_0i^2/d_0i <= 4*E_0, where E_0=x^T B_0 x/2 <= S_0. With locally bounded capacities, a nonnegative absolutely continuous a starting at zero cannot become positive while S_0 remains positive and this passive regular description holds. Gronwall's inequality gives this boundary invariance; a multiplicative rule a_dot=a*f with locally bounded f is its familiar special case. At S_0=0, division is invalid and this argument does not restrict zero-cost activation. Nor does it select activation there. Additional stored variables, inputs or hybrid nodal resets change the premises and require their own balance.

This is not a general obstruction to a new bridge between active regions. With positive existing strengths and regular resultants at both ports, the fully weighted candidate has a smooth a=0 limit. Its storage has the form S_base+a*c, where c is the new edge cost in brackets above. Existing nodal loss can be positive at zero. The instantaneous passive budget permits a_dot(0)*c <= Diss(0), including positive onset when strict slack is available. This proves feasibility of a budget, not an independently selected derivative, candidate or time. It also does not transfer to native unweighted phase support.

6. Autonomous contact is possible, but its constitutive completion is open

Retain the fully weighted extension in section 4. Consider two components with positive existing strengths and regular phase resultants, one supplied potential bridge {u,v} of variable dimensionless weight a>=0, and fixed other weights, capacities, coefficients and clock. The potential pair is part of the declared model before its transport weight becomes positive. Use the same state-dependent edge cost and total loss as above, and define a port-local loss allocation

\[ c=\tfrac12(x_u-x_v)^2+\beta[1-\cos(\theta_v-\theta_u)],\qquad D_i=e\nu_iq_i^2/d_i,\qquad \mathcal D=\sum_iD_i,\qquad \mathcal B=D_u+D_v\le\mathcal D. \]

The port gradients consume their current weighted neighborhoods. Unlike a global-loss allocation, mathcal B does not read an unrelated disconnected component. It still supplies a chosen allocation and reads both endpoints of the candidate pair; locality relative to this potential pair does not derive candidate access or the allocation itself.

Two complete local autonomous countermodels

Combine the same nodal rows with either of the two postulated support rows

\[ \dot a=f_1=\frac{\mathcal B}{\beta+c},\qquad\text{or}\qquad \dot a=f_2=\frac{\beta\mathcal B}{(\beta+c)^2}. \]

These are autonomous equations for (X,a), not an external invocation or thresholded diagnostic. On a regular positive-background chart they are smooth, nonnegative at the boundary and nonsingular because beta>0. Local existence is the ordinary smooth-ODE result, stopped at any first exit from that chart. No global continuation or later recovery is asserted. If mathcal B(0)>0, each produces a(t)>0 for sufficiently small t>0. An initially zero allocation alone does not prove permanent inactivity: subsequent nodal evolution can change it.

More generally, a locally Lipschitz joint law X_dot=F(X,a), a_dot=mathcal H(X,a) preserves a=0 if mathcal H(X,0)=0 on the entire admitted boundary. The restricted solution (X(t),0) then solves the full equations, so local uniqueness enforces invariance. A zero derivative only at the initial state does not meet this hypothesis. This statement concerns the proposed support row, not the phase metric H.

Both retain the stated storage and obey its full balance:

\[ \begin{aligned} \dot S_1&=-(\mathcal D-\mathcal B) -\frac{\beta\mathcal B}{\beta+c}\le0,\\ \dot S_2&=-(\mathcal D-\mathcal B) -\mathcal B\left[1-\frac{\beta c}{(\beta+c)^2}\right] \le-(\mathcal D-\mathcal B)-\tfrac34\mathcal B\le0. \end{aligned} \]

Here beta*c/(beta+c)^2<=1/4. More generally a_dot=(mathcal B/beta)*h(c/beta) is passive for any smooth nonnegative h with r*h(r)<=1 for r>=0. The two displayed laws choose h(r)=1/(1+r) and h(r)=1/(1+r)^2. Having no additional numerical parameter does not remove this functional constitutive freedom.

Both laws respect relabeling of the entire declared candidate, exchange of its endpoints, common form shifts and common phase rotations. Under the complete form/clock conversion x_new=lambda*x, t_new=b*t, beta_new=lambda^2*beta, the unit owner gives mathcal B_new=lambda^2*mathcal B/b and c_new=lambda^2*c. Consequently f_new=f/b, as required for dimensionless a. These restrictions do not distinguish the two laws.

An exact first-contact discriminator

Take base edges (0,1) and (2,3) with unit conductance, candidate (1,2), a(0)=0, forms (0,1,0,0), phases zero, unit capacities and e=w=1/2, beta=1. Both components have positive strengths and capacities; the left component initially supplies loss and the right is at equilibrium. The exact initial quantities and predictions are

\[ c=\tfrac12,\quad \mathcal D=1,\quad\mathcal B=\tfrac12,\qquad \begin{array}{c|cc} &f_1&f_2\\\hline \dot a(0)&1/3&2/9\\ \dot S(0)&-5/6&-8/9\\ \ddot x_2(0)&1/6&1/9\\ \ddot\theta_2(0)&-1/(6\pi)&-1/(9\pi) \end{array} \]

Both receiver rates initially vanish. On this preparation their derivatives with respect to the new weight are partial_a F_x2=e and partial_a F_theta2=-w/(beta*pi). The old nodal-flow contribution to their acceleration is zero, so multiplication by the respective a_dot(0) gives the displayed values. Thus the two autonomous models predict

\[ x_2^{(1)}(t)-x_2^{(2)}(t)=t^2/36+O(t^3),\qquad \theta_2^{(1)}(t)-\theta_2^{(2)}(t)=-t^2/(36\pi)+O(t^3). \]

This is a local analytic prediction of the explicitly weighted completions, not a trajectory of the unit-only executor or proof of their physical status. It establishes constructive feasibility of contact and different nodal consequences under the same symmetry, units and passive budget. The full system already contains a latent dependency through its support row before the realized transport is positive; it does not create interaction from an unspecified empty substrate. In contrast, a complete locally unique system whose full state remains factorized as X_L_dot=F_L(X_L) and X_R_dot=F_R(X_R) cannot acquire cross-component causal dependence under that unchanged law. Introducing the candidate row revises that premise.

Fixed-support dynamics leaves support storage undetermined

There is a deeper freedom than choosing the allocation function. With the selected storage and other coordinates held fixed, partial_a S=c>=0. A pure scalar gradient descent a_dot=-M*c, M>=0, weakens an existing edge; its projection onto a>=0 cannot leave zero. This statement concerns that additional gradient premise, not all passive contact laws: the preceding countermodels already permit growth funded by simultaneous nodal loss.

The fixed-support nodal equations cannot distinguish

\[ \widetilde S(X,a)=S(X,a)+\Psi(a) \]

from S by differentiating in X. For every held a the nodal gradients, field and storage decay rate are identical. Once a varies, however,

\[ \dot{\widetilde S}=-\mathcal D+[c+\Psi'(a)]\dot a. \]

For example, the independently chosen, bounded-below potential Psi(a)=beta*(a^2/2-a) gives support gradient c+beta*(a-1). Writing k=c+Psi'(a), a supplied smooth nonnegative scalar mobility M defines the projected gradient rule

\[ \dot a=\begin{cases} -M k,&a>0,\\ \max(0,-M k),&a=0. \end{cases} \]

In this dimensionless-conductance chart, [M]=1/([x]^2[t]). Where the gradient row is active, the full balance is S_tilde_dot=-mathcal D-M*k^2; on the projected boundary hold it is -mathcal D. With Psi=0, the nonnegative cost forbids positive boundary drive. The displayed quadratic instead grows from zero when c<beta and M>0. The number one and the shape of this potential encode a constitutive choice even though beta already appears in the nodal model. The resulting threshold is not derived by the fixed-support equations. This example revises the chosen storage and kinetics; it does not invalidate the conditional bounds for Psi=0. The positive-onset demonstration has an ordinary smooth field locally where c<beta; no globally smooth projected boundary field is asserted. Equivalently, adding the irrelevant constant beta/2 gives Psi_plus=beta*(a-1)^2/2: a=0 has excess beta/2 above this potential's minimum. That available difference is an added relation-storage premise, not energy obtained by changing the reference zero. This example remains within the regular positive-background extension and does not repair the first-neighbor normalization boundary.

Equilibrium elimination does not supply the missing negative gradient. On an a-independent hidden domain, suppose the finite reduced storage is S_eff(y,a)=inf_h [S_base(y,h)+a*c(y,h)], with c>=0. For a_2>=a_1, every hidden-state value is nondecreasing, so taking the infimum preserves S_eff(y,a_2)>=S_eff(y,a_1). If a differentiable interior minimizing branch h*(y,a) exists, stationarity and the chain rule give partial_a S_eff=c(y,h*)>=0. The same derivative holds for the value along any differentiable interior stationary branch, without asserting that it realizes the infimum. Thus eliminating equilibrium hidden coordinates alone cannot produce a negative support-gradient drive from this storage. Moving constraints, new a-dependent terms, nonequilibrium memory and nodal-loss-funded contact lie outside this statement. The controlled hidden reduction already distinguishes static elimination from exact finite-capacity dynamics; no additional closure or evolution law follows from this monotonicity proof.

The freedom in Psi and its mobility is the support-dependent version of the undetermined potential and mobility in the variational owner. It exposes a concrete missing premise: whether a relation carries intrinsic storage or work beyond nodal mismatch, and how that quantity couples to nodal reorganization. Pure nodal evolution at fixed support cannot answer it. Further progress requires an independently justified restriction on this support sector, not tuning an allocation until links appear. Neither a unique activation law nor a new weighted runtime is installed by these countermodels. The constitutive support controls separate native weighted-form observations, exact proposed support budgets and the conditional receiver jets; they do not execute a weighted trajectory.

7. Contact and internal reorganization belong to one joint state

The connection's effect on its endpoints is already part of the admitted problem. In the continuous completion write, with held capacity,

\[ \dot X=F(X,a),\qquad \dot a=\mathcal H(X,a),\qquad X=(x,\theta). \]

Changing a changes the weighted gradients, normalization and phase geometry consumed by F; changing X changes the cost and local loss consumed by the candidate support laws in section 6. These are two directions of dynamical dependence. They do not establish a symmetric kinetic coefficient or select mathcal H. Holding the nodal law fixed in that comparison did not freeze the nodal state or its rates.

On a smooth admitted chart the chain rule gives

\[ \ddot X=D_XF\,F+(\partial_aF)\,\mathcal H. \]

The first term retains internal nodal evolution; the second records the instantaneous effect of contact growth on acceleration. The first-contact witness above already has nonzero receiver acceleration from this second term even though its initial receiver rates vanish. This is an alteration of the subsequent nodal trajectory, not an instantaneous jump in X. The result does not show that contact must alter every coordinate at every state. For example, common form and phase give q=g=0 on either compatible support; the nodal field remains zero. Capacity is held by this model and cannot be assigned a new evolution merely because a connection appears.

The storage also retains this joint dependence. With S_tilde=S_base(X)+a*c(X)+Psi(a), its exact balance is

\[ \dot{\widetilde S} =\bigl[\nabla_X S_{\rm base}+a\nabla_Xc\bigr]\cdot\dot X +\bigl[c+\Psi'(a)\bigr]\dot a. \]

In particular, partial_a grad_X S_tilde=grad_X c: the contact contribution changes the nodal storage gradients too. The existing form/phase exchange cancellation already uses these gradients at the current contact weight. Reorganization and contact work must therefore be accounted for in one balance. This identity supplies no extra reservoir and does not remove the undetermined Psi or kinetics. If a further reciprocal transfer is proposed, its change to the nodal rows must be specified and checked in this same balance; counting the same work twice would not constitute a mechanism.

Conditional kinetic-reciprocity audit

On the regular chart of the fully weighted extension, write S_tilde=S_base(X)+a*c(X)+Psi(a) and k=partial_a S_tilde=c+Psi'(a). The mixed derivative identity is

\[ \partial_a\nabla_X\widetilde S=\nabla_X k=\nabla_Xc. \]

This is reciprocity of the chosen state potential. It does not determine the kinetic matrix. In particular, the existing nodal rows admit a conditional block-diagonal completion: retain their form/phase exchange operator and add only a nonnegative scalar support mobility m. The resulting support row is a_dot=-m*k in the interior (with the declared projection at a=0). Its contribution to the storage rate is \(-m k^2\); the nodal rows remain unchanged and contribute their existing \(-\mathcal D\). Thus the joint rate is \(-\mathcal D-m k^2\) when the support row is active. This completion demonstrates compatibility, not a selected law. With the current Psi=0, k=c>=0, so this gradient row cannot give positive onset at the zero boundary. A different boundary drive requires an additional support-storage or kinetic premise, as in the conditional countermodel above.

By contrast, if one additionally requires an Onsager-symmetric cross mobility between X and a, its support component contributes a term proportional to -nabla_X S_tilde in a_dot, while the same cross block contributes a term proportional to -k in X_dot. A nonzero reciprocal kinetic block therefore changes the admitted nodal rows on a general open state domain; preserving those rows requires its net nodal contribution to vanish there, or requires explicitly revised nodal equations. An antisymmetric reversible cross block has its own paired contributions and the same row-preservation obligation. Neither type of cross block follows from partial_a grad_X S_tilde=grad_X k, from passivity alone, or from the two-way dependence of F and H. The storage choice, diagonal mobilities, cross kinetics and any reversible exchange remain separate premises. This conditional audit does not select an activation law or imply a regime transition.

For hybrid contact the corresponding object is a complete reset (U,X)->(U_plus,X_plus). The full-reset owner already separates nodal reorganization from support work in the same action. A support-only event can have X_plus=X while changing subsequent rates, as the one-bridge interface shows. Actual UM and RA-then-UM controls instead retain their changes to nodal state. Neither event contract selects its own invocation.

Finally, a primitive scalar node and a composite NFR have different state cards. The former contains only the declared coordinates of its model. The latter can reorganize through its internal fine nodes, their form/phase configuration and any admitted support evolution. A regional mean is not generally sufficient: the composition and memory results retain internal degrees of freedom or their justified memory. This is the existing route for investigating internal deformation, not evidence of new unmodeled coordinates inside every scalar node.

These results make joint node/contact evolution an explicit requirement for the next admission. The audit below classifies what could supply the missing nonzero boundary drive; it derives none. The execution plan owns what remains.

Zero-boundary audit: what can drive a first contact

Keep the regular weighted extension, one supplied candidate {u,v} and held capacities. The storage rate is S_tilde_dot=-mathcal D+k*a_dot with k=c+Psi'(a), c>=0 and mathcal D>=0: the nodal part supplies a cost and a budget, neither of which is a drive. At an uncoupled equilibrium (X*,0), each component has q=Bx=0 and g=0; hence F=0 and mathcal D=0.

Passive onset needs a nonpositive support force. Passivity at that state gives k* a_dot(0)<=0. Boundary admission also requires a_dot(0)>=0. Thus k*>0 forces a_dot(0)=0, and positive instantaneous onset requires k*<=0, i.e. Psi'(0)<=-c(X*). If c(X*)>0, this requires a negative relation-storage slope within this unchanged balance. At c=0, however, Psi=0 already gives k=0: no negative storage is necessary for a zero-cost rate, and the inequality selects none. A zero initial rate alone does not exclude higher-order onset away from a full equilibrium.

On the regular a=0 boundary near X*, with k>=k_0>0, bounded capacities and positive background strengths, loss is quadratic in q. A passive row therefore obeys a_dot<=mathcal D/k=O(|X-X*|^2). The rows f_1,f_2 of section 6 are of this kind: mathcal B<=mathcal D vanishes at equilibrium, so they preserve it and open contact only while the port nodes still carry a form gradient. At k*=0, passivity neither forbids nor selects growth.

A derived relation coordinate has no independent drive. Let a=A(X)>=0 with closed evolution X_dot=F(X,A(X)). The chain rule gives exactly a_dot=grad A . F, so no support row remains to choose; A reads both endpoints across components, a declared cross-component dependence. If A is differentiable at an interior zero, that zero minimizes A, so grad A=0 and a_dot=0. If A is twice differentiable there, then a_ddot=F^T(grad^2 A)F>=0; this may also vanish and does not guarantee onset. A positive first-order rate requires leaving this smooth-interior-minimum hypothesis, for example a kink with a_dot=A'(X;F) a one-sided derivative. If along an outward family A'(X;F)>=b*epsilon, b>0, while mathcal D=O(epsilon^2) and k>=k_0>0, passivity fails for sufficiently small epsilon. This is a directional scaling condition, not a universal exclusion of every nonsmooth gate.

Exact witness. Take forms (1+s,1,0,0) with s<0, zero phases, unit capacities, e=w=1/2, beta=1 and Psi=0. The native a=0 fields give c=1/2, mathcal D=s^2 and c_dot=s/2, so the cost falls. With threshold kappa=c(0), the linear gate A=(kappa-c)_+/kappa has a_dot(0+)=|s| and support work |s|/2; the quadratic gate A=((kappa-c)_+/kappa)^2 has a_dot(0)=0 and right acceleration a_ddot(0+)=2*s^2; this gate is C1, not C2 at its threshold. The linear budget |s|/2<=s^2 holds exactly for |s|>=1/2: the first-contact fixture (s=-1) passes and a smaller internal gradient does not. These are first jets of the closed derived evolution; they select neither gate.

Boundary storage slope is not a bistability theorem. Retain the projected gradient row above and a C2 potential, with kappa=-Psi'(0). At zero weight and component consensus, S_base=0. If c(X*)>kappa, continuity gives a local constrained storage minimum, neutral along the component offset family. If c(X*)<kappa, increasing a at fixed X lowers storage. With m(X*,0)>0 the support rate is then positive: this point is not an equilibrium of the joint system, rather than an equilibrium proved unstable. With m=0 even that rate conclusion fails. Equality requires higher-order information.

If Psi has an interior global minimizer a_*>0 with the additional strict inequality Psi(a_*)<Psi(0), the completely uniform connected state attains that lower storage. Merely having a positive minimizer does not imply the strict inequality. Nonincrease of a semidefinite storage and two static minima alone establish neither two attracting regimes nor a bifurcation: positivity of the kinetics, invariant neighborhoods, neutral directions and asymptotic behavior must be checked. The existing controls evaluate slopes, storage values and a Hessian, not those dynamical obligations.

Candidate source	It must still supply	At an uncoupled equilibrium
Nodal rows: cost c, loss mathcal D	Nothing	Cost obstructs and loss only funds: no drive
Passive allocation of loss	The allocation rule	Rate bounded by zero when k>0; zero-cost case separate
Gradient row with relation storage	Psi'(0) and a mobility	Positive rate for c(X*)<kappa only with positive boundary mobility
Smooth derived coordinate	The function A	No first-order onset at an interior zero; higher orders need checking
Kinked derived coordinate (gate)	A and its threshold	Fails under the stated linear-drive/quadratic-loss scaling
Hybrid reset	Candidate, time, reset and budget	Existing event contracts

This audit derives no first contact. It reduces the missing information to a relation-storage slope, an allocation, a gate threshold or an event rule; the nodal rows supply none of them and none is installed. The constitutive support controls check symbolic jets and represented storage/Hessian controls, including a numerical spectrum. They execute no weighted trajectory.

Composite internal storage reduces to the elimination obstruction

A composite NFR carries internal fine nodes, and the sufficient interface card retains their state behind each port. One might expect that large internal storage, such as a protected winding, could supply the negative slope the scalar node lacks. It does not, for a first contact at an uncoupled equilibrium: that internal storage is exactly the hidden-coordinate domain of the elimination argument in section 6.

Write the composite internal coordinates as h and the external partner state as y. A single supplied candidate adds a*c(h,y) with the nonnegative port cost c=\tfrac12(x_u-x_v)^2+\beta(1-\cos(\theta_v-\theta_u))>=0; the internal graph, and hence the domain of h, does not depend on a. The reduced storage is S_eff(y,a)=\inf_h[S_int(h)+a*c(h,y)]. By the envelope identity at a differentiable interior minimizer h*(a),

\[ \left.\frac{\partial S_{\rm eff}}{\partial a}\right|_{a=0}=c\bigl(h^*(0),y\bigr)\ge 0, \]

because the implicit variation dh*/da is annihilated by internal stationarity. The slope reads only the ports, at the composite's own internal equilibrium h*(0). Internal reorganization can fund contact only while the chosen allocation has loss available. For the port-local allocation this needs nonzero endpoint form gradients. Those nodal gradients vanish at equilibrium; they are not the support derivative c, which may remain positive.

For a protected winding, use that fixed sector as the hidden domain, or retain only the stationary-branch identity. A local minimum is not automatically the global infimum over an unrestricted phase domain.

At a stationary internal preparation its first-order internal storage variation vanishes; this does not make its whole stored cost constant during deformation. A winding is retained along continuous paths avoiding its branch boundary. Changing it can involve a phase reset on unchanged support or a path leaving that protected chamber, not necessarily deletion of an internal cycle. The complete-reset accounting retains both state and support effects. Nonequilibrium dynamics leaves the stationary-elimination premise, but need not change the hidden-state domain.

Exact witness. Take two disjoint acute winding-one rings as composites and one supplied unit port bridge, read through the shared reset observer. On an n-ring the equal-gap winding state has gap 2\pi/n<\pi/2, local phase cost 1-\cos(2\pi/n) per edge with \cos(2\pi/n)>0, so it is a strict local minimum of phase storage modulo common rotation in the fixed winding class. Its value n(1-\cos(2\pi/n)) differs with n (5(1-\cos 72^\circ) against 7(1-\cos(360^\circ/7))), yet with aligned ports the first-contact storage change is exactly zero for both. Offsetting the port phase by \delta gives change 1-\cos\delta, and an EPI contrast r at the ports gives r^2/2; both are nonnegative and read only the ports, never the internal winding. These are detached storage snapshots of the port-cost jump, not a weighted trajectory or an occurrence law.

This is the composite branch of the stationary, fixed-hidden-domain obstruction, not an exclusion of every composite mechanism. Moving constraints, nonequilibrium internal dynamics and new support-dependent storage terms must be assessed with their own complete laws.

Nonequilibrium memory retains preparation and evolving boundaries

Static elimination assumed the internal coordinate sits at its equilibrium. A genuinely distinct branch keeps a slow internal coordinate h out of equilibrium. Its exact evolution retains the initial state,

\[ h(t)=e^{A_ht}h_0+\int_0^t e^{A_h(t-s)}R_h(y(s),h(s))\,ds, \]

through the hidden-memory owner. The initial state must be retained, but a zero h0 need not stay zero: the nonlinear source and moving boundary can generate hidden state. The existing zero-hidden-state counterexample already makes this distinction.

For the fast-mediator chart, E_exc=S_full-S_eff=u^2+2*beta*cos(delta/2)*(1-cos v)>=0 on its stated domain. This is excess over the instantaneous frozen-boundary reconstruction, not necessarily over a full joint equilibrium. Along the full law, E_exc_dot=-mathcal D-grad(S_eff) dot F(y,z), so total storage loss alone does not fix the sign of the excess derivative.

A static P3 control makes this concrete without a new solver. Put the mediator at node 1, forms (1,1/2,0), phases zero, capacities (1,3,2) and e=w=1/2, beta=1. The same two-edge midpoint decomposition has zero initial excess. Native rates give u_dot=-1/8, v_dot=1/(8*pi) relative to the moving endpoint means, while total storage rate is -3/8. Consequently E_exc_dot(0)=0 and E_exc_ddot(0)=(1+1/pi^2)/32>0. Hidden excess can grow while total storage falls; its presence is not always an independently supplied nonzero hidden preparation. The shared-field control keeps represented rates distinct from this ideal derivative identity.

The prepared P2 storage/loss check remains a valid special case, not a theorem of monotone hidden memory. None of these rewrites alone selects a support row, but neither do they exclude a separately justified internal mechanism.

Declared forcing changes work, not the support gradient

The live Gamma registry supplies dx/dt=nu_f*p+Gamma. If only this form source is added to the weighted comparison, the chain rule becomes S_tilde_dot=-mathcal D+q^T Gamma+k*a_dot. Source work can have either sign; moving form at zero capacity does not by itself fund contact. In particular q=0 gives zero instantaneous source work despite a possible nonzero form rate. Gamma changes the work balance, not the storage derivative k=c+Psi'(a).

The source law and provenance must be declared, whether it is prescribed externally or depends on retained state. The unforced relational executor rejects nonzero Gamma; this forced comparison is not installed. Neither a source nor its budget alone selects support evolution or event occurrence.

The origin of relation storage is a nonselection classification

The reviewed branches expose remaining premises; they are not an exhaustive classification of all possible support laws. A negative support slope, work available to pay positive support cost, and a rule selecting occurrence are different quantities. At an uncoupled equilibrium the unchanged unforced balance has vanishing nodal loss; its zero-cost case remains separate.

Mechanism reviewed	Remaining obligation
Relation-storage potential	Storage slope and kinetics; negative slope is not required at zero cost
Loss allocation	An allocation of contemporaneous loss, not a negative support gradient
Derived relation coordinate	Chain-rule closure, domain and any gate premise
Stationary composite reduction	Nonnegative slope on the fixed hidden domain; moving constraints are outside this proof
Nonequilibrium memory	Initial state and boundary-driven dynamics; hidden excess need not decay monotonically
Declared forcing/Gamma	Signed source work and a support law; forcing alone does not determine a storage slope
Hybrid reset	Candidate, time, reset and budget via the event contracts

The distinct autonomous countermodels in section 6 prove nonselection by the compared premises, not impossibility of autonomous continuous formation. The event-law comparison retains different support and occurrence hypotheses; placing both results in a table does not exclude every future completion. Primitive formation and independent selection of its law remain open. The execution plan owns the next bounded admission, not a theorem that these premises are exhaustive.

A possible regime transition is an open interpretation

The passage from separately evolving patterns to a maintained coupled organization could be investigated as a transition of the joint dynamics. This meaning of transition is distinct from changing the angular phase theta of a node. The current smooth positive-onset examples establish neither a critical threshold nor a change of stability or a maintained coupled regime. A weight becoming positive is not sufficient evidence for those claims.

Once a joint law is justified, identify its uncoupled and coupled invariant states or persistent regimes and the declared preparation or model parameter being varied. A stability change or coexistence of distinct regimes would provide a concrete mechanism to examine; a threshold assigned to a diagnostic would not derive it. The gradient row of the zero-boundary audit locates a boundary-force sign change for a supplied potential, not by itself a bifurcation or two attracting regimes. Its kappa is declared and the projected joint dynamics needs the separate stability analysis above. A finite-network dynamical bifurcation, if established, would retain its own scope rather than establish a physical thermodynamic transition. This is an interpretation criterion for the same contact study, not a new campaign or an additional phase-transition law.

8. Reuse and the remaining foundation

The normalized-neighbor owner, support derivative and phase geometry supply the existing calculations. Their fixed-active-edge derivative deliberately excludes birth. The shared pressure controls check the zero-strength and positive-background limits on five execution paths. The relational executor remains unit-support execution; none of the weighted alternatives above is installed. Existing attachment controls and actual operator reset controls exercise distinct hybrid cases without selecting their occurrence.

The foundational obligation is to specify which relation is being formed, its zero state, candidate access, normalization and work accounting before deriving its evolution. Current nodal dynamics already derives effective relations on supplied fine support. Continuous creation of primitive relations requires a justified completion at the boundary, while hybrid creation requires a justified event mechanism. These alternatives remain revisable; neither negative result proves that TNFR can never supply such a completion.