Native mediated pattern dynamics¶

Native one-, two- and three-port mediator reduction, fast-limit scope, memory clocks and the frozen causal-response control; no boundary source is installed.

Part of Native pattern reduction and memory. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

9. A nodal intermediary mediates joint form/phase interaction¶

Full state and conditional recovery¶

Replace the direct 0--5 bridge between the prepared C5 rings by 0--10--5. All twelve edges have unit conductance. The eleven-node state retains signed form and primitive phase at every node, including mediator 10. Hold capacity nu>0 on both rings and mu>0 on the mediator, with fixed e,w,beta>0 and the same structural clock. Here e,w are the effective normalized channel coefficients. The law is the existing RelationalExchangeModel, not an additional coupling or memory equation.

At uniform form, winding-one phases theta_i=2*pi*(i mod 5)/5 on each ring and theta_10=0, all sine sums vanish and every edge cosine is positive. Set c=cos(2*pi/5), r=1+2*c. Port phase metrics are H_p=pi*r, and the mediator metric is H_m=2*pi. The existing local recovery theorem therefore applies to this connected graph: two common-offset freedoms and twenty stable transverse directions, with a sufficiently small acute recovery neighborhood. The two rings can have a joint restoring geometry through an intermediary without a direct ring-to-ring edge. This is local recovery of a prepared geometry on supplied support, not capture from disconnected support, a global formation theorem or a physical bound-state identification.

Derived memory, with two indispensable hidden coordinates¶

Write u,v for form and lifted phase deviations. Retain all twenty ring coordinates as y, and hide only z_m=(u_10,v_10). Define

\[ D_0=\begin{pmatrix}-e&-w/\pi\\w/(\beta\pi)&0\end{pmatrix},\qquad B_0=-\tfrac12D_0,\qquad C_0=\begin{pmatrix}e/3&w/(\pi r)\\-w/(\beta\pi r)&0\end{pmatrix}. \]

The Jacobian of the admitted full field has hidden row z_m_dot=mu*D_0*z_m+mu*B_0*(z_0+z_5). Each visible port receives nu*C_0*z_m; its other instantaneous terms remain in the full visible block A. Here z_0,z_5 denote the two form/phase port pairs within y, not closed states of the rings. Let B inject nu*C_0 at both port rows, and let C read both port pairs with mu*B_0. Exact tangent elimination gives

\[ \dot y(t)=Ay(t)+B e^{\mu D_0t}z_m(0) +\int_0^t B e^{\mu D_0(t-s)}C y(s)\,ds. \]

The cross-region and self-memory port blocks are identical:

\[ \boxed{\quad K_\mu(t)=\mu\nu C_0 e^{\mu D_0t}B_0.\quad} \]

This kernel follows from the declared field; no delay, fitted gain or primitive pulse is introduced. It is exact for the tangent model. Nonlinear continuation retains nonlinear hidden forcing, as in section 3, and cannot substitute this fixed convolution for the full engine. The initial hidden-state term remains necessary even when a particular experiment prepares it to zero.

Since det(C_0)=w^2/(beta*pi^2*r^2)>0, both mediator coordinates influence the observed ring rates. For the supplied full-state Jacobian J and coordinate observation O, rank(O)=20 and rank((O;OJ))=22. Thus a twenty-coordinate instantaneous all-state linear closure is impossible; retaining both hidden coordinates or their exact memory is necessary. This reuses the invariant-row argument, rather than introducing another state-minimality criterion.

An exact nonlinear pressure chart retains the moving boundary¶

The degree-two mediator also admits an exact change of coordinates beyond the tangent approximation. Retain the full visible ring state and the same fixed support, coefficients and capacities. On a continuous local phase lift, retain full-field admission, including all ring edges, and write

\[ \bar x=\frac{x_0+x_5}{2},\qquad \bar\theta=\frac{\theta_0+\theta_5}{2},\qquad \delta=\theta_5-\theta_0,\qquad u=x_m-\bar x,\qquad v=\theta_m-\bar\theta. \]

\[ z_m=2\cos(\delta/2)e^{-iv},\qquad H_m=2\pi\cos(\delta/2)\operatorname{sinc}(v)>0, \qquad g_m=-v/\pi,\qquad q_m=2u. \]

Consequently its evaluated pressure, not a stale stored observation, obeys

\[ p_m=-eu-\frac w\pi v,\qquad v=-\frac\pi w(p_m+eu). \]

At fixed visible state, (u,p_m) is therefore an invertible replacement for the two mediator coordinates. It preserves their initial information; it does not remove the node, its capacity or either incident edge. Applying the unchanged full field and differentiating the moving endpoint averages gives

\[ \dot u=\mu p_m-\dot{\bar x},\qquad \dot v=\frac{\mu w u} {\beta\pi\cos(\delta/2)\operatorname{sinc}(v)} -\dot{\bar\theta},\qquad \dot p_m=-e\dot u-\frac w\pi\dot v. \]

These are exact nonlinear coordinate identities throughout the admitted interval. The endpoint rates come from the retained full network; they are not new external forcing or a closed two-port law. In particular, with held mu, pressure history determines the contrast only after retaining its initial value and the endpoint motion:

\[ u(t)=u(0)+\mu\int_0^t p_m(s)\,ds -[\bar x(t)-\bar x(0)]. \]

Substituting this expression and v=-pi*(p_m+e*u)/w into the pressure row gives a nonlinear memory representation with moving-boundary terms. Dropping u(0), either endpoint-rate term, or the visible phase contrast changes the model. This is the same closure principle as the P2 pressure chart, with the additional boundary motion required by the mediator's environment.

The nonzero determinant of C_0 above also rules out an exact smooth scalar mediator summary for this full visible-rate observation on an open neighborhood of the reference. At fixed visible state, the two rates at one port have derivative nu*C_0 with respect to the two mediator coordinates, of rank two; any smooth factorization through one scalar has rank at most one. This is local minimality for the stated observation, not a claim about restricted one-dimensional preparations, lossy approximations or histories. The finite nonlinear compensation witness exhibits the corresponding failure of pressure alone inside a recovery domain. The coordinate identities themselves also hold at mu=0, where the mediator's absolute state freezes but its endpoint-relative coordinates can still change. Positive-capacity recovery remains a separate theorem. The native controls check both moving-boundary rows and an independent pressure derivative at nonuniform visible states, including a frozen mediator. They test these coordinate identities, not a reduced nonlinear trajectory solver.

A controlled nonlinear reduction when the mediator is fast¶

Hold the ring capacities at one and vary only the positive mediator capacity mu, with the same unforced law, support, clock and effective coefficients. This is a time-scale regime of the existing law, not a new interaction. Let y retain all twenty visible ring coordinates and let z=(u,v) be the endpoint-relative mediator coordinates just defined. Reconstruct the full state by x_m=bar(x)+u, theta_m=bar(theta)+v and evaluate the native visible field F(y,z). Set b(y,z)=(dot(bar(x)),dot(bar(theta))) using its port rows. Then

\[ \dot y=F(y,z),\qquad \dot z=\mu f_\delta(z)-b(y,z),\qquad f_\delta(u,v)= \begin{pmatrix} -eu-wv/\pi\\ wu/[\beta\pi\cos(\delta/2)\operatorname{sinc}(v)] \end{pmatrix}. \]

Both F and b are independent of mu: visible capacities are held, and this relational pressure does not consume neighbor capacity. The unique frozen-boundary fast equilibrium is z=0. The candidate reduced equation is dot(y_*)=F(y_*,0), with the same visible initial state. At finite mu, however, dot(z)=-b(y,0) on z=0; the midpoint constraint is generally not invariant. Its validity as an approximation requires the estimate below, not merely instantaneous storage minimization.

The reduced field retains the actual interface. Write q_ring for the form gradient from each ring's internal edges alone. At the reconstructed midpoint,

\[ \widetilde q_0=q_{{\rm ring},0}+\frac{x_0-x_5}{2},\qquad \widetilde q_5=q_{{\rm ring},5}+\frac{x_5-x_0}{2}, \]

with all other visible gradients unchanged. The relative phase resultants at the two ports are

\[ \begin{aligned} \widetilde Z_0&=e^{i(\theta_1-\theta_0)}+ e^{i(\theta_4-\theta_0)}+e^{i\delta/2},\\ \widetilde Z_5&=e^{i(\theta_6-\theta_5)}+ e^{i(\theta_9-\theta_5)}+e^{-i\delta/2}. \end{aligned} \]

Retain the original port degree three, the other visible degrees two, and the native definitions g_tilde=Arg(Z_tilde)/pi and H_tilde=pi*|Z_tilde|*sinc(Arg(Z_tilde)). Thus the visible rows are dot(x_i)=-e*q_tilde_i/d_i+w*g_tilde_i and dot(theta_i)=w*q_tilde_i/(beta*H_tilde_i). In particular, this is not the law on a ten-node graph with an ordinary unit 0--5 edge: that edge would give the full form contrast and a full-angle neighbor phasor. The native eleven-node field evaluated at the midpoint already supplies the required reduction; no second pressure implementation is needed.

Its storage is also inherited, rather than fitted:

\[ \begin{aligned} S_{\rm eff}={}&\frac12\sum_{\{i,j\}\in E_{\rm ring}}(x_i-x_j)^2 +\frac{(x_0-x_5)^2}{4}\\ &+\beta\left[\sum_{\{i,j\}\in E_{\rm ring}} (1-\cos(\theta_i-\theta_j))+2(1-\cos(\delta/2))\right],\\ \dot S_{\rm eff}={}&-e\sum_{i\ne m}\frac{\widetilde q_i^2}{d_i}\le0. \end{aligned} \]

To obtain the last identity, both hidden storage derivatives vanish at the midpoint, so differentiating its reconstruction contributes no extra term. The remaining derivatives and form/phase exchange cancellation are the native ones. The half-angle potential belongs to the declared |delta|<pi lift; it is not a globally defined replacement cosine edge on the phase circle. The selected phase lift remains part of its domain. Away from the midpoint, the exact storage difference is

\[ S_{\rm full}(y,z)-S_{\rm eff}(y) =u^2+2\beta\cos(\delta/2)(1-\cos v)\ge0. \]

A fixed nonzero hidden initial state can therefore carry finite excess storage even when its visible trajectory effect becomes small. Its initial layer loss is retained by the full law; the reduction does not claim uniform storage or instantaneous-rate convergence at time zero.

Uniform attraction for the default coefficients. The following explicit bound uses e=w=1/2, beta=1; it is not a uniform claim over all coefficients. For |delta|<=1/2, |v|<=1/4, write f_delta(z)=A_0*z+R_delta(z) with

\[ A_0=\begin{pmatrix}-1/2&-1/(2\pi)\\1/(2\pi)&0\end{pmatrix},\qquad P=\begin{pmatrix}2&\pi\\\pi&2+\pi^2\end{pmatrix}. \]

Direct multiplication gives A_0^T*P+P*A_0=-I. The leading minor and determinant of P-I are one, while trace(P)=4+pi^2<14, hence I<P<14I. Moreover,

\[ \cos(\delta/2)\operatorname{sinc}(v)\ge \frac{31}{32}\frac{95}{96}=\frac{2945}{3072},\qquad \|R_\delta(z)\|\le\frac{127}{6\cdot2945}\|z\|. \]

Here cos(t)>=1-t^2/2, sinc(t)>=1-t^2/6 on the stated intervals and pi>3 suffice. Therefore 2*||P||*127/(6*2945)<1/2, and the fast field obeys

\[ 2z^TPf_\delta(z)\le-\tfrac12\|z\|^2. \]

This common quadratic bound does not differentiate a moving P(delta) and holds for arbitrary admitted changes of the visible phase contrast. It uses positive form damping: the e=0 periodic boundary does not justify the same attracting reduction.

Domain and continuation hypotheses. Fix a finite horizon T on which the reduced solution y_* exists. Choose constants d>0, 0<rho<1/4 such that its closed visible tube of radius d, together with W(z)=sqrt(z^T*P*z)<=rho, is compactly contained in a smooth full-state chart. Require all reconstructed ring edges to remain acute and |delta|<=1/2 there. The mediator edges are then also acute, because |v+-delta/2|<=1/2<pi/2. In this fixed tube choose bounds, independent of mu,

\[ \|b(y,z)\|\le B,\qquad \|F(y,z)-F(y_*(t),0)\| \le L(\|y-y_*(t)\|+\|z\|). \]

Smoothness supplies finite such bounds on an admitted tube; no numerical values are inferred merely from a trajectory. Its margin must be justified for the reduced solution, rather than assuming the full solution stays in the domain whose preservation is to be proved. Near the aligned winding reference, its strict acute margins and smooth reduced field provide such a neighborhood for sufficiently close preparations on any fixed finite horizon, by ordinary local existence and continuous dependence.

Initial layer and visible error. Up to the first exit from this tube, the previous quadratic inequality and |b|<=B give

\[ D^+W\le-\frac\mu{56}W+14B,\qquad W(t)\le e^{-\mu t/56}W(0) +\frac{784B}{\mu}(1-e^{-\mu t/56}). \]

The upper right derivative formulation includes W=0. Since ||z||<=W, comparison of the two visible equations and Gronwall's inequality imply

\[ \boxed{\quad \sup_{0\le t\le T}\|y(t)-y_*(t)\| \le\frac{Le^{LT}}{\mu}\,[56W(0)+784BT].\quad} \]

For W(0)<rho, choose mu so that 784B/mu<rho and the displayed visible bound is strictly below d. The estimate for W is a convex combination of two values below rho. Both estimates therefore prevent a first exit; compact smooth continuation extends the solution through T and validates the bounds on that whole interval. These conditions are sufficient, not optimal, and do not supply a numerical cutoff without actual tube bounds.

A fixed small hidden initial state is allowed. Its visible contribution is included in the 56W(0)/mu term, while the hidden state retains an initial layer of decay exp(-mu*t/56). It is not uniformly small at time zero: for mu>=1 it becomes O(1/mu) after a time of order log(mu)/mu. Thus finite-memory influence can be approximated in this regime without denying the exact hidden-state obstruction at fixed capacity. Neither arbitrary initial environments nor infinite-horizon errors are covered.

This is an ideal continuous-law result. Its field evaluation reuses detached evaluate_relational_exchange at the reconstructed midpoint and projects the visible rows. The native static controls can check its field substitution and storage identities; they do not certify the tube constants or a reduced trajectory. Fixed-step Euler is not justified as mu grows: resolving the fast initial layer needs a separate numerical step-size analysis. No new solver, primitive edge, occurrence law or physical identification follows from eliminating this already supplied mediator.

Two fast intermediaries compose through their inherited interface¶

Replace the intermediary path by 0--10--11--5, retaining the same two C5 rings and all thirteen unit edges. There are twenty visible coordinates and four hidden coordinates. Hold ring capacities at one and initially give both intermediaries capacity mu>0, with e=w=1/2, beta=1. The unforced law and clock are unchanged. Choose a local acute path lift and write X=x_5-x_0, delta=theta_5-theta_0. Frozen-endpoint hidden equilibrium is

\[ x_{10}^*=x_0+X/3,\quad x_{11}^*=x_0+2X/3,\qquad \theta_{10}^*=\theta_0+\delta/3,\quad \theta_{11}^*=\theta_0+2\delta/3. \]

Let u=(x_10-x_10^*,x_11-x_11^*) and v=(theta_10-theta_10^*,theta_11-theta_11^*). For z=(u_1,u_2,v_1,v_2), set

\[ M=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\qquad \begin{aligned} C_1&=\cos(\delta/3+v_2/2)\operatorname{sinc}(v_1-v_2/2),\\ C_2&=\cos(\delta/3-v_1/2)\operatorname{sinc}(v_2-v_1/2). \end{aligned} \]

The hidden metrics are H_10=2*pi*C_1, H_11=2*pi*C_2, the form gradient is M*u, and the phase source is -M*v/(2*pi). Thus the exact fast map is

\[ f_\delta(z)= \begin{pmatrix} -(e/2)Mu-(w/(2\pi))Mv\\ (w/(2\beta\pi))\operatorname{diag}(C_1^{-1},C_2^{-1})Mu \end{pmatrix}. \]

The moving-boundary system again has dot(y)=F(y,z) and dot(z)=mu*f_delta(z)-b(y,z). Here b consists of the derivatives of the four interpolated coordinates above, for example dot(x_10^*)=(2*dot(x_0)+dot(x_5))/3. All these endpoint rates come from the retained visible field. The hidden equilibrium is unique in this chart: the phase rows first require M*u=0, and the form rows then require M*v=0.

Exact static composition and reduced balance. Evaluating the full native field at this lift gives

\[ \widetilde q_0=q_{{\rm ring},0}-X/3,\qquad \widetilde q_5=q_{{\rm ring},5}+X/3, \]

and port resultants equal to their two internal ring phasors plus exp(i*delta/3) at port 0 and exp(-i*delta/3) at port 5. Retain original port degree three and all native metric/phase-source definitions. The storage and its derivative under the induced visible law are

\[ S_{\rm eff}=S_{\rm rings}+X^2/6+3\beta(1-\cos(\delta/3)),\qquad \dot S_{\rm eff}=-e\sum_{i\ne10,11}\widetilde q_i^2/d_i\le0. \]

Both hidden storage gradients vanish at the lift, so the same envelope chain rule used for one intermediary proves the balance. Hidden initial storage is not removed from the full system. Its form excess is u^T*M*u/2; its phase excess is

\[ \beta[3\cos(\delta/3)-\cos(\delta/3+v_1) -\cos(\delta/3+v_2-v_1)-\cos(\delta/3-v_2)]. \]

The latter is nonnegative when the three lifted gaps remain acute, by strict convexity of 1-cos on that interval and their fixed sum. A fixed hidden initial displacement can therefore again carry finite initial-layer storage.

Successive static elimination gives the same result if its inherited interface is retained. Eliminating node 10 first gives x_10=(x_0+x_11)/2, theta_10=(theta_0+theta_11)/2. The remaining gradient is q_11=(x_11-x_0)/2+(x_11-x_5), still with original degree two, while its phase resultant is

\[ e^{-i(\theta_{11}-\theta_0)/2}+e^{i(\theta_5-\theta_{11})}. \]

Setting both remaining rows to zero yields the same thirds; reversing the order does too. By contrast, replacing the eliminated path by an ordinary unit 0--11 edge would produce the false interpolation x_10=x_0+X/4, x_11=x_0+X/2. Reconstructing the original graph gives q_11=-X/4, and the analogous phase preparation has hidden gradient sin(delta/4)-sin(delta/2), generally nonzero. Static elimination therefore composes through the inherited field, not through an arbitrary replacement graph. This equality does not by itself interchange finite-capacity dynamics or establish nested initial layers for successively separated time scales.

The same static statement holds for a finite path of ell unit-conductance edges, with only degree-two interior nodes and a specified path lift. Here ell counts fine links; it is not the separate graph length attribute or an inferred physical distance:

\[ S_{\rm path}^{\rm eff}(X,\delta) =\frac{X^2}{2\ell}+\beta\ell(1-\cos(\delta/\ell)). \]

Each gap is delta/ell in the selected acute sector. Joining lengths ell_1,ell_2 and minimizing their common boundary places that boundary at x_A+ell_1*(x_B-x_A)/(ell_1+ell_2), with the same lifted phase interpolation. The phase condition follows from the injectivity of sine on the acute interval. Energy and endpoint messages therefore agree with length ell_1+ell_2, associatively. The message retains length, lift, endpoint form contrast, adjacent phase and original node degrees. Different allowed lift sectors cannot be silently identified using endpoint phases modulo full turns. This is a static series result, not a dynamical estimate uniform in path length or a result for branching interiors.

For example, a three-edge path whose endpoint circular gap is 3*pi/4 admits both lifted gaps pi/4 on every edge and -5*pi/12 on every edge. Its two hidden phase pairs are respectively (pi/4,pi/2) and (-5*pi/12,-5*pi/6) relative to the left endpoint. Both have zero hidden phase source and acute fine edges, yet their adjacent port phasors differ. With uniform form and the maintained ring twists, donor pressure has opposite signs. Thus length and circular endpoint phases alone are insufficient; static composition must retain the selected lift sector. This control is outside the small-delta quantitative neighborhood used next.

Joint fast attraction and a finite-horizon bound. At the default coefficients and delta=v=0, the joint fast Jacobian and a common quadratic matrix are

\[ J_0=\begin{pmatrix}-M/4&-M/(4\pi)\\M/(4\pi)&0\end{pmatrix},\qquad P_4=\begin{pmatrix}2I&\pi I\\\pi I&(2+\pi^2)I\end{pmatrix}. \]

They satisfy I<P_4<14I and J_0^T*P_4+P_4*J_0=-diag(M,M)/2<=-I/2. For |delta|<=1/2, |v_1|,|v_2|<=1/8, the cosine arguments have magnitude at most 11/48 and the sinc arguments at most 3/16. Hence

\[ C_i\ge\frac{4487}{4608}\frac{509}{512} =\frac{2283883}{2359296},\qquad 0\le C_i^{-1}-1\le\frac{75413}{2283883}<\frac1{28}. \]

Since ||M||=3 and pi>3, the nonlinear remainder obeys ||f_delta(z)-J_0*z||<=||z||/112. Consequently 2*z^T*P_4*f_delta(z)<=-||z||^2/4. This controls the four hidden coordinates jointly; independent single-node decay estimates are not substituted for the coupled proof.

Reuse the preceding finite-horizon tube and first-exit construction, now with W=sqrt(z^T*P_4*z), rho<1/8, and bounds B,L for this twelve-node reconstruction. On its admitted interval,

\[ \begin{aligned} W(t)&\le e^{-\mu t/112}W(0) +\frac{1568B}{\mu}(1-e^{-\mu t/112}),\\ \sup_{0\le t\le T}\|y(t)-y_*(t)\| &\le\frac{Le^{LT}}\mu[112W(0)+1568BT]. \end{aligned} \]

Require W(0)<rho, 1568B/mu<rho and a visible error bound strictly below the tube radius d. These close the same first-exit argument. The three intermediary edge gaps are acute throughout the box, with magnitude at most 5/12; the tube must separately retain all ring-edge and visible-state margins. Thus the reduction has an O(1/mu) visible error with its hidden initial layer retained, on a fixed finite horizon and admitted neighborhood. Neither a fixed-step Euler certificate nor an infinite-horizon bound follows.

Fixed positive unequal capacity ratios also admit a local stability certificate. For capacities mu*a_1,mu*a_2, put D=diag(a_1,a_2), D_4=diag(D,D) and P_a=P_4*D_4^-1. The fast map is D_4*f_delta, and exactly 2*z^T*P_a*D_4*f_delta=2*z^T*P_4*f_delta<=-||z||^2/4. Moreover I/a_max<P_a<14I/a_min. The same proof therefore gives constants depending on these fixed ratios, with its hidden tube rescaled to preserve ||z||<=sqrt(a_max)*sqrt(z^T*P_a*z)<1/8. They are not uniform as a ratio vanishes or becomes unbounded. The explicit 112,1568 bounds above are for equal capacities only.

The native static controls check the reconstructed field, inherited balance and the wrong-edge control. They do not evaluate a reserved trajectory or supply numerical tube constants. No additional pressure owner, topology mutation or reduced executor is introduced: composition is performed on the already supplied nodal system.

A branching intermediary induces a collective three-port interaction¶

Take three C5 rings on nodes 0,...,14, with ports A=0, B=5, C=10, and connect one intermediary m=15 to all three ports. All eighteen edges have unit conductance. Retain all thirty visible form/phase coordinates, unit visible capacities, a held positive intermediary capacity, and the same unforced relational law with e=w=1/2, beta=1. The support is supplied; stationary elimination does not assert that the full system is stationary.

For the three port phases define

\[ Z=e^{i\theta_A}+e^{i\theta_B}+e^{i\theta_C}=Re^{i\Psi},\qquad \bar x=(x_A+x_B+x_C)/3. \]

Work in a local chart with R>0 and all three gaps alpha_p=theta_p-Psi strictly acute, and retain admission of every ring edge. A nonzero resultant alone does not guarantee this acute-star premise. The unique conditional hidden equilibrium is

\[ x_m^*=\bar x,\qquad \theta_m^*=\Psi. \]

Indeed the hidden phase row first requires q_m=0, fixing the form average; the form row then requires zero phase source. In the admitted chart this selects the circular resultant direction. The antipodal stationary point of the phase storage is outside this domain. Both hidden storage gradients vanish at the admitted lift.

Inherited field and storage. At each port the hidden contribution to the form gradient is x_p-bar(x), and its relative neighbor phasor is exp(i*(Psi-theta_p)). Add these to the port's two internal ring contributions and retain degree three, the native phase source and the native phase metric. Other visible rows are unchanged. Consequently the full sixteen-node native field at the conditional lift supplies the induced visible law without another pressure formula. Its storage is

\[ S_{\rm eff}=S_{\rm rings} +\frac16\sum_{p<q}(x_p-x_q)^2+\beta(3-R),\qquad \dot S_{\rm eff}=-e\sum_{i\ne m}\widetilde q_i^2/d_i\le0. \]

The envelope chain rule again removes the hidden reconstruction derivative because its storage gradients vanish there. This is a closed conditional field on the full visible state, not a closed state of three rigid rings or an exact finite-capacity elimination of hidden history.

The phase storage is not a sum of independent pair interactions. Define V=3-R. Even allowing arbitrary three-times differentiable pair functions of their two absolute phases, plus one-port terms, a representation

\[ V=V_{AB}(\theta_A,\theta_B)+V_{AC}(\theta_A,\theta_C) +V_{BC}(\theta_B,\theta_C)+\sum_p V_p(\theta_p) \]

would require partial_A partial_B partial_C V=0 throughout its domain. But at (theta_A,theta_B,theta_C)=(0,0,t), with 0<t<pi/4, direct differentiation gives

\[ \begin{aligned} \partial_A\partial_B V &=-\frac{(2+\cos t)^2}{(5+4\cos t)^{3/2}},\\ \partial_A\partial_B\partial_C V &=-\frac{2\sin t(1-\cos t)(2+\cos t)}{(5+4\cos t)^{5/2}}\ne0. \end{aligned} \]

These preparations lie in the acute-star domain and can be arbitrarily close to alignment. Thus no independent-pair identity holds on an open neighborhood of that reference. Allowing a nominal pair to consume the third port's state would instead declare a collective interaction.

Low-order agreement does not remove this obstruction. For local differences Delta_pq=theta_p-theta_q, set S_k=sum_pairs Delta_pq^k and P_6=product_pairs Delta_pq^2. Expansion about alignment gives

\[ V=\frac{S_2}{6}-\frac{S_4}{216}-\frac{S_6}{19440} +\frac{P_6}{648}+O(\|\Delta\|^8). \]

Here S_4=S_2^2/2 and S_6=S_2^3/4+3*P_6. General pair potentials can match the quadratic and quartic terms; the displayed genuinely mixed storage term first occurs at sixth order. The particular cosine surrogate sum_pairs(1-cos(Delta_pq))/3 already fails at fourth order. This order statement concerns storage, not every component of the nonlinear field.

The actual port rate is collective too. Keep all forms zero, hold ring A in its winding-one reference, and rotate rings B and C by independent s,t. Let a=2*cos(2*pi/5), so 0<a<1, and define

\[ \Psi(s,t)=\arg(1+e^{is}+e^{it}),\qquad g(\psi)=\arg(a+e^{i\psi}),\qquad \dot x_A=\frac w\pi g(\Psi(s,t)). \]

The complete state and both internal neighbors of A remain fixed in this comparison. At s=t>0 sufficiently small, with Q=5+4*cos(t),

\[ \Psi_s=\Psi_t=\frac{2+\cos t}{Q},\qquad \Psi_{st}=\frac{2(2+\cos t)\sin t}{Q^2}>0, \]

while

\[ g'(\psi)=\frac{1+a\cos\psi}{1+a^2+2a\cos\psi}>0,\qquad g''(\psi)=\frac{a(1-a^2)\sin\psi}{(1+a^2+2a\cos\psi)^2}>0 \]

at the resulting positive Psi. Hence partial_s partial_t dot(x_A)=(w/pi)*(g''*Psi_s*Psi_t+g'*Psi_st)>0. An additive rate depending separately on (A,B) and (A,C), plus A alone, has zero mixed derivative under these same variations. This rules out that independent-pair field representation, even without requiring a pairwise storage. Shared state-dependent normalizations that read the third ring would not satisfy the proposed independence contract.

Near alignment, this mixed rate derivative is 2*w*t*(1+3*a)/(27*pi*(1+a)^3)+O(t^3). Thus the native visible field already has a collective cubic contribution, although the storage's first genuinely mixed contribution is sixth order. The inherited phase metric and readout prevent those two order statements from being interchanged.

A controlled fast regime is available locally. To justify a dynamical approximation, set u=x_m-bar(x), v=theta_m-Psi(y), z=(u,v) and give the mediator capacity mu. The exact moving-boundary rows are

\[ \dot z=\mu \begin{pmatrix}-eu-wv/\pi\\3wu/[\beta\pi R\operatorname{sinc}(v)]\end{pmatrix} -\begin{pmatrix}\dot{\bar x}\\\dot\Psi\end{pmatrix},\qquad \dot\Psi=\frac1R\sum_{p=A,B,C}\cos(\alpha_p)\dot\theta_p. \]

The phase-boundary velocity is a weighted circular-mean derivative, not an arithmetic average. Its port rates are native visible rates. At the default coefficients, require |alpha_p|<=1/4 and |v|<=1/4 in a compact admitted tube about a reduced visible trajectory. Then

\[ \frac R3\operatorname{sinc}(v)\ge \frac{31}{32}\frac{95}{96}=\frac{2945}{3072}. \]

The same two-dimensional matrix P from the fast-mediator theorem therefore gives exactly its common quadratic estimate and 56,784 bounds. Apply its first-exit argument with newly justified bounds B,L, hidden radius rho<1/4, and visible margin d for this sixteen-node reconstruction. The boundary velocity and visible field remain independent of mu, so the visible trajectory error is O(1/mu) on the fixed finite horizon, including the hidden initial-layer contribution. Neither the earlier graph's tube constants nor its preparation are silently reused. At finite capacity the conditional lift is not generally invariant.

The hidden storage excess is exactly 3*u^2/2+beta*R*(1-cos(v)); a fixed nonzero hidden initial state can carry finite storage into that layer. Its loss and initial rate discrepancy are not erased by the visible approximation. The native mediation controls check the induced rows, storage and mixed-response discriminator without a new response campaign. The result derives a collective conditional interaction from the supplied nodal law and support; it installs no new connection rule or reduced executor.

Capacity sets the memory clock, and the kernel need not be positive¶

Return here to the native two-port model at the start of section 9, with c=cos(2*pi/5) and its original port quantity r=1+2*c. This r is distinct from the resultant ratio rho=R/k above. For that two-port single-intermediary model, the hidden eigenvalues solve s^2+mu*e*s+mu^2*w^2/(beta*pi^2)=0. They have negative real parts for the positive premises above. Default e=w=1/2, beta=1 gives two negative real roots; an oscillating mediator is not required. For dimensionless s>0, K_(s*mu)(t)=s*K_mu(s*t) retains the same ring capacity nu. Hence

\[ \int_0^\infty K_\mu(t)\,dt=\frac\nu2 C_0\qquad(\mu>0), \]

because B_0=-D_0/2. Changing mediator capacity changes the transient memory while leaving this integrated tangent kernel unchanged. This is not an invariance of full trajectories or final nonlinear form offsets. At mu=0 the mediator freezes and transmits no donor-induced change from identical mediator initial states. The positive-capacity integral limit is not uniform as mu tends to zero; it cannot be assigned to the frozen case.

The phase-to-phase cross entry satisfies

\[ [K_\mu(0)]_{vv}=-\frac{\mu\nu w^2}{2\beta\pi^2r}<0, \qquad \int_0^\infty[K_\mu(t)]_{vv}\,dt=0. \]

Continuity and exponential decay force a positive contribution at some later lag. Thus even the overdamped default has a signed memory entry. This describes feedback from past phase deviations in a chosen tangent chart, not alternating physical attraction, a receiver trajectory reversal or an autonomous bond. Pure-diffusion positive-kernel results cannot be transferred to this joint law.

A prospective onset discriminator in the actual nonlinear field¶

Prepare only donor form u_0(0)=epsilon!=0; all other form deviations and all phase deviations, including the mediator's, are zero. Direct differentiation at this preparation gives, for the mediated receiver,

\[ \dot u_5(0)=\dot v_5(0)=0,\qquad \ddot u_5(0)=\epsilon\mu\nu \left[\frac{e^2}{6}-\frac{w^2}{2\beta\pi^2r}\right],\qquad \ddot v_5(0)=-\frac{\epsilon\mu\nu ew}{2\beta\pi r}. \]

These initial derivatives are exact for the nonlinear ideal field as well: the receiver's initial form gradient is zero, so its phase-metric derivative does not enter the phase acceleration. The full subsequent memory remains nonlinear. A direct 0--5 bridge instead gives initial receiver rates e*nu*epsilon/3 and -w*nu*epsilon/(beta*pi*r). Removing the intermediary path leaves the independent receiving ring with no donor response.

The distinction is first-order versus second-order onset, not a finite waiting time before transmission. The phase contribution competes with diffusion in the form acceleration: at e=w=1/2, beta=1 it reduces but does not reverse the positive form signal for epsilon>0. The displayed coefficient changes sign below beta=3*w^2/(e^2*pi^2*r); this is a conditional family boundary, not a reason to fit beta after a response. Varying only mediator capacity predicts proportional initial accelerations, including a frozen-mediator null.

Shared implementation and evidence scope¶

derive_coordinate_memory centralizes the detached exact block split of any supplied fixed generator z_dot=J*z. It retains visible/hidden coordinate order, the four blocks and K(0)=B*C; it does not assume diffusion, positivity, stability or a memory cutoff. Rationalized trigonometric entries describe the supplied rational matrix, not exact real pi or the ideal C5 cosine. derive_linear_observation remains the independent owner of invariant-row closure.

The mediator controls compare the analytic joint Jacobian and prepared response against the production field. Static finite-difference checks are complemented by a closed-form two-step receiver control at dt=1/32, distinct from the reserved horizon below. Neither supplies an ODE error enclosure. Model defaults, ordered nodes 0,...,10, unit edges, held capacities, explicit donor amplitude and binary64 arithmetic are retained in the tests. No stochastic preparation, new event selector or change to the nodal executor is involved. The mechanism is conditional mediated interaction with local recovery; primitive support origin and a finite capture prediction remain separate obligations in the sole execution plan.

Orientation sensitivity: winding sign is invisible at a single symmetric port¶

The magnetic binding comparison asks whether a pattern's orientation changes its interaction. The present one-port-per-ring geometry has an exact limitation. Reflect the donor ring by R=(1 4)(2 3), fixing port 0, mediator 10 and the entire receiving ring. This is an automorphism of the supplied graph. With capacities and all other node data transformed consistently, the ideal full field is equivariant:

\[ F(Rx,R\theta)=R F(x,\theta). \]

For the uniform-capacity preparation above, this reflection maps the donor's positive twist to its negative twist modulo full phase turns, while leaving a donor-port form impulse and the receiving preparation unchanged. Uniqueness on the admitted smooth domain then gives z_minus(t)=R*z_plus(t). Mediator and receiving-ring histories are identical throughout their common admitted interval. This is a nonlinear symmetry statement, not merely the fact that the tangent coefficients contain the even quantity cos(kappa).

Consequently, the current experiment cannot distinguish equal from opposite winding signs using its receiver response. Assigning those signs magnetic polarities would introduce a physical meaning that this interface cannot resolve. Winding itself depends on a declared cycle orientation; no spatial magnetic moment follows from its integer value. Nor does a signed memory entry establish magnetic attraction or repulsion.

This does not rule out orientation-sensitive TNFR interactions under other admitted preparations or structures. The existing hidden form-orientation control already shows nonlinear dependence on relative form/phase orientation. An asymmetric internal preparation, two separately distinguished contacts or a justified geometric observation can remove the independent reflection symmetry. Their state, support and observation must be declared before such a claim is tested; new contacts also require fresh phase-domain admission. They are not an automatic extension of the single-port result. This comparison does not replace the finite mediator-capacity prediction in the sole queue.

10. A finite causal-response test of the effective connection¶

Influence has an onset order, not a selected activation time¶

For the section 9 donor preparation, compare the perturbed receiver with the unperturbed equilibrium at the same supplied support and capacities. Smoothness and the exact initial acceleration imply

\[ \Delta\theta_5(t)= -\frac{\epsilon\mu\nu ew}{4\beta\pi r}\,t^2+O(t^3). \]

With positive coefficients and nonzero epsilon, this difference is nonzero for every sufficiently small positive time. The first derivative vanishes at zero; a finite waiting interval does not follow. A threshold-based time of detection would be an observation policy, not an autonomous support event. The unperturbed equilibrium itself generates no signal. At zero mediator capacity, equal mediator initial states stay equal and the receiving dynamics are independent of the donor. Removing the fine path also removes influence.

This defines the useful causal question without adding a primitive edge: does an intervention in one region change another through the retained intermediary? The memory law answers it conditionally. Joint geometric recovery has its separate positive-capacity theorem; capture from a distant preparation, support birth and physical binding do not follow from a nonzero response alone.

A frozen mediator can organize both rings without coupling their changes¶

The zero-capacity control is stronger than an absent signal at equilibrium. With mediator state (x_m,theta_m) held fixed, each ring is a separate system with one anchored boundary. Its acute reference is uniform form x_m and the same ring twist rotated by theta_m. Write B_g,K_g for the grounded form Laplacian and cosine-weighted phase Hessian, including the anchoring edge. Both are positive definite. On each ring, A=N*D^-1 and M=N*H_*^-1 remain positive diagonal, with degrees including that edge. The local Jacobian is

\[ J_g=\begin{pmatrix} -eAB_g&-wMK_g\\ (w/\beta)MB_g&0 \end{pmatrix}. \]

Storage coordinates transform this into [[-D_cal,-C_cal],[C_cal^T,0]], where D_cal=e*B_g^(1/2)*A*B_g^(1/2)>0 and C_cal=(w/sqrt(beta))*B_g^(1/2)*M*K_g^(1/2) is invertible. The real-part argument in the existing local recovery proof then makes this grounded Jacobian Hurwitz. Each ring has local exponential recovery to its fixed-boundary reference, without an internal common-offset freedom. This is a separate anchored theorem, not an extension of the whole-network strictly-positive-capacity theorem to a zero capacity.

Nevertheless, the two systems consume no changing state from each other. Perturbing one cannot alter the other when mediator and receiver preparations are held fixed. Thus common geometry and individual restoration can occur without mutual influence. A claim of a collective bond must retain this common-boundary control as well as its geometry and recovery observations.

A symmetric port prediction has no quadratic amplitude error¶

Reflect both rings around their ports, fixing nodes 0, 5 and 10, and call the permutation R. Let Q=diag(R,R) act on the 22 form/phase deviations from the ideal reference. Simultaneous form/phase inversion and graph equivariance give, on the admitted lifted chart,

\[ G(-Qz)=-QG(z),\qquad -R\theta_*=\theta_*\pmod {2\pi}. \]

The donor direction a=(e_0,0) is reflection-even. Each simultaneous Euler map inherits the symmetry, so its N-step state satisfies z_N(-epsilon)=-Q*z_N(epsilon). A linear observation O of the receiver port or mediator obeys OQ=O. Smooth dependence on the initial amplitude therefore yields, for a fixed admitted grid,

\[ Oz_N(\epsilon)=\epsilon O(I+hJ_\mu)^Na+O(\epsilon^3). \]

The full 22-coordinate state need not have cubic error: internal reflection-odd modes can carry quadratic corrections. This result specifies the approximation order near zero; it supplies neither a numerical cubic remainder constant at a chosen amplitude nor an exact-ODE error bound. Native represented reference phases can have tiny residual drift, which must be recorded rather than subtracted from the response after evaluation.

Frozen finite-executor protocol¶

Use the existing law with e=w=1/2, beta=1, ring capacities one, donor form epsilon=1/64, and otherwise the section 9 equilibrium. Fix mediator capacities mu=0,1,2, horizon T=1/4, and 64 and 128 simultaneous Euler steps. These two grids are numerical controls of the same preparation, not independent physical trials or validated continuous-time enclosures.

The memory predictor advances its exact tangent blocks with old-state values:

\[ y_{n+1}=y_n+h(Ay_n+Bh_n),\qquad h_{n+1}=h_n+h(Cy_n+Dh_n). \]

The hidden initial deviation is zero in this preparation. The memory-omitting control retains y_{n+1}=y_n+hAy_n; its receiving ring remains at zero. It is a deliberate ablation, not a proposed closed law for arbitrary states. For positive mediator capacity, a stronger instantaneous comparison sets 0=Cy+Dh, hence h=-D^-1*C*y=(z_0+z_5)/2. Its visible generator is A-B*D^-1*C. Both the capacity factor and its inverse cancel, so this approximation predicts identical responses at mu=1 and mu=2 on every matched grid. It also replaces the prepared zero hidden state with its instantaneous equilibrium and need not match the initial onset. It is not an exact transient reduction and is undefined by this inverse at mu=0. The capacity intervention therefore discriminates retained memory from a fixed instantaneous connection even when their integrated kernel agrees. Observe the receiver port pair (x_5,theta_5-theta_{*,5}), retaining the full state and mediator endpoint as additional evidence.

Before native execution, freeze each tangent prediction and its source/runtime provenance. The prospective decision requires positive-capacity endpoint and mu=2 minus mu=1 intervention errors in maximum norm no greater than 1% of their respective predicted signals plus 1e-12. Each predicted signal must exceed 100e-12; the frozen-mediator receiver must stay within 1e-12. Also require held support/capacities, the declared fresh-pressure path, acute margin at least pi/20, and actual work-balance residual at most 1e-12. These are finite experiment decision tolerances, not emergent constants. A pass would test the derived interaction without fitting an edge or kernel; it would not promote the tolerance to a proved nonlinear or solver error bound.

Reserved response and decision¶

The mediator instrument uses the shared coordinate-memory decomposition and Euler arithmetic for the prediction, and step_relational_exchange for the full nonlinear response. The prediction and source archive were frozen before the reserved response. All declared gates passed on both grids without changing the preparation, law, coefficients or decision tolerances.

At 128 steps, the receiver results were:

Mediator capacity	Predicted form	Observed form	Predicted phase deviation	Observed phase deviation	Pair maximum error
1	`1.4859487561e-5`	`1.4859487442e-5`	`-1.0748178383e-5`	`-1.0748178470e-5`	`1.18910e-13`
2	`2.8689674324e-5`	`2.8689674024e-5`	`-2.0729626866e-5`	`-2.0729627404e-5`	`5.37972e-13`

The capacity intervention had predicted maximum norm 1.3830186762e-5 and observed norm 1.3830186582e-5; its pair error was 4.51169e-13. Instantaneous elimination predicted exactly zero intervention difference. The omitted-memory control predicted zero receiver response. The frozen mediator's observed receiver norm stayed below 2.41e-18 on both grids, consistent with its null prediction and materialized reference drift.

These values describe the retained finite executor comparison. In particular, the largest positive-capacity receiver error relative to its predicted pair was 1.88e-8, but full-state tangent errors reached 7.22e-7, consistent with retaining a different approximation scope for hidden internal shape. The largest 64-to-128-grid full-state difference was 1.884e-6; this is not a continuous-time error certificate. No further refinement or amplitude campaign is needed for the frozen decision.

The minimum observed acute margin was 0.31220729, the largest instantaneous work-balance residual was 2.51e-18, and the largest Euler storage-step defect was 5.52e-9. These are different quantities. The static rounded reference had maximum rate 1.95e-17; that drift was not subtracted. Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1, Windows AMD64, binary64 states, ascending node order, sorted unit edges and no randomness. The archive retains all 606 TNFR Python source files plus this producer; dependency versions are recorded separately. The record tests check frozen evidence without rerunning its trajectories.

This closes the bounded F4 question: a derived intermediary memory predicts a finite response and its capacity intervention better than the stated memory-omitting controls. It establishes neither a new primitive edge nor a unique law of physical binding. A ring's response can be causally linked to another without a direct edge, while the fine paths carrying that interaction remain premises of the model.