Capture of three acquired C9 organizations on an oriented contact cycle

Both prescribed contact cycles capture every member of the original independently acquired source family. The equal control converges to the aligned winding-one organization. The unequal control instead converges to a uniquely determined shifted registration on the same marked support. All three internal windings remain one throughout the complete fine evolution. These conclusions retain the original residual coordinates; none of the sources is reset to its nominal equilibrium.

This is a conditional full-state result for the supplied sine law, held unit capacities and externally supplied attachment. It adds source-to-target capture to the exact inherited network law. The contact amplitudes and cycle registration remain in that law: convergence does not remove their chart obstruction. The result neither identifies a physical interaction nor supplies the occurrence or energy source of the attachment event. The NFR ontology permits a joined identity to retain those boundary data.

Complete fine law and actual carried sources

Retain the 27 labelled nodes \((r,\ell)\), in order \(9r+\ell\), with \(r=0,1,2\) and \(\ell=0,\ldots,8\), and put \(\alpha=2\pi/9\). Each constituent has its unit-conductance C9 support. The supplied oriented contact cycle is \(e=(0,1),(1,2),(2,0)\). At an oriented contact \((r,t)\) add the undirected edges

\[ (r,\ell)--(t,\ell)\quad\hbox{of weight }1-s_e,\qquad (r,\ell)--(t,\ell+1)\quad\hbox{of weight }s_e, \tag{1} \]

with indices modulo nine. The two fixed controls are

\[ s^{\rm eq}=(1/2,1/2,1/2),\qquad s^{\rm un}=(1/4,1/2,3/4). \tag{2} \]

Every node has cross conductance two and total weighted degree four. There are 27 internal and 54 added positive edges in either control.

Use signed real form \(x\), primitive phase with the specified continuous lifts \(\theta\), the actual weighted Laplacian \(L_f\), and the same structural clock \(\tau=(1023/1024)t\). With \(\gamma=1/(1023\pi)\), the complete post-contact law is

\[ \begin{split} S_i(\theta)&=\sum_{j\sim i}w_{ij}\sin(\theta_j-\theta_i),\\ x'&=\tfrac14[-L_fx+\gamma S(\theta)],\qquad \theta'=\tfrac{\gamma}{4}L_fx,\\ H(x,\theta)&=\tfrac12x^TL_fx+ \sum_{\{i,j\}}w_{ij}[1-\cos(\theta_j-\theta_i)],\\ H'&=-\tfrac14\|L_fx\|_2^2,\qquad (\mathbf1^Tx)'=(\mathbf1^T\theta)'=0 . \end{split}\tag{3} \]

Primes mean derivatives in \(\tau\); both evolution rows acquire the factor \(1023/1024\) in original time. Capacities remain one. The pressure in the nodal form row is independently specified by (3), and no additional input or live source is present after attachment. This detached weighted comparison retains the execution boundary in the network-law owner; it is not the native unique-neighbor phase channel or an operator sequence.

Take three independent copies of the original sine class-one ramp preparation. Each original coordinate error is at most \(10^{-10}\), with separate exact zero sums for form and phase errors on each C9. The shared isolated recovery is consumed at its original acquisition time \(100\), uninterrupted dwell \(10^{13}\), and handoff \(\tau_h=100+10^{13}\). It supplies

\[ \begin{split} \Theta_{r,\ell}&=\alpha(\ell-4),\qquad \theta(\tau_h^-)=\Theta+y^{\rm src},\\ \|x_r(\tau_h^-)\|_2&\le\varepsilon,\qquad \|y^{\rm src}_r\|_2\le\varepsilon,\qquad \varepsilon=10^{-70},\\ \mathbf1^Tx_r(\tau_h^-)&=\mathbf1^Ty^{\rm src}_r=0 . \end{split}\tag{4} \]

All actual coordinates and their fixed lifts pass unchanged through the event. Only support changes. Independence means that every combination of the three admitted source members is included; no favorable cross-constituent correlation is imposed.

The recovery proof is used at sine deformation zero. Matching equilibria or tangents of another constitutive law would not admit its nonlinear recovery here. The earlier two-object and three-object contacts have their own different supports and are not source or work premises of (1).

The joined stationary registration

The registered family has

\[ x_{r,\ell}=X_r,\qquad \theta_{r,\ell}=\Phi_r+\alpha(\ell-4). \tag{5} \]

The retained phase origins have common mean zero. Write

\[ m_e=1-s_e+s_e e^{i\alpha}=\rho_e e^{i\phi_e},\qquad \Delta_e=\Phi_t-\Phi_r,\qquad J_e=\rho_e\sin(\Delta_e+\phi_e). \tag{6} \]

The exact collective law uses the cycle incidence matrix \(B\), positive at each tail, and the unweighted three-node Laplacian \(L=BB^T\):

\[ X'=\tfrac14[-LX+\gamma BJ],\qquad \Phi'=\tfrac{\gamma}{4}LX . \tag{7} \]

In particular, equal currents on the three oriented contacts give zero net current at every fine node, because each node has the same two port weights. The internal sine currents cancel pairwise. Thus the following targets are stationary for all 54 fine evolution rows, not merely for their constituent means.

For the equal control,

\[ x_*=0,\qquad \Phi_*=(0,0,0),\qquad \theta_*=\Theta,\qquad J_e=\tfrac12\sin\alpha . \tag{8} \]

The nonzero edge-current terms have zero divergence. This balance is not an independently identified transport of a physical conserved quantity.

For the unequal control put

\[ \begin{split} \rho_h&=\sqrt{1-\tfrac38(1-\cos\alpha)},\qquad \rho_l=\cos(\alpha/2),\\ \phi&=\arg(3/4+(1/4)e^{i\alpha}),\\ F(a)&=\rho_h\sin a-\rho_l\sin(\pi/3-2a). \end{split}\tag{9} \]

There is a unique root \(a\in[347/1000,348/1000]\). Indeed, the endpoint signs are negative and positive, respectively, and

\[ F'(a)=\rho_h\cos a+2\rho_l\cos(\pi/3-2a)>0 \tag{10} \]

throughout that interval. Set \(b=\pi/3-2a\) and

\[ \begin{split} \Delta_*&=(a-\phi,\ b-\alpha/2,\ a-\alpha+\phi),\\ P&=(0,\Delta_{*,01},\Delta_{*,01}+\Delta_{*,12}),\qquad \Phi_*=P-\tfrac13(\mathbf1^TP)\mathbf1,\\ x_*&=0,\qquad \theta_{*,r,\ell}=\Phi_{*,r}+\alpha(\ell-4). \end{split}\tag{11} \]

The three differences sum to zero. Their registered arguments are \((a,b,a)\), so (9) gives the common current \(\rho_h\sin a=\rho_l\sin b\). The mean-zero choice in (11) retains the original common origin; it does not independently reset the three frames.

For orientation, these quantities are approximately

\[ \begin{split} a&=0.3470833235,\quad b=0.3530309041,\\ \Delta_*&=(0.1780325616,\ 0.0039650537,\ -0.1819976153),\\ \Phi_*&=(-0.1200100590,\ 0.0580225026,\ 0.0619875563). \end{split}\tag{12} \]

The proof and implementation use outward rational enclosures of (9)--(11), not these displayed decimals. The unequal aligned state has \(X'=\gamma\sin\alpha\,Bs/4\ne0\); it is a source reference that moves, not the target.

Orient each fine edge consistently with its internal cycle or supplied contact, and choose a fixed integer \(n_e\) so that

\[ \delta_e(\theta)=\theta_j-\theta_i+2\pi n_e \tag{13} \]

equals \(\alpha\) on each target internal edge, and equals \(\Delta_{*,e}\) or \(\Delta_{*,e}+\alpha\) on the respective contact ports. This handles the seams of the lifted C9s. The integers remain fixed; \(\delta_e-\delta_{*,e}=y_j-y_i\) for \(y=\theta-\theta_*\). Both controls satisfy the strict bound

\[ |\delta_{*,e}|<p_*:=22/25 \quad\hbox{for every fine edge}. \tag{14} \]

The target has winding one in all three original marked C9s. Its phase Hessian satisfies

\[ \nabla^2 U(\theta_*)\succeq\cos(22/25)L_f, \]

so, using the full-support gap proved below, its nonconstant eigenvalues are greater than \( (2/5)\cos(22/25)>0 \). This certifies the local target geometry; source entry still requires the full barrier argument.

Actual support work and initial excess

Let \(L_c\) be the added-support Laplacian. Its maximal degree is two, so \(0\preceq L_c\preceq4I\). The unchanged source incurs the exact support work

\[ W=\tfrac12x^TL_cx+ \sum_{\text{added }e}w_e[1-\cos\delta_e(\theta)] . \tag{15} \]

At \(x=0,\theta=\Theta\), both controls have the same nominal work

\[ W_* =9\sum_es_e(1-\cos\alpha) =\tfrac{27}{2}(1-\cos\alpha)>0 . \tag{16} \]

For unequal weights the added potential gradient at \(\Theta\) is not zero. It is, however, constant within each constituent. Therefore its linear term against \(y^{\rm src}\) vanishes by the three separate zero-sum constraints in (4). This source correlation is essential; a collection of independent coordinate bounds without the exact zero means would not justify dropping the first variation.

All added edge angles along the segment from \(\Theta\) to the actual source have magnitude below \(\alpha+2\varepsilon<1\), so their cosine Hessians are positive. Their global absolute second derivatives are at most one. Since \(\|x\|_2^2,\|y^{\rm src}\|_2^2\le3\varepsilon^2\), Taylor's formula and the bound on \(L_c\) give

\[ \boxed{W_*\le W\le W_*+12\varepsilon^2.} \tag{17} \]

The strictly positive nominal work must be externally supplied. Previous continuous loss is not credited as a reservoir, and neither passivity nor autonomous event occurrence follows. The exact correction \(12\varepsilon^2\) is retained separately from the irrational enclosure of \(W_*\), rather than being erased by the interval grid.

Let \(H_*=H(0,\theta_*)\) and \(E_{\rm nom}=H(0,\Theta)-H_*\). For the equal control \(E_{\rm nom}=0\). For the unequal control, counting all fine copies gives

\[ E_{\rm nom} =9\left[ 2\rho_h\cos a+\rho_l\cos b -\tfrac32(1+\cos\alpha) \right],\qquad 0<E_{\rm nom}<27/100 . \tag{18} \]

Its value is approximately \(0.26115033\). The inequality is certified by enclosing the root before evaluating (18), independently of any evolved response. The target energy retains both the internal baseline and the contact constants; no energy offset is discarded from (15).

For the complete graph \(0\preceq L_f\preceq8I\). The complete potential gradient at \(\Theta\) is still constant within each constituent, so the same zero-sum cancellation applies. Consequently the actual initial excess, including all fine residuals, satisfies

\[ E_{\rm in}:=H(x(\tau_h^+),\theta(\tau_h^+))-H_* \le E_{\rm nom}+24\varepsilon^2 . \tag{19} \]

This calculation is performed at the actual carried source, without setting its internal deviations to zero or moving it to (11).

Full-state trapping and winding retention

The required barrier is not restricted to the registered family (5). On the global mean-zero leaf define the fine phase cell

\[ \mathcal D=\{(x,y):\mathbf1^Tx=\mathbf1^Ty=0,\quad |\delta_e(\theta_*+y)|<2\ \hbox{for every fine edge}\}. \tag{20} \]

The bound \(2<\pi\) retains every winding, but some individual edge Hessians can be negative in this cell. We therefore prove a storage bound relative to the admitted target directly; no global convexity or acute-sector theorem is being assumed.

For \(V(z)=1-\cos z\), \(|t|\le p_*=22/25\), and \(|z|\le2\), Taylor's integral formula gives

\[ \begin{split} V(z)-V(t)-\sin t\,(z-t) &=(z-t)^2\int_0^1(1-v)\cos((1-v)t+vz)\,dv\\ &\ge c_*(z-t)^2,\qquad c_*>3/20, \end{split}\tag{21} \]

where

\[ c_*= \frac{\cos(22/25)-\cos2-(28/25)\sin(22/25)} {(28/25)^2}>3/20 . \tag{22} \]

For the inequality, the absolute interpolation argument is at most \((1-v)p_*+2v\), and cosine decreases with absolute argument on \([0,2]\). Integrating that lower cosine bound gives (22). Outward rational trigonometric bounds verify the last strict inequality; \(c_*\) is approximately \(0.1515229882\).

The target is a full fine-node critical point. Thus summing the linear terms in (21), with their actual weights and edge orientations, gives zero. Throughout the closed cell,

\[ E:=H-H_*\ge \tfrac12x^TL_fx+\tfrac3{20}y^TL_fy . \tag{23} \]

This includes all 52 nonconstant fine directions. The Laplacian dominates the Cartesian support consisting of the unit C9 edges and conductance \(1/4\) matching contacts between every pair of constituents. The nonzero gap of that support is

\[ \min\{2-2\cos\alpha,\ 3/4\}>2/5 . \tag{24} \]

The additional shifted and matching conductance is positive semidefinite. Therefore, on the global mean-zero leaf,

\[ E\ge \tfrac3{50}\bigl(\|x\|_2^2+\|y\|_2^2\bigr). \tag{25} \]

The strict gap in (24) gives strict inequality for a nonzero state in the weaker bound (25).

To admit the unequal source, a Euclidean radius alone is unnecessarily weak. Retain the actual edge correlations through effective resistance. For a fine edge \(e=\{i,j\}\), define the topology-only number

\[ R_e=(e_j-e_i)^TL_f^+(e_j-e_i). \tag{26} \]

Weighted Cauchy--Schwarz on the zero-mean quotient gives

\[ (y_j-y_i)^2\le R_e\,y^TL_fy . \tag{27} \]

For the two rational conductance supports (2), exact rational arithmetic gives

\[ \begin{array}{c|c} \text{control}&\max_{\text{fine edges}}R_e\\ \hline s^{\rm eq}&217703/419823\\ s^{\rm un}&743902545880/1319494263111 \end{array} \qquad R_e<3/5 . \tag{28} \]

These finite algebraic certificates can be reproduced without spectral rounding. Form the 27-by-27 rational Laplacian from (1), remove the row and column of node 26 to obtain the positive definite grounded matrix \(A\), and solve \(Az=b\) exactly for each reduced incidence column \(b\). Then \(R_e=b^Tz\); the residual \(Az-b=0\) verifies the calculation. Simultaneous cyclic rotation of all three C9s leaves nine edge orbits, so one representative of each orbit suffices. The maxima occur on shifted contacts, on every contact in the equal case and on \((0,1)\) in the unequal case. The argument uses the complete weighted graph, not a marginal bound assigned separately to each constituent.

At a first phase-cell boundary some \(|\delta_e|=2\), so (14) implies \(|y_j-y_i|\ge2-p_*=28/25\). Equations (23), (27) and (28) then give the full-state boundary barrier

\[ \boxed{E_{\partial\mathcal D} \ge \frac{3}{20}\frac{(28/25)^2}{3/5} =\frac{196}{625}.} \tag{29} \]

All actual sources begin in \(\mathcal D\), since their edge angles at attachment are at most \(\alpha+2\varepsilon<1\). Their excess (19) satisfies

\[ E_{\rm in}\le E_{\rm nom}+24\varepsilon^2 <196/625 . \tag{30} \]

The exact loss law (3) forbids reaching (29). Every complete fine trajectory from (4) therefore stays in \(\mathcal D\) for all future unforced time. The actual dynamics, not just its collective restriction, retains winding one in every constituent. All means and edge correlations used in the barrier belong to the same state.

The resulting explicit identity bound is

\[ \|x(\tau_h+T)\|_2^2+ \|\theta(\tau_h+T)-\theta_*\|_2^2 \le \frac{50}{3}(E_{\rm nom}+24\varepsilon^2), \qquad T\ge0 . \tag{31} \]

For the equal control the strict gap improves this to joint norm less than \(20\varepsilon\). For the unequal control the displacement from the original aligned source to the shifted target is part of the same full-state excess; it is not treated as a negligible residual.

Uniqueness in the cell and convergence

The target is the only critical phase geometry on the trapped mean-zero leaf. For any edge values \(|z|\le2\), \(|t|<22/25\),

\[ \sin z-\sin t =2\cos((z+t)/2)\sin((z-t)/2). \tag{32} \]

Both \(|z+t|\) and \(|z-t|\) are below \(72/25<\pi\). Therefore \((z-t)(\sin z-\sin t)>0\) whenever \(z\ne t\). For any other phase geometry in the closed cell, connectedness and the fixed mean imply that at least one \(y_j-y_i\ne0\). Hence

\[ y^T\bigl(\nabla U(\theta_*+y)-\nabla U(\theta_*)\bigr) =\sum_e w_e(y_j-y_i) [\sin\delta_e(\theta_*+y)-\sin\delta_{*,e}]>0 . \tag{33} \]

Since \(\nabla U(\theta_*)=0\), another critical point is impossible. This is strict radial monotonicity relative to the target, not a claim that the Hessian is positive everywhere in the cell.

The vector field (3) is globally Lipschitz in lifted coordinates. The trapped sublevel is compact on the fixed mean leaf by (25) and has positive distance from the phase boundary by (29)--(30). On it, zero dissipation requires \(L_fx=0\), hence \(x=0\). An invariant trajectory in that set must also have \(S(\theta)=0\), because otherwise the form row immediately leaves \(x=0\). Equation (33) then forces \(\theta=\theta_*\). The invariant-set argument applied to (3) proves

\[ (x(\tau_h+T),\theta(\tau_h+T))\longrightarrow(0,\theta_*) \quad\text{as }T\longrightarrow\infty . \tag{34} \]

No quantitative settling time is asserted. The theorem admits every declared carried source to the target's full-state basin; it is stronger than local stability of a supplied equilibrium, but applies only to these two supports and their declared source family.

Retained identity, implementation and independent controls

After contact, only the global form and lifted phase means in (3) remain conserved. The three constituent means generally move. The target shift in (11), the fine residuals, and the contact-cycle registration all belong to one complete state and law. They cannot be combined as independent marginal budgets or erased by resetting local phase origins.

The exact six-coordinate family (5) remains a useful restriction, but the reached sources need not lie on it. The full-state proof above retains their internal coordinates until convergence. It does not supply an exact memory-free quotient for arbitrary off-family states, a quantitative transient response, autonomous support formation, constitutive uniqueness or a physical identification.

The separate assembly-work obstruction shows that even simultaneous internal reorganization cannot make either attachment passive from these recovered sources while retaining acute winding-one child endpoints. That endpoint class is narrower than the transient cell used here; the result neither removes this supplied-work capture theorem nor transfers its capture certificate to arbitrary reset endpoints.

The capture implementation rebuilds the selected rational support, target enclosures, edge resistance certificate, original source recovery, support work and full-state barrier. The shared target and barrier owner has no source or entry premise. It re-admits the exact support before reusing its pure grounded resistance solve in a bounded two-entry cache; edge ordering is restored for each caller. Target, barrier and source verdicts are not cached. This recovered-source consumer and the separate connected-source acquisition each admit their own preparation and reached-state bounds before applying it. It does not accept incoming report verdicts as replacement premises. The independent controls check the fine-node target balances, exact rational support certificates, residual work and energy, retained source restrictions and invalid raw inputs. These are static algebraic and analytic certificates; no source producer, numerical trajectory or reserved response is rerun.