Interaction at a locally unprepared C9 receiver

The declared two-ramp source acquires winding one in both prepared C9s by structural time 200. The initially unprepared receiver has winding zero throughout a much longer certified interval: its first possible branch crossing is later than structural time 16,000,000. Nevertheless, the actual nonlinear current at time 200 drives its mixed form-phase coordinate in the direction of the donors' winding profile. Thus the fast form layer does not copy the missing preparation into the receiver; any eventual acquisition needs a later nonlinear phase mechanism.

This is a complete-law mechanism and delay result. It alone does not establish eventual receiver acquisition, continued donor retention or joined full-state capture. The subsequent frozen enclosure certifies donor loss for this source while the receiver retains zero winding throughout its validated interval. Later acquisition or regeneration remains unresolved. The delay is a proved consequence of the frozen source and law, not a response horizon chosen after a trajectory. This mechanism assessment evaluates no trajectory or reserved response.

Complete source and its preparation cost

Retain the 27 labelled nodes, three unit C9s and oriented contacts \((0,1),(1,2),(2,0)\) from the joint-acquisition model. Only its equal control is admitted: each contact joins offsets zero and one with weight one half. All fine edges are present from preparation; each node has weighted degree four. Capacities are held at one and \(\tau=(1023/1024)t\), \(\gamma=1/(1023\pi)\). With the actual weighted Laplacian \(L\), the complete detached law is

\[ S_i(\theta)=\sum_jw_{ij}\sin(\theta_j-\theta_i),\qquad x'=\tfrac14[-Lx+\gamma S(\theta)],\qquad \theta'=\tfrac\gamma4Lx . \tag{1} \]

There is no input, state reset, support event or evolving conductance. The law is the supplied weighted sine comparison, not native unique-neighbor phase pressure or registered operator execution.

Put \(q_j=j-4\), \(\alpha=2\pi/9\), and

\[ m=\frac{2046}{9}\left(\frac{355}{113}\right)^2,\qquad Q_{r,j}=\begin{cases}q_j,&r=0,1,\\0,&r=2.\end{cases} \tag{2} \]

The full preparation is

\[ x(0)=mQ+\eta_x,\quad \theta(0)=\eta_\theta,\quad |\eta_{x,r,j}|\le r_x=10^{-10},\quad |\eta_{\theta,r,j}|\le r_\theta=10^{-10},\quad \sum_j\eta_{x,r,j}=\sum_j\eta_{\theta,r,j}=0 . \tag{3} \]

Every allowed combination of errors is retained. The zero sums in (3) are separate initial constituent premises; only the global form and lifted-phase means remain conserved during the connected evolution. All 54 fine coordinates and the boundary state remain part of the model. Winding is the sum of principal oriented phase differences around a marked C9, divided by \(2\pi\), away from a branch boundary. All three constituents initially have winding zero. The donors start with form ramps, not previously acquired objects.

The unchanged complete storage and loss are

\[ H=\tfrac12x^TLx+\sum_{\{i,j\}}w_{ij}[1-\cos(\theta_j-\theta_i)], \qquad H'=-\tfrac14\|Lx\|_2^2 . \tag{4} \]

The two internal form ramps contribute \(72m^2\). Their mutual contact contributes \(18m^2\), and each contact to the zero-form receiver contributes \(30m^2\). Consequently

\[ H(mQ,0)=150m^2=72m^2+78m^2,\qquad \|Q\|_2^2=120,\qquad \|LQ\|_1=117 . \tag{5} \]

The last identity follows from the three actual Laplacian blocks

\[ \begin{split} (LQ)_0&=(-27/2,-7/2,-5/2,-3/2,-1/2,1/2,3/2,5/2,17),\\ (LQ)_1&=(-17,-5/2,-3/2,-1/2,1/2,3/2,5/2,7/2,27/2),\\ (LQ)_2&=(7/2,6,4,2,0,-2,-4,-6,-7/2). \end{split} \tag{6} \]

Since \(0\preceq L\preceq8I\), all sources satisfy

\[ 150m^2-117mr_x\le H(x(0),\theta(0)) \le150m^2+117mr_x+108(r_x^2+r_\theta^2). \tag{7} \]

The nominal cost is approximately \(755122024.11569\) in the model's structural storage units. Removing one local ramp therefore does not remove its contact cost. Neither initial storage nor later dissipation supplies an autonomous preparation or event reservoir.

The full fast layer has a different endpoint reference

Set

\[ A=L/4,\quad a=\gamma x,\quad \epsilon=\gamma^2,\quad F(\theta)=S(\theta)/4,\quad z=\theta+a . \tag{8} \]

The exact complete rows are

\[ a'=-Aa+\epsilon F(\theta),\qquad \theta'=Aa,\qquad z'=\epsilon F(\theta). \tag{9} \]

The connected-support proof gives a mean-zero gap \(g=1/10\) for \(A\), while \(|F_i|\le1\) and \(\|F\|_2\le M=\sqrt{27}\) globally. Hence

\[ \|a(\tau)\|_2\le \gamma e^{-\tau/10}(m\sqrt{120}+r_x\sqrt{27}) +10\epsilon\sqrt{27}. \tag{10} \]

Define its time-200 upper bound

\[ B=\gamma e^{-20}(m\sqrt{120}+r_x\sqrt{27}) +10\epsilon\sqrt{27}. \tag{11} \]

The norm bound (10) is at most \(B\) for every \(\tau\ge200\). The appropriate reference phase is now

\[ P=\alpha Q=(\alpha q,\alpha q,0), \tag{12} \]

which is flat on the receiver. Integrating the exact mixed row gives

\[ \|\theta(200)-P\|_2\le b, \quad b=\sqrt{120}|\gamma m-\alpha| +\sqrt{27}(r_\theta+\gamma r_x)+B +200\epsilon\sqrt{27}<1/5000 . \tag{13} \]

In particular, \(B\) is approximately \(0.00000504649136123\), \(B/\gamma<1/50\), and \(b\) is approximately \(0.00010696036\). The rational ramp mismatch is retained rather than setting \(355/113=\pi\).

Each donor's principal internal edge difference at time 200 is \(\alpha\) plus an error of magnitude at most \(\sqrt2b\), with its one seam integer fixed as in the winding-one target. Their sums equal \(2\pi\), so both donor windings are one. The receiver's real edge differences have magnitude at most \(\sqrt2b<\pi\), telescope to zero and give winding zero. These are actual full-state conclusions for every source in (3). In particular this state has not entered the all-winding-one joined capture cell. The earlier three-ramp source certificate cannot replace the missing receiver source in (12).

No receiver branch crossing throughout the certified delay

The receiver's winding is zero throughout the initial layer as well as at its endpoint. To prove this without a trajectory, split \(A=A_I+A_C\) into internal-cycle and contact Laplacians divided by four. On the equal support they are commuting convolution operators on \(\mathbb Z_3\times\mathbb Z_9\). Their heat kernels are nonnegative and preserve constants.

Let \(h_\tau\) be the internal C9 heat kernel. Its nonconstant Fourier modes decay at least as fast as \(e^{-\tau/10}\), whence

\[ |h_\tau(j)-1/9|\le(8/9)e^{-\tau/10}. \tag{14} \]

The contact heat kernel started in region two has total mass

\[ p(\tau)=\tfrac23(1-e^{-3\tau/4}) \tag{15} \]

in the two donor regions. This is the exact marginal three-region heat law; it does not assume independence between region and offset. Aggregate those donor offsets into a nonnegative C9 kernel of mass \(p\). Since the forward difference of the ramp is \(\Delta q=1-9\delta_{\mathrm{seam}}\), (14) bounds every receiver edge of the unforced transported source by

\[ |\Delta(e^{-A\tau}Q)_{2,j}| \le8p(\tau)e^{-\tau/10}<4 . \tag{16} \]

For completeness, the maximum of \(e^{-\tau/10}(1-e^{-3\tau/4})\) is \((15/17)(2/17)^{2/15}<3/4\). The strict inequality is the integer comparison \(4\,20^{15}<17^{17}\).

Variation of constants in (9) gives the exact full-law identity

\[ \theta(\tau)=\theta(0)+(I-e^{-A\tau})a(0) +\epsilon\int_0^\tau (I-e^{-A(\tau-v)})F(\theta(v))\,dv . \tag{17} \]

Use (16) for the nominal receiver source, Markov contraction for the initial errors, and \(|F_i|\le1\) for both integral terms. For every receiver edge and \(0\le\tau\le200\),

\[ |\Delta\theta_{2,j}(\tau)| <4\gamma m+2r_\theta+4\gamma r_x+4\epsilon\tau \le4\gamma m+2r_\theta+4\gamma r_x+800\epsilon<\pi . \tag{18} \]

The upper bound is approximately \(2.79260473077\). Thus no receiver edge can have crossed a principal-phase boundary in that layer.

For later times the integrated mixed row in (9), the zero nominal receiver value of \(z(0)\), and (10) instead give

\[ |\Delta\theta_{2,j}(\tau)| \le2(r_\theta+\gamma r_x)+\sqrt2B+2\epsilon\tau, \qquad \tau\ge200 . \tag{19} \]

Define the derived threshold

\[ T_*:=\frac{\pi-2(r_\theta+\gamma r_x)-\sqrt2B} {2\epsilon}. \tag{20} \]

It is approximately \(16224447.0054\), and outward bounds certify \(T_*>16000000\). Combining (18) and (19), every member of (3) retains receiver winding zero on \([0,T_*)\). At \(T_*\) the strict edge estimate ends; the result does not claim that a crossing occurs then, or at any later time. No corresponding long-window donor-retention conclusion follows from this receiver bound.

Nonlinear interaction and a controlled slow comparison

The delay is not a symmetry cancellation of all receiver interaction. At the reference (12), direct evaluation on the oriented fine support gives, with \(q=q_j\),

\[ \begin{split} S_{0,j}(P)&=\tfrac12\sin\alpha-\sin(\alpha q),\\ S_{1,j}(P)&=-\tfrac12\sin\alpha-\sin(\alpha q),\\ S_{2,j}(P)&=(1+\cos\alpha)\sin(\alpha q). \end{split} \tag{21} \]

For example, the receiver sees donor phases at offsets \(j-1,j,j+1\), with weights one half, one and one half. This gives the last row of (21), not a prescribed replacement of its own phase by a donor ramp. The receiver current is positive for \(j=5,\ldots,8\), negative for \(j=0,\ldots,3\), and zero at the central reference node.

These noncentral signs also hold for the actual state at time 200. The sine Lipschitz bound and (13) imply

\[ |S_i(\theta(200))-S_i(P)|\le4\sqrt2 b . \tag{22} \]

The smallest noncentral reference magnitude is \((1+\cos\alpha)\sin(\pi/9)>3/5\), while \(4\sqrt2b<1/500\). Thus the exact row \(z'=\epsilon S/4\) at the end of the layer drives each noncentral receiver coordinate with the sign of its desired donor ramp. This is a statement about the mixed-coordinate rate. The actual phase row remains \(\theta'=Aa\); (21) is not its instantaneous replacement.

The nominal preparation respects the graph involution \((r,j)\mapsto(1-r\pmod3,8-j)\) combined with sign reversal of both form and lifted phase. The full law preserves that symmetry. The all-winding-one target respects it as well, so this symmetry does not exclude the desired outcome. General errors in (3) need not obey that nominal symmetry.

For an exact description of the slow mechanism put \(s=\epsilon\tau\), retaining the same phase lifts and form. Equation (17) becomes

\[ \theta(s)=\theta_0+(I-e^{-As/\epsilon})a_0 +\int_0^s(I-e^{-A(s-v)/\epsilon})F(\theta(v))\,dv . \tag{23} \]

This is the actual nonlinear memory equation with its original form source. Equivalently,

\[ z_s=F(z-a),\qquad a_s=-\epsilon^{-1}Aa+F(z-a),\qquad \theta=z-a . \tag{24} \]

The reference phase-descent law \(y_s=F(y)\) is an approximation to (23), not an independently exact closure of (1).

A finite-window comparison can be admitted without discarding the initial layer. Let \(y(0)=P\), \(K=2\), \(M=\sqrt{27}\), \(k=g/\epsilon\), and

\[ A_0=\gamma(m\sqrt{120}+r_x\sqrt{27}),\qquad d_0=\sqrt{120}|\gamma m-\alpha| +\sqrt{27}(r_\theta+\gamma r_x). \]

The global Lipschitz constant \(K\) follows from \(\|DF\|_2\le\|L\|_2/4\le2\). Subtract the integral equation for \(y\) from (23), use the mean-zero heat gap on the source and on \(F\), and apply the scalar Gronwall resolvent. This gives the full-source bound

\[ \|\theta(s)-y(s)\|_2 \le d_0e^{Ks}+A_0e^{-ks} +\frac{KA_0+M}{k+K}(e^{Ks}-e^{-ks}). \tag{25} \]

Indeed the pre-resolvent error is bounded by \(d_0+A_0e^{-ks}+(M/k)(1-e^{-ks})\); evaluating its convolution with \(Ke^{Ks}\) gives (25). Form retains its own bound (10). A future phase comparison must keep both errors, the fixed source and clock, and actual joined-capture margins. A large right-hand side is an unavailable sufficient estimate, not a failure of acquisition. No reduced trajectory or infinite-time interchange is used here.

Regional boundary work and the remaining capture obligation

For a marked region let \(L_Ix\) and \(S_I\) contain its internal cycle terms, while \(L_Cx\) and \(S_C\) contain its incident contact terms. Restrict these vectors to the region and write \(\ell=L_Ix\), \(b_C=L_Cx\), \(u=S_I\), \(v=S_C\). Its internal storage

\[ H_r=\tfrac12x_r^TL_{I,r}x_r+ \sum_{\{i,j\}\subset r}[1-\cos(\theta_j-\theta_i)] \]

satisfies the exact balance

\[ H_r'=-\tfrac14\|\ell\|^2 -\tfrac14\ell\cdot b_C +\tfrac\gamma4(\ell\cdot v-u\cdot b_C). \tag{26} \]

To verify it, use \(\nabla_{x_r}H_r=\ell\), \(\nabla_{\theta_r}H_r=-u\), and both actual rows in (1). The internal reciprocal terms cancel. The remaining contact terms are the regional boundary work; they have no fixed sign and are not an externally supplied forcing. Adding regional storage alone omits contact storage. Including that contact storage recovers the complete loss identity (4).

Equations (21)--(26) specify a causal route for later phase-mediated change while retaining its form memory and work. They do not yet show that the receiver reaches winding one before a donor loses its own identity, or that the complete state reaches the joined trapping sublevel. The subsequent frozen full-source enclosure resolves that transfer question negatively on its validated interval: both donors lose winding while the receiver retains zero winding. The regional retention theorem turns (26) into a conditional all-prefix work budget and derives its exhaustion at each donor's first boundary contact. Neither the three-ramp certificate nor an equilibrium tangent supplies an alternative two-ramp retention result. Later acquisition or regeneration remains unresolved.

The implementation re-admits primitive error allowances, rebuilds the support and complete source cost, and evaluates the preparation, fast-layer, delay and current-sign inequalities with shared exact rational and outward interval owners. It reuses the prepared form-cost and transit kernels also consumed by the three-ramp assessment. The slow comparison (25) is a proved method for a future admitted consumer; this assessment does not evaluate it or a reduced trajectory. Its frozen conclusion uses the original errors in (3); different admitted errors require fresh inequalities. Failed sufficient bounds remain unavailable. The execution plan is the sole owner of the next research task.