Joint acquisition on existing fine support

Every member of the declared phase-flat preparation acquires winding one in all three marked C9 constituents by structural time \(T=200\), for both fixed connected supports. Its complete form and phase state enters the joined target's trapping sublevel at that deadline and subsequently converges to the equal or unequal stationary registration. The initial whole-network storage is below the frozen ceiling \(10^9\).

This is conditional joint formation on supplied fine support under the declared reciprocal sine law. It does not attach previously recovered children, create an edge, or supply an autonomous preparation mechanism. The assembly-work obstruction remains valid for its different late-attachment experiment. The present proof retains the substantial initial form storage, including the contact contribution. Storage is measured in this model's structural units, without a laboratory energy identification.

Complete law, preparation and identity

Use the 27 labelled nodes \((r,j)\), ordered as \(9r+j\), where \(r=0,1,2\) and \(j=0,\ldots,8\). Each constituent has its unit-weight C9 edges. For the oriented contacts \(e=(0,1),(1,2),(2,0)\), include the undirected edges

\[ (r,j)--(t,j)\ \text{with weight }1-s_e,\qquad (r,j)--(t,j+1)\ \text{with weight }s_e, \tag{1} \]

where indices are modulo nine. The two controls are

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

All 81 positive edges are present at preparation and remain unchanged. Every node has weighted degree four. The contacts and node markings are supplied model data; the law does not select either support.

Let \(L_f\) be the actual weighted Laplacian, let \(x=\mathrm{EPI}\) be signed real form, and retain primitive circular phases with continuous lifts \(\theta\). Capacities are held at one, the structural clock is \(\tau=(1023/1024)t\), and \(\gamma=1/(1023\pi)\). The complete law is

\[ \begin{split} S_i(\theta)&=\sum_{j\sim i}w_{ij}\sin(\theta_j-\theta_i),\\ x'&=\tfrac14[-L_fx+\gamma S(\theta)],\qquad \theta'=\tfrac\gamma4L_fx,\\ H(x,\theta)&=\tfrac12x^TL_fx+ U(\theta),\qquad U(\theta)=\sum_{\{i,j\}}w_{ij}[1-\cos(\theta_j-\theta_i)],\\ H'&=-\tfrac14\|L_fx\|_2^2 . \end{split}\tag{3} \]

Primes denote structural time. Both rows acquire the factor \(1023/1024\) in original time. Equation (3) independently supplies pressure and the companion phase law. There is no forcing, live source, state reset, support event or conductance evolution. This is the detached weighted comparison from the network-law owner, separate from native unique-neighbor phase pressure and registered operator execution.

Set

\[ q_{r,j}=j-4,\qquad m=\frac{2046}{9}\left(\frac{355}{113}\right)^2, \qquad \alpha=\frac{2\pi}{9},\qquad \Theta=\alpha q . \tag{4} \]

The fresh connected-source declaration is

\[ \begin{split} x(0)&=mq+\eta_x,\qquad \theta(0)=\eta_\theta,\\ |\eta_{x,r,j}|&\le r_x=10^{-10},\qquad |\eta_{\theta,r,j}|\le r_\theta=10^{-10},\\ \sum_j\eta_{x,r,j}&=\sum_j\eta_{\theta,r,j}=0\quad(r=0,1,2). \end{split}\tag{5} \]

Every combination satisfying these bounds and correlations is included. The rational \(355/113\) in the preparation is retained exactly; it is not replaced by \(\pi\). The common initial lifted phase origin is zero. Although these primitive ramps also appear in the isolated-source work, none of its acquisition, recovery or residual verdicts is consumed here.

Antisymmetry of the sine currents gives \(\mathbf1^TS(\theta)=0\). Therefore the global form and lifted phase means remain zero. Constituent means need not remain zero after connected evolution starts. All 54 fine coordinates remain in the proof; a six-coordinate registered family is not used as a transient quotient.

For each marked C9, winding means the sum of its oriented principal circular phase differences divided by \(2\pi\), when no edge is on the branch boundary. At preparation all actual differences have magnitude at most \(2r_\theta<\pi\) and telescope to zero. Each constituent therefore starts at winding zero. The acquisition question is a change of actual circular geometry, followed by retention and convergence; the node labels alone do not provide that result. The retained constituent and boundary state matches the NFR ontology.

Whole-network preparation cost

The deadline \(T=200\), original-time deadline \(204800/1023\), and initial storage ceiling \(10^9\) are fixed before evaluating the certificate. The phase-flat nominal source has zero phase storage. On one C9 the form ramp has eight differences of magnitude \(m\) and one of magnitude \(8m\), giving storage \(36m^2\). Matching contacts have zero nominal form difference. Each shifted contact contributes \(36s_em^2\). Thus both controls have the same complete nominal cost

\[ H(mq,0)=108m^2+36m^2\sum_es_e=162m^2 . \tag{6} \]

In particular, \(54m^2\) of this cost is already on the contact edges. It cannot be omitted by treating preparation as three isolated objects.

The exact residual accounting can use the actual Laplacian. Write \(s_{\mathrm{out},r}\) and \(s_{\mathrm{in},r}\) for the two incident oriented contact weights. Direct evaluation gives

\[ (L_fq)_{r,j}= \begin{cases} -9-s_{\mathrm{out},r}-8s_{\mathrm{in},r},&j=0,\\ s_{\mathrm{in},r}-s_{\mathrm{out},r},&1\le j\le7,\\ 9+8s_{\mathrm{out},r}+s_{\mathrm{in},r},&j=8. \end{cases}\tag{7} \]

Consequently

\[ \kappa_s:=\|L_fq\|_1 =54+18\sum_es_e+7\sum_r|s_{\mathrm{in},r}-s_{\mathrm{out},r}| =\begin{cases}81,&s=s^{\mathrm{eq}},\\88,&s=s^{\mathrm{un}}. \end{cases}\tag{8} \]

The degree bound gives \(0\preceq L_f\preceq8I\). Expanding the form quadratic, bounding its linear term by \(m\kappa_s r_x\), and using \(1-\cos z\le z^2/2\) for the phase term yields

\[ 162m^2-m\kappa_sr_x \ \le H(x(0),\theta(0))\le 162m^2+m\kappa_sr_x+108(r_x^2+r_\theta^2)<10^9 . \tag{9} \]

The nominal value is approximately \(815531786.0449456\). Even the larger linear correction is less than \(1/50000\); all bounds in (9) are exact rational calculations. This is supplied preparation storage. The later loss in (3) is not an inferred preparation source or an event reservoir.

Global complete-law transport to the fixed deadline

The graph contains the Cartesian support formed by the three unit C9s and matching contacts of weight \(1/4\) between each constituent pair. Its nonzero Laplacian gap is

\[ \min\{2-2\cos\alpha,3/4\}>2/5. \tag{10} \]

The remaining contact conductance is positive semidefinite. Thus \(A=L_f/4\) has gap greater than \(1/10\) on the global mean-zero space. This is a full-support gap, with no requirement that the three constituent means remain separately zero. Degree four also implies the global forcing bound

\[ |S_i(\theta)|\le4,\qquad \|S(\theta)\|_2\le4\sqrt{27} \tag{11} \]

for every lifted phase configuration. Neither (10) nor (11) assumes an acute sector or a winding class.

Variation of constants in the actual form row, together with \(\|q\|_2=\sqrt{180}\), gives

\[ \begin{split} \|x(T)\|_2&\le b_x,\\ b_x&=e^{-20}\bigl(m\sqrt{180}+r_x\sqrt{27}\bigr) +10\gamma\sqrt{27}<1/50 . \end{split}\tag{12} \]

The forcing integral is bounded by \(\int_0^T e^{-(T-v)/10}\,dv\le10\). It includes the coupled phase history rather than substituting an externally prescribed phase.

Adding the two evolution rows after scaling the form row by \(\gamma\) gives an exact integrated relation,

\[ (\theta+\gamma x)'=\frac{\gamma^2}{4}S(\theta),\qquad \theta(T)=\theta(0)+\gamma x(0)-\gamma x(T) +\frac{\gamma^2}{4}\int_0^T S(\theta(v))\,dv . \tag{13} \]

It retains both original errors and the rational ramp mismatch. Therefore

\[ \begin{split} \|\theta(T)-\Theta\|_2&\le b_\theta,\\ b_\theta&=\sqrt{180}|\gamma m-\alpha| +\sqrt{27}(r_\theta+\gamma r_x) +\gamma b_x+200\gamma^2\sqrt{27}<1/5000 . \end{split}\tag{14} \]

The mismatch is independently evaluable from the supplied preparation:

\[ \gamma m-\alpha =\frac{2}{9\pi}\left[\left(\frac{355}{113}\right)^2-\pi^2\right]. \tag{15} \]

Outward rational enclosures give \(b_x\) approximately \(0.01623004820\) and \(b_\theta\) approximately \(0.00010725580\). The strict rational thresholds in (12) and (14), not rounded displayed decimals, are used below.

The reusable prepared-transit kernel applies with scaled form \(a=\gamma x\), generator \(A=L_f/4\), feedback \(\gamma^2\), forcing bound \(\sqrt{27}\), gap bound \(1/10\), and decay bound enclosing \(e^{-20}\). Its phase remainder is measured from the actual \(\theta(0)+a(0)\); (14) separately adds the initial errors and the difference from \(\Theta\). This transfers a proved method, with every coefficient and source bound re-admitted for (3).

The vector field is globally Lipschitz. Equations (12)--(14) are analytic bounds for the uninterrupted full flow and require no numerical trajectory, step-size assumption or replacement of a reached state by its nominal reference. In particular, no all-window acute certificate is asserted. The winding change from zero necessarily involves a principal-phase branch crossing before the deadline; these global estimates permit it.

Actual entry, winding and eventual joined registration

Reuse only the targets and conditional barrier from the full-state capture theorem. The target for the equal control is \((0,\Theta)\). For the unequal control it is \((0,\theta_*)\), with nonzero constituent origins and the unique current-balance root declared there. In both cases the target has global lifted mean zero, all fine registered edge gaps have magnitude below \(22/25\), and all three internal windings equal one. Its complete stationarity is established on all fine nodes.

Let \(H_*=H(0,\theta_*)\), and retain the corresponding fixed seam integers in each lifted edge difference \(\delta_e\). The prior target calculation gives

\[ E_{\mathrm{nom},s}=H(0,\Theta)-H_* =0\quad(s^{\mathrm{eq}}),\qquad 0<E_{\mathrm{nom},s}<27/100\quad(s^{\mathrm{un}}). \tag{16} \]

The barrier proof applies on the common mean-zero leaf to the full cell

\[ \mathcal D_s=\{(x,\theta): |\delta_e(\theta)|<2 \text{ for every fine edge}\} . \tag{17} \]

Its every boundary point satisfies \(H-H_*\ge196/625\). Internally,

\[ H-H_*\ge\tfrac12x^TL_fx +\tfrac3{20}(\theta-\theta_*)^TL_f(\theta-\theta_*). \tag{18} \]

That proof uses the exact weighted-support resistance certificate and a Bregman bound relative to the actual target. The cell need not have a positive Hessian everywhere. These are target and geometry premises; the old independently recovered source is not a premise of (17)--(18).

Write \(z=\theta(T)-\Theta\). Equations (12)--(14) imply that every fine edge at \(T\), using the fixed seam integers, has gap of magnitude at most \(\alpha+\sqrt2b_\theta<1\). Hence the actual state lies inside (17). On each marked internal C9 its principal gaps are \(\alpha+z_{r,j+1}-z_{r,j}\), with the seam treated by the same integer as the target. Summation gives exactly \(9\alpha=2\pi\). All three actual windings at the deadline are therefore one.

For the energy entry test it is essential to retain the first variation at \(\Theta\). The unequal aligned reference is not stationary, and there is no separate constituent-mean cancellation at \(T\). Direct fine-node evaluation gives

\[ g_s:=\|\nabla U(\Theta)\|_2=\|S(\Theta)\|_2 =3\sin\alpha\sqrt{\sum_r(s_{\mathrm{out},r}-s_{\mathrm{in},r})^2} =\begin{cases}0,&s^{\mathrm{eq}},\\ \sqrt{27/8}\sin\alpha,&s^{\mathrm{un}}. \end{cases}\tag{19} \]

The global absolute quadratic remainder is at most \(\tfrac12z^TL_fz\le4\|z\|_2^2\). Thus the complete reached excess has the sufficient bound

\[ \begin{split} E_T&:=H(x(T),\theta(T))-H_*\\ &\le E_{\mathrm{nom},s} +4b_x^2+4b_\theta^2+g_sb_\theta <196/625 . \end{split}\tag{20} \]

For example, the coarse common bound \(g_s\le4\sqrt{27}<24\) and the rational thresholds in (12), (14) already imply

\[ E_T<\frac{27}{100}+\frac4{2500} +\frac4{25000000}+\frac{24}{5000} =\frac{1727501}{6250000}<\frac{196}{625}. \tag{21} \]

Using the actual values in (19) gives excess upper bounds below \(11/10000\) for the equal control and \(263/1000\) for the unequal control. Both inequalities cover every initial source in (5).

The dissipation law now prevents escape through the boundary of (17). The conditional full-state theorem consequently retains all three windings for every \(\tau\ge200\) and proves

\[ (x(\tau),\theta(\tau))\longrightarrow(0,\theta_*) \quad\text{as }\tau\longrightarrow\infty . \tag{22} \]

Its invariant-set argument applies because the trapped sublevel is compact on the conserved mean leaf, zero dissipation forces \(x=0\), and the only invariant critical phase point in the cell is \(\theta_*\). No numerical settling time is claimed. At the deadline the fine state need not belong to the exactly registered family; the residual coordinates remain part of the same convergence proof.

Implementation, controls and scope

The joint-acquisition certificate re-admits primitive error bounds, reconstructs both actual weighted supports, accounts for (6)--(9), and uses shared prepared-transit and target/barrier owners for (12)--(22). The source-independent target and barrier owner rebuilds its geometry from primitive weights and supplies no acquisition verdict. Shared outward dyadic128 intervals retain the irrational coefficients; exact rationals retain the preparation, conductance and error data. Other nonnegative error allowances require their own freshly evaluated inequalities; the frozen source verdict is not transferred to them. Failed sufficient inequalities remain unavailable and do not demonstrate nonformation.

The independent controls check full fine support and degree, exact ramp cost and residual coefficients, the complete-law transport bounds, initial and final winding conditions, the unequal first variation, mean premises and invalid raw inputs. They also guard the reuse boundary: the old isolated recovery and its \(10^{-70}\) endpoint are not inputs to this new preparation.

The theorem establishes acquired constituent winding, full-state capture and eventual joined registration under one unchanged supplied law and connected support. These are distinct from autonomous edge birth, selection of a preparation or a constitutive law, collective operator admission, a finite-reservoir mechanism, scale selection and physical identification. The same support also admits the zero-form, phase-flat stationary state; the result does not assert that every preparation forms the organization. The execution plan alone assigns the next research question.