Full-state capture and collective registration of the organized source¶
Declared source, capture question and disconnected control¶
This admission continues the complete organized-source port without changing its law, actual source, attachment or preparation history. The question is whether the acquired objects survive their interaction and approach a common joined registration. The control distinguishes a property of the connected organization from identities already present in its parts.
| Item | Fixed choice and scope |
|---|---|
| Fine state and support | Original 45 nodes 9*c+j, five marked C9s, four host matching interfaces and the nine donor contacts (j,36+j); all 90 unit edges retained |
| Complete joined law | x_tau=(-Lx+gamma*S(theta))/4, theta_tau=gamma*Lx/4, gamma=1/(1023*pi), with actual joined support; uniform quarter factor in both rows, held unit capacities, no later input or event |
| Clock and lifts | tau=(1023/1024)*t; theta[c,j]=psi[j]+delta[c,j], psi[j]=2*pi*(j-4)/9, continuous offsets and fixed closing-edge turns; transform both rows under clock changes |
| Host source | The unchanged actual acquired four-child family at s=4; four independent original class-one ramp preparations, each with separate zero-sum form/phase errors bounded by 1e-10; retain the reached forms, phases and every fine residual |
| Donor source | Independent original class-one ramp with the same error allowances; actual recovery under the quarter-factor isolated law, with formation time 200 and dwell 2*10^13 in its elapsed tau clock; translate only its reached phase origin by 1/8 |
| Source information | Re-admit all ten primitive allowances before rebuilding the two source owners; retain all 80 relative original error directions, source storage and declared preparation start times without additional waiting |
| Attachment | Supplied support-only event, with unchanged coordinates and the actual positive contact work retained separately from prior storage and loss; no autonomous occurrence or funding mechanism |
| Conserved leaf | T_x=sum_c mean_j(x[c,j])=0, T_phi=sum_c mean_j(delta[c,j])=1/8; these are sums of child means, not raw node totals |
| Candidate joined target | x*=0, theta*[c,j]=psi[j]+1/40; establish stationarity in all fine rows and its admitted identity, not only in the carrier restriction |
| Trapping domain | Full forms and lifted phases on the conserved leaf with every fixed-branch internal or matching edge angle strictly between -pi/2 and pi/2; retain the fixed internal cycle sums 2*pi |
| Source inclusion | Derive the complete reached-state excess H45-H45(target) from host entry bounds, donor recovery and fine errors; an old pre-acquisition upper bound or nominal carrier alone is insufficient |
| Required joined result | Strict source-to-boundary storage separation, all-future retention of all five marked windings, and asymptotic convergence to the target; a failed sufficient margin means unavailable capture, not demonstrated identity loss |
| Disconnected control | At the same actual initialized source omit only the nine donor contacts; keep the original host support, donor cycle, capacity, uniform quarter factor, clock and all reached coordinates. Compare the separate conserved component means with the proposed joined alignment; this alternative preparation is not a later edge-removal event |
| Collective property question | Can joined phase registration be derived for the connected whole while being excluded for the unchanged disconnected control? Child winding already exists before attachment; joint registration alone is not a new winding, a new primitive law or physical identification |
| Horizon and numerical boundary | All elapsed post-attachment tau>=0 for the conditional theorem; no trajectory, reserved finite readout or settling deadline. Use exact rational source arithmetic and the shared outward rational trigonometric enclosures at their existing 128-bit dyadic precision |
The proof must include fine transverse directions, not project the reached source onto the exact ten-coordinate family. Disconnected component means and connected totals belong to different supports and must not be combined as one invariant. Preserve the previous source-port declaration and result.
Continuous retention of both original parent registrations, a quantitative
recovery time, further changes of constitutive law, spontaneous attachment
and general behavior at larger complexity remain separate questions. New
collective properties are allowed, but each requires its own observation,
domain and control; complexity alone does not establish their existence.
The declaration above was committed as e7e7ce6e before this assessment.
The following result concerns the complete reached source on its actual
attached support. It uses no post-attachment trajectory.
1. Full lifted cell, stationary target and storage¶
Write \(\alpha=2\pi/9\), \(\psi_j=\alpha(j-4)\), and \(\theta_{c,j}=\psi_j+\delta_{c,j}\). On every oriented internal cycle edge use the continuous angle
This includes the fixed closing-edge turn. On a matching edge between children \(c,b\) at slot \(j\), use \(a_{cb,j}=\delta_{b,j}-\delta_{c,j}\). The declared cell is the intersection of all 90 strict inequalities \(|a_e|<\pi/2\), on the leaf
All 90 fine coordinates are retained; the two linear conservation laws remove only their common origins. This affine phase cell is convex. Its closure has \(|a_e|\le\pi/2\), and every boundary point has at least one saturated edge. It is the closure of the open cell: mixing with the strict target below approaches every closed-cell point from the interior. The five marked cycle sums are fixed by (1), rather than inferred from selected phase snapshots.
For the actual 90-edge Laplacian and sine current, the complete law is
The coefficients remain uniform despite unequal node degrees. The first row supplies the pressure at held unit capacity. Both rows, and the storage derivative below, acquire the factor \(1023/1024\) in original time \(t\); the phase row is not exempt from this clock change.
Set
The two internal sine currents cancel at every fine node of this target; all matching currents vanish separately. Thus \(S(\theta^*)=0\), and both rows in (3) vanish. All internal angles are \(\alpha\), all matching angles are zero, and (2) holds. This verifies a fine-state equilibrium with five winding-one cycles, independent of carrier closure.
The storage and exact identities are inherited from the source-port balance:
There is no later input or support event. The given vector field is globally Lipschitz in finite real lifts, so every finite initial state has a unique complete solution. The proof below concerns this supplied continuous law, not a numerical time step or a native operator execution.
2. Every acute boundary face has a positive excess¶
The function \(f(a)=1-\cos a\) is convex on \([-\pi/2,\pi/2]\). Jensen's inequality and each exact cycle sum in (1) give \(\sum_j f(a_{c,j})\ge V_9\) on the whole closed cell. The five cycle edge sets are disjoint. All other edge costs and the form storage are nonnegative. In particular, \(H-H_*\ge0\).
If an internal edge reaches \(+\pi/2\), the other eight angles of that cycle sum to \(3\pi/2\). Its cycle storage is therefore at least
If it reaches \(-\pi/2\), the remaining angles sum to \(5\pi/2\) and its lower bound is \(1+8[1-\cos(5\pi/16)]>b_9\). These arguments cover both signs of every one of the 45 internal edges. No feasibility of an independently minimized face is required: a lower bound valid on the larger cycle domain is valid on the actual full face.
If instead any of the 45 matching edges reaches either acute boundary, that edge contributes one, while the five internal cycles still contribute at least \(5V_9\). Other matching edges can only add storage. Consequently all 180 signed faces satisfy
The shared outward 128-bit rational trigonometric bounds give \(6/25<b_9-V_9<1\), hence \(\beta_{\rm face}>6/25\). For orientation, this excess is approximately \(0.24264308965044\). The strict rational inequality, not the displayed decimal, is used below. This is a complete boundary lower bound, not a claim that a full-support boundary minimizer attains it. Additional cycle correlations may strengthen it but are unnecessary for this source.
3. Re-admitting the actual reached family and its complete storage¶
Rebuild the attachment source from all ten primitive allowances. The host retains its reached state at \(s=4\) and its moving reference
Here \(\bar\gamma\) and \(\bar\eta\) are the outward upper bounds on \(\gamma\) and \(\gamma^2\), respectively, using the shared interval square for the latter. The overlined gaps are the freshly rebuilt host entry bounds. The original host proof supplies \(R,Z\) for the four independent recovered children. They bound all actual fine deviations; (8) does not set them to zero or require them to match between children.
The independent donor retains \(x^D=r_D\), \(\delta^D=1/8+z_D\), with separate exact zero means of \(r_D,z_D\) and \(\|r_D\|_2,\|z_D\|_2\le q=10^{-70}\). Its quarter-factor isolated dynamics is the original half-factor dynamics at half speed in both rows. Thus its formation time 200 and recovery dwell \(2\cdot10^{13}\) are the already declared transformed durations, not a new simulation or an additional waiting period. All five original families, and their 80 relative initial directions, remain in this source.
Initial cell membership follows on every edge: internal angles lie in \([\alpha-2\max(Z,q),\alpha+2\max(Z,q)]\subset(0,\pi/2)\); host matching angles have absolute value at most \(\max(\overline B,\overline\Delta)+2Z<\pi/2\); donor-contact angles satisfy the strictly positive acute bounds from the fresh attachment. The event changes no coordinate, so this is the actual post-event state.
For the host's four internal cycles the reference forms are constant and the reference phase gradients cancel at each node. The block cycle Laplacian has norm at most four. The internal form storage and the Taylor remainder of the internal phase storage are therefore at most \(2R^2\) and \(2Z^2\), respectively. This phase bound uses cancellation of the whole cycle's first variation, rather than treating each edge as individually stationary.
There are 18 host matching edges with reference gap \(B\), and 18 with reference gap \(\Delta\). On each such edge the actual form gap has magnitude at most \(\bar\gamma A d+2R\), and its phase gap at most \(d+2Z\), for the corresponding upper bound \(d\). Using \(1-\cos a\le a^2/2\) yields the source-inclusive host excess
This estimate needs no childwise residual mean conservation after host assembly. Its nominal part is \(9[1+(\bar\gamma A)^2](\overline B^2+\overline\Delta^2)\); all remaining nonnegative cross terms and squares are retained exactly, including terms below the trigonometric interval grid. The donor excess is at most \(4q^2\) by the same internal-cycle calculation.
Write \(b=(\overline B+\overline\Delta)/2\), and retain the actual new-edge work upper bound
The last strict inequality already follows with the coarse rational bounds in (8), \(\bar\gamma<1/3000\), \(\bar\eta<1/9000000\), and \(q=10^{-70}\): the resulting upper bound is less than \(0.082533\). The calculator uses the actual outward host entry bounds, preserving the stronger source association. An old upper bound before host acquisition is not substituted for (9).
Equation (10) adds the donor's complete reached storage and the actual contact work to the reached host storage. The separately admitted positive work \(1/16<H_B<1/10\), five primitive preparation costs, original host assembly work and prior dissipation keep their source owner's accounting. Past loss funds none of these supplied events. Neither (9) nor (10) changes a source coordinate or introduces a work reservoir.
4. Permanent winding retention and full-state convergence¶
Combining (7) and (10) gives a strict whole-source separation:
Suppose a member of the admitted family first reached the cell boundary. By (5), its storage could not exceed its initial value, while every boundary face requires at least \(H_*+\beta_{\rm face}\). This contradicts (10)--(11). The strict gap also excludes accumulation at the boundary. Hence every source member remains in the acute fixed-branch cell for all elapsed post-attachment \(\tau\ge0\), and all five marked windings remain exactly one.
For convergence, the support is connected. On the fixed zero-form-mean leaf, bounded \(x^TLx\) bounds the form coordinates. Bounded fixed-branch edge angles and the fixed lifted mean similarly bound all phase offsets. The trapped storage sublevel is thus compact and separated from the phase boundary. On its interior the phase Hessian is the weighted graph Laplacian with edge weights \(\cos a_e>0\). It is positive definite on the fixed-mean tangent space. Strict convexity therefore makes the stationary phase configuration (4) the only critical point in this cell and mean leaf.
In the zero-loss set of (5), \(Lx=0\); the form mean then forces \(x=0\), and the phase row vanishes. To remain in that set, the form row further requires \(S(\theta)=0\). Its largest invariant subset is consequently the single equilibrium (4). The invariance principle on the compact trapped sublevel proves, for every independently perturbed member of the original prepared family,
The common phase origin follows from the retained raw total \(9/8\) divided by 45. The result concerns every fine coordinate; it is not an inference from five observed means or the ten-coordinate restriction. It reuses the method of the acute-sector capture theorem with the present support, uniform quarter mobility, source and all edge faces proved explicitly. No global settling time or finite readout is supplied.
5. A collective registration distinguished by the control¶
Define the lifted relative-registration observation
It is invariant under a common change of phase origin and retains the specified lifts. At the actual initial source, the host mean is zero and the donor mean is \(1/8\), so \(\rho(0)=1/8\) for the whole family. For the connected law, (12) gives \(\rho(\tau)\to0\).
In the disconnected control only the nine donor contacts are omitted. Both complete rows keep their uniform quarter factor, the same structural clock, held capacities and every reached coordinate. Edge cancellation now separately conserves each component's form sum and lifted phase sum. The host mean is therefore zero forever, the donor mean is \(1/8\) forever, and
Thus this control cannot converge to any state with a common child phase origin, including (4). No disconnected convergence theorem or finite response calculation is needed for the obstruction. This alternative starts from the same reached coordinates on its own declared support; it is not a later removal event with an unaccounted storage jump.
The joined whole approaches common relative registration that the unchanged separate components cannot approach together. This is a conditional collective property caused by the admitted interaction. The child windings already existed; their permanent retention is a separate conclusion. The result supplies no new primitive state variable or fundamental law, and does not make fine means an exact autonomous reduction. Continuous retention of the two earlier parent registration neighborhoods, autonomous contact selection, physical identification and emergence at other sizes remain outside this theorem.
6. Shared implementation and verification boundary¶
The detached capture owner re-admits primitive source allowances through the existing attachment owner, rebuilds reached host storage with the fine residual terms, covers both classes of acute boundary face and compares their outward bounds. Conditional candidate bounds remain distinct from an admitted original source. Invalid primitives are rejected, and an unavailable source or nonpositive sufficient margin cannot produce a capture verdict. No cached report supplies a premise.
The independent controls check the full fine target, the cycle-face Jensen bound, source-inclusive storage and exact residual corrections, the two different conserved-mean structures, primitive admission and unavailable outcomes. They verify the implementation of the formulas; the boundary, retention, convergence and control-obstruction proofs are those above. The calculation installs no support or event and runs no trajectory or frozen producer.