Work required to assemble recovered organizations

Status: conditional endpoint obstruction for the three recovered sine winding-one C9 objects and the two fixed oriented contact supports. Every simultaneous form/phase reset that retains their acute winding-one internal geometry requires positive net storage work. Allowing internal deformation and arbitrary relative phase origins does not remove that requirement. This is a lower-bound theorem, not the sharp minimum, an event selector or a claim about physical binding. The research program alone assigns subsequent work.

The acquired network capture theorem already establishes convergence after a supplied, unchanged-state contact. Its positive contact work is a separate premise. The present result tests whether simultaneous internal reorganization can pay that cost for the actual recovered children, using the reset-accounting principle from support events. The weighted sine storage below is its own model; native unique-neighbor phase storage and registered operator execution are not substituted for it.

Complete endpoint model and retained identity

Retain nodes \((r,j)\), ordered as \(9r+j\), with \(r=0,1,2\) and \(j=0,\ldots,8\). Each marked child retains all nine original unit-conductance cycle edges. Write \(\alpha=2\pi/9\), \(x=\mathrm{EPI}\in\mathbb R^{27}\), and let \(\theta\) denote primitive circular phase. Capacity is held at one. The before support \(E_0\) is the disjoint union of the three cycles. The after support \(E_s\) additionally contains, for each oriented contact \(e=(r,t)\in\{(0,1),(1,2),(2,0)\}\),

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

All indices are modulo nine, all added edges are undirected, and

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

No old edge is removed or reweighted. Each final node has weighted degree four. The complete support-dependent storage, with its absolute edge baselines retained, is

\[ H_E(x,\theta)=\frac12\sum_{\{i,j\}\in E}w_{ij}(x_i-x_j)^2 +\sum_{\{i,j\}\in E}w_{ij}[1-\cos(\theta_j-\theta_i)] . \tag{3} \]

The phase weights in (3) are actual conductances. A comparison that replaces them with unique-neighbor weights changes this model.

This is detached endpoint accounting for the same sine model as the capture owner. That owner declares the complete continuous laws, the structural clock \(\tau=(1023/1024)t\), pressure, held capacity and loss. No finite duration is assigned to a reset, and no new continuous row is introduced. The units of work below are the declared structural storage units of (3); there is no laboratory energy identification.

The initial endpoint is an actual member of the original three-source family, after acquisition time \(100\) and uninterrupted recovery dwell \(10^{13}\). At the unchanged handoff \(\tau_h=100+10^{13}\), the shared recovery at sine deformation zero gives

\[ \begin{split} \Theta_{r,j}&=\alpha(j-4),\qquad \theta^-_{r,j}=\Theta_{r,j}+y^-_{r,j},\\ \|x^-_r\|_2&\le\varepsilon,\qquad \|y^-_r\|_2\le\varepsilon, \qquad \varepsilon=10^{-70},\\ \mathbf1^Tx^-_r&=\mathbf1^Ty^-_r=0 . \end{split}\tag{4} \]

The three copies have independent original coordinate errors at most \(10^{-10}\), with their declared separate zero means. No source is reset to \((0,\Theta)\), no new dwell is selected, and no favorable correlation between children is imposed.

Admit hypothetical finite after coordinates \((x^+,\theta^+)\) subject to the following endpoint identity restriction. For each marked child let \(\delta^+_{r,j}\) be the principal circular gap from node \((r,j)\) to \((r,j+1)\). Require

\[ -\pi/2<\delta^+_{r,j}<\pi/2,\qquad \sum_{j=0}^8\delta^+_{r,j}=2\pi . \tag{5} \]

Thus every retained child is internally strictly acute and has winding one. Its internal shape, absolute and relative phase origins, and every finite signed form may change. Endpoint means need not equal the source means. Proving an obstruction in this larger class also covers any subclass that conserves them. New contact gaps need not be acute.

Condition (5) is an explicit endpoint class, not a consequence of winding alone or a declaration that any such state was acquired dynamically. In particular, the capture theorem permits nonacute transient gaps in its larger trapping sector. Its no-reset capture certificate is not transferred to arbitrary endpoints here. An admitted algebraic reset is not yet a registered operator: live preconditions, collective descent and occurrence remain separate obligations of the NFR ontology and operator contracts.

Common form translation, common phase rotation and simultaneous cyclic relabeling preserve (3)--(5). Separate child phase rotations preserve internal storage but generally change contact storage; they are retained coordinates, not independently removable gauges of the joined model.

The actual source has only a small internal excess

Put

\[ V(z)=1-\cos z,\qquad B_0=27V(\alpha). \tag{6} \]

On the acute interval, \(V''(z)=\cos z>0\). Jensen's inequality and the winding constraint in (5) give, separately for each child,

\[ \sum_{j=0}^8V(\delta^+_{r,j}) \ge9V\!\left(\frac19\sum_j\delta^+_{r,j}\right) =9V(\alpha). \tag{7} \]

Thus every after endpoint has old-support storage at least \(B_0\), including its nonnegative form contribution. Equality in phase requires all internal gaps to be \(\alpha\).

The source in (4) is also in that acute winding sector. Let \(L_0\) be the Laplacian of the three unit cycles. Its largest eigenvalue is at most four. The internal phase gradient vanishes at \(\Theta\); this is a fine-node identity because each node has opposite sine currents from its two cycle neighbors. Taylor's integral formula and \(\nabla^2 U\preceq L_0\) give

\[ \begin{split} H_{E_0}(x^-,\theta^-)-B_0 &\le\tfrac12(x^-)^TL_0x^- +\tfrac12(y^-)^TL_0y^-\\ &\le2\|x^-\|_2^2+2\|y^-\|_2^2 \le12\varepsilon^2. \end{split}\tag{8} \]

Combining (7) with (8),

\[ \boxed{B_0\le H_{E_0}(x^-,\theta^-)\le B_0+12\varepsilon^2.} \tag{9} \]

All residual coordinates remain in this bound. The exact rational term \(12\varepsilon^2\) must remain separate from an outward interval for the irrational baseline \(B_0\); rounding it away would incorrectly replace a reached state with the ideal minimum.

Every retained winding forces positive contact storage

Consider one oriented contact \(e=(r,t)\) and one node index \(j\). Set \(a=\theta^+_{t,j}-\theta^+_{r,j}\), interpreted modulo a turn, and \(\delta=\delta^+_{t,j}\). The two port contributions are

\[ \begin{split} C_{e,j} &=(1-s_e)V(a)+s_eV(a+\delta)\\ &=1-\operatorname{Re}\{e^{ia}(1-s_e+s_e e^{i\delta})\}\\ &\ge 1-|1-s_e+s_e e^{i\delta}|=:f_{s_e}(\delta). \end{split}\tag{10} \]

Seam integers do not change the cosines or the phasor identity. This inequality minimizes over one source phase independently; it remains a lower bound when the actual 27 phases impose further correlations.

For \(u=s(1-s)\) and \(0<s<1\), write

\[ \rho_s(z)=\sqrt{1-2u(1-\cos z)},\qquad f_s(z)=1-\rho_s(z). \tag{11} \]

On \([-\pi/2,\pi/2]\), \(\rho_s^2\ge1-2u\ge1/2\), so the derivatives are regular and

\[ f_s''(z)=\frac{u\cos z}{\rho_s(z)} +\frac{u^2\sin^2z}{\rho_s(z)^3}>0 . \tag{12} \]

The endpoint positivity follows from the second term; at zero the first term is positive. Applying Jensen to the nine target-child gaps gives

\[ \sum_{j=0}^8 C_{e,j} \ge\sum_j f_{s_e}(\delta^+_{t,j}) \ge9f_{s_e}(\alpha)=9[1-\rho_{s_e}(\alpha)]. \tag{13} \]

Every contact has its own target child with winding one, so summing these three inequalities is valid on the same complete endpoint. Define

\[ G(s)=9\sum_{e}[1-\rho_{s_e}(\alpha)]. \tag{14} \]

The added form storage is nonnegative, hence (7) and (13) imply the full-state lower bound

\[ \boxed{H_{E_s}(x^+,\theta^+)\ge B_0+G(s).} \tag{15} \]

No invariant registered-family assumption or collective tangent reduction appears in this argument.

A simpler independent lower bound follows from \(1-\sqrt{1-z}\ge z/2\) for \(0\le z\le1\):

\[ G(s)\ge 9V(\alpha)\sum_es_e(1-s_e)>0. \tag{16} \]

For the two fixed controls,

\[ \begin{array}{c|c|c} &G(s)&\text{simpler lower bound}\\ \hline s^{\rm eq}&27[1-\cos(\pi/9)]&\tfrac{27}{4}V(\alpha)\\ s^{\rm un}&9[3-2\sqrt{1-\tfrac38V(\alpha)}-\cos(\pi/9)] &\tfrac{45}{8}V(\alpha) \end{array} . \tag{17} \]

The two values of \(G\) are approximately \(1.62829924\) and \(1.35048908\), respectively. These decimals are explanatory; the certificate uses outward rational trigonometric and square-root bounds. The same enclosures give \(V(\alpha)>1/5\). Thus even the simpler bounds give \(G(s^{\rm eq})>27/20\) and \(G(s^{\rm un})>9/8\), far above the exact source excess in (9).

Equations (14)--(17) are not claims of a sharp joined minimum. Simultaneous equality in the child Jensen bounds requires uniform internal twist. At such a state, equality in every contact's phasor bound would require \(\Phi_t-\Phi_r=-\arg(1-s_e+s_e e^{i\alpha})\) modulo a turn. The left sides sum to zero around the three-contact cycle, while the sum of those arguments is \(\pi/3\) for both controls, as established by the network compatibility theorem. Thus the separately relaxed contact minima are incompatible. The positive lower bound remains valid without solving the sharper coupled minimization.

Required work and passive path obstruction

The exact storage jump for a simultaneous reset and attachment is

\[ W_{\rm event}=H_{E_s}(x^+,\theta^+)-H_{E_0}(x^-,\theta^-). \tag{18} \]

It includes both internal reorganization and all added-edge costs. The counterfactual algebraic decomposition

\[ \begin{split} W_{\rm event}={}&H_{E_0}(x^+,\theta^+)-H_{E_0}(x^-,\theta^-)\\ &+H_{E_s}(x^+,\theta^+)-H_{E_0}(x^+,\theta^+) \end{split}\tag{19} \]

does not impose an execution order. Equations (9) and (15) show that contemporaneous internal release is insufficient:

\[ \boxed{W_{\rm event}\ge G(s)-12\varepsilon^2>0.} \tag{20} \]

In particular, the internal term can release at most \(12\varepsilon^2\), whereas the new contacts cost at least \(G(s)\). The source has already relaxed close to its minimum within the retained acute winding sector. The obstruction applies to every original admitted source and every after endpoint in (5), not merely a nominal target or one attempted reset.

The unchanged-state attachment from the capture theorem costs approximately \(3.15840002\), with its own exact residual correction. Equation (20) allows a reset to reduce that cost but excludes making it storage-passive under the declared endpoint restrictions. It neither constructs an optimal reset nor certifies capture from a proposed reset endpoint.

The finite hybrid balance also excludes an evasion by passive staging. Suppose every continuous segment has its separately admitted storage law \(H'=-D\), \(D\ge0\), every event uses the actual complete storage jump \(J_k\), and all segments and jumps use the same storage normalization (3). For any such finite hybrid path from (4) to an endpoint in (5),

\[ H_{\rm final}-H_{\rm initial} =-\int D\,d\tau+\sum_kJ_k, \qquad \sum_kJ_k\ge G(s)-12\varepsilon^2+\int D\,d\tau>0. \tag{21} \]

Consequently no path with every event passive, \(J_k\le0\), has these endpoints. Even permitting individually positive jumps while demanding a nonpositive total jump budget cannot satisfy (21). If independently declared supplied-work budgets obey \(J_k\le W_k\), their sum must also obey the same positive lower bound. Past loss is not an available reserve without a separately specified storage coordinate and reservoir law.

This uses telescoping accounting, not a theorem admitting an arbitrary intermediate support or trajectory. Each intermediate law, regularity domain and event must be admitted separately. The result does not supply a schedule, finite stopping time, non-Zeno proof or autonomous occurrence argument. Temporary edge removal cannot evade the endpoint inequality if the declared final support is restored and the same complete storage is used throughout. Permanently removing or reweighting original edges changes the final-support premise instead.

Deletion and limits of the conclusion

At any final endpoint in (5), removing all added contacts without changing the coordinates changes storage by

\[ H_{E_0}(x^+,\theta^+)-H_{E_s}(x^+,\theta^+) =-H_{\rm contacts}(x^+,\theta^+)\le-G(s)<0. \tag{22} \]

The three marked cycles and their acute winding-one endpoint identities remain. This is frozen-state endpoint accounting, not a post-deletion recovery theorem or an assertion that the preceding acquired trajectory is internally acute at a specified finite time.

Thus held-support convergence does not establish storage binding against an admitted contact-deletion action. A negative deletion jump also does not select that action's occurrence. Both attachment and detachment still require an event model; no physical binding energy is identified here.

The obstruction retains its declared acute endpoint identity class, late recovered source, original internal edges, two fixed contact controls and weighted sine storage. It does not exclude attachment earlier during acquisition, endpoints outside the acute class, an identity change, removal or reweighting of original edges, or a separately justified storage or reservoir law. Those would change premises and require their own source, event and capture analysis. No such extension is executed here.

Implementation and independent controls

The assembly-work certificate re-admits primitive inputs, rebuilds the selected weighted support and original source recovery, and derives the contact lower bounds from the shared contact-amplitude owner. It retains the exact source excess separately from interval-rounded nominal terms and makes source-dependent claims only when that recovery is certified. Incoming report verdicts are not premises.

The independent controls exercise the actual support, nonuniform acute winding endpoints, finite form/phase reset accounting, strict work signs, original residual scope, deletion, and invalid or unavailable source inputs. These are analytic and static checks. No original source producer, trajectory, reserved response or physical dataset is regenerated.