Regional work and retention of organized winding

The unchanged reciprocal sine law admits a sufficient regional work budget for retaining an acquired C9 winding. It also imposes a necessary work cost on any first passage of the initially winding-zero receiver. These are conditional complete-state statements. The actual two-ramp source fails the donor-retention budgets on the interval ending at its certified loss; the budgets do not change that evaluated result or establish an alternative acquisition route.

The ontology requires the retained constituent and boundary state when discussing a collective identity. Here that identity is the integer winding on each marked C9, with its continuous phase lift, fine form and incident contacts retained. It is not a complete description of an NFR or a physical identity. The research program alone owns subsequent research priorities.

Complete law, clock and regional balance

Use exactly the receiver source and support: three unit C9s with the fixed reciprocal contacts of offsets zero and one, each of weight one half, on the oriented region pairs \((0,1),(1,2),(2,0)\). Every node has weighted degree four. Write

\[ A=L/4,\qquad S_i(\theta)=\sum_jw_{ij}\sin(\theta_j-\theta_i), \qquad \gamma=1/(1023\pi). \]

In the declared structural clock \(\tau=(1023/1024)t\), the full law is

\[ x_\tau=\tfrac14[-Lx+\gamma S(\theta)],\qquad \theta_\tau=\tfrac\gamma4Lx. \tag{1} \]

Capacities are held at one; support and conductances are held fixed. There is no forcing, reset or support event. This is the detached weighted sine comparison, not native relational execution or a registered operator word. The nodal identity alone does not select (1).

Retain all 54 form and phase coordinates. Common form translation and common phase rotation preserve (1); global means are conserved. Regional means need not be. The circular phase is represented by a continuous lift for the following balances, whose energies and branch-normal rates do not depend on a common phase origin. Storage and integrated work are in this model's nondimensional structural storage units. None has an admitted laboratory-energy interpretation.

For one region \(r\), let \(L_{I,r}\) be its unit-cycle Laplacian. Restrict the internal and contact forces to that region:

\[ \ell_r=L_{I,r}x_r,\qquad b_r=(L_Cx)_r,\qquad s_r=(S_I(\theta))_r,\qquad c_r=(S_C(\theta))_r. \]

Its internal phase potential and complete internal storage are

\[ U_r(\theta_r)=\sum_{j=0}^8 [1-\cos(\theta_{r,j+1}-\theta_{r,j})],\qquad H_r=\tfrac12x_r^TL_{I,r}x_r+U_r, \tag{2} \]

with indices modulo nine. Define the nonnegative internal loss and signed boundary power by

\[ D_r=\tfrac14\|\ell_r\|^2,\qquad P_r=-\tfrac14\ell_r\cdot b_r +\tfrac\gamma4(\ell_r\cdot c_r-s_r\cdot b_r). \tag{3} \]

The regional balance then reads

\[ \frac{dH_r}{d\tau}=-D_r+P_r. \tag{4} \]

Indeed, \(\nabla_xH_r=\ell_r\) and \(\nabla_\theta H_r=-s_r\); applying both rows of (1) cancels the internal sine exchange. Boundary power has no fixed sign. It is determined by the retained neighboring state, rather than an independently chosen external input.

For a declared release time \(a\), put

\[ \mathcal W_r(\tau)=\int_a^\tau P_r(v)\,dv,\qquad \mathcal D_r(\tau)=\int_a^\tau D_r(v)\,dv\ge0. \]

The exact all-prefix identity is

\[ H_r(\tau)=H_r(a)+\mathcal W_r(\tau)-\mathcal D_r(\tau). \tag{5} \]

Changing to elapsed slow time \(u=\gamma^2(\tau-a)\) multiplies both instantaneous powers by \(\gamma^{-2}\); the integrated work and loss in (5) are unchanged. Omitting this clock factor changes the budget.

A first-boundary barrier for winding zero and one

Fix the principal-branch integers at release, and continue the registered oriented gaps

\[ \delta_j(\tau)=\theta_{r,j+1}(\tau)-\theta_{r,j}(\tau)-2\pi k_j \tag{6} \]

until their first contact with \(\{-\pi,\pi\}\). Before that time, \(\delta_j\in(-\pi,\pi)\), and \(\sum_j\delta_j=2\pi w\) is the constant winding sum. At the first boundary contact all gaps lie in \([-\pi,\pi]\), and at least one is \(\pi\) or \(-\pi\). Winding at the boundary itself is not used.

For every \(|d|\le\pi\), concavity of sine on \([0,\pi/2]\) gives

\[ 1-\cos d=2\sin^2(|d|/2)\ge 2d^2/\pi^2. \tag{7} \]

Suppose first that \(w=1\). If one boundary gap is \(+\pi\), the remaining eight sum to \(\pi\). By Cauchy--Schwarz and (7),

\[ U_r\ge2+\frac2{\pi^2}\frac{\pi^2}{8}=\frac94. \tag{8} \]

If the chosen boundary gap is \(-\pi\), the other eight sum to \(3\pi\), giving the stronger bound \(U_r\ge17/4\). Simultaneous boundary contacts cause no problem: select any one of them in the same argument. Since the form part of (2) is nonnegative, every first boundary contact from a winding-one cell obeys

\[ H_r\ge U_r\ge B,\qquad B=9/4. \tag{9} \]

For winding zero, a boundary gap \(\pm\pi\) leaves a sum \(\mp\pi\) for the other eight. Equations (8)--(9) therefore hold for its first boundary contact too.

The number \(9/4\) is a conservative lower bound, not an asserted sharp barrier or a universal TNFR constant. The theorem uses the marked unit C9 and the cosine potential in (2); different weights, cycles or constitutive potentials require their own bounds. Reaching this storage is necessary for a first branch passage, but does not prove that passage occurs.

Reserves for the original full source family

Use the unchanged preparation with two original form ramps, nominal zero phase and independent per-coordinate errors bounded by \(10^{-10}\), with the separately declared regional zero sums at preparation. The slow-admission theorem rebuilds, at \(a=400\),

\[ \|x(a)\|_2\le X_0,\qquad \|\theta(a)-P\|_2\le b_0, \quad P=\alpha(q,q,0),\quad \alpha=2\pi/9,\quad q_j=j-4. \tag{10} \]

Here \(X_0\) is approximately \(0.016168003\), and \(b_0\) is approximately \(0.000207559164\). They are consequences of the original source, not new release preparations. Donor winding one is certified from \(\tau=200\) through release, while the receiver still has winding zero.

For the unit cycle, \(0\preceq L_{I,r}\preceq4I\). The Hessian of \(U_r\) is a cosine-weighted cycle Laplacian and satisfies \(\nabla^2U_r\preceq L_{I,r}\preceq4I\) at every lifted phase. Both the winding-one twist \(\alpha q\) and the zero phase are critical points of the internal potential. Taylor's integral remainder thus gives

\[ U_r(\vartheta+e)\le U_r(\vartheta)+2\|e\|_2^2 \quad\text{for }\vartheta\in\{\alpha q,0\}. \tag{11} \]

There is no omitted linear term here: it vanishes at these internal critical points. This does not assert that \(P\) is a critical point of the complete contact potential.

Each regional restriction of the errors in (10) has norm at most the full-state bound. Consequently, simultaneously for both donors,

\[ H_r(a)\le\overline H_D :=9(1-\cos\alpha)+2X_0^2+2b_0^2,\qquad r=0,1, \tag{12} \]

and for the receiver

\[ H_2(a)\le\overline H_R:=2X_0^2+2b_0^2. \tag{13} \]

The associated guaranteed reserves are

\[ R_D=B-\overline H_D>143/1000,\qquad R_R=B-\overline H_R>2249/1000. \tag{14} \]

Their outward evaluations are approximately \(0.143877\) and \(2.249477\), respectively. The implementation rebuilds the source, support and outward trigonometric bounds before asserting positivity. All original error directions remain; no nominal symmetry is imposed. The separate regional upper bounds reuse full-state norm budgets and must not be mistaken for an exact additive partition of the initial storage or of its uncertainty.

Conditional retention and necessary receiver work

Consider a donor starting in the release winding-one cell with (12). If, for every \(\tau\in[a,T]\),

\[ \mathcal W_r(\tau)-\mathcal D_r(\tau)<R_D, \tag{15} \]

then it cannot touch a branch boundary on that interval. Otherwise its first contact \(\sigma\le T\) would satisfy (9), whereas (5), (12) and (15) imply \(H_r(\sigma)<B\), a contradiction. This proves continuous winding-one retention, including the already admitted time-200 to release bridge.

A certified upper bound for the supremum of the net work over the whole interval is sufficient when it is strictly smaller than \(R_D\). Because each prefix includes release, this supremum is nonnegative. An endpoint-only budget cannot exclude an earlier passage and return. Equality at the barrier is inconclusive. The stronger all-prefix condition \(\mathcal W_r<R_D\) is sufficient because \(\mathcal D_r\ge0\). Likewise, a proved pointwise bound \(P_r\le D_r\) on a domain reached by the actual source would make its net work nonincreasing. None of these power restrictions follows from complete-system dissipation alone.

Conversely, if a donor first touches its branch boundary at \(\sigma\), then the exact balance gives

\[ \mathcal W_r(\sigma)-\mathcal D_r(\sigma) =H_r(\sigma)-H_r(a)\ge B-H_r(a)\ge R_D, \tag{16} \]

or \(\mathcal W_r(\sigma)\ge R_D+\mathcal D_r(\sigma)\). This is a condition at first contact, not a claim that the threshold was already crossed strictly before contact. A touch without a change in winding also satisfies it. Budget failure alone proves no loss.

For the receiver, any first exit from its winding-zero branch cell needs

\[ \mathcal W_2(\sigma)-\mathcal D_2(\sigma)\ge R_R, \qquad \mathcal W_2(\sigma)\ge R_R+\mathcal D_2(\sigma). \tag{17} \]

The receiver's necessary net-work bound exceeds the upper bound sufficient to certify donor retention through (15). This compares the declared reserves; it neither makes (15) necessary for retention nor specifies the actual power ratios. Equation (17) is only a necessary precursor: it does not select the crossing direction, prevent a return, or prove joined capture.

The joint question is whether the actual contact dynamics can satisfy both donor all-prefix conditions while delivering a receiver passage and the separate complete-state capture conditions. The inequalities alone are neither an existence proof nor an incompatibility proof. In particular, the receiver need not receive its net work by increasing donor internal storage. Contact storage must remain in the accounting. If

\[ H_C=\tfrac12x^TL_Cx+ \sum_{\text{contact }\{i,j\}}w_{ij}[1-\cos(\theta_j-\theta_i)], \qquad \mathcal D_{\rm all}(\tau) =\int_a^\tau\tfrac14\|Lx(v)\|^2\,dv, \]

then \(H=\sum_rH_r+H_C\) and the complete loss identity imply

\[ \sum_r[\mathcal W_r(\tau)-\mathcal D_r(\tau)] =H_C(a)-H_C(\tau)-\mathcal D_{\rm all}(\tau). \tag{18} \]

Internal losses do not add to complete loss without the contact cross terms. The already supplied contact state, its own storage change and complete loss all constrain work distribution. Loss is not an event reserve, and this continuous balance supplies no contact-occurrence law.

Why a phase-only cell does not enforce retention

For an internal oriented edge let \(b=e_j-e_i\). Its registered gap in (6) has exact normal velocity

\[ \delta_\tau=\gamma b^TAx. \tag{19} \]

At any fixed boundary phase, both forms \(x=cb\) and \(x=-cb\), \(c>0\), are finite signed forms with zero global mean. On this support, the two endpoint degrees are four and the internal edge weight is one, so

\[ b^TAb=\tfrac14(4+4+2)=5/2, \qquad \delta_\tau=\pm5\gamma c/2. \tag{20} \]

Thus the same boundary phase admits both inward and outward normal motion under the unchanged law. The amplitude \(c\) can be arbitrarily small; an unsigned small-form assertion alone does not select the sign. Boundary phases with only that gap on the boundary exist, for example one gap \(\pi\) and eight gaps \(\pi/8\) for winding one. They can be represented on the global mean-zero phase leaf. Smoothness then also provides nearby interior states with an outward passage.

Consequently a winding cell extended over all admitted mean-zero forms is not forward invariant. This is a complete-law obstruction to a phase-only universal retention argument, not a claim about which boundary states the original source actually reaches. It does not invalidate a full-state storage sublevel, a source-specific tube or an independently proved contact-work budget. Retaining form and boundary state is essential even when a reduced phase reference approximates one source accurately.

Consequence of the preserved donor-loss evidence

The frozen enclosure already proves that both donors lose winding one for every original admitted source, while the receiver stays winding zero throughout the validated interval. It retains both donor identities through elapsed slow time \(37/4\), and establishes both losses by \(19/2\). Each donor therefore has a first boundary contact in that interval, possibly at a different time for different sources or donors.

Combining that preserved conclusion with (16) proves that each donor's actual accumulated boundary work reaches at least its guaranteed reserve plus its internal accumulated loss at its first contact. Accordingly, neither donor satisfies (15) over the complete evaluated interval. This is a derived necessary-work consequence of existing full-family evidence. It is not a newly measured work integral or a new trajectory evaluation; the frozen response is neither rerun nor rewritten.

Receiver retention in winding zero does not conversely establish that its net work stayed below (17): the barrier is necessary for passage, not sufficient. Nor does a failure of the donor sufficient budget prove that every preparation or every compatible TNFR law loses its donors. The three-ramp theorem retains its different source and its valid conditional acquisition result.

The source-rebuilding assessment in _organized_regional_retention.py evaluates the three release-storage bounds and reserves. Its optional supplied donor net-work prefix bound tests only the conditional inequality; it does not supply a measured history or establish its premise. The companion stored-state work observer rebuilds the regional powers from primitive form and phase and checks the instantaneous partition balance. A state observation is not an integrated work certificate. The observer's optional constitutive parameter is admitted by the shared composition-compatible law; zero preserves the sine calculation. Its own-law extension and distinct geometric barriers belong to the constitutive comparison. The source assessment above remains sine-only and supplies no changed-law release bound.

These results identify the missing mathematical obligation: control of the actual distribution of contact work, together with directional receiver passage and full-state capture. They provide no new forcing, operator event, constitutive selection or physical identification.