Autonomous exchange with a finite auxiliary reservoir¶

Fixed joint model and prospective question¶

The collective construction still supplies its input as a function of time. This separate question fixes one autonomous joint law and a finite reservoir preparation before assessing its response. A valid balance alone will not establish registration loss or return. Failure of the declared response remains an admissible result.

Item	Fixed choice
Network source	The same four independently uncertain recovered C9 children, each with original form and phase allowances `1e-10`, carried through unchanged acquisition to actual `z4=(x4,theta4)` at absolute slow time `s=4`; retain every reached form, phase and velocity
Network support and capacity	Matching 36-node support, all 72 unit edges, held unit capacities; no support or phase event during the new flow
Auxiliary state	One signed real form coordinate `r`, with held unit capacity in the normalized `tau` clock and the pressure law specified below. No phase coordinate is consumed for this auxiliary element
Reservoir preparation	Independently supplied `r0 in [12/5-1e-10,12/5+1e-10]`; added reservoir storage `H_r=r^2/2` has budget three. This budget excludes the network's already accounted acquired storage
Complete connected law	`x_tau=(-Lx+gammaS(theta))/4+br`, `theta_tau=gammaLx/4`, `r_tau=-b^TLx-lambdar`, with `gamma=1/(1023pi)`, `b[g,a,j]=gamma(2a-1)/12`, `lambda=gamma^2/2`
Clock	`tau=(1023/1024)t`, absolute slow time `s=gamma^2tau/2`, elapsed slow time `h=s-4`; transform all three rows under a clock change
Storage and work	Complete joint storage `H_joint=H(x,theta)+r^2/2`, with the existing `H=x^TLx/2+sum_edges(1-cos(gap))`; network input work `integral (Lx)^Tbr d(tau)` must be supplied by the finite reservoir, with its remaining storage and loss retained
Disconnected control	The same actual network and reservoir preparation, but `b=0` and unchanged `lambda`; the network then follows its original unforced law. Zero initial reservoir alone is not this control
Registration-loss readout	At `h=1/2` (absolute `s=9/2`), every intraparent same-slot circular gap must exceed `1/4` in magnitude in BOTH connected parents; both disconnected parents must retain registration
Return deadline	At `h=8` (absolute `s=12`), both connected parents must have every intraparent same-slot circular gap strictly below `1/4` in magnitude
Retained identity	Every marked C9 must retain winding one throughout `0<=h<=8` in both branches
Decision	Source-inclusive loss/return discrimination with finite supplied reservoir work and retained children, or a scoped obstruction to the fixed criterion. A failed sufficient estimate remains unavailable rather than proving failure

The network and reservoir are initialized jointly at the start of this comparison; their subsequent equations contain no time profile, timed release or encoded copy of the earlier cubic input. Initializing this coupling and preparing the reservoir are supplied premises, not autonomously selected events. The existing 72-edge network Laplacian is unchanged: b*r is an added constitutive exchange channel, not an unaccounted new graph edge.

Use the existing normalized structural form, phase, storage and time units. The new coordinate has the form scale that makes r^2/2 a storage in the same units as H; the coefficients in its supplied pressure and network input have the rate units required by the three displayed rows. No laboratory units, mass, particle or thermal-bath identification is supplied by this choice.

This port excites the shared child-index mode of both parents, not the prior one-parent phase map. A reference may be parameterized by each actual r0 in its frozen interval; no midpoint substitution discards reservoir uncertainty. Actual and reference reservoir states subsequently respond to their own networks, so a new joint-state error proof is required. The preceding identical-input comparison does not apply unchanged.

The auxiliary coordinate is not claimed to be an acquired or formed NFR mode. Its origin, pressure, storage, marked coupling and preparation remain supplied constitutive assumptions; the joint balance can restrict consistency without selecting a unique law. No source refinement, coefficient scan or numerical trajectory is activated by this analytic question. The specified readouts and budgets will not be moved after assessing the response.

The declaration above was committed as 9a0071de before this analytic assessment. The fixed connected branch does not meet its registration-loss criterion: both parents still have every same-slot gap below the threshold at the reserved half-unit readout. The proof below also admits the finite reservoir balance, retained children and registered deadline. It does not infer absence of a disruption at every other time.

1. Complete joint balance, clocks and reference restriction¶

The original structural clock is absolute: the new comparison starts at \(\tau_0=8/\gamma^2\), corresponding to \(s=4\). A subscript zero below denotes this initial state, and elapsed slow time is \(h=\gamma^2(\tau-\tau_0)/2=s-4\). Write \(q=Lx\). The autonomous equations are globally well posed on finite intervals: the sine map is smooth and globally Lipschitz, and every other term is linear in the declared state. The global form mean and lifted phase mean are conserved because \(\mathbf1^Tb=0\). No auxiliary phase is needed. With unit auxiliary capacity in the normalized clock, its independently supplied pressure is \(p_r=-b^TLx-\lambda r\).

The complete storage derivative is

\[ \begin{split} H_\tau&=-\tfrac14\|Lx\|_2^2+(Lx)^Tbr,\\ (r^2/2)_\tau&=-r b^TLx-\lambda r^2,\\ (H+r^2/2)_\tau&=-\tfrac14\|Lx\|_2^2-\lambda r^2. \end{split}\tag{1} \]

Thus the network's signed net input work through any finite time is

\[ W_{\rm net} =\frac{r_0^2-r(\tau)^2}{2}-\lambda\int_{\tau_0}^\tau r(u)^2du \le\frac{r_0^2}{2} \le\frac{(12/5+10^{-10})^2}{2}<3. \tag{2} \]

The network's initial acquired storage remains part of the joint preparation. It is not included in the three-unit added-reservoir budget. Equation (2) allows the network to charge the reservoir; it is not a bound on cumulative positive-only supply. The finite reservoir is an actual coordinate of this supplied model, not a reinterpretation of previously dissipated storage.

Every row transforms under \(\tau=e t\), \(e=1023/1024\), by multiplication by \(e\). Under elapsed slow time \(h=\gamma^2(\tau-\tau_0)/2\), the complete equations become

\[ x_h=\frac{-Lx+\gamma S(\theta)}{2\gamma^2} +\frac{2b}{\gamma^2}r,\qquad \theta_h=\frac{Lx}{2\gamma},\qquad r_h=-\frac{2b^TLx}{\gamma^2}-r. \tag{3} \]

These are structural clocks. The rate units of \(b,\lambda\), pressure and storage are those fixed in the declaration; no laboratory clock is inferred.

The exact child-constant restriction retains all four forms \(X_c\), four lifted offsets \(\Phi_c\), and \(r\). Let \(\mathcal L\) be the square Laplacian, \(\beta_c=\gamma\sigma_a/12\), and \(\mathcal S_c=\sum_{d\sim c}\sin(\Phi_d-\Phi_c)\). Then

\[ X_\tau=\frac{-\mathcal LX+\gamma\mathcal S(\Phi)}4+\beta r,\qquad \Phi_\tau=\frac{\gamma\mathcal LX}4,\qquad r_\tau=-9\beta^T\mathcal LX-\lambda r. \tag{4} \]

The factor nine in the reservoir row counts the fine nodes of each child; the network denominator remains four. Internal C9 sine currents cancel, and matching contacts preserve the winding carrier. Both facts hold for arbitrary child offsets in this invariant family.

For the acquired reference write

\[ x^*_{g,a,j}=\frac{\sigma_g U+\sigma_a V}{2},\qquad \theta^*_{g,a,j}=\psi_j+\frac{\sigma_g\Delta+\sigma_a B}{2}. \tag{5} \]

The auxiliary input preserves this separated subfamily. With \(\varepsilon=\gamma^2\), \(u=U/\gamma\) and \(v=V/\gamma\), its exact slow equations are

\[ \begin{gathered} \Delta_h=u,\qquad \varepsilon u_h=-u-\sin\Delta,\\ B_h=v,\qquad \varepsilon v_h=-v-\sin B+r/3,\qquad r_h=-6v-r. \end{gathered}\tag{6} \]

In particular \(b^TLx^*=3\gamma V\). All acquired reference velocities are retained. At the new initial time, the acquisition proof gives \(0<B_0<1/64\) and \(-(1+2\varepsilon)B_0\le v_0\le0\); neither coordinate is reset. The reference reservoir starts at the particular actual \(r_0\) in the frozen interval. Uniform bounds below retain that entire interval.

The reference storage, apart from its constant internal-cycle contribution, is \(9\varepsilon(u^2+v^2)+18(1-\cos\Delta)+18(1-\cos B)+r^2/2\). Its slow derivative is \(-18(u^2+v^2)-r^2\). These balances check the coefficients in (6); they alone do not establish the reserved response. Replacing the middle equation by \(v=-\sin B+r/3\) would instead be a singular reduced model, and is not used in the proof.

2. A new joint-state error estimate¶

Let \(R,Z<10^{-60}\) be the freshly rebuilt acquisition's full form and phase error bounds at absolute slow time four. They retain the original 64 fine source directions. Compare each actual connected state with the reference (5) initialized with its same actual \(r_0\). Write the form, phase-lift and reservoir differences as \(\xi,\zeta,\rho\). Initially \(\|\xi_0\|_2\le R\), \(\|\zeta_0\|_2\le Z\), \(\rho_0=0\). The independent reservoir uncertainty has not been replaced by its midpoint or folded into a smaller fine-state allowance.

The deviations satisfy

\[ \xi_\tau=-L\xi/4+\gamma\Delta S/4+b\rho,\qquad \zeta_\tau=\gamma L\xi/4,\qquad \rho_\tau=-b^TL\xi-\lambda\rho, \quad \Delta S=S(\theta)-S(\theta^*). \tag{7} \]

The two network inputs differ whenever the reservoir errors differ. Thus the previous identical-input theorem cannot be invoked. Instead use the positive comparison quantity

\[ \mathcal V=\tfrac12\xi^TL\xi+\tfrac12\rho^2+4\|\zeta\|_2^2. \tag{8} \]

It is an error estimate, not an additional physical reservoir. The global form-error mean stays zero because the coupling vector is mean zero. The fixed graph has \(\lambda_2(L)>1/3\), \(\|L\|_2\le8\), and \(\|\Delta S\|_2\le8\|\zeta\|_2\), globally in the lifted phases. The reciprocal reservoir terms cancel exactly:

\[ \begin{split} \mathcal V_\tau &=-\tfrac14\|L\xi\|_2^2-\lambda\rho^2 +\tfrac\gamma4(L\xi)^T(\Delta S+8\zeta)\\ &\le\frac{\gamma^2}{16}\|\Delta S+8\zeta\|_2^2 \le16\gamma^2\|\zeta\|_2^2 \le4\gamma^2\mathcal V. \end{split}\tag{9} \]

Since \(\mathcal V_0\le4(R^2+Z^2)\), slow time gives \(\mathcal V(h)\le\mathcal V_0 e^{8h}\). For either horizon \(H\in\{1/2,8\}\), the convenient rational bounds are

\[ Z_H=(R+Z)3^{4H},\qquad R_H=6Z_H,\qquad Q_H=4Z_H, \tag{10} \]

and throughout that interval \(\|\zeta\|_2\le Z_H\), \(\|\xi\|_2\le R_H\), \(|\rho|\le Q_H\). Here \(e<3\), the graph gap controls the mean-zero form, and the integer constants conservatively dominate the square-root coefficients from (8). In particular \(Z_{1/2}<2\cdot10^{-59}\) and \(Z_8<4\cdot10^{-45}\).

The same estimate applies to the disconnected branch with \(b=0\). Its reference and actual reservoir both decay as \(r_0e^{-h}\), while the network follows the original unforced law. No assertion that a connected reservoir initialized at zero stays zero is required.

3. Cooperative bounds for the fixed connected response¶

The following comparison treats the finite \(\varepsilon\) equations directly. Define

\[ p=r+6B,\qquad y=v+4B. \tag{11} \]

Then (6) is equivalent to

\[ B_h=y-4B,\qquad \varepsilon y_h=-(1-4\varepsilon)y +(2-16\varepsilon)B-\sin B+p/3,\qquad p_h=6B-p. \tag{12} \]

Its off-diagonal derivatives are nonnegative: in particular the derivative of the second numerator in \(B\) is at least \(1-16\varepsilon>0\). The nonnegative orthant is invariant, using \(\sin B\le B\) for \(B\ge0\). The initial state lies there because \(y_0\ge[4-(1+2\varepsilon)]B_0>0\) and \(p_0=r_0+6B_0>0\). Standard componentwise differential comparison therefore applies; no prior assumption that \(r\) stays positive is needed.

Put

\[ d=1/64,\qquad c=9/10,\qquad r_+=12/5+10^{-10}, \qquad \begin{split} \bar B(h)&=d+\tfrac14(1-e^{-4h}),\\ \bar y(h)&=1+4d,\\ \bar p(h)&=6d+\tfrac32+(r_+-2)e^{-h}+\tfrac12e^{-4h}. \end{split}\tag{13} \]

These coordinates dominate \((B_0,y_0,p_0)\). The first and third equations in (12) hold exactly. For the second, the required residual is

\[ \begin{split} &\varepsilon\bar B_{hh}+\bar B_h+\sin\bar B+2\bar B-\bar p/3\\ &\quad=\sin\bar B+(1/3-4\varepsilon)e^{-4h} -(r_+-2)e^{-h}/3\\ &\quad\ge cd+c/4+(1/3-c/4-4\varepsilon)e^{-4h} -(r_+-2)e^{-h}/3>0. \end{split}\tag{14} \]

Indeed \(0<\bar B\le17/64\), and \(1-(17/64)^2/6>c\) gives \(\sin\bar B\ge c\bar B\). Both \(1/3-c/4-4\varepsilon\) and \(cd+c/4-(r_+-2)/3\) are strictly positive for the unchanged \(\varepsilon<10^{-7}\). Thus \(0\le B(h)\le\bar B(h)\le17/64\).

At the frozen half-unit readout, \(e^2<8\) gives the useful strict margin

\[ B(1/2)\le\frac1{64}+\frac14(1-e^{-2})<\frac{15}{64}. \tag{15} \]

This evaluates the declared candidate, without a parameter scan or a replacement of its deadline. It is a genuine upper bound below the desired disruption threshold, not the failure of a lower-bound method.

For the later registered deadline, the preceding global bound admits \(\sin B\ge cB\) along the actual reference. Replace the second row's sine term by that lower linear bound. The resulting linear system is again cooperative. A supersolution is

\[ \widehat B(h)=\tfrac13e^{-h/5},\qquad \widehat y(h)=(4-1/5)\widehat B(h),\qquad \widehat p(h)=\tfrac52e^{-h/5}. \tag{16} \]

Its initial values dominate the reference, in particular \(p_0<r_++6/64<5/2\). Its first and third rows are exact, and its second-row residual is \((\varepsilon/25+1/5)\widehat B>0\). Therefore

\[ 0\le B(8)\le\tfrac13e^{-8/5}<\frac1{12}, \tag{17} \]

where \(e^{8/5}>4\) follows already from its first four Taylor terms. No small-\(\varepsilon\) replacement of the acquired velocity was used in either comparison.

4. Actual identities, control and decision¶

In the reference, both parents have the same internal child-registration gap \(B\). The actual deviation of any same-slot gap is at most \(2Z_H\). At the reserved readout, (15) and the new joint estimate give

\[ |\theta_{g,1,j}-\theta_{g,0,j}|_{\rm circ} \le\frac{15}{64}+2Z_{1/2}<\frac14 \quad\text{for every }g,j. \tag{18} \]

The displayed bound is within the principal circular branch. Both connected parents therefore retain registration at the time when the frozen criterion requires every one of their gaps to exceed \(1/4\). The complete original source and the full reservoir interval are covered. This excludes the specified registration-loss discrimination for this fixed candidate.

For the disconnected control, the original acquired reference's unforced cone gives \(0<B(h)\le B_0<1/64\). Applying (10) shows its actual parents retain registration at the same readout, and also at the deadline. This is the declared \(b=0\) control with the same preparation; it is not a claim about a connected zero-initial-reservoir preparation.

Equation (17) gives, for the connected branch at the deadline,

\[ |\theta_{g,1,j}-\theta_{g,0,j}|_{\rm circ} \le\frac1{12}+2Z_8<\frac14. \tag{19} \]

This establishes registration at that deadline. Since the prescribed loss criterion failed, it is not reported as a demonstrated loss-and-return cycle. The proof has not excluded a transient threshold crossing at every other time between the two readouts.

In either reference branch, each child remains a winding carrier plus a uniform offset under its exact inherited dynamics. The full error bound confines every actual oriented internal edge gap, including the closing edge with its retained integer lift, to

\[ [\alpha-2Z_8,\alpha+2Z_8]\subset(0,\pi/2),\qquad \alpha=2\pi/9. \tag{20} \]

Their sum stays \(2\pi\), proving winding one for all four original children through the full eight-unit elapsed interval in both branches.

The result is therefore a source-inclusive negative decision for the frozen registration-loss readout, with an admitted autonomous balance, finite net reservoir work, child retention and registered deadline. No source allowance, coupling coefficient, preparation or readout has been moved. A different port, reservoir, horizon or complete law is a distinct question and is not excluded by this result.

5. Evidence, reuse and scientific scope¶

The detached implementation admits all eight network-source allowances and the reservoir allowance before rebuilding the original source and assessing the strict response margins. The joint comparison treats the two reservoir responses as part of the state; an old identical-input verdict is not reused. The scalar response bounds are prospective consequences of the fixed law, not sampled trajectories. An unavailable source or sufficient margin is reported separately from the proved negative decision.

The independent controls check the full and inherited rows, factor-nine reservoir reaction, all clock conversions, storage power, finite reservoir budget, joint-error cancellation, comparison supersolutions and circular identity margins. The original full fine uncertainty and independent reservoir interval remain explicit throughout.

The auxiliary scalar, its pressure and storage, the reciprocal coupling and their joint preparation remain supplied premises. A successful balance does not derive their occurrence or identify the scalar as a formed NFR. Conversely failure of this fixed response does not invalidate the existing organized objects, supplied-input constructions or every autonomous source mechanism. No unique fundamental law, material property or physical measurement bridge has been selected.