Continuous form actuation of the acquired-parent phase event¶
Fixed realization question¶
The restoring-response theorem admits a supplied hybrid phase map and its subsequent unforced response. This question tests a different premise: finite-duration realization of that same endpoint map through an added form input. The preceding work-balance review motivates an obstruction test; this is an analytic admission, not a blind experiment or a new response trajectory.
| Item | Fixed choice |
|---|---|
| Actual source | The same four independently uncertain acquired C9 children, with original form and phase allowances 1e-10, carried to the actual state z4=(x4,theta4) at slow time s=4; no nominal reset or smaller error family |
| Target | The actual mapped endpoint T(z4): restore every form to x4, shift every phase of child (0,0) by -1/4 and child (0,1) by +1/4, and return all other phases to their original declared lifts |
| Fixed substrate | Original matching 36-node support, all 72 unit edges, held unit capacities, signed real forms and primitive circular phases with their continuous lifts |
| Complete forced law | dx/dtau=(-Lx+gamma*S(theta))/4+Gamma_tau, dtheta/dtau=gamma*Lx/4; no phase forcing, support event, capacity change or intermediate hybrid jump |
| Input class | Any integrable Gamma_tau on the finite interval, with absolutely continuous form and the stated rows almost everywhere; this includes continuous inputs and excludes impulsive resets. No amplitude or bandwidth restriction is used to manufacture an obstruction |
| Clock and duration | tau=(1023/1024)*t, gamma=1/(1023*pi), s=gamma^2*tau/2; elapsed slow duration h=1, so elapsed tau=2/gamma^2. Transform the input along with both rows |
| Storage and input work | Retain H=0.5*x^T*L*x+sum_edges(1-cos(gap)); supplied signed work is W_in=integral (Lx)^T*Gamma_tau d(tau), including the actual endpoint storage change and continuous loss |
| Budget | W_in<=3 structural storage units. A bound on total positive supplied work would be at least as restrictive; no preceding loss is a reservoir |
| Path scope | Allow all paths in this input class when testing impossibility. If realization is not excluded, a constructive certificate must additionally establish the required child identities and endpoint conditions |
| Decision | A source-inclusive obstruction for every admitted form input, or an admitted realization with its full path and work. An unavailable sufficient estimate is unresolved |
The target is the exact hybrid-map endpoint of the state at s=4, reached
after one elapsed slow unit. It is not the unforced evolution of that endpoint
over the same interval. Prescribed phase lifts make the displacement explicit;
any extension to a circular-only endpoint must account for possible winding
increments separately. The earlier hybrid theorem and its frozen evidence
remain unchanged. Failure under this duration, budget and input channel does
not rule out another duration, work supply or independently admitted phase law.
The declaration above was committed as be66de2d before its certificate was
assessed. The obstruction below concerns that unchanged input class, endpoint,
duration and budget. No input is fitted or simulated.
1. Complete forced rows, regularity and clocks¶
Let the local actuation interval start at the actual acquired state \(z_4=(x_4,\theta_4)\). Use elapsed structural time \(\tau\) and write \(T_\tau=2h/\gamma^2\). On the unchanged matching support, the forced initial-value problem is
Admit \(\Gamma_\tau\in L^1([0,T_\tau];\mathbb R^{36})\). The sine currents are smooth and bounded on the fixed finite graph; the form diffusion is linear. Solutions have absolutely continuous form, and the phase row gives a continuously differentiable real lift because form is continuous. The equations and storage balance hold almost everywhere. On this compact interval the state is bounded, the loss is finite and the input power is integrable. These regularity premises include continuous inputs and measurable inputs with finite integral. They do not admit a Dirac impulse or an intermediate state reset.
The obstruction does not restrict which nodes may be forced, the input's amplitude or bandwidth, or whether it was obtained through open-loop design or feedback. Every resulting input and trajectory in the stated class is covered. Retaining the live form source is essential: it is not absorbed into an unforced certificate or reconstructed as a pressure after observing motion.
For \(e=1023/1024\), the original clock obeys \(\Gamma_t=e\Gamma_\tau\). In elapsed slow time \(\sigma\), with \(d\sigma/d\tau=\gamma^2/2\), put \(\Gamma_\sigma=2\Gamma_\tau/\gamma^2\). Both rows become
The local interval is \(0\le\sigma\le h\); its absolute slow-time origin is the already reached \(s=4\). Input and loss integrals transform with their clocks. These are structural clocks, without a laboratory-unit identification.
2. Actual endpoint storage and input-work balance¶
Let \(k\in\mathbb R^{36}\) be the fixed lifted displacement:
The target is exactly \((x_4,\theta_4+k)\). In particular, every form must return to its actual starting value, and every unaffected lifted phase must return to its original value. Evolving the hybrid endpoint freely for one slow unit would give a different target and is not substituted here.
The complete storage and its gradients are
Substituting (1), the reciprocal terms cancel, leaving
Hence every admitted realization of the specified endpoint satisfies the exact accounting identity
The endpoint term is the actual hybrid-map storage jump. It is independent of the path chosen to realize that endpoint. The previous event-work proof rebuilds it from the original uncertain acquired source:
Here \(B_4,\Delta_4\) are the original reference's reached phase gaps, and \(Z_{\rm pre}\) bounds the actual phase error at that time. The signed linear correction is retained: acquired child-mean errors are not reset to zero. Fresh source admission and these static work bounds suffice; no verdict about the later restoring trajectory is required.
The input work in (5) is signed. If some part of an input extracts storage, that contribution remains negative in the balance. Total positive supplied work is at least signed work, so a bound on the former cannot bypass the obstruction established for the latter.
3. A necessary loss for the prescribed phase transport¶
The unchanged phase row fixes \(Lx=4\theta_\tau/\gamma\). Substitution into the loss, followed by the integral Cauchy-Schwarz inequality, gives
Equivalently, in the slow clock, (2) gives \(\mathcal D=2\int_0^h\|\theta_\sigma\|_2^2d\sigma\). The same inequality is then \(\mathcal D\ge2\|k\|_2^2/h\). The cancellation of \(\gamma\) reflects the prescribed slow duration; it does not identify a physical constant or eliminate clock dependence.
This bound is global. It assumes no small-angle approximation, invariant family, monotone path, controlled trajectory, or intermediate child identity. Even paths that temporarily lose all the organizational identities must pay this loss if they satisfy the phase law and the prescribed endpoints. Thus proving the impossibility does not first require a retention certificate for every candidate path.
Both common modes are harmless to the bound. The lifted phase mean is conserved because \(\mathbf1^TL=0\), and the displacement (3) respects that requirement. Arbitrary common form motion or a common component of the input does not change \(Lx\), phase velocity or the loss. Returning every form also requires the net common form input to vanish, but imposing that additional necessary condition cannot lower (7).
4. Decision for the fixed one-unit realization¶
Combining the exact endpoint work and the necessary loss yields
For the frozen \(h=1\), (6) alone already gives
The last margin is \(1/4-18Z_{\rm pre}>0\) for the complete original source. The detached certificate may use the sharper outward lower endpoint of (6) plus \(9/4\), while preserving the same decision and budget. Consequently no admitted form input can realize the fixed actual endpoint within one slow unit and three structural work units. This is an obstruction for the entire declared input class and original uncertain family, not a failed controller, a solver failure or the absence of one attempted path.
There is therefore no need to construct a forcing or evolve a driven path for this gate. An unavailable original-source or endpoint-work admission would instead leave the decision unavailable; cached status labels alone are not evidence for (9).
5. Equality, alternative lifts and limits of the conclusion¶
Equality in (7) requires a constant lifted phase velocity almost everywhere, \(\theta_\tau=k/T_\tau\). Since \(x\) is continuous and the phase row is unchanged, this also requires \(Lx(0)=4k/(\gamma T_\tau)\), with the same condition at the final endpoint. It is not enough to draw a straight phase path while choosing unrelated endpoint forms.
For the acquired source, equality is impossible. Define the mixed form projection \(m(x)=\frac1{36}\sum_{g,a,j}(2g-1)(2a-1)x_{g,a,j}\). Its graph mode has Laplacian eigenvalue four. The acquired reference has \(m(x^*)=0\), and its actual fine form deviation gives \(|m(x_4)|\le R_{\rm pre}/6\). In contrast, the constant-velocity condition at \(h=1\) would require
whose magnitude exceeds that source-error allowance. This nonattainment does not establish an additional uniform positive gap above (7). No amplitude or bandwidth bound has been imposed, and no sharp infimum or optimal input has been constructed. The necessary bound, without such an optimization claim, already resolves the fixed budget.
The protocol specifies continuous lifts. If one separately weakens only the endpoint phase condition to equality modulo \(2\pi\) at every marked node, each continuous path instead has displacement \(k+2\pi n\) for an integer vector \(n\). Since every \(|k_i|\le1/4<\pi\),
The conserved lifted mean additionally requires \(\sum_i n_i=0\). The periodic storage endpoint is unchanged. Thus extra full turns cannot reduce this particular work-duration lower bound. This argument concerns the same complete nodewise circular endpoint; targeting only a winding, parent mean, approximate endpoint or another observation is a different problem and inherits no such conclusion without its own admission.
The obstruction leaves the hybrid map and its restoring-response theorem valid under their original premises. It also leaves open a longer duration, a different work budget, an admitted phase input or another complete law. Passing a necessary bound in one of those alternatives would not certify a realizing path, retained child identity or autonomous event occurrence.
6. Evidence and ownership¶
The detached assessor rebuilds the eight primitive source allowances and the actual acquired-state evidence before computing the fixed displacement, endpoint-work enclosure and loss lower bound. The shared signed event-work calculation retains the source's linear uncertainty correction. No incoming restoring-response verdict, force callback, numerical path or frozen response is needed to establish the obstruction.
The independent controls check the forced storage identity, clock transformations, the exact displacement norm, source association, signed work correction and strict rejection margin. They validate the implementation of the conditional theorem; a finite collection of force samples would not prove its universal input quantifier. Structural work and duration retain their declared units, without a sensor bridge, material interpretation or uniquely selected law.
The separate constructive-actuation theorem uses a frozen four-unit path to establish a sufficient realization with the same source, target, input channel and net work budget. Its actual path and child-retention proof are additional evidence; the one-unit obstruction above and its frozen protocol remain unchanged.