Constructive continuous realization of the acquired-parent phase event¶
Fixed path and realization question¶
The one-unit obstruction remains unchanged. This separate analytic question fixes one longer path before assessing its work and identity bounds. Its purpose is to prove existence under the same input channel and budget, not to optimize the obstruction or select a controller from a scan.
| Item | Fixed choice |
|---|---|
| Source | The same four independently uncertain acquired C9 children, each with original form and phase allowances 1e-10, carried to actual z4=(x4,theta4) at absolute slow time s=4; retain every fine coordinate and actual reached velocity |
| Target | Exactly T(z4)=(x4,theta4+k), with k[0,a,j]=(2*a-1)/4 and all other entries zero; restore every form and every unaffected phase lift |
| Complete law | Fixed matching 36-node support, all 72 unit edges, held unit capacities; x_tau=(-Lx+gamma*S(theta))/4+Gamma_tau, theta_tau=gamma*Lx/4, with gamma=1/(1023*pi) |
| Duration and clock | Elapsed slow duration h=4; tau=(1023/1024)*t, s=gamma^2*tau/2, so elapsed T_tau=2*h/gamma^2. Path parameter v runs from zero to one during this interval |
| Prescribed path | The cubic path (1) below, including both actual endpoint phase velocities; no nominal-state or zero-velocity replacement |
| Input | The explicit finite form input induced by that path and the initial complete state, declared before its response. No phase input, support/capacity change or intermediate state jump |
| Work budget | Signed net work W_in=integral (Lx)^T*Gamma_tau d(tau)<=3, retaining the actual endpoint storage rise plus all continuous loss. Positive-only supply, input amplitude and bandwidth are different obligations |
| Identity | Every marked C9 retains winding one throughout the forced path; parent registration may be lost as required by the target |
| Release | Set Gamma=0 at absolute s=8 without a state jump; match the exact endpoint and autonomous law before transferring the earlier restoring response by elapsed-time translation |
| Decision | A full-source realization with exact endpoints, retained children, admitted input and work, or an explicit obstruction. A failed sufficient bound for this chosen path is unavailable |
Write c=mean(x4), r=x4-c*1, and let L^+ be the inverse of L
on its mean-zero subspace, zero on the constant mode. For the supplied
displacement, L^+k=(2*a-1)/16-(2*g-1)*(2*a-1)/32 at node (g,a,j).
With b=h*L*x4/(2*gamma), fix
This is a supplied input recipe depending on initial complete state and marked targets. Its all-node actuation authority and initial-state knowledge are premises. It does not infer pressure from an evaluated response, realize a named operator automatically, select its occurrence, or establish optimal work. No numerical trajectory, duration scan or source refinement is activated.
The declaration above was committed as 2f0eab07 before the bounds below
were assessed. The duration remains four slow units and the signed input-work
budget remains three. The construction supplies one state-dependent recipe,
not one input independent of the initial state: for every admitted actual
z4, that state's prescribed input realizes its own exact target.
1. Complete rows, regularity and exact endpoints¶
Index the nodes by \((g,a,j)\in\{0,1\}^2\times\{0,\ldots,8\}\), and write \(\sigma_g=2g-1\), \(\sigma_a=2a-1\). On the fixed matching support, the child and mixed vectors obey
Thus the expression for \(L^+k\) in the declaration is exact. It does not require a numerical inverse, an inferred support or removal of a fine-state coordinate. The connected graph's sole Laplacian kernel is the constant mode; \(\mathbf1^Tk=0\) admits the inverse on its complement.
The endpoint identities for the fixed polynomials are
Consequently (1) gives \((x(0),\theta(0))=(x_4,\theta_4)\) and \((x(1),\theta(1))=(x_4,\theta_4+k)\), for every complete actual initial state. Every form, including fine errors and the common mode, returns to its starting value. Every unaffected phase lift also returns exactly. The endpoint phase velocities are the inherited ones, not zero velocities: \(\theta_\tau(0)=\theta_\tau(T_\tau)=\gamma Lx_4/4\).
Indeed \(d v/d\tau=1/T_\tau=\gamma^2/(2h)\), so
The last row of (1) then supplies the form row exactly. Its derivative is explicit:
The path is polynomial and the input is smooth on the finite forcing interval, since the fixed sine currents are smooth. In particular the input is finite and integrable. The resulting initial-value problem has a unique solution, and direct substitution proves that it is this prescribed path. No solver residual or measured-response reconstruction is used in (5). All-node actuation and exact initial-state knowledge are supplied premises; the recipe does not give a controller using only parent observations.
Both means are retained: \(\mathbf1^T\theta(v)=\mathbf1^T\theta_4\), \(\mathbf1^Tx(v)=\mathbf1^Tx_4\), and \(\mathbf1^T\Gamma_\tau=0\). For the original acquired source the form mean is zero, but it has not been discarded from the realization. Under the original clock \(e=1023/1024\), the input is \(\Gamma_t=e\Gamma_\tau\); under elapsed slow time \(\sigma=\gamma^2\tau/2\), it is \(\Gamma_\sigma=2\Gamma_\tau/\gamma^2\). Transforming both rows therefore gives the same input-work integral in all three clocks.
2. Exact continuous loss and the original-source work budget¶
Retain the complete storage \(H=\tfrac12x^TLx+\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]\). The input balance applies to this smooth realization:
Here \(\Delta H=H(x_4,\theta_4+k)-H(x_4,\theta_4)\) is the same actual endpoint storage change as in the hybrid theorem. No preceding loss is credited as a work reservoir. Since \(Q'=1-P'\), integration gives
Thus the loss is known exactly for every supplied actual state:
This includes the cost of matching the actual endpoint velocities. It is not the unattainable constant-phase-velocity substitution from the earlier necessary bound, and no minimum-work claim is made.
Fresh acquisition admission supplies the same reference at slow time four, with
The bars denote outward bounds, not a changed law. These are the original 64 independent fine source directions, after the same isolated recovery and acquisition; neither the allowances nor a dwell time is reduced. In particular the actual four child means and forms need not equal the reference.
Using \(\|L\|_2\le8\), orthogonality of the two reference modes and \(\|\sigma_g\|_2=\|\sigma_a\|_2=6\), a convenient uniform bound is
The admitted bounds \(B_4<1/64\), \(\Delta_4<1/500\), \(A<1001/1000\) and \(\gamma>1/5000\) already imply \(M<1/4\) at the frozen \(h=4\). Since \(\|k\|_2=\sqrt{9/8}<9/8\), (7) gives the sufficient upper bound
The source-dependent endpoint work has the already proved signed correction
The correction is linear and signed; acquired child means have not been reset to recover an inapplicable quadratic remainder. Equations (10)-(11) give, uniformly over the complete admitted original source,
The detached assessor can retain the sharper outward version of (9)-(11). The coarse rational margin already proves the frozen signed net budget. It does not establish a bound of three on total positive supplied work, input amplitude or bandwidth. A negative power segment, if present, is accounted for as extraction in the signed balance; no actuator or reservoir realizing such exchange has been derived.
3. Whole-path child identities with every fine direction retained¶
Write \(r_4=x_4-x_4^*\), \(z_4=\theta_4-\theta_4^*\), and apply the same recipe to the reference in (8). Both \(k\) and \(b^*=hLx_4^*/(2\gamma)\) are constant within each C9 child. Therefore the reference path has the original winding carrier \(\psi_j=(2\pi/9)(j-4)\) plus a time-dependent uniform child offset. Its internal circular gaps remain exactly \(\alpha=2\pi/9\).
The actual path difference is exactly
The source form error is globally mean zero, so no extra common term enters (13). The exact polynomial bounds are \(\max|Q|=1/(6\sqrt3)<1/10\) and \(-1/2\le Q'\le1\). Consequently, throughout the full forcing interval,
This is an exact source-dependent path comparison, not an unforced stability estimate applied to a forced trajectory. In particular the long structural duration does not license resetting errors to a symmetric family.
Every oriented child gap on its original winding branch lies in \([\alpha-2Z_{\rm path},\alpha+2Z_{\rm path}]\subset(0,\pi/2)\). This includes the closing C9 edge with its original integer lift correction. No gap crosses a circular branch boundary, and the sum of the continued principal gaps remains \(2\pi\). All four winding-one child identities are therefore retained for every actual source state along the entire path. The parent registration is allowed to fail, as required by the target; no preservation of its identity throughout the intervention is claimed.
4. Release into the already admitted autonomous response¶
Forcing starts at absolute slow time four and ends at absolute time eight. Set \(\Gamma=0\) at the latter instant, with no state jump. The input may have a finite discontinuity when switched on or off; the state remains continuous and the phase velocity remains continuous because form does. Such a switch remains within the integrable input contract.
By (3), the release state is exactly the previous theorem's actual post-event state \(T(z_4)\), not an interval-reset substitute or a state whose parent means merely agree. The fixed support, capacities, complete sine law and clocks are identical. Uniqueness of the autonomous flow then identifies every subsequent state with the restoring response translated by four slow units. Its finite child retention and parent recovery apply through absolute time twelve; its actual-state eight-coordinate prediction applies at absolute time nine. No additional acquisition interval or tighter source is required for this handoff.
The previous paired no-kick comparison, if used, must likewise be initialized from its original actual state \(z_4\) at the translated release origin. It is not automatically a comparison against a second system evolving freely from absolute time four throughout this forcing interval. No such new chronological control is proved here. Nor does translating a finite theorem establish indefinite maintenance or autonomous event occurrence.
5. Evidence, implementation and scope¶
The detached implementation re-admits the eight original primitive source allowances and rebuilds the acquired state and release premises. Its path arithmetic evaluates (1) on admitted complete state; it does not install an input in a live network or certify that arbitrary supplied vectors belong to the acquired family. Conditional path/work bounds and source-dependent admission remain separate. A failed source or sufficient estimate is unavailable, not a proof of nonreachability.
The independent controls check the exact polynomial integrals, endpoint and phase-row identities, signed storage balance, circular child branches and whole-source bounds. The result is an explicit mathematical realization under supplied full-node actuation and complete initial-state knowledge. It preserves the one-unit impossibility result, without claiming optimal duration, optimal work, a registered operator's runtime admission, a selected natural forcing law or a laboratory actuator and clock.
The separate collective actuation theorem uses eight initial child means and four uniform input channels. Its endpoint is approximate, and its same-input error, actual work and perturbed release require new bounds. That reduction leaves the exact construction above intact.