---
title: "Common-Mode Shielding During Finite Population Changes"
author: "Guillaume Bolduc"
version: "v0.3"
format: "Derived Markdown edition"
source_tex_sha256: "df6a1bf4ea8e94b90e1754781e0b6fb31c6b6f3c812085b4ad5d822024b2e4fe"
---
# Common-Mode Shielding During Finite Population Changes
**Author:** Guillaume Bolduc
**Version:** v0.3
**Status:** certified revised frozen source frame; mathematical CLEAN STOP
> This Markdown edition was mechanically derived from the selected LaTeX source. LaTeX mathematics and any complex layout commands are intentionally retained where a safe Markdown equivalent was not available. The exact source binding is SHA-256 `df6a1bf4ea8e94b90e1754781e0b6fb31c6b6f3c812085b4ad5d822024b2e4fe`.
## Abstract
We study a nominal admission controller with a nonempty protected cohort and finitely many shielded newcomers. Complete coherent residual tables centered on the protected cohort cancel all shielded physical residuals from its mean acceleration. Protected departures and gated promotions have explicit position and velocity coordinate maps. Each population or role event invalidates the preceding residual epoch and uses a bounded zero-correction waiting law. With finitely many born incarnations, at most one promotion per incarnation, the stated delayed-priority contracts, and the parent factored-gain small-age condition, a final continued shield phase captures every remaining newcomer. A finite group release is then admissible, and final relative geometry recovers with position and velocity retained at the switch. Uniform protected-relative and event-seam bounds require an explicit family of uniformly bounded histories, service, forcing, event counts and stability margins; capture time and newcomer effort may grow with initial distance. We derive the final group-release mean response and distinguish summable limiting shifts from summable whole-response envelopes. Proofs concern exact nominal dynamics. New finite rational checks corroborate selected algebra only; no online release detector, collision, actuator, radio, rigid-body or perpetual-churn result is established.
**Proposed IEEE reference:** G. Bolduc, “Common-Mode Shielding During Finite Population Changes,” *HORTRAME Research*, HORTRAME Systems, internal preprint, ver. 0.3, Oct. 2026. Public locator and human acceptance are pending; no DOI is asserted.
\begin{IEEEkeywords}formation control, delayed feedback, population changes, staged admission.\end{IEEEkeywords}
## Source frame and model
The starting point is the single-newcomer shield [citation: r1]. Its delayed relative argument comes from the complete coherent snapshot construction and E1--E2 recovery theorem in [citation: r2]. The priority input uses the actual causal delayed process of [citation: r3], rather than interpreting the synchronous fixed-epoch matrix of [citation: r4] as an asynchronous algorithm. Source identities are bound in the review package to the seven originals in the owner-supplied archive. In particular, the supplied [citation: r2] is the `cb0d7247\allowbreak{}\ldots{}` revision; older parent hashes quoted inside the addenda remain historical citations, not an assertion of byte equality.
Let a finite set of physical bodies have fixed offsets \(r_k\in\mathbb R^3\). Their active incarnations form \(A=I\mathbin{\dot\cup}J\), where \(I\ne\varnothing\) is protected and \(J\) is shielded. Each body has at most one active incarnation. An incarnation enters \(J\) once, may promote to \(I\) at most once, and may leave; no demotion or incarnation reuse occurs inside an episode. The initial active set and number of arrivals are finite. A new incarnation can be assigned to an existing finite-state body after its preceding incarnation's terminal retirement; its identity is fresh and its law changes without a plant reset. A retired body remains in physical state under the declared separate law \(\dot x=v,\ \dot v=-Cv+d_{\rm ret}\), stops active priority/control operations, and rejects later packets addressed to the retired incarnation. This law is no collision or parking guarantee.
Every surviving body retains position and velocity at events. New physical states are finite, not reset to the incumbent center. Offsets and gains do not change during the episode. Events have a locally finite total order, including an internal order for equal physical timestamps; a finite batch may be one explicitly defined combined departure/promotion event. A correct, sufficiently preannounced record lets every active controller apply the new physical partition and waiting law coherently at the declared activation time. This is a counted timing/cohort service, not instantaneous detection of an unannounced failure. The resulting protected set must be nonempty. Newly arriving incarnations cannot be directly promoted.
The priority process has ordered locally applied objective supports, positive normalized rows when initialized, and fixed per-objective anchor vectors. \(Z_{S_k}\) lists those anchors in participant k's applied support order and \(q_k=w_kZ_{S_k}\). Support maps and transported priority history obey [citation: r3,r4]. A newborn without coherent positive bootstrap has no ready priority row and remains under the waiting law; it is not assigned a fictitious row. The first complete new-epoch residual table additionally requires coherent positive goal/support bootstrap for every active source within the service budget. Promotions alone change shield roles, not objective semantics. Anchors are bounded independently of newcomer starting positions; the diameter of their common hull supplies a bound \(Q\) on \(\|q_k-\bar q_I\|\) wherever the initialized goals are used. Bodies and objectives need not have the same count in a fixed-support calculation.
Write
$$
p_k=x_k-r_k,\quad h_k=p_k-q_k,\quad
\bar y_B=|B|^{-1}\sum_{i\in B}y_i,
$$
$$
\xi_I=\bar p_I,\quad \eta_i=p_i-\xi_I\ (i\in I),\quad
e_j=p_j-\xi_I\ (j\in J),
$$
$$
\widetilde q_i=q_i-\bar q_I,\qquad g_j=q_j-\bar q_I.
$$
Means, relative coordinates and goal metadata belong to the stated shield epoch. Ordinary disturbances are locally integrable; goals are locally bounded measurable signals. Positions are continuously differentiable and velocities locally absolutely continuous between ordinary membership/role events, with equations holding almost everywhere. No velocity impulse branch is used here.
### Exact gain and priority contracts
Retain the parent gains
$$
K=ab,\qquad C=a+b,\qquad a,b>0.
$$
The position impulse kernel of the stable undelayed relative operator is
$$
g(t)=\begin{cases}(e^{-at}-e^{-bt})/(b-a),&a\ne b,\\t e^{-at},&a=b,\end{cases}\quad t\ge0.
$$
It has \(\|g\|_{L^1}=1/K\) and
$$
G_v=\|g'\|_{L^1}=2\max_{t\ge0}g(t),\qquad \kappa=KH G_v<1.
$$
For \(a=b\), \(G_v=2/(ae)\). Merely \(K,C>0\) does not imply this factored-gain proof domain: \(K=C=1\) has \(C^2-4K=-3\). A general underdamped delayed theorem is not claimed.
On a final objective/physical support, the actual priority law has stochastic coefficient rows, current self-weights at least \(\zeta>0\), all positive used off-diagonal weights at least \(\zeta\), used ages at most \(D_p\), and strongly connected unions of actually consumed influence in every \(B_p\) action indices. Initial augmented histories are actual positive final-support rows, and packets used by the recurrence must exist. The locally finite embedding of infinitely many actions in physical time tends to infinity, so \(k(t)\to\infty\). The common-column argument of \cite[Theorem 2]{r3}, \cite[P1]{r2} gives a common row \(\mu\) and hence \(\widetilde q_i,g_j\to0\) for shared anchors. No arithmetic-mean or schedule-independent endpoint is inferred. A physical exponential rate would additionally need a lower action-density bound, which this article does not claim.
## Coherent service and shield law
Each complete origin table binds the shield epoch, protected and active ordered sets, priority support/goal version, source time, and every source incarnation. Its entries are the actual same-time \(h_k(\sigma)\). At use, every active controller uses the same complete committed table, never a mixture of epochs or incomplete origins. Reception may be asynchronous; application is coherent. Actual sampling/commits are locally finite and cover the future after bootstrap, with
$$
0\le t-\sigma(t)
\chi_k(\sigma)=h_k(\sigma)-\bar h_I(\sigma),\qquad k\in A.
$$
During coherent capture use
$$
\dot x_k=v_k,\qquad \dot v_k=-K\chi_k(\sigma(t))-Cv_k+d_k(t).
$$
If a service provides full-A centered values \(m_k=h_k-\bar h_A\), then \(\chi_k=m_k-\bar m_I\) exactly. This identity still requires the complete coherent origins and known protected identities; it supplies no uncounted centroid oracle.
### Lemma 1. Modal cancellation
For every eligible table, \(\sum_{i\in I}\chi_i=0\), \(\chi_i=\eta_i-\widetilde q_i\), and \(\chi_j=e_j-g_j\). Consequently
$$
\ddot\xi_I+C\dot\xi_I=\bar d_I,
$$
$$
\ddot\eta_i+C\dot\eta_i=-K\eta_i(\sigma)+K\widetilde q_i(\sigma)+d_i-\bar d_I,
$$
$$
\ddot e_j+C\dot e_j=-Ke_j(\sigma)+Kg_j(\sigma)+d_j-\bar d_I.
$$
\begin{IEEEproof}
Subtracting the protected residual mean gives the two coordinate identities. Its protected sum is zero. Average the protected body equations, then subtract their mean from each active equation. Equal gains and coherent use are essential. No limit argument is used.
\end{IEEEproof}
The unpromoted bodies' absolute residuals have no direct term in the protected mean acceleration. They can still affect priority values or a motion-dependent schedule; those indirect effects are bounded or excluded only by the relevant stated contract.
### Theorem 1. Capture on a continued fixed shield epoch
Suppose a shield epoch is continued indefinitely with finite \(I\ne\varnothing,J\), the service and gain contracts above, finite actual histories, final-support priority convergence, and bounded differential disturbances \(d_k-\bar d_I\) tending to zero for every \(k\in A\). Then \(\eta_i,\dot\eta_i\to0\) and \(e_j,\dot e_j\to0\). This is a continued-shield statement; it does not describe shield coordinates after an actual release terminates that law.
\begin{IEEEproof}
Choose an actual analysis cut \(T_0\) after bootstrap and the priority final-support cut, late enough that the whole \([T_0-H,T_0]\) controller history and every used source have the stated fixed epoch/support. Rebase time for analysis; retain actual states, eligible stored frames and inherited clock phase. Nothing is reset.
Each coordinate \(y=\eta_i\) or \(e_j\) obeys \(\ddot y+C\dot y=-Ky(\sigma)+f\), where \(f=Kq_{\rm rel}(\sigma)+d_{\rm rel}\) is bounded and tends to zero. Rewrite the stable undelayed operator as
$$
\ddot y+C\dot y+Ky=K[y(t)-y(\sigma(t))]+f(t).
$$
If \(V_-\) bounds actual history velocity and \(H_v,H_x\) are the suprema of the homogeneous velocity/position responses from actual cut data, then on every finite horizon
$$
\begin{aligned}
\sup_{t\ge-H}\|\dot y(t)\|&\le V:=\frac{V_-+H_v+G_v\|f\|_\infty}{1-\kappa},\\
\sup_{t\ge0}\|y(t)\|&\le H_x+HV+\|f\|_\infty/K.
\end{aligned}
$$
Indeed \(\|y(t)-y(\sigma)\|\le H\sup\|\dot y\|\); apply variation of constants with \(g'\), combine history/current suprema, and solve the small-gain inequality. Position follows using \(g\). The bounds do not depend on the horizon. Let \(V_\infty=\limsup\|\dot y\|\). Splitting convolutions into a finite prefix and a late tail yields \(V_\infty\le\kappa(V_\infty+\delta)\) for every \(\delta>0\), because \(\sigma(t)\ge t-H\) and the exogenous input vanishes. Hence velocity and delay error vanish; the position convolution then tends to zero. Priority convergence supplies the vanishing goal input.
\end{IEEEproof}
An incumbent's bound uses its protected-relative history, not a shielded newcomer's capture history. A newcomer's own bound, effort and capture time may depend on its finite initial displacement. A finite common position endpoint requires a separate mean-forcing integrability condition; vanishing differential forcing alone does not supply it.
## Events, coordinate seams and causal cuts
At time \(T\), let \(L=I\setminus I'\) be departing protected incarnations and \(P=I'\setminus I\subseteq J\) promoted pre-event incarnations. Then \(I'=(I\setminus L)\cup P\ne\varnothing\). Births enter \(J'\), and shielded departures change only \(J\), unless combined with another declared event.
### Lemma 2. Exact protected-frame map
At the event, with no position/velocity reset,
$$
\begin{aligned}
\Delta\xi&=\xi_{I'}-\xi_I=\frac{\sum_{j\in P}e_j-\sum_{\ell\in L}\eta_\ell}{|I'|},\\
\Delta v&=\bar v_{I'}-\bar v_I=\frac{\sum_{j\in P}\dot e_j-\sum_{\ell\in L}\dot\eta_\ell}{|I'|}.
\end{aligned}
$$
Surviving protected, remaining shielded and promoted coordinates respectively subtract \(\Delta\xi\) and \(\Delta v\) from their old relative position and velocity. Promoted coordinates change from \(e_j\) to \(\eta_j'=e_j-\Delta\xi\).
\begin{IEEEproof}
In the new protected sum retain \(n\xi_I-\sum_{\ell\in L}p_\ell+\sum_{j\in P}p_j\), where \(n=|I|\). Substitute \(p_\ell=\xi_I+\eta_\ell\), \(p_j=\xi_I+e_j\), and \(n-|I'|=|L|-|P|\). Common terms cancel. The same calculation on velocities proves the second identity. Coordinate subtraction gives the remaining statements.
\end{IEEEproof}
Thus
$$
\begin{aligned}
\|\Delta\xi\|&\le\frac{\sum_{j\in P}\|e_j\|+\sum_{\ell\in L}\|\eta_\ell\|}{|I'|},\\
\|\Delta v\|&\le\frac{\sum_{j\in P}\|\dot e_j\|+\sum_{\ell\in L}\|\dot\eta_\ell\|}{|I'|}.
\end{aligned}
$$
These algebraic bounds contain no direct unpromoted-newcomer term. Uniform independence from original distances additionally requires Section 5's family. Adding/removing only shielded bodies has \(\Delta\xi=\Delta v=0\). Ungated promotion with \(e_j=Ru\) creates seam \(Ru/(n+1)\), so arbitrary immediate promotion has no uniform distance bound.
An old centered table is not automatically valid after departure: old corrections \((a,-a)\) sum to zero on two incumbents, but retaining the first correction after the second departs gives nonzero protected mean \(a\). New centering of the singleton gives zero. This is a failure of cohort reuse, not a physical position jump.
### New epoch and waiting law
Advance the physical plant to the ordered event, check the promotion premise, apply the declared cohort/role change, increment the shield epoch, and invalidate every preceding committed correction and pending commit. Clear their eligibility, not their historical evidence. Preserve physical state and priority state under its own protocol. Recompute neither a mixed-epoch table nor an old correction implicitly.
Use \(\dot x_k=v_k,\ \dot v_k=-Cv_k+d_k\) for active bodies until the first complete post-event table. Its actual source time must be at or after \(T\), under the new epoch, partition and goal metadata. Later sources satisfy the same eligibility and age rules; older tables cannot replace a newer committed table. Wrong incarnations/sets/epochs, incomplete tables and inconsistent source metadata are rejected; identical duplicates add no second origin term. Priority prefixes and shield epochs are distinct identifiers.
The nominal service also applies the age bound to pending-table eligibility and release certification. A relevant pending table is one still eligible to become the committed correction or enter a release certificate in the live epoch. At time \(t\), its actual source time \(s\le t\) must satisfy \(t-s
\begin{aligned}
\ddot\xi_{I'}+C\dot\xi_{I'}&=\bar d_{I'},\\
\ddot\eta_i+C\dot\eta_i&=d_i-\bar d_{I'},\\
\ddot e_j+C\dot e_j&=d_j-\bar d_{I'}.
\end{aligned}
$$
No shielded absolute residual enters this mean law. Zero correction changes estimator/controller state but never resets the plant. It trades a finite capture pause for safe epoch semantics; it does not erase causal influence already stored in the physical or priority state.
## Finite-event capture and release
A promotion gate uses thresholds \(\varepsilon,\nu>0\). In one unchanged shield epoch, it requires every proposed \(j\in P\) to satisfy \(\|e_j(s)\|\le\varepsilon\), \(\|\dot e_j(s)\|\le\nu\) throughout \([T-H,T]\), after coherent bootstrap. The current correction and every relevant pending pre-event source, as defined in Section 3, lie in that safe epoch/window. Permanently ineligible historical tables are not inputs to the certificate and cannot block this gate. Promotion then invalidates all preceding corrections and pending eligibility as in Section 3. A gate is an existential or externally certified condition: residual origins \(h=x-r-q\) alone do not provide a coherent current physical/velocity history. No online detector is implemented or inherited. A causal detector needs its own observation, communication, error and certificate contract.
### Theorem 2. Conditional finite-event continuation
Suppose initial active incarnations and arrivals are finite, each incarnation follows the one-promotion/no-demotion policy, departures are terminal, every event leaves \(I\ne\varnothing\), and promotions satisfy the gate. Assume the coherent causal service, finite actual initial states, ordinary forcing/goal regularity, and the gain contract. After the last exogenous event and any actual optional promotions, assume final-support priority and differential-forcing premises of Theorem 1. Then every finite prefix of the supplied admissible execution has finite nominal plant state, surviving position/velocity is continuous, Lemmas 1--2 and the waiting identities hold in their respective phases, and a continued final shield epoch captures its remaining newcomers. If any remain, a finite final group release can be selected. With no later events and the corresponding final ordinary service, its actual final relative geometry recovers without a plant reset.
\begin{IEEEproof}
The number of born incarnations is finite. Each can promote once and depart once, so promotions and departures add only finitely many events; they are not excluded merely because exogenous events ended. Between prescribed locally finite sampling/commit/priority events, the nominal plant with held corrections is linear with locally integrable forcing. Event maps are finite and have nonzero protected denominators; induction establishes finite-prefix continuation for the supplied causal execution. This statement does not establish well-posedness of an unspecified state-dependent event or sensing rule.
After the actual last optional event, either no newcomer remains and the law becomes ordinary after bounded bootstrap, or a fixed shield epoch remains. Choose its actual complete-history/priority cut and apply Theorem 1 to the continued shield law. Convergence and the finite newcomer set supply a finite \(S\) after which all \(e_j,\dot e_j\) satisfy the thresholds at every time. Choose a finite \(T>S+H\), after coherent bootstrap and any required preannouncement. The whole \([T-H,T]\) is safe. The current used source has age below \(H\); every relevant pending source does too by Section 3's explicit eligibility rule. All these sources therefore lie in the safe window. Any older retained table is permanently ineligible for both use and certification; local finiteness alone is not used to establish this fact. Select one final promotion \(P=J\), which invalidates all pre-event correction eligibility. This is a permitted choice, not a claim that an unspecified online policy detects it.
After this last release, set \(I^+=A,J^+=\varnothing\), invalidate pre-event corrections, wait at most \(T_s\), and resume the ordinary full-A centered law. After another actual complete-history cut its relative equations are the parent form. Since all pre-release \(d_k-\bar d_I\to0\), averaging those differences gives \(d_k-\bar d_A\to0\). Role promotion does not change the final priority support, so shared goals still agree. The parent/Theorem 1 argument on the final ordinary coordinates proves their recovery from the actual retained history.
\end{IEEEproof}
For prescribed causal source/priority/event inputs the nominal continuation is determined by its linear flow and ordered updates. Arbitrary coupled state-dependent service rules still require their own existence/uniqueness contract. Finiteness of population changes alone supplies neither priority connectivity nor physical recovery.
Partial promotion is different from final ordinary release. With \(P\subsetneq J\), set \(I'=I\cup P,J'=J\setminus P\), start the new waiting epoch, then center every active correction on \(I'\). The remaining shielded bodies continue their tracking law. Applying ordinary full-A centering instead would let their residuals enter the new protected mean.
## Uniform protected-relative and event-seam bounds
“No direct residual term” and “uniformly independent of original distance” are distinct. For a uniform claim fix a family with: a finite uniform bound \(M\) on all birth/departure/promotion events; a finite uniform active population bound; protected initial relative positions/velocities and actual initial history bounds independent of newcomer distances; common fixed \(a,b,H,T_s\) with \(\kappa<1\); a shared uniform anchor-hull bound \(Q\); uniform differential disturbance bound \(D\) in every relevant epoch; fixed gate thresholds; and valid coherent source/priority contracts. Newcomer initial positions/velocities are arbitrary finite values and may vary. Timing and priority schedules may vary, including capture times growing with distance, provided these uniform conditions remain true. The compared family does not gain a hidden distance-dependent service or forcing budget.
### Proposition 1. Uniform finite-event relative envelopes
Under this family, protected relative positions/velocities in shield and waiting phases, and the protected event-coordinate seams, have finite bounds depending on these declared data and \(M\), independent of the shielded initial distances. The bound is not uniform as \(M\to\infty\).
\begin{IEEEproof}
Induct over epochs. At a protected-set event, Lemma 2 bounds the shift using the preceding protected envelope, gate errors and the nonzero \(|I'|\). Every surviving or promoted new protected coordinate is therefore uniformly bounded; no unpromoted far coordinate enters this step.
Fix a reference time scale \(\tau_0>0\) and \(Y=\|y\|+\tau_0\|\dot y\|\) for a protected relative coordinate. During the first \(W=T_s+H\) of an epoch, every used source is post-event, or correction is zero. With \(F=\tau_0(KQ+D)\) and \(\Lambda=1/\tau_0+C+\tau_0K\), the integral equations bound the running supremum by Gronwall:
$$
\sup_{0\le u\le w}Y(u)\le (Y(0)+Fw)e^{\Lambda w},\qquad 0\le w\le W.
$$
The same bound applies to an earlier preempted epoch. It uses post-event sources only and invents no newcomer pre-birth history. If the epoch continues past \(W\), its complete actual length-H history is uniform. Theorem 1's parent small-gain bound, with \(\|f\|_\infty\le KQ+D\), then supplies a uniform envelope for arbitrarily long remaining shield flow. Homogeneous-response suprema depend continuously and linearly on the bounded cut data. A subsequent event applies the first step again. At most \(M\) iterations produce finite constants. This proves the claim, including waiting and short epochs.
\end{IEEEproof}
The same argument applies after final full release: Lemma 2 bounds all newly protected cut data through the gate, and full-A differential forcing has a bound at most twice a bound relative to old I. A uniform finite common *physical excursion* additionally needs control of mean forcing and accumulated mean/coordinate responses. It is not implied by the relative envelope. A common force can generate unbounded translation while every relative error is zero.
For \(M\) protected events, define their actual nonnegative bounds \(c_r^x,c_r^v\) by Lemma 2's right sides. Then \(\sum_r\|\Delta\xi_r\|\le\sum_r c_r^x\) and likewise for velocity. These are coordinate-seam totals; they are not a sum of each body's physical displacement.
## Final group-release mean response
At a final release \(P=J\) with \(n=|I|\), \(p=|P|\), no simultaneous departure and no later cohort/role event, let \(\xi_+=\bar p_{I\cup P}\). Continue the old protected analytical free mode from its actual \(\xi_I(T),\bar v_I(T)\) under future \(\bar d_I\), calling it \(\xi_I^{\rm ref}\). It is a comparison trajectory, not an injected control input. Both the post-event zero-correction wait and the subsequent coherent ordinary law have zero mean restoring force over \(I\cup P\).
### Theorem 3. Exact mean seam and envelope
Put \(y=\xi_+-\xi_I^{\rm ref}\), \(E=\sum_{j\in P}e_j(T)\), \(V=\sum_{j\in P}\dot e_j(T)\), and \(F_P(s)=\sum_{j\in P}(d_j(s)-\bar d_I(s))\). Then
$$
\begin{aligned}
y(t)={}&\frac{E}{n+p}+\frac{1-e^{-C(t-T)}}{C(n+p)}V\\
&+\frac1{C(n+p)}\int_T^t(1-e^{-C(t-s)})F_P(s)\,ds.
\end{aligned}
$$
If \(F_P\in L^1([T,\infty))\), then
$$
\begin{aligned}
\sup_{t\ge T}\|y(t)\|&\le B_{\rm seam},\\
B_{\rm seam}&:=\frac{\|E\|+\|V\|/C+\|F_P\|_{L^1}/C}{n+p}\\
&\le\frac{p\varepsilon+p\nu/C+\|F_P\|_{L^1}/C}{n+p}.
\end{aligned}
$$
and \(y_\infty=(E+V/C+\int_T^\infty F_P/C)/(n+p)\). This is a finite difference limit even if the two means separately drift. Finite physical endpoints additionally require the parent mean-forcing condition.
\begin{IEEEproof}
Averaging both laws gives \(\ddot y+C\dot y=F_P/(n+p)\), with initial data \(E/(n+p),V/(n+p)\) from Lemma 2. Solve the damped velocity equation and integrate. The position-force kernel is bounded by \(1/C\); dominated convergence under \(L^1\) proves the limit and the triangle inequality proves the envelope.
\end{IEEEproof}
For an actual original protected body after release, \(x_i=r_i+\xi_++\eta_i^+\). Its deviation from the continued old free formation includes both \(y\) and \(\eta_i^+\); a mean seam alone does not bound that body's whole path. Final relative recovery removes the latter term only in the limit. With zero differential mean force, exact zero limiting seam needs \(E+V/C=0\); waiting until arbitrary positive thresholds hold does not ensure that equality.
### Proposition 2. Limiting shifts versus whole responses
For a declared sequence of such mean responses with release times \(T_r\), define each \(y_r\) on the common domain \(\mathbb R\) by setting it to zero for \(tN}y_r(t)\|\le\sum_{r>N}b_r\). For the distinction take zero forcing, \(y(T)=a\ne0\), \(\dot y(T)=-Ca\). The exact solution is \(a e^{-C(t-T)}\): its limiting shift is zero while its trajectory is nonzero. Thus a whole-response bound by the sum of limiting bounds fails already for one response.
\end{IEEEproof}
This proposition concerns a defined sum of comparison responses. It is not a theorem that an arbitrary infinitely changing cohort decomposes into those responses, nor an infinite-event capture guarantee. A bound \(s_r\le s>0\) alone gives no convergence of a limiting-shift series; constant positive scalar shifts are a counterexample to that inference, not a claim that every admission execution realizes them.
## Verification and scientific position
The initial 0.1 manuscript reports 10,000 centering, 10,000 transition and 5,000 telescoping checks without its script, seed or result package. Those remain unverified inherited reports; they are not authenticated by reproducing the same counts. This revision supplies **new** COSET exact rational checks with seed 20261007: 10,000 centering/cancellation cases including changed shielded positions, 10,000 protected-set position/velocity transitions, and 5,000 fixed-position telescoping sequences. Separate controls detect stale centering, partial-promotion misuse, zero-limit/nonzero-transient seams and structural epoch eligibility; valid fresh-centering and matching-epoch contrasts pass. The code and machine-readable results identify their exact revisions. These finite algebraic tests are not delayed-flow simulation, complete transport/gate implementation, physical observation or general proof. TORSION separately reports 4,000 centering, 4,000 transitions and 2,000 seam checks at its frozen reviewer frame; those remain attributed peer evidence with declared shared sources.
The previous simulation test matrix was a proposal, not a completed suite. Second arrival, protected/shielded departure, combined departure/promotion, stale frames, distance sweeps and finite mixed histories are relevant future nominal implementation checks. No existing simulator is changed or certified by this article. A later implementation must map stored epochs, partitions, source ages, pending commits, actual physical state and priority prefixes to the stated service. Its observation certificate and state-dependent event rules need their own bridge.
The arithmetic mean projection and stable convolution machinery are standard, not novelty claims. The closest internal connection is [citation: r1]'s shield algebra/capture/release argument; [citation: r2]'s service and delayed-recovery mechanism and [citation: r3]'s causal priority history are actual dependencies. [citation: r4] supplies support-map semantics, not delayed convergence. The sampled latch [citation: r5] is a different common-mode system and is excluded from composition; [citation: r6]'s conditional-query/memory results are context only. The fixed-epoch averaging note [citation: r7] provides the comparator's admissibility, threshold and scope mechanisms rather than a shield proof.
External work already studies delay-robust consensus under broader regimes [citation: r8] and open double-integrator formation reconfiguration with barrier-based safety, a formation manager and switching conditions [citation: r9]. The relevant model/reconfiguration sections were inspected for bounded comparison. Those results are not imported to establish this article's safety or service; this article's particular protected-centering and coherent waiting construction retains different assumptions. No first-publication, general superiority or exhaustive novelty conclusion is asserted.
## Preservation, limits and next boundary
The old shield embeds algebraically when \(|J|=1\) and \(I\) is fixed: its full-centered newcomer correction is \((n+1)m_j/n=h_j-\bar h_I\), and its protected correction is \(m_i-\bar m_I\). With the same coherent post-bootstrap inputs and initial history, the laws coincide, so that phase's cancellation and conditional relative recovery are preserved. The new zero-correction epoch wait differs from [citation: r1]'s retained incumbent correction prefix, and the final release reacquires its table. Whole pre-bootstrap/post-switch trajectories are therefore not declared identical to the predecessor. This is a conditional extension of the shield guarantees with an explicit service/waiting trade-off, not an unqualified stronger realization of every predecessor branch; the optional impulse and exact-rigidity branches are not imported.
**Proved:** modal cancellation, exact cohort-coordinate maps, conditional fixed/final recovery, uniform finite-event protected-relative envelopes under the declared family, and the final mean-response formulas and summability distinction.
**Assumed:** exact nominal equal-gain plant; fixed offset/anchor identities; correct coherent service, event ordering and source eligibility, including permanent pending-age expiry and older-source supersession for use and certification; bounded age/bootstrap, actual priority packet/weight/connectivity contracts and physical embedding; factored-gain small-age stability; finite lifecycle events; stated disturbance/history and uniform-family bounds; an externally satisfied release gate.
**Surviving guarantees and trade-offs:** no plant reset, no direct unpromoted residual in protected mean feedback, controlled gated coordinate seams, and final relative recovery survive. Waiting, complete origins and finite-lifecycle bookkeeping pay for changed cohorts. No unchanged full-trajectory comparison, finite common endpoint without mean integrability, or whole-motion bound from limiting seams survives as an unconditional claim.
**Next attack:** a causal observation/certificate interface for release is the nearest missing implementation bridge. Composition with the sampled latch, infinite-event tracking/summable budgets, finite precision and physical approach feasibility remain distinct deferred problems. No new theorem or experiment for those problems is started by naming them.
No collision avoidance, actuator/speed/energy bound for arbitrary far newcomers, distance-independent capture deadline, arbitrary radio delivery, Byzantine membership, rigid-body flight, historical priority restoration, gain/offset edits or perpetual-churn convergence is established. Exact geometry and nominal feasibility do not establish real-world safety. Final acceptance is scoped to the two permanent AI reviewers' exact artifact frame; human final acceptance and any dissemination remain separate.
## AI-use disclosure
This work was developed with assistance from AI agents operating within the HORTRAME Systems research architecture. Depending on the research stage, these agents contributed to literature analysis, mathematical formulation, proof reconstruction, adversarial review, counterexample search, simulation and code analysis, verification design, reproducibility checks, revision, and manuscript preparation. AI-agent outputs were treated as research artifacts subject to verification and correction rather than as independent authority. G. Bolduc is the human author responsible for the manuscript, its claims, source selection, and publication decisions. AI-agent review does not constitute independent human peer review, proof-assistant certification, experimental validation, or publication acceptance.
COSET studied the seven supplied originals, derived the initial assessment and fresh exact checks, and revised the manuscript. TORSION separately studied the same originals and froze its initial assessment before comparing findings, then reviewed affected corrections. Their shared model/source/protocol lineage limits independence. These statements record actual performed contributions; final-frame acceptance is recorded separately rather than inferred here.
## References
1. G. Bolduc, *A Common-Mode Shield for Live Admission in Delayed Open Priority Networks*, internal addendum, ver. 01, 6 Oct. 2026. Supplied source SHA-256 `7e6ee10b\allowbreak{}455312d5\allowbreak{}9351981a\allowbreak{}781418ef\allowbreak{}cb1e1dd4\allowbreak{}982aaba8\allowbreak{}9dd4a84a\allowbreak{}e05c03db`.
1. G. Bolduc, *Translation-Equivariant Disturbance Recovery in Delayed Open Priority Networks*, supplied preprint, ver. 05, 2026. Source SHA-256 `cb0d7247\allowbreak{}80404a8b\allowbreak{}af8dbf35\allowbreak{}562e9fa4\allowbreak{}4df46ae3\allowbreak{}4b0647f7\allowbreak{}5ec81f02\allowbreak{}7b805c68`; coupled section E1--E2 and P1.
1. G. Bolduc, *Exact Restoration and Historical Memory in Delayed Priority Networks with Three-Dimensional Motion*, supplied Markdown original, ver. 0.6, 3 Oct. 2026. SHA-256 `06d74f66\allowbreak{}d823e5a6\allowbreak{}b5c8d03b\allowbreak{}b431a0a2\allowbreak{}b67389ab\allowbreak{}27249457\allowbreak{}55a5dad9\allowbreak{}0e02f9a2`; Section IV, Lemma 2 and Theorem 2.
1. G. Bolduc, *Priority-State Dynamics in Open Multi-Agent Systems: Dynamic Membership and History-Dependent Consensus Endpoints*, supplied full preprint, ver. 07, front matter dated 6 Oct. 2026. Source SHA-256 `3cb6fae7\allowbreak{}de2b7ec8\allowbreak{}a48d1aaf\allowbreak{}18306293\allowbreak{}abe98438\allowbreak{}de361e7e\allowbreak{}f9c3149a\allowbreak{}520915b8`. Historical bibliography dates remain distinct.
1. G. Bolduc, *A Sampled Common-Translation Latch for Nominal Double Integrators*, addendum, ver. 01, 6 Oct. 2026. SHA-256 `105792f1\allowbreak{}15b1f6f5\allowbreak{}3e6155a4\allowbreak{}debb1ec6\allowbreak{}4718d01d\allowbreak{}c5bf6f1f\allowbreak{}33455fe6\allowbreak{}677d7be6`.
1. G. Bolduc, *Conditional Queries with Finite Historical States and Periodic Replacement*, ver. 09, correction 01, 4 Oct. 2026. SHA-256 `7e23573e\allowbreak{}5e676d46\allowbreak{}928049d2\allowbreak{}23d49c0e\allowbreak{}1d4ef2e3\allowbreak{}bb973ce5\allowbreak{}66c1c51e\allowbreak{}42ed7ccb`.
1. G. Bolduc, *Admissibility, Mean, and Spectral Contraction of the Fixed-Epoch Averaging Matrix*, ver. 01, 6 Oct. 2026. SHA-256 `6fe31be4\allowbreak{}76634112\allowbreak{}a22b3211\allowbreak{}58324e94\allowbreak{}b3bade87\allowbreak{}9906f0be\allowbreak{}12dec4e9\allowbreak{}7ba58561`.
1. A. V. Proskurnikov and G. C. Calafiore, *Delay Robustness of Consensus Algorithms: Beyond the Uniform Connectivity (Extended Version)*, arXiv:2105.07183v1, 2021. [Primary source](https://arxiv.org/html/2105.07183v1).
1. P. \c{S}ekercio\u{g}lu, N. De Carli and D. V. Dimarogonas, *Safe Formation Control of Open Multi-Robot Systems with Connectivity-Preserving Reconfiguration*, arXiv:2609.29939v1, 24 Sept. 2026. [Primary source](https://arxiv.org/html/2609.29939v1).