Composition, interaction and formation of coherent regions¶

This collection separates sufficient regional observations, supplied support events, mediated interaction and native source/receiver formation constraints. Sections 1–7 below own first-variation closure, nonlinear counterexamples and the regional phase-response budget. The other chapters retain their proofs, complete-law premises, implementations and checks.

Chapters and scope¶

Mathematical question	Owner
Which regional observations close, and what information do they lose?	Composition and response, §§1–7
What can a supplied attachment or reset establish?	Support events and synchronization, §§8–13
When does existing support mediate a restoring connection?	Effective connections and detachment, §§14–16
Which source preparations and native-law domains admit the intended target?	Formation barriers and boundary access, §§17–23
Which results instead use the smooth-sine complete law?	Sine pattern dynamics

The native-law owner specifies the complete equations and phase-domain admission. The smooth-sine comparison changes that law; a shared storage identity does not transfer a trajectory or theorem. Supplied support, instantaneous compatibility, formation and maintenance are separate obligations. The execution plan alone assigns the next task.

1. Fixed model and observation¶

Use the existing single-bridge preparation: unit cycles (0,1,2,3,4) and (5,6,7,8,9), bridge (0,5), held unit capacities, uniform form and theta_*[i]=theta_*[i+5]=2*pi*i/5 for i=0,...,4. Both rings have winding one. This is the earlier interaction graph, not the two-bridge formation graph. The reference coefficients are e=w=1/2, beta=1; the calculation below allows e>=0, w,beta>0, without extending the recovery theorem to its lossless boundary.

Put kappa=2*pi/5, c=cos(kappa)=(sqrt(5)-1)/4>0 and h=1+2c. For form deviations u and local lifted phase deviations v, reuse the proved equilibrium derivative:

\[ J=\begin{pmatrix}-eD^{-1}B&-wH^{-1}K\\ (w/\beta)H^{-1}B&0\end{pmatrix},\qquad H=\pi\operatorname{diag}(h,2c,2c,2c,2c,h,2c,2c,2c,2c). \]

Here B is the unit-support Laplacian, D its full degrees (three at ports, two elsewhere), and K the phase Hessian with ring weights c and bridge weight one. This matrix is the exact mathematical Jacobian at the ideal equilibrium; a rounded phase preparation need not be an exact equilibrium of materialized arithmetic.

For either species z, let m_L,m_R be its arithmetic ring means. Fix the three observations

\[ O_3z=(\mu,p_L,p_R) =(m_L-m_R,\ z_0-m_L,\ z_5-m_R). \]

The six-output observation is O=diag(O_3,O_3) on (u,v). It retains relative regional means and both port contrasts while discarding the two common offsets. The means are declared observations, not transport-weighted conserved totals or a redefinition of primitive EPI/phase. The full tangent state has 20 coordinates, or 18 after those common symmetries.

2. Six observed coordinates do not close¶

Define hidden regional form shapes

\[ \eta_L=(0,1,-1,-1,1,0,0,0,0,0)^T,\qquad \eta_R=(0,0,0,0,0,0,1,-1,-1,1)^T. \]

They have zero mean and zero port value, so O_3*eta_L=O_3*eta_R=0. Nevertheless, for (u,v)=(eta_L,0),

\[ O_3\dot u=(-e/15,\ 11e/15,\ 0), \]

\[ O_3\dot v= \left(\frac{w}{5\beta\pi ch},\ -\frac{w(10c+1)}{5\beta\pi ch},\ 0\right). \]

The zero state and this hidden state therefore have identical outputs but different output derivatives. Scaling the hidden state makes the witness arbitrarily small. No matrix G can satisfy O*J=G*O. The phase response already excludes closure even if e=0; this is not solely a diffusion effect.

3. Ten coordinates are necessary and sufficient at first variation¶

Independently reflect each ring through its port: swap 1<->4,2<->3, or 6<->9,7<->8. The matrices B,D,H,K, hence J, commute with these reflections. For each species the even subspace consists of ring values (a,b,d,d,b), dimension six over both rings. Removing its common offset leaves five coordinates. The odd subspace has four coordinates per species. Its eight joint directions are unobserved and remain unobserved under J. Together with the two global offsets they bound the observable rank by ten.

On the even quotient, ker O consists precisely of eta_L,eta_R independently in form and phase: four hidden coordinates. Observe just the derivatives of the two form-port and two phase-port contrasts. Their matrix on those four directions is

\[ \begin{pmatrix} a&0&b&0\\0&a&0&b\\-d&0&0&0\\0&-d&0&0 \end{pmatrix},\qquad a=\frac{11e}{15},\quad b=\frac{w(10c+1)}{5\pi h},\quad d=\frac{w(10c+1)}{5\beta\pi ch}. \]

Its determinant is

\[ b^2d^2= \frac{w^4(10c+1)^4}{625\beta^2\pi^4c^2h^4}>0. \]

Thus one derivative recovers all four missing even directions:

\[ \operatorname{rank}O=6,\qquad \operatorname{rank}\begin{pmatrix}O\\OJ\end{pmatrix}=10,\qquad \operatorname{rank}\begin{pmatrix}O\\OJ\\OJ^2\end{pmatrix}=10. \]

The last equality follows from the invariant ten-dimensional annihilator of odd modes and offsets, not from a floating rank threshold. Any autonomous linear state retaining O must retain OJ, hence needs at least ten coordinates. Their invariant span supplies a sufficient state. Minimality concerns linear observations retaining these fixed outputs for every tangent state, not arbitrary nonlinear encodings or selected trajectories.

A natural choice adds two pair contrasts per species:

\[ s_L=(z_1+z_4-z_2-z_3)/2,\qquad s_R=(z_6+z_9-z_7-z_8)/2, \]

giving C_5z=(mu,p_L,p_R,s_L,s_R) and C=diag(C_5,C_5). A right inverse T chooses means m_L=mu/2,m_R=-mu/2 and lifts each ring as

\[ z_{\rm port}=m+p,\quad z_{\rm near}=m-p/4+s/2,\quad z_{\rm far}=m-p/4-s/2. \]

Then C*T=I, C*J=(C*J*T)*C and O=(O*T)*C. The effective state consists of internal shape and relative-offset coordinates. It is not one scalar EPI/phase pair per ring and is not already an emergent canonical node.

4. The same ten coordinates fail as a nonlinear closed state¶

The distinction matters even arbitrarily close to equilibrium. Choose u=epsilon*eta_L, epsilon>0, with right-ring form zero. Compare

\[ \theta^A=\theta_*,\qquad \theta^B=\theta_*+t(e_1-e_4),\qquad 0<t<\pi/10. \]

Here t labels a supplied phase displacement, not elapsed model time. All ten natural observations agree: the phase perturbation changes neither ring mean, port value nor pair average. Both states remain strictly acute. The form gradient at port zero is q_0=-2*epsilon; all right-ring form gradients vanish, so all right-ring phase rates vanish. The relative resultant at port zero is positive real in both states, but its phase metric changes from pi*h to pi*[1+2*cos(kappa+t)].

The observable combination mu_v+p_L_v=v_0-mean_R(v) has derivative theta_dot_0-mean_R(theta_dot): changing from deviations to the displayed absolute phase lifts adds only a fixed reference constant. Its exact rate difference is

\[ \Delta\dot y=-\frac{2w\epsilon}{\beta\pi} \left[\frac1{1+2\cos(\kappa+t)}-\frac1{1+2\cos\kappa}\right]<0. \]

This is nonzero for arbitrarily small positive epsilon,t, with expansion

\[ \Delta\dot y=-\frac{4w\epsilon\sin\kappa}{\beta\pi h^2}\,t +O(\epsilon t^2). \]

The omitted odd phase geometry modifies the already existing resultant metric; its interaction with form changes the observed model rate. No new force, parameter or pressure channel is needed. Consequently no autonomous nonlinear function of this C alone reproduces all nearby output rates. Every minimal ten-dimensional linear realization has this same row space and therefore the same failed fibers. A different nonlinear observation, additional state or an explicitly derived memory remains a separate question.

This is not a claim that every pure odd perturbation becomes visible. A special pure-odd family can remain invariant through combined sign/reflection symmetry. The counterexample uses mixed even form and odd phase. Ordinary graph reflection also reverses the prepared winding, so graph relabeling alone cannot justify discarding perturbations around this fixed lock.

5. Nonlinear regional response from the existing phase metric¶

The unweighted rate retains geometry that the cut alone loses¶

Return to any admitted fixed simple unit support, with held nonnegative capacities and available positive phase metrics H_i. The connected model's phase row is already

\[ \dot\theta_i=k a_i q_i,\qquad q=Bx,\quad a_i=\nu_i/H_i,\quad k=w/\beta>0. \]

The mobility a combines supplied capacity with the phase-dependent metric; it is a derived coefficient, not another primitive variable or an independently selected feedback. These are consequences of the specified relational closure, not a proof that the nodal identity uniquely selects that closure.

For a fixed nonempty region R, let n=|R| and let the outward cut retain the full graph's neighbors. Symmetry cancels its internal form differences:

\[ Q_R=\sum_{i\in R}q_i =\sum_{\substack{i\in R,\ j\notin R\\\{i,j\}\in E}}(x_i-x_j). \]

Define population averages and covariance, including the divisor n:

\[ \bar a_R=\frac1n\sum_Ra_i,\qquad \bar q_R=Q_R/n,\qquad \operatorname{Cov}_R(a,q) =\frac1n\sum_R(a_i-\bar a_R)(q_i-\bar q_R). \]

Expanding the centered product gives the exact identity

\[ \boxed{\quad S_R:=\sum_R\dot\theta_i =k\left[\bar a_R Q_R+n\operatorname{Cov}_R(a,q)\right]. \quad} \]

The arithmetic regional mean rate is S_R/n. Replacing each a_i by its regional mean discards the displayed covariance; it is exact only when that term vanishes at the state in question. This does not make its future value predictable from the regional mean or cut. Unlike the already implemented weighted phase-cut balance, the arithmetic rate retains correlation between local form response and local phase mobility.

An edge identity separates internal and boundary contributions¶

Pair the two orientations of each internal edge, counting that edge once. The same quantity satisfies the discrete Green identity

\[ \sum_Ra_iq_i =\sum_{\{i,j\}\in E(R)}(a_i-a_j)(x_i-x_j) +\sum_{\substack{i\in R,\ j\notin R\\\{i,j\}\in E}} a_i(x_i-x_j). \]

Consequently

\[ n\operatorname{Cov}_R(a,q) =\sum_{\{i,j\}\in E(R)}(a_i-a_j)(x_i-x_j) +\sum_{\substack{i\in R,\ j\notin R\\\{i,j\}\in E}} (a_i-\bar a_R)(x_i-x_j). \]

The correction contains both internal mobility/form alignment and variation of mobility along the boundary. It is therefore not generally a purely internal source. Both contributions are signed; neither is a new pressure channel, an assigned regional energy derivative or a dissipation theorem. The shared cut owner supplies the existing outward orientation; no induced subgraph normalization or second definition of the cut is needed.

Exact represented accounting and a bound without a chosen tolerance¶

For a captured engine field, hats below denote exact rational readings of materialized scalars. Retain hat q=B*hat x from the exact work owner before its separate float materialization; compute hat a_i=hat nu_i/hat H_i and hat k=hat w/hat beta in rational arithmetic. Let hat r_i be the captured phase rate. The represented-rate residual is

\[ \delta_R=\sum_R(\widehat r_i-\widehat k\widehat a_i\widehat q_i), \qquad \widehat S_R=\widehat k \left[\overline{\widehat a}_R\widehat Q_R +n\operatorname{Cov}_R(\widehat a,\widehat q)\right]+\delta_R. \]

This equality retains actual rate rounding rather than declaring it zero. It does not bound trigonometric materialization errors in H, errors relative to the ideal irrational equilibrium, or errors of an evolved trajectory. The read-out is a model-rate observation at one captured state, not a measured temporal derivative or evidence that a step occurred.

With population variance Var_R(a)=sum_R(a_i-abar_R)^2/n, Cauchy--Schwarz gives

\[ \left[n\operatorname{Cov}_R(a,q)\right]^2 \le n^2\operatorname{Var}_R(a)\operatorname{Var}_R(q). \]

Thus the exact represented report can check the squared bound

\[ \left[\widehat S_R-\delta_R -\widehat k\overline{\widehat a}_R\widehat Q_R\right]^2 \le\widehat k^2n^2 \operatorname{Var}_R(\widehat a)\operatorname{Var}_R(\widehat q). \]

No tolerance, fitted coefficient or rounded square root is needed. Uniform mobility or uniform q makes the correction vanish instantaneously; zero covariance can also occur without either uniformity. The bound still consumes the internal state, so it does not supply an autonomous reduced law.

Zero capacity gives a_i=0 without division by capacity; it does not remove the phase-domain requirement H_i>0. For a singleton region the covariance is zero. For full support the outward cut is empty and Q_R=0, but S_R=k*n*Cov_R(a,q) may be nonzero. Common phase rotation is a symmetry, not a proof of conservation of the arithmetic phase sum. State-dependent weighted phase rates likewise do not integrate to a conserved weighted phase total without additional terms. These signed instantaneous rates establish neither persistent drift nor a monotone structural clock. Phase sums along a trajectory require continuous lifts; branch jumps in stored wrapped angles are not the derivatives in these identities.

The same ten-coordinate witness changes the regional mean response¶

In section 4's paired-ring witness the complete left form gradient is q_L=epsilon*(-2,3,-2,-2,3). Both bridge endpoints have form zero, hence Q_L=Q_R=0 in both states, and all right phase rates vanish. With sinc(t)=sin(t)/t, continuously extended by sinc(0)=1, the left metrics are

\[ H_0=\pi[1+2\cos(\kappa+t)],\qquad H_1=H_4=2\pi c\operatorname{sinc}(t), \]

\[ H_2=H_3=2\pi\cos(\kappa-t/2)\operatorname{sinc}(t/2). \]

These follow directly from each neighbor-phasor resultant. They give

\[ \overline{\dot\theta}_L(t)=\frac{k\epsilon}{5\pi}F(t),\qquad F(t)=-\frac2{1+2\cos(\kappa+t)} +\frac3{c\operatorname{sinc}(t)} -\frac2{\cos(\kappa-t/2)\operatorname{sinc}(t/2)}. \]

The formula holds throughout the stated acute interval. The exact relation c*(1+2c)=1/2 yields

\[ F(0)=2,\qquad F'(0)=\sin\kappa\left(\frac1{c^2}-\frac4{h^2}\right) =4\sqrt5\sin\kappa>0. \]

Here the derivative of F is with respect to the supplied phase displacement, not evolution time. Consequently

\[ \Delta\overline{\dot\theta}_L =\frac{4k\epsilon\sqrt5\sin\kappa}{5\pi}\,t+O(\epsilon t^2)>0 \]

for sufficiently small positive t and epsilon>0. This local sign conclusion does not assert a sign theorem over the whole acute interval. The port-rate change from section 4 is negative, while this regional mean-rate change is positive. Identical ten-coordinate observations and identical zero cuts thus coexist with different aggregate responses. Here the boundary current itself vanishes, so the entire correction is internal. The existing geometric mobility explains the missing response without completing a nonlinear closure or introducing a new evolution mechanism.

6. The ten coordinates and their current rates still do not close¶

Let F(z) denote the complete admitted vector field on form and local phase deviations, and keep the same linear observation C. The proposed augmented observation is

\[ Z(z)=(Cz,CF(z)),\qquad \dot Z(z)=(CF(z),C\,DF(z)F(z)). \]

Its second block is a model-derived rate at a supplied full state, not an independently measured history. A closed autonomous law for Z on an open neighborhood would require the displayed derivative to agree whenever Z agrees. The following exact pair violates that condition arbitrarily close to the prepared equilibrium.

Hidden form orientation changes the metric's next response¶

Put s=sin(kappa)>0 and introduce the left reflection-odd form direction

\[ \zeta_L=(0,1,0,0,-1,0,0,0,0,0)^T. \]

Keep phases exactly theta_* in both states and choose

\[ x^+=\epsilon\eta_L+\delta\zeta_L,\qquad x^-=\epsilon\eta_L-\delta\zeta_L,\qquad \epsilon,\delta>0. \]

Their right forms vanish; in deviation coordinates z^+=(x^+,0) and z^-=(x^-,0). At this phase lock g=0, so both vector-field rows depend exactly linearly on form:

\[ \dot x=-eD^{-1}Bx,\qquad \dot\theta=kH^{-1}Bx. \]

The two forms differ by a reflection-odd direction, and these fixed linear maps preserve that parity. Consequently Cz^+=Cz^- and CF(z^+)=CF(z^-) exactly: their complete augmented observations Z coincide.

Their left gradients are

\[ q_L^\pm=\epsilon(-2,3,-2,-2,3) \ \pm\ \delta(0,2,-1,1,-2), \]

while q_R=0. In particular q_0=-2*epsilon and H_0=pi*h coincide. At the lock the relative resultant is positive real. Since the derivative of sinc vanishes at zero, differentiation of the existing phase metric at any node gives

\[ \dot H_i=-\pi\sum_{j\sim i} \sin(\theta_{*,j}-\theta_{*,i})(\dot\theta_j-\dot\theta_i). \]

The bridge contributes zero sine at the port. Thus

\[ \dot H_0=-\pi s(\dot\theta_1-\dot\theta_4),\qquad \dot H_0^\pm=\mp\frac{2k\delta s}{c}, \]

because H_1=H_4=2*pi*c. The form row gives q_dot=-e*B*D^-1*q at this state; its port value is the same q_dot_0=5*e*epsilon for both preparations. Differentiating the phase row therefore yields the exact difference

\[ \Delta\ddot\theta_0 =-\frac{kq_0}{H_0^2}\left(\dot H_0^+-\dot H_0^-\right) =-\frac{8k^2\epsilon\delta s}{\pi^2ch^2}<0. \]

Every right-ring phase acceleration agrees: q_R=0, the change of form velocity at the donor port is zero, and hence the change of q_dot_R is zero. In particular the receiving port acceleration is -2*e*k*epsilon/(3*pi*h) in both states, and the other right accelerations vanish. The retained observable mu_v+p_L_v consequently has exactly the nonzero acceleration difference displayed above. Both supplied states lie in the strictly acute domain, and choosing epsilon,delta arbitrarily small places the pair in every open neighborhood of the equilibrium. No trajectory, Taylor remainder estimate or rank threshold is needed for this obstruction.

Equal scalar budgets do not remove the ambiguity¶

The two forms are reflections of one another on the left ring, while the primitive phase winding is held fixed. Their Dirichlet form energies and continuous losses e*sum_i(q_i^2/d_i) agree by that graph symmetry; their phase storage is identical. For each complete ring, and for full support, the scalar phase-response summaries in section 5 also agree: mean mobility, cut, mean q, covariance, both variances, the squared bound and the summed current phase rate. Mobility is reflection-even at the lock and q is reflected, so the relevant sums are unchanged. This statement does not apply to arbitrary subsets or erase the different retained nodal gradients. In particular, direct evaluation gives E_D=5*epsilon^2+2*delta^2 and loss=e*((43/3)*epsilon^2+5*delta^2) for both states.

There is no contradiction with reflection equivariance of the full model: a simultaneous reflection of form and phase would reverse the prepared left winding. This pair reverses only the hidden form orientation relative to that fixed winding. Its different acceleration follows from the already present phase metric, not from a new primitive variable, force or physical spin interpretation.

Scope of the negative result¶

At equilibrium the derivative of the augmented observation is the vertical stack DZ=[C; C*J] and has rank ten, since CJ=G*C. This is an equilibrium rank statement, not a claim about generic nearby rank. The augmented vector has twenty entries, already exceeding the eighteen-dimensional full-state offset quotient; its entry count alone establishes neither compression nor predictive sufficiency. The equal-fiber counterexample, independently of those counts, rules out an autonomous law for this Z on a full open neighborhood of the equilibrium.

The proof concerns the ideal smooth law and exact trigonometric lock. Rounded runtime preparations need not be exact locks, and finite differences do not convert their near-equalities into these exact fibers. The model-rate and acceleration conclusions also do not authenticate measured temporal evidence. Pure odd invariant families and other specially restricted domains remain separate from the open-domain obstruction.

This closes the selected state-plus-rate test without extending a derivative ladder. Retain the full engine state for nonlinear prediction unless a different state reduction or a memory construction is justified separately. The counterexample does not prove that every reduction is impossible or that the full quotient is mathematically minimal; diffusion-only memory results cannot supply the missing joint-law justification.

The subsequent joint memory derivation retains that initial hidden state and its nonlinear forcing. Its prepared-even finite-time approximation theorem is separate from an exact autonomous closure of this observation.

7. Shared computation and evidence boundaries¶

derive_linear_observation now owns exact invariant-row selection for a supplied rational generator z'=Jz. It returns minimal rows, lift, output map and reduced generator with checked CT=I, CJ=GC, O=DC, complete rank progression and an operational rank-call budget. The existing affine diffusion realization delegates to it, retaining its x'=-Ax+b sign, original basis, API and resource diagnostics. No diffusion reference is fabricated for this joint law. Matrix product and inverse arithmetic share a mathematical owner, with historical import paths preserved.

The static instrument checks the displayed full matrices, natural coordinates and exact rank on explicit rational coefficient probes. Its console example declares c=inverse_pi=1/3, e=w=1/2, beta=1, using Python Fraction arithmetic, fixed node/output order and no randomness or trajectory. Those rational values are not the ideal trigonometric constants. The proof above uses their actual irrational values through positivity and symmetry; rational probes do not supply that proof or a certified runtime approximation.

Controls also compare the independently assembled ideal Jacobian with static native-field finite differences and check the nonlinear witness. Floating tolerances in those implementation checks are distinct from exact rank, nonlinear identities or certified error bounds. No existing research artifact is rewritten.

The same instrument's analyze_state_rate_obstruction reuses the graph and Jacobian assembly to check section 6 on declared rational c,s,inverse_pi. It retains complete deviation states, rates, accelerations, their projections and equal storage/loss. At the lock, F=J*(x,0) but acceleration also includes the derived metric-variation term; applying J twice alone would miss the obstruction. Zero even or odd amplitude gives an inconclusive pair, not a closure theorem. The native controls use epsilon=1/32, delta=1/64, e=w=1/2, beta=1, fixed node order 0,...,9, and binary64 arithmetic. Central probes of the field at z +/- h*F(z), h=2^-10, check the acceleration with relative tolerance 2e-7 and absolute tolerance 2e-12. They do not call an integrator or claim a derivative of rounded machine arithmetic. These tolerances check consistency with the ideal derivative, not certified ODE error. The existing SDK's full centered form vectors distinguish the pair, so no state was added to the engine to recover the missing information.

The engine's exact phase_mobility and phase_rate_rounding_defect retain the arithmetic used by its phase row. The shared detached observer consumes them and the existing cut to expose region.phase_response, including its covariance and squared bound. The SDK delegates to that owner and the exact exporter retains the new fields; no report changes the law. The API contract owns field names and availability. Independent static controls check the analytic path case, heterogeneous capacity, exact gradients before floating materialization, edge accounting, zero-capacity and lossless limits, sign reversal, partitions and the mixed-state witness. They add no trajectory.

The result supplies a useful reduced local description and identifies its precise failure beyond first variation. It establishes neither autonomous fractal nesting, nonlinear regional closure nor physical material constituents.

Section link directory¶

Published section links retain their original targets; each entry routes to the responsible proof chapter.

8. One supplied bridge: sufficient interface and endpoint-only obstruction¶

Definitions, proof and controls are maintained in this chapter.

9. Support-event budgets and the missing occurrence law¶

Definitions, proof and controls are maintained in this chapter.

10. Passive bridge relocation with a shared recovery domain¶

Definitions, proof and controls are maintained in this chapter.

11. Support-law closure: restrictions do not select occurrence¶

Definitions, proof and controls are maintained in this chapter.

12. Nodal reorganization and connection in one action¶

Definitions, proof and controls are maintained in this chapter.

13. Precontact rhythm, phase agreement and sustained locking¶

Definitions, proof and controls are maintained in this chapter.

14. Connection mechanisms: mediation, reinforcement and primitive birth¶

Definitions, proof and controls are maintained in this chapter.

15. Sufficient conditions for a restoring effective link¶

Definitions, proof and controls are maintained in this chapter.

16. Formation-enabling support and later pattern detachment¶

Definitions, proof and controls are maintained in this chapter.

17. Storage admission for a formed source and an initially uniform receiver¶

Definitions, proof and controls are maintained in this chapter.

18. Regular-domain geometry of the fixed source/receiver preparation¶

Definitions, proof and controls are maintained in this chapter.

19. A bounded continuous response of the fixed regular preparation¶

Definitions, proof and controls are maintained in this chapter.

20. Fixed-IVP continuation and the remaining target budget¶

Definitions, proof and controls are maintained in this chapter.

21. A sharp collective energy envelope and transition barrier¶

Definitions, proof and controls are maintained in this chapter.

22. Complete reflected regular equilibria and full-network stability¶

Definitions, proof and controls are maintained in this chapter.

23. Finite-time zero-resultant access in the reflected law¶

Definitions, proof and controls are maintained in this chapter.

Smooth-sine proof owner¶

Sections 24–32 are maintained in Smooth-sine pattern dynamics. The links below preserve the original section and subsection targets.

24. Equilibrium geometry of components joined by bridges¶

Definitions, proof and controls are maintained in the smooth-sine owner.

25. Cycle periods constrain joint composition¶

Definitions, proof and controls are maintained in the smooth-sine owner.

26. Storage below every sector face forces capture¶

Definitions, proof and controls are maintained in the smooth-sine owner.

27. Prepared form can acquire a nonzero phase sector¶

Definitions, proof and controls are maintained in the smooth-sine owner.

28. Exact form, phase and memory representations¶

Definitions, proof and controls are maintained in the smooth-sine owner.

29. A controlled fast-form and slow-phase comparison¶

Definitions, proof and controls are maintained in the smooth-sine owner.

30. Full-state capture from controlled phase geometry¶

Definitions, proof and controls are maintained in the smooth-sine owner.

31. A fixed original preparation budget forces consensus at small ratio¶

Definitions, proof and controls are maintained in the smooth-sine owner.

32. Equal original budgets do not determine the acquired geometry¶

Definitions, proof and controls are maintained in the smooth-sine owner.