Sine relative state, recovery and conservative identity¶

Exact relative coordinates and bracket, whole-set recovery with a live environment, and the separate zero-loss protection and recurrence result.

Part of Native pattern reduction and memory. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

18. Relative pattern state without discarding internal dynamics¶

Return to the explicitly supplied normalized-sine law. Fix a finite connected simple unit graph with at least two nodes, its node identities, held capacities nu_i>=0, coefficients e>=0,w>0,beta>0 and one clock, with no forcing, clipping or support events. These are model premises, not an inferred origin of support. The comparison-law owner establishes its smooth complete field and storage identity; native trajectory records remain evidence of the native law.

An exact quotient by two common origins¶

Choose any node r as reference. Define

\[ u_i=x_i-x_r,\qquad z_i=\exp\!\bigl(i(\theta_i-\theta_r)\bigr),\qquad u_r=0,\quad z_r=1. \]

The remaining coordinates lie in R^(n-1) x (S^1)^(n-1). This is a global circular quotient, not a global choice of real angle. It identifies precisely one common form translation and one common phase rotation. It retains all other nodal coordinates, including hidden or intermediary nodes, and requires the same support, capacities, coefficients and clock. No regional averaging, node deletion, phase-magnitude projection or memory truncation occurs.

Put k_i=nu_i/d_i, a=w/pi, b=w/(beta*pi) and compute from the relative state

\[ \begin{aligned} q_i&=\sum_{j\sim i}(u_i-u_j),& S_i&=\sum_{j\sim i}\operatorname{Im}(z_j\overline z_i),\\ V_i&=k_i(-e q_i+aS_i),& \Omega_i&=b k_iq_i. \end{aligned} \]

These are the original nodal form and phase rates: all consumed differences are unchanged by the quotient. Differentiation gives the exact closed induced law

\[ \boxed{\dot u_i=V_i-V_r,\qquad \dot z_i=i(\Omega_i-\Omega_r)z_i.} \]

The phase row preserves |z_i|=1. A real relative lift rho_i=theta_i-theta_r instead obeys rho'_i=Omega_i-Omega_r, with rho_r=0. Such lifts are legitimate on a declared chart or along a continuous trajectory, but their independent full-turn representatives are not extra physical coordinates.

Conversely, a quotient solution and initial common origins reconstruct the original solution through

\[ \dot x_r=V_r,\qquad \dot\theta_r=\Omega_r,\qquad x_i=x_r+u_i,\qquad \exp(i\theta_i)=\exp(i\theta_r)z_i. \]

Thus holding the reference coordinate at zero inside the original nodal equations would change the model whenever its rates are nonzero. Subtracting the reference rows is essential. A zero-capacity node has V_i=Omega_i=0 in the original coordinates, but its relative coordinates can move when the reference moves. If a zero-capacity node is chosen as reference, both reconstruction rates vanish and the remaining relative rates equal their original rows. This special case does not justify freezing a general active reference.

The storage is a function of this complete relative state:

\[ E(u,z)=\frac12\sum_{\{i,j\}}(u_i-u_j)^2+ \beta\sum_{\{i,j\}} \bigl[1-\operatorname{Re}(z_j\overline z_i)\bigr], \qquad \dot E=-e\sum_i k_iq_i^2. \]

Common-reference velocity contributes no work because sum_i q_i=sum_i S_i=0. Removing the two origins therefore preserves the actual exchange and dissipation, rather than replacing them with a diagnostic score.

The same complete law has a dissipative Hamiltonian representation¶

On the full form/phase coordinates, the preceding storage has gradient grad(E)=(q,-beta*S). For the held diagonal matrix K=diag(nu_i/d_i) and b=w/(beta*pi), define

\[ J=\begin{pmatrix}0&-bK\\ bK&0\end{pmatrix},\qquad R_{\!d}=\begin{pmatrix}eK&0\\0&0\end{pmatrix}. \]

Direct multiplication, using beta*b=w/pi, gives exactly

\[ (\dot x,\dot\theta)=(J-R_{\!d})\nabla E. \]

Here J is constant and skew, so its coordinate derivatives vanish and its bracket satisfies the Jacobi identity. The angular coordinate fields are well defined on the phase torus; no global real phase lift is needed for the bracket of smooth periodic functions. Zero capacities make this Poisson structure degenerate rather than invalid. R_d is positive semidefinite, and E'=-grad(E)^T R_d grad(E)=-e*q^T K q recovers the exact loss.

This is an unforced dissipative Hamiltonian instance of the declared complete sine model. Network energy storage, interconnection and dissipation already have an established general framework; see van der Schaft and Maschke, Port-Hamiltonian systems on graphs. That reference supplies context, not a proof of this particular TNFR identity, which follows from the multiplication above. In contrast to an auxiliary Hamiltonian analogy, the displayed matrices reproduce both consumed evolution rows. The result remains restricted to held support and capacities and this pressure/phase law: it neither gives every TNFR runtime a Poisson structure nor identifies the storage with physical energy. Such a representation alone establishes no novelty, physical validation or uniquely selected microscopic law.

For strictly positive capacities this tensor also explains the form invariant within the selected law. Put rho=K^-1*1, W=sum_i rho_i and Q_x=rho^T*x. Its Hamiltonian vector field is

\[ J\nabla Q_x=(0,b\mathbf1). \]

Thus Q_x/b generates common phase rotation. Rotation invariance of the storage gives {Q_x,E}=0; the dissipative term also contributes zero, since rho^T eKq=e*sum_i q_i=0. This recovers conservation of Q_x from the same complete field. It does not derive conservation from phase symmetry alone: choosing the constant reciprocal tensor already fixes an additional constitutive structure that the native Arg law does not share.

On a continuous real phase lift define Q_theta=rho^T*theta. Then

\[ J\nabla Q_\theta=(-b\mathbf1,0),\qquad \{Q_x,Q_\theta\}=-bW\ne0. \]

The lifted quantity generates a common form translation with the displayed sign, but is not a globally single-valued observable on the phase torus. Neither quantity is a Casimir: their vector fields are nonzero, even though both are conserved by this particular storage and loss. These identities are conditional interpretations of the existing bracket, not independent selection of the bracket, pressure or a physical charge. The static matrix controls check the generators, their nonzero mutual bracket and both consumed rows.

Reconstruction from conserved means when every capacity is positive¶

For nu_i>0, set omega_i=d_i/nu_i and W=sum_i omega_i. Reciprocity gives

\[ \sum_i\omega_iV_i=-e\sum_iq_i+a\sum_iS_i=0,\qquad \sum_i\omega_i\Omega_i=b\sum_iq_i=0. \]

Hence the weighted form mean M_x and the weighted mean M_theta of a chosen continuous phase lift are constant. Reconstruction can then be written algebraically:

\[ x_r(t)=M_x-\frac1W\sum_i\omega_i u_i(t),\qquad \theta_r(t)=M_\theta-\frac1W\sum_i\omega_i\rho_i(t). \]

The lifted mean is not a single-valued scalar on the phase torus. Changing initial phase representatives changes its representation consistently, without changing the reconstructed circular state. These formulas are unavailable at zero capacity; the previous direct reconstruction remains valid.

There are no nonzero uniformly drifting relative equilibria when every capacity is positive. A stationary quotient would require V_i=v and Omega_i=omega for every node. The two weighted identities force v=omega=0. The phase row then gives q=0 and the form row S=0. Thus the stationary relative patterns are exactly uniform form with a critical phase geometry. A common pulse or uniform rotation is not generated by merely changing the reference.

What observation uncertainty cancels, and what remains¶

At one common actual time, suppose reported form coordinates are y_i=x_i+C_x+epsilon_i and consistently lifted phases are psi_i=theta_i+C_theta+eta_i. The shared offsets can be unknown and can vary between synchronous captures. Taking relative coordinates first cancels them exactly:

\[ y_i-y_r=u_i+\epsilon_i-\epsilon_r,\qquad \psi_i-\psi_r=\rho_i+\eta_i-\eta_r. \]

If only separate bounds |epsilon_i|<=delta_i are supplied, the sharp independent-error radius for an anchored form difference is delta_i+delta_r; the phase statement is identical in its admitted lift. There is no basis for subtracting this uncertainty merely because the common origin was removed.

The shared reference error also creates dependence between relative coordinates. For an edge i--j, subtract before interval evaluation: its error is epsilon_i-epsilon_j, with sharp radius delta_i+delta_j. Subtracting two separately expanded anchor boxes would produce the valid but unnecessarily broad radius delta_i+delta_j+2*delta_r. An explicit common-error model permits algebraic cancellation; matching marginal error sizes alone do not prove such dependence. Circular uncertainty must retain its phase-set or justified lift interpretation, including any branch ambiguity.

Synchronous means the same actual time, not merely matching nominal timestamps. With different acquisition jitter, both the nodal state and a moving sensor reference may be evaluated at different times. Then C(t_i)-C(t_r) need not vanish. Node-speed bounds control the former error; an independently justified reference-drift bound is needed for the latter. The prior sample budget bounds the declared coordinate and clock, not an undeclared moving instrument reference.

Likewise, the existing hidden-state inverse consumes original nodal rates, not rates relative to a moving node. Supplying V_i-V_r,Omega_i-Omega_r unchanged to that inverse can produce a consistent but wrong reconstruction. Consider a three-node star with ports r,p of unit capacity and one hidden node h:

\[ (x_r,x_p,x_h)=(0,1,2),\qquad (\theta_r,\theta_p,\theta_h)=(0,\delta,0),\qquad \sin\delta\ne0. \]

Its original port rows are

\[ (V_r,\Omega_r)=(2e,-2b),\qquad (V_p,\Omega_p)=(e-a\sin\delta,-b). \]

The anchored rows are therefore (0,0) at r and (-e-a*sin(delta),b) at p. Those are exactly the absolute port rows of a different hidden state x_h=0,theta_h=0 with the same visible coordinates. The observation geometry has rank two, so the original inverse can identify this phantom form instead of the actual value 2. The issue is the missing reference velocity, not loss of dynamics in the complete quotient. Restore the reference rows with their evidence or derive a separately admitted relative inverse; a relabeled rate dictionary does neither.

Only one global origin of each kind is redundant on this connected model. Removing independent origins from several coupled regions would discard their relative offsets, which drive interaction across the connecting edges. Likewise, omitting a hidden state, changing capacity or losing a supplied boundary input is not this symmetry. The previous causal-memory and hidden-state requirements remain in force under a change of reference.

Winding is retained information, not an unconditional invariant¶

The quotient determines every edge phase ratio z_j*conjugate(z_i). Away from antipodal edges, it therefore determines the principal phase increment on each oriented edge and the winding of each supplied oriented cycle. Common rotation changes none of them. At an antipodal ratio -1 the two limiting increments +pi and -pi meet: a principal-increment winding cannot be continuously assigned through that boundary.

The sine law itself remains smooth there. Consequently an antipodal crossing can change winding without a singular nodal state, deleted edge or new event. A snapshot winding and positive initial margin alone do not establish its indefinite preservation. Uncertain edge sets touching that boundary require an unresolved winding or separate branch evidence. Temporal preservation needs an invariant region, such as the sufficient local recovery domain below.

A whole-set sufficient recovery criterion under the sine law¶

One concrete maintained-pattern identity is the common-origin orbit of an exact critical phase geometry theta_* with uniform form. Assume now e,w,beta>0 and every held nu_i>0. Require S(theta_*)=0 exactly and all reference principal edge increments delta_*,ij strictly acute. A small computed residual alone does not establish this premise.

Let L be the unit graph Laplacian, lambda_2(L)>0 its spectral gap, and P=I-11^T/n. In a declared continuous deviation lift from theta_*, put

\[ Z^2=\|Pu\|^2+\|P(\rho-\rho_*)\|^2,\qquad \mathcal E=E(u,z)-\beta V_\phi(\theta_*). \]

This norm uses the declared model coordinates; it is not a universal physical metric or a diagnostic definition of identity. With m=min_edges(pi/2-|delta_*,ij|), choose

\[ 0<r<\frac m{\sqrt2},\qquad c_r=\min_{\{i,j\}}\cos(|\delta_{*,ij}|+\sqrt2r)>0,\qquad \kappa_r=\frac{\lambda_2(L)}2\min(1,\beta c_r). \]

Let U be an admitted state uncertainty set in this relative chart, with all its internal coordinates retained and the declared fixed support, coefficients and capacities unchanged. The sufficient conditions are

\[ \boxed{\sup_{U}Z<r,\qquad \sup_{U}\mathcal E<\kappa_r r^2.} \]

They imply that every actual state in U stays in this phase chart and converges to the reference geometry modulo common origins. The set need not be a singleton; correlated uncertainty may be conservatively enclosed by a larger set if that entire enclosure passes the inequalities.

The geometric barrier kappa_r does not depend on the positive capacity values. With support and coefficients still fixed, the same criterion consequently holds for each held capacity vector in an independently admitted strictly positive family. This covers capacity uncertainty without choosing its midpoint. It does not supply a uniform decay rate as capacities approach zero, and it does not admit a zero capacity by continuity of a positive-capacity theorem.

Proof of trapping and convergence. Inside Z<=r, each edge's phase-deviation difference is bounded by sqrt(2)*r. The line segment from the reference remains acute, and its cosine Hessian is bounded below by c_r L on the common-phase quotient. Criticality removes the linear phase-storage term. Taylor's formula and the graph spectral gap give

\[ \mathcal E\ge \frac{\lambda_2(L)}2\|Pu\|^2+ \frac{\beta c_r\lambda_2(L)}2 \|P(\rho-\rho_*)\|^2 \ge\kappa_r Z^2. \]

The exact sine loss mathcal E'=-e*q^T K q<=0, with K=diag(nu_i/d_i)>0, prevents a first exit through Z=r. The trajectory remains in a compact interior quotient sublevel. On its largest invariant zero-loss subset, q=0, hence Omega=0. Preserving q=0 further requires LKS=0, so KS=c*1. But sum_i S_i=0 and K>0 give c*sum_i(1/k_i)=0; therefore c=0 and S=0. The positive phase Hessian on this convex local chart makes theta_* its unique critical phase geometry modulo rotation. LaSalle's principle thus gives convergence to that orbit.

Local exponential recovery follows from the same sine rows, rather than from a transferred native trajectory. Choose orthonormal columns R spanning 1^perp and define B=R^T L R>0, A=R^T K R>0 and C=R^T Hess(V_phi)(theta_*) R>0. The quotient Jacobian is

\[ J=\begin{pmatrix}-eAB&-aAC\\ bAB&0\end{pmatrix}. \]

In storage coordinates it is similar to [[-D,-H],[H^T,0]] with D=e*B^(1/2)*A*B^(1/2)>0 and H=(a/sqrt(beta))*B^(1/2)*A*C^(1/2) invertible. For an eigenvector (v,w), Re(lambda)*(|v|^2+|w|^2)=-v^*D*v. A zero real part would force v=0 and then H*w=0, contradicting a nonzero eigenvector. All quotient eigenvalues have negative real part; smoothness gives local exponential recovery. The weighted-mean reconstruction then also determines the finite limiting common origins.

The positive-dissipation and positive-capacity premises matter. At e=0 a nonzero local excess storage is conserved and cannot converge to its zero value at the target. Two inactive nodes with distinct frozen forms can preclude uniform-form recovery however small their difference. Positive stiffness or the above acute criterion is sufficient, not a classification of every stable sine geometry or its full basin.

This theorem closes a bounded mechanism question: irrelevant common origins can be removed exactly, while all sufficiently small admitted relative perturbations in the stated domain recover under the supplied nodal law. It does not prove spontaneous entry into that domain, select the law, create support or identify a pattern with matter. The inequalities are a mathematical whole-set criterion; their application to a concrete uncertain pattern requires outward admission of every premise, as in the cycle family below.

Computable whole-set recovery for an exact cycle twist¶

Supply an ordered cycle (v_0,...,v_(n-1)) containing every node exactly once, n>=3, whose consecutive and closing edges equal the complete support. Extra edges, a selected cycle inside a larger network and an unspecified orientation do not meet this contract. Let the target winding ell be an integer with 4*|ell|<n and define

\[ \alpha=\frac{2\pi\ell}{n},\qquad \theta^*_{v_j}=j\alpha\pmod{2\pi}. \]

This is an exact symbolic target, including consensus ell=0. Each node has neighboring increments +alpha,-alpha, so S_i(theta*)=sin(alpha)+sin(-alpha)=0 identically. Its Hessian is cos(alpha)*L and cos(alpha)>0. No rounded phase vector or small numerical residual is used to recognize an equilibrium. The cyclic Fourier modes of the unit Laplacian have eigenvalues 2-2*cos(2*pi*j/n), giving the exact nonzero gap

\[ \lambda_2(L)=2-2\cos(2\pi/n). \]

Chart choice and cancellation before interval evaluation¶

The caller supplies one integer turn m_j per observed phase lift; zero turns are an explicit permissible choice. Define the deviation lift h_j=theta_(v_j)+2*pi*m_j-j*alpha. Turns choose a chart for the same circular data; they are not automatically fitted to force admission and do not reduce an observation error.

For original synchronous observations with independent residual bounds epsilon_i around their nominal coordinates, every pair satisfies

\[ \begin{aligned} X_{ij}&= (x^{\rm nom}_{v_j}-x^{\rm nom}_{v_i}) +[-\delta_i-\delta_j,\delta_i+\delta_j],\\ H_{ij}&= (\theta^{\rm nom}_{v_j}-\theta^{\rm nom}_{v_i})+ \pi\left[2(m_j-m_i)-\frac{2\ell(j-i)}n\right] +[-\epsilon_i-\epsilon_j,\epsilon_i+\epsilon_j]. \end{aligned} \]

Here delta_i denotes the form residual radius; epsilon_i denotes the phase residual radius. Common form and phase origins cancel exactly. The entire rational coefficient of mathematical pi is formed before outward interval evaluation; separately materializing target angles would introduce avoidable cancellation error.

The exact identity ||Pf||^2=(1/n)*sum_(i<j)(f_j-f_i)^2 gives the sufficient squared norm bound

\[ Z^2\le Z_+^2:= \frac1n\sum_{i<j} \left(\sup X_{ij}^2+\sup H_{ij}^2\right). \]

Pairwise interval bounds can be conservative because the same residual appears in several pairs, but they never add an artificial reference-node error. They retain every node and certify the whole declared observation class rather than its midpoint.

Quadratic excess storage without subtracting twist energy¶

Orient every cycle edge from v_j to v_(j+1 mod n) and let d_j=h_(j+1 mod n)-h_j. The closing edge has the same circular target increment alpha, even though its target real-lift difference is -(n-1)*alpha. Its full-turn difference must be retained when computing d_(n-1). The deviations telescope exactly: sum_j d_j=0.

Since every target edge has the same sine, the entire linear term in phase excess storage cancels:

\[ \begin{aligned} V_\phi(\theta)-V_\phi(\theta_*) &=\sum_j[\cos\alpha-\cos(\alpha+d_j)-\sin\alpha\,d_j],\\ \cos\alpha-\cos(\alpha+d)-\sin\alpha\,d &=d^2\int_0^1(1-s)\cos(\alpha+sd)\,ds. \end{aligned} \]

This is an exact integral remainder for either sign of d. It avoids subtracting two nonzero twist energies and has quadratic rather than linear sensitivity to small interval widths.

Let H_j enclose the edge deviation d_j and set

\[ c_j^+= \sup\cos\!\left(\alpha+\operatorname{hull}(0,H_j)\right). \]

In computation, the cosine interval's upper endpoint is an outward upper bound for this supremum. The segment contains alpha, so c_j^+>=cos(alpha)>0, and c_j^+<=1. Therefore

\[ \mathcal E\le Q_+:= \frac12\sum_j\sup X_{j,j+1}^2+ \frac\beta2\sum_j c_j^+\sup H_j^2. \]

Using c_j^+=1 remains safe but less informative. No individual Taylor term is presumed independent of another; their upper bounds enclose the complete sum. This upper-bound inequality is valid before basin admission. Nonnegative excess storage follows only after the appropriate acute neighborhood is established, not from clipping a possibly negative general excess to zero.

For supplied r>0, the cycle constants are

\[ m=\frac\pi2-|\alpha|,\qquad c_r=\cos(|\alpha|+\sqrt2r),\qquad \kappa_r=\frac{\lambda_2(L)}2\min(1,\beta c_r). \]

Outward pi, square-root and cosine bounds provide certified lower endpoints. Admission requires sqrt(2)*r<m, a strictly positive resolved lower bound kappa_-<=kappa_r, and

\[ \boxed{Z_+^2<r^2,\qquad Q_+<\kappa_-r^2.} \]

Positive dissipation and every strictly positive held capacity remain separate hypotheses. Failure to resolve these sufficient inequalities is unavailability of this certificate, not proof that the state cannot recover or that every point in its uncertainty set leaves the basin.

Original observation sets and propagated boxes are different inputs¶

The preceding pair formulas use the original per-node residuals once. Their conclusion covers that synchronous observation set. A Cartesian outer box from full-state propagation generally contains additional states because residual correlations have been lost. It must be checked in full, even if the tighter original set passed.

For such a propagated box, form each X_ij by subtracting the two full endpoint form intervals, and each H_ij by subtracting the two full phase intervals followed by the same exact target/turn coefficient. Apply the identical norm and remainder proof to these wider pair intervals. Use the original full endpoint boxes, before another subtraction of the moving reference; do not reuse the source's smaller edge bounds or its earlier storage bound.

The report's actual validated endpoint time identifies the state being certified. A partial forecast can therefore admit recovery starting at that earlier validated time, but cannot silently claim certification at an unreached requested time. Original observations provide exact held capacities. A full solver box may instead carry its final node's held capacity as an augmented interval: its whole interval must be strictly positive, while the other held capacities satisfy the same condition. The positive-family extension above then applies without selecting a capacity midpoint. A box including zero is outside this recovery theorem.

An explicitly deformed C5 observation class that passes¶

Choose the positively oriented unit C5, target ell=1, e=w=1/2,beta=1 and unit capacities. Supply no extra phase turns, r=1/16 and dyadic nominal observations

\[ x^{\rm nom}=(1/1024,-1/1024,0,0,0),\qquad \theta^{\rm nom}_{v_j}=5j/4. \]

Every node has form and phase residual radius 1/4096; both common origins remain arbitrary. These observations are not the exact target: the target increment remains alpha=2*pi/5, not 5/4. No trajectory or generated response is needed to admit this class.

The elementary bounds 25/8<pi<22/7 give 0<alpha-5/4<1/140. Put

\[ A=\frac1{140}+\frac1{2048},\qquad B=\frac1{35}+\frac1{2048},\qquad F=\frac5{2048}. \]

All form-pair magnitudes are at most F and all phase-deviation pair magnitudes at most B. Hence Z_+^2<=2*(B^2+F^2)=2184757/1284505600<1/256=r^2. The four nonclosing phase-edge deviations have magnitude at most A; the closing one has magnitude at most B.

Every phase interpolation segment lies between 31/25 and pi/2. The cosine upper Taylor bound at 31/25 gives c_j^+<=1-(31/25)^2/2+(31/25)^4/24<1/3 as an analytic upper bound. Consequently

\[ Q_+\le\frac52F^2+\frac{4A^2+B^2}{6} =\frac{59951}{308281344}<\frac1{2560}. \]

Furthermore lambda_2(L)>1, and alpha+sqrt(2)*r<377/280<27/20. The alternating lower bound cos(27/20)>=1-(27/20)^2/2+(27/20)^4/24-(27/20)^6/720>1/5 gives c_r>1/5 and kappa_r*r^2>1/2560. All strict recovery conditions therefore hold for this complete uncertain class. An implementation still records its own outward computed margins; the analytic inequalities are an independent admission proof, not substituted fixture values.

This result proves conditional recovery of an already supplied geometric identity despite nonzero relative observation uncertainty. It does not create the target winding, establish spontaneous entry into this neighborhood, select the microscopic law, add support or identify the cycle with a physical constituent.

19. Recovering patterns with a live retained intermediary¶

Two locally recoverable regions can belong to a single recoverable interacting geometry under the supplied sine law. The statement here retains the complete fine support and the intermediary's form, phase and capacity. It neither replaces the intermediary by a direct edge nor derives when a new edge appears.

An exact critical target on the full eleven-node support¶

Take cycles (0,1,2,3,4) and (5,6,7,8,9) and the two additional edges 0--10 and 5--10. Node h=10 is the intermediary, and nodes 0,5 are the ports. There are eleven nodes and twelve unit edges; each port has degree three and the intermediary degree two. For alpha=2*pi/5, specify

\[ \theta^*_{j}=\theta^*_{5+j}=j\alpha\pmod{2\pi} \quad(0\le j<5),\qquad \theta^*_{10}=0. \]

At an ordinary ring node the incoming sine terms are sin(alpha)+sin(-alpha)=0. The same cancellation holds at a port, and its additional intermediary edge contributes sin(0)=0. Both intermediary edges have zero phase difference. Therefore S(theta*)=0 exactly on the entire graph, not just in either isolated cycle. Uniform form completes a stationary state.

The phase Hessian assigns ring edges weight cos(alpha)>0 and the intermediary edges weight one. It is positive on the full common-phase quotient, including perturbations that move the two rings against each other or change the intermediary. Independent regional stability would not by itself prove this full-network result. For strictly positive held capacities and e,w,beta>0, Section 18 therefore supplies local exponential recovery of this complete geometry.

More generally, the shared exact phase-cycle reconstruction can admit a supplied rational-turn target on the full graph. It checks integral cycle periods, strict acute edges and symbolic cancellation of opposite sine terms. Only these algebraic facts are reused here. An unproved sine residual, an approximate equilibrium or that owner's separate dynamical interpretation cannot substitute for the full sine-law hypotheses.

For an exactly critical target with possibly different edge angles alpha_e, the excess-storage cancellation extends directly: sum_e sin(alpha_e)*(h_j-h_i)=grad(V_phi)(theta*)^T h=0. The quadratic remainder bound in Section 18 consequently applies edge by edge with its own target angle. No uniform twist or equal-capacity premise is needed for that cancellation.

An independent full-graph spectral-gap bound¶

For every unordered node pair choose a path P_ij with length l_ij. Cauchy--Schwarz and the pairwise centering identity give

\[ \begin{aligned} \|Pv\|^2 &=\frac1n\sum_{i<j}(v_j-v_i)^2\\ &\le\frac1n\sum_e C_e(v_{e^+}-v_{e^-})^2,\qquad C_e=\sum_{\{i,j\}:e\in P_{ij}}l_{ij}. \end{aligned} \]

Thus lambda_2(L)>=n/max_e C_e. This is a finite path-counting proof, not a numerical eigensolver assumption. For the stated support, the shortest paths give:

Edges, with the same pattern on the other ring	Path load `C_e`
`0--10` and `5--10`	`121`
`0--1` and `0--4`	`57`
`1--2` and `3--4`	`34`
`2--3`	`5`

For example, distances from a ring's nodes to its port are (0,1,2,2,1). The bridge's paths from that ring to the intermediary contribute 6+5=11; paths between the rings contribute 5*6+25*2+5*6=110, giving 121. Hence

\[ \lambda_2(L)\ge\frac{11}{121}=\frac1{11}. \]

With beta=1 and r=1/16, the earlier elementary cosine bound still gives c_r>1/5, because the largest target edge angle is 2*pi/5. The whole-network trapping barrier therefore satisfies kappa_r*r^2>1/28160. This analytic lower bound is independent of the engine's general rational quotient-gap certificate; a more conservative computed gap remains valid if its own margins pass.

A finite uncertain class within the interacting basin¶

Use default e=w=1/2,beta=1 and unit capacities. For both rings supply the dyadic nominal phases theta_j^nom=theta_(5+j)^nom=1287*j/1024; supply intermediary phase zero. The nominal form is x_0=1/4096 and zero elsewhere. Every node has form and phase residual radius 1/65536, and the two common origins may be arbitrary. Target turns remain exactly j/5 on the rings and zero at the intermediary. No phase turns are added to the nominal observations.

The elementary bounds 157/50<pi<22/7 imply |1287/1024-2*pi/5|<1/1024. Define

\[ R=\frac1{32768},\qquad A=\frac{33}{32768},\qquad B=\frac{129}{32768},\qquad F=\frac9{32768}. \]

Here R bounds a difference of two residual errors, A bounds each nonclosing ring-edge phase deviation, B bounds each closing one and every pair's phase deviation, and F bounds every form difference. The two port/intermediary phase deviations are at most R. The pairwise identity gives

\[ Z_+^2\le5(B^2+F^2) =\frac{41805}{536870912}<\frac1{256}=r^2. \]

All ring-edge interpolation segments lie between 31/25 and pi/2, so the independent Taylor bound used above supplies a cosine upper bound 1/3. Bridge segments can use the bound one. Three form edges touch the perturbed donor port; the other nine have only residual error. Therefore

\[ \begin{aligned} \mathcal E &\le\frac{3F^2+9R^2}{2} +\frac{4A^2+B^2}{3}+R^2\\ &=\frac{3563}{536870912} <\frac1{28160}<\kappa_r r^2. \end{aligned} \]

The entire synchronous uncertainty class is consequently within the full interacting recovery basin. The same argument covers any strictly positive held capacity vector; it gives no uniform recovery time as a capacity approaches zero. A computed certificate must retain its own full-graph gap and error bounds. In particular, the general quotient-gap owner is not claimed to return the stronger path bound 1/11; its separately computed uncertainty and barrier margins can still establish admission.

Exact nonlinear donor--intermediary--receiver onset¶

A separate, exactly specified preparation makes causality transparent. Use the exact target phases, nominal forms x_0=epsilon>0 and x_i=0 otherwise, ring capacities one and intermediary capacity mu>=0. This is not the preceding dyadic phase-observation class. All statements in this paragraph concern the actual nonlinear rows at this finite preparation, not a fitted small-amplitude derivative.

Write a=w/pi and b=w/(beta*pi). At time zero, q_h=-epsilon and S_h=0, giving

\[ \dot x_h=\frac{\mu e\epsilon}{2},\qquad \dot\theta_h=-\frac{\mu b\epsilon}{2}. \]

Every receiver-ring node initially has zero form and phase rate. At its port 5, differentiating the three incident contributions gives

\[ \dot q_5=-\dot x_h=-\frac{\mu e\epsilon}{2},\qquad \dot S_5=\dot\theta_h=-\frac{\mu b\epsilon}{2}. \]

The latter uses the intermediary edge cosine one and zero initial phase rates at both receiver neighbors. Retaining a general positive receiver-port capacity nu_5 for this identity yields

\[ \boxed{ \ddot x_5=\frac{\nu_5\mu\epsilon}{6}(e^2-ab),\qquad \ddot\theta_5=-\frac{\nu_5\mu e b\epsilon}{6}.} \]

The donor capacity does not enter this leading two-edge response: the intermediary already consumes the donor's initial form contrast. For mu>0,e>0 the receiver phase thus obeys

\[ \theta_5(t)-\theta_5(0)= -\frac{\nu_5\mu e b\epsilon}{12}t^2+O(t^3). \]

It is strictly negative for all sufficiently small positive times. The exact smooth law supplies this local onset; no numerical horizon, time-series evaluation or fitted remainder is asserted. At the default coefficients e^2-ab=(1-1/pi^2)/4>0, the initial receiver form curvature is positive as well.

If mu=0, the intermediary instead remains exactly at its initial form and phase. The unperturbed receiver twist with that fixed boundary is an exact stationary solution, so uniqueness makes the receiver stay unchanged for all time. This is a declared capacity intervention on the same existing support. It separates a causally active intermediary from a frozen one, not one unknown physical law from another.

Influence followed by recovery and a retained common-form record¶

For the exact-phase preparation, the full quotient norm and initial excess storage are Z^2=10*epsilon^2/11 and mathcal E=3*epsilon^2/2. Taking epsilon=1/4096 places them strictly below the same r=1/16 trapping conditions. Thus for every fixed mu>0 this causal response coexists with full-network recovery: both ring geometries and their intermediary recover to the supplied target. The perturbation is not required to destroy the pattern to transmit an effect.

The conserved weighted form mean gives an additional exact endpoint statement. The two rings have total degree 22 and unit capacity; the intermediary contributes weight 2/mu. Only the degree-three donor port initially carries form epsilon. Hence the limiting uniform form is

\[ x_i(\infty)=\frac{3\epsilon}{22+2/\mu} \quad\hbox{for every node and each fixed }\mu>0. \]

For mu=1 this is epsilon/8. The initial exact target phase has its target weighted lift mean, so the limiting common phase shift is zero in this local lift. The frozen-intermediary receiver instead keeps form zero forever. The positive-capacity comparison therefore supplies both a transient causal signal and a lasting common-form offset under the permanently retained support.

This offset is a conserved coordinate record; it is not a new winding, a new constituent or an independently identified physical memory. It lies in the common-origin coordinate removed by the relative quotient: distinguishing it requires the retained preparation frame or another admitted reference, not the final internal pair differences alone. It tends to zero as mu tends to zero. Neither the recovery theorem nor this limit supplies a uniform finite response time near that boundary.

A symmetry obstruction to forming the receiver twist¶

Recovery of the supplied receiver identity does not establish formation from a flat receiver. Reuse the exact automorphism mechanism from the orientation audit, now acting only on the receiver: R=(6 9)(7 8) fixes its port 5, the intermediary and all donor nodes. If paired receiver capacities are equal, this permutation preserves the full sine law.

A flat receiver phase and uniform receiver form are fixed by R, regardless of the donor or intermediary state. Equivariance and uniqueness keep that receiver reflection exact under the endogenous single-port drive. A winding-one or winding-minus-one receiver twist is not reflection-fixed; a common rotation cannot repair this because the port is fixed. Such an initial state therefore cannot converge to either of those receiver identities. Away from antipodal edges, its principal-increment receiver winding remains zero by paired edge cancellation.

This is a consequence of an already justified symmetry, not a new simulation campaign or a claim that asymmetric formation is impossible. A meaningful formation test must declare information that breaks this symmetry and satisfy the relevant storage/transition budget before evaluating a trajectory. Changing an initial state or held capacity supplies that information; it does not derive its origin. No new pressure term or operator selector is installed by the present recovery and causality results.

23. Conservative phase identity with nonlinear recurrence¶

Retain the law and distinguish trapping from recovery¶

Take the complete isolated unit cycle C_n, n>=3, with declared orientation and strictly positive held capacities. Use the complete normalized-sine law with e=0,w,beta>0 and no input, event, clipping or support change:

\[ \dot x=aKS(\theta),\qquad \dot\theta=bKLx,\qquad K=\operatorname{diag}(\nu_i/2),\quad a=w/\pi,\quad b=w/(\beta\pi). \]

The total storage E=x^TLx/2+beta*V(theta) is conserved. The existing acute recovery proof separates a geometric first-exit barrier from an attracting dynamics argument. Only the barrier is reused here. The latter requires e>0; it is not a recovery theorem for the present conservative law.

Choose an integer k with 4*abs(k)<n and the exact critical target

\[ \theta_{*,j}=\frac{2\pi k j}{n},\qquad \alpha=\frac{2\pi k}{n},\qquad E_*=\beta n(1-\cos\alpha). \]

Opposite sine currents cancel at each node, and all target edge increments are strictly acute. The nontrivial case n=5,k=1 is the winding-one identity under study; the same derivation also covers the other declared acute cycle twists.

A genuine phase chart with common origins retained¶

Let P=I-11^T/n and write the centered form as u=Px. Represent nearby circular phases by

\[ \theta=\theta_*+c\mathbf1+v\pmod{2\pi},\qquad v\in\mathbf1^\perp,\quad c\in\mathbb R/2\pi\mathbb Z. \]

Here c is one common circular phase origin, not a separately adjustable origin at each node. Choose r>0 such that

\[ |\alpha|+\sqrt2\,r<\frac\pi2. \]

The representation is injective for ||v||<r. To see this, suppose two centered deviations and common origins give the same phase state. Subtracting any two node equations gives

\[ (v_i-v_j)-(v'_i-v'_j)=2\pi(m_i-m_j) \]

for integers m_i. Its left-hand side has magnitude less than 2*sqrt(2)*r<2*pi, so all the integers agree. Centering then forces v=v' and equality of the common origins on the circle. The linear differential of the chart is invertible, so its image is open in the phase torus. The strict radius bound also supplies an injective chart on a slightly larger neighborhood of its closure. There is no phase-seam exit hidden inside the admitted ball.

Define the full relative norm in the same model coordinates as the recovery owner,

\[ Z^2=\|u\|^2+\|v\|^2. \]

For Z<=r the oriented wrapped edge increments are alpha+v_(j+1)-v_j, including the closing edge. They remain strictly acute, and their sum is 2*pi*k. Every state in this chart therefore has the same declared oriented winding k.

The coercive barrier is independent of dissipation¶

Let

\[ \lambda_2=2-2\cos(2\pi/n),\qquad c_r=\cos(|\alpha|+\sqrt2r)>0,\qquad \kappa_r=\frac{\lambda_2}{2}\min(1,\beta c_r)>0. \]

The phase segment from theta_* to theta_*+v remains acute throughout the closed radius ball. Criticality cancels its linear storage term, and its Hessian is bounded below by c_r L. The graph spectral gap thus gives

\[ \mathcal E:=E-E_* \ge\frac{\lambda_2}{2}\|u\|^2 +\frac{\beta c_r\lambda_2}{2}\|v\|^2 \ge\kappa_r Z^2 \qquad(Z\le r). \]

Choose an independently declared excess ceiling and weighted-mean interval,

\[ 0<h<\kappa_r r^2,\qquad m_{\rm lo}<m_{\rm hi}, \]

where

\[ m_\rho(x)=\frac{\sum_i\rho_i x_i}{\sum_i\rho_i},\qquad \rho_i=\frac{2}{\nu_i}>0. \]

The proposed full-state family is

\[ \boxed{ \mathcal U=\left\{ Z^2<r^2,\quad \mathcal E<h,\quad m_{\rm lo}<m_\rho(x)<m_{\rm hi} \right\},} \]

with the common phase origin c free on its circle. The initial radius condition cannot be omitted: a small energy relative to E_* outside this chart does not identify this phase pattern.

For every member of U, conservation of mathcal E prevents a first radius exit. At such an exit the coercive bound would require mathcal E>=kappa_r*r^2>h. The argument works in both time directions because energy and the weighted form mean are exactly conserved. The chart is retained, and within it

\[ Z(t)^2\le\frac{\mathcal E(0)}{\kappa_r} <\frac h{\kappa_r}<r^2 \qquad\text{for all real }t. \]

Every state in the family therefore retains acute edges and winding k indefinitely under the continuous declared law. This is trapping, not convergence to theta_*. For a nonzero excess energy, convergence to that zero-excess target is impossible under exact conservation.

Open positive-volume family and almost-everywhere motion¶

The family is open in the full 2n-dimensional state manifold. Indeed the phase chart is locally invertible, and form can be parameterized by (m_rho,u) with u perpendicular 1:

\[ x=\left(m_\rho-\frac{\rho^\mathsf Tu} {\rho^\mathsf T\mathbf1}\right)\mathbf1+u. \]

This is an invertible linear coordinate change. Uniform form with mean strictly inside the interval and the exact target phase lies inside every displayed strict inequality; a full open neighborhood does too. Thus the family has positive ambient product volume, not merely volume within a synchronized or fixed-energy surface.

Its volume is finite. Positive mean weights and the radius give

All forms therefore lie in a bounded box, and all phases lie on their compact torus. The open family need not itself be compact; its closure in the retained chart and form box is compact. The strict energy barrier keeps its trajectories away from the chart's radius boundary, while the conserved mean keeps each trajectory at its own interior mean value. The smooth sine flow is complete and maps U onto itself in both time directions.

The full nonlinear recurrence theorem now applies to this invariant finite-volume family. Its divergence is zero, so almost every state of U, in ambient Lebesgue form times circular Haar phase measure, returns arbitrarily close to itself along arbitrarily late times. The equilibrium subset has measure zero, since positive capacity forces equilibrium form to be uniform. Consequently almost every member has nonstationary recurrent motion while retaining this phase identity.

The quantifiers differ: every admitted member is trapped with fixed winding; only almost every member is certified recurrent by this measure theorem. Neither result selects a common period, return time, oscillation amplitude, physical frequency or attracting orbit. The prescribed support and target identity are not formed by this admission test.

Relative observations and the absolute mean slab are different evidence¶

An uncertain source can certify the shape conditions if its entire declared relative-state set satisfies

\[ \sup Z^2<r^2,\qquad \sup\mathcal E<h. \]

Reuse the existing owner for centered pairwise norms, exact target turns and the phase-energy Taylor remainder. Original observations retain their correlated common-origin-plus-residual set; a forecast uses its full validated endpoint box at the actual validated time. A requested future time is not certified merely by a partial forecast. No midpoint, redefined phase origin or changed law supplies a missing strict bound.

These shape inequalities prove trapping for every compatible state even if the common form origin is unspecified. However, SineRelativePattern permits an arbitrary common form shift. Its entire observation set therefore cannot be declared inside a finite absolute m_rho interval by substituting the nominal mean. The finite-volume family and relative shape membership must be reported separately. Absolute-family membership requires independent mean information, or an explicitly conditional intersection of the observation set with the declared mean slab.

The same separation applies to recurrence: neither relative trapping, absolute slab membership nor an error box makes its selected true state an almost-everywhere random draw. Singular preparations, fixed-energy subsets and finite numerical grids require separate evidence. A stationary target member has only trivial recurrence; the family theorem does not turn it into a pulse.

This combines two existing mechanisms under compatible conservative hypotheses: an all-state phase barrier and nonlinear volume-preserving recurrence. It establishes a supplied coherent identity compatible with persistent recurrent motion, not spontaneous formation, microscopic selection of zero loss or fractal inheritance.