Sufficient unordered pair state

Realizable pair invariants, internal-state reconstruction, unequal-capacity correlations and support-symmetry boundaries; global cancellation uses the declared fixed conservative doubled C5.

Part of Coarse-graining, coherence geometry and bridge results. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

7. Nonlinear sine inheritance with retained internal coordinates

This result uses the complete normalized-sine relational law, with e=0, w,beta>0, fixed unit support, strictly positive held capacities, one common structural clock, and no input, clipping or operator event. It extends the existing synchronized replica construction by retaining the coordinates transverse to that construction. It does not transfer the native resultant-pressure law or the independently supplied phase law used in earlier sections to this model.

7.1. Supplied support and an invertible description

Let a finite connected simple unit base graph have degrees d_i>0. Replace each node i by the ordered pair (i,+),(i,-). For every base edge {i,j} retain all four fine edges {(i,s),(j,t)}, with s,t in {+,-}, and no other edge. In particular, there is no edge within a pair. The fine degree is 2*d_i. Both members of pair i have the same held capacity nu_i>0; different pairs may have different capacities. The support and partition are supplied, not produced by the dynamics.

Use the exact invertible form coordinates and declared continuous phase lifts

\[ x_{i,\pm}=X_i\pm u_i,\qquad \theta_{i,\pm}=\Theta_i\pm\delta_i . \]

Here X,Theta describe the paired means and u,delta their internal differences. No fine node or coordinate has disappeared. Near synchronized phases, choose the pair chart |delta_i|<pi/2, so the two phases are not antipodal and Theta_i is their unambiguous circular midpoint. Beyond a chosen chart, real-lift averages are not a globally defined circular observation; another lift may change the midpoint by pi. The equations below hold on supplied continuous lifts, with the circular state obtained by projection, rather than by repeatedly guessing a phase unwrapping.

Put a=w/pi, b=w/(beta*pi) and Delta_ji=Theta_j-Theta_i. The fine nodal rows are

\[ \dot x_{i,s} =\frac{a\nu_i}{2d_i} \sum_{j\sim i}\sum_{t=\pm1} \sin(\Theta_j+t\delta_j-\Theta_i-s\delta_i), \]

\[ \dot\theta_{i,s} =\frac{b\nu_i}{2d_i} \sum_{j\sim i}\sum_{t=\pm1} (X_i+s u_i-X_j-t u_j). \]

Taking their half-sum and half-difference gives the complete nonlinear retained-coordinate law

\[ \boxed{\begin{aligned} \dot X_i &=\frac{a\nu_i}{d_i}\cos\delta_i \sum_{j\sim i}\cos\delta_j\sin\Delta_{ji},\\ \dot\Theta_i &=\frac{b\nu_i}{d_i}\sum_{j\sim i}(X_i-X_j),\\ \dot u_i &=-\frac{a\nu_i}{d_i}\sin\delta_i \sum_{j\sim i}\cos\delta_j\cos\Delta_{ji},\\ \dot\delta_i&=b\nu_i u_i . \end{aligned}} \]

The identities sin(A+B)+sin(A-B)=2*sin(A)*cos(B) and the cancellation of the two neighbor form differences prove these rows directly. They are not a tangent approximation or a fitted effective pressure. In particular, the coarse phase row is autonomous in X, but the coarse form row still consumes the internal phases.

7.2. Internal phase geometry modulates inherited coupling

The mean fine phasor in a pair is exactly

\[ Z_i=\frac{e^{i\theta_{i,+}}+e^{i\theta_{i,-}}}{2} =e^{i\Theta_i}R_i,\qquad R_i=\cos\delta_i . \]

In the nonantipodal midpoint chart, R_i>0 is the magnitude of this resultant. The inherited form row and its amplitude derivative are

\[ \dot X_i=\frac{a\nu_i}{d_i} \sum_{j\sim i}R_iR_j\sin(\Theta_j-\Theta_i), \qquad \dot R_i=-b\nu_i u_i\sin\delta_i. \]

Thus the factor R_i*R_j is calculated from retained internal phase geometry. It is a dynamical contribution to the paired response, not a threshold that activates a control action. Its evolution still requires internal information. Using this resultant does not justify dropping u_i or the phase difference sign, and no general closure in (X,Theta,R) is claimed.

The original fine support remains fixed. The varying factor neither creates a new edge nor derives an occurrence time for one. Moreover, Theta_dot still contains the unmodified base Laplacian and the original degree d_i. Replacing every adjacency or degree by R_i*R_j would change that row and define another model. This is an inherited phase-to-form interaction with internal state, not the unchanged bare coarse law on an arbitrarily weighted graph.

7.3. Exact synchronized inheritance and storage scaling

If u=delta=0 initially, their two rows vanish. Smooth uniqueness makes this synchronized submanifold invariant. The remaining rows are exactly the original conservative sine law on the base graph, with the same nu_i,beta,w and clock. This is inheritance on a supplied invariant submanifold, not synchronization from a general preparation.

Write

\[ E_c(X,\Theta)=\frac12\sum_{\{i,j\}}(X_i-X_j)^2 +\beta\sum_{\{i,j\}}[1-\cos(\Theta_j-\Theta_i)] . \]

Summing over the four fine edges associated with each base edge yields the exact identity

\[ \boxed{ E_f=4E_c+ 2\sum_i d_i u_i^2+ 4\beta\sum_{\{i,j\}}\cos(\Theta_j-\Theta_i) [1-\cos\delta_i\cos\delta_j] .} \]

For the form term, the four squared differences sum to 4*((X_i-X_j)^2+u_i^2+u_j^2). For phase, their four cosines sum to 4*cos(Theta_j-Theta_i)*cos(delta_i)*cos(delta_j). These two identities give the displayed formula and, on synchronization, E_f=4*E_c. Neither capacity nor the clock has been rescaled to obtain the factor four.

Both full fine storage and synchronized coarse storage are conserved in their respective conservative laws. Off the synchronized submanifold, E_c is not independently conserved in general: the other retained terms exchange storage with it. Their displayed phase contribution is nonnegative in a local acute coarse chart, but not for arbitrary coarse phase differences. Thus the algebraic decomposition alone is not a global positive split into separate reservoirs.

7.4. A nearby exact obstruction to autonomous block means

Take the base C5, ordered 0,1,2,3,4, with alpha=2*pi/5, constant X and Theta_j=j*alpha. First prepare u=delta=0. This is an exact equilibrium of the fine and coarse laws. Now keep the same X,Theta,u=0 and change only delta_0=epsilon, with every other delta_i=0 and

\[ 0<|\epsilon|<\pi/2-\alpha. \]

The fine edges remain strictly acute, and the entire preparation can be arbitrarily close to the synchronized target. Nevertheless its exact coarse form rates are

\[ \dot X_1=\frac{a\nu_1}{2}\sin\alpha(1-\cos\epsilon)>0,\qquad \dot X_4=-\frac{a\nu_4}{2}\sin\alpha(1-\cos\epsilon)<0, \]

with X_dot=0 at the other three nodes. All coarse phase rates remain zero at this instant. The first preparation had all these rates zero. The two states therefore have identical observed (X,Theta) and different derivatives. No autonomous first-order vector field on those block means can reproduce every fine state in any neighborhood of this target. This is a failure of that proposed coarse closure, not a failure of the complete retained-coordinate nodal law.

The all-state closure obstruction also holds for any connected base graph in the stated domain. Choose a node j, set its coarse phase to a small gamma!=0 and all other coarse phases to zero, and compare delta=0 against a preparation with only delta_j=epsilon!=0. At every neighbor i of j the coarse form defect is (a*nu_i/d_i)*(cos(epsilon)-1)*sin(gamma)!=0. The two preparations have the same coarse state; choosing |gamma|+|epsilon|<pi/2 keeps the comparison within the pair chart with all fine phase gaps acute. This general obstruction does not require that either preparation be a critical target.

Internal form can also be hidden from an instantaneous rate comparison. At the same target, prepare delta=0 and only u_k=U!=0. Define the coarse form defect relative to the base sine row by

\[ D_i=\frac{a\nu_i}{d_i}\sum_{j\sim i} (\cos\delta_i\cos\delta_j-1)\sin\Delta_{ji}. \]

Initially D=0 and D_dot=0. For a neighboring node i, however,

\[ \ddot D_i(0) =-\frac{a\nu_i}{2} \sin(\Theta_k-\Theta_i)(b\nu_kU)^2 . \]

This follows by differentiating the exact cosine factors twice and using delta_k_dot=b*nu_k*U; their value and first derivative vanish in the defect at the preparation. It is an exact field-jet statement, not a trajectory or a tangent replacement of the nonlinear law. Internal form changes internal phase and hence the later coarse response even when the initial coarse rate defect vanishes.

7.5. Trapping permits persistent internal organization

For this full ten-node fine graph, the lifted twist is itself an exact acute critical target. On pair-constant vectors its combinatorial Laplacian is twice the C5 Laplacian; on pair-antisymmetric vectors it is 4*I. Thus its smallest positive eigenvalue is

\[ \lambda_{2,f}=\min(2\lambda_{2,C5},4)=5-\sqrt5>0 . \]

Every target edge has cosine cos(alpha)>0. The existing local first-exit proof therefore applies to the full fine graph. With centered fine form and target phase deviations of combined squared norm Z_f^2, choose r>0 such that alpha+sqrt(2)*r<pi/2 and set

\[ \kappa_f=\frac{5-\sqrt5}{2} \min\{1,\beta\cos(\alpha+\sqrt2r)\}. \]

If initially

\[ Z_f^2<r^2,\qquad E_f-4E_{*,C5}<\kappa_f r^2, \]

conservation and the strict boundary coercivity exclude a first exit. The argument is valid in both time directions. It keeps both internal and collective coordinates in the local tube: in the declared chart,

\[ Z_f^2= 2\|PX\|^2+2\|P(\Theta-\Theta_*)\|^2 +2\|u\|^2+2\|\delta\|^2,\qquad P=I-\frac15\mathbf1\mathbf1^T . \]

This is a sufficient trapping condition on the full state, not a sum of five independent pair certificates. All target cycles retain their winding inside the acute tube. The previous small-epsilon obstruction can be chosen within such a tube, so persistent fine organization and failure of block-mean closure are compatible.

The transverse linearization at the synchronized twist is

\[ \dot u_i=-a\nu_i\cos\alpha\,\delta_i,\qquad \dot\delta_i=b\nu_i u_i . \]

Its eigenvalues are +/-i*nu_i*sqrt(a*b*cos(alpha)). These conditional tangent modes describe internal exchange, not decay or a nonlinear period certificate. The full conservative flow preserves volume. The existing nonlinear recurrence theorem implies that almost every point outside the synchronized submanifold in an admitted finite-volume trapped family cannot converge to it: near returns to its initial state retain a positive distance from that closed submanifold. Conservative trapping therefore does not establish attracting synchronization. This does not assert recurrence of a selected moving state or supply an observed return time.

The same support also distinguishes an unstable geometry without changing the law. The uniform C5 twist with winding two is another exact critical target, with edge cosine cos(4*pi/5)<0. Its transverse equation is

\[ \ddot\delta_i =-ab\nu_i^2\cos(4\pi/5)\,\delta_i , \]

so the tangent has a positive real eigenvalue nu_i*sqrt(-a*b*cos(4*pi/5)). The smooth full sine equilibrium is therefore locally unstable. In contrast, winding one's positive edge cosine supports the preceding full-state trapping domain. This is a conditional geometric distinction under identical support and coefficients: losing the winding-two organization does not mean that its storage is dissipated. With e=0 the total storage remains constant. It also does not prove that every perturbation of the unstable target follows the same transition or ends in the winding-one family.

For the different law e>0, decay and target recovery must use the existing positive-loss theorem and local recovery hypotheses. The conservative recurrence argument no longer applies, and the conservative transverse modes cannot be relabeled as its attractors.

7.6. Meaning and limits of the scale result

The paired description preserves a larger organization together with its internal fine patterns. It does not require that the fine patterns vanish when a collective description is introduced. Conversely, grouping two nodes does not by itself establish either constituent or the group as an autonomously formed NFR; their preparation, identity and support remain explicit premises here.

Exact same-law inheritance is proved on synchronization. Near that submanifold, the exact larger description retains internal coordinates, and their emergent resultant amplitudes modify collective interactions. This supplies a specific relation between organization and internal exchange. It does not establish generic autonomous coarse closure, spontaneous hierarchical formation, a scale-independent fractal dimension, a universal pulse, or a microscopic physical identification.

The detached replica observer retains the complete captured graph and independently compares fine half-sum/half-difference rates with the factored expressions. It uses exact represented form values and supplied phase-lift integers, and retains interval bounds for trigonometric quantities. Zero-containing arithmetic residuals are checks of that capture, distinct from the all-state identities proved above. The replica tests separate synchronized inheritance, hidden internal motion, nonlinear coarse nonclosure and the two target geometries. The observer does not certify an arbitrary capture as lying in the local trapping domain or execute a trajectory.

8. An exact collective state for unordered pairs

Keep the complete conservative sine law, complete two-replica support, strictly positive held paired capacities and nonantipodal pair chart of section 7. There are no external inputs or node-selected events. Swapping the two members of any pair is then an exact symmetry: it preserves the support, capacities and complete nodal rows. This section removes only those independent swap labels. Internal form and phase information remain in the collective description.

8.1. Realizable invariants and their fibers

For each pair define

\[ R_i=\cos\delta_i,\qquad U_i=u_i^2,\qquad Q_i=u_i\sin\delta_i . \]

The symbol Q_i here denotes an internal form-phase correlation, not nodal pressure p=DeltaNFR. Their units are respectively 1, [x]^2 and [x]. Each is unchanged by the pair swap (u_i,delta_i)->(-u_i,-delta_i). The retained coarse coordinates X_i,Theta_i are unchanged as well.

Their exact realized domain is

\[ \boxed{\quad 0<R_i\le1,\qquad U_i\ge0,\qquad Q_i^2=U_i(1-R_i^2). \quad} \]

Necessity follows from |delta_i|<pi/2 and the sine-cosine identity. For sufficiency and uniqueness of the unordered pair:

• If 0<R_i<1, choose delta_i=arccos(R_i)>0 and u_i=Q_i/sqrt(1-R_i^2). The constraint gives u_i^2=U_i. The other possible lift is its simultaneous negative, exactly the pair swap. This also covers U_i=Q_i=0 with nonzero phase separation.
• If R_i=1, the chart forces delta_i=0 and the constraint forces Q_i=0. The choices u_i=+sqrt(U_i) and u_i=-sqrt(U_i) are the same unordered pair.
• At R_i=1,U_i=Q_i=0 there is just one lift, u_i=delta_i=0.

Thus (X,Theta,R,U,Q) separates all fine states in this chart up to the independent pair swaps. Together with the supplied support and held capacities, it specifies the unordered fine configuration. In particular, the fully coherent phase value R_i=1 does not imply synchronized form: U_i>0 remains distinguishable.

This is a quotient by a finite symmetry group, not a reduction in the generic number of continuous degrees of freedom. Each internal pair has two degrees of freedom, represented by three invariants with one constraint. At the synchronized tip (R,U,Q)=(1,0,0), the gradient of Q^2-U*(1-R^2) vanishes and the swap fixes the original state. These invariants are not an ordinary smooth coordinate chart at that tip. Their algebraic singularity must not be interpreted as an extra physical degree of freedom, a dynamical instability or failure of the original smooth nodal law.

8.2. Closed dynamics with internal state retained

Let

\[ A_i=\frac{a\nu_i}{d_i} \sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i), \qquad c_i=b\nu_i . \]

These are calculated from the retained state and declared coefficients, with units [A]=[x]/[t] and [c]=1/([x][t]). They introduce no new constitutive parameter. The original internal rows are u_i_dot=-A_i*sin(delta_i) and delta_i_dot=c_i*u_i. Differentiating the invariants gives

\[ \boxed{\begin{aligned} \dot X_i&=\frac{a\nu_i}{d_i} \sum_{j\sim i}R_iR_j\sin(\Theta_j-\Theta_i),\\ \dot\Theta_i&=\frac{b\nu_i}{d_i} \sum_{j\sim i}(X_i-X_j),\\ \dot R_i&=-c_iQ_i,\\ \dot U_i&=-2A_iQ_i,\\ \dot Q_i&=-A_i(1-R_i^2)+c_iU_iR_i . \end{aligned}} \]

Every consumed quantity is determined by the unordered collective state. These are exact nonlinear quotient rows. Unlike block means alone, the augmented state closes because it retains the internal information that caused the earlier obstruction. No hidden initial condition was set to zero, fitted to a response or erased.

The constraint is an exact first integral of these rows:

\[ \begin{aligned} \frac{d}{dt}[Q_i^2-U_i(1-R_i^2)] ={}&2Q_i[-A_i(1-R_i^2)+c_iU_iR_i]\\ &+2A_iQ_i(1-R_i^2)-2c_iU_iR_iQ_i=0 . \end{aligned} \]

This algebra alone does not prove realizability: the same equality also has nonphysical solutions such as R_i>1,U_i<0. The inequalities, phase chart and lifting argument are essential.

8.3. Evolution, boundaries and lift independence

Start with a realizable collective state and choose any fine lift supplied by section 8.1. Evolve the original smooth fine nodal law. On every time interval for which all pairs remain in |delta_i|<pi/2, its invariants solve the displayed collective rows and remain realizable.

Conversely, those rows define a smooth vector field in the ambient (X,Theta,R,U,Q) variables; there is no division by U, Q or 1-R^2 in them. Local uniqueness therefore makes every collective solution from that initial state equal to the invariant image of the fine solution. Choosing the other fine lift merely performs fixed pair swaps, which are symmetries of the full law. It yields the same collective solution. This proves dynamical sufficiency even at a tip where the inverse invariant coordinates are not differentiable.

Two boundary controls make the role of the retained variables explicit:

\[ \begin{array}{ll} R_i=1,\ U_i>0: &\dot Q_i=c_iU_i,\quad \ddot R_i=-c_i^2U_i<0,\\[2mm] U_i=0,\ 0<R_i<1: &\dot Q_i=-A_i(1-R_i^2),\quad \ddot U_i=2A_i^2(1-R_i^2)\ge0 . \end{array} \]

The first state has aligned primitive phases but a form difference that immediately changes its internal correlation and subsequently its resultant amplitude. The second can acquire a form difference from phase separation when A_i!=0. At the tip R_i=1,U_i=Q_i=0 all three internal derivatives vanish. That internal tip remains invariant even when the means or neighboring internal states evolve.

No further chart guarantee follows just from the collective equality constraint. At R_i=0 the primitive phases are antipodal, their mean phasor vanishes, and its circular midpoint is not uniquely defined. The original fine law remains smooth there. The chosen collective chart must end or be replaced by a separately justified chart; continuing a signed R or clipping it to zero is not an automatic continuation of this unordered observation. The present result is exact up to that chart boundary unless an independent trapping condition excludes it. For the fixed conservative unit doubled C5, Section 29 provides a global invariant representation through the boundary; it does not extend this midpoint chart or its input contract.

8.4. Storage and identity pass to the exact quotient

The full fine storage becomes the invariant function

\[ \boxed{ \mathcal H(X,\Theta,R,U)= 2\sum_{\{i,j\}}(X_i-X_j)^2+ 2\sum_i d_iU_i+ 4\beta\sum_{\{i,j\}} [1-R_iR_j\cos(\Theta_j-\Theta_i)] . } \]

This is exactly E_f from section 7, not an additional Hamiltonian or an independently calibrated energy. Its conservation follows by lifting the collective rows to the same conservative law, or by direct differentiation using those rows. It retains internal form storage even at R_i=1. On the realized domain every displayed contribution is nonnegative. Setting R=1,U=Q=0 recovers 4*E_c and the supplied synchronized same-law sector.

The previous full ten-node acute-twist barrier can also be expressed without the swap labels. In its declared target chart,

\[ Z_f^2= 2\|PX\|^2+2\|P(\Theta-\Theta_*)\|^2 +2\sum_i[U_i+(\arccos R_i)^2], \qquad P=I-\frac15\mathbf1\mathbf1^T . \]

This equals the fine centered norm because delta_i^2=(arccos(R_i))^2 in the admitted pair chart. Thus the existing sufficient conditions Z_f^2<r^2 and H-4*E_*,C5<kappa_f*r^2 admit exactly the same trapping conclusion when read through these invariants. In particular,

\[ U_i<r^2/2,\qquad R_i>\cos(r/\sqrt2)>0 \]

throughout such a trapped trajectory. The acute radius condition in section 7 implies r/sqrt(2)<pi/2. Consequently that independent full-state barrier keeps the unordered pair chart valid for all time; the constraint by itself did not do so. This rewrites a proved barrier rather than creating another approximation, recovery criterion or trajectory calculation.

8.5. What has and has not been reduced

The result removes a redundant choice of member labels while retaining the complete unordered pair, including its internal form variance, phase separation and their signed correlation. The collective object therefore still contains its smaller constituents and their dynamics. It is not a claim that a pair has become a single fundamental scalar node with the original bare nodal state space.

The complete bipartite support, partition, equal paired capacities and phase chart remain assumptions. Unequal member capacities, selective forcing, member-specific events or another fine support can break the swap symmetry and require another observation model. No autonomous partition formation, generic fractality, unlimited hierarchical closure, material particle or microscopic law selection follows from this finite symmetry quotient.

The existing replica observer exposes the invariant state of an admitted fine capture and compares its directly calculated derivatives with these closed rows. Its exact represented U and interval R,Q retain the fine capture and arithmetic provenance; an interval constraint residual containing zero does not admit every tuple in that surrounding rectangular box. The same test owner checks pair-swap invariance, boundary controls, direct fine-to-collective rates and storage. This is not an API for accepting arbitrary collective tuples, reconstructing a chosen numerical lift, or running a trajectory.

8.6. The collective state inherits the same Poisson structure

The exact quotient inherits more than a closed vector field. Use the same-law Poisson tensor, with bracket convention dot f={f,E_f}. Each fine member has degree 2*d_i, hence

\[ \{x_{i,\pm},\theta_{i,\pm}\} =-\frac{b\nu_i}{2d_i}. \]

All other fine coordinate brackets vanish. The mean/half-difference change therefore gives

\[ \{X_i,\Theta_i\}=\{u_i,\delta_i\}=-\kappa_i,\qquad \kappa_i=\frac{b\nu_i}{4d_i}>0, \]

with zero brackets between distinct pairs and between a pair's mean and internal coordinates. The factor four follows from both the fine degree and the two factors of one half in the coordinate change. Holding support and paired capacities fixed makes each kappa_i constant.

The chain rule for R=cos(delta), U=u^2, Q=u*sin(delta) gives

\[ \boxed{ \{R_i,U_i\}=-2\kappa_iQ_i,\qquad \{R_i,Q_i\}=-\kappa_i(1-R_i^2),\qquad \{U_i,Q_i\}=-2\kappa_iU_iR_i . } \]

Together with {X_i,Theta_i}=-kappa_i, these are all the nonzero independent brackets. They descend under every pair swap: simultaneously negating u_i,delta_i preserves their constant bracket and leaves the invariant functions unchanged.

Jacobi follows by pushforward of the fine bracket on the invariant algebra. It can also be checked without choosing a lift. Set

\[ \mathcal C_i=Q_i^2-U_i(1-R_i^2). \]

In the ordered coordinates (R_i,U_i,Q_i), the internal tensor is J_ab=-kappa_i*epsilon_abc*partial_c C_i. A three-dimensional bracket of this form satisfies Jacobi because its defining vector -kappa_i*grad C_i has zero curl. Moreover J*grad C_i=0, so C_i is a Casimir: its bracket with every smooth retained function vanishes, not only with this storage. The direct sum with the constant mean blocks is again Poisson.

For a realized nontip state the internal tensor has rank two. If R_i<1, its (R_i,Q_i) entry is nonzero; if R_i=1 and U_i>0, its (U_i,Q_i) entry is nonzero. At the tip (R_i,U_i,Q_i)=(1,0,0) it has rank zero. The independent mean block retains rank two there. This is the finite-symmetry quotient's singular internal stratum, not an additional continuous degree of freedom or a singularity of the fine nodal law.

The Hamiltonian is exactly the already derived H=E_f in section 8.4. Its nonzero coordinate derivatives are

\[ \begin{aligned} \partial_{X_i}\mathcal H&=4\sum_{j\sim i}(X_i-X_j),\\ \partial_{\Theta_i}\mathcal H &=-4\beta R_i\sum_{j\sim i}R_j\sin(\Theta_j-\Theta_i),\\ \partial_{R_i}\mathcal H &=-4\beta\sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i),\\ \partial_{U_i}\mathcal H&=2d_i,\qquad \partial_{Q_i}\mathcal H=0 . \end{aligned} \]

Using a=beta*b, the mean brackets reproduce the exact X,Theta rows, while the internal brackets give

\[ \begin{aligned} \{R_i,\mathcal H\}&=-4\kappa_i d_iQ_i=-c_iQ_i,\\ \{U_i,\mathcal H\} &=-8\kappa_i\beta Q_i\sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i) =-2A_iQ_i,\\ \{Q_i,\mathcal H\} &=-4\kappa_i\beta(1-R_i^2) \sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i) +4\kappa_i d_iU_iR_i\\ &=-A_i(1-R_i^2)+c_iU_iR_i . \end{aligned} \]

Thus all five collective rows, the internal constraint and storage conservation share one inherited structure. No Hamiltonian, pressure channel or feedback controller is added.

The polynomial Poisson identity also exists outside the realized set; it does not admit those states. The inequalities, lifting and chart continuation conditions of sections 8.1–8.3 remain essential. In particular, a Casimir value or a numerical residual cannot establish realizability, and R=0 still requires a different collective chart. Likewise, a recurrence argument must retain the fine-state measure or a justified quotient measure: the constrained invariants are not independent variables carrying full-dimensional Lebesgue volume. The e=0 law, supplied support and equal held member capacities remain premises; this identity selects neither microscopic loss nor a physical interpretation of storage.

8.7. Temporal resultant information recovers the unordered internal state

The same closed rows yield an exact observation consequence, without introducing another variable or law. Retain the section 8 premises and the structural clock. Suppose the means X,Theta, all neighboring resultant amplitudes R and the held coefficients are independently known. Thus c_i=b*nu_i>0 and A_i=(a*nu_i/d_i)*sum_j R_j*cos(Theta_j-Theta_i) are known; only relative neighboring mean phases are consumed by A_i. For observations of an admitted smooth fine trajectory,

\[ \dot R_i=-c_iQ_i,\qquad \ddot R_i=c_iA_i(1-R_i^2)-c_i^2U_iR_i. \]

Consequently, everywhere in the local chart R_i>0,

\[ \boxed{ Q_i=-\frac{\dot R_i}{c_i},\qquad U_i=\frac{c_iA_i(1-R_i^2)-\ddot R_i}{c_i^2R_i}. } \]

These observations recover the exact unordered internal state, not the constituent labels or a preferred signed lift. Held capacity is essential: if c_i varied, the second derivative would also contain -dot(c_i)*Q_i. The formula does not estimate an unknown capacity, select a clock, or identify a laboratory sensor with this resultant.

For 0<R_i<1, the Casimir additionally gives the first-derivative formula

\[ U_i=\frac{\dot R_i^2}{c_i^2(1-R_i^2)}. \]

That expression becomes singular at aligned phases. The second-derivative formula remains regular there: at R_i=1, any admitted trajectory has dot R_i=Q_i=0 but

\[ \boxed{\qquad U_i=-\ddot R_i/c_i^2.\qquad} \]

Thus a perfectly aligned instantaneous pair can conceal a form imbalance that its resultant curvature distinguishes. The neighbor term vanishes at this exact boundary, although it is needed for the general second-derivative reconstruction. This reuses the boundary dynamics of section 8.3; zero first derivative alone did not imply a stationary internal state.

The inverse retains admission and observation limits. Its output must satisfy 0<R_i<=1, U_i>=0 and the exact Casimir constraint. Small c_i or proximity to R_i=0 amplifies uncertainty in the second-derivative inverse; the first-derivative inverse instead loses conditioning near R_i=1. Independent intervals for state and derivatives need not come from one joint trajectory. Numerical differentiation requires its own sampling, timing and remainder bounds; neither interval overlap nor this exact identity supplies them. There is no arbitrary noisy-tuple admission, raw-data reconstruction API or replacement of retained internal state by the single value R.

13. Unequal constituent capacities: persistence and its symmetry boundary

Keep the same complete conservative sine law and doubled-cycle support. Change only the supplied held capacities: each constituent now has its own nu_(i,+)>0 and nu_(i,-)>0. Define

\[ v_i=\frac{\nu_{i,+}+\nu_{i,-}}2,\qquad \eta_i=\frac{\nu_{i,+}-\nu_{i,-}}2,\qquad v_i>|\eta_i|. \]

The letter v_i in this section is mean capacity, not the centered fine-form vector used in Section 12. Both v_i and eta_i are held parameters. The same form/phase coordinates x_(i,+/-)=X_i+/-u_i, theta_(i,+/-)=Theta_i+/-delta_i remain invertible on supplied lifts. The pair chart is still |delta_i|<pi/2. No new force, state law or capacity evolution is postulated.

13.1. Exact asymmetric mean and internal rows

For a base degree d_i define the normalized neighboring sums

\[ S_i=\frac1{d_i}\sum_{j\sim i}R_j\sin(\Theta_j-\Theta_i), \quad C_i=\frac1{d_i}\sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i), \quad G_i=\frac1{d_i}\sum_{j\sim i}(X_i-X_j), \quad R_i=\cos\delta_i . \]

For the doubled C5, d_i=2. Before multiplying by the individual capacity, the two fine form rows are a*(R_i*S_i-/+sin(delta_i)*C_i), and the two fine phase rows are b*(G_i+/-u_i). Their exact half-sums and half-differences therefore give

\[ \boxed{\begin{aligned} \dot X_i&=a(v_iR_iS_i-\eta_i\sin\delta_i\,C_i),\\ \dot\Theta_i&=b(v_iG_i+\eta_i u_i),\\ \dot u_i&=a(\eta_iR_iS_i-v_i\sin\delta_i\,C_i),\\ \dot\delta_i&=b(\eta_iG_i+v_i u_i). \end{aligned}} \]

The additional terms are consequences of multiplying the existing fine rows by unequal capacities. Setting eta_i=0 recovers Section 7, even if v_i differs between pairs. Variation between equal member pairs and inequality within a pair are thus distinct changes. No rounded near-equality can replace the exact equality used in the earlier invariant-tip theorem.

The inherited signed Poisson bracket makes the same distinction. With k_i=b/(4*d_i) its nonzero independent entries are

\[ \{X_i,\Theta_i\}=\{u_i,\delta_i\}=-k_i v_i,\qquad \{X_i,\delta_i\}=\{u_i,\Theta_i\}=-k_i\eta_i . \]

Different pairs have zero brackets. These entries follow by taking half-sums/differences of {x_(i,+/-),theta_(i,+/-)}=-b*nu_(i,+/-)/(2*d_i). The bracket is constant for held capacities and hence satisfies Jacobi. Its determinant in each four-coordinate block is k_i^4*(v_i^2-eta_i^2)^2>0. The same full fine storage from Section 7 generates all four displayed rows, including the cross terms. Equal capacities eliminate those cross terms; they do not supply the underlying storage balance.

13.2. Collective trapping and ambient recurrence survive

The exact storage still satisfies E_f_dot=0. The full field still has zero divergence: each fine form row consumes phase, and each fine phase row consumes form, with all capacities held. Its conserved form mean is now

\[ m_\rho(x)= \frac{\sum_{i,s}\rho_{i,s}x_{i,s}}{\sum_{i,s}\rho_{i,s}}, \qquad \rho_{i,s}=\frac{2d_i}{v_i+s\eta_i}>0 . \]

In retained coordinates this is

\[ \boxed{ m_\rho= \frac{\displaystyle\sum_i \frac{d_i(v_iX_i-\eta_i u_i)}{v_i^2-\eta_i^2}} {\displaystyle\sum_i \frac{d_i v_i}{v_i^2-\eta_i^2}}.} \]

The ordinary mean is not a substitute. For the same winding-one target the fine storage, spectral gap and acute geometry do not depend on capacity. Retain r,kappa_f,E_* from Section 12, choose 0<epsilon_*<kappa_f*r^2 and a finite open interval m_lo<m_hi, and define

\[ \mathcal U_\rho= \{Z_f^2<r^2,\ E_f-E_*<\varepsilon_*, \ m_{\rm lo}<m_\rho<m_{\rm hi}\}. \]

For each supplied positive held capacity list this is an open invariant family with finite positive fine volume. The same energy argument prevents radius exit in both time directions. To verify the volume claim explicitly, use centered form z=P_f x and write

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

This is an invertible form coordinate change. Positive rho makes m_rho a convex average, so

\[ |x_j-m_\rho| \le\max_k|x_j-x_k| \le\sqrt2\,\|z\|<\sqrt2r . \]

Thus the closure of U_rho lies in a bounded form box times the phase torus; target neighborhoods prove positive volume. The smooth two-sided flow preserves the family and its fine ambient measure. The existing recurrence proof gives almost-everywhere nonstationary recurrence there, with its original selected-state and singular-measure limitations. This family includes synchronized tips: deleting them would generally destroy invariance, as the next subsection shows.

These conclusions apply separately to each fixed positive capacity list. They do not install a time-varying capacity law, give a common return deadline, or transfer the equal-capacity internal-circulation claim. The singular boundary at a zero constituent capacity is outside this theorem.

13.3. Exact counterexamples to tip invariance and pointwise activity

At a synchronized tip u_i=delta_i=0 the asymmetric internal rows reduce to

\[ \dot u_i=a\eta_i S_i,\qquad \dot\delta_i=b\eta_i G_i . \]

They need not vanish. The tip and its complement are therefore not generally invariant. This is a structural obstruction, not a numerical threshold effect.

An explicit preparation also shows that a nontip pair can have zero instantaneous internal velocity. Take v_i=1 for all pairs, eta_0=eta with 0<eta<1, and all other eta_i=0. Use exact target phases Theta_i=i*alpha and delta_i=0. Prepare

\[ X_0=\epsilon,\quad X_{i\ne0}=0,\qquad u_0=-\eta\epsilon,\quad u_{i\ne0}=0,\qquad \epsilon>0. \]

All S_i=0 and G_0=epsilon. Hence

\[ \dot u_0=\dot\delta_0=0,\qquad u_0\ne0,\qquad \dot\Theta_0=b\epsilon(1-\eta^2)>0 . \]

The pair's old invariant rates R_dot,U_dot,Q_dot all vanish at this instant, although its mean phase moves. This also makes the normalized signed internal angular velocity zero, excluding the earlier strict positive angular lower bound. The storage excess and full norm are exactly

\[ E_f-E_*=4(1+\eta^2)\epsilon^2,\qquad Z_f^2=\left(\frac85+2\eta^2\right)\epsilon^2 . \]

For example eta=1/2,epsilon=1/1024 gives capacities (nu_(0,+),nu_(0,-))=(3/2,1/2) and forms (x_(0,+),x_(0,-))=(epsilon/2,3*epsilon/2), with excess 5*epsilon^2 and norm squared 21*epsilon^2/10. Its conserved weighted mean is 5*epsilon/16, rather than the ordinary mean epsilon/5. The target phases in this proof are symbolic angles; substituting rounded radians would not prove the exact zero derivative.

The zero internal rate at pair zero also has an exactly represented phase control. Choose rational h near alpha and raw pair means (0,h,2h,-2h,-h), with respective full-turn lifts (0,0,0,1,1). The two neighboring phases of pair zero are +h,-h, so their sine currents cancel exactly. The same form preparation still has u_0_dot=delta_0_dot=0 without claiming that the entire phase state is critical. Taking rational h arbitrarily close to alpha preserves any strict neighborhood admission of the symbolic preparation by continuity.

Both the asymmetry eta and preparation size epsilon can be arbitrarily small. For any fixed admitted radius and excess ceiling, sufficiently small epsilon puts the state in the collective trapping family; a common form translation places its weighted mean inside the chosen slab without changing any displayed rate or bound. Thus collective geometric persistence can coexist with an internal stall arbitrarily close to the symmetric regime.

For a separate tip-crossing control keep the same preparation but set u_0=0. Now delta_0_dot=b*eta*epsilon!=0 at the tip. Smooth local uniqueness gives nontip states just before and after this instant on the same full solution. Choosing the preparation inside U_rho retains those states in its two-sided invariant family. A nontip preparation can consequently reach the tip at a finite time.

Neither control proves permanent internal cessation, loss of the collective winding, or absence of later turns. They disprove the specific all-state no-tip-entry and nonzero-internal-velocity statements outside their equal-member-capacity premises. They do not contradict the surviving full-state recurrence theorem.

13.4. The unordered state must retain capacity correlations

Swapping only (u_i,delta_i) while fixing unequal labeled capacities is no longer a symmetry. For example at delta_i=0, the two forms u_i and -u_i have the same old R,U,Q but different mean phase rates through eta_i*u_i. Even retaining the signed parameter eta_i does not restore closure of those old observed coordinates.

For an unordered pair, interchange the whole constituent, including its held capacity. This exact relabeling sends (eta_i,u_i,delta_i) to (-eta_i,-u_i,-delta_i) while preserving v_i,X_i,Theta_i. The missing correlations can be retained as

\[ P_i=\eta_i u_i,\qquad T_i=\eta_i\sin\delta_i,\qquad E_i=\eta_i^2 . \]

Here E_i is squared capacity contrast, not the total fine storage E_f. Together with R_i,U_i,Q_i these quantities identify the exact joint swap orbits on the pair chart. Their realizable domain is

\[ 0<R_i\le1,\quad U_i,E_i\ge0,\quad v_i>0,\quad E_i<v_i^2, \]

\[ Q_i^2=U_i(1-R_i^2),\quad P_i^2=E_iU_i,\quad T_i^2=E_i(1-R_i^2),\quad P_iT_i=E_iQ_i . \]

Equivalently the symmetric matrix

\[ \begin{pmatrix} E_i&P_i&T_i\\ P_i&U_i&Q_i\\ T_i&Q_i&1-R_i^2 \end{pmatrix} \]

is positive semidefinite of rank at most one: it is the outer product of (eta_i,u_i,sin(delta_i)) with itself. If E_i>0, choose eta_i=sqrt(E_i), u_i=P_i/eta_i and delta_i=asin(T_i/eta_i); its only other lift is the simultaneous sign reversal. If E_i=0, then P_i=T_i=0 and the earlier R,U,Q lifting applies. This is a finite symmetry quotient with retained internal information, not fewer continuous dynamical degrees of freedom.

The exact forward equations require no division by eta_i:

\[ \boxed{\begin{aligned} \dot X_i&=a(v_iR_iS_i-T_iC_i),\\ \dot\Theta_i&=b(v_iG_i+P_i),\\ \dot R_i&=-b(v_iQ_i+T_iG_i),\\ \dot U_i&=2a(P_iR_iS_i-v_iQ_iC_i),\\ \dot Q_i&=a(R_iT_iS_i-v_i(1-R_i^2)C_i) +bR_i(P_iG_i+v_iU_i),\\ \dot P_i&=a(E_iR_iS_i-v_iT_iC_i),\\ \dot T_i&=bR_i(E_iG_i+v_iP_i),\\ \dot E_i&=\dot v_i=0 . \end{aligned}} \]

These are exact derivatives of the realized invariants, so their constraints, the same fine storage and their lift are preserved for as long as the pair chart persists. The displayed algebra is not permission to evolve arbitrary incompatible interval tuples; domain preservation comes from the fine lift and smooth uniqueness. On U_rho the full-state barrier keeps the chart for all time. At E_i=0 the correlations vanish and the equal-capacity closure is recovered without a singular forward limit.

Their joint consistency can also be checked without expanding separate squared constraints. The vector z_i=(eta_i,u_i,sin(delta_i)) obeys z_i_dot=M_i*z_i along the full solution, where

\[ M_i=\begin{pmatrix} 0&0&0\\ aR_iS_i&0&-av_iC_i\\ bR_iG_i&bR_iv_i&0 \end{pmatrix}. \]

Consequently its Gram matrix obeys (z_i*z_i^T)_dot=M_i*(z_i*z_i^T)+(z_i*z_i^T)*M_i^T. This is an exact consistency identity with coefficients supplied by the same retained state, not a separately prescribed linear model.

13.5. Weighted centers remove direct cross terms, not internal information

One useful coordinate check uses lambda_i=eta_i/v_i and the inverse-capacity weighted pair centers

\[ \widehat X_i=X_i-\lambda_i u_i,\qquad \widehat\Theta_i=\Theta_i-\lambda_i\delta_i,\qquad \nu_{{\rm eff},i}=\frac{v_i^2-\eta_i^2}{v_i} =\frac{2\nu_{i,+}\nu_{i,-}}{\nu_{i,+}+\nu_{i,-}} . \]

Because capacities are held, the exact rows give

\[ \dot{\widehat X}_i=a\nu_{{\rm eff},i}R_iS_i,\qquad \dot{\widehat\Theta}_i=b\nu_{{\rm eff},i}G_i . \]

This explains the harmonic-mean capacity and cancellation of direct cross terms in weighted coordinates. It does not give an autonomous law for those means: G_i still consumes X_i=widehat(X)_i+lambda_i*u_i and S_i consumes Theta_i=widehat(Theta)_i+lambda_i*delta_i and neighboring R_j. The weighted phase center is only a local lifted coordinate; unequal noninteger phase weights do not define a global circular average. Nor does a change of mean coordinates undo the actual signed internal stall in Section 13.3.

The symmetry boundary therefore refines the persistent-pattern claim: positive held capacities preserve the conservative storage, local collective identity and ambient recurrence result, while the stronger all-pair circulation theorem requires its stated symmetry. Capacity correlations restore an exact unordered description of the asymmetric law; they do not restore that stronger theorem, select capacities, or prove autonomous formation of the graph or material constituents.

13.6. Shared engine scope

The scale owner exposes assess_sine_replica_capacity and SineReplicaCapacityAssessment. It reuses one full capture and the shared support/lift coordinates, then compares the four signed rows and eight capacity-aware invariant rows against the fine pushforward. Its storage gradients and two signed Poisson coefficients retain the cross terms. Constraint residuals check that capture's realizable image; the reader does not admit arbitrary independent invariant tuples. Capacity contrasts refer to the captured represented values, rather than precision discarded before that capture.

Optional SineReplicaCapacityFamily evidence uses the shared acute barrier, exact degree/capacity mean weights and a declared radius, excess ceiling and open mean slab on the complete ordered doubled C5. It includes tips and distinguishes family admission, source trapping, source membership and ambient almost-everywhere recurrence. It supplies neither an all-time internal-activity certificate nor an angular-circulation certificate for unequal capacities.

The earlier equal-capacity readers retain their exact capacity requirements. The replica tests separately check fine-row factorization, capacity-state swaps, constraints, weighted conservation and the symmetry-breaking controls. Neither reader changes the model, adjusts a capacity, evolves a trajectory or promotes a finite residual check to recurrence of the captured state.

18. Which attachments permit an unordered pair-state law?

18.1. Equivalence concerns the entire member state on a fixed graph

Keep the complete conservative normalized-sine law on a fixed finite simple connected unit graph, with a common held capacity nu>0, w,beta>0 and no input or event. Let an explicit partition put every fine node in a two-member pair. Define G as the finite group generated by independently exchanging the two members of each pair. Each permutation acts on both form and circular phase. The support, capacities and clock stay fixed.

On the pair chart of Section 8, the retained coordinates (X,Theta,R,U,Q) identify exactly these swap orbits. They retain the internal member information up to interchange; they are not merely pair means or a reduction in continuous dimension. The all-state condition for removing member labels from the evolution is

\[ \boxed{F(Pz)=P F(z)\quad\hbox{for every }z \hbox{ and every independent pair swap }P.} \]

Equivariance suffices: smooth uniqueness gives Phi_t(Pz)=P Phi_t(z), so the orbit of a state has a well-defined future orbit. It is also necessary for the exact all-state orbit quotient. On the dense set where no pair consists of identical member states, the finite action is free. Locally the quotient is invertible up to its finitely many separate lifts. If two such lifts have the same quotient flow, continuity forces their initial relative permutation to remain the same for sufficiently small time. Differentiating at zero gives the displayed identity. Continuity of the field then extends it to the fixed strata. The singular invariant chart at a tip does not remove this obligation.

This is different from simultaneous graph/state relabeling. Relabeling A to P A P^T while also relabeling z always describes the same unlabeled fine system. Here A is held unchanged; the question is whether exchanging member states alone can lose predictive information. Nor does failure of the all-state criterion exclude a separately proved special invariant subfamily or a smaller joint-swap symmetry.

18.2. Necessary and sufficient fixed-support condition

Write L_rw=I-D^{-1}A for the normalized Laplacian of this unit graph. The complete field is

\[ \dot x=a\nu D^{-1}S(\theta),\qquad \dot\theta=b\nu L_{\rm rw}x,\qquad a=w/\pi,\quad b=w/(\beta\pi). \]

Since b*nu>0, equivariance of the phase row for every form already requires

\[ L_{\rm rw}P=P L_{\rm rw}. \]

For distinct nodes p,q, its matrix entry is -1/d_p exactly when an edge is present, and zero otherwise. Permutation commutation therefore preserves this nonzero pattern: P is an automorphism of the fixed adjacency. Conversely, a graph automorphism preserves degrees and changes neither the linear neighbor sum nor the nonlinear sum sum_j sin(theta_j-theta_i) after the same state permutation. It consequently commutes with both complete rows.

Thus, under the stated capacity and support hypotheses,

\[ \boxed{\begin{split} \text{independent pair-state quotient for all states} &\ \Longleftrightarrow\ \text{every independent pair swap is a graph automorphism}\\ &\ \Longleftrightarrow\ \text{each interpair block is complete or empty.} \end{split}} \]

The last line allows an edge within any pair. More explicitly, members p,q are interchangeable precisely when N(p) minus {q} = N(q) minus {p}; adjacency between p and q is itself unchanged by their transposition. For two different pairs, independent exchange of their row and column members forces all four entries of their 2-by-2 adjacency block to agree. Each block is therefore either all four cross-edges or no edge. The presence or absence of each within-pair edge is unrestricted by the swap symmetry.

Positivity and the fixed unit graph are substantive premises here. A frozen zero-capacity row need not reveal its attachments through the phase field. Unequal capacities require the joint capacity-state treatment in Section 13, and a weighted graph requires the corresponding weighted equivariance test. Neither case is covered by simply dropping a hypothesis.

Equal neighbor counts alone are insufficient. For example, pairs (0,1) and (2,3) with internal edges 0--1,2--3 and cross-edges 0--2,1--3 form an equitable partition: each member has one neighbor in each relevant block. Exchanging only 0,1 changes the cross-edge pattern. The full unordered member-state quotient fails even though a linear mean reduction can have matching neighbor counts. A purely diffusive or linear quotient certificate cannot replace the complete-law condition.

18.3. Internal edges preserve symmetry but change the inherited rows

The preceding criterion is broader than the existing replica runtime's no-internal-edge admission. To see the distinction directly, let epsilon_i in {0,1} mark an internal pair edge, let d_i count adjacent base pairs, and put D_i=2*d_i+epsilon_i. Connected support gives D_i>0. Use the same signed coordinates as Section 7 and define

\[ R_i=\cos\delta_i,\qquad J_i=\sum_{j\sim i}R_j\sin(\Theta_j-\Theta_i),\qquad C_i=\sum_{j\sim i}R_j\cos(\Theta_j-\Theta_i),\qquad q_i=\sum_{j\sim i}(X_i-X_j). \]

Exact averaging and differencing of the fine rows give

\[ \boxed{\begin{aligned} \dot X_i&=\frac{2a\nu}{D_i}R_iJ_i,& \dot\Theta_i&=\frac{2b\nu}{D_i}q_i,\\ \dot u_i&=-\frac{2a\nu}{D_i} \sin\delta_i\,(C_i+\epsilon_iR_i),& \dot\delta_i&=\frac{2b\nu}{D_i}(d_i+\epsilon_i)u_i. \end{aligned}} \]

Indeed the optional internal edge contributes -epsilon_i*sin(2*delta_i) to the plus-member sine sum and 2*epsilon_i*u_i to its form-gradient sum, with the opposite contributions for the minus member. The degree used to normalize every fine row changes at the same time.

For epsilon_i=0 these expressions recover Section 7. For epsilon_i=1 the same finite swap quotient still exists, but its internal restoring term and normalization differ. An internal edge also adds 2*u_i^2+2*beta*sin(delta_i)^2 to the fine storage. It is therefore incorrect either to reject its mathematical swap symmetry or to apply the old no-internal-edge formulas unchanged. The current strict replica consumer keeps its original admission; the larger symmetry test does not install these broader rows as a runtime law.

18.4. Frozen one-edge control: identical retained state, different future

Return to the ten-node preparation of Section 16, with d=u=1/8, beta=w=nu=1 and structural pairs (0,1),(2,3),(4,5),(6,7),(8,9). Remove exactly edge (0,8) in the detached preparation. Compare two states on this same nineteen-edge graph: the original state and the one obtained by exchanging both x and theta of nodes 0 and 1. No support-removal event or change in its energy balance is being executed.

The full degree of node 8 is now three, with neighbors {1,6,7}, whereas node 9 has degree four and neighbors {0,1,6,7}. The original forms imply

\[ (L_fx)_8=u,\qquad (L_fx)_9=0,\qquad \dot\theta_8=\frac{u}{3\pi},\qquad \dot\theta_9=0. \]

The retained phase mean of pair (8,9) consequently has rate

\[ \dot\Theta_{(8,9)}=\frac{u}{6\pi} =\frac1{48\pi}. \]

After the state-only swap, x_1=u instead of -u; the two neighbor sums at node 9 still cancel. Therefore

\[ (L_f\widetilde x)_8=-u,\qquad (L_f\widetilde x)_9=0,\qquad \dot{\widetilde\Theta}_{(8,9)} =-\frac{u}{6\pi}=-\frac1{48\pi}. \]

The derivative difference is exactly -1/(24*pi). These are the complete-law phase rows with the actual changed degrees, not a perturbative estimate or an inherited base-row formula.

Nevertheless, every retained unordered pair coordinate is initially identical in the two states. Only pair (0,1) has changed its signed lift: u_0 and delta_0 both negate. Its means, R=cos(delta), U=u^2 and Q=u*sin(delta) are unchanged; all other pairs are untouched. The phase-mean-rate difference in an untouched pair is therefore an exact obstruction to any autonomous law on these retained unordered coordinates alone for this support.

The lost information is concrete: which member state of pair (0,1) is attached to node 8. Keeping the graph fixed does not recover that state-to-attachment correlation after the member labels have been discarded. Equal capacities do not remove it.

18.5. What a larger collective description must retain

An observed pair can remain a useful organization even when its members are not independently interchangeable under the law. The phase-only observer, the support symmetry and the future closure are separate claims. A failed symmetry test neither erases an observed grouping nor proves that it immediately disappears.

The same nineteen-edge source also passes the existing analytic pairing-window reader with the unchanged independent error radii 2^-20 and window [1/128,1/64]. Its center phase-rate numerators are (1/8,-1/8,1/8,-1/8,0,0,0,0,1/24,0); the general full-field remainder from Section 17 retains this changed rate and does not assume pair (8,9) stays synchronized. Every original pair remains the strict mutual nearest choice throughout the admitted box/window, while strict replica support is rejected. This supplies a finite-time instance of observed organization without the proposed fully unordered dynamical closure. It is a reuse of the same bound, not a support event, retuned budget or trajectory experiment.

On asymmetric support, retain the constituent states together with their incidence information, or quotient only by actual symmetries of that joint data. Correlations between member state and external attachments cannot in general be reconstructed from pair means or R,U,Q. A more compressed sufficient description needs its own proof; the counterexample does not select a universal new macro variable or force law.

This result establishes an information boundary under a supplied network. It neither selects that network, generates a microscopic connection nor establishes autonomous birth of a collective constituent. Exact finite-symmetry reduction retains internal dynamics and removes only labels whose interchange truly preserves the complete field.

18.6. Shared support reader and state-only evidence

assess_sine_pair_support_symmetry in the existing scale owner consumes one complete sine comparison, an explicit pair partition and, optionally, a caller-selected witness_pair. Its domain keeps zero form loss and common positive held capacity. It checks external neighbor equality directly, retains internal pair edges and every interpair edge count, and records a nonzero exact phase-row matrix witness for each failed swap. No sampled state or tolerance decides the all-state criterion.

The optional SinePairSwapWitness exchanges form and phase on the selected pair while keeping the captured graph and capacities unchanged. It reuses the shared form-gradient, phase-rate-numerator and sine-rate owners, retains both full fields, and compares pair phase-mean rates. Exact equality of the raw member-state multisets establishes the same swap orbit without testing equality of trigonometric enclosures. Phase means here use the supplied continuous real lifts locally; the report does not infer a global circular mean chart.

SinePairSupportSymmetryAssessment keeps this concrete countercontrol separate from its support-wide verdict and from the existing strict replica admission. A symmetric graph with an internal pair edge can pass the first test while remaining outside the no-internal-edge consumer. Conversely, a chosen state with vanishing witness rates does not restore all-state closure on asymmetric support.

The support-symmetry contract and independent replica tests cover the complete/empty criterion, internal edges, equitable but noninterchangeable members, exact degree-sensitive countercontrol and graph/state relabeling distinction. The reader neither mutates support nor installs a broader autonomous collective runtime.

19. Sufficient collective state on asymmetric support

19.1. Remove only the independently redundant member labels

Keep the fixed simple connected unit support, common held capacity nu>0, conservative normalized-sine law and explicit pair partition of Section 18. Write S for the pair indices whose individual member transposition is a graph automorphism, and O for the remaining pair indices. The graph determines this distinction through the exact external-neighbor criterion. It does not depend on an instantaneous synchronization score or an observed small rate.

For every pair retain the means X_i,Theta_i and the same declared nonantipodal chart

\[ x_{i,\pm}=X_i\pm u_i,\qquad \theta_{i,\pm}=\Theta_i\pm\delta_i, \qquad |\delta_i|<\pi/2. \]

Theta_i denotes the circular midpoint with a supplied local continuous lift; adding the same full turn to its members changes that lift, not the circular state. The retained internal coordinates are

\[ \boxed{ \begin{cases} (u_i,\delta_i), & i\in O,\\ (R_i,U_i,Q_i)=(\cos\delta_i,u_i^2,u_i\sin\delta_i), & i\in S. \end{cases}} \]

For an ordered pair, the plus/minus entries remain associated with their named incident edges. The sign is this supplied member ordering, not a physical orientation selected by the model. For an unordered pair, 0<R_i<=1, U_i>=0 and Q_i^2=U_i*(1-R_i^2) are essential realizability conditions. The full support, pair membership, capacity, coefficients and clock remain part of the specification.

Let H be the finite group generated by the individual swaps in S. Two fine circular states in this chart have the same mixed description if and only if they lie in the same H orbit. Indeed each ordered pair reconstructs its assigned member states directly. For an unordered pair with 0<R<1, choose

\[ \delta=\arccos R>0,\qquad u=Q/\sqrt{1-R^2}. \]

The other lift negates both coordinates and is exactly its permitted swap. At R=1, delta=0, Q=0 and u=+sqrt(U) or -sqrt(U) give the same permitted orbit. At the synchronized tip (R,U,Q)=(1,0,0) there is one fine lift. These pairwise choices prove both reconstruction and exact separation of the product orbits. The sign of Q is necessary: erasing it would in general identify states outside a permitted swap orbit.

Every generator of H preserves the full field by Section 18, so these orbits have a well-defined future while the chart is valid. This is sufficient state for the declared law, not a reduction in the generic number of continuous degrees of freedom. Each pair still has its two internal degrees of freedom. Nor is H asserted to be the entire graph automorphism group: a simultaneous exchange of several otherwise asymmetric pairs may be another symmetry, but finding or quotienting such joint permutations is outside this result.

19.2. Inherited rates use the actual fine attachments

For any representative reconstructed above, evaluate the original rows

\[ f_v=\frac{a\nu}{D_v}\sum_{k\sim v}\sin(\theta_k-\theta_v), \qquad h_v=\frac{b\nu}{D_v}\sum_{k\sim v}(x_v-x_k), \qquad a=w/\pi,\quad b=w/(\beta\pi), \]

where D_v is its actual fine degree. For every pair the means obey X_dot=(f_plus+f_minus)/2 and Theta_dot=(h_plus+h_minus)/2. For each ordered pair retain also

\[ \dot u_i=(f_{i,+}-f_{i,-})/2,\qquad \dot\delta_i=(h_{i,+}-h_{i,-})/2. \]

For an unordered pair differentiation gives

\[ \dot R_i=-\sin\delta_i\,\dot\delta_i,\quad \dot U_i=2u_i\dot u_i,\quad \dot Q_i=\sin\delta_i\,\dot u_i+ u_i\cos\delta_i\,\dot\delta_i. \]

These expressions are independent of the choice of permitted fine lift. Equivariance exchanges the two rates when the member states are exchanged; their means and the three differentiated invariants are unchanged. Rates of untouched ordered members also stay the same. The complete-bipartite replica formulas are not being applied to an incomplete cross-block.

The closure can also be made explicit without an inverse square root. For i in S, let T_i be the common external neighbor set of its members, m_i=|T_i|, and let epsilon_i indicate the optional internal edge. Their common fine degree is D_i=m_i+epsilon_i>0. Define from retained state

\[ J_i=\sum_{k\in T_i}\sin(\theta_k-\Theta_i),\qquad C_i=\sum_{k\in T_i}\cos(\theta_k-\Theta_i),\qquad q_i=\sum_{k\in T_i}(X_i-x_k), \]

\[ A_i=\frac{a\nu}{D_i}(C_i+2\epsilon_iR_i),\qquad c_i=\frac{b\nu}{D_i}(m_i+2\epsilon_i). \]

There is no hidden sign choice in these sums. If a member of a different unordered pair belongs to T_i, its other member belongs as well, since that pair's own swap is a graph automorphism. Its combined contributions to J_i,C_i,q_i are therefore

\[ 2R_j\sin(\Theta_j-\Theta_i),\qquad 2R_j\cos(\Theta_j-\Theta_i),\qquad 2(X_i-X_j). \]

Any ordered neighbor contributes its retained assigned form and phase individually. The same both-or-neither property lets an ordered receiver evaluate its neighboring unordered pairs using 2R_j*sin(Theta_j-theta_v) and 2*(x_v-X_j). Thus all required sums are functions of the mixed state and support.

Exact averaging of the external sine terms gives R_i*J_i; their half-difference is -sin(delta_i)*C_i. An internal edge adds -sin(2*delta_i) to the plus-member sine row and 2*u_i to its form-gradient row. Consequently the unordered rows are exactly

\[ \boxed{\begin{aligned} \dot X_i&=\frac{a\nu}{D_i}R_iJ_i,& \dot\Theta_i&=\frac{b\nu}{D_i}q_i,\\ \dot R_i&=-c_iQ_i,& \dot U_i&=-2A_iQ_i,\\ \dot Q_i&=-A_i(1-R_i^2)+c_iU_iR_i. \end{aligned}} \]

This includes symmetric pairs attached to only one member of an ordered neighboring pair, as well as internal edges. In the special complete/empty interpair support of Section 18.3, substituting m_i=2*d_i recovers its formulas. The more general rows follow from the same microscopic field, rather than an added interaction.

19.3. Realizability, synchronized tips and the chart boundary

The explicit mixed rows are smooth functions of the ambient retained variables in a local midpoint chart. They contain no division by U, Q or 1-R^2. Their derivative of the unordered constraint Q^2-U*(1-R^2) vanishes identically, with the same cancellation as Section 8.2. More strongly, lift any realizable initial state, evolve the original smooth fine law, and project it. This produces a realizable solution of the mixed rows. Local uniqueness of their smooth ambient field implies that every solution from that mixed initial state is this projection, including at a singular invariant tip. Thus the inequalities and the full realizable image are preserved for as long as the phase chart remains valid; the algebraic constraint alone would not have proved this.

For an unordered pair at R=1,U=Q=0, all internal derivatives vanish and the tip remains invariant. This follows from its actual swap symmetry, even when neighboring ordered members differ. At R=1,U>0, the nonzero Q_dot=c_i*U retains the form-driven departure from phase alignment. By contrast, an ordered pair at u=delta=0 need not remain synchronized: unequal attachments can give unequal fine rows immediately. Equality of a pair's instantaneous member states is not a substitute for graph symmetry.

At |delta_i|=pi/2 the circular midpoint becomes ambiguous; for an unordered pair its resultant magnitude is zero. The chosen mixed chart then ends even though the original sine field remains smooth. No new global chart or trapping theorem on arbitrary asymmetric support is supplied here. In numerical admission, a chart margin not certified strictly positive requires refusal, not clipping, sign selection or a tolerance-based assertion of nonantipodality.

19.4. The retained receiver control now discriminates the two states

On the fixed nineteen-edge preparation in Section 18.4, the unordered indices are S={1,2,3}, while O={0,4} retains the attachment-bound member states of (0,1) and (8,9). For the original source, pair (0,1) has

\[ (X_0,\Theta_0,u_0,\delta_0) =(0,1/16,1/8,-1/16). \]

Its state-only swap on the same graph instead has u_0=-1/8 and delta_0=1/16. The mixed initial states are now distinct. The original receiver (8,9) has u_4=delta_4=0 but

\[ \dot\Theta_4=\dot\delta_4=\frac{1}{48\pi}; \]

both signs reverse for the swapped source. Its equal initial member states therefore do not make its ordered internal coordinate redundant. These derivatives reproduce the already known fine-row control; they are not new reserved observations. The new result is the sufficient state and rate construction that retains the cause of their distinction without changing the underlying law.

Independently swapping any subset of pairs 1,2,3 changes neither the mixed circular state nor its rates. A simultaneous relabeling of the graph, fine state and ordered pair identities only relabels the description. Reversing an asymmetric pair's declared ordering while tracking its incidence reverses u,delta and their rates; exchanging its states on a fixed incidence assignment is instead the different fine model state examined by the control. No absolute orientation or extra force is inferred from this bookkeeping.

19.5. Shared observer and scope

assess_sine_mixed_pair_state in the scale owner combines the existing exact support-symmetry assessment with the shared pair-lift chart and complete fine sine rates. It records each pair's ordered or unordered mode, retained coordinates, reconstruction stratum and pushforward rates. The captured fine state remains available as provenance; an unordered pair's signed representative is not an additional retained mixed coordinate.

The theorem concerns exact realized states. Trigonometric interval enclosures in a numerical report are neither exact invariant values nor a joint uncertainty inverse for arbitrary proposed collective data. The reader does not reconstruct a new fine state from independently supplied R,U,Q boxes, silently promote overlap to equality, or integrate an autonomous macro-node runtime. The existing strict replica admission remains separate, and none of its persistence, pulse or recurrence certificates are broadened merely by accepting this mixed description.

The mixed-state contract and independent replica tests cover the full-node rates, the fixed receiver distinction, allowed swaps, simultaneous relabeling, chart refusal and synchronized tips. The unchanged finite observation window from Section 18.5 does not become an all-time grouping theorem. The construction removes a specific information loss under supplied support; it neither creates connections nor proves spontaneous permanent constituents or identifies physical matter.

19.6. A regular phase/rate chart of the existing unordered state

The phase/motion observation has an exact sufficient-state specialization. Use the complete three-node star 0--1,0--2, unit held capacities, fixed unit support, e=0,w=beta=1, no inputs or events, and the conservative sine law. All primes below use tau=t/pi. This support is not the complete replica graph; its coefficients must come from its own nodal rows.

Remove the common form and phase origins and write the two leaf states as

\[ x_\pm-x_0=X\pm u,\qquad \theta_\pm-\theta_0=\Theta\pm\delta,\qquad |\delta|<\pi/2. \]

With R=cos(delta), U=u^2, Q=u*sin(delta), the simultaneous leaf swap removes only a redundant label. As in Sections 8 and 19.1, realizability requires 0<R<=1, U>=0 and Q^2=U*(1-R^2). The six fine coordinates lose two common origins and a discrete label, leaving four continuous degrees of freedom. Neither U nor Q is an additional primitive.

The root rows are x_0'=R*sin(Theta) and theta_0'=-X; the leaf rows give u'=-cos(Theta)*sin(delta) and relative phase rates 2X+/-u. Their exact pushforward is therefore

\[ \boxed{ X'=-2R\sin\Theta,\quad \Theta'=2X,\quad R'=-Q,\quad U'=-2Q\cos\Theta,\quad Q'=UR-(1-R^2)\cos\Theta. } \]

These are Section 19.2's symmetric pair with one common external neighbor, together with that root's actual motion. The constraint is conserved by direct differentiation. The full fine storage is

\[ H=X^2+U+2-2R\cos\Theta,\qquad H'=0. \]

The common degree-weighted form mean is conserved. Given it and an initial common phase origin, the full representative is recoverable: if the mean is m, then x_0=m-X/2, x_+/-=m+X/2+/-u, while the root phase obeys theta_0'=-X. The relative description has not erased a needed external reference for comparison with another system.

Observable inverse. Let z_+/-=exp(i*(Theta+/-delta)) and retain the raw incident sums, without the pair normalization used in Section 8,

\[ Z=z_++z_-=2Re^{i\Theta},\qquad M=(2X+u)z_++(2X-u)z_-=e^{i\Theta}(4XR+2iQ). \]

On the strict regular domain 0<|Z|<2, these four real observations reconstruct the existing quotient:

\[ R=|Z|/2,\quad \Theta=\arg Z,\quad X=\tfrac12\operatorname{Re}(M/Z),\quad Q=R\operatorname{Im}(M/Z),\quad U=\frac{Q^2}{1-R^2}. \]

Choosing either sign of delta reconstructs its matching u; the two lifts differ by precisely the permitted leaf swap. Conversely, every complex pair (Z,M) in this domain has a real fine lift. Rational observable coordinates can require irrational unit phasors; exact observation admission does not restrict the inverse to rational fine coordinates.

This is a faithful local representation of four degrees of freedom, not their dimensional reduction. The inverse of the internal motion also reuses Section 8.7: in the moving midpoint frame the normalized pair moments are Z_pair=R, M_pair=i*c*Q, so R_dot=-c*Q. Its first/second derivative reconstructions retain the known held c, neighboring state and complete law. Raw incident sums, normalized pair phasors and node-relative neighbor resultants share the chain rule but are different observations.

An exact rational field, without inverse angles. Put n=|Z|^2, a+i*b=M/Z, U=b^2*n/(4-n), P=Z/conj(Z) and Z2=Z^2-2P. Here P=z_+*z_- and Z2=z_+^2+z_-^2. Differentiating the full phase row gives each relative acceleration r_j'=-sin(delta_j)-(3/2)*Im(Z). The identities sum(r_j^2*z_j)=(U-a^2)*Z+2*a*M and sum(sin(delta_j)*z_j)=(Z2-2)/(2i) yield

\[ \boxed{ Z'=iM,\qquad M'=\frac{i}{2}(Z2-2)-\frac32\operatorname{Im}(Z)Z +i[(U-a^2)Z+2aM]. } \]

Storage in this chart is H=a^2/4+U+2-Re(Z). The field is smooth on its domain. Fine-law uniqueness and the inverse establish exact local closure until the trajectory leaves the chart, not an all-time domain guarantee.

The two singular boundaries lose different information.

• At coincident leaf phases R=1, Q=0, the observation loses U. With all phases zero, forms (0,0,0) and (0,1,-1) both give Z=2,M=0, but M'=0 and M'=2i. The retained pair state remains sufficient; Section 8.7's U=-R'' at this boundary explains the missing information.
• At antipodal phases z,-z, one has Z=0,M=2u*z. If M!=0, it fixes the unordered phase pair and its associated form difference. The missing scalar is the relative mean form X, not simply the pair orientation: M'=-2*Im(z)*z+4i*X*M. For phases (0,0,pi), forms (-1/2,1,0) and (1/2,0,-1) have the same weighted form mean zero, H=13/4,Z=0,M=1, but M'=4i and M'=-4i. Adding scalar storage does not resolve the sign. If also M=0, orientation is lost as well: zero form with leaf phasors (1,-1) and (i,-i) gives the same observation but M'=0 and -2i.

These are observation failures, not singularities of the fine law or evidence for a new forcing channel. Near either boundary the inverse is ill-conditioned; tolerances, clipping or a noisy derivative estimate cannot replace joint state/observation admission. Section 28 retains the analogous distinction between a zero collective resultant and continuing fine motion.

The shared phase-response owner implements derive_sine_star_moment_closure using exact represented-real moment pairs and rejects the singular domain. Its independent controls evaluate the full nodal rows, both singular witnesses, conserved storage, leaf exchange, origin symmetry and representation limits. The contract specifies the clock, SDK projection and detached evaluation scope.

Finally, the inverse uses geometry and the phase row, not the sine form current. Other supplied common smooth odd periodic edge currents, applied uniformly with the same phase row, preserve the removed symmetries and also have a pushforward on this chart, with their own vector field and storage. Closure here therefore does not select sine, a universal pressure law, persistent identity, autonomous grouping or a physical constituent.

29. Global unordered pair state through phase cancellation

Keep Section 28's current and support: fixed unit doubled C5, consecutive pairs {2a,2a+1} for a=0,...,4, held unit capacities, conservative normalized sine, e=0,w=beta=1, structural time t and no inputs or events. All four fine cross edges join each adjacent pair, and each fine degree is four. The following is a global representation of this same law, including zero resultants. It does not select a law, create a grouping or change support.

29.1. Derived coordinates and exact realizability

For each pair let z_+/-=exp(i*theta_+/-) and define

\[ X=\frac{x_++x_-}{2},\quad u=\frac{x_+-x_-}{2},\quad Z=\frac{z_++z_-}{2},\quad D=\frac{z_+-z_-}{2}, \]

\[ \boxed{P=z_+z_-=Z^2-D^2,\qquad U=u^2,\qquad W=uD.} \]

The retained state is (X,Z,P,U,W). Here X is real form, U has squared form units, W has form units, and Z,P are dimensionless circular-phase observations. None is a complex EPI or an independent constitutive parameter. Exchanging the two members sends (u,D) to (-u,-D) and preserves all five quantities. The complex product P does not require a branch or a midpoint.

Writing n=|Z|^2, the necessary and sufficient realized domain is

\[ \boxed{ X\in\mathbb R,\quad |P|^2=1,\quad Z=P\overline Z,\quad n\le1,\quad U\ge0,\quad W^2=U(Z^2-P). } \]

Necessity follows from unit fine phasors. For sufficiency note first that Z^2=n*P. If U>0, choose u=sqrt(U) and D=W/u; if U=0, the constraint forces W=0, and choose either square root of D^2=Z^2-P. In both cases

\[ D^2=-(1-n)P,\qquad |D|^2=1-n,\qquad \operatorname{Re}(Z\overline D)=0. \]

The last equality follows because (Z*conj(D))^2=-n*(1-n). Hence z_+/-=Z+/-D are unit phasors and x_+/-=X+/-u are a fine lift. The two choices differ only by the simultaneous member swap. This also proves that equal invariant states correspond exactly to that swap orbit, including U=0, coincident phases and antipodal phases. In particular |W|^2=U*(1-n) is a derived consistency identity, not an extra premise.

At n=1, D=W=0 but U still distinguishes unequal forms. At Z=0, P retains the antipodal orientation even when U=W=0. The invariant description has eight real components with constraints and the same four continuous degrees of freedom per pair as the fine state. At swap-fixed states it is not an ordinary smooth coordinate chart. Common phase and form origins remain retained; only redundant member labels are removed.

29.2. The inherited polynomial field

Let neighboring pair indices be read modulo five and define derived quantities

\[ B_a=\frac{Z_{a-1}+Z_{a+1}}2,\qquad h_a=X_a-\frac{X_{a-1}+X_{a+1}}2. \]

The actual fine rows give pi*x_dot_+/-=Im(conj(z_+/-)*B_a) and pi*theta_dot_+/-=h_a+/-u_a. Differentiating the retained coordinates yields

\[ \boxed{\begin{aligned} \pi\dot X_a&=\operatorname{Im}(\overline Z_a B_a),\\ \pi\dot Z_a&=i(h_a Z_a+W_a),\\ \pi\dot P_a&=2i h_a P_a,\\ \pi\dot U_a&=2\operatorname{Im}(\overline W_a B_a),\\ \pi\dot W_a&=i\left[h_aW_a+U_aZ_a+ \frac{(Z_a^2-P_a)\overline B_a-(1-|Z_a|^2)B_a}{2}\right]. \end{aligned}} \]

For example, W_dot=u_dot*D+u*D_dot, and the form contribution satisfies D*Im(conj(D)*B)=[(1-n)*B-D^2*conj(B)]/(2i). This is the last row's source. No division by Z, U, resultant magnitude or a phase angle occurs. The rows are real polynomial functions of the retained components. Their coefficients come from the same fine degree, capacity and law, not from the three-node star or an imposed oscillator.

The previously observed Y=<x*z> is exactly Y=X*Z+W. Thus its contribution to Z_dot is retained, but its own sufficient continuation also needs P,U. On the regular chart, Z=R*exp(i*Theta), P=exp(2i*Theta) and W=i*Q*exp(i*Theta). Substitution recovers all Section 8 rows with unit coefficients, including pi*R_dot=-Q. The old chart is a useful local description of this state, not a different dynamics.

29.3. Storage, current and global continuation

The fine graph storage, expressed without a midpoint, is

\[ F=2\sum_{\{a,b\}\in C_5}(X_a-X_b)^2+4\sum_a U_a,\qquad V=4\sum_{\{a,b\}\in C_5} [1-\operatorname{Re}(\overline Z_a Z_b)],\qquad H=F+V. \]

This is the expansion of sum_fine_edges (x_i-x_j)^2/2 and sum_fine_edges [1-cos(theta_i-theta_j)]. No storage term has been added. Both components are nonnegative on the realized set, H_dot=0, and sum_a X_a is conserved. The individual component rates can be nonzero.

For the directed neighbor contribution b->a, retain the existing mean-form normalization I_ba=Im(conj(Z_a)*Z_b)/(2*pi). If G_a=pi*Z_dot_a, then

\[ \pi^2\dot I_{b\to a} =\tfrac12\operatorname{Im}(\overline G_a Z_b+\overline Z_a G_b). \]

This formula is valid at cancellation. The degree/capacity weighted cut contribution is still 8*I_ba; the phasor representation does not change the regional ledger or interpret form current as energy current.

To establish continuation, lift any admitted invariant state to the fine state just constructed. The smooth sine law is equivariant under each permitted pair swap, so either lift projects to the same invariant solution. Local uniqueness of the ambient polynomial field makes that projection the unique solution from the realized datum, even at singular strata of the invariant set. Realizability is preserved by this argument, not inferred from a small constraint residual. On connected C5 the conserved H and sum X bound all forms and U; |Z|<=1, |P|=1 and |W|^2=U*(1-|Z|^2) bound the remaining coordinates. Consequently the realized solution continues for every finite positive or negative time. This is neither global stability of a chosen pattern nor a claim that arbitrary unconstrained ambient initial data describe fine nodes.

29.4. Why cancellation needs the retained phase product

On the same doubled C5, put all forms zero and all surrounding pair phasors equal to one. Compare one selected pair (z_+,z_-)=(1,-1) with (i,-i). Both have X=Z=U=W=Y=0 there, the same surrounding retained state, full storage H=8, all block currents zero and Z_dot=0. Only P differs: it is respectively -1 and +1. The last row gives

\[ \pi\dot W=0\quad\hbox{versus}\quad -i,\qquad \pi^2\ddot Z=0\quad\hbox{versus}\quad 1. \]

Thus (X,Z,Y) or (X,Z,U,W) alone is not globally sufficient. This counterexample uses the complete original law, not an added force or hidden event. Likewise, coincident phases with different U have the same X,Z,P,W and the same initial Z_dot, but different subsequent motion, already in Z_ddot; changing the sign of W at fixed nonzero Z,U,P changes the form/phase association and Z_dot.

The raw second circular moment is z_+^2+z_-^2=4Z^2-2P. Its information can therefore be required by a global collective state even though the sine current consumes first phasor moments. This does not introduce a second-harmonic pressure channel or select a fundamental interaction law.

29.5. Implementation and numerical scope

The shared pair owner provides derive_sine_global_pair_state from ten exact unit phasors and signed forms, and evaluate_sine_global_pair_state from the five invariant tuples after full realizability admission. Both use the same inherited field. Outputs retain exact numerators of the original-clock rates, storage components and directed current observations; SDK export preserves their rational values. The contract specifies ordering, admission and the distinction from graph capture.

Eliminating a retained pair instead reuses the existing nonlinear mediator identity twice. Its members have no mutual edge and each has the same four visible neighbors. Conditional on those neighbors' histories, use k=4,mu=w=beta=1,e=0 for each hidden member, retaining both initial states, original visible degrees and both form/phase contributions. The zero-loss factor is F_0(t)=t; the identity introduces neither decaying memory nor a time-scale separation. This reuse does not justify replacing the hidden preparation by a stationary state or transferring a different support's approximation bounds.

These evaluators provide neither a time integrator nor a numerical trajectory certificate. A finite Euler update of constrained invariants need not remain realizable; no projection or tolerance repair is supplied. Existing regular graph/scale adapters retain their own input and phase domains. General support, unequal capacities, native Arg pressure, occurrence of grouping and physical identification require separate results. The independent controls differentiate all fine rows, test the boundary strata, full storage, symmetry, realizability and exact export without replaying a frozen experiment.

29.6. Observable internal state at cancellation

Retain exactly Section 29's complete law, support and unit coefficients. Observe all pair interfaces (X,Z) with their ordering and clock; hold the surrounding retained preparation fixed when comparing two selected-pair states. Primes in this subsection use tau=t/pi, so Z'=pi*Z_dot. The derivatives below are exact causal left derivatives from an already observed interval, or separately supplied exact evidence. Their existence as mathematical quantities does not make them available from noisy finite samples.

At an instant with selected Z=0, put v=Z' and a=Z''. Realizability and the inherited rows give

\[ \boxed{W=-iv,\qquad U=|v|^2,\qquad a=2ihv+\frac{B+P\overline B}{2}.} \]

Indeed Z'=iW, |W|^2=U and W^2=-UP at cancellation; differentiating Z'=i(hZ+W) gives the last identity. Consequently:

• If v!=0, the first derivative recovers P=v^2/|v|^2, as well as U,W. The supplied second derivative must satisfy the last identity above.
• If v=0, then U=W=0. When B!=0, the second derivative recovers P=(2a-B)/conj(B). This remains informative when a=0: the unique compatible orientation is P=-B/conj(B).
• If v=B=0, the second derivative must be zero. Every unit P is compatible with these local observations. This is a two-derivative ambiguity, not yet a statement about a whole history.

Every recovered P must have unit norm. These are conditional inverse results for the supplied law, not an independent validation of that law. In particular, the reconstruction does not infer pressure by fitting the evaluated response.

Delayed neighborhood information. Suppose v=0 and the first nonzero neighborhood derivative is B^(m), with all lower derivatives zero. Repeated differentiation of the inherited rows gives

\[ Z^{(k)}=0\ (0\le k\le m+1),\qquad Z^{(m+2)}=\frac{B^{(m)}+P\overline{B^{(m)}}}{2},\qquad \boxed{P=\frac{2Z^{(m+2)}-B^{(m)}}{\overline{B^{(m)}}}.} \]

To see the derivative order directly, choose a unit d with d^2=-P_0 and put H(tau)=integral h, with integral starting at the observation instant. The rotating-frame fine lift can be written

\[ e^{-iH}Z=id\sin\eta,\quad u=\eta',\quad b=e^{-iH}B,\quad \eta''=\operatorname{Im}(\overline d b)\cos\eta, \qquad \eta_0=\eta'_0=0. \]

Multiplication by exp(-iH) preserves the first nonzero derivative order of B. Thus eta and Z vanish through order m+1; differentiating at order m+2 gives the displayed identity. Also X-X_0=O((tau-tau_0)^(2m+3)) from X'=Im(conj(Z)B), so mean form reveals no earlier orientation information. In the original clock the inverse numerator is 2*pi^2*d_t^(m+2)Z-d_t^m B, divided by conj(d_t^m B).

The finite autonomous field is real analytic on every realized trajectory. Either this finite order exists or B vanishes identically on the connected trajectory. A finite zero jet does not establish the latter without an independently proved sufficient-order or invariance argument. No uniform sufficient derivative order is established here. Analyticity also prevents exact silence on an open time interval followed by spontaneous reactivation on the same unchanged trajectory. Delayed revelation above concerns a finite zero jet at one instant. Comparing differently prepared autonomous neighborhoods is a separate experiment, not a later intervention silently added to the law.

Persistent silence and indistinguishability are different. The same lift and uniqueness of the fine equations imply, on an interval containing the initial instant,

\[ \boxed{Z\equiv0\quad\Longleftrightarrow\quad b+P_0\overline b\equiv0\quad\Longleftrightarrow\quad b(\tau)\in\mathbb R d.} \]

This tests the actual evolving neighborhood, not an externally held source. A nonzero b confined to a fixed real line permits exactly one silent orientation. A b not confined to such a line permits none. When b=0 identically, all antipodal orientations remain silent. The primitive phases may still rotate: P=exp(2iH)*P_0. Silence is not full equilibrium.

If identical interface histories become nonzero in Z anywhere, realizability gives P=Z/conj(Z) there; their common h transports this equality back to the initial instant. Thus distinct indistinguishable initial orientations must have Z=0 throughout. Two such orientations require B=0 identically: subtracting B+P_1*conj(B)=0 and B+P_2*conj(B)=0 proves it, since their phase products retain the same nonzero difference up to exp(2iH). Conversely, if one such trajectory has B=0, changing only its initial P preserves the selected (X_0,0) and the entire surrounding evolution. The surrounding rows consume that pair only through X,Z; uniqueness supplies this converse. Hence permanently unobservable orientation is an exact environmental restriction, not a generic consequence of a zero snapshot resultant.

Whole-network witnesses. The following preparations use the actual doubled C5, with pair 0 selected and omitted within-pair differences zero:

• For an invisible continuum, set every form to zero, pair 0 to (d,-d), pairs 1 and 2 to (1,1), and pairs 3 and 4 to (-1,-1). Every unit d modulo sign gives a stationary solution with identical interfaces and identical surrounding retained state. Nonzero surrounding resultants do not preclude their cancellation at the selected pair.
• For an invisible moving family, make every pair antipodal and assign equal within-pair forms with arbitrary means X_a. Then Z=U=W=B=0 throughout, the means stay constant, and P_a=exp(2ih_a*tau)*P_a(0) can rotate.
• For delayed revelation, take pair 0 as (d,-d) with both forms zero and Z_1=Z_2=Z_3=1,Z_4=-1. With X_1=2 and all other means zero, B_0=0,B'_0=i, and Z'''_0=i(1-P_0)/2. Alternatively, all means zero and U_1=1 give B_0=B'_0=0,B''_0=-1/2 and Z''''_0=-(1+P_0)/4.
• Initial alignment alone is insufficient: choose pair 0 as (1,-1), all surrounding phasors one, X_1=2, and other means zero. Here B_0=1, Z'_0=Z''_0=0, but Z'''_0=2i.

These are exact preparations and derivative consequences, not formation, attraction or robustness theorems. They establish neither material screening nor a quantum state. The shared observe_sine_pair_cancellation evaluator implements the two-derivative inverse and rejects inconsistent evidence. Its unavailable branch preserves the higher-order/persistent distinction; it does not certify either from a finite zero record. See its contract, usage and independent fine-row controls. The finite exchange result perturbs the stationary invisible family above and compares two retained orientations under the same complete law. It supplies an exact stationary control and an analytic finite bound on actual directed form transfer; the phase-derivative distinction alone is not used as the finite claim. The receiver readout instead bounds one neighboring mean-form endpoint with declared positive preparation and readout errors. It discriminates two specified orientation families without reconstructing arbitrary hidden state or consuming phase derivatives as observations. The joint law/orientation obstruction retains that fixed-law result while testing a different premise: an unknown constitutive coefficient can compensate orientation in this one endpoint.