Phase and joint form-phase grouping

State-derived pair observations, fully acute equilibrium geometry, finite grouping windows and joint-identity episodes; observation does not create support.

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.

14. State-derived phase partners and independent law admission

The pair partition in Sections 7–13 specifies which collective description is being tested. This section asks a separate question: can the phase state itself identify those pairs without consuming the proposed partition, support edges, capacities, forms or a label convention? On the protected doubled-C5 family the answer is yes. On a general snapshot the same observation must be allowed to abstain, and a detected partition still needs independent support and complete-law admission.

14.1. A branch-independent comparison of circular distances

For any two fine nodes define their circular distance and squared chord,

\[ d_{\mathbb S^1}(\theta_p,\theta_q) =\min_{n\in\mathbb Z}|\theta_q-\theta_p+2\pi n| \in[0,\pi], \]

\[ D_{pq}=|e^{i\theta_q}-e^{i\theta_p}|^2 =2[1-\cos(\theta_q-\theta_p)] =4\sin^2\!\left(\frac{d_{\mathbb S^1}(\theta_p,\theta_q)}2\right). \]

The function d -> 2*(1-cos(d)) is strictly increasing on [0,pi]. Thus comparing squared chords gives exactly the same nearest-partner ordering as comparing circular distances. It requires no inverse trigonometric function, common phase origin, branch cut or choice of real phase lifts.

This observation is invariant under common circular rotation, integer full-turn changes of phase representatives, simultaneous reversal of phase orientation, and permutation of node labels. The returned partition is an unordered set of unordered pairs; serialization order is not an extra physical choice. These are mathematical invariances of the admitted phase state. Adding a large common floating-point offset can discard previously represented phase differences; such a changed capture is not an exact symmetry control.

14.2. The protected tube separates true partners

Use the exact winding-one doubled-C5 target and phase chart of Section 12,

\[ \theta=\theta_*+c\mathbf1+h\pmod{2\pi},\qquad \|h\|\le Z_f<r,\qquad \alpha=\frac{2\pi}{5},\qquad \alpha+\sqrt2r<\frac\pi2 . \]

For the two members p,q of one target pair their target phases coincide. Cauchy–Schwarz therefore gives

\[ d_{\mathbb S^1}(\theta_p,\theta_q) \le |h_q-h_p| \le\sqrt2\,\|h\| <\sqrt2r<\frac\pi{10}. \]

For nodes in different target pairs the target circular distance is either alpha or 2*alpha. The triangle inequality on the circle, applied after canceling the common origin, gives

\[ d_{\mathbb S^1}(\theta_p,\theta_q) \ge d_{\mathbb S^1}(\theta_{*,p},\theta_{*,q}) -d_{\mathbb S^1}(h_p,h_q) >\alpha-\sqrt2r>\frac{3\pi}{10}. \]

Every node consequently has exactly one nearest phase neighbor: its other constituent. Each choice is mutual. The inequalities also give the strict chord separation

\[ \max_{\text{within pair}}D_{pq} <2[1-\cos(\sqrt2r)] <2[1-\cos(\alpha-\sqrt2r)] <\min_{\text{between pairs}}D_{pq}. \]

The target and partition appear here to prove recovery, not as inputs to the nearest-partner observation. The proof applies equally to the reversed winding. It consumes no capacity equality. In the unequal-positive-held-capacity family U_rho of Section 13, the same full-state barrier preserves the phase tube for every real time. Hence the phase-only observation identifies the same constituent pairs throughout the full trajectory, including possible internal stalls or tip crossings. This is an all-state identification consequence of geometric trapping, independent of almost-everywhere recurrence.

No common frequency, internal angular monotonicity or threshold for declaring a link was needed. The numerical observer need not consume r or either angular threshold: it only compares the actual phase distances.

14.3. A prospective nearest-partner certificate and its abstentions

For a finite captured phase list, let [L_pq,U_pq] enclose each squared chord. A sufficient certificate that q is the unique nearest neighbor of p is

\[ U_{pq}<L_{pk}\qquad \text{for every }k\ne p,q . \]

When each node has one such certified neighbor, retain the full partition only if those choices are mutual. A complete mutual map has no fixed points and partitions the nodes into disjoint pairs. This does not require selecting a distance cutoff, minimizing a global matching cost or fitting a grouping scale.

Strict comparison is essential. Equal distances do not specify one partner, and overlapping enclosures do not prove either equality or unique ordering. Odd cardinality, unmatched nodes and nonmutual nearest choices prevent a complete pair partition. In those cases the observation abstains; it does not break ties using node names, greedily rematch unused nodes or borrow a partition from graph topology.

For example, the four exact phases (0,1/10,3/10,1) lie within one short circular arc and each has a unique nearest neighbor. The first two choose one another, the third chooses the second and the fourth chooses the third. The choices do not define a complete mutual pairing. Three coincident phases instead give actual nearest-neighbor ties. Both are admissible circular observations, but neither justifies an arbitrary choice of pairs.

The geometric theorem establishes exact separation throughout its admitted family. A fixed-precision implementation still must enclose the captured distances and prove its strict inequalities; numerical range limits or wide enclosures can cause abstention. Conversely, an unambiguous match outside that family is only an instantaneous observation. It supplies no future partner guarantee by itself.

14.4. Phase pairing does not certify the paired support

After the phase observation, the proposed pairs must independently satisfy the existing complete-replica support contract: every fine node occurs once, there are no within-pair edges, each active base edge has all four cross-edges, and the remaining law, capacity and phase-chart hypotheses hold. This check may inspect edges; the preceding observation may not. An asymmetric-capacity consumer must retain the correlations of Section 13 rather than silently invoking an equal-capacity theorem.

A state-only permutation supplies an explicit discriminating control. Start with the actual doubled-C5 support and give both members of pair i the same phase phi_i, using five distinct circular phases. Now exchange only the phase attributes of nodes (0,-) and (1,-). Leave graph edges, node labels, capacities and forms untouched. The unique observed zero-distance pairs become

\[ \{(0,+),(1,-)\},\quad \{(1,+),(0,-)\}, \]

with the other three pairs unchanged. Every node still has one unique mutual phase partner. Yet both new pairs contain live edges, because the original base edge 0--1 carried all four fine cross-edges. They fail the no-within-pair-edge contract immediately.

This control works with exact distinct represented phases such as phi_i=1287*i/1024. It does not claim they equal the symbolic critical target. A phase-only permutation is not a graph relabeling; relabeling the entire graph and all attributes would preserve admission. Replacing the observed pairs by the old graph twin classes would hide the intended incompatibility and is not an allowed repair.

After successful support admission, a target-dependent family report also needs an explicit order and orientation of the five observed pairs, and declared lifts of the captured phases. Nearest pairing alone fixes neither a particular cyclic starting pair nor a chosen winding sign. Those declarations must be recorded before applying the target's norm/storage certificate; they are not inferred by quietly reordering phases until a certificate succeeds.

The result therefore distinguishes identification, law admission and persistence. It recognizes an organization already present in the state, verifies whether the supplied support permits its retained collective description, and uses a separate invariant-tube certificate for future identification. Conservation protects the admitted phase geometry; it is not a new force creating a pair, an edge or a material constituent.

14.5. Observation and admission interfaces

The scale owner provides observe_phase_pairs(nodes=..., phases=...) and PhasePairObservation. This standalone observation reads only the labeled phase list. Its squared-chord bounds, certified nearest indices, strict margins and abstention reasons retain the numerical evidence. The 64-node limit bounds quadratic arithmetic work; it is neither a physical length scale nor a grouping threshold. Exact ties recognized from equal absolute raw phase gaps are sufficient tie controls, not an exhaustive classification of every circular tie.

assess_sine_state_pairing and SineStatePairingAssessment then reuse one graph capture. The phase projection supplies the observed candidate; the existing capacity-aware owner independently checks its support, law and supplied phase lifts. A caller's pair_order must agree with that observed unordered partition, and is mandatory for target-dependent family evidence. Without family bounds, capture order merely arranges the result for display. No automatic unwrapping or topological substitution is performed.

The wrapper can preserve a valid phase proposal while reporting its independent collective-law admission as rejected. When its optional full-state geometry certifies trapping, the separation theorem justifies all_time_pairing_persistence_certified. This flag does not certify formation, internal circulation or recurrence of the selected state. Membership in a smaller declared excess/mean family remains separately reported by the reused owner. The state-pairing contract owns the public admission and reporting details.

The replica tests separate phase-only comparisons, exact and unresolved ambiguities, nonmutual choices, complete relabeling, independently refused support and protected-family admission. These are detached observations and conditional reports; neither interface evolves the graph.

15. Exhaustive fully acute equilibria on the doubled cycle

15.1. Complete equations and the sector being classified

Take the fixed complete doubled-C5 graph with its twenty unit edges and no other edges. Its structural pairs have identical neighbor sets; pair i is adjacent to both constituents of pairs i-1 and i+1 modulo five. This is a fact about the supplied support, not a partition inferred by the phase observer.

Keep e=0, a=w/pi>0, b=w/(beta*pi)>0, arbitrary strictly positive held fine capacities and no input or event. With the fine combinatorial Laplacian L_f, define

\[ K=\operatorname{diag}(\nu_p/4)>0,\qquad S_p(\theta)=\sum_{q\sim p}\sin(\theta_q-\theta_p). \]

The complete law is

\[ \dot x=aKS(\theta),\qquad \dot\theta=bKL_f x . \]

Full equilibrium requires both rows to vanish. A vanishing form rate or instantaneous sine balance alone is not this condition. Restrict the classification to configurations in which every actual fine-edge principal phase gap lies in (-pi/2,pi/2). Equivalently, every edge cosine is strictly positive. This is an explicit sector restriction; it does not follow merely from the model being conservative.

The exact classification is

\[ \boxed{ x_{i,+}=x_{i,-}=m,\qquad \theta_{i,+}=\theta_{i,-} =c+\frac{2\pi k i}{5}\pmod{2\pi}, \quad k\in\{-1,0,1\},} \]

where m is any real form origin and c any common circular phase origin. Neither constant form, member-phase equality nor the integer winding is assumed in deriving this list.

15.2. Zero phase rates force uniform form

At full equilibrium, b*K*L_f*x=0. Both b and every diagonal entry of K are positive, so L_f*x=0. The fine graph is connected; equivalently,

\[ x^\mathsf TL_f x=\sum_{\{p,q\}\ {\rm fine\ edge}}(x_p-x_q)^2=0 \]

forces equality along every edge and hence x=m*1. Capacity heterogeneity affects the motion away from equilibrium but cannot change this kernel argument. Strict positivity is essential; the proof is not transferred to frozen zero-capacity rows.

15.3. Member-phase equality follows without a pair chart

For a structural pair i form the common neighbor phasor

\[ Z_i=\sum_{q\in N_i}e^{i\theta_q}. \]

Both constituents have exactly this neighbor set. For either member p,

\[ e^{-i\theta_p}Z_i =\sum_{q\in N_i}\cos(\theta_q-\theta_p) +i\sum_{q\in N_i}\sin(\theta_q-\theta_p). \]

The zero form rate and positive capacity imply that its imaginary part vanishes. All four incident edge cosines are strictly positive, so its real part is positive. Thus Z_i!=0 and

\[ e^{i\theta_p}=\frac{Z_i}{|Z_i|}. \]

Applying this equation to the other member proves equality of their phases on the circle. No midpoint, phase unwrapping, equal-capacity assumption or preselected synchronized pair state was used. This complex-number identity concerns the sine critical equations; it does not invoke the separate native resultant-pressure runtime.

The common member phase may now be denoted Theta_i. Its coincident-member chart is a consequence of equilibrium and acuteness.

15.4. Acute sine balance and circular closure give exactly three windings

Let gamma_i be the principal phase increment from pair i to i+1. Every gamma_i belongs to (-pi/2,pi/2). Each fine node has two neighbors in each adjacent pair, so its remaining sine-balance equation is

\[ 2\{\sin\gamma_i-\sin\gamma_{i-1}\}=0 . \]

Sine is strictly increasing on the acute interval. Hence every gamma_i equals the same gamma. Circular closure gives

\[ \prod_{i=0}^4e^{i\gamma_i}=1,\qquad 5\gamma=2\pi k\quad(k\in\mathbb Z). \]

The strict acute bound becomes 4*abs(k)<5. Its only integer solutions are k=-1,0,1, and reconstructing the consecutive phases gives the claimed family.

Conversely, choose any of those integers, any m,c and any positive held fine capacities. Uniform form gives zero phase rates. At every fine node the two forward sine terms and two backward sine terms cancel exactly, giving zero form rates. All fine edge gaps are 0 or +/-2*pi/5 and are strictly acute. Thus every listed state is a full equilibrium and every fully acute equilibrium is listed. The converse uses exact angles, not small computed residuals.

These are three signed winding families relative to a declared base orientation. Reversing that orientation interchanges +1 and -1. The two twists also have identical edge cosines and storage,

\[ E_{*,k}=20\beta[1-\cos(2\pi k/5)] . \]

Consensus has zero storage and the two twists have equal positive storage. These facts do not select an equilibrium's occurrence under the conservative law. They are not three physical species or spatial pentagons. The support specifies relational adjacency, while the classified geometry is a pattern of circular phases on that support.

15.5. Exact families, captured phases and observer abstention are distinct

Consensus is a valid full equilibrium although every node has nine equally close phase neighbors. The phase-only observer therefore abstains rather than selecting the support's structural pair classes. There is no contradiction: the classification uses support to prove which complete equilibria exist; the observer asks what phase alone identifies. At either exact nonzero twist, the only zero-distance neighbor is the other constituent, so the phase observation identifies the pairs. Nearby protected captures can also have unique partners without being critical states.

There is a useful exact arithmetic boundary for the current capture format. All captured raw radians are rational numbers: integers, Fractions and finite binary values have that property. If a captured adjacent phase difference represented an exact nonzero twist, it would have to satisfy

\[ \theta_q-\theta_p=2\pi(n+k/5),\qquad n\in\mathbb Z,\quad k=\pm1 . \]

The left side is rational. The right side is irrational because n+k/5 is a nonzero rational and pi is irrational. Thus no such raw-radian capture is exactly a nonzero member of this critical family. Exact symbolic turns and rounded radians are different inputs. Similarly, captured rational phases equal on the circle must have identical raw values: a nonzero rational difference cannot equal 2*pi*n.

It follows that, within a certified fully acute captured sector, exact full equilibrium is equivalent to exact uniform captured form and exact equality of all captured raw phases. This yields separate valid source verdicts:

• Nonuniform captured form excludes full equilibrium on this connected positive-capacity support, independently of any acute-sector decision.
• Uniform form and identical captured raw phases certify exact consensus.
• Uniform form, certified fully acute edges and nonidentical captured raw phases exclude exact critical membership by the preceding classification and arithmetic argument.
• Uniform form outside, or not certified inside, the acute sector does not receive an equilibrium classification from this theorem.

The third verdict does not measure the size of a response, exclude proximity to a protected target, or imply numerical or dynamical instability. A rounded twist can be extremely close to equilibrium while failing exact membership. No tolerance promotes that approximate state to the symbolic critical family.

15.6. The strict sector boundary and what has not been selected

Acuteness cannot be removed from the classification. As an exact symbolic control, set uniform form, give one constituent phase pi, and give every other fine node phase zero. Every edge sine is zero, so both complete rows vanish for any positive capacities. The two members of the selected structural pair have different phases, and its incident pi gaps have negative cosine. This is an equilibrium outside the classified sector. Replacing pi by a rounded binary value would not be the same exact counterexample.

The theorem classifies neither nonacute equilibria nor arbitrary nonstationary organizations. The pulse and protected families studied earlier are not required to be equilibria at every time. Existing acute-barrier results can be reused at the listed targets with their own hypotheses; this enumeration supplies no attraction theorem, global stability classification or choice among consensus and twists. It also supplies no route from an arbitrary preparation into the two-sided invariant protected family.

What has been derived is the complete list of fully acute critical phase geometries conditional on this law and supplied support. The mechanism selecting a preparation, changing support or producing a particular organization remains a separate question.

15.7. Shared classification and captured-state evidence

The scale owner exposes assess_sine_replica_equilibria and SineReplicaEquilibriaAssessment. It retains one full source capture and validates the declared structural pair order against the complete support, with arbitrary positive fine capacities. Classification does not require a supplied source pair-phase chart or a successful phase-only grouping.

The integer condition 4*abs(k)<5 generates the candidate windings. Each SineReplicaEquilibriumTarget reuses the phase geometry owner for exact-turn reconstruction and symbolic odd-sine cancellation. Only those algebraic properties are consumed; its separate phase-law interpretation is not transferred to the complete model.

The report keeps this exhaustive symbolic classification separate from the captured edge-cosine enclosures, exact form/phase equalities and source equilibrium verdict. A nonacute source does not invalidate the conditional family theorem, and a small source residual does not prove target membership. Nonuniform form provides a global exclusion through the phase row; otherwise an uncertified acute sector remains outside the captured-state classification unless exact consensus is present.

The replica tests check independent fine-row cancellations, the complete integer list, capacity independence, exact consensus, rounded nonzero twists and the nonacute boundary. The reader does not choose a winding, repair a capture, evolve a trajectory or replace the independent phase-only observer with topological pair recovery.

16. Local onset and loss of a support-compatible observed grouping

16.1. Differentiate the observation using the complete law

The squared-chord observation of Section 14 is

\[ D_{pq}=2[1-\cos(\theta_q-\theta_p)] . \]

Along a smooth solution of the complete sine law its exact derivative is

\[ \boxed{ \dot D_{pq} =2\sin(\theta_q-\theta_p)(\dot\theta_q-\dot\theta_p).} \]

This formula is circular and does not differentiate an arbitrary wrapping branch. The phase observer itself still consumes only phase. Its rate, however, needs the existing complete phase law. For fine degree d_p, write the exact rate numerator

\[ n_p=\frac{w}{\beta}\frac{\nu_p}{d_p}(L_f x)_p, \qquad \dot\theta_p=\frac{n_p}{\pi}. \]

Then

\[ \dot D_{pq} =\frac{2}{\pi}\sin(\theta_q-\theta_p)(n_q-n_p). \]

The form, capacities, actual neighbors and structural clock enter through n. They cannot be reconstructed from the phase-distance snapshot alone. These are rates of the existing observation, not a new force, controller or event law.

For a node p comparing partners q and r, define the preference margin

\[ M_{p;q,r}=D_{pr}-D_{pq}. \]

A positive margin favors q. If it is zero at the preparation, a strictly positive derivative proves that this particular tie is resolved toward q for all sufficiently small positive times. To infer a complete pairing, every tied choice and every other competitor must also be accounted for.

16.2. One frozen represented preparation

Use the fixed complete unit doubled-C5 support with structural pairs

\[ (0,1),\ (2,3),\ (4,5),\ (6,7),\ (8,9). \]

Each adjacent base-pair connection contains all four fine edges and there are no other edges. Keep e=0, beta=1, all held capacities one, the structural clock, and no input, clipping or event. The normalized zero-loss repository reference has w=1; retaining b=w/pi>0 in the formulas also displays the clock factor.

Freeze d=u=1/8 and the exact initial values

\[ \theta(0)=(0,d,2d,3d,1,1,2,2,3,3), \]

\[ x(0)=(u,-u,u,-u,0,0,0,0,0,0). \]

These dyadic values are represented exactly. Their d,u symbols name the fixed preparation, not scanned parameters. This state is not an equilibrium or an assumption of fully acute fine edges: for example the live edge 0--8 has gap 3>pi/2. The full smooth sine law applies; the native resultant-pressure law and the acute equilibrium classification are not substituted for it.

Each structural pair has zero form sum. Every fine node's four neighbors consist of two complete adjacent pairs, whose total form is therefore zero. Since the fine degree is four,

\[ L_f x(0)=4x(0),\qquad \dot\theta(0)=b\,x(0) =(\omega,-\omega,\omega,-\omega,0,0,0,0,0,0), \quad \omega=bu>0 . \]

These are the actual initial phase rows of the full nonlinear system. Form continues to evolve through xdot=a*K*S(theta); no frozen form or straight-line phase extrapolation is being treated as a trajectory.

16.3. All initial competitors and the two transverse ties

The full phase span is 3<pi, so every initial circular distance is the ordinary absolute difference of these displayed radians. Put F(s)=2*(1-cos(s)) for 0<=s<=pi. The complete initial nearest sets and closest outsider distances are:

Node or nodes	Initial nearest set	Nearest circular distance	Smallest distance to an outsider
0	{1}	d	2d
1	{0,2}	d	2d
2	{1,3}	d	2d
3	{2}	d	2d
4,5	The other member of (4,5)	0	1-3d=5d
6,7	The other member of (6,7)	0	1=8d
8,9	The other member of (8,9)	0	1=8d

Every comparison against an outsider has strictly positive chord margin at least

\[ g_0=F(2d)-F(d)=2(\cos d-\cos2d)>0 . \]

For the last three pairs their outsider margins are larger: F(5d)>F(2d)>g_0 or F(8d)>F(5d). Thus the table checks all competitors, not just the adjacent ties. At time zero the complete phase-pair observer abstains because nodes 1 and 2 each have two equally near choices.

The exact chord rates for the three tied-length gaps are

\[ \dot D_{01}(0)=\dot D_{23}(0)=-4\omega\sin d,\qquad \dot D_{12}(0)=4\omega\sin d . \]

Define the two margins favoring the structural partners,

\[ M_1=D_{12}-D_{10},\qquad M_2=D_{21}-D_{23}. \]

They obey

\[ M_1(0)=M_2(0)=0,\qquad \dot M_1(0)=\dot M_2(0)=8bu\sin d>0 . \]

For the normalized w=1 reference and frozen u=d=1/8, the common margin derivative is exactly sin(1/8)/pi. Positivity follows from 0<1/8<pi, not from a small sampled trajectory or a fitted tolerance.

16.4. A complete local transition follows from smoothness

All finitely many distances and preference margins are smooth along the actual solution. The strict outsider margins therefore remain positive on some two-sided neighborhood of time zero. For the two ties,

\[ M_j(t)=8bu\sin(d)\,t+o(t)\qquad(j=1,2). \]

There exists a common epsilon_t>0 such that the two margins are positive for 0<t<epsilon_t and negative for -epsilon_t<t<0, while all outsider comparisons retain their initial signs.

For every sufficiently small positive time, the unique nearest map is consequently

\[ 0\leftrightarrow1,\qquad 2\leftrightarrow3,\qquad 4\leftrightarrow5,\qquad 6\leftrightarrow7,\qquad 8\leftrightarrow9 . \]

This is a complete mutual matching. It equals the actual structural pair partition, so its independent support check succeeds: no matched pair contains an edge, and adjacent matched blocks have all four cross-edges. Initial within-pair phase gaps are d,d,0,0,0<pi; continuity also preserves their local pair charts after shrinking the same unspecified neighborhood if necessary. The retained collective law thus applies to this observed grouping.

For every sufficiently small negative time, the first four choices instead are

\[ 0\longmapsto1,\qquad 1\longmapsto2,\qquad 2\longmapsto1,\qquad 3\longmapsto2 . \]

The remaining three pairs retain their unique mates. Every node has a unique nearest neighbor, but the choices of 0 and 3 are not mutual. No complete mutual-nearest partition exists. A greedy rematching would change the observation contract rather than complete this proof.

The conclusion concerns exact mathematical matching at every sufficiently small time on either side. Fixed arithmetic enclosures may abstain at times extremely close to the tie, where strict margins are too small to resolve. The theorem supplies neither a numerical value of epsilon_t nor a certified finite endpoint, forecast or sampled trajectory. A first derivative alone cannot supply any of those.

There is also a qualitative robustness consequence for full initial states, with the law, support and capacities held fixed. Choose any two nonempty closed time windows strictly inside the proved negative and positive intervals. On each window, every node's chosen nearest partner is separated from every competitor by a strictly positive margin. The minimum of these finitely many continuous margins on the compact windows is positive. Joint continuity of the smooth flow in time and initial state therefore gives an open neighborhood of the supplied full preparation for which all those inequalities persist on both windows. Every preparation in this neighborhood has the same nonmutual backward choices and complete support-compatible forward matching.

This does not preserve the exact simultaneous tie at time zero. Locally the two initial tie equations reduce to theta_0-2*theta_1+theta_2=0 and theta_1-2*theta_2+theta_3=0, two independent conditions. Perturbations may separate their crossing times; there is no single common transverse section asserted for the whole neighborhood. Neither the windows nor the preparation radius have numerical bounds here. The argument establishes robustness of the strict observations away from the crossing, not an implemented finite-window certificate.

16.5. Form reversal controls the direction with the same phase geometry

Now reverse every form coordinate at the preparation while preserving the exact phases, capacities and support:

\[ \widetilde x(0)=-x(0),\qquad \widetilde\theta(0)=\theta(0). \]

The initial phase observation, its exact ties and all outsider margins are unchanged. Linearity of the phase row in form gives dot(theta_tilde)(0)=-dot(theta)(0). Both tied margin derivatives are therefore -8bu*sin(d)<0. Complete mutual pairing holds locally on the negative side and nonmutual choices locally on the positive side for this control.

This reversal is also an exact identity of the complete conservative flow. If (x(t),theta(t)) solves xdot=a*K*S(theta), thetadot=b*K*L_f*x, then

\[ \widetilde x(t)=-x(-t),\qquad \widetilde\theta(t)=\theta(-t) \]

solves the same equations with the reversed initial form. Direct differentiation proves both rows. Smooth uniqueness identifies it with the control solution, so its entire local phase-distance history is the original history reversed in time. This relies on e=0 and the absence of time-directed forcing or events; it is not a transfer to a positive-loss runtime.

The two supplied states thus have identical instantaneous phase geometry and opposite local directions of grouping change. Form is necessary predictive information here. Conservation does not choose one direction, and the emergence of a distance ordering is not creation of a law that causes it.

16.6. What has appeared and what has not

This result proves local onset and local loss of observability of a support-compatible collective grouping under the existing unforced fine dynamics. The nodes and all twenty edges were already present. No edge is added, no operator event is selected, and no constituent capacity is changed.

Nor is this first entrance into the earlier protected family. That family is invariant in both time directions, so a point outside it cannot enter it at a finite time. In particular the tie preparation cannot lie in the protected winding-one tube, whose strict separation already identifies partners at every time. The present local interval therefore supplies no acquisition of permanent trapping, long-term pair identity, attracting waveform or material constituent.

It does establish a concrete same-law mechanism: retained form fixes phase velocity, phase velocity changes nearest-distance margins, and those margins can resolve or destroy a complete compatible observation. A finite observation window or a stronger formation claim would require its own independently declared prediction and additional proof. Section 17 supplies the former at an explicitly frozen budget.

16.7. Implementation and evidence

assess_sine_pairing_transition in the shared scale owner retains one complete sine comparison and the existing phase-only observation. It consumes SineExchangeComparison.phase_rate_numerators() from the comparison owner; the same exact rational phase row also serves resultant kinematics. The report records chord-rate bounds, exact nearest groups, strict outsider margins and separate backward/forward conclusions.

For an exact tied group, it preserves the common factor 2*sin(abs(gap))/pi and compares the exact rational coefficients sign(gap)*(n_q-n_p). A resolved factor sign and strict coefficient order can certify a split without subtracting independently rounded derivative enclosures. Overlapping distances do not establish a tie; zero or unresolved factors and tied first derivatives remain unavailable. Complete nearest choices are required before a matching or nonmutuality conclusion, and a complete matching still receives its own fixed-support admission.

The public contract keeps certified_time_horizon unavailable. The report does not assign a numerical robustness radius, install the phase observation as a controller or evolve the graph. The independent replica tests check the frozen complete-field rates, all competing partners, the form-reversal control and the distinction between a local mathematical ordering and finite-precision observation.

17. Whole-box prediction throughout a fixed observation window

17.1. The frozen question and complete state

Retain the full conservative normalized-sine law, fixed unit doubled-C5 support, held capacities one and beta=w=1 from Section 16. No input, clipping, event or native resultant-pressure row is added. Let theta* and x* be the exact ten-node preparations there: d=u=1/8,

\[ \theta^*=(0,d,2d,3d,1,1,2,2,3,3),\qquad x^*=(u,-u,u,-u,0,0,0,0,0,0). \]

For each sigma in {+1,-1}, admit the entire closed preparation box

\[ \mathcal B_\sigma= \left\{(x_0,\theta_0): |x_{0i}-\sigma x_i^*|\leq\rho,\quad |\theta_{0i}-\theta_i^*|\leq\rho\quad\hbox{for every }i \right\}, \qquad \rho=2^{-20}. \]

The phase errors are specified in the declared real lifts. The flow and final observation remain circular. All twenty errors are independent; form sums, pair means, phase equality within the last three pairs and the two initial nearest ties are not constraints on either box. The support and capacities are exact. Reversing form exchanges the two boxes, including their uncertainty sets.

Freeze the same future structural-time window for both predictions,

\[ t_-=1/128,\qquad T=1/64,\qquad t_-\leq t\leq T. \]

The claim is the original complete mutual matching for B_+ and the nonmutual nearest map of Section 16 for B_-, for every source in the relevant box and every time in this closed window. The budgets are fixed before their inequalities are evaluated.

17.2. A global full-field remainder, without a reduced trajectory

The complete fine rows are

\[ \dot x_i=\frac1{\pi d_i} \sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \dot\theta_i=\frac1{\pi d_i} \sum_{j\sim i}(x_i-x_j), \qquad d_i=4 . \]

They imply the global bounds

\[ |\dot x_i|\leq\frac1\pi,\qquad |\ddot\theta_i| \leq\frac1{\pi d_i}\sum_{j\sim i} (|\dot x_i|+|\dot x_j|) \leq\frac2{\pi^2}. \]

These bounds use the actual evolving form and all neighbors, and hold at arbitrary phase configurations. In particular they do not freeze the phase-to-form feedback or assume fine-edge acuity. The smooth field has complete solutions: form grows at most linearly on each finite time interval and the phase row then has a finite integral.

At the center preparation L_f x*=4*x*. An arbitrary initial form error in the box changes the phase rate by at most 2*rho/pi, including both the node's own error and its neighbors' errors. Hence the exact integral remainder gives, for every source and 0<=t<=T,

\[ \boxed{\left| \theta_i(t)-\theta_i^*-\frac{\sigma x_i^*}{\pi}t \right| \leq E(t):=\rho+\frac{2\rho}{\pi}t+\frac{t^2}{\pi^2}.} \]

The three terms retain initial phase error, initial form error and full nonlinear acceleration, respectively. The central affine expression is a reference used in an enclosure; it is not asserted to be a nonlinear trajectory. No local Taylor solver, state truncation or omitted internal coordinate enters this bound.

The same calculation has a reusable conservative form on any admitted fixed unit support with held nonnegative capacities. Put a=w/pi, b=w/(beta*pi), and let the independent initial form and phase radii be r_xi,r_thetai. For node i define

\[ M_i=a\nu_i,\qquad V_i=b\nu_i\left(r_{xi}+\frac1{d_i}\sum_{j\sim i}r_{xj}\right), \qquad B_i=b\nu_i\left(M_i+\frac1{d_i}\sum_{j\sim i}M_j\right). \]

If v_i is the complete phase rate at the center capture, then

\[ |\theta_i(t)-\theta_{0i}^{\rm center}-v_i t| \leq r_{\theta i}+V_i t+\tfrac12B_i t^2 \quad(t\geq0). \]

The center rate uses the shared phase-law numerator, not a fitted response. This extension requires e=0 and the declared absence of inputs/events; the form bound is not transferred unchanged to a positive-loss law.

17.3. Exact rational margins for the whole window

Use the rational bounds 25/8<pi<22/7, with the coarser lower bound 3<pi to simplify the remainder. Since E(t) increases for nonnegative time, every phase error on [0,T] is at most

\[ \overline E =\rho+\frac{2\rho T}{3}+\frac{T^2}{9} =\frac{8483}{301989888}. \]

Write omega=u/pi. The central first-four phase gaps, in node order, are

\[ d-2\sigma\omega t,\qquad d+2\sigma\omega t,\qquad d-2\sigma\omega t. \]

Each actual gap differs from its central value by at most 2*Ebar. Their common strictly positive lower bound is

\[ d-\frac{2uT}{3}-2\overline E =\frac{18669277}{150994944}>0 . \]

Thus the order of the first four phases is unchanged for either box throughout the window. More generally the total real phase span is at most

\[ 3+\frac{2uT}{3}+2\overline E =\frac{453189923}{150994944} <\frac{25}{8}<\pi . \]

All phase separations in this proof are consequently also their circular distances; no hidden wrapping transition can alter the nearest ordering.

For the two initially tied choices, the difference between the undesired and desired distances, with desired chosen according to the sign sigma, is bounded below by

\[ \begin{aligned} m_{\rm tie} &=4u\,t_-\,\frac7{22}-4\overline E\\ &=\frac{938879}{830472192} >\frac1{1024}>0. \end{aligned} \]

The 4*Ebar term is the worst error of the three-coordinate combination at each tie, retaining the repeated central node's coefficient two. The bound applies even when a source in the box has no exact initial tie.

For the first four nodes, every competitor outside the singleton or tied group in Section 16 starts at distance at least 2*d, whereas the chosen partner starts at distance d. Bounding the two central motions and the two gap errors gives

\[ m_{\rm outside} =d-\frac{4uT}{3}-4\overline E =\frac{9232093}{75497472} >\frac1{1024}. \]

For each of the last three pairs, its internal distance can be as large as 2*Ebar; those members have not been kept synchronized. Every outsider starts at distance at least 5*d. Their preference margin is at least 5*d-2*u*T/3-4*Ebar, which is larger than m_outside. These inequalities account for every competitor of every node, not just the two central ties.

It follows that every selected nearest distance has uniform preference margin greater than g=1/1024. This also gives a lower bound for the actual squared-chord observer. If circular distances satisfy 0<=s<t<=pi and t-s>=g, then

\[ \begin{aligned} F(t)-F(s) &=4\sin\!\left(\frac{t+s}{2}\right) \sin\!\left(\frac{t-s}{2}\right)\\ &\geq4\sin^2(g/2)=2(1-\cos g)>0, \qquad F(z)=2(1-\cos z). \end{aligned} \]

Indeed the midpoint angle lies between (t-s)/2 and pi-(t-s)/2, so its sine is at least the sine of the half-separation. Thus the whole-box prediction has a strict quantitative chord margin, not only an ordering of unwrapped angles.

17.4. The two reserved predictions and their limits

For every initial state in B_+ and every t in [1/128,1/64], the exact nearest map is

\[ 0\leftrightarrow1,\quad 2\leftrightarrow3,\quad 4\leftrightarrow5,\quad 6\leftrightarrow7,\quad 8\leftrightarrow9. \]

All choices are strict and mutual. The complete fixed-support admission independently succeeds for this partition. Its within-pair phase gaps also remain below pi, so the retained collective chart is available. This does not make its internal differences zero.

For every initial state in B_- on the same window, the nearest indices are

\[ (1,2,1,2,5,4,7,6,9,8). \]

All choices are strict, but 0->1 and 3->2 are not mutual. There is no complete mutual-nearest matching under the declared observation contract, even though the support itself is unchanged. Both conclusions cover independent perturbations of every fine coordinate; neither follows from a center trajectory or a single evaluated endpoint. Some sources in these boxes may already have their window's nearest map at time zero. The whole-box conclusion is therefore a uniform future observation prediction, not a claim that every source undergoes one simultaneous onset at a specified time.

The original local reader still supplies only an existential neighborhood. The present theorem instead proves its separately frozen finite-window prediction using explicit global remainders. It gives no conclusion outside [1/128,1/64], no observation-noise budget beyond the stated preparation uncertainty, and no eventual capture by the earlier two-sided invariant family. A numerical observer still has to resolve a margin with its own arithmetic. The result is finite organizational robustness under a supplied complete law, not new support, a controller, permanent formation or physical identification.

17.5. Shared consumer and numerical evidence

assess_sine_pairing_window in the existing scale owner consumes a single comparison capture, mandatory full vectors of form and phase error radii, and an explicit 0<=window_start<window_end. It requires zero form loss and retains the captured support, capacities and structural clock. The source box is supplied analysis evidence; the reader does not authenticate a laboratory preparation.

SinePairingWindowAssessment records the initial intervals, the per-node form-speed, initial phase-rate-error and acceleration upper bounds, the resulting remainders and the complete phase window. Pair differences reuse the exact shared rate-numerator difference before applying the common time interval. This preserves information that subtraction of independent whole-node phase windows would lose. Where the absolute gap enclosure lies below mathematical pi, monotonicity gives tight chord bounds from certified scalar cosine endpoints. Elsewhere the shared interval cosine is used without inferring a new phase lift.

Only strict comparison with every competitor certifies a node's nearest choice throughout the box and window. The report then distinguishes complete mutual matching, certified nonmutuality and unavailable bounds, followed by separate paired-support admission where a complete matching exists. Wide intervals do not refute a physical trajectory or the proposed grouping. The existing local reader keeps its absent numerical horizon.

The window contract and independent replica tests retain these distinctions, the frozen two-box control, all competing nodes, invalid domains and unresolved-bound controls. This is a full-field analytic enclosure on the existing owner, with no trajectory producer, finite-difference fit or enlarged solver budget.

24. Joint form-phase identification during autonomous exchange

24.1. A storage-defined observation, not an extra dynamical rule

For the fixed positive storage scale beta, define

\[ J_{ij}=(x_i-x_j)^2+2\beta[1-\cos(\theta_i-\theta_j)], \qquad d_\beta(i,j)=\sqrt{J_{ij}}. \]

The map z_i=(x_i,sqrt(beta)*cos(theta_i),sqrt(beta)*sin(theta_i)) embeds the form and circular phase into Euclidean three-space, with d_beta(i,j)=||z_i-z_j||. Thus d_beta is a metric on this state space: it vanishes precisely for equal form and equal circular phase, and it inherits the triangle inequality. J is its square, not itself a metric. Both terms use the existing configured storage scale in the same declared, currently nondimensional coordinate chart. If form units are assigned, beta must carry their square for d_beta to have form units; this bookkeeping does not establish a laboratory bridge for the law's coefficients or clock. The scale is inherited from the declared storage, not fitted to identify a desired partition or proved to be a unique physical geometry.

The observation is invariant under node relabeling, common form shifts, common phase rotations and independent full-turn changes of phase representatives. It consumes form and phase; held support and capacity remain separate model premises. As in Section 14, an inferred pair requires unique mutual nearest partners. A tie must remain a tie. Such instantaneous identification does not establish sufficient state for a future law or select a new support or hierarchy event.

24.2. Exact reference identity survives a phase-only collision

Keep the complete conservative normalized-sine law and the synchronized K2,2 preparation of Section 23: common positive capacity, fixed support, equal initial forms and a supplied lift 0<|Delta_0|<pi. The full trajectory retains identical members within each pair. For every time,

\[ \boxed{J_{\rm within}=0,\qquad J_{\rm cross}=D(t)^2+2\beta[1-\cos\Delta(t)] =2E_c=4\beta\sin^2(\Delta_0/2)=:J_*>0.} \]

This is an identification consequence of the existing inherited storage identity, not another conservation postulate. Every node has its other constituent as its unique nearest partner throughout the entire reference motion. The observation can recover that partition without reading its labels or the supplied support; those premises are used to prove the claim, not to resolve an observed distance tie.

The P2 libration crosses Delta=0. At that instant all four phases coincide, so every phase-only distance is zero and phase-only pairing must abstain. But then D^2=2E_c>0: the joint observation retains exactly the same strict pair distinction. At the form turning points D=0, the nonzero phase separation instead supplies it. Form and phase exchange their contributions without erasing the joint distinction. At Delta_0=0 with equal forms the reference has J_*=0, four identical observed states and no uniquely identified pairing. This degenerate reference is not certified by continuity from positive amplitude.

24.3. A full-field perturbation bound over a declared window

Now allow independent preparation errors in the form and phase of all four nodes. The perturbed state need not preserve either pair's internal synchronization. Both trajectories use the same K2,2 graph, common capacity nu>0, beta,w>0, e=0, clock, and no input, event or clipping. Specify real phase lifts for the initial comparison and continue them by the smooth phase row.

The error norm is taken in the eight real coordinates

\[ Z=(x_0,\ldots,x_3,\sqrt\beta\theta_0,\ldots, \sqrt\beta\theta_3),\qquad r_0^2=\|Z(0)-Z_*(0)\|_2^2. \]

For componentwise preparation radii f_i,p_i>=0, the sufficient initial bound is r_0^2<=sum_i(f_i^2+beta*p_i^2). Each component can vary independently within its declared interval. A common-origin or phase-lift choice must be specified before this bound is evaluated; small circular distance alone is not a declaration of small error in an arbitrary lift.

Put c=w*nu/(pi*sqrt(beta)). On this degree-two regular graph the Jacobian of the full fine field in the scaled real lifts is

\[ DF(Z)=c\begin{pmatrix}0&-L_{\cos}(\theta)/2\\L/2&0\end{pmatrix}, \]

where L_cos is the symmetric edge Laplacian with signed coefficients cos(theta_j-theta_i). Both L/2 and L_cos/2 have Euclidean operator norm at most two. For example the absolute row sum of the signed matrix is at most two after division by the degree, and its symmetry bounds the spectral norm. Equivalently -L <= L_cos <= L as quadratic forms and the largest eigenvalue of the K2,2 Laplacian is four. The block Jacobian norm is therefore at most L_*=2c, uniformly in form and phase.

This proves a global Lipschitz bound in the scaled lift coordinates, and hence global continuation and the full-field difference estimate

\[ \|Z(t)-Z_*(t)\|_2\le r_0e^{L_*t}\le r_0e^{L_*T}=:r_T \qquad(0\le t\le T). \]

It does not assert that the embedded circular-state vector field has an equally bounded Jacobian independent of form. The embedding is used only for its observation error:

\[ \|z_i-z_{*,i}\|^2 =(x_i-x_{*,i})^2+ 4\beta\sin^2[(\theta_i-\theta_{*,i})/2] \le(x_i-x_{*,i})^2+\beta(\theta_i-\theta_{*,i})^2. \]

For any two nodes the reverse triangle inequality and ||e_i-e_j||<=sqrt(2)*sqrt(||e_i||^2+||e_j||^2) give

\[ |d_\beta(i,j)-d_{\beta,*}(i,j)|\le\sqrt2\,r_T. \]

Writing s=sqrt(J_*), throughout the whole window we consequently have

\[ \max_{\rm within}d_\beta\le\sqrt2r_T,\qquad \min_{\rm cross}d_\beta\ge\max(0,s-\sqrt2r_T). \]

In particular the following strict, prospective inequality guarantees the same mutual nearest pairs for every preparation in the error ball or declared component box and every time in the window:

\[ \boxed{J_*>8r_0^2\exp\!\left( \frac{4w\nu T}{\pi\sqrt\beta}\right).} \]

The inequality is sufficient, not necessary. A failed or unresolved bound cannot by itself prove loss of identity. Zero uncertainty gives the exact reference result. For nonzero uncertainty the estimate establishes a finite-window guarantee, not attraction, nonlinear orbital stability or permanent identification. The held graph and capacities also do not emerge from this error estimate.

24.4. A fixed full-exchange control and genuine degeneracies

Before evaluating the finite report, fix the source to K2,2 with C=1/2, phases (0,0,1/4,1/4), beta=nu=1, e=0 and effective w=1. Set T=10 in its structural clock and give each form and phase component radius 10^-6. Then r_0^2<=8*10^-12 and J_*=2*(1-cos(1/4)). These are preparation specifications, not fitted errors or observed response curves.

The existing P2 period bound gives T_pulse<=pi^2/cos(1/8)<10; for example pi<22/7 and cos(1/8)>=127/128 already give the strict upper bound 61952/6223<10. Thus the chosen window includes one full reference exchange, including its phase-only collision. The strict identity inequality can also be checked without trajectory evaluation: J_*>1/40, while its right-hand side is less than 64*10^-12*3^14<1/40. These conservative inequalities use sin(y)>=2y/pi for 0<=y<=pi/2, 3<pi<22/7, and exp(40/pi)<exp(14)<3^14. A finite interval report can retain sharper margins without changing the declared source or budget.

The zero-amplitude control has no reference separation. The separately declared large-error control, with every radius 1/8 and the same nonzero-amplitude reference, includes a preparation with all forms 1/2 and all phases 1/8. All four observed node states then coincide at the initial time. The entire box therefore cannot have the strict pair-identification property, independently of how sharp an error bound is. This explicit counterexample must not be generalized to every preparation budget for which the sufficient inequality fails.

24.5. Shared observation and window admission

observe_joint_pairs(nodes=..., forms=..., phases=..., storage_scale=...) in the scale owner observes the actual signed forms and circular phases. It reuses strict mutual nearest-partner admission and may abstain; it does not read a proposed partition or use support to resolve a tie.

assess_sine_joint_pairing_window separately admits the complete K2,2 reference and its supplied pairs, phase lifts, form_error_bounds, phase_error_bounds and window_end. The returned initial_scaled_error_squared, lipschitz_upper and propagated_scaled_error_squared_upper retain the full-field error provenance. identification_budget_margin applies the strict squared inequality; the report also requires strict interval nearest-partner comparisons. reference_identity_certified is separate from whole_window_identity_certified. A numerical abstention cannot erase the exact reference theorem or become evidence of actual perturbed failure.

The report's reference_period_bounds reuse the existing P2 period owner; window_covers_reference_period concerns that reference only, not a period or a recurrence deadline for perturbed trajectories. No graph is advanced, and no snapshot is represented as the evolved endpoint. The independent algebra controls check the circular metric identity, conserved reference separation, complete scaled-lift Jacobian and both observation-error factors.

25. The frozen phase transition does not change the joint pairing

25.1. Compare two observations of the same complete preparation

Keep the exact doubled-C5 graph, law, clock and the two form-reversed sources from Sections 16–17, with e=0, w=beta=nu=1 and d=u=1/8:

\[ \theta^*=(0,d,2d,3d,1,1,2,2,3,3),\qquad x^*=(u,-u,u,-u,0,0,0,0,0,0). \]

For sigma=+1,-1, retain all twenty independent initial radii rho=2^-20 around (sigma*x*,theta*) and the original observation window [1/128,1/64]. The supplied structural pairs are (0,1),(2,3),(4,5),(6,7),(8,9). No parameter, horizon, support, pressure law or event schedule is adjusted for the joint observation. The K2,2 constant-separation result is not applied to this graph.

The joint observation of Section 24 is J_pq=(x_q-x_p)^2+beta*D_pq, where D_pq=2*(1-cos(theta_q-theta_p)) is the existing phase-only chord. Differentiation along the complete fine field gives

\[ \boxed{\dot J_{pq}= 2(x_q-x_p)(\dot x_q-\dot x_p) +2\beta\sin(\theta_q-\theta_p)(\dot\theta_q-\dot\theta_p).} \]

Both rows are necessary. In particular a measured or computed phase rate alone does not determine this derivative. The sine form row depends on the full neighbor phase currents; the phase row depends on the full form gradients. These are observation rates, not new terms in either evolution law.

25.2. A different joint partition is already strictly present

Write F(s)=2*(1-cos(s)). At either source center,

\[ J_{02}=J_{13}=F(2d),\qquad J_{01}=J_{12}=J_{23}=4u^2+F(d). \]

The closest opposite-form competitors are farther by

\[ h_0=4u^2+F(d)-F(2d)\ge d^2=1/64>0. \]

Indeed u=d and F(2d)-F(d)=integral_d^(2d) 2*sin(s) ds <=3d^2. The remaining first-four comparison 0--3 has the same form cost and a larger phase gap 3d. Every node outside those four is at phase distance at least 5d from them. Here F(2d)<=4d^2 while F(5d)>10d^2, using the chord bounds below. The last three pairs have exactly zero internal joint distance and strictly positive distance to every outsider. All phase gaps lie below pi. Thus every competitor has been checked, and the unique mutual joint pairs at both centers are

\[ \boxed{\mathcal C= \{(0,2),(1,3),(4,5),(6,7),(8,9)\}.} \]

The form-reversed source has the same squared form differences, so this observation is unchanged. All its joint-distance rates reverse: the initial form rates stay unchanged, whereas form differences and phase rates change sign. In particular Jdot_02=Jdot_13=0 at both centers, because their form differences and phase-rate differences are zero. These zero derivatives do not create a tie: their nearest margins are strictly positive. More generally the exact conservative reversal of Section 16 gives J_tilde(t)=J(-t); it does not imply that an initially strict nearest relation must change at time zero.

25.3. The same joint pairs persist on the complete frozen boxes

Reuse the full-field phase bound of Section 17. For every source in either box and 0<=t<=T=1/64,

\[ |\theta_i(t)-\theta_i^*-(\sigma x_i^*/\pi)t|\le E, \quad E=\rho+2\rho T/3+T^2/9 =\frac{8483}{301989888}. \]

The same complete field has |xdot_i|<=1/pi. Integration, including each independent initial form error, therefore gives the additional bound

\[ \boxed{|x_i(t)-\sigma x_i^*|\le B:=\rho+T/3 =\frac{16387}{3145728}.} \]

This bound does not freeze form or assume a common form error within any pair. The phase enclosure uses the original central phase rates and nonlinear remainder; it is not an extrapolated trajectory. The span estimate from Section 17 remains below mathematical pi on this entire interval, so absolute phase gaps equal circular gaps.

For 0<=s<=pi, the elementary chord inequalities are

\[ \frac25s^2\le\frac4{\pi^2}s^2\le F(s)\le s^2, \]

where the first inequality uses pi^2<10. They follow from 2y/pi<=sin(y)<=y on 0<=y<=pi/2. For the desired first-four pairs (0,2),(1,3), the reference forms and central phase rates coincide within each pair. Consequently

\[ J_{\rm desired}\le U_1:=4B^2+(2d+2E)^2. \]

Every remaining first-four competitor has initial form difference 2u and phase gap at least d. Independently bounding both coordinates gives

\[ J_{\rm competitor}\ge L_1:=(2u-2B)^2+ \frac25\left(d-\frac{2uT}{3}-2E\right)^2. \]

Both lower gap bounds inside the squares are strictly positive. They must not be squared without that check. Their exact margin is

\[ L_1-U_1= \frac{33345408988471}{37999121855938560} >\frac1{2048}>0. \]

For a first-four node and an outsider among the last six, the phase gap is at least g_2=5d-u*T/3-2E>0, giving J_competitor>=(2/5)*g_2^2. Its margin over U_1 is greater than L_1-U_1. For each of the last three desired pairs, J_desired<=U_2:=4B^2+4E^2; every outsider again has phase gap at least g_2, giving a still larger margin. This includes other members of the last six, whose initial interpair gap is at least one.

Thus all eighty directed nearest-partner comparisons have the required strict sign. For every source in either frozen box and every time 0<=t<=1/64, the joint nearest map is

\[ \boxed{(2,3,0,1,5,4,7,6,9,8).} \]

In particular this proves the new observation on the original frozen window [1/128,1/64]. The same estimates also cover its preparation and intervening interval: the identified pairs cannot disappear and reappear before that window. This is an analytic corollary of the unchanged budgets, not a replacement or retiming of the earlier phase-only forecast.

25.4. Observation change, collective state and formation are distinct

Under phase-only observation the positive box has the structural matching on its window, while the negative box has the nonmutual nearest map proved in Section 17. Under the joint observation both boxes already have the same strict partition C initially and retain it throughout the interval just proved. Therefore that specific phase-only onset or loss is not onset or loss of joint pair identity under the specified joint criterion.

This comparison neither invalidates the old phase-only theorem nor declares a uniquely correct physical NFR metric. It identifies the different information retained by the two observations. Neither distance ordering creates support, selects an operator or proves permanent formation.

The joint pairs also do not automatically inherit the structural pair law. The proposed pairs (0,2) and (1,3) contain existing fine edges, so they fail the strict no-internal-edge replica admission. More decisively, even the broader independent-swap criterion of Section 18 fails. For (0,2),

\[ N(0)\setminus\{2\}=\{3,8,9\},\qquad N(2)\setminus\{0\}=\{1,4,5\}. \]

Their interchange does not preserve the fixed support. Proximity in the joint observation consequently cannot discard their distinct attachment identities; an unordered pair state alone is not the generic exact dynamical quotient. The mixed-state and boundary-current owners retain the relevant internal and environmental information. Recognition of a stable finite-window pattern and sufficiency of its proposed collective evolution remain separate obligations.

25.5. A projection of the existing window, not another forecast

SinePairingWindowAssessment.joint_observation() in the shared scale owner returns SineJointPairingProjection from the existing capture, error box and declared window. It combines the actual form-speed tube with the already computed phase chord bounds; it neither rereads an evolved graph nor changes the original phase-only result. The source source_joint_distance_rate_bounds consume both captured fine rows.

initial_observation, initial_box_candidate_pairs and candidate_pairs refer respectively to the exact center, the whole initial box and the whole reported window. same_pairing_as_initial_box compares the last two only when both are certified complete pairings. Equality there does not establish persistence across an unobserved intervening gap; the stronger result in Section 25.3 has its own analytic proof. support_admission_status retains the existing strict replica admission and must not be read as an exhaustive symmetry test. The independent algebra and rational controls check all competing nodes, the full-row reversal, the source support obstruction and the fixed-box bound without evaluating a trajectory.

26. Acquiring a joint pairing at its exact observation boundary

26.1. A separately declared critical preparation

Keep the unit doubled-C5 support of Sections 16-17, its ten labeled nodes, and the complete conservative normalized-sine law with held e=0, w=beta=nu_i=1. There are no inputs, events or clipping. Every fine node has degree four, and the full rows are

\[ \dot x_i=\frac{S_i}{4\pi},\qquad \dot\theta_i=\frac{(Lx)_i}{4\pi},\qquad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i). \]

Use the same phases and the same joint observation J as Section 25, but declare a new mathematical preparation, chosen by an exact observation boundary rather than a search over responses:

\[ d=\frac18,\quad F(s)=2(1-\cos s),\quad u_c=\frac12\sqrt{F(2d)-F(d)}>0, \]

\[ \theta^*=(0,d,2d,3d,1,1,2,2,3,3),\qquad x^\sigma=\sigma(u_c,-u_c,u_c,-u_c,0,0,0,0,0,0), \quad\sigma\in\{+1,-1\}. \]

The positive square root exists because F is strictly increasing on (0,pi). This source is supplied, not autonomously selected. It does not replace the earlier u=1/8 source, independent error boxes, fixed horizon or evaluated evidence. In particular its exact trigonometric form amplitude is not a rounded graph attribute.

At either preparation, the five distances

\[ J_{01}=J_{12}=J_{23}=J_{02}=J_{13}=J_*:=F(2d) \]

are exactly equal, because 4*u_c^2=F(2d)-F(d). The complete initial nearest sets are

\[ \{1,2\},\quad\{0,2,3\},\quad\{0,1,3\},\quad\{1,2\}, \quad\{5\},\{4\},\{7\},\{6\},\{9\},\{8\}. \]

All other comparisons have a strict gap. For (0,3) the excess over J_* is F(3d)-F(d). Between a first-four node and a last-six node the phase gap is at least 5d, so the excess is at least F(5d)-F(2d). The last three synchronized pairs have distance zero; their outsiders have phase gap at least 5d. The chord inequalities from Section 25 give

\[ F(3d)-F(d)\ge\frac{13}{5}d^2,\qquad F(5d)-F(2d)\ge6d^2,\qquad F(5d)\ge10d^2. \]

These are uniform positive margins for every initially untied competitor. Consequently the only local ordering question is how the five tied edges separate under the complete law.

26.2. Both rows determine the crossing direction

At the positive preparation Lx=4x, so theta_dot=x/pi. Write the first four sine sums as

\[ \begin{aligned} S_0&=\sin2d+\sin3d+2\sin3,\\ S_1&=\sin d+\sin2d+2\sin(3-d),\\ S_2&=-\sin2d-\sin d+2\sin(1-2d),\\ S_3&=-\sin3d-\sin2d+2\sin(1-3d). \end{aligned} \]

Apply the full joint-distance derivative from Section 25:

\[ \dot J_{ij}=2(x_j-x_i)(\dot x_j-\dot x_i) +2\sin(\theta_j-\theta_i)(\dot\theta_j-\dot\theta_i). \]

Define the three dimensionless coefficients

\[ \begin{aligned} A&=5\sin d-\sin3d+2[\sin(3-d)-\sin3],\\ B&=2\sin d-2\sin2d+2[\sin(1-2d)-\sin(3-d)],\\ C&=5\sin d-\sin3d+2[\sin(1-3d)-\sin(1-2d)]. \end{aligned} \]

The tied-edge rates at the positive source are exactly

\[ \boxed{\frac\pi{u_c} (\dot J_{01},\dot J_{12},\dot J_{23},\dot J_{02},\dot J_{13}) =(-A,B,-C,0,0).} \]

All three coefficients are strictly positive, by elementary inequalities:

• The triple-angle identity gives 5*sin(d)-sin(3d)=2*sin(d)+4*sin(d)^3>0. Since pi/2<3-d<3<pi, sine decreases between 3-d and 3; the additional difference in A is positive.
• The same identity and the sine-difference formula give

[ C=4\sin(d/2)[\cos(d/2)-\cos(1-5d/2)]+4\sin^3d>0. ]

Here d/2=1/16<1-5d/2=11/16<pi, so both displayed terms are positive. This retains the form/phase cancellation in the smallest tied-edge response; the phase row alone cannot justify its sign. * sin(3/4)>=3/4-(3/4)^3/6=87/128>2/3. Also 0<pi-23/8<1/3, using 3<pi<22/7, and hence sin(23/8)=sin(pi-23/8)<1/3. Finally sin(2d)-sin(d)<=d by the derivative bound for sine. Therefore B>2*(2/3-1/3-d)=5/12>0.

No ordering between A, B and C is needed. Reversing all forms leaves the form row unchanged and negates the phase row. Both terms of every J_dot therefore reverse, while every initial J stays unchanged.

26.3. One-sided acquisition and qualitative full-state robustness

The field and the joint distances are smooth. A tied margin with a strictly positive derivative becomes positive on a sufficiently short positive-time interval. There are finitely many comparisons, and the initially untied ones have the positive margins just proved. Thus some tau>0 exists such that, for every 0<t<tau, the positive source has nearest map

\[ \boxed{(1,0,3,2,5,4,7,6,9,8),} \]

whereas the negative source has nearest map

\[ \boxed{(2,2,1,1,5,4,7,6,9,8).} \]

For the positive source, edges 01 and 23 decrease while 12 increases and 02,13 have zero first derivative. This resolves every tied choice in favor of the original structural pairs. For the reversed source, 12 decreases while 01,23 increase; nodes 0 and 3 select the zero-rate alternatives 2 and 1. This map is not a complete mutual pairing. The last three pairs remain the unique nearest choices by their initial strict gaps.

The exact conservative reversal (x(t),theta(t)) -> (-x(-t),theta(-t)) also shows that the positive orbit has the nonmutual map on -tau<t<0. It crosses from that region, through the declared ties, into the structural mutual-pairing region. This is local acquisition of the specified joint observation under the complete autonomous law, not merely a phase-distance transition or a choice among initially unique joint pairs.

The assertion has qualitative robustness in all twenty fine coordinates. For any compact interval [a,b] with 0<a<b<tau, all desired margins along either reference are uniformly positive. Continuous dependence of the full flow yields an open initial-state neighborhood preserving its respective nearest map throughout [a,b]. There is no restriction to synchronized perturbations, pair sums or the original one-dimensional form family. The exact simultaneous ties at time zero need not survive a perturbation.

More strongly, choose any small positive a<tau on the positive reference. Its state at -a has a strict nonmutual map and its state at +a has the strict structural map. Both strict endpoint observations persist for an open neighborhood of that earlier state, by the same continuous-dependence argument. Thus nearby full states also undergo a change between these observation regions, although their individual tie-crossing times and order need not coincide. These are existential intervals and neighborhoods; none is a certified numerical horizon, error radius or permanent lifetime.

26.4. What is acquired, and what remains supplied

The positive partition passes the strict replica support contract: its pairs have no internal edges, neighboring blocks have complete unit bipartite support, capacities are common, and every within-pair interchange is a graph automorphism. Their initial absolute within-pair phase half-gaps are d/2,d/2,0,0,0, all strictly within the local midpoint chart; tau may be shortened to keep that admission throughout the crossing. The exact collective-state result of Sections 7 and 18-19 therefore applies with its retained internal form/phase variables. The first two pairs are not synchronized at preparation, so their newly strict observation does not authorize discarding those internal variables. The negative map has no complete mutual pairing to admit as such a partition.

The support symmetry and its sufficient quotient existed before the crossing. The dynamics makes that partition strictly recognizable by the supplied joint observation; it does not create the support, the quotient degrees of freedom, an operator-selection law or a physical constituent. No invariant prepared family has been entered from outside, and no attraction or permanent maintenance is asserted. The result is consequently compatible with the conservative formation boundary stated at the beginning of this note.

No new execution law or observer is necessary. The existing joint observation and full-row derivative owners already retain the required information. Their represented-real admission must not relabel a rounded u_c as the exact source, infer an equality from overlapping intervals, or promote this local theorem to a finite-window certificate. Any such numerical claim would require its own prospectively declared enclosure and budget. The independent symbolic controls check the exact critical relation, complete fine-row coefficients and every tied ordering; the proof above supplies their inequality and local-flow scope. The outward full-field controls separately bound the symbolic amplitude, all initially untied comparisons and the complete joint rates through shared rational kernels. Their rounding control demonstrates why a represented approximation is not the exact tie.

27. Repeated finite episodes of joint identification

This is a consequence of the local crossing in Section 26 and the existing full-state invariant-volume recurrence theorem. It uses the same complete conservative normalized-sine law: fixed unit doubled-C5 support, e=0, held nu_i=w=beta=1, structural time, and no forcing, events, clipping or changed mobility. All twenty fine coordinates are retained, with phases on the torus. No trajectory is evaluated and no recurrence assertion is inferred from a finite numerical return.

27.1. Identity, loss and a common acquisition interval

Let Phi_t denote the complete flow on M=R^10 x (R/(2*pi*Z))^10. Define two open subsets of that state space:

• P consists of states whose strict joint nearest map is the structural matching (1,0,3,2,5,4,7,6,9,8).
• N consists of states whose strict joint nearest map is (2,2,1,1,5,4,7,6,9,8), the nonmutual map in Section 26.

Both predicates use exactly the joint distances J_ij already defined. Each is a finite conjunction of strict continuous inequalities J_i,p(i)<J_ij for all other candidates j. Hence P and N are open and disjoint. Identity here means membership in P; its loss means nonmembership. Visiting N establishes a stronger, strict alternative observation, rather than an unavailable or inconclusive numerical report.

Write z_c for the positive exact critical preparation. Section 26 gives tau>0 such that Phi_t(z_c) lies in N for -tau<t<0 and in P for 0<t<tau, with the local pair midpoint chart admitted on these intervals. Choose a>0 sufficiently small, with a<tau/4, and set

\[ z_-:=\Phi_{-a}(z_c). \]

Then z_- lies strictly in N, while its entire image over the positive interval [2a,3a] lies strictly in P. Compactness of that time interval and continuous dependence on all fine coordinates give an open neighborhood U of z_- with compact closure and a bounded open matching neighborhood V such that

\[ \boxed{\overline U\subset N,\qquad \Phi_s(\overline U)\subset V\subset P \quad\text{for every }s\in[2a,3a].} \]

The neighborhoods can be chosen inside the admitted pair chart during these initial and matching windows. The compact image tube has a common strictly positive nearest-partner margin, although no numerical value is claimed. The same a,U,V work for every preparation in U, including independent perturbations of all twenty coordinates. Every such state is nonstationary already, since a stationary state cannot move between the disjoint sets N and P.

The midpoint chart is an additional collective-coordinate admission on these windows, not part of the identity predicate P. A longer matching episode may leave that chart without losing its joint-distance matching.

27.2. One finite invariant ambient family

The recurrence theorem needs a finite invariant measure, not just a local crossing. Its hypotheses can be admitted here with an explicit loose slab. There are twenty unit edges, and every fine degree is four. The conserved storage and weighted mean therefore take the forms

\[ E=\frac12\sum_{\{i,j\}\in\mathcal E}(x_i-x_j)^2 +\sum_{\{i,j\}\in\mathcal E}[1-\cos(\theta_j-\theta_i)], \qquad m=\frac1{10}\sum_i x_i. \]

For the closed form box |x_i|<=1/4, with arbitrary circular phases,

\[ E\le\frac12\,20\left(\frac12\right)^2+2\,20 =\frac{85}{2}<43,\qquad |m|\le\frac14<\frac12. \]

Thus the whole box times the phase torus lies strictly inside

\[ \mathcal S=\{z:E(z)\le43,\ -\tfrac12\le m(z)\le\tfrac12\}. \]

At the critical source, u_c<d=1/8, since 4*u_c^2=F(2d)-F(d)<4d^2. By choosing a smaller if necessary, then shrinking U, its compact closure lies in |x_i|<1/4. Therefore overline(U) lies in the interior of S. The form box itself need not be invariant: S is the invariant family. These numerical ceilings are analysis bounds on supplied preparations, not new parameters of the law.

The source theorem applies without a change of model. This is a finite connected simple unit graph with positive held capacities; both complete rows use the same constant degree/capacity mobility. Storage and the weighted form mean are conserved. The full divergence vanishes because the form row depends only on phase and the phase row only on form. Consequently S is compact, the flow exists in both time directions, and it preserves finite positive product measure

\[ d\mu=dx_0\cdots dx_9\,d\mathrm{Haar}_{\mathbb T^{10}}. \]

No bounded interval of unwrapped phase lifts or unproved measure on a fixed-energy surface is being introduced. Since U is full-dimensional and open with compact closure, 0<mu(U)<infinity.

27.3. Infinitely many distinct finite matching episodes

Fix the single sampling increment h=4a and write T=Phi_h. First take any z in U that returns to U infinitely often under T. This is an explicit deterministic premise: there are integers

\[ 0=n_0<n_1<n_2<\cdots,\qquad n_k\longrightarrow\infty, \qquad \Phi_{n_kh}(z)\in U. \]

Put r_k=n_kh. Each return is a state of the same autonomous system; it is not a reset or a fresh draw of initial data. The common transit gives

\[ \Phi_{r_k}(z)\in N,\qquad \Phi_t(z)\in V\subset P\quad \text{for every }t\in[r_k+2a,r_k+3a]. \]

Also r_(k+1)-r_k>=h=4a>3a. Hence each guaranteed matching window lies strictly between two nonmutual returns. Define the matching-time set

\[ \mathcal I_z=\{t>0:\Phi_t(z)\in P\}. \]

It is open. Let I_k be the connected component containing [r_k+2a,r_k+3a]. Since neither bounding return lies in P,

\[ \boxed{ [r_k+2a,r_k+3a]\subset I_k\subset(r_k,r_{k+1}). } \]

These components are pairwise distinct, bounded intervals, each of duration at least a>0. There are infinitely many of them. Every return to the open set N also has a nonempty time neighborhood in N, so the selected episodes are separated by actual intervals of strict alternative identification, not merely an unresolved tie at one instant.

More generally every positive-time component of I_z is bounded: an unbounded component would contain all sufficiently late times and contradict the unbounded sequence of returns to N. Only the selected infinite subfamily has the common lower duration bound a; no such bound is asserted for other episodes. This argument supplies no upper lifetime bound common to preparations, bound on the next return or exact period. The constants a,U,V and the recurrence times have existence proofs here, not numerical certificates.

Everything so far follows for each state satisfying the stated return premise. The existing finite-measure recurrence theorem, applied to the fixed map T and measurable subset U of S, supplies that premise for mu-almost every state in U.

Thus, for almost every preparation in this nonempty open full-state acquisition neighborhood, the structural joint observation repeatedly appears, persists for a positive interval, is lost, and reappears. It does not eventually become permanent for those preparations. This is a scoped lifetime result for a noninvariant observation; it neither contradicts nor enters the separately protected two-sided invariant identity families.

27.4. Measure, interpretation and integration boundaries

The exceptional set is null in the stated ambient twenty-dimensional product measure. An initial probability law supported in U and absolutely continuous with respect to that measure inherits the result with probability one. The dynamics has not selected such a preparation law. No pointwise recurrence conclusion follows for the exact critical source, its earlier point z_-, an individual captured graph, a fixed-energy or fixed-mean preparation, or a synchronized or finitely sampled family solely from their inclusion in S or U.

The recurring identity is the same specified matching, not a proof that nodes, edges, quotient coordinates or physical constituents are repeatedly created and destroyed. Its support and sufficient state remain supplied; only the joint observation changes. The conclusion does not establish a common pulse, periodic waveform, attraction, dissipation or a clock selected by synchronization. It does not transfer to the native argument-pressure law, positive-loss sine dynamics or another reciprocal mobility without their own measure and lifetime premises.

No additional runtime or report is needed. The existing recurrence reader admits a family and explicitly leaves individual nonstationary recurrence unavailable; the joint observer reports a captured state's distances rather than unknown future return times. The recurrence controls check complete-law invariants, finite-family admission and that chosen-state boundary. The ambient-family integration control independently checks the declared box/slab and existing family API without asserting a numerical recurrence sequence. The joint crossing controls and outward bounds check the local mechanism reused here. Finite tests validate these premises and their implementation, not the infinite recurrence sequence itself.