Pair geometry under alternative reciprocal mobility

The distinct complete positive-mobility law, grouping discriminator and protected relative geometry; old waveform, mean and volume claims do not transfer automatically.

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.

20. Constitutive scope of the local grouping mechanism

20.1. The common work identity leaves a mobility law to specify

Reuse the reciprocal mobility class on fixed simple connected unit support. Let L_f be its combinatorial Laplacian, q=L_f*x, and S_i=sum_{j~i} sin(theta_j-theta_i). In the conservative sector the complete unforced rows are

\[ \boxed{\dot x_i=w\nu_i m_i(\theta)S_i,\qquad \dot\theta_i=(w/\beta)\nu_i m_i(\theta)q_i.} \]

Here w,beta>0, capacities are held and strictly positive, and the structural clock is fixed. For the present local result the supplied mobility functions are finite, strictly positive, phase-only and locally Lipschitz on a circular neighborhood of the preparation. These are sufficient hypotheses for a unique differentiable local solution, not conditions inferred from an observed response. A model must specify its mobility functions and their domain; positivity is not a complete law. The frozen comparison below uses smooth functions of incident phase differences, preserving global phase-origin and graph-relabeling symmetries.

The nodal pressure is the independently supplied map p_i=w*m_i*S_i. The common storage is

\[ E=\frac12\sum_{\{i,j\}}(x_i-x_j)^2+ \beta\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]. \]

Its gradients are grad_x E=q and grad_theta E=-beta*S. Consequently each node's two exchange work terms cancel:

\[ q_i\dot x_i-\beta S_i\dot\theta_i =w\nu_i m_i q_iS_i-w\nu_i m_i S_iq_i=0. \]

No derivative of the mobility is needed for this identity. It does not select m_i=1/(pi*d_i), establish an invariant volume, or give all candidates the same trajectory or weighted means. The existing conditional classification selects normalized sine pressure only after adding its stated pairwise-superposition or phase-independent-response premise and normalization. That selection theorem is not being assumed to hold for every positive reciprocal mobility.

20.2. Positive reciprocal response preserves the local direction of grouping

Retain exactly Section 16's twenty-edge doubled-C5 preparation: d=u=1/8, phases (0,d,2d,3d,1,1,2,2,3,3), forms (u,-u,u,-u,0,0,0,0,0,0), common capacity nu=1, and w=beta=1. Keeping k=w*nu/beta>0 visible below makes the common rate factor explicit. Every fine degree is four and the prepared neighbor form sums vanish, so q=4*x. Write m_i for the chosen mobility evaluated at this initial phase state. The complete phase row gives

\[ (\dot\theta_0,\dot\theta_1,\dot\theta_2,\dot\theta_3) =4ku\,(m_0,-m_1,m_2,-m_3),\qquad \dot\theta_i=0\quad(i=4,\ldots,9). \]

Keep the same circular squared-chord observation D_ij=2-2*cos(theta_j-theta_i) and the two initially tied preference margins

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

Differentiating the chords using the actual phase row, rather than assuming a phase clock, gives

\[ \boxed{\begin{aligned} \dot M_1(0)&=8ku\sin d\,(m_0+2m_1+m_2)>0,\\ \dot M_2(0)&=8ku\sin d\,(m_1+2m_2+m_3)>0. \end{aligned}} \]

For example, M1_dot=2*sin(d)*(theta_dot2-2*theta_dot1+theta_dot0); the second expression is 2*sin(d)*(2*theta_dot2-theta_dot1-theta_dot3). The signs follow solely from this preparation, positive capacity and mobility, and 0<d<pi. Equal mobilities are unnecessary.

Section 16.3 already checks every other potential partner at these same phases. Its strictly positive outsider margins remain positive on some sufficiently small neighborhood of time zero for each admitted complete field. The two nonzero margin derivatives resolve the ties. Thus the original preparation has the same complete mutual matching (0,1),(2,3),(4,5),(6,7),(8,9) for all sufficiently small positive times, and the same nonmutual choices of Section 16.4 for all sufficiently small negative times. The observed organization does not require the particular constant sine mobility.

Complete form reversal retains the mobility values, negates q and every initial phase rate, and reverses both margin derivatives. It therefore reverses the local direction of the observation change. More strongly, because the complete conservative form row is phase-only and its phase row is linear in form, (-x(-t),theta(-t)) is the solution with reversed initial form. This exact local time-reversal identity uses the same phase-only mobility law in both rows. Merely assuming positive state-dependent mobility without its form-reversal symmetry would not justify that identity, although the displayed initial sign argument still has its own pointwise version.

These are local conclusions for each specified law. An arbitrarily large positive common multiplier already changes its clock speed, and the class has no shared numerical Lipschitz or acceleration budget. No uniform positive time interval follows from the strict initial signs alone. In particular, Section 17's uncertainty box and window [1/128,1/64] are not transferred to another mobility. Support-compatible observed pairs likewise do not license use of the sine-specific collective rate, pulse or recurrence formulas.

20.3. A frozen relative-rate discriminator

Use only the two previously admitted comparison members

\[ m_i^{(\epsilon)}(\theta) =\frac{1+\epsilon(S_i/d_i)^2}{\pi d_i}, \qquad \epsilon\in\{0,1\}. \]

Both complete rows change together when the mobility changes. The same source, support, held capacities, coefficients and structural clock are retained. No parameter is fitted. Before evaluating the alternative, fix the dimensionless initial-rate observation

\[ \boxed{\mathscr D= \frac{\dot M_1(0)-\dot M_2(0)} {\dot M_1(0)+\dot M_2(0)}.} \]

The positive source has a strictly positive denominator; the form-reversed control has a strictly negative one. Both are valid denominators. A generic numerical consumer must still report the ratio unavailable whenever its denominator enclosure includes zero. The raw margin derivatives remain separate evidence.

A common positive clock rescaling multiplies both derivatives by the same factor and leaves mathscr D unchanged. Adding a common phase-origin velocity also changes neither derivative, because their phase-rate coefficient sums are zero. Therefore distinct values exclude an identification by common clock rescaling and common phase drift at this source. They do not exclude arbitrary state redefinitions or establish an independent physical measurement bridge.

Put s_i=S_i/4 at the frozen preparation. The preceding exact expressions give

\[ \mathscr D_\epsilon= \frac{\epsilon(s_0^2+s_1^2-s_2^2-s_3^2)} {8+\epsilon(s_0^2+3s_1^2+3s_2^2+s_3^2)}. \]

Hence mathscr D_0=0 exactly, without comparing rounded intervals. To evaluate the other member independently, the required fine currents are explicitly

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

The shared outward rational trigonometric arithmetic at d=1/8 certifies the strict bound

\[ \boxed{\frac1{500}<\mathscr D_1<\frac1{400}, \qquad \mathscr D_0=0.} \]

Rounded displays of the enclosed values are:

Fixed mobility	M1_dot(0)	M2_dot(0)	mathscr D
epsilon=0	0.0396852001938463	0.0396852001938463	Exactly 0
epsilon=1	0.0417943681701375	0.0415967934157799	Approximately 0.00236925293520505

The rational strict interval, rather than those rounded decimal displays, establishes separation. Form reversal negates both raw margin derivatives and leaves their ratio unchanged. Its exact control follows algebraically; no new trajectory or observation window was evaluated.

The result separates two questions. Both laws create the same local nearest-partner ordering from the chosen preparation, so observing only that qualitative ordering cannot select between them. Their relative rates nevertheless differ on the same complete state. Reciprocal work cancellation and the qualitative mechanism are broader than the selected quantitative constitutive response.

20.4. Shared comparison and evidence boundary

SineExchangeComparison.with_current_squared_mobility(epsilon=...) in the comparison owner retains the original capture and declares the alternative mobility, both complete rows and their storage-work evidence. The assess_sine_pairing_mobility reader in the scale owner evaluates two explicitly supplied chord-margin triples and their rate contrast, sum and available ratio. Its triple order is (observer, alternative, preferred), so (1,2,0) and (2,1,3) represent exactly M1 and M2 above.

The reader uses the shared fine capture, phase-current arithmetic and actual phase rows. Exact cancellation is retained where the law and gap identities provide it; overlapping intervals are not promoted to equality. The generic report evaluates specified instantaneous margins, not a theorem about every possible preparation. The positive-class conclusion is the conditional argument of Section 20.2.

The constitutive comparison contract and independent replica tests retain both frozen controls, invalid-domain and availability cases, full-node rates and explicit law provenance. Native Arg dispatch and the selected sine runtime remain separate. No finite-horizon forecast, unchanged invariant volume, periodic pulse, microscopic law selection or physical identification follows from this static comparison.

20.5. Recursive inheritance and the equilibrium tangent do not select one law

The existing composition comparison already distinguishes equal replication from independent neighbor superposition. Its counterfamily and the frozen discriminator above give a stronger combined scope statement: exact recursive inheritance and the same small-signal exchange can coexist with different complete dynamics. The following is a corollary of those admitted laws, not a new constitutive model or evidence that a hierarchy forms itself.

Take any finite connected simple undirected unit base graph with at least two nodes, positive held capacities nu_i, and fixed w,beta>0. Keep e=0 and either existing member epsilon=0 or epsilon=1. For an integer k>=1, replace each base node i by k constituents (i,r), put no edges within a fiber, and replace each base edge by all k^2 cross edges. Supply synchronized fiber states x_(i,r)=X_i, theta_(i,r)=Theta_i and capacity nu_(i,r)=nu_i. With q=L X and S_i=sum_j sin(Theta_j-Theta_i) on the base, direct neighbor counting gives, for every base preparation,

\[ d_{i,r}^{\rm fine}=k d_i,\qquad q_{i,r}^{\rm fine}=k q_i,\qquad S_{i,r}^{\rm fine}=k S_i,\qquad m_{i,r}^{(\epsilon),\rm fine}=\frac1k m_i^{(\epsilon)}. \]

Consequently both actual fine rows are the copied base rows:

\[ \dot x_{i,r}=w\nu_i m_i^{(\epsilon)}S_i,\qquad \dot\theta_{i,r}=(w\nu_i/\beta)m_i^{(\epsilon)}q_i. \]

Uniqueness preserves synchronization; no capacity or clock rescaling is needed. Every base edge contributes the same storage on each of its k^2 copies, so the full storage satisfies E_fine=k^2 E_base on this invariant submanifold. Applying an integer ell>=1-fold construction to the result is, up to relabeling, the (k*ell)-fold construction, with storage factor (k*ell)^2. This proves finite iterative same-law inheritance for both candidates. The support, fiber partition and synchronized preparation remain supplied; it proves neither attraction to these fibers nor generic off-fiber closure, autonomous hierarchical formation or a fractal dimension. Section 7 retains the internal variables required away from synchronization for the sine law.

The common equilibrium tangent is equally insufficient for selection. At any full equilibrium with q=0,S=0, the difference between these complete fields is

\[ \dot x_i^{(\epsilon)}-\dot x_i^{(0)} =\frac{\epsilon w\nu_i}{\pi d_i^3}S_i^3,\qquad \dot\theta_i^{(\epsilon)}-\dot\theta_i^{(0)} =\frac{\epsilon w\nu_i}{\beta\pi d_i^3}S_i^2q_i. \]

Both corrections are cubic in a joint form/phase perturbation, including at nonconsensus critical targets. Let H_* be the cosine phase Hessian there and K=diag(nu_i/d_i). Their full Jacobian is therefore identical:

\[ J_*= \begin{pmatrix} 0&-(w/\pi)K H_*\\ (w/(\beta\pi))K L&0 \end{pmatrix}. \]

When H_* is positive on the common-phase quotient, the existing conservative tangent argument gives the same free oscillatory modes, with the common origins treated separately. The same prescribed infinitesimal input and observation also have the same linear response. This does not transfer a finite-amplitude periodic family, its period or its transverse stability. In particular, the bounded positive-frequency work-port peak theorem requires positive loss; it is not a theorem about a bounded forced response at e=0.

Nevertheless, the already frozen non-equilibrium comparison in Section 20.3 gives mathscr D_0=0 and 1/500<mathscr D_1<1/400. Thus the candidates have identical recursive inheritance and equilibrium linearizations but different relative rates that a common clock rescaling cannot remove. No frozen response is rerun for this conclusion. To select a microscopic law one still needs a justified additional restriction or an independent discriminator; calling either prepared inheritance or an oscillatory tangent "fractal resonance" cannot supply that restriction.

The shared comparison tests check both admitted mobilities on a nonregular-degree base with unequal per-fiber capacities and successive two- and three-fold replication. They retain every fine row and check complete rates and storage against independent values; this is implementation evidence for the corollary, not its proof.

20.6. Universal collective-mean closure forces a constant circular current

The sine counterexample has a stronger class-level counterpart. Let j be a bounded real kernel on the full phase circle. Consider the additive form source

\[ \dot x_i=\frac{w\nu}{\pi d_i}\sum_{k\sim i}j(\theta_k-\theta_i), \qquad w,\nu>0. \]

Hold capacity, coefficients, support and clock fixed, with no inputs or events. Use this form row, optionally with an additional linear form-diffusion term, which vanishes on the uniform-form preparations below. No other form source is added. The phase row remains separately supplied; the necessary form-row test below does not select it.

Replace both vertices of K2 by pairs and its edge by all four cross-edges, giving K2,2. Pair membership and every attachment are known and unchanged. Suppose an autonomous collective state retains only each pair's arithmetic form mean X, circular mean direction Theta, and held capacity, discarding internal dispersion. Require this description to reproduce the form-mean rate for every phase center and every sufficiently small internal dispersion. It need not be assumed in advance that the reduced rate has the original kernel's formula; the synchronized preparation already fixes that value.

Give all four nodes the same form. Prepare the first pair at (Theta_A,Theta_A) and compare the second pair at (Theta_B,Theta_B) with (Theta_B+delta,Theta_B-delta), where |delta|<delta_0<pi/2. Both preparations have the same retained state. In particular the second pair's mean phasor is exp(i*Theta_B)*cos(delta), whose direction remains Theta_B because cos(delta)>0. This uses a regular circular mean and compatible local lifts, not an angle assigned at a zero resultant. The first pair's two constituents have identical neighbor lists, so its mean-rate equality, with Delta=Theta_B-Theta_A, requires

\[ \boxed{\quad j(\Delta+\delta)+j(\Delta-\delta)=2j(\Delta) \quad(\Delta\in\mathbb R/2\pi\mathbb Z,\ |\delta|<\delta_0). \quad} \]

Classification without an extra regularity assumption. Fix any allowed delta. Applying this identity successively at Delta+n*delta gives

\[ j(\Delta+n\delta)=j(\Delta) +n\,[j(\Delta+\delta)-j(\Delta)]\qquad(n\in\mathbb Z). \]

All arguments are circular. Boundedness of j therefore forces the bracket to vanish. Every sufficiently small phase translation leaves j unchanged; subdividing an arbitrary phase displacement into finitely many such translations shows that j is constant on the circle. No continuity, differentiability or measurability was used. If the kernel is also odd, that constant is zero. Conversely a constant kernel passes this source-rate test, but this alone does not establish closure of the other evolution rows.

Thus no nontrivial bounded odd additive circular current admits the proposed universal state of collective means and held capacities. The obstruction is already an instantaneous source mismatch; choosing a phase evolution cannot repair it. It concerns loss of the resultant's length, a stronger compression than the first-moment information premise, which retains the full complex resultant and degree. Replacing sine by a different nonconstant kernel does not remove this information requirement.

The result does not forbid closure on a synchronized invariant family, restricted phase domains, retained internal coordinates or exact memory. For example a kernel that is affine on a restricted lifted interval can pass its local midpoint tests without being globally constant. A capacity or clock that consumes the discarded dispersion adds information and needs its own law; it is outside the held, common-clock premise. Native Arg pressure is nonadditive and is not classified here. Nothing creates a primitive edge or selects a physical particle, support or interaction law.

The existing global pair state and state on asymmetric support retain the information needed by their declared sine laws; their coordinates are derived from the constituents rather than added primitive causes. The static information controls reuse exact rational constituent phasors to compare equal form/phase means with different resultant lengths and different sine/cubic sources. They check independent full-neighbor sums and the shared reader's scope, not the universal classification or future trajectories. No new runtime law or frozen response is introduced.

21. Protected relative geometry without an assumed invariant volume

21.1. Quotient only the common origins of the actual complete law

Retain the exact twenty-edge doubled-C5 support and winding-one target of Section 12, common positive held capacity and the conservative current-squared mobility of Section 20. The comparison uses only its already declared members epsilon=0 and epsilon=1, with w=beta=nu=1. The identities below display the existing positive coefficients and apply to each fixed finite epsilon>=0; no parameter search is needed. All constituent forms, phases and fixed connections remain in the model. No input, support event or clock change is added. Reflecting the target gives the same result for winding minus one: the cosine storage, spectral gap and radius condition use the same absolute target increment.

Let n=10, d=4, P=I-11^T/n, L=L_f and

\[ S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad m_i(\theta)=\frac{1+\epsilon(S_i/d)^2}{\pi d}, \qquad M=\operatorname{diag}(m_i). \]

Constant common form translation and common circular phase rotation are exact symmetries: neither changes q=Lx, S, M or either complete rate row. In Section 12's target chart write

\[ x=\bar x\mathbf1+v,\qquad \theta=\theta_*+c\mathbf1+h\pmod{2\pi},\qquad v,h\perp\mathbf1. \]

The retained relative state is (v,h), with eighteen continuous coordinates. Only the two common origins are discarded. Pairwise form differences and circular phase differences, including internal member state, remain determined. The common phase origin c has a local continuous lift, rather than a globally defined real mean on the phase torus.

The induced field in the same structural clock is exactly

\[ \boxed{\dot v=w\nu P M(h)S(h),\qquad \dot h=(w\nu/\beta)P M(h)Lv.} \]

Here S(h) and M(h) mean their evaluation at theta_*+h; the omitted common phase cancels from every gap. No projected mobility or scalar effective capacity replaces these matrix products. The removed origins obey the independently determined rows

\[ \dot{\bar x}=\frac{w\nu}{n}\mathbf1^TMS,\qquad \dot c=\frac{w\nu}{\beta n}\mathbf1^TMLv. \]

Their initial values and these integrals reconstruct the full state as long as the chart is valid. Discarding them therefore limits the claim to relative structure; it does not prove that the original means are constant. Projecting out their computed motion also does not reparametrize time or supply an independently chosen phase clock.

21.2. The existing storage barrier protects the relative pattern

The exact same fine storage descends to the quotient:

\[ E(v,h)=\tfrac12 v^TLv+ \beta\sum_{\{i,j\}}[1-\cos(\theta_{j,*}-\theta_{i,*}+h_j-h_i)]. \]

Since q=Lv and S both have zero sum, differentiating using the projected field cancels the exchange terms exactly:

\[ \dot E=w\nu q^TPMS-w\nu S^TPMq=0. \]

The projections act trivially on the left vectors q,S; the remaining cancellation uses the symmetry of the same diagonal M in both rows. Conservation holds along the changed law itself, not along a substituted sine trajectory.

Keep every radius, phase-lift and storage hypothesis from Section 12:

\[ \alpha=2\pi/5,\quad E_*=20\beta(1-\cos\alpha),\quad 0<r,\quad \alpha+\sqrt2r<\pi/2, \]

\[ c_r=\cos(\alpha+\sqrt2r)>0,\qquad \kappa_f=\frac{5-\sqrt5}{2}\min(1,\beta c_r),\qquad 0<\eta_*<\kappa_f r^2. \]

The symbol eta_* is the existing excess-storage ceiling, renamed here to distinguish it from the mobility parameter. Put Z^2=||v||^2+||h||^2 and mathcal E=E-E_*. The unchanged target Hessian and fine spectral gap give the same geometric bound

\[ \mathcal E\ge\kappa_f Z^2\qquad(Z\le r). \]

This inequality concerns the storage and target chart; its derivation does not contain the mobility. Define the relative family

\[ \boxed{\mathcal V= \{(v,h):Z^2<r^2,\quad \mathcal E<\eta_*\}.} \]

Every state in V is protected for all positive and negative times under the declared mobility. If there were a first exit at Z=r, the same conserved excess would have to satisfy both mathcal E<eta_* and mathcal E>=kappa_f*r^2, a contradiction. More explicitly,

\[ \boxed{Z(t)^2\le\frac{\mathcal E(0)}{\kappa_f} <\frac{\eta_*}{\kappa_f}<r^2\qquad(t\in\mathbb R).} \]

There is no finite-time escape hidden in this statement. On the phase torus |S_i|<=d and 1/(pi*d)<=m_i<=(1+epsilon)/(pi*d). In particular every full form rate is bounded by w*nu*(1+epsilon)/pi. The full form can grow at most linearly on a finite time interval, and the phase rates are at most linear in that finite form. Smooth continuation therefore exists in both time directions. Alternatively, the relative field is smooth on a neighborhood of the compact trapped closure; its two origin rows stay bounded there and can be integrated separately.

The family V is open, has finite positive eighteen-dimensional volume, and has compact closure inside the valid relative chart. Its invariance does not require a bounded interval for the absolute form mean. Every fine edge remains acute in its target-compatible lift, and every cycle retains the target winding. Section 14's phase-separation argument therefore continues to identify the same five constituent pairs. These consequences use the proved geometry, not the sine-specific rates or an invariant-volume assumption.

The individual pair swaps remain exact symmetries of this particular mobility: they permute degrees, currents and both rows together. Consequently each synchronized tip set is invariant. Smooth uniqueness also prevents a state outside such a set from reaching it in finite time. Removing all five tip sets gives the corresponding open invariant family V_*, still of finite positive relative volume. This fact alone does not establish a nonzero internal velocity, an angular circulation bound or a repeating waveform; those are separate dynamical questions.

The conclusion is conditional maintenance of an already admitted organization. The earlier local-transition preparation is not thereby placed inside this protected family. Since the family is invariant in both time directions, it is not a finite-time capture target for states outside it under this same complete autonomous law. This uses the two-sided invariance argument, not the unproved invariant measure for positive epsilon. A measure-based obstruction to asymptotic capture requires separate recurrence hypotheses.

21.3. Mean motion and divergence change although storage does not

The original weighted-mean calculation exposes the difference without choosing a new response preparation. More generally, on the admitted unit support with held positive capacities put rho_i=d_i/nu_i and W=sum_i rho_i. For the same mobility family, the old weighted form mean and a locally lifted phase mean satisfy

\[ \boxed{\begin{aligned} \frac{d}{dt}\frac{\sum_i\rho_i x_i}{W} &=\frac{w\epsilon}{\pi W}\sum_i\frac{S_i^3}{d_i^2},\\ \frac{d}{dt}\frac{\sum_i\rho_i\theta_i}{W} &=\frac{w\epsilon}{\beta\pi W} \sum_i\frac{S_i^2q_i}{d_i^2}. \end{aligned}} \]

The constant-mobility terms vanish because sum S_i=sum q_i=0. The remaining terms are actual drift, not a fitted correction to a conserved quantity. The weighted phase expression still depends on declared continuous lifts; it is not a global scalar observable on the phase torus. For the doubled-C5 coefficients frozen above these two right-hand sides become respectively epsilon*sum(S_i^3)/(640*pi) and epsilon*sum(S_i^2*q_i)/(640*pi).

Write C_i=sum_{j~i} cos(theta_j-theta_i), so partial_theta_i S_i=-C_i. The divergence of the full field is

\[ \boxed{\operatorname{div}F =-\frac{2w\epsilon}{\beta\pi} \sum_i\frac{\nu_i q_i S_i C_i}{d_i^3}.} \]

The form row has no form derivative; the displayed term is the diagonal phase derivative of the changed phase row. This is also the Euclidean divergence of the induced relative field. To see this without treating constrained coordinates as independent, choose a constant linear basis for the relative form and locally lifted phase coordinates and append their two common origins. The complete field is independent of the two origin values. Their two diagonal derivative entries are zero, leaving precisely the relative-coordinate trace. The same argument applies to node-reference differences instead of the centered basis.

For epsilon=0 both mean drifts and the divergence vanish identically. For epsilon=1 these identities fail even arbitrarily near the existing protected target, not merely outside its acute domain. An analytic local check makes this explicit without a new numerical preparation. Put c_alpha=cos(alpha)>0 and let H be any nonzero centered direction. At h=s*H,

\[ S=-s c_\alpha LH+O(s^2),\qquad C=d c_\alpha\mathbf1+O(s). \]

Taking v=-s*H in the same open family gives

\[ \operatorname{div}F =-\frac{2w\nu\epsilon c_\alpha^2}{\beta\pi d^2} s^2\|LH\|^2+O(s^3), \]

which is nonzero for sufficiently small nonzero s when epsilon>0. For the form mean, L maps the centered subspace invertibly onto itself. That subspace contains vectors Q with sum Q_i^3!=0. Choose the direction with LH=Q; then

\[ \begin{aligned} \dot{\bar x} &=-\frac{w\nu\epsilon c_\alpha^3}{n\pi d^3} s^3\sum_i Q_i^3+O(s^4),\\ \dot c &=-\frac{w\nu\epsilon c_\alpha^2}{\beta n\pi d^3} s^3\sum_i Q_i^3+O(s^4). \end{aligned} \]

Both are likewise nonzero locally. These are directional proofs about the existing open family, not a new selected-amplitude experiment or radius search. Thus the old absolute-mean slab and ordinary-volume proof cannot simply be inherited. A zero drift or divergence at a particular captured state would not restore either all-state identity.

21.4. What is still required for an almost-everywhere return claim

The relative trapping theorem concerns every admitted initial state. Recurrence requires a separate measure argument. For epsilon=0, the relative field is divergence-free; ordinary relative volume restricted to V or V_* is finite and invariant. The existing finite-measure proof then gives relative-state recurrence for almost every point with respect to that measure. Full-state sine recurrence retains the additional absolute-mean and phase-torus premises of its owner.

For epsilon=1, nonzero Euclidean divergence prevents the same volume proof, but does not disprove recurrence or exclude a different invariant density. To recover the comparable almost-everywhere statement on the relative family, one sufficient missing object is a finite invariant measure equivalent to its natural relative volume. For a positive differentiable density varrho(v,h), its exact obligation is

\[ \operatorname{div}_{v,h}(\varrho F_{\rm rel})=0, \qquad 0<\int_{\mathcal V}\varrho\,dv\,dh<\infty, \]

with equivalence to the preparation measure and compatibility with the complete two-sided flow. Positivity and smoothness on a neighborhood of the compact closure would supply useful sufficient finiteness and equivalence conditions. A singular invariant measure concentrated at the stationary target exists, but its recurrence says nothing about almost every initial state in the open family.

Even a density depending on phases alone requires a genuine integrability check. Let varrho(theta) be global-phase-shift invariant, write

\[ B_i=\partial_{\theta_i}\log m_i,\qquad a_i=\partial_{\theta_i}\log\varrho. \]

Because q=Lx spans the centered subspace, invariance for every relative form would require a common value

\[ \nu_i m_i(a_i+B_i)=\gamma(\theta)\quad\hbox{for every }i. \]

The shift condition sum_i a_i=0 fixes the only candidate gradient:

\[ \boxed{a_i=-B_i+ \frac{\sum_j B_j}{\sum_j1/(\nu_jm_j)}\frac1{\nu_i m_i}.} \]

This one-form must be an exact gradient on the relative chart, with the necessary positivity, normalization and any extension conditions. No such solution is established here. A product of reciprocal nodal mobilities cannot be assumed to solve it: differentiating that product also introduces neighboring mobilities' derivatives. Moreover failure of this phase-only ansatz would not exclude a density depending on both relative form and phase. The general equation above, rather than an unproved invariant-volume label, is the remaining obligation.

This theorem therefore certifies relative geometry, while leaving the alternative's comparable almost-everywhere recurrence claim unavailable. It neither labels that dynamics nonrecurrent nor imports the old internal-pulse period or circulation theorem. Relative return, if subsequently proved, would itself not ensure return of discarded common origins or provide a deadline for a selected state.

21.5. Shared balance and admission evidence

The existing mobility comparison retains its own complete-law provenance. Its relative_balance(reference_node=...) reader uses captured fine rows, phase currents and cosine sums to report reference-node coordinate rates, weighted-mean drift and full/relative divergence. Exact constant-mobility cancellations remain distinct from finite state-specific interval enclosures.

assess_sine_mobility_geometry in the scale owner reuses the actual doubled-cycle support, symbolic target, supplied phase lifts and shared source storage/barrier admission. It separates an admitted relative family, a trapped captured source and membership in the stricter family with synchronized tips removed. The same geometry bounds do not relabel the alternative as the original law. The report retains recurrence as unavailable when its invariant-measure obligation is unresolved, rather than turning that absence into a negative trajectory verdict.

The source capture is not treated as the exact symbolic twist merely because rounded phases are close to it. No new radius, uncertainty budget, preparation scan or frozen response is used. The relational contract and independent replica tests own the executable admission and balance checks. Protected relative organization remains distinct from autonomous formation, physical identification and selection of the microscopic mobility.