Retained organized identity under sufficiently weak external contact¶
1. A controlled change of contact strength¶
The unit-contact problem leaves stationary stability in the original complete marked class unresolved. This result changes one supplied model parameter: all 45 external spokes have the same positive weight eta. It proves stationary existence and all-future identity retention for a declared weak-contact range, including the original acquired and observed source family. It is not a solution of the unit-weight stationary problem. An exact stationary continuation also identifies a strictly stable full-state branch and its collective fold. The larger branch interval gives local stability of that branch; the acquired-source inclusion retains its own smaller interval. Within that source interval, the full fine residuals decay exponentially and the actual collective rows approach the six-node wheel law with an explicitly bounded remainder. The original all-origin source family cannot, however, consist entirely of preparations converging to locally asymptotically stable equilibria: its retained relative-phase circle forces at least one exceptional origin for every fixed source member. At the fixed strength eta=1/100000, a separate transient estimate certifies five explicit donor-origin intervals selecting the five attractive relative registrations, while retaining every original source error and the complete observation history.
| Item | Declared model and scope |
|---|---|
| Coordinates and graph | The same 54 nodes (c,j), five host children c=0,...,4, donor c=5, and slots j=0,...,8; retain all signed forms and circular phases |
| Old support | All 99 old edges have weight one: six internal C9 cycles and nine host rim C5 cycles |
| Changed support parameter | Every same-slot host-donor spoke has weight eta>0; eta is dimensionless and distinct from the constitutive deformation parameter used elsewhere |
| Complete law | Both weighted Laplacian and weighted sine current consume the same eta. Held unit capacities, uniform quarter factors and the smooth full-circle phase domain remain supplied |
| Clock | Prime denotes the existing tau=(1023/1024)t clock; both rows transform together |
| Means | Work on the zero total form leaf and any declared conserved lifted phase mean, including the original donor origin beta |
| Marked identity | Six internal winding-one cycles, nine old rim winding-one cycles, and the original 32 independent zero mixed host periods; new spoke loops receive no prescribed winding |
| Generic entry | The initial old edges lie in their original acute fixed-branch cell and the full weighted storage is below the barrier in section 2 |
| Source interval | 0<eta<=1/100000, with all twelve original source allowances, all 96 relative directions and every supplied donor origin beta |
| Certified source basins | At fixed eta=1/100000, the five donor-origin intervals of half-width 3/5 centered at 2kpi/5 select the corresponding strict relative minima; the intervening gaps are not completely classified |
| Events and execution | The contact event changes weights/support only, with coordinates unchanged. No later forcing or event is supplied. This is a detached complete-law theorem, not native operator admission or an occurrence rule |
| Evidence boundary | Analytical inequalities and static algebraic controls; no stationary search, trajectory, source-producer replay, response fit or new physical record |
Let \(E_{\rm old}\) and \(E_{\rm sp}\) denote the old edges and spokes. Define
Here w=1 on old edges and w=eta on spokes. Changing only the sine term, only the form exchange or only a recorded storage cost would define a different model. Reciprocity gives the actual complete storage law
Eta is supplied contact strength, not a selected law, inferred physical constant or control inferred from dissipation. Eta=0 is a disconnected reference used in the proof; every dynamical conclusion below requires strictly positive eta.
2. The old marked cell retains its complete boundary barrier¶
Use the original lifted branches: internal closing edges and host rim closing edges receive their fixed 2*pi addition. Every old edge gap lies in (-pi/2,pi/2). The resulting cell preserves the complete marked class, including the zero mixed host periods, not just fifteen winding labels. Spokes are unrestricted.
Put
The strict rational lower bound is the existing cycle-face inequality. Its method applies here because the 99 old unit edges are partitioned into six C9 and nine C5 cycles. Convexity of 1-cos on the closed acute interval bounds their total phase cost below by \(V_*\).
On a positive face of an n-cycle, one angle equals pi/2 and the remaining angles sum to 3pi/2, giving \(b_n\) by Jensen. A negative internal face instead has remaining mean 5pi/16 and a larger cost. A negative rim face is empty: one angle -pi/2 and four at most pi/2 cannot sum to 2*pi. These cases cover all 198 signed old-edge faces: 54 positive and 54 negative internal faces, 45 positive rim faces and 45 empty negative rim faces. No joint attainability of separate cycle minima is assumed.
The weighted spoke potential and the full form storage are nonnegative. Consequently every actual boundary state satisfies
The old graph has two components. After fixing an origin in each, its closed acute cell is a bounded polytope of relative lifts. Restoring the relative host-donor origin on the circle makes its closure compact modulo common rotation. A fixed lifted mean does not require pretending that this relative origin is fixed.
Entry and retention theorem. If the initial old phases belong to this cell and \(H_\eta(0)<V_*+B\), no future old edge reaches an acute face under (1). A first exit would contradict (2)--(4). The strict energy margin separates the retained phase sublevel from all old faces. All six internal and nine rim windings and all original mixed host periods therefore persist for every future time.
For every fixed eta>0 the complete graph is connected. Thus the form part of \(H_\eta\) bounds all forms on the zero-mean leaf. Together with the compact circular phase description this makes the retained sublevel precompact. This is an all-future trapping argument for the full state; no carrier restriction, acute-spoke premise or phase reset is used.
3. Positive contact admits stationary minima and a stable retained set¶
Consider the reference phase circle
It lies strictly inside the old cell for every circular beta. The fifth roots of unity sum to zero, so its unweighted spoke potential is exactly 45, independently of beta. Therefore
For every
the reference circle lies below the boundary barrier. Minimizing \(V_\eta\) on the compact closed old cell, including its relative-origin circle, then gives an interior minimum: its value is at most (6), whereas every boundary value is at least \(V_*+B\). At that minimum all full phase derivatives vanish and the full relative phase Hessian is positive semidefinite. With x=0 it is a stationary point of both rows in (1).
This proves existence in the original complete marked class. It does not make the reference circle itself stationary when eta>0, select an origin, or prove a strictly positive Hessian. The minimum may be degenerate.
Let \(\mathcal M_\eta\) be the compact set of phase minima just obtained, modulo common rotation. The equilibrium set with x=0 and phase in \(\mathcal M_\eta\) is Lyapunov stable in the relative circular phase space with mean-zero forms. Equivalently one may use its full periodic preimage on a fixed-mean lift leaf; a single compact choice of lifted representatives is not asserted to be invariant. Subtract its minimum storage value from \(H_\eta\). Within the trapping cell this difference is nonnegative and vanishes exactly on that set: connectedness makes its form contribution zero only at x=0. Compactness gives a positive storage separation from the complement of any neighborhood of the set inside a bounded retained sublevel. Continuity provides a smaller initial neighborhood below this separation, and (2) prevents exit. This is set stability, without an assumed isolated or strictly positive stationary point.
There is a nonempty open set of initial full states below the barrier, since (6) is strict. Every such state admitted inside the cell retains the original identity. The actual acquired preparation requires its own inclusion, supplied in section 5.
4. The weighted complete law still converges to one equilibrium¶
The complete-contact convergence method must be re-admitted for (1). This section holds throughout \(0<\eta\le1\), independently of the narrower trapping condition (7). The latter is included because B<5 already follows from (3). Edgewise comparison gives
The positive lower bound is for each fixed eta. No uniform positive gap is asserted as eta approaches zero.
Set \(k=\gamma^2/16\), \(U_\eta=kV_\eta\), \(g_\eta=\nabla U_\eta\), \(P_\eta=L_\eta^{-1}\) on the relative space, and \(v=\theta'=\gamma L_\eta x/4\). Differentiating the actual second row gives
The Hessian of \(V_\eta\) is the actual weighted cosine Laplacian, between \(-L_\eta\) and \(L_\eta\) in Loewner order. The same proof function \(F_\eta=E_\eta+g_\eta^Tv/8\) therefore obeys, with all coefficients now consuming eta,
These estimates use k<1/16 and (8). \(F_\eta\) is an analytic proof function, not a substituted storage or law.
Bounded form energy, compact circular phase space and the unchanged autonomous loss identity put every omega-limit point at \(v=g_\eta=0\). The potential and \(F_\eta\) are real analytic. The finite-chart analytic gradient inequality and finite-length argument in the linked owner therefore apply with \(P_\eta\) and \(\ell_\eta\): every finite zero-form-mean initial state has one limiting equilibrium, x tends to zero, and the original continuous phase lifts have finite total variation and converge. This derives lift boundedness rather than assuming it. The graph metric \(P_\eta\) carries no identification with physical mass.
In particular every state trapped by section 2 converges to an equilibrium strictly inside the same old acute cell, retaining all its marked information. The limit is not selected and need not belong to \(\mathcal M_\eta\); convergence by itself does not imply that the limiting Hessian is positive semidefinite. No convergence rate uniform in eta, finite settling deadline or statement at eta=0 follows.
5. The original acquired and observed source enters the weak-contact trap¶
Keep the complete outer acquisition, sequential observation word and independent donor preparation. The five host sources and donor retain all twelve original allowances in [0,1e-10], their separate zero-sum errors and all 96 relative directions. Every beta is supplied at donor primitive preparation. No reached coordinate, residual, clock, command or waiting interval is changed. Only the subsequently supplied spoke weight is eta.
The retained acquisition inequalities give, before the observation word,
Use the new analytic upper bound \(\bar\gamma=1/3000>\gamma\), not a re-evaluated archived interval endpoint. Over the original word of duration T=1/5000, put
The degree-four host heat contraction bounds every form by \(B_o\), including all fifteen supplied phase events. Forms never jump. The full form row then bounds each net form change by \(d_x\). The phase row bounds each continuous phase drift by m. The final commanded phase offset is exactly zero, while that drift and the form change remain.
Each of the 90 host edges consequently has net form-gap change at most \(2d_x\) and phase-gap change at most 2m. Its initial and final form gaps have magnitude at most \(2B_o\). The square difference identity and the global Lipschitz constant one of 1-cos give the endpoint bound
This bound compares actual endpoints. It neither cancels the paid work of intermediate events nor treats a final inverse command as restoration of state. All earlier acquisition work, event work and continuous losses retain their original accounting.
The recovered donor bounds give form and relative-phase norms at most q=1e-70. Its isolated twist has zero phase gradient and internal Laplacian norm below four, hence
The phases at contact also lie in the required old acute cell. This follows separately from the original source and observation-history inclusions. With zeta=1e-6, A=7/5 and \(\alpha=2\pi/9\), the final host rim gaps have magnitude at most \(A+2\zeta+2m<\pi/2\), and the host internal gaps lie in \([\alpha-2\zeta-2m,\alpha+2\zeta+2m]\subset(0,\pi/2)\). The donor internal gaps lie in \([\alpha-2q,\alpha+2q]\subset(0,\pi/2)\). The observation word already preserves the original branches and mixed periods. Energy alone is not used to infer this initial phase admission.
The new support event leaves coordinates unchanged. Its complete weighted storage jump is
No old edge changes weight. Since \(B_o+q<11/1000\), for every \(0<\eta\le1/100000\) and every beta,
This is supplied positive contact work, not an event funded by past loss. Its bound does not select when the contact occurs.
Combining the actual endpoint estimates gives
Here time zero is the unchanged contacted endpoint, not a state reset. The full joined form total is zero, and the joined lifted phase total is the original 9*beta. The interval for eta lies inside (7), because B>7/500.
Thus the entire original outer-one source family, every original allowance tuple and every donor origin remains in the original old acute cell for all future time under this weak-contact law. All fifteen winding-one identities and all 32 independent zero mixed host periods persist. The full forms tend to zero and the continuous phase lifts converge to one retained-identity equilibrium. Sections 5.1--5.2 below show that this limit has the uniform internal carrier profile. Its collective origins remain unspecified; a possible degenerate or saddle limit is not excluded by source convergence alone. Section 6.5 separately classifies the limits and excludes degeneracy in this energy sector.
These are new analytical consequences of the retained source inequalities under the explicitly changed contact model. No frozen producer was replayed, no archived response was re-evaluated, and no original source or response flag is amended. In particular the original unit-contact response thresholds and horizons do not become weak-contact predictions.
5.1. The source sublevel makes all fine residuals decay exponentially¶
Retain \(0<\eta\le1/100000\), the original old acute cell and \(H_\eta-V_*<3/250\). These premises hold throughout every trajectory in section 5 by the actual storage identity. The result below covers any admitted state satisfying those same premises.
Put \(\alpha=2\pi/9\). The premises give a stronger bound on each internal gap. With one C9 gap fixed at s, Jensen on the other eight gaps gives the constrained cycle cost
Here is an exact certificate for the strict constant. Put \(C_4(y)=1-y^2/2+y^4/24\) and \(C_6(y)=C_4(y)-y^6/720\). The alternating cosine bounds on [0,1] and \(157/50<\pi<22/7\) give
the last inequality is rational arithmetic. On the feasible acute interval, \(F_9\) is strictly convex with minimum at alpha, and \(F_9(-\pi/3)>F_9(\pi/3)\). All other old cycles contribute at least their own \(V_n\), while form storage and spoke potential are nonnegative. A gap with absolute value at least pi/3 would therefore contradict the full storage sublevel. Every actual internal gap satisfies \(|\kappa_{c,j}|<\pi/3\), throughout the future trajectory and at its limit. The rim gaps retain their previously proved acute admission.
Define the exact child means and fine residuals using the original compatible lifts. With \(\psi_j=\alpha(j-4)\), write
P subtracts each child's mean, so its range has 48 coordinates. It commutes with \(L_\eta\), because the same weighted wheel is used in every slot. The lower bound in (17b) follows from the internal C9 factor: \(\lambda_2(L_9)=2-2\cos(2\pi/9)>2/5\). For example, \(\pi>3\) and the cosine upper polynomial at 2/3 already prove this rational bound. The upper bound follows from (8).
Compare the actual phases with \(\theta^0_{c,j}=\psi_j+\Phi_c\) along the straight compatible lifted segment \(\theta^0+s q\), 0<=s<=1. Internal gaps interpolate between alpha and the actual gaps, remaining strictly inside (-pi/3,pi/3). Each carrier rim gap is the average of its nine actual rim gaps on the same closing-edge convention. Every rim gap along the segment is therefore acute. These facts do not require the segment's storage to lie below the source sublevel.
On a fine direction, the internal Hessian contributes more than \(\lambda_2(L_9)/2\), and the rim Hessian is nonnegative. The spoke Hessian is bounded below by \(-\eta L_{\rm sp}\); its largest possible negative contribution is \(6\eta\), since the spoke graph is nine disjoint five-leaf stars. Thus the projected Hessian along this whole segment is bounded below by \(1/5-6\eta>19/100\) and above by 10. Moreover \(P\nabla V_\eta(\theta^0)=0\) for every set of child origins, because its internal currents cancel and its wheel currents are constant in j. Consequently the actual fine gradient obeys
Projecting both actual evolution rows gives the exact fine equations
The matrix \(\mathcal A\) can vary with time and with the collective state. No derivative bound on it is assumed. Instead use the fixed positive quadratic function
This is an auxiliary proof function, not a replacement storage or a measurement map. Young's inequality bounds the absolute cross term by \(m\|q\|^2/2+(200\gamma^2/m)\|z\|^2\). Since \(\gamma<1/3000\), one has \(2/5-200\gamma^2/m>m/2\). Also \(\mathcal F\le\|q\|^2+\|z\|^2\), \(\|q\|^2\le4\mathcal F\), and \(\gamma^2\|z\|^2\le2\mathcal F\). It follows that
The rate is in the declared tau clock and is uniform over the supplied weak-contact interval. It controls only the fine residuals; the complete collective convergence rate need not remain uniform as eta tends to zero. At each fixed positive eta, section 4 still gives one limiting full equilibrium. Equation (17f) makes its internal phases exactly the uniform alpha profile and its forms constant within each child. Full equilibrium and zero total form then make all forms zero. The surviving collective phase origins are not selected by this argument.
5.2. The averaged wheel law has an exponentially bounded remainder¶
Let \(L_{\mathrm W,\eta}\) and \(S_{\mathrm W,\eta}\) be the Laplacian and sine current of the six-node weighted wheel, with rim weight one and spoke weight eta. The actual child means in (17b) obey
These rows follow by averaging the complete law: internal form and phase currents cancel over each cycle. Their residual is exactly
The mean-zero fine residuals cancel the linear Taylor term on every wheel edge; the sine remainder is at most one half the squared gap. Summing both incident nodal contributions gives the first bound in (17h). The second uses \(L_{\mathrm W,\eta}\le L_{\mathrm W,1}\le6I\). This reuses the constituent-memory distinction: hidden fine state can affect the collective rows quadratically, while here its decay and the resulting error are controlled for the actual contacted preparation.
The set q=z=0 is an invariant full dynamical carrier with arbitrary child-constant forms a. On it, (17g) is exactly the weighted wheel law. It is broader than the stationary x=0 carrier defined in section 6.1. For the whole original source family the fine residuals approach this set exponentially, and the exact collective equations acquire a vanishing remainder. This is an asymptotic reduction with an explicit error bound, not exact closure of the transient mean state.
In particular (17h) does not prove that the actual means shadow one autonomous wheel trajectory for all future times, or that they select the strictly stable branch of section 6. The initial fine state remains in \(\mathcal F(0)\), and a collective unstable direction can still amplify a small accumulated error. No coarse event law, controller, autonomous source selector or physical observation follows.
6. An exact stable carrier branch ends at a collective fold¶
The preceding energy argument proves existence without identifying the minimum. A separate exact stationary construction identifies a branch, its full-state stability and the way that stability ends. Eta is a parameter comparing different autonomous laws (1), not a time-dependent input. This construction neither changes nor follows the acquired source.
6.1 Full fine-coordinate rigidity near a stable carrier¶
Put \(\alpha=2\pi/9\). A carrier is a state
Its internal sine currents cancel pairwise. Consequently it is stationary precisely when the six origins are stationary for the weighted wheel potential
Here rim indices are modulo five. The full potential on the carrier is \(6V_9+9V_{\mathrm w,\eta}\). Let \(M_{\mathrm w,\eta}\) be the six-node Hessian of (19). In slot-first tensor order the full phase Hessian is
Thus full positive semidefiniteness is equivalent to wheel positive semidefiniteness. Strict positivity modulo the one common rotation is equivalent in the same way. If the wheel Hessian is positive semidefinite, each of the 48 phase directions with zero mean within every child obeys
This gives a local statement about the full stationary equations, not only their linearization. In a local phase chart write a nearby state as (18) plus six child-mean-zero residuals. For any origins and eta, the projection of the carrier's phase gradient onto those 48 residual directions is exactly zero: the remaining wheel currents are constant in j. At a positive-semidefinite stationary carrier, (21) makes the derivative of these projected equations with respect to the residuals invertible. The implicit function theorem makes the zero residual their unique nearby solution. Every nearby full stationary state is therefore also a carrier, including when the wheel Hessian itself is degenerate. The form is zero at every equilibrium on the zero-form-mean leaf because \(L_\eta\) is connected for eta>0.
In particular, the carrier subset is both relatively open and closed inside the stationary positive-semidefinite locus. Closedness is the closed circular identity (18); openness is the preceding normal uniqueness. Any connected subset of that locus meeting a carrier remains carrier. This does not exclude other disconnected, nonseparable stationary sets.
A marked rim cut cannot provide a positive-semidefinite escape from this carrier locus. If rim neighbors a and b differ by pi, use the wheel vector \(e_a-e_b\), replicated in every slot. Its internal differences vanish, while its two spoke terms cancel because the spoke cosines are opposite. The rim edge a--b contributes -4; the other two incident rim edges contribute at most 2 together. Hence
This bound holds for every eta>=0 and does not require stationarity. The carrier's internal gaps remain alpha. Thus neither a marked cut nor a fine-coordinate bifurcation permits stationary positive-semidefinite continuation from the carrier to another organization.
6.2 The reflected stationary equations have an exact lower branch¶
Ground the donor origin at zero and take the rim origins, in their original cyclic order, to be
Set \(a=1+\eta\), \(c=\cos v\), and \(r=|a+e^{iv}|\). The potential restricted to (23) is
The u equation is \(\sin(v-u)=a\sin u\). Its locally minimizing solution is \(u=\arg(a+e^{iv})\), so
The remaining stationary equation and the defining identity for r give
Reflection makes the fixed rim and donor equations vanish; the two remaining equations occur in opposite pairs. These are consequently the full wheel stationary equations, not merely a constrained critical point condition.
Let \(a_*>1\) be the unique root
Uniqueness follows since \(a^4/(a+1)^3\) is strictly increasing for a>0. Substitution of the two rational endpoints gives opposite signs. For \(1<a<a_*\), f has two positive roots. Use the smaller one, denoted \(r_-(a)\). It continues from \(r_-(1)=(\sqrt5-1)/2\), where (23) is the uniform outer twist. At the endpoint the two roots meet at \(r_*=\sqrt{(a_*+1)/3}\). On the lower branch
The cubic's positive critical point is a strict local minimum; its value is negative exactly before (27), which proves the two-root assertion. The formulas above also give \(-29/30<c<0\). Thus (25) has \(0<u<\pi/2<v<\pi\), with u<v. The five positive principal rim gaps are \(u,v-u,2\pi-2v,v-u,u\). They stay strictly below pi and sum to 2pi, including at the fold. All six internal windings, all nine rim windings and all 32 mixed host periods remain those of the original marked class. For the original closing-edge branch convention, use the compatible rim lifts \((0,u,v,2\pi-v,2\pi-u)\), with the fixed 2pi addition on the edge from child four to child zero. The large \(2\pi-2v\) gap is between the two reflected v origins. A common rotation matches any declared conserved phase mean.
6.3 Full relative stability up to the endpoint¶
Grounding the donor removes the common phase rotation. The wheel Hessian splits into ordinary reflection-odd and reflection-even directions. For the odd vector \((0,p,q,-q,-p;0)\), write its quadratic form as \(2(q,p)O(q,p)^T\). For the even vector \((t,p,q,q,p;0)\), write it as \(2(q,p,t)E(q,p,t)^T\). On the stationary branch,
The identity \(B_1+aA=r\) supplies the middle diagonal. Substitution of (25)--(26), retaining the cubic rather than an approximate root, gives
At a=1, E is the positive multiple \(\cos(2\pi/5)\) of the three-vertex path Laplacian. It has two positive eigenvalues and one zero eigenvalue. Its determinant is strictly positive for every 1<a<=a_* because r increases from the positive zero of \(r^2+r-1\). Continuity and absence of a zero eigenvalue make E positive definite throughout this interval. The lower diagonal of O is r>0. Its determinant is positive before the fold, and zero at the fold, where O has exactly one zero eigenvalue and one positive one.
Equations (20)--(21) therefore prove strict positivity of all 53 relative full phase directions for \(0<\eta<\eta_*\). This gives a locally asymptotically stable equilibrium of the actual two-row law: the strict local storage minimum traps a sufficiently small neighborhood, and section 4 supplies convergence to the unique local equilibrium. All forms and all fine phase directions are included. No source convergence to this particular equilibrium or uniform attraction rate is inferred.
At eta=eta_* the full relative Hessian has exactly one zero direction, which is collective and reflection-odd. Its other 52 eigenvalues are positive; the 48 fine directions retain (21).
6.4 The fold is unstable and has no nearby stationary bypass¶
Eliminate u using (25), and call the resulting potential \(\widehat{\mathcal V}_a(v)\). Its first derivative is
At (27), its first and second derivatives vanish, whereas
For example, differentiating the bracket in (31) gives \(\sin v(a^2/r^3-2)\); at the double root \(a^2=2r^3\). One further derivative gives (32). The full potential is nine times this wheel potential plus a constant. The nonzero cubic term supplies arbitrarily close zero-form states of strictly lower storage, so the fold is not a local minimum.
It is also Lyapunov unstable, without a center-manifold assumption. The positive fine block in (21) first makes every nearby equilibrium a carrier. The positive E block in (29) then forces every nearby wheel equilibrium into (23): phase inversion combined with rim reflection preserves the potential, and uniqueness in the transverse implicit equations sets the reflection-breaking coordinates to zero. The first balance has u derivative \(a\cos u+\cos(v-u)=r>0\), so its minimizing elimination is locally unique. Also \(dr/dv=-a\sin v/r\ne0\), and the bracket in (31) is exactly \(f(r,a)/(ar)\). Thus (26) is locally equivalent to the full remaining stationary equation. At the fold \(f_r=0\), \(f_{rr}=6r_*>0\), and \(f_a=2a_*-r_*>0\). Thus at fixed eta_* the equilibrium is locally isolated modulo rotation; on the eta>eta_* side there is no nearby full equilibrium. If a trajectory from one of the nearby lower-storage states remained forever in a sufficiently small closed neighborhood, section 4 would force convergence to this isolated equilibrium, contradicting nonincrease of storage. Arbitrarily small perturbations therefore leave that neighborhood.
These same implicit equations give the complete local stationary fold, including all 54 nodes: two reflected carrier equilibria meet, the lower-root branch has positive relative Hessian, and the other local branch has one negative collective direction. No nearby fine or reflection-breaking stationary branch continues stability through the endpoint. This is a local classification of this branch, not a global classification of the remaining nonseparable sector.
The endpoint occurs without a marked winding cut. It also lies beyond the original acute rim chart. Indeed \(r_*>83/100\) and \(a_*<27/25\) give
Hence \(c_*>-1/\sqrt2\), so the rim gap between v and -v is \(2\pi-2v_*>\pi/2\), while remaining less than pi. Stability persists through some nonacute rim configurations and then fails collectively; leaving the acute energy chart is not itself loss of marked identity or loss of stability.
6.5 The source energy range has five minima and five saddles¶
The source limit can now be classified without selecting its destination. Fix \(0<\eta\le1/100000\), and consider a wheel equilibrium with the original acute winding-one rim and
Every actual limit from section 5 satisfies (33a): its internal profile is the exact carrier, its forms vanish, and its full excess storage is less than 3/250. Its nine identical rim contributions therefore give \(9(V_{\rm rim}-V_5)<3/250\), since the spoke storage is nonnegative.
Finite-interval classification. Modulo common phase rotation there are exactly ten labeled wheel equilibria satisfying these premises. Five are strict relative minima and five have exactly one negative relative phase direction. Their full 54-node carriers have, respectively, 53 positive relative phase directions, or 52 positive directions and one negative direction. The former are locally asymptotically stable for the actual complete law; the latter are hyperbolic saddles with one unstable direction. No exceptional zero eigenvalue occurs in this positive eta interval.
The proof needs the small collective restoring term rather than an uncontrolled first-order approximation. Ground the donor at zero and write compatible rim lifts as
Here t is the mean host origin relative to the donor; the centered uniform rim profile has mean zero. Put \(b=2\pi/5\), \(c_b=\cos b\), \(s_b=\sin b\), let D be the forward rim incidence, and set \(A=c_bD^TD\), \(G=A^+\). The projected host stationary equations are
The sine on the right is a five-vector. G already removes its mean. A row of G is the cyclic permutation of \((2/5,0,-1/5,-1/5,0)/c_b\). Consequently
These bounds use the exact \(c_b=(\sqrt5-1)/4\). They give a uniform complex-eta construction with real t. Take \(R=3/200\), \(r_0=1/100\), and center (33c) at \(u^{(1)}_c=-\eta\sin(t+b(c-2))/\lambda_1\). On \(|\eta|\le R\), \(\|u^{(1)}\|_\infty\le U_1=141/4000\) and \(\|Du^{(1)}\|_\infty\le D_1=207/5000\). The complex ball \(\|u-u^{(1)}\|_\infty\le r_0\) therefore satisfies \(\|u\|_\infty<U_0=23/500\), \(\|Du\|_\infty<D_0=31/500\). The fixed point is constrained to the complex zero-mean subspace.
Here are explicit rational bounds for contraction and existence. Define
On the positive arguments used here these bound, respectively, \(\cosh z-1\), \(\sinh z\), and \(\sinh z-z\), by their convergent power series. Also \(e^z\le1/(1-z)\). With \(g_0=259/100\), \(g_1=389/100\), \(c_0=31/100>c_b\), and \(s_0=24/25>s_b\), direct rational arithmetic gives
The first line bounds the derivative of the map in (33c): \(|\cos(b+d)-c_b|\le s_b\sinh|d|+c_b(\cosh|d|-1)\), and \(\|D\|_\infty=2\). The second bounds its displacement at \(u^{(1)}\), using the exact linearized equation for that center. Thus the map contracts this ball into itself. Its unique solution \(u(t,\eta)\) is holomorphic in eta on the disk, real for real eta, and smooth in real t. It is a projected stationary shape, not yet a stationary wheel: the remaining mean equation is
If \(W(t,\eta)=V_{\mathrm w,\eta}(t+b(c-2)+u_c;0)\), the projected equations give \(\partial_tW=T\). The uniform complex bound is \(|T|\le5R/(1-U_0)<2/25=:M_0\). Cyclic relabeling makes T periodic with period 2*pi/5; phase inversion combined with rim reflection makes it odd in t. Changing eta's sign is equivalent to replacing t by t+pi. Finally, the coefficient of eta^n in T is a trigonometric polynomial of degree at most n. This follows inductively from (33c): the linear forcing has degree one, and multiplication in every Taylor coefficient adds its harmonic degrees and perturbation orders together. Only sine harmonics \(\sin(5kt)\) with \(5k\le n\) and parity \(5k\equiv n\pmod2\) can occur. Orders below five and order six therefore vanish exactly.
The coefficient at order five follows from the existing exact branch, without assigning a numerical root. Put \(\varphi=(1+\sqrt5)/2\), and \(\ell=(2,2,1)^T\) in the reflection-even coordinates (q,p,t) of (29). The host mean of that vector is \(\ell^T(q,p,t)/5\). Eliminating its two mean-zero directions gives
After a cyclic rim relabeling, the reflected solution of section 6.2 has compatible origins \((-v,-u,0,u,v)\), mean zero, and belongs to the locally unique shape branch. At eta=0, its r equals \(\varphi^{-1}\), \(dr/d\eta=\varphi\), and \(d(r^2+r-1)/d\eta=\sqrt5\varphi\). Equation (30) consequently gives \(\det E=(25\varphi^8/8)\eta^5+O(\eta^6)\). The denominator in (33g) tends to \(25/(4\varphi^2)\). Thus \(W_{tt}(0,\eta)=25\varphi^{10}\eta^5+O(\eta^6)\), which fixes the only possible fifth-order torque coefficient:
The remainder can be bounded relative to \(\sin(5t)\), including arbitrarily near its zeros. Cauchy's coefficient estimate gives \(\|T_n\|_\infty\le M_0/R^n\). Each sine Fourier coefficient of \(T_n\) has magnitude at most \(2M_0/R^n\). Since \(|\sin(5kt)|\le k|\sin(5t)|\), the quotient of the nth coefficient is bounded by \(M_0m(m+1)/R^n\), where \(m=\lfloor n/5\rfloor\). Use \(m(m+1)\le n(n+5)/25\) and put \(h=\eta/R\). Summing the absolutely convergent series from order seven gives
The last strict bound is rational arithmetic at eta=1/100000; the positive-coefficient series increases with eta. The quotient extends continuously through every zero of sin(5t), by the same uniformly convergent sine-polynomial quotients. Hence T has exactly the ten zeros \(t=k\pi/5\), k=0,...,9, on the relative-origin circle. Its derivative is strictly positive at even k and strictly negative at odd k. This is a finite-interval conclusion, not just a leading-order prediction.
It remains to exclude other low-energy shapes outside the contraction ball. Every acute winding-one C5 gap is positive: its other four gaps are strictly below pi/2 and their sum is less than 2*pi. With one gap fixed at s, Jensen bounds the rim cost below by \(F_5(s)=1-\cos s+4[1-\cos((2\pi-s)/4)]\). This function increases above b. Put \(C_8(y)=1-y^2/2+y^4/24-y^6/720+y^8/40320\). The cosine upper bound on [0,3/2], \(\pi>157/50\), and \(c_b>309/1000\) give the exact certificate
Thus every gap of a candidate satisfying (33a) is less than 29/20, where \(\cos(29/20)>1/10\) follows from the sixth-order cosine lower polynomial. The constructed shape also has all gaps in (0,29/20), by its bound \(\|Du\|_\infty<D_0\). The straight compatible segment from any candidate to the constructed shape with the same mean t stays in that interval. On its zero-mean host directions the full wheel potential has Hessian at least \((1/10)L_5-\eta I>0\), since \(\lambda_2(L_5)>1\). Two projected stationary shapes cannot occur on such a strictly convex segment. Every admitted candidate must therefore be the constructed shape. Conversely, the holomorphic bound and u(t,0)=0 give \(\|u(t,\eta)\|_\infty\le(U_0/R)\eta=(46/15)\eta\) by Schwarz's lemma, so all ten constructed equilibria have \(V_{\rm rim}-V_5\le10\|u\|_\infty^2<100\eta^2<1/750\). The classification covers exactly the stated energy sector.
The four host shape directions are strictly positive. Their Schur complement is \(W_{tt}\), so (33i) proves the asserted wheel inertia. The full fine phase block stays positive even for the saddles: acute rim weights contribute nonnegatively, the internal block contributes \(\cos(2\pi/9)\lambda_2(L_9)>1/5\), and the negative spoke part is bounded below by \(-6\eta I\). Equation (20) thus leaves the single negative direction collective. To check full dynamical hyperbolicity, the linearized relative phase equation has the symmetric pencil \(s^2L_\eta^{-1}+sI/4+\gamma^2M_\eta/16\). At s=i*omega, its imaginary part excludes nonzero imaginary roots, and invertibility of the phase Hessian excludes s=0. For positive real s the pencil increases strictly from a matrix with one negative direction to a positive matrix, giving exactly one unstable eigenvalue; a nonreal eigenvalue with nonnegative real part is excluded by the imaginary part of its scalar quadratic identity. All remaining dynamical eigenvalues have negative real part.
This identifies the possible destinations, including their stability types. It does not identify the destination of any specified acquired source or replace the decaying fine-state remainder by exact transient wheel closure.
6.6 An exact antipodal saddle retains positive contact weights¶
An explicit member of the classified saddle family provides a static control on the sign and model boundary. Set \(a=1-\eta\) and let \(r\) be the unique zero of f in (26) between 3/5 and \(r_*^0=(\sqrt5-1)/2\). Indeed, \(f(3/5,999/1000)=14601/10^6>0\), f increases with a on this interval, and \(f(r_*^0,a)=(a-1)(a+1-r_*^0)<0\). Also f decreases strictly with r on this interval, since \(r_*^0<5/8\) and \(f_r<3(5/8)^2-1999/1000<0\).
Use (25)--(26) to define c, u and v, then take the actual rim origins \((-v,-u,0,u,v)\) and donor origin pi. The formal value a-1=-eta only constructs these phases. Shifting the donor by pi changes every spoke sine and cosine sign, making the phase equations exactly those of the actual positive-eta wheel. Both evolution rows still use the positive weighted \(L_\eta\) in (1); no negative-weight flow, capacity change or clock change is invoked.
The root bounds imply \(-5/6<c<-3/4\). Hence A and B_1 in (25) are positive and \(\cos(2v)>1/8\): all five principal rim gaps remain acute and sum to 2*pi. The six internal profiles remain the uniform alpha twist, and all mixed host periods are zero. A common rotation matches any declared conserved phase mean.
In the actual phase Hessian, (29)--(30) apply with this formal a: O is positive definite, while det E<0 and never vanishes for a<1. Its two positive directions at a=1 persist throughout this interval. The wheel therefore has exactly one negative relative direction; the positive fine block and full dynamical classification follow as in section 6.5. This is an index-one saddle of the positive-contact law.
It also lies in the full retained source sublevel. Along this smooth stationary family the phase-gradient contribution to the storage derivative vanishes, leaving
At eta=0 its storage is V_*. This construction exhibits a saddle in the admitted region; it does not assert that a specified actual source converges to it. The following argument establishes exceptional actual source capture without assigning its origin or saddle label.
7. The all-origin source family cannot select only attractive equilibria¶
Fix any \(0<\eta\le1/100000\). Fix an actual admissible host and donor source member: all twelve allowances, every primitive form and phase error in all 96 relative directions, the complete acquisition, donor recovery and sequential observation history remain fixed. Vary only the primitive donor origin \(\beta\) modulo \(2\pi\), as in the original registration family. No new source, event or response is evaluated.
All-origin selection obstruction. For every such fixed member, at least one donor origin converges to one of the five hyperbolic saddles classified in section 6.5, modulo common phase rotation. The limit still retains the complete old marked identity and has the uniform internal carrier profile, by sections 4--5. The theorem does not locate this exceptional origin. It does not exclude open basins of the five strict relative minima.
7.1 The actual prepared circle persists at every finite time¶
Use the original contact phase representation, whose equation (2) reads
Here \(H\) includes the reached host state and all observation backaction; \(z_D\) is the actual recovered donor residual. Neither is replaced by an ideal profile. Isolated donor rotation leaves its complete relative preparation unchanged, so all forms and both residual arrays are independent of beta. In particular the two endpoints beta=0 and beta=2pi have identical circular states. Their continuous lifts differ by 2pi on the nine donor vertices and by zero on the host vertices.
The complete field (1) is unchanged by this integer vertex phase shift: every sine is unchanged, and the phase row depends only on forms. Uniqueness therefore preserves that same endpoint difference at every finite time. Continuous dependence on prepared initial data makes
a continuous lifted family for each finite \(\tau\ge0\). The circular relative phase \(e^{if_\tau(\beta)}\) consequently covers the entire unit circle, with degree one. This follows directly from the endpoint identity and the intermediate value theorem; no origin scan or limiting interchange is used. In particular there is no common finite time at which the entire family lies in a neighborhood restricting this relative phase to a proper circular arc.
This statement is about a circle of separately prepared trajectories. It does not say that any one trajectory keeps rotating: each continuous phase lift converges by section 4. The joined lifted mean is beta/6 and is conserved for each trajectory. To compare equilibria across beta, work on the zero-total-form space with circular phases modulo a common rotation. The projection in (35) is well-defined on that quotient. A fixed lifted-mean chart describes the same local relative stability, but there is no single fixed-mean lift convention turning (34) into an ordinary closed loop of real coordinates.
7.2 Open attractive basins cannot cover this source circle¶
Suppose every beta in this prepared circle converged to a locally asymptotically stable relative equilibrium. Each such equilibrium has an open basin of attraction. Basins of distinct equilibria are disjoint, because section 4 gives a unique limit. Their inverse images under the continuous preparation map would form disjoint open sets covering the connected beta circle. There can therefore be only one nonempty such set: every preparation would converge to the same relative equilibrium \(e_*\).
Choose a neighborhood U of \(e_*\) in which the relative phase used in (35) lies in one proper circular arc. Local asymptotic stability supplies a smaller neighborhood V whose forward trajectories remain in U. For each prepared state, convergence gives a finite entrance time into V. By continuous dependence, a neighborhood of that prepared state also lies in V at that same time, and all its later states stay in U. Compactness of the prepared circle gives a finite subcover. The largest of those finitely many entrance times consequently puts the entire evolved circle in U. This contradicts (35).
Thus at least one origin has a limit that is not locally asymptotically stable. No continuity of the infinite-time limit map, uniform convergence rate, finite equilibrium enumeration or symmetry of the observation word was assumed. In particular an exact reflection of an ideal profile cannot identify this exceptional actual preparation: the full ordered observation history and arbitrary original errors remain present.
The topological argument itself needs no equilibrium enumeration. Combining it with the independently proved low-energy classification in section 6.5 gives the sharper conclusion: all actual limits belong to the ten classified equilibria, so every limit that is not locally asymptotically stable is one of the five index-one hyperbolic saddles. At least one actual origin therefore lies in a saddle's stable set.
A prospective selection statement must consequently allow exceptional origins or supply an independent restriction to a certified basin. This obstruction does not determine their number or measure, certify any specified beta's destination, or establish that all remaining origins reach the known stable branch. The separate source-basin certificate below supplies explicit sufficient origin intervals at one fixed contact strength. Its intervening gaps remain unclassified beyond the existence of exceptional origins in each gap.
8. Explicit acquired-source basins at a fixed weak contact strength¶
Fix eta=1/100000 in both complete evolution rows (1), with the original held capacities, clock, acquisition, observation word and unchanged contacted coordinates. Retain all twelve original allowances and all 96 relative error directions. The following is an analytical source selection theorem; no trajectory, source producer or reserved response is evaluated. In this section t denotes the relative origin used in (33b), while tau remains the evolution clock.
Source-basin certificate. For k=0,...,4, every actual preparation whose donor origin belongs to the circular interval
converges to the corresponding strict relative minimum from section 6.5. The inequality uses the lift of beta nearest the displayed center. All old marked identity survives as already proved in section 5. The conserved full lifted mean fixes the remaining common phase rotation. These five symmetry-related labeled registrations are not five species or different physical objects. The source is neither reset to a stationary shape nor made exactly reflection symmetric.
8.1 Uniform shape regularity holds along the actual trajectory¶
Use the exact child means in (17b), and let a bar with subscript H denote the average over the five host children. With \(b_c=2\pi(c-2)/5\), define
Here u is exactly the projected stationary shape constructed in (33c)--(33e). The host coordinates h and p have zero mean. Since the actual nine rim cycles retain their acute compatible branches, their averaged gaps are also acute winding-one gaps. Jensen across slots, the six internal cycle lower bounds and nonnegative form and spoke storage give, at every future time,
Every averaged rim gap is consequently in (0,29/20), by the same constrained cycle inequality (33j). The constructed u also has its rim gaps in that interval. The straight compatible segment from h to u retains those bounds, without requiring that segment itself to lie in the full storage sublevel.
Let \(P_H=I-\mathbf1\mathbf1^T/5\), and let g be the projected host potential gradient at the actual mean phases. The projected gradient at u vanishes. Hence
Indeed the segment's host Hessian is bounded below by \((1/10)L_5-\eta I>(9/100)I\); its norm is below \(\|L_5\|+\eta<5\). Differentiating the projected stationary equation gives the last bound, because its partial t derivative is \(\eta P_H\cos(t+b+u)\), of norm at most sqrt(5)*eta. Thus the regularity estimate applies during the actual transient, not merely near its final equilibrium.
8.2 The exact mean and shape rows retain the fine-state error¶
Write \(F_0=\mathcal F(0)\) for the fine Lyapunov function in (17e) and retain \(\rho=19\gamma^2/800\). Let r be the actual wheel remainder in (17h), and set \(r_H^0=P_Hr_H\). The donor row admits the sharper channel-specific bound
The first step uses cancellation of the linear fine gap and the quadratic sine remainder; the second uses the star Laplacian norm six. In particular this donor forcing carries eta. It must not be replaced by an unweighted global remainder.
The actual donor current is \(T(t,\eta)+\delta\), where T is (33f) and
These equations follow by separately averaging the five host rows and the donor row, then subtracting the host mean. They retain both full evolution rows, the moving shape and the actual fine-state forcing.
Define an auxiliary shape function and a corrected origin by
The correction chi is retained state, not a change of the phase observation or an assumed instantaneous form relaxation. In the derivative of \(\mathcal E^2\), the unforced fixed-shape terms are
With \(m=9/100\), Young's inequality gives \(2\gamma|p^Tg|\le(m/2)\|e\|^2+(50\gamma^2/m)\|p\|^2\). Since \(K>I\), \(\gamma<1/3000\) and \(\mathcal E^2\le\|e\|^2+\|p\|^2\), these terms give \(\mathcal E'\le-9\gamma^2\mathcal E/1600\). The additional moving-shape term has magnitude at most \(\|e+\gamma p\|\|u_t\||t'|\); (38)--(40) bound its contribution to \(\mathcal E'\) by \(28\eta^2|\chi|\). The fine forcing contributes at most \(\gamma^2\|r_H^0\|\). Consequently, with upper right derivatives at vanishing norms,
Here (33i) gives \(|T|<1000\eta^5\): \(600<5\varphi^{10}<800\), with the upper bound following from \(\varphi<13/8\), and the torque remainder is below 200.
8.3 The original preparation controls the entire transient¶
The separate host and donor total forms are zero before contact and through their independent preparation and observation histories. The host phase total is zero; every observation jump has zero total. Therefore the actual contacted source has exactly
Coarse existing source inclusions already suffice for the needed norm bounds. With zeta=1e-6 and m from (12), subtracting each child's mean gives \(|q_{c,j}(0)|<2(\zeta+m)<1/400000\). Also \(|z_{c,j}(0)|<2B_o<21/1000\), so \(|\gamma z_{c,j}(0)|<7/10^6\). The donor's retained bounds are smaller. Thus
For the second inequality, the rim energy bound and its Hessian \((1/10)L_5>I/10\) along the segment to the uniform rim give \(\|h(0)\|^2<2/75\), hence \(\|h(0)\|<1/6\). Schwarz's bound from section 6.5 gives \(\|u(t,\eta)\|<7\eta\). The exactly zero host mean form and (12) give \(\|p(0)\|<\sqrt5 B_o<1/40\). Substitution in (40), with \(\gamma<1/3000\), proves (42). These are bounds on the reached source; they impose no new restriction on its original uncertainty.
Set \(\mathcal X=\mathcal E+20\eta|\chi|\) and \(\lambda=\gamma^2/200\). At the declared fixed eta, \(9/1600-30\eta^2>1/200\), \(\eta>\gamma^2/20\), and \(8/3+8\eta^2<3\). Equation (41) therefore implies
The coefficient 160 is exactly \(3\gamma^2/(\rho-\lambda)\), and \(\chi(0)=0\) removes any initial relative-form term. The last line follows by integrating the second inequality in (41) using the uniform bound on \(\mathcal E\). This comparison controls the moving shape and accumulated source transient. It does not substitute the stationary shape for the actual wheel trajectory.
8.4 An inward torque prevents passage to a neighboring basin¶
First consider \(|\beta|\le3/5\). In the upper strip \(3/5\le y\le5/8\), (43) gives \(2.999995<5t<3.125005\). Put \(v=\pi-5t\). The rational bounds \(157/50<\pi<22/7\) give \(0.014995<v<1\); therefore \(\sin(5t)=\sin v\ge5v/6>1/100\). The strictly positive torque quotient (33i) then gives \(T(t,\eta)>4\eta^5\). Meanwhile (40) and (43) bound the nondecaying part of \(|\delta|\) by
It cannot reverse the inward torque in this strip. Only the decaying part can move y outward there; its entire future adverse displacement is bounded by
If y first reached 5/8, take its last preceding crossing of 3/5. Between those times it stays in the strip, so integrating (40) would give \(1/40\le J<1/1000\), a contradiction. The lower strip gives the same conclusion with reversed signs, because T is odd. Thus
The complete flow converges by section 4, its fine coordinates vanish by section 5.1, and section 6.5 classifies every possible limit. The only classified relative origin allowed by (45) is t=0, a strict minimum. The period 2*pi/5 of T repeats the same argument about each even center. Together with \(y(0)=-\beta\), this proves (36), including its destination's minus sign.
The five certified source intervals are disjoint and have total angular length six on the donor-origin circle. This is angular coverage, not a probability without a separately supplied distribution. Each of the five gaps between neighboring intervals contains at least one origin converging to a saddle: otherwise the connected interval joining the two certified endpoints would be covered by disjoint open attractive basins, although its endpoints have different destinations. This uses the existing complete limit classification, not continuity of the infinite-time destination map. No exact exceptional count, location or classification of every remaining origin follows.
The estimate is restricted to the fixed eta=1/100000. In particular the comparison \(\eta>\gamma^2/20\) was used; the proof does not silently extend to the whole source-strength interval or to eta=1. It introduces no event, controller, autonomous contact selector, laboratory clock or physical identification. The certified basins belong to the supplied complete law and the unchanged actual history.
9. Meaning and remaining boundary¶
The positive weak-contact interval proves that the same graph topology and complete marked identity are compatible with stationary existence and indefinite maintenance. Quantitative contact strength matters: assigning the same eta consistently in both evolution rows and storage preserves enough of the old energy barrier to admit the actual reached sources. The supplied coupling has not been selected by TNFR's nodal identity or by a physical observation.
Within the original source interval, the same energy margin forces exponential relaxation of all fine form and phase residuals. The actual source limits are consequently uniform internal carriers, and the collective wheel rows have the controlled decaying remainder (17h). Every collective limit belongs to the classified set of five strict minima and five hyperbolic saddles modulo common phase rotation. At eta=1/100000, section 8 certifies which strict minimum is selected on five explicit donor-origin intervals, uniformly over the original source uncertainties and observation history. Each intervening gap contains at least one saddle-converging origin; its remaining source destinations are not classified by the certificate. Uniform strict attractor selection over the original entire origin circle is therefore impossible, even though every member retains its identity and converges.
The exact continuation identifies a strict full-state stationary branch and its collective fold at eta_* in (7/100,2/25). This is a critical coupling for that branch, without a universal transition claim. Local fine-coordinate rigidity and the cut witness exclude two possible stable bypasses; they do not remove distant stationary components.
The unit-weight problem at eta=1 remains outside the retained branch. Its finite common-host exclusions, complete-flow convergence and remaining nonseparable stationary feasibility question are unchanged. No unit-contact stable target or unit-contact winding loss has been established. The execution plan owns that unresolved dependency.
The result gives no autonomous contact selector, event reservoir, laboratory unit bridge or identification with physical matter. Its reusable mechanisms are the full old-edge barrier, explicit new-edge work, carried source-history estimate, analytic convergence, full fine relaxation and its collective error bound, stationary normal rigidity, the exact collective fold, the all-origin attractive-selection obstruction and the source-basin comparison retaining the fine transient under the actual positive weighted law.