Organized identity under continued external contact

Declared geometry and actual-source admission

This gate studies continued contact after the relative-origin comparison. It separates stationary geometry, stability, source inclusion and loss of a marked identity. No finite response, target label or carrier restriction supplies the others.

The separate weak-contact theorem changes every spoke conductance to a supplied positive strength eta in both complete evolution rows and storage. It proves all-future retention of the actual outer-one family for 0<eta<=1e-5. This distinguishes topology from contact strength; it changes neither the unit-strength results below nor their unresolved outer-one maintenance question.

Item	Fixed scope before source assessment
Complete law	Keep the 54-node unit support C9 square W6, all 108 fine form/phase coordinates, held unit capacities, x_tau=(-Lx+gammaS(theta))/4 and theta_tau=gammaLx/4, gamma=1/(1023pi), tau=(1023/1024)t. No input or event after attachment
Stationary geometry	Classify the aligned outer-origin target and the antipodal-donor stationary control on the full fine state, including both conserved means. Retain the twelve-coordinate carrier separately
Organized carrier question	Determine whether any carrier stationary state can retain the closed acute outer winding-one chart, and what any obstruction implies for continued carrier motion. Acute-chart exit, winding loss and full-fine source loss are separate claims
Full-fine boundary	Retain all six internal winding-one cycles and all nine old outer cycles as marked identities. New wheel triangles constrain reuse of an all-edge acute chart; they do not alone exclude a stable state with some nonacute edges
Actual control source	Rebuild all twelve original allowances in [0,1e-10] and the whole sequential observation history. For a separately declared control choose the outer-zero host and donor primitive relative origin beta=0. Keep every fine residual, all 96 relative preparation directions and exact zero joined means; no reset, target substitution or extra waiting
Control acceptance	Prove the actual control lies in the joined acute target cell with total initial excess above 6*V9 below 1/100 and strictly below every cell boundary. If admitted, establish all-future marked winding retention and asymptotic convergence; no finite settling deadline
Work	Count actual new attachment work and all earlier supplied preparation and observation costs separately. A sharper present-storage bound must not erase historical costs or treat dissipated storage as an event reserve
Other registrations and outer-one source	Keep the original family and every beta available as parameters. The beta=0 control does not imply capture of another origin or the outer-one source. Carrier statements do not replace full-fine source inclusion; report unresolved cases explicitly
Numerical boundary	Exact rational primitive admission and existing dyadic128 outward arithmetic. No trajectory, source-producer replay, root search, relative-origin scan, precision tuning or new physical-data evaluation

The geometric proof may classify or obstruct a proposed retained organization. The actual-source control has the fixed acceptance criterion above before its assessment. Conditional maintenance and a scoped obstruction both count as informative outcomes. Neither identifies a fundamental physical property; the plan retains property-first physical comparison and its independent bridge.

The declaration above was committed as 4c4addf1 before the actual-source assessment. The following proofs classify the fixed support and complete law; they evaluate no source trajectory, trial registration or equilibrium search.

1. Complete state, identities and stationary equations

Use node order (c,j), with five host constituents c=0,...,4, external constituent c=5, and internal slots j=0,...,8. The support is the product of C9 and the wheel W6: six internal cycles, five outer ring edges per slot, and five spokes per slot. Its 144 unit edges are those already owned by the external-contact geometry. No edge, capacity, phase row or clock is changed here. Write

\[ \alpha=2\pi/9,\qquad \psi_j=\alpha(j-4),\qquad \gamma=1/(1023\pi),\qquad S_i(\theta)=\sum_{b\sim i}\sin(\theta_b-\theta_i). \]

With the unnormalized Laplacian L, the complete law and storage are

\[ \begin{split} x'&=(-Lx+\gamma S(\theta))/4,& \theta'&=\gamma Lx/4,\\ H(x,\theta)&=\tfrac12x^TLx+ \sum_{\{i,b\}}[1-\cos(\theta_b-\theta_i)],& H'&=-\tfrac14\|Lx\|_2^2. \end{split}\tag{1} \]

Prime denotes the declared tau clock. Both global form and lifted phase means are conserved. Connectedness implies that a stationary state has constant form and satisfies S(theta)=0. On the declared zero-form-mean leaf its form is zero. A stationary relative phase cannot acquire a nonzero common angular velocity: its conserved lifted mean forbids it.

The six marked internal winding observations and nine old outer winding observations use oriented principal phase gaps on their fixed cycles. An acute chart requires its declared gaps to lie in (-pi/2,pi/2); crossing its boundary does not by itself change a winding number. Changing a principal winding requires reaching a branch cut at an absolute gap pi. New spoke triangles are part of the support but carry no newly prescribed identity. The complete state retains every fine deviation and both means.

2. Two full-state stationary controls and their complete tangents

Aligned constituent origins

For a common origin Phi and common form m, the aligned state

\[ x^*_{c,j}=m,\qquad \theta^*_{c,j}=\psi_j+\Phi \tag{2} \]

is stationary: opposite internal currents cancel and every wheel current vanishes. It retains six internal windings one and nine outer windings zero. The global means choose m and Phi; these are not independently damped modes.

Let L9 denote the C9 Laplacian and K the W6 Laplacian. In the chosen node order, the unweighted form Laplacian and phase-storage Hessian at (2) are

\[ L=I_6\otimes L_9+K\otimes I_9,\qquad M_+=\cos\alpha\,I_6\otimes L_9+K\otimes I_9. \tag{3} \]

Both are positive on the common-mode complement. This is a full-fine statement, including perturbations that leave the twelve-coordinate carrier.

Antipodal external constituent

Keep all five host origins at Phi and the external origin at Phi+pi. The spoke currents still vanish, so this state is also stationary. Its internal and old outer windings agree with (2), but its spoke cosines are negative. Write K_minus for the wheel Laplacian with ring weights one and spoke weights minus one. Its phase Hessian is

\[ M_-=\cos\alpha\,I_6\otimes L_9+K_-\otimes I_9. \tag{4} \]

The stationary label and the same marked windings do not imply the same stability. If a particular conserved lifted mean is required, subtract the appropriate common phase from all 54 coordinates; this changes neither current nor tangent.

Exact joint mode classification

The internal Fourier values are

\[ \lambda_9(k)=2-2\cos(2\pi k/9),\qquad k=0,\ldots,8. \]

The wheel has a common mode, a host-versus-external mode proportional to (1,1,1,1,1,-5), and four ring Fourier modes with external entry zero. Their simultaneous eigenvalues are:

Wheel mode	Form value mu	Aligned phase value kappa	Antipodal phase value kappa
Common	0	0	0
Host versus external	6	6	-6
Ring Fourier r=1,...,4	3-2cos(2pi*r/5)	3-2cos(2pi*r/5)	1-2cos(2pi*r/5)

Each pair r,5-r contributes two real ring directions. For any product mode, define lambda=lambda9(k)+mu and h=cos(alpha)*lambda9(k)+kappa. The complete two-row tangent is

\[ \begin{pmatrix}\delta x\\\delta\theta\end{pmatrix}' =\frac14 \begin{pmatrix}-\lambda&-\gamma h\\\gamma\lambda&0\end{pmatrix} \begin{pmatrix}\delta x\\\delta\theta\end{pmatrix}, \qquad s^2+\frac\lambda4s+\frac{\gamma^2\lambda h}{16}=0. \tag{5} \]

For the aligned target, all 53 noncommon product modes have positive lambda and h. Their 106 temporal eigenvalues have negative real parts. The remaining two neutral directions are exactly the conserved form and phase means. Thus (2) is locally asymptotically stable on each fixed-mean leaf; the acute storage argument below supplies a nonlinear version.

For the antipodal target, all nine host-versus-external product modes have h=cos(alpha)*lambda9(k)-6<0, since lambda9(k)<=4 and cos(alpha)<1. Each has one positive and one negative temporal root. The four ring modes have positive antipodal phase values (3-sqrt(5))/2 and (3+sqrt(5))/2, each twice. The other 44 relative product modes therefore contribute 88 stable directions. Altogether there are nine unstable, 97 stable and two neutral directions. This full-state stationary control is a saddle, not a retained-source capture certificate.

3. Triangles obstruct reuse of a wholly acute winding-one cell

At any fixed slot j, let d_c be the principal host ring gap from c to c+1 and sigma_c the principal spoke gap from host c to the external node. The oriented triangle periods are integers

\[ n_c=\frac{d_c+\sigma_{c+1}-\sigma_c}{2\pi},\qquad \sum_{c=0}^4n_c=\frac1{2\pi}\sum_{c=0}^4d_c. \tag{6} \]

These are topological identities for the actual fine state, with principal branches away from their cuts. Outer winding one makes their sum one, so some triangle has a nonzero period. Its three absolute principal gaps cannot all be strictly below 2*pi/3. In particular a wholly acute joined chart cannot retain outer winding one. If all five old ring gaps remain acute, that triangle additionally forces at least one spoke gap to have absolute value at least 3*pi/4: its two spoke magnitudes must sum to at least 2*pi-|d_c|, which exceeds 3*pi/2 in the strict chart.

This holds independently at each of the nine slots and requires no carrier assumption. It explains why the earlier positive-edge-cosine storage trap cannot simply be reused on the enlarged support. It does not establish that the complete phase Hessian is indefinite: negative edge weights can be compensated by the rest of a graph. Nor does it prohibit equilibria with some nonacute edges or prove loss of an actual source winding.

4. No stationary acute outer winding on the exact carrier

The invariant carrier retains all twelve coordinates,

\[ x_{c,j}=X_c,\qquad \theta_{c,j}=\psi_j+\Phi_c, \qquad X'=(-KX+\gamma S_{W_6}(\Phi))/4,\quad \Phi'=\gamma KX/4. \tag{7} \]

It is a justified restriction of (1), not a replacement for the independently perturbed source. Its exact storage and loss are

\[ H_{12}=6V_9+\tfrac92X^TKX+ 9\sum_{\{c,b\}\in W_6}[1-\cos(\Phi_b-\Phi_c)], \qquad H_{12}'=-\tfrac94\|KX\|_2^2, \tag{8} \]

where V9=9*(1-cos(alpha)).

Closed acute-cell obstruction

Suppose a carrier stationary state has outer winding one and five directed principal ring gaps d_j in [-pi/2,pi/2]. Their sum is 2*pi; consequently none can be negative, since the other four have sum at most 2*pi. Set the hub phase to zero, and denote the ring phases modulo 2*pi by phi_j. The stationary ring equations are

\[ \sin\phi_j=\sin d_j-\sin d_{j-1}. \tag{9} \]

Choose a maximal gap d_m. Since sine is increasing on [0,pi/2], its first endpoint has nonnegative sine and its second endpoint nonpositive sine. The positive oriented gap is at most pi/2, so these endpoints straddle the hub's antipodal phase. Choose representatives

\[ \phi_m=\pi-u,\qquad \phi_{m+1}=\pi+v, \qquad u,v\ge0,\quad u+v=d_m. \]

Adding the two adjacent relations in (9) gives

\[ \sin d_{m-1}+\sin d_{m+1} =2\sin d_m-\sin u-\sin v \le\sin d_m\le1, \tag{10} \]

where sin(u)+sin(v)>=sin(u+v) for these angles. For two angles a,b in [0,pi/2], sin(a)+sin(b)<=1 implies a+b<=pi/2. For completeness, at fixed sum s>=pi/2 the concave sine sum has minimum 1+sin(s-pi/2) at an endpoint; equality at sum pi/2 requires one angle zero and the other pi/2. Equation (10) therefore bounds the sum of the two gaps adjacent to d_m by pi/2.

The maximal gap and the other two gaps are each at most pi/2. Since the five gaps sum to 2*pi, every bound must be an equality. Thus four gaps are pi/2 and one is zero. At the two coincident endpoints of that zero gap, (9) would require the same phase to have sine minus one and plus one, respectively. This is impossible. Hence no carrier stationary state lies in the closed acute outer-winding-one cell. Reflection of all phases proves the corresponding winding-minus-one result.

The argument uses the actual unit ring and spoke weights. It neither changes the interface nor identifies a constitutive coefficient from a desired retained organization.

Dynamical consequence and its precise boundary

Every carrier trajectory starting in this cell leaves its closure after some finite time. To prove this, suppose instead it remained there for all future times. Conserved mean form and (8) bound X. The phase torus is compact, and the closed acute winding cell stays away from principal branch cuts, so the trajectory has a nonempty compact omega-limit set. Storage is nonincreasing. Its largest invariant zero-loss subset requires KX=0; the phase row then vanishes, and invariance of the form row requires S_W6(Phi)=0. The closed-cell obstruction excludes that set, contradicting indefinite retention. The same reasoning applies for every supplied hub origin and every finite initial carrier form.

This supplies no uniform exit deadline. Leaving an acute cell at absolute gap pi/2 is not changing winding at a branch cut of absolute gap pi. Nor does carrier exit establish exit of every actual fine-state source: off the carrier, internal C9 torques enter the stationary equations and cannot be discarded from (9). Broader nonacute winding-one equilibria and their possible attraction require their own classification.

5. A full-fine obstruction to stable retention near the carrier

The previous section's stationary equations are restricted to the carrier. A different argument applies to the entire fine phase Hessian, even when internal currents do not cancel. Let

\[ M(\theta)=B\,\operatorname{diag}(\cos\delta_e)B^T, \qquad D(\theta)=\sum_{54\ \mathrm{internal\ edges}} [1-\cos(\delta_e-\alpha)]. \tag{11} \]

Here B is any consistently oriented incidence matrix. Orient each internal cycle in increasing slot order; its closing gap retains the declared +2*pi winding representative. The cosine in D makes any equivalent representative harmless. D measures departure of the internal edge gaps from the carrier twist. It is not a distance between complete states and does not replace their form or phase coordinates.

Suppose all nine old outer cycles have strictly acute winding one. With an arbitrary common reference beta, put

\[ \chi_{c,j}=\theta_{c,j}-\psi_j-\beta, \qquad v_{c,j}=\sin\chi_{c,j},\quad w_{c,j}=\cos\chi_{c,j}. \]

For every edge, the squared differences of these two vectors sum to 2*(1-cos(chi_b-chi_a)). Their two phase quadratic forms therefore obey

\[ v^TMv+w^TMw =2\sum_e\cos\delta_e\,[1-\cos(\chi_b-\chi_a)]. \tag{12} \]

For same-slot wheel edges, the demodulated difference is the actual gap. For internal edges it is the actual gap minus alpha, modulo 2*pi. Since cos(delta_e)<=1 and every bracket is nonnegative, the internal contribution to the sum on the right before its factor two is at most D. This inequality does not assume positive internal cosines.

To bound a wheel, let its five acute ring gaps be d_i. Winding one makes them positive and their sum 2*pi. Put c5=cos(2*pi/5) and C=sum_i cos(d_i). Concavity of cosine and the quadratic-mean inequality give

\[ 0<C\le5c_5<5/2,\qquad \sum_i[\cos d_i-\cos^2d_i] \le C-C^2/5\le5c_5(1-c_5). \tag{13} \]

The hub's antipodal point is bracketed by two consecutive ring vertices. Their hub-relative phases are pi-a and pi+b, with nonnegative a,b and a+b<pi/2. The two corresponding spoke contributions cos(sigma)-cos(sigma)^2 sum to less than minus two, since cos(a)+cos(b)>1 and cos(a)^2+cos(b)^2>1. Each of the remaining three spoke contributions is at most 1/4. Thus every wheel contributes less than minus d, where

\[ d=\frac54-5c_5(1-c_5) =\frac{5(7-3\sqrt5)}8>0. \tag{14} \]

Combining the nine wheels and the internal edges yields the strict bound

\[ v^TMv+w^TMw<2[D(\theta)-9d]. \tag{15} \]

Consequently D<=9d excludes a positive-semidefinite phase Hessian in this acute outer-winding-one domain. At least one of v,w has negative quadratic form. Subtracting its global mean preserves that form because M*1=0, so the negative direction belongs to the conserved-phase-mean tangent space. All fine coordinates are included; no carrier substitution or assumed cancellation of internal torques enters this test.

At a stationary state, negative phase curvature supplies a positive growth root for the complete two-row dynamics. On the zero-mean subspace L is invertible. Eliminating the form perturbation gives

\[ L^{-1}\delta\theta''+\tfrac14\delta\theta' +\frac{\gamma^2}{16}M\delta\theta=0. \tag{16} \]

The symmetric matrix polynomial P(s)=s^2*L^{-1}+s*I/4+gamma^2*M/16 has a negative eigenvalue at s=0 and is positive definite for sufficiently large positive s. Continuity therefore supplies a positive s with singular P(s), and its kernel gives an exponentially growing complete tangent mode. Such a stationary state is linearly unstable on the fixed-mean leaf.

This is a local stationary obstruction. A negative Hessian at a moving source is not a positive Lyapunov exponent for that trajectory, nor proof of future identity loss. A stable stationary retained outer-winding-one organization must either have D>9d, involving changed internal gaps, or lie outside the old strictly acute outer chart. Existence or acquisition of either alternative remains unproved. Instability on the full relative state space also does not prove that the 96-dimensional prepared family intersects every growing direction.

For the actual outer-one source at attachment, let Z be its freshly rebuilt full-phase error about the carrier and m_o the continuous per-node phase drift bound carried through the entire observation word. The commands are constant within each child and their final sum is zero; no commanded internal edge difference is retained. Host internal edge deviations from alpha are bounded by 2*(Z+m_o), while the external constituent's deviations are bounded by 2*q, with q=1e-70. Thus all original source errors and all relative origins satisfy the initial bound

\[ D(\theta(0))\le90(Z+m_o)^2+18q^2. \tag{17} \]

This checks that the actual initial outer-one family lies inside the negative-curvature domain when the right side is below 9d. It is neither a stationary-state substitution nor a future bound on D along its motion.

5.1. Winding triangles and full-domain necessary curvature tests

The analytical mechanism admission declared at 0aec0fe4 asks whether phase-Hessian witnesses can cover the complete source-compatible marked class. The following argument removes the acute outer-gap hypothesis from a necessary defect bound and excludes every common internal profile. It does not exclude all nonseparable stationary states. All results use (1), the original 144 unit edges and the complete phase Hessian; no source, trajectory, optimizer or numerical certificate is evaluated.

Put g(a)=cos(a)-cos(a)^2. For one wheel, define

\[ F_W=\sum_{e\in W_6}g(\delta_e). \tag{17a} \]

Winding-triangle bound. If the five rim edges avoid their principal cuts and have winding one, then F_W<=-1/4, regardless of the hub origin or any acute-gap assumption.

First avoid spoke cuts as well. Orient each spoke triangle consistently with its rim edge. Their principal windings sum to the rim winding, because oppositely oriented spokes cancel. Each triangle winding is -1, 0 or 1, so at least one is 1. Its three principal increments a,b,c are strictly between zero and pi and sum to 2*pi: a nonpositive increment could not complete that sum with the other two strictly below pi.

For such a triangle, g(a)+g(b)+g(c)<=-2. Here is a direct proof that also admits boundary limits. Relabel so c=max(a,b,c)>=2*pi/3, and put p=cos(c/2) and z=cos((a-b)/2). Then 0<=p<=1/2 and p<=z<=1. The sum is

\[ 2(1-2p^2)z^2-2pz-4(1-p^2)^2. \tag{17b} \]

It is convex in z, so its maximum is at an endpoint. At z=p it equals -2-8(p^2-1/2)^2; at z=1 it equals -2+2p[2p(1-p^2)-1]. Both are at most -2, since 2p(1-p^2)<=1. Each of the seven remaining wheel edges has g<=1/4. Thus F_W<=-2+7/4=-1/4. If a spoke is at a cut, perturb only the hub origin to avoid all spoke cuts and pass to the limit. The marked rim winding is unchanged and the cosine expression is continuous.

For the complete fine state, orient internal gaps in increasing slot order and define

\[ \begin{split} W&=\sum_{90\ \mathrm{wheel\ edges}}g(\delta_e),& A&=\sum_{54\ \mathrm{internal\ edges}}\cos\delta_e,\\ B_2&=\sum_{54\ \mathrm{internal\ edges}}\cos^2\delta_e,& C_2&=\sum_{54\ \mathrm{internal\ edges}}\cos\delta_e\sin\delta_e. \end{split} \tag{17c} \]

For m=0,...,8, take the real vectors v_m(c,j)=sin(theta_(c,j)-m*alpha*j) and w_m(c,j)=cos(theta_(c,j)-m*alpha*j), with alpha=2*pi/9. The phase reference is periodic on C9, including its closing edge. The squared-difference identity used in (12) gives exactly

\[ \frac{v_m^TMv_m+w_m^TMw_m}{2} =W+A-B_2\cos(m\alpha)-C_2\sin(m\alpha). \tag{17d} \]

There is no assumption of stationarity or a separable phase profile here. For nine winding-one rims, (17a) implies W<=-9/4. Positive semidefiniteness of M consequently requires

\[ A-\max_{m=0,\ldots,8} [B_2\cos(m\alpha)+C_2\sin(m\alpha)]\ \ge\ \frac94. \tag{17e} \]

For m=1, the internal contribution is sum_internal cos(delta_e)[1-cos(delta_e-alpha)]<=D, with D from (11). Thus D<9/4 excludes a positive-semidefinite full phase Hessian, even for nonacute outer gaps and reflection-breaking states. The non-strict boundary D=9/4 is not excluded by this bound. At a stationary point the negative direction gives complete-law instability by (16). At a moving state it remains an instantaneous curvature statement, not a future identity-loss verdict. The original acute-domain theorem and frozen source assessment retain their own stated bounds.

5.2. No stable stationary contact with a common internal phase profile

Suppose the full phase state has the form theta_(c,j)=psi_j+Phi_c modulo a turn for all six children, and its rim origins have winding one. The shared profile psi need not be uniform, acute or a stationary internal cycle. Choose vectors constant within each child: v_(c,j)=sin(Phi_c) and w_(c,j)=cos(Phi_c). Internal edges contribute exactly zero. The nine wheels have identical gaps, hence

\[ v^TMv+w^TMw=18F_W\ \le\ -\frac92. \tag{17f} \]

At least one vector has negative quadratic form. Subtracting its global mean preserves that form and places it on the full relative phase leaf. Consequently no stationary state with a shared internal phase profile and retained outer winding one can be stable under (1). This excludes arbitrary common-profile distortions and arbitrary rim origins, without a reflection or acute-gap premise. It is broader than the helical carrier and does not require the constituent internal torques to vanish separately.

A possible stable stationary target must therefore be genuinely nonseparable: some relative constituent phase difference must depend on the internal slot. Such dependence is a necessary property, not a supplied new law or evidence that a stable target exists. It does not establish formation, capture, a bound state or a physical constituent.

5.3. An exact blind point of the nine fixed phase-pair sums

The necessary tests (17d) do not form a complete Hessian certificate. An exact counterexample remains inside the original complete product marked class, including its mixed cycles and all true cut margins.

Set psi_0=0 and give its nine successive internal gaps the values

\[ (\pi/3,\pi/3,\pi/3,\pi/3,\pi/3,\pi/3,\pi/3,-\pi/6,-\pi/6). \tag{17g} \]

Their sum is 2*pi; all lie strictly inside the acute interval. For hosts put Phi_c=(c-2)*pi/3, c=0,...,4, and put Phi_D=0 for the donor. Use theta_(c,j)=psi_j+Phi_c and zero forms, with a common phase translation if needed to retain the declared lifted mean. The five rim gaps are four copies of pi/3 and the closing 2*pi/3. The internal closing branch is -1, as is the rim closing branch; all other marked branches are zero. Every mixed marked square telescopes to zero. Thus this is in the original complete class, not merely a configuration sharing fifteen winding labels.

Per wheel the rim contributes 1/4 to F_W, and the spokes contribute -1, giving W=-27/4 over all nine wheels. Per internal child the sums in (17c) are

\[ A_0=\frac72+\sqrt3,\qquad B_0=\frac{13}4,\qquad C_0=\frac{5\sqrt3}4,\qquad \sqrt{B_0^2+C_0^2}=\frac{\sqrt{61}}2. \tag{17h} \]

For every one of the nine m values, the half pair sum in (17d) is at least -27/4+6*(A_0-sqrt(61)/2)>1/4. The strict inequality follows exactly from sqrt(3)>5/3 and sqrt(61)<8, which give A_0-sqrt(61)/2>7/6. No floating-point trigonometric evaluation is needed.

Nevertheless let u equal one on host child zero and zero elsewhere, then subtract its global mean. Internal edges again contribute zero. At that host the two rim cosines are 1/2 and -1/2, and the spoke cosine is -1/2, so u^T M u=-9/2. The full Hessian is not positive semidefinite. The state is explicitly nonstationary: at slot j=1 the equal adjacent internal sine currents cancel, while S_(0,1)=sqrt(3)/2.

This example defeats the nine pair-sum tests, not every individual sine or cosine vector, their linear combinations or the full quadratic form on their span. A positive pair sum can hide a negative individual direction. It also does not refute a stronger theorem restricted to stationary states. The new common-profile test (17f) correctly rejects this same configuration.

The pair tests and common-profile obstruction alone do not admit a chamber-wide stationary certificate. The following stationarity arguments further restrict the nonseparable sector without discarding full relative curvature or replacing it by more evaluations of these nine sums.

5.4. Every stationary PSD candidate has a nonzero aligned resultant

For a stationary phase state, define

\[ z_i=e^{i\theta_i},\qquad R_i=\sum_{b\sim i}z_b,\qquad r_i=\Re(\overline z_iR_i)=\sum_{b\sim i}\cos(\theta_b-\theta_i). \tag{17i} \]

Every node has odd degree, five on a host and seven on the donor. For odd d, the maximum of sum a_k^2 on [-1,1]^d with sum a_k=0 is d-1. Indeed a vertex of this polytope has at most one nonsaturated coordinate. The other d-1 coordinates sum to an even integer, so the remaining coordinate must be zero; the saturated coordinates balance. Convexity puts a maximum at a vertex. Apply this to the stationary sine currents: sum sin(delta_ib)=0 implies sum cos(delta_ib)^2>=1 at every node.

Suppose the full phase Hessian M is positive semidefinite. Edgewise cos(delta)<=1 gives M<=L. The C9 Laplacian is bounded by 4I, while W6 is a subgraph of K6 and its Laplacian is bounded by 6I. Thus 0<=M<=10I, and spectral calculus gives M^2<=10M. Since M_ii=r_i,

\[ r_i^2+1\ \le\ (M^2)_{ii}\ \le\ 10r_i, \qquad r_i\ge5-2\sqrt6>\frac1{10}. \tag{17j} \]

Stationarity also makes conj(z_i)*R_i real. Consequently R_i=r_i*z_i with the strict bound (17j), and every such candidate obeys the regular fixed-point equation z_i=R_i/|R_i|. This excludes zero resultants and antipodal alignment, without a separable, acute or reflection premise. It is a necessary condition, not a proof of existence, uniqueness or full Hessian positivity. The angular fixed-point derivative still has signed weights cos(delta_ib)/r_i; their row sum is one, but negative entries prevent inference of a maximum principle or contraction.

Nor can unrestricted stress positivity replace tangent-Hessian positivity. With adjacency matrix A and T=diag(r)-A, stationarity gives T*z=0, but 1^T*T*1=-2*V(theta). Requiring T to be PSD would force zero potential and synchronization. The stable aligned-twist control (2) already has T*1=(2*cos(alpha)-2)*1<0, although its M is positive on the full relative leaf by (3). This rejects that stronger certificate, not the control.

5.5. Stationary comparison with a cyclically shifted profile

Use lifts in the original complete product marked chamber: internal and rim closing branches are -1, and all other marked branches are zero. Let kappa_(c,j) be the forward principal internal gap, and define

\[ \theta^+_{c,j}=\begin{cases} \theta_{c,j+1}-\alpha,&j<8,\\ \theta_{c,0}+2\pi-\alpha,&j=8, \end{cases} \qquad d_{c,j}=\theta^+_{c,j}-\theta_{c,j} =\kappa_{c,j}-\alpha. \tag{17k} \]

This is the slot automorphism followed by a common phase rotation, with the compatible lift retained. It preserves the complete law, every child lifted mean and the marked branch vector. Internal and rim principal gaps are cyclically shifted. Thus a stationary theta gives a stationary theta+ with a permutation-conjugate Hessian M+. The straight lifted segment theta+t*d, 0<=t<=1, stays in the original marked chamber: each marked gap interpolates between two values strictly inside the same cut bounds. This segment is a comparison of configurations, not a solution trajectory.

For each oriented edge, let a and b be its gaps at theta and theta+ using the same lift/branch convention at both endpoints. Set h=(b-a)/2 and mu=(a+b)/2. Include all 144 edges. On unmarked spokes retain these compatible lifted gaps; independently wrapping the two endpoints would change the secant. On every edge 2h=d_right-d_left=kappa_right-kappa_left, hence |h|<pi. Stationarity at both endpoints and the edge quadratic form give

\[ \begin{split} 0&=\sum_e(\sin b_e-\sin a_e)(b_e-a_e) =4\sum_e\cos\mu_e\,h_e\sin h_e,\\ \frac{d^TMd+d^TM^+d}{2} &=4\sum_e\cos\mu_e\,h_e^2\cos h_e\ \ge\ 0 \quad\text{if }M\succeq0. \end{split} \tag{17l} \]

Subtracting these identities gives the necessary inequality

\[ \sum_e\cos\mu_e\,h_e(\sin h_e-h_e\cos h_e)\ \le\ 0. \tag{17m} \]

The factor h*(sin h-h*cos h) is strictly positive for 0<|h|<pi: it is even, and the derivative of sin h-h*cos h is h*sin h>0 for positive h in that interval. Therefore a PSD stationary candidate must either have a changing edge with cos(mu)<0, or satisfy cos(mu)=0 on every changing edge. In the latter case (17l) has zero quadratic form at both PSD endpoints, so Md=0. It also gives sin a=sin b separately on every edge. The next argument excludes this entire flat alternative, including Hessians with additional zero eigenvalues.

5.6. The flat alternative requires nine exceptional slots in six children

Suppose the flat alternative holds. Equality of each edge's sine current before and after the shift means its current is independent of the slot. Each internal C9 then has a common positive current: zero current would force all its cut-free gaps to zero, and negative current would force all gaps negative, both contradicting winding one. Write its acute solution as a in (0,pi/2]; each gap is either a or pi-a. If there are m obtuse gaps, their sum is 9a+m*(pi-2a)=2*pi. For m>=2 this sum is at least 2*pi+5a>2*pi. Thus the only profiles are

\[ \begin{array}{ll} \text{uniform:}&\kappa_j=2\pi/9\quad\text{for every }j,\\ \text{one obtuse gap:}&\kappa_{k_c}=6\pi/7,\qquad \kappa_j=\pi/7\quad(j\ne k_c). \end{array} \tag{17n} \]

Adjacent children cannot have different types. On a wheel edge connecting such children, at the eight ordinary internal slots the successive compatible wheel gaps differ by +/-5*pi/63. There are two consecutive ordinary slots, so three consecutive wheel gaps would have equal sine and nonzero constant spacing epsilon=+/-5*pi/63. This is impossible: sin t=sin(t+epsilon)=sin(t+2epsilon) first implies sin(t+epsilon)*(1-cos epsilon)=0 and then sin epsilon=0. The wheel is connected, so all six children have the same type.

If all are uniform, they share an internal profile and section 5.2 gives a negative relative Hessian direction. In the other case, d from (17k) has the same two values in every child: its high value occurs only at slot k_c, and the difference between the two values is q=5*pi/7. Use Md=0 at node (c,k_c-1). Its only internal neighbor with high d is the forward neighbor, across an edge of weight cos(pi/7)>0; its internal contribution to Md is -q*cos(pi/7), which is nonzero. Cancellation requires a wheel neighbor whose exceptional slot is k_c-1; otherwise all wheel differences of d vanish. At (c,k_c+1), the unique internal high neighbor is backward across the obtuse edge. Its contribution is +q*cos(pi/7), so a wheel neighbor with exceptional slot k_c+1 is also required. These arguments need no sign assumption on the wheel weights.

The nonempty set of occupied exceptional slots is therefore closed under both neighboring slots modulo nine. It must contain all nine slots, but six children can occupy at most six. This contradiction excludes the flat PSD alternative exactly, without a strict-positivity assumption.

Consequently every stationary candidate with a PSD full phase Hessian in the original complete marked class has a changing edge with strictly negative midpoint cosine in (17k). This is negative curvature of one changing edge potential at the comparison midpoint. Neither negative total segment curvature nor a negative candidate-Hessian direction has been proved; other edges can contribute positively. It is not an assertion about time evolution. Symmetry and stationarity alone have not forced a common profile.

For an independent nonstationary control, the phase-pair blind point (17g) has exactly two changing internal gaps per child under (17k), while its wheel gaps are unchanged. Its stationarity secant is

\[ \sum_e(\sin b_e-\sin a_e)(b_e-a_e) =3\pi(\sqrt3+1)>0. \tag{17o} \]

It therefore fails the first identity in (17l), although all nine pair sums pass. The stationary conditions add information missing from those sums; this example is not evidence about existence of a stationary target.

The remaining sector is regular, genuinely nonseparable and has negative midpoint cosine on a changing edge. Neither that necessary property nor the resultant bound certifies stability or excludes this sector. The next reduction retains all its phase directions while separating donor relaxation from the remaining host curvature.

5.7. The donor block is strictly positive at every PSD candidate

Partition the full phase Hessian into 45 host and nine donor coordinates:

\[ M=\begin{pmatrix}A&C\\C^T&D\end{pmatrix},\qquad C_{(c,j),k}=-\cos(\sigma_{c,j})\,\mathbf1_{j=k},\qquad C^TC=\operatorname{diag}(s_j),\quad s_j=\sum_{c=0}^4\cos^2(\sigma_{c,j}), \tag{17p} \]

where sigma_(c,j)=theta_(D,j)-theta_(c,j). A is the principal block of the joined Hessian, including spoke diagonal terms; it is not the Hessian of the isolated host. D similarly includes donor internal edges and spoke diagonal terms. No internal edge cosine is assumed positive.

In the original outer-one marked class, every s_j>0. Otherwise all five host phases at that slot would equal the donor phase plus or minus pi/2. Opposite choices at adjacent rim nodes would place their marked edge at a forbidden cut. Equal choices around the connected rim give winding zero. Both alternatives contradict the admitted cut-free winding-one rim. Thus C has full column rank, even though individual spoke cosines may vanish.

If M is PSD, D is PSD. A nonzero y in ker D would give (0,y)^T*M*(0,y)=0, hence M*(0,y)=0 and Cy=0, a contradiction. Therefore D is strictly positive at every PSD candidate, including those with a degenerate full relative Hessian. Moreover, the donor block of M^2<=10M gives the quantitative restriction

\[ D^2+\operatorname{diag}(s_j)\preceq10D, \qquad D\succeq\frac1{10}\operatorname{diag}(s_j)\succ0. \tag{17q} \]

These are curvature statements on the actual phase state and require no stationarity assumption. If D has a negative direction, extending it by zero host coordinates gives a negative full direction. If D is PSD but singular, take Dy=0, y!=0. Since A<=10I and Cy!=0, the explicit vector v=(-Cy/10,y) has v^T*M*v<=-||Cy||^2/10<0. Thus failure of strict donor positivity excludes full PSD in this marked class; it is not merely a failed sufficient test. Subtract the common mean to obtain a relative witness. At a stationary point (16) then gives complete-law instability; at a moving point it gives only instantaneous curvature.

Let t_j=sum_c cos(sigma_(c,j)) and Q=sum_j t_j. Since D*1_D=t, (17q) also gives

\[ Q\ \ge\ \frac1{10}\sum_j(t_j^2+s_j)>0. \tag{17r} \]

At stationarity, summing the donor equations additionally gives sum_(c,j) sin(sigma_(c,j))=0. Hence a PSD stationary candidate has strictly positive real aggregate host-donor phase overlap. This is a derived phase balance, not an identified physical observable or a new interaction law. Its quantitative bound depends on the actual phase state.

5.8. Exact donor relaxation and the remaining 44 relative directions

For any admitted phase state with D>0, define its Schur complement

\[ K=A-CD^{-1}C^T. \tag{17s} \]

For arbitrary host variation h and donor variation b, completing the square gives the exact identity

\[ \begin{pmatrix}h\\b\end{pmatrix}^{\!T} M\begin{pmatrix}h\\b\end{pmatrix} =h^TKh+(b+D^{-1}C^Th)^TD(b+D^{-1}C^Th). \tag{17t} \]

Thus M>=0 if and only if K>=0. The nine donor directions contribute strictly positive curvature; all negative and additional zero directions are preserved by K. Since M*1=0, D*1_D=-C^T*1_H and K*1_H=0. Strict positivity of the full Hessian on 53 relative phase directions is therefore equivalent to positivity of K on all 44 host-relative directions. An exact negative K witness h lifts to (h,-D^-1*C^T*h) with the same quadratic form; global recentering does not change it. This is a congruence and inertia identity, not equality of eigenvalues, dynamical modes or decay times.

Together with section 5.7 this is a complete pointwise curvature criterion on the original marked class: test strict donor positivity, then retain the full K. Checking only host means would discard 40 host internal directions. The inequality K<=A shows that allowing the donor to respond can only lower the quadratic cost compared with holding it fixed. D need not have nonpositive off-diagonal entries, and K need not be a Laplacian with positive conductances. Neither matrix defines a supplied pressure law.

There is also an exact local stationary interpretation. At any point with S_D=0 and D>0, the implicit function theorem gives a unique local donor branch b=phi(a) solving S_D(a,b)=0, with phi'(a)=-D^-1*C^T. Define the reduced potential W(a)=V(a,phi(a)). On this branch,

\[ \nabla W=-S_H(a,\phi(a)),\qquad \nabla^2W=K. \tag{17u} \]

Full stationary points correspond exactly to critical points of W in this local chart. After shrinking it, continuity gives D>=kappa*I for some kappa>0 throughout its donor fibers, and V(a,b)>=W(a)+(kappa/2)*||b-phi(a)||^2. Local minima, including degenerate ones, therefore correspond; strict minima correspond after quotienting the common phase. No numerical neighborhood or global donor branch has been certified by this existence argument.

The donor coordinates remain represented by phi(a) and their nine residual directions. The full form/phase dynamics is still (1): this stationary graph need not be invariant under that flow. In particular, S_D=0 neither removes donor form diffusion nor freezes its phase row. The reduction supplies no autonomous 44-coordinate TNFR evolution, source inclusion, selected event or acquisition result.

5.9. Collective compensation must include the donor response

For a subset U of host nodes, let h be its indicator and put b_j=sum_{c:(c,j) in U} cos(sigma_(c,j)). Formula (17t) gives the exact donor-relaxed cut test

\[ h^TKh=\sum_{e\in\mathrm{host\ cut}(U)}\cos\delta_e +\sum_j b_j-b^TD^{-1}b. \tag{17v} \]

Every PSD candidate must make this nonnegative for every U. The last term retains the joint response of all nine donor coordinates, rather than treating their compensation separately or holding them fixed. These cuts are useful necessary tests, not a substitute for full K positivity.

The exact nonstationary blind point (17g) is a control for that distinction. Its spoke cosine sum is one at every slot and its internal cosines are positive, so D=M_internal+I>=I. Nevertheless, for the indicator h of host child zero, h^T*A*h=-9/2 and C^T*h=(1/2)*1_D. Since D*1_D=1_D, the relaxed donor variation is -(1/2)*1_D and

\[ h^TKh=-\frac92-\frac94=-\frac{27}4<0. \tag{17w} \]

Positive donor curvature therefore does not establish full stability or stationarity. This example remains outside equilibrium; the calculation is a static algebraic control, not a new candidate evaluation.

The reduction proves how to account exactly for compensating curvature; it does not prove that the remaining sector contains a stable target or that K must always have a negative direction at its critical points. The following reconstruction removes donor ambiguity at full stationarity, while retaining every admissible host state and possible degenerate minimum. Neither reduction identifies a physical force.

5.10. Host balance uniquely reconstructs a stationary donor

The local branch in (17u) solves only the nine donor equations. Different such branches can exist at the same fixed host state. Full stationarity also requires all 45 host equations, which determine at most one circular donor state in the original outer-one class, without any Hessian premise.

Write the host phases as h_(c,j), the donor phases as d_j, and let y_j=(cos d_j,sin d_j)^T. Compute the host-only torque T on the 45 internal host edges and 45 rim edges, excluding all donor edges. At each slot define

\[ T_{c,j}=\sum_{b\sim_H(c,j)}\sin(h_b-h_{c,j}),\qquad E_j(c,:)=(\sin h_{c,j},-\cos h_{c,j}). \tag{17x} \]

The host equation is T_(c,j)+sin(d_j-h_(c,j))=0, hence E_j y_j=T_j. The five-row matrix E has rank two. Otherwise all five host phases would belong to one antipodal pair: an adjacent change between the two members would put a marked rim edge at a forbidden cut, while no change would give winding zero. Both contradict the cut-free winding-one rim. By the two-column Cauchy--Binet identity,

\[ G_j=E_j^TE_j\succ0,\qquad \det G_j=\sum_{c<e}\sin^2(h_{c,j}-h_{e,j})>0,\qquad y_j=G_j^{-1}E_j^TT_j. \tag{17y} \]

Thus every full stationary completion has exactly this donor phasor. The formula is globally defined on the admitted host class and uses no argument of a host resultant, acute-gap condition, reflection symmetry or donor positivity. It does not provide a uniform conditioning bound: the Gram determinant can approach zero near excluded rim cuts.

This construction is covariant under a common phase rotation. For its two-dimensional rotation matrix R, E_j becomes E_j R^T, T is unchanged, and reconstructed y becomes R y. Integer changes of phase representatives leave the circular reconstruction unchanged. Uniqueness is circular, not uniqueness of unwrapped lifts. Any subsequent use of lifted means still requires the declared marked branches and global phase mean.

5.11. Exact completion guards and their curvature boundary

Begin with all 45 host phases in the original complete host marked class: five internal windings, nine rim windings and all 32 mixed-cycle periods, with their cut-free branch admission. Formula (17y) is an exact reformulation of full stationary feasibility only with these additional guards:

1. Every full residual E_j y_j-T_j vanishes. Solving the normal equations alone does not ensure these five equations at each slot.
2. Every y_j^T y_j=1. Do not normalize the reconstructed vector: doing so changes the host balance, and a zero vector defines no donor phase.
3. The nine donor phasors have cut-free internal winding one in the original donor marked class. No new winding condition is imposed on the unmarked spokes, and the earlier numerical search margin is not a hypothesis here.
4. The donor rows vanish. With cross(u,v)=u_x v_y-u_y v_x, they are

\[ \operatorname{cross}(y_j,y_{j+1}) -\operatorname{cross}(y_{j-1},y_j)+\sum_{c=0}^4 T_{c,j}=0. \tag{17z} \]

For necessity, host balance gives cross(y_j,(cos h_(c,j),sin h_(c,j)))=T_(c,j), which fixes the plus sign in (17z); the other conditions follow from a full admitted stationary state. Conversely, the unit-circle guard provides donor phases, the full linear residual gives all host rows, and (17z) gives all donor rows. The two marked components are precisely the host product and donor C9, so their separate admissions cover the original marked class. Set x=0 on the zero-form-mean leaf; choose compatible lifts and a common phase rotation to impose the declared global phase mean. This proves equivalence with a full equilibrium of (1), modulo the stated circular/lift distinction.

At an admitted completion, sections 5.7--5.8 still require D>0 and the full K>=0 for PSD curvature. Strict K positivity on all 44 host-relative directions gives the positive full relative Hessian and local exponential stability; a negative full direction gives stationary instability by (16). Degenerate PSD is not a sufficient stability verdict. Reconstruction has not solved any of these curvature conditions or proved that a completion exists for a given host state.

For example, fix the uniform host internal and rim twists h_(c,j)=alpha*(j-4)+2*pi*(c-2)/5. They belong to the original host class, but both opposite internal and rim currents cancel: T_j=0 and G_j=(5/2)*I. Formula (17y) gives y_j=0, so no donor shape or relative origin can complete this host state to equilibrium. In contrast, the aligned outer-zero host of section 2 has rank-one E and admits both the aligned and antipodal stationary donors. The winding-one premise is essential to uniqueness. The analytic reflected saddle provides a nontrivial completion in the admitted class, while its negative curvature shows why exact reconstruction is not a stability certificate.

The connection to the local reduced potential is also exact. If two donor-stationary branches at the same host state have phasors y and z, their reduced host gradients differ by E_j(y_j-z_j) at slot j. Therefore the squared gradient difference is sum_j (y_j-z_j)^T G_j (y_j-z_j), strictly positive for distinct circular donors. Such branches cannot share a host critical point. This does not join their local potentials into one global W.

The remaining problem uses 45 host phases, or 44 modulo common rotation, subject to the complete residual, circle, donor and marked-class guards. No independence or equation-rank count for those constraints is assumed. This is stationary feasibility, not an autonomous host evolution: (1) still consumes all 108 form and phase coordinates. Neither a source capture result nor a physical identification follows.

5.12. Donor deformation alone cannot stabilize a common host profile

The common-profile obstruction of section 5.2 initially assumes all six children share one internal profile. Full stationarity supplies a stronger restriction: it is enough to assume this for the five hosts, while allowing an arbitrary donor profile at the outset.

Suppose h_(c,j)=psi_j+Phi_c. Write the common host internal torque as I_j=sin(kappa_j)-sin(kappa_(j-1)), the rim torque as R_c=sin(Phi_(c-1)-Phi_c)+sin(Phi_(c+1)-Phi_c), and put beta_j=d_j-psi_j. Host stationarity requires I_j+R_c+sin(beta_j-Phi_c)=0. Subtract the equations at slots j and k:

\[ (I_j-I_k)+(\sin\beta_j-\sin\beta_k)\cos\Phi_c -(\cos\beta_j-\cos\beta_k)\sin\Phi_c=0. \tag{17aa} \]

A cut-free winding-one rim has at least three distinct circular phases: with only two, every transition has a reverse transition whose principal gap cancels it, and an antipodal transition is forbidden. Three distinct points on a circle are not collinear, so the columns 1, cos Phi_c, sin Phi_c have rank three. Equation (17aa) forces I to be constant across slots and beta to be constant modulo a turn. Summing I over the internal cycle gives zero, hence every I_j=0.

Consequently the donor shares the same circular internal profile as the hosts. Section 5.2 then supplies a negative full phase-Hessian direction. No stationary state with a common five-host profile can have PSD full curvature, even if the donor was allowed to deform independently. This does not require acute edges or reflection symmetry. A stable stationary candidate in the original class must have genuinely different internal profiles among the hosts themselves, not merely a changed donor.

The execution plan retains the unresolved feasibility and curvature question in that sector, including nonacute states, broken reflection and degenerate host minima. No actual-source fate or global exclusion has been established.

5.13. The stationary reconstruction has 36 regular compatibility conditions

The 44 relative host coordinates in section 5.11 are subject to the exact reconstruction guards. Their local geometry can be determined at a full equilibrium; counting the displayed scalar equations alone would not do so. Let Y(h) be the 18-component reconstruction (17y), and define its residual and circle guards by

\[ \mathcal R(h)=E(h)Y(h)-T(h),\qquad \mathcal U_j(h)=\tfrac12\bigl(Y_j(h)^TY_j(h)-1\bigr), \qquad E_{(c,j),(k,a)}=E_j(c,a)\mathbf1_{j=k},\quad G=E^TE. \tag{17ab} \]

Here E is a 45-by-18 matrix with host rows in the original (c,j) order and two phasor columns per donor slot; a=1,2. All derivative identities below are evaluated at an admitted equilibrium. For a host variation v, differentiation must include the changing E as well as T. The derivative of the host-only torque is minus its own Hessian, whereas dot(E)*y supplies the spoke diagonal contribution. Thus dot(T)-dot(E)*y=-A*v. Differentiating the normal equations and using T=E*y gives

\[ \dot Y=-G^{-1}E^TAv,\qquad \dot{\mathcal R}=(I-P_E)Av,\qquad \dot{\mathcal U}_j=y_j^T\dot Y_j,\qquad P_E=EG^{-1}E^T. \tag{17ac} \]

The residual derivative has only 27 possible independent components, since rank(E)=18. Let J denote rotation through pi/2 in the phasor plane. A circle-tangent variation is uniquely dot(Y)_j=J*y_j*b_j, where b is a nine-component donor phase variation. Direct substitution gives E*diag(J*y_j)=C, with the negative spoke-cosine sign in (17p). Consequently the complete linearized residual and circle guards satisfy

\[ \dot{\mathcal R}=0,\quad\dot{\mathcal U}=0 \quad\Longleftrightarrow\quad Av+Cb=0\ \text{for a unique }b. \tag{17ad} \]

Uniqueness uses the full column rank of C, proved in section 5.7. This is a condition on all fine host coordinates, including their internal modes. It does not replace the remaining donor equilibrium equations.

Suppose first that A>0. Then (17ad) has exactly the nine-dimensional solution space v=-A^-1*C*b; the Jacobian of the complete guard map has rank 36. These are 36 locally independent conditions within the displayed 45 residual and nine circle equations, not 54 independent restrictions. The host implicit function theorem gives an actual local branch h=psi(d) solving all host rows, with psi'=-A^-1*C. Since C is injective, its projection to the host coordinates is a local embedding. The exact reconstruction and circle guards describe precisely that local image: near the equilibrium their unit donor phasors recover the same donor phase chart, and the full residual gives all host rows. Thus the admitted host-compatible states form a smooth nine-dimensional local manifold, or eight dimensions after quotienting the common phase. All marked cuts remain avoided after shrinking the chart; no numerical radius is provided by this argument.

Strictly positive full relative curvature implies A>0: every nonzero vector (v,0) is different from the common phase direction. The regular manifold therefore covers every candidate with a strictly positive full relative Hessian. Merely PSD curvature does not yet admit this conclusion.

5.14. All remaining regular curvature lies along compatible variations

On the host-stationary branch define the local potential and its Hessian

\[ U(d)=V(\psi(d),d),\qquad \nabla U=-S_D(\psi(d),d),\qquad \nabla^2U=N:=D-C^TA^{-1}C. \tag{17ae} \]

Host balance removes the chain-rule term involving the host gradient. Consequently full equilibria correspond exactly to critical points of U in this chart. Host directions transverse to the branch have positive curvature, as the exact identity

\[ \begin{pmatrix}v\\b\end{pmatrix}^{\!T}M \begin{pmatrix}v\\b\end{pmatrix} =(v+A^{-1}Cb)^TA(v+A^{-1}Cb)+b^TNb \tag{17af} \]

shows. After shrinking the chart, positive host curvature also gives a uniform local fiber bound V(h,d)>=U(d)+(kappa/2)*||h-psi(d)||^2 for some kappa>0. Local minima correspond, including minima degenerate along U; strict minima correspond modulo common rotation. The phase stationarity Jacobian has rank 45+rank(N). Since A*1_H=-C*1_D, one has N*1_D=0. Strict positivity of N on its eight donor-relative directions is therefore equivalent to strict positivity of M on all 53 relative directions. Its 45 other directions are retained by the positive host term in (17af). A negative N direction gives a negative full witness (-A^-1*C*b,b). Additional null directions require further nonlinear analysis; they do not by themselves establish a stationary family.

There is a sharper connection to the earlier 44-direction host Hessian K. If both A>0 and D>0, the host space splits as

\[ \mathbb R^{45}=\ker C^T\ \oplus\ \operatorname{range}(A^{-1}C). \tag{17ag} \]

The first space has dimension 36. On it K has the positive quadratic form of A, and its K cross term with the second space vanishes. The second space is exactly the tangent space (17ad) of the compatibility manifold. Put R=C^T*A^-1*C>0. Pulling K back to this nine-dimensional space gives

\[ (A^{-1}C)^TK(A^{-1}C) =R-RD^{-1}R=N-ND^{-1}N. \tag{17ah} \]

This matrix has the same inertia as N. Indeed Z=D^-1/2*N*D^-1/2<I, since D-N=R>0; the congruent matrix Z-Z^2 has eigenvalue z*(1-z), with the same sign as z for every z<1. Thus the negative index and additional nullity of K are fully represented on the compatible tangent; the other 36 directions are positive. The matrices are not isospectral, and their quadratic forms need not have the same sign on each identical coordinate vector. A K witness uses the donor-relaxed lift from (17t); an N witness uses (17af).

This is a local stationary reduction, not an eight-coordinate temporal law. A moving solution of (1) need not preserve the host-balanced branch. Host-branch uniqueness gives the local covariance psi(d+chi*1_D)=psi(d)+chi*1_H. The complement 1_D^perp is a convenient gauge for curvature, not a separately conserved donor mean or the exact pullback of the full phase-mean leaf. A common rotation restores the declared global mean without changing the quadratic forms. The construction provides no global branch, admitted neighborhood size or source inclusion.

5.15. Singular host curvature retains extra unresolved directions

The regular admission A>0 cannot be omitted. At a PSD full Hessian, let r=dim ker(A). For any n in ker(A), PSD implies M*(n,0)=0, hence C^T*n=0. Equivalently, range(C) is contained in range(A). With the Moore--Penrose inverse, all solutions of (17ad) are therefore

\[ v=-A^\dagger Cb+n,\qquad n\in\ker A. \tag{17ai} \]

The two terms are independent and b has nine independent components. Thus the linearized host-compatible space has dimension 9+r, and the Jacobian rank of the residual and circle guards is exactly 36-r. Every nonzero (n,0) is an additional full phase-Hessian null direction whose donor component vanishes; it is not the common phase direction. These are genuine neutral directions of the complete linearized law at equilibrium. Subtracting the full common mean retains their independent nongauge null directions on the fixed-mean leaf, but may change the donor common angle; the donor relative angles remain unchanged. They need not integrate to curves of equilibria. Guard rank loss alone proves neither a nonlinear manifold of that dimension nor instability. Possible degenerate host minima remain in the unresolved domain. If A instead has a negative direction, extending it by a zero donor variation directly gives a negative full direction.

An independent algebraic control uses the known aligned outer-zero state of section 2. For internal mode m and host rim mode r, write z_m=cos(alpha)*lambda9(m) and lambda5(r)=2-2*cos(2*pi*r/5). The host-block eigenvalues and the donor-reduced eigenvalues are

\[ \lambda_A(m,r)=z_m+1+\lambda_5(r)>0,\qquad \lambda_N(m)=z_m+5-\frac5{z_m+1} =\frac{z_m(z_m+6)}{z_m+1}. \tag{17aj} \]

A donor mode b induces the host-balanced variation v_(c,j)=b_j/(z_m+1) in every child. N has exactly eight positive relative modes and the common zero mode, so (17af) retains all 53 positive relative directions of that control. Its outer winding is zero and its E matrices have rank one: it checks the host-relaxation algebra and exact coefficients, not the rank-36 theorem or existence of a target in the outer-one class.

The remaining regular question is whether the admitted local potentials U have an acceptable critical minimum in the original complete marked class. The singular PSD host sector requires separate higher-order or instability arguments. Neither sector has been decided here. The execution plan owns that unresolved dependency; no further numerical campaign follows from this compatibility geometry.

5.16. Finite neighborhoods of common host profiles exclude PSD equilibria

The common-host obstruction has a quantitative extension that allows an arbitrary donor shape. Write the actual host and donor phasors as z_(c,j)=exp(i*h_(c,j)) and w_j=exp(i*d_j). Choose any common reference profile p_j on the unit circle and five reference origins q_c=exp(i*Phi_c) whose cut-free rim has winding one. Suppose

\[ \max_{c,j}|z_{c,j}-p_jq_c|\le\varepsilon,\qquad B_\Phi(c,:)=(1,\Re q_c,-\Im q_c),\qquad \sigma=\sigma_{\min}(B_\Phi)>0. \tag{17ak} \]

The last inequality follows from the three-point argument in section 5.12. This is a chord-distance bound between circular phases, so no small-angle lift or inverse trigonometric branch is consumed. The profile p need not be uniform or acute. The actual complete marked-class admission remains a separate premise; proximity alone does not prove its windings or mixed periods.

Define beta_j=w_j*conj(p_j), still a unit phasor. Let I_j be the internal torque of p at slot j and R_c the rim torque of q at host c. The reference host current with the actual donor is I_j+R_c+Im(beta_j*conj(q_c)). For unit phasors, changing either endpoint by chord distance at most epsilon changes its edge sine or cosine by at most epsilon. Each host has four host neighbors and one donor neighbor. The difference between its actual current and this reference current is therefore at most 4*(2*epsilon)+epsilon=9*epsilon.

At a full stationary state the actual host currents vanish. Consequently there are residual vectors r_j with ||r_j||_infinity<=9*epsilon such that

\[ B_\Phi \begin{pmatrix}I_j\\ \Im\beta_j\\ \Re\beta_j\end{pmatrix} +R=r_j, \qquad |\beta_j-\beta_0| \le\frac{18\sqrt5}{\sigma}\,\varepsilon. \tag{17al} \]

The bound follows by subtracting the slot-zero equation: the right side has Euclidean norm at most 18*sqrt(5)*epsilon, and the two phasor components are part of the coefficient vector controlled by sigma. It uses all five host equations and does not assume donor coherence.

Now set the donor reference origin to beta_0 and use two vectors constant within each child: v contains the real parts of (q_0,...,q_4,beta_0), and u contains their imaginary parts, repeated over all nine slots. Their internal-edge differences vanish exactly, even when the actual internal profiles are nonacute. At the full common-profile reference, section 5.1 gives the pair quadratic form 18*F_W<=-9/2. On each rim edge the actual cosine differs from its reference by at most 2*epsilon. On each spoke the difference is at most (1+18*sqrt(5)/sigma)*epsilon, by (17al). The squared pair difference on any edge is at most four. There are 45 rim edges and 45 spokes, so

\[ \begin{split} v^TMv+u^TMu &\le-\frac92+ \left(540+\frac{3240\sqrt5}{\sigma}\right)\varepsilon,\\ \varepsilon &<\frac{9/2}{540+3240\sqrt5/\sigma} \quad\Longrightarrow\quad M\not\succeq0. \end{split} \tag{17am} \]

At least one vector is then a negative full phase-Hessian witness. Subtracting its common mean preserves its quadratic form, making it a relative witness; (16) gives instability of the complete stationary law. More quantitatively, if Delta is the positive magnitude of the upper bound's negative right side, then lambda_min(M|relative)<=-Delta/54: the two centered squared norms sum to at most 54. Equality at the displayed radius is not an instability certificate.

Thus a possible stable stationary target requires finite host deformation away from each fixed common-profile reference. The explicit radius depends on its origins through sigma; this singular value has no positive uniform bound as reference rims approach marked cuts. The estimate supplies no trajectory, source inclusion or claim that a target exists outside it.

5.17. A uniform reference rim has a finite equilibrium obstruction

For the uniform reference rim Phi_c=2*pi*(c-2)/5, the rim torque R is zero and B_Phi^T*B_Phi=diag(5,5/2,5/2). For any reference internal profile p and any actual donor, the reference host-current vector at each slot therefore has squared norm 5*I_j^2+5/2. This uses only |beta_j|=1, with no stationarity assumption. The current perturbation bound used above gives

\[ \max_c|S_{c,j}| \ge\frac1{\sqrt2}-9\varepsilon \quad\text{at every slot }j. \tag{17an} \]

Indeed the reference Euclidean norm is at least sqrt(5/2), its change is at most 9*sqrt(5)*epsilon, and the maximum component is at least the Euclidean norm divided by sqrt(5). Hence epsilon<1/(9*sqrt(2)) excludes equilibrium altogether, not just PSD equilibrium. No donor shape, relative origin or change to the common internal reference profile removes this defect within the stated bound. This extends the zero-vector reconstruction example in section 5.11 to arbitrary reference profiles and a finite open neighborhood. It is a static current obstruction and also gives the following conditional complete-flow exit result.

Fix this uniform q and 0<=epsilon<1/(9*sqrt(2)). Allow any common p at each state and arbitrary donor phases, and define the closed tube by min_p max_(c,j)|z_(c,j)-p_j*q_c|<=epsilon. The minimum over the compact profile torus makes this a closed phase set. By (17an) it contains no state with S=0.

Under the unforced complete law (1), no trajectory can remain in that closed tube for all future time. On the fixed zero-form-mean leaf, H is nonnegative and its form term bounds x, since L is positive on relative forms. The sine current is bounded and the law has only linear form dependence, so its forward solutions exist globally; their form/phasor states are precompact. The loss identity gives a nonempty invariant omega-limit set inside Lx=0. On the fixed mean leaf this is x=0, so the phase row is zero. Invariance then requires S=0 in the form row. If a trajectory stayed in the closed tube, its omega-limit set would lie there as well, contradicting (17an). Therefore a trajectory initially inside must leave it at some finite time.

This conclusion uses the unchanged autonomous law, support and loss identity, with no subsequent forcing or events. It gives no explicit exit time and prevents neither later return nor retention of marked windings. Its source premise is checked separately below. The initial profile p may vary across slots and need not be a carrier profile. Marked cuts do not stop the full flow, and tube exit is not winding loss. This exit argument does not apply merely from exclusion of PSD equilibria, such as the broader gap below; an unstable equilibrium can still belong to an invariant limit set.

The retained outer-one source must leave the uniform-rim tube

The original prepared source satisfies that premise using its existing complete bounds. The outer acquisition estimate retains the finite carrier endpoint v_c=4*(c-2)/pi, with |Phi_c-v_c|<0.000022083; it does not replace that endpoint by the uniform twist. The observation-history transport gives the actual contact phases h_(c,j)=alpha*(j-4)+Phi_c+e_(c,j)+drift_(c,j), with |e_(c,j)|<=Z<7*10^-70 and the whole-word drift bounded by m=2*g*(1/100+g/5000)/5000 for any valid upper bound g on gamma. All commanded offsets have cancelled, while this continuous drift remains.

Take p_j=exp(i*alpha*(j-4)) and the uniform q from (17an). The distance from the actual source is bounded by

\[ \max_{c,j}|z_{c,j}-p_jq_c| \le \left(\frac8\pi-\frac{4\pi}5\right)+f_\theta+Z+m <\frac1{25}<\frac1{9\sqrt2}. \tag{17ao} \]

The inequalities require no new numerical assessment. Since pi^2<10 and pi>157/50, the endpoint mismatch is positive and less than 702/19625<9/250. Use f_theta<23*10^-6 and Z<10^-8. For the drift proof choose the independent analytic majorant g=1/3000, valid because gamma=1/(1023*pi)<1/3069; the displayed monotone formula then gives m<10^-8. Their sum is strictly less than 1/25. Finally 9*sqrt(2)<25, so this closed tube is below the no-equilibrium threshold. Phase-to-chord conversion uses |exp(i*a)-exp(i*b)|<=|a-b|.

The registered source family retains all twelve original allowances in [0,10^-10], all 96 relative source directions and every primitive donor origin beta. That isolated donor rotation leaves the host endpoint unchanged; attachment changes no coordinate. Consequently every member of the original outer-one family, at every beta, must leave the closed uniform-rim tube of radius 1/25 after some finite unforced contact time under (1).

This is an analytical corollary of the existing source inequalities and the complete-flow argument, not a replay or amendment of their frozen assessments. It supplies no uniform exit-time bound, winding loss, terminal state or physical identification. The complete-flow convergence proof below separately strengthens departure to permanent exclusion and gives a single, still unidentified, limiting equilibrium. Leaving this tube may change the host origins while keeping a common internal profile; it does not by itself force departure from the whole common-host family considered next. No response record or source-assessor flag is changed.

5.18. Compactness gives a uniform but uncomputed reorganization gap

The dependence on sigma in (17am) leaves a separate question: could PSD stationary states approach common host profiles while their reference rims approach cuts? The cut limits can be included without treating them as admitted targets.

Let K_closed be the closure, in the compact 54-phasor torus, of the original complete marked phase class. Let E_PSD consist of its stationary states with M>=0. Both stationarity and PSD are closed conditions, so E_PSD is compact. Let C_common be the compact set of phases with z_(c,j)=p_j*q_c for all five hosts, with arbitrary unit p, q and arbitrary donor phasors. We show that these two compact sets are disjoint.

At any supposed intersection, the reference rim q lies in the closure of the cut-free winding-one rim class. If it has at least three distinct phasors, the rank-three argument (17aa) still applies, even at marked cuts: stationarity forces the donor to share p with a constant relative origin. The wheel inequality of section 5.1 extends to this rim closure by continuity. The common-profile pair then has quadratic form at most -9/2, contradicting PSD.

The remaining case has at most two distinct rim phasors. A one-point rim has winding zero in an open neighborhood without cuts, so cannot be such a limit. The same holds for two nonantipodal points: every transition between them has a reverse transition, their principal gaps cancel, and all rim edges stay away from cuts. Thus the only possible two-point limit is an antipodal partition, with nonempty host groups P and N at q_plus and -q_plus. Let k be the number of rim edges crossing that partition; k>=2. Put Q=sum_j Re(w_j*conj(p_j*q_plus)). The indicators of all nodes in the respective host groups, with zero donor entries, give exact cut forms

\[ \mathbf1_P^TM\mathbf1_P=-9k+|P|Q,\qquad \mathbf1_N^TM\mathbf1_N=-9k-|N|Q. \tag{17ap} \]

Internal edges do not cross either cut, each crossing rim cosine is -1, and the spoke terms have the displayed opposite signs. If Q is nonpositive choose P; otherwise choose N. The chosen form is at most -9k<=-18, again contradicting PSD. This boundary control requires no donor profile assumption and no stationarity equation. Recentring its indicator preserves the negative relative form.

Define the distance of the actual hosts from the entire common-profile family by

\[ d_{\rm common}(z_H) =\min_{p\in(S^1)^9,\ q\in(S^1)^5} \max_{c,j}|z_{c,j}-p_jq_c|. \tag{17aq} \]

The minimum is attained, this distance is continuous, and it vanishes exactly on the host projections of C_common. Since C_common allows every donor, disjointness gives a number delta_*>0 such that every state in E_PSD satisfies d_common>=delta_*, if E_PSD is nonempty. If it is empty, the exclusion is vacuous. In particular every possible stable stationary target in the original marked class must be separated by a finite host deformation from all common host profiles, including sequences whose reference rims approach cuts. Additional Hessian zero modes are included in this PSD statement.

This is an existence proof of a positive separation, with no value or computable certificate for delta_star supplied. It proves neither that E_PSD is nonempty nor that the remaining genuinely different host profiles are unstable. The explicit reference-dependent bound (17am), the uniform-rim defect (17an), and this global uncomputed gap have different conclusions. Only the uniform-rim result has the conditional complete-flow exit consequence established above. None supplies actual-source capture, general common-profile-tube exit or physical identification. The execution plan retains the stationary feasibility and full stability decision beyond these exclusions.

5.19. Every complete unforced contact trajectory converges to one equilibrium

The same fixed sine law gives a global convergence statement without a marked-class, acute-gap or positive-Hessian premise. Every finite initial state with zero form mean has a global forward solution, finite total phase variation and a single limiting equilibrium. The limit can depend on initialization; this is not uniqueness of the stationary configuration. It does not decide which equilibrium is reached or whether a stable stationary target exists in the original marked class.

Write mathcal H=1^perp, and restrict L to this relative space. Set lambda=lambda_min(L|mathcal H)>0 and mu=||L||<=10. At the conserved lifted phase mean, phase can be represented on the compact flat torus mathcal H/(2*pi*(Z^54 intersect mathcal H)). This quotient retains the given continuous lift locally; no boundedness of that lift is assumed. Let q denote this circular relative phase, let V be the 144-edge phase potential in (1), and put

\[ k=\frac{\gamma^2}{16}<\frac1{16},\qquad U(q)=kV(q),\quad g=\nabla U,\quad P=L^{-1}|_{\mathcal H},\qquad v=q'=\frac\gamma4Lx. \tag{17ar} \]

Since grad V=-S, differentiation of the actual phase row gives the equivalent damped mechanical system and its energy identity

\[ Pv'+\frac14v+g=0,\qquad E=\frac12v^TPv+U=kH,\qquad E'=-\frac14\|v\|^2. \tag{17as} \]

Here P is a mathematical metric on the declared coordinates; it is not an identification of physical mass. The nonnegative energy bounds x and v on the fixed form-mean leaf. The smooth vector field has bounded phase currents and only linear form dependence, so solutions exist for all forward time and their form/circular-phase states are precompact. The invariance argument used in section 5.17 gives a nonempty compact omega-limit set consisting entirely of equilibria: zero loss requires v=0, and invariance of that condition in (17as) requires g=0. In particular v->0, g->0 and E->E_infinity.

To prove convergence to one point rather than just to this stationary set, use the auxiliary real-analytic proof function

\[ F(q,v)=E(q,v)+\frac18g(q)^Tv. \tag{17at} \]

F is not a new model law, physical storage or installed diagnostic. The declared H and its loss identity remain unchanged.

The phase Hessian satisfies ||Hess U||<=k*mu: the signed-cosine Laplacian M obeys -L<=M<=L. Along the complete law,

\[ \begin{split} F'&=-\frac14\|v\|^2+ \frac18\left(v^T(\nabla^2U)v-rac14g^TLv-g^TLg\right),\\ \frac14|g^TLv|&\le\frac12g^TLg+\frac1{32}v^TLv,\\ F'&\le-\frac{17}{128}\|v\|^2-\frac1{16}g^TLg \le-c\big(\|v\|^2+\|g\|^2\big),\qquad c=\min\left\{\frac{17}{128},\frac\lambda{16}\right\}>0. \end{split} \tag{17au} \]

The constant uses only k<1/16 and mu<=10, including nonacute states and marked-cut crossings. F is bounded below on the orbit, decreases, and has limit E_infinity, since its extra term tends to zero. Every point of the omega-limit set is a critical point of F at that same value.

The local analytic gradient inequality therefore applies to F in a coordinate chart around each such point. This is the analytic method for damped second-order systems developed by Haraux and Jendoubi (1998), DOI 10.1006/jdeq.1997.3393; the local gradient-inequality and finite-length argument is also stated in Begout, Bolte and Jendoubi, On damped second-order gradient systems, arXiv version 5. Their bounded Euclidean-trajectory hypothesis is not assumed for a phase lift here. Compactness of the circular omega-limit set supplies a finite chart cover. Taking the minimum of its positive exponents and enlarging its constant gives, for every sufficiently late time,

\[ (F-E_\infty)^{1-\rho} \le C\|\nabla F\| \le C_1(\|v\|+\|g\|),\qquad 0<\rho\le\tfrac12. \tag{17av} \]

Indeed the orbit eventually stays in every neighborhood of its compact omega-limit set. In combining the finitely many inequalities one can also require 0<=F-E_infinity<1. The last bound follows directly from grad_q F=g+(Hess U)*v/8 and grad_v F=P*v+g/8.

If F reaches its limiting value at a finite time, (17au) makes the state stationary thereafter. Otherwise (17au)--(17av) imply

\[ -\frac{d}{d\tau}(F-E_\infty)^\rho \ge c_1(\|v\|+\|g\|),\qquad \int_0^\infty(\|v\|+\|g\|)\,d\tau<\infty \tag{17aw} \]

for a positive constant c1 at sufficiently late times; the finite initial interval contributes a finite amount. Thus the phase has finite total variation. Its original continuous lift is Cauchy and converges in the fixed-mean affine space. This proves boundedness of the lift as a consequence, rather than using it as a premise. Equation (17as) also gives integrable v', hence integrable x'=(4/gamma)*P*v'. The full state converges to a single point with x=0 and S(theta)=0.

The limit may be a saddle, a degenerate equilibrium or a marked-cut state. The theorem excludes persistent nonstationary limiting motion under (1), while allowing transient motion, connecting orbits and damped oscillations. Even finite phase variation can coexist with infinitely many marked-cut crossings if the limit lies on a cut. If the limit is cut-free, all marked windings are eventually constant; their values are not selected by this argument. No convergence rate, finite settling time, source-to-target capture or physical identification is supplied.

There is a stronger departure consequence for the already admitted outer-one source. Define its distance from the fixed uniform reference rim, with arbitrary common profile and donor, by d_uniform=min_p max_(c,j)|z_(c,j)-p_j*q_c| as in section 5.17. The current floor (17an) forces every stationary state to satisfy d_uniform>=1/(9*sqrt(2)). Consequently the limiting distance of every continued-contact trajectory obeys

\[ d_{\rm uniform}(\theta_\infty)\ge\frac1{9\sqrt2}. \tag{17ax} \]

In particular the entire original source family, at every donor origin, eventually stays outside the closed tube of radius 1/25 that initially contains it by (17ao). A finite last-exit time exists but is not bounded here. This permanent exclusion already follows from the compact omega-limit argument and the no-equilibrium gap: arbitrarily late returns would give a stationary accumulation point inside that tube. It does not require the analytic finite-length step. The latter separately proves convergence of the complete state and its continuous lifts.

These statements retain the unchanged complete law, support, capacity and clock, with no later input or event. They reuse the existing source inclusion rather than evaluating a new trajectory or modifying a frozen assessment. Permanent departure from this particular uniform-rim tube does not establish winding loss or departure from every common-host profile. Stable stationary feasibility in the full original marked class, and the actual source's eventual marked identity, remain unresolved.

6. Actual aligned outer-zero control after the full observation word

The local target in (2) becomes a maintenance result only with an actual source inclusion argument. This section uses the separately declared outer-zero control and primitive external origin beta=0. It retains the original twelve allowances, recovered fine states and all five sequential observation blocks. The source owner supplies the actual residual control, not the exactly stationary zero-error target. No coordinate is reset after observation and no settling interval is inserted.

Let r and z be the freshly rebuilt full-form and full-phase error upper bounds from the host source, and q=1e-70 the external source bound. Let g be the upper bound for gamma and T the complete observation-word duration. At the beginning of that word the actual control has ||x_H||_infinity<=r and phase offsets from psi of magnitude at most z. During observation the host has degree four and both rows retain their uniform quarter factor. The diffusion semigroup contracts the maximum norm, and |S_i|<=4, so throughout the complete word

\[ B=r+gT,\qquad \|x_H\|_\infty\le B. \tag{18} \]

All fifteen supplied phase events leave form unchanged. Their accumulated commanded offsets sum to zero at the end of the word, but the continuous phase drift survives. Because |theta_i'|<=2*g*B, the actual reached host phase offsets satisfy

\[ Z_c=z+2gBT,\qquad |\theta_{H,c,j}-\psi_j|\le Z_c. \tag{19} \]

This bound accounts for motion during every commanded phase offset; it does not interpret the final inverse command as restoration of state. The separately recovered external constituent retains form and phase offset magnitudes at most q. Thus define

\[ b=\max(B,q),\qquad e=\max(Z_c,q). \tag{20} \]

The declared beta=0 and preserved zero means put the actual joined source on the same mean leaf as x*=0, theta*_{c,j}=psi_j. Each internal gap is in [alpha-2e,alpha+2e]; ring and spoke gaps have absolute value at most 2e. Initial inclusion in the joined acute target chart follows by checking the strict margins

\[ \alpha-2e>0,\qquad \pi/2-\alpha-2e>0,\qquad \pi/2-2e>0. \tag{21} \]

The first margin is a useful initial property, not an assertion that all internal edge gaps remain positive forever.

Present storage and new attachment work

Every form edge difference has magnitude at most 2b. The 144 edges give form storage at most 288*b^2. The target phase gradient vanishes, so the sum of first-order phase terms is zero. Every edge phase perturbation has magnitude at most 2e, and the absolute second derivative of its potential is at most one. Taylor's integral remainder therefore bounds the complete phase excess by 144*(2*e)^2/2=288*e^2. Consequently

\[ H(0)-6V_9\le U_0:=288(b^2+e^2). \tag{22} \]

This is the storage of the actual reached source, including the observation history through (18)--(19). It is not the sum of historical preparation expenditures. For the new 45-edge attachment event alone, 1-cos(a)<=a^2/2 gives the separate bound

\[ 0\le W_B\le\frac{45}{2}\big[(B+q)^2+(Z_c+q)^2\big]. \tag{23} \]

All earlier acquisition, outer attachment and observation events retain their own signed work and expenditure accounting. A small present excess or new attachment cost neither erases these costs nor assigns dissipated storage to an undeclared event reservoir.

Full-state trap and convergence

On the joined acute cell, each of the six internal C9 cycles has winding one and phase cost at least V9 by convexity. All 90 wheel edges have nonnegative phase cost. An internal acute boundary costs at least b9-V9 above the six-cycle minimum, where

\[ b_9=1+8[1-\cos(3\pi/16)]. \tag{24} \]

This is the smaller positive-boundary value. A negative internal face leaves the other eight gaps summing to 5*pi/2 and has the larger bound 1+8*(1-cos(5*pi/16)). A wheel-edge acute boundary costs one while the six internal cycles retain their minima. Hence every feasible acute cell boundary has storage excess at least

\[ \mathcal B=\min\{b_9-V_9,1\}>0. \tag{25} \]

No claim that the separate minimizing cycle configurations can be realized simultaneously is needed: these are lower bounds on every actual boundary point. The phase-space cell uses the branches inherited from the aligned target. Within it all edge cosines are positive, so the phase Hessian is positive definite on the zero-mean leaf. Its lifted cell is convex, and the target is its unique stationary phase point.

If the rebuilt actual source satisfies (21) and U0<min(1/100,Bcal), nonincrease of H forbids a first exit from the cell. Use the preserved closed sublevel H-6V9<=U0<Bcal on the fixed-mean leaf inside the target cell. Its closure cannot meet the cell boundary because every boundary point costs at least Bcal. Connectedness bounds form through its quadratic storage and phase through fixed-branch edge differences. The sublevel is therefore compact. Its largest invariant zero-loss set requires Lx=0 and S(theta)=0; it is the single aligned target. The invariance principle therefore proves all-future retention of six internal windings one and nine old outer windings zero, and full-state convergence to the target. It supplies no finite settling time. The proof does not extend source inclusion to beta different from zero or to the outer-one preparation.

7. Fixed actual-source assessment and research boundary

The declared exact-rational rebuild at the maximum of all twelve original allowances certifies source admission and the following conservative bounds:

\[ \begin{split} B&<6.224\cdot10^{-8},& Z_c&<7.746\cdot10^{-15},\\ U_0&<1.116\cdot10^{-12}<1/100,& \mathcal B&>0.2426,\\ W_B&<8.714\cdot10^{-14},& D(\theta_1(0))&<1.395\cdot10^{-16}<8/5<9d. \end{split}\tag{26} \]

The three initial cell margins are strictly positive. All original source directions survive the argument; arbitrary correlation is permitted, and smaller admitted allowances satisfy the same monotone bounds. The actual beta=0 outer-zero control therefore passes the frozen criterion with its whole observation history. It retains all fifteen marked windings for all future time and converges to the aligned target under the supplied continued-contact law. The statement is about an entire admitted source family, not a fitted nominal trajectory.

For the outer-one family, (26) certifies only initial membership in the negative phase-curvature domain, uniformly over the supplied relative origin. Its nonstationary future is not replaced by an equilibrium or an infinitesimal perturbation calculation. The exact carrier cannot remain in the old acute chart indefinitely; a stable full-fine stationary organization retaining that chart would require D to exceed 9d. The existence, acquisition and maintained identity of such a substantially reorganized fine state remain unresolved. An outer gap becoming nonacute would still require a separate winding-loss check. The subsequent full-domain curvature result supplies the stronger necessary condition D>=9/4 without the acute outer-gap restriction and excludes every common internal phase profile. It does not extend the frozen source calculation into a future-retention certificate.

The separate uniform-rim source corollary uses the retained endpoint and observation-history inequalities to prove eventual exit of the full outer-one family from a specified uniform-rim neighborhood, at every donor origin. This is a full-fine departure result, with neither an exit-time bound nor a winding-loss conclusion; the frozen assessment (26) and its flags keep their original scope.

These results separate maintained constituent identity, loss of a convenient capture chart, stationary instability and actual-source identity loss. They do not justify another series of short-horizon refinements or transfer the beta=0 control to another registration. Any following change of organization, interface, state restriction or complete law needs its own admission. The execution plan owns that decision and the property-first physical comparison; none of the present bounds identifies mass, charge or another fundamental physical property.

The reflection classification adds a separate obstruction without an acute-gap or small-defect premise. Its static reflection sector includes nonuniform internal profiles; no stable stationary state there retains all fifteen winding-one identities. An explicit nonacute saddle shows that stationary existence remains distinct from stability. Neither result supplies the actual outer-one source's fate.

8. Detached implementation and verification boundary

The maintenance owner reuses the existing complete contact support and exact interval arithmetic. Its geometric derivation retains the separate form and phase eigenvalues and both complete tangent rows, the two conserved neutral directions and the constant in the full-fine curvature obstruction. Repeated spectral coefficients are evaluated once within each derivation, without caching source premises or verdicts. Its source assessor admits all twelve original primitive allowances and freshly rebuilds the source and observation evidence. It accepts no incoming report as a replacement for those premises and advances no trajectory.

Source inclusion, attachment work, initial defect-domain membership and the beta=0 capture verdict retain distinct meanings. A passing initial defect-domain flag is not a future instability, retention or loss verdict. The original external-contact and registration reports retain their earlier finite scope and frozen evidence.

The dedicated tests exercise complete support and tangent modes, both stationary controls, primitive admission, source rebuilding and scoped source-bound verdicts. Source regeneration and trajectory execution remain forbidden in this detached proof path. Numerical checks verify implemented arithmetic and wiring; the analytic arguments above establish the conditional mathematical claims and their domain boundaries.

The later analytical extension in sections 5.1-5.19 uses exact identities and counterexamples. Static regression checks separately reconstruct the stationary controls' neighbor resultants and the blind point's lifted secant from edge sums. These are arithmetic controls, not a stationary search or a proof by sampled configurations. The extension does not change the earlier assessor's threshold, flags or frozen evidence, and adds no numerical decision API. Its necessary curvature, resultant and slot conditions must not be exposed as sufficient stability or maintenance verdicts.

The donor-relaxation controls independently assemble the joined Hessian blocks, verify (17t) and the exact negative witness (17w), and check all 44 positive host-relative modes at the known aligned outer-zero control. That control tests the Schur algebra; its different rim winding is not evidence of an outer-one target. The shared root/Hessian certificate continues to admit its original full boxes and is not replaced by an uncertified donor branch or a collection of cut tests.

The stationary-reconstruction controls independently build host torques and two-column Gram matrices. They retain the full residual and unit-circle guards, test rotation covariance and reconstruct the existing analytic reflected saddle in the reflection tests. That saddle remains unstable; its reconstruction supplies neither a new search nor a source evaluation. The globally unique circular completion does not change the full-box certificate's admission or numerical budget.

The compatibility-manifold argument differentiates those exact guards and retains their dependent equations. Centered differences of the host-only reconstruction at the existing reflected saddle check (17ac)--(17ad); that saddle's negative host direction rejects the positive-A premise. Host-relaxation controls check (17af), (17ah) and (17aj) at the aligned outer-zero state, whose rank-one E is explicitly outside the winding-one reconstruction theorem. Neither local rank nor an infinitesimal null direction replaces the nonlinear stationary guards or the original full-box certificate.

The host-reorganization controls check the current-error estimate, the uniform-rim residual floor, the phase-pair comparison and the antipodal boundary cuts using independently assembled edge sums. These static controls do not sample or compute the compact uniform gap. The conditional exit proof uses the complete law and its exact loss identity; it neither runs a trajectory nor re-evaluates any frozen source or response. The actual-source corollary combines the already proved complete-source bounds with a separate analytic majorant for the same observation drift. Its uniform-rim tube is distinct from the broader PSD-only exclusion.

The contact-convergence controls independently assemble the full edge rows, solve the relative Poisson relation, and check the mechanical energy and auxiliary-function derivatives at static states. They include a nonstationary state at rest where the original loss vanishes but the auxiliary derivative is strictly negative. These arithmetic controls establish neither finite trajectory length nor convergence; section 5.19 owns those analytic claims and their compactness and gradient-inequality premises.