Stationary obstruction on the proposed maintained-parent UM support

The full support predicted by default parent-target UM has no stationary state in the closed acute edge-branch chamber containing its hypothetical endpoint. A certified boundary minimum of the bare phase potential proves this exclusion on all 21 fine coordinates. The argument does not assume that an unknown stationary state preserves constituent symmetry. It also gives a same-storage, same-means comparison path from the predicted source to the acute boundary.

These are analytical results for the supplied conservative complete law. No UM event, trajectory, autonomous selection or physical identification is provided. Nonacute equilibria and nonstationary organizations are not excluded. The research program alone selects further work; this conditional geometry does not activate it.

1. Complete law, source and every edge branch

Re-admit the retained THOL successor with the source consumer, then rebuild its ideal default UM reset. No earlier producer or event is replayed. The chamber owner retains all 21 nodes, 53 edges, actual predicted conductances and capacities, structural clock and declared phase lifts. The child is index 20 (2_sub_0); the parent is index 2. The actual retained child remains isolated: this successor support is still hypothetical.

On a continuous segment hold support, symmetric positive conductances and positive capacities fixed, with zero forcing and the weighted-form/bare-phase law:

\[ \dot x=M S_U(\theta),\qquad \dot\theta=M L_Wx,\qquad M_{ii}=\frac{\nu_i}{\pi d_i}>0, \]

\[ H=Q_W(x)+V_U(\theta),\quad Q_W=\frac12\sum_{\{i,j\}\in U}W_{ij}(x_j-x_i)^2,\quad V_U=\sum_{\{i,j\}\in U}[1-\cos(\theta_j-\theta_i)]. \]

Its conserved lifted means have weights rho_i=d_i/nu_i. Conductances and capacities remain in this law and its means; only the stationary phase equations below are independent of their positive values.

Orient every edge (i,j) with i<j. For its integer branch k_ij, define delta_ij=theta_j-theta_i-2*pi*k_ij. Exactly eight branches are one:

(0,8), (0,9), (1,8), (1,9),
(10,18), (10,19), (11,18), (11,19).

All other branches are zero. Let C require |delta_ij|<pi/2 on every edge, and let C_bar replace these by closed inequalities. Both domains are convex in real lifts; common phase translation is the only gauge being removed. The code constructs a deterministic 20-edge spanning tree and all 33 fundamental cycle rows, retaining the complete branch class. Two original winding labels alone would discard part of this information.

Rational interval admission gives an ideal minimum acute margin approximately 0.0106079513764, and greater than 0.010607951375 under the preceding conditional 2^-40 per-changed-output allowance. Thus the proposed source is strictly interior. The allowance is an assumption about a future reset, not a bound on an unevaluated implementation. Both conductance and capacity positivity are rechecked on that envelope.

2. Stationarity and strict monotonicity

Since M is invertible, stationarity requires L_W x=0 and S_U(theta)=0. Connected positive support implies constant form. With g=grad V_U=-S_U, the remaining question is whether g=0 anywhere in C_bar.

For any two points y,z in that chamber, edge summation gives

\[ (z-y)^T[g(z)-g(y)] =\sum_{e\in U}(\delta_e(z)-\delta_e(y)) [\sin\delta_e(z)-\sin\delta_e(y)]. \]

Sine is strictly increasing on the entire closed interval [-pi/2,pi/2]. Consequently the displayed expression is positive unless z-y is a common constant. This proves uniqueness modulo translation if a stationary point exists; it does not establish existence. Zero edge cosines at the boundary do not invalidate this strict monotonicity argument.

The weighted conserved means also rule out a nonzero rigid common drift as a relative equilibrium: a common rate with zero weighted sum must vanish. This observation does not exclude motion of internal relative coordinates.

3. A certified boundary witness on all fine nodes

Use five real parameters u=(p,q,t,b,v) to construct a witness y:

Fine nodes	Lifted phase
0, 1	0
2	p
3	q
4, 5	2pi-3t
6, 7	2pi-2t
8, 9	2*pi-t
10, 11	b
12, 13	b+v
14, 15	b+pi/2
16, 17	b+pi
18, 19	b+3*pi/2
20	p

Write this affine map as y=A*u+pi*c. It defines a proposed witness, not a reduced complete dynamics or an assumption on every stationary point. Solve the five restricted potential equations

\[ F(u)=A^T g(Au+\pi c)=0. \]

The certificate owner admits a fixed rational center, radius and proposed inverse, independently recomputes interval F and its Jacobian from all 53 edges, and certifies a contraction u -> u-R*F(u) strictly into that box. The inverse is a witness to check, not an assumed exact inverse. Pi and transcendental functions use the shared outward rational interval arithmetic. Numerical root probes are only a way to propose that witness; they supply no existence theorem.

The fixed center, in (p,q,t,b,v) order, is

(1082587343106, 1211889604568, 1438506301057,
 113581483716, 891083831083) / 2^40,

with radius r=2^-24. Recomputed bounds give eta<4.423e-13, contraction factor <3.076e-6 and image radius <6.256e-13<r. Thus the witness root exists inside the declared box; this is not an assertion that its rational center is an exact root. The contraction bound also makes R*J invertible at the center, hence the square matrix R is invertible. A fixed point of u-R*F(u) therefore has F(u)=0; an arbitrary proposed preconditioner is not assumed injective.

The existing unit-support stationary kernel cannot be applied unchanged: its independent free-node coordinates differ from this affine face, whose Jacobian is A^T Hess(V) A. The local certificate reuses shared interval and exact-linear arithmetic without replacing the full phase equations.

At the enclosed root all 12 edges on the three complete B-chain blocks {14,15}->{16,17}->{18,19}->{10,11} have forward gaps exactly pi/2. Their exact values follow from the affine identities, not a tolerance check. Every other edge remains strictly acute. The child-parent gap is exactly zero.

Full fine-node cancellation now gives more than restricted stationarity. The two A-chain interior pairs 6,7 and 8,9 have zero gradient by equal incoming/outgoing sine terms. The child also has zero gradient. Therefore F_p=g_2, F_q=g_3, and F_t=-6*g_4, with g_4=g_5. The B-chain interior pairs 16,17 and 18,19 cancel in the same way. Then F_v=2*g_12 and F_b=2*(g_10+g_12+g_14). Finally sum_i g_i=0 forces g_0=g_1=0.

Thus the exact root's full gradient has only four nonzero entries:

\[ g_{10}=g_{11}=-a,\qquad g_{14}=g_{15}=a, \qquad a>0, \]

where

\[ a=2(\cos v-1)+\cos(p-b). \]

Its certified enclosure lies inside (0.0145091,0.0145097). Every nonactive edge has acute margin greater than 0.1971598. Strict positivity and all nonactive margins are interval conclusions over the entire root box. In particular this witness is not an equilibrium.

4. Full closed-chamber exclusion

For arbitrary z in C_bar, sum the upper acute inequalities over all 12 forward edges of the three B-chain blocks. Intermediate node terms cancel, leaving

\[ z_{14}+z_{15}-z_{10}-z_{11}\ge\pi. \]

Equality holds at y. Hence

\[ g(y)^T(z-y) =a[(z_{14}+z_{15}-z_{10}-z_{11})-\pi]\ge0. \]

Convexity of V on C_bar proves that y is its global minimum modulo translation. If some distinct z were stationary, strict monotonicity would require (z-y)^T(0-g(y))>0, contradicting this nonnegative normal product. A common translate of y has the same nonzero gradient. Therefore there is no stationary phase state anywhere in the full closed acute chamber.

This is a full-coordinate separating certificate, including the acute boundary and asymmetric candidates. It does not infer a global obstruction from a failed numerical root finder, a restricted ansatz, or negative curvature at one candidate outside the chamber.

5. Storage and both means do not trap this source in the acute chamber

Let (x_0,theta_0) be the ideal hypothetical endpoint, with actual positive weights and capacities. The same argument applies to any admitted conditional endpoint. Its form storage Q_0=Q_W(x_0) is strictly positive: forms are unchanged by UM and the parent and child have different forms on their new positive-conductance edge. Write h=H(x_0,theta_0) and m_rho=(rho^T x_0)/(rho^T 1). Fresh interval arithmetic gives Q_0>0.0105 for both the ideal source and the conditional envelope. Each envelope member uses its own fixed conductances, capacities and weighted means in the following construction.

Translate y by a common phase so that its weighted phase mean equals the source mean, and set theta(s)=(1-s)*theta_0+s*y, 0<=s<=1. This segment stays strictly acute until its boundary endpoint. Convexity and minimality give V(theta(s))<=V(theta_0)<h. Define

\[ x(s)=m_\rho 1+ \sqrt{\frac{h-V(\theta(s))}{Q_0}}\,(x_0-m_\rho 1). \]

The multiplier is at least one. The path begins at the proposed source, retains both weighted means and has exactly H(x(s),theta(s))=h. All 12 active gaps reach their acute faces transversely. A sufficiently short extension crosses them while the radicand remains positive, by continuity and the strict margin h-V(y)>=Q_0>0.

This comparison path establishes that the connected component of the storage-and-means level containing the source is not separated from the acute boundary. Those conserved quantities alone cannot provide the proposed acute trapping certificate. The parameter s is not structural time; this path is not a solution, a pressure reconstruction or a prediction of an actual future boundary crossing.

6. Scope and verification

The retained source, predicted support and complete law fix this result. It excludes an acute stationary identity and the stated storage-barrier route. It establishes neither instability of every organization nor loss of principal winding. An acute face has gap pi/2; a principal phase cut has gap pi. Those are different boundaries of different identities. The separate principal-chamber theorem now excludes nonacute stationary storage minima and extends the invariant-shell comparison to principal cuts. Nonstationary maintenance still needs its own argument. Positive capacity rescaling changes rates and means but cannot repair the present stationary phase obstruction.

Verification combines branch and cycle controls, interval witness controls and independent full-support checks. The latter recompute fine currents and static comparison identities without executing UM, THOL, a trajectory or a frozen producer. High-precision checks are independent numerical controls; the exact inequalities and analytical cancellations carry the theorem. Earlier frozen artifacts remain unchanged.