Stationary and energy geometry of the actual sine contact support

1. Frozen analytical question and complete marked chamber

This declaration precedes numerical response evaluation. It fixes the next analytical question after actual contact: whether the admitted joined support permits a stationary organization in the original complete acute chamber. It does not evolve the contacted source or replace its complete law.

• Retain the twenty labelled nodes A_(k,s),B_(k,s), with k=0,...,4 and s in {+,-}. Each old doubled C5 has all four edges between successive pairs. Add the actual internal A_0 edge and the twelve actual edges from A_0 to B_0,B_1,B_4. There are 53 edges. Retain their generated positive conductances, held post-word capacities, zero forcing and the original structural clock from the contact protocol.
• Use exactly the conservative weighted-form/bare-phase completion with e=0,w=beta=1: x_dot=M*S_U(theta), theta_dot=M*L_W*x, where M_ii=nu_i/(pi*d_i)>0 and d_i is the bare degree. Actual support and conductances remain held, including member-to-boundary association. No pair synchronization is assumed.
• Orient every old cycle block from pair k to pair k+1 mod 5. For each of its four fine edges set branch b_e=0 when k<4 and b_e=-1 on the closing block 4 -> 0. Orient the internal edge from A_(0,+) to A_(0,-) and set its branch to zero. Orient all cross edges from A to B: their branches are zero for B_0,B_1 and one for B_4. These orientations and branches cover every support edge.
• With real phase lifts define delta_e=theta_head-theta_tail-2*pi*b_e. The declared chamber requires -pi/2<delta_e<pi/2 for every one of the 53 edges. Quotient only the common phase translation, using sum theta=sum theta_* if a representative is needed, where theta_*(A_(k,s))=theta_*(B_(k,s))=2*pi*k/5. Keep all fine coordinates and the full branch vector, rather than only the two constituent winding observations.
• Derive the full stationary equations first. Use exact current balances, strict acute-angle monotonicity and the connected-chamber method from marked-cycle information. Any pair reduction must follow from those full equations and their phase-domain symmetries. No fixed-weight dynamical pair closure is assumed.
• The bounded outcome is either a proved stationary target with a strict relative-storage minimum or a proved obstruction in this exact chamber. No numerical root finder, optimizer, trajectory, parameter search or precision retry is authorized by this declaration. Static checks may independently verify the support, branch inventory and algebraic identities; they cannot replace the proof.

Stationary exclusion in this chamber would not imply winding loss, the absence of nonstationary organizations, or the absence of stationary states in other phase chambers. A stationary candidate would not establish source entry, attraction, maintenance from the actual preparation or a new NFR.

2. Full stationary equations and source chamber admission

Let U,W,nu retain the actual contact values. The bare graph is connected, all conductances are positive, and all held capacities are positive. Every node has positive bare degree. Consequently the diagonal mobility M is invertible and the full twenty-node stationary equations are exactly

\[ L_Wx=0,\qquad S_U(\theta)=0. \]

The first equation implies constant form because

\[ x^TL_Wx=\sum_{\{i,j\}\in U}W_{ij}(x_j-x_i)^2, \]

and positivity propagates equality along the connected support. No conductance has been replaced by one, averaged across a pair or discarded. The remaining stationary phase problem depends on bare support. This separation holds at stationarity under this complete law; it does not make the full weighted evolution independent of member-to-boundary association.

Define the bare phase potential and its gradient by

\[ V_U(\theta)=\sum_{e\in U}(1-\cos\delta_e),\qquad \nabla V_U=-S_U. \]

The common phase direction is neutral. The conserved lifted means with weights rho_i=d_i/nu_i exclude a nonzero rigid common form or phase drift: sum rho_i*x_dot_i=sum rho_i*theta_dot_i=0. Thus an equilibrium modulo these two common translations is an ordinary stationary state. A nonstationary organized motion with changing internal coordinates is a different possibility.

The declared chamber is nonempty: the common reference theta_* has old forward gaps alpha=2*pi/5, internal and zero-pair contact gaps zero, and cross gaps +alpha,-alpha at B pairs one and four. All are strictly acute. The actual post-word family belongs to the same chamber. The existing contact phase bound gives |theta_i(0)-theta_*,i|<=a_0=5/1536. Every old or cross reference gap therefore changes by at most 2*a_0=5/768, and

\[ \alpha+\frac5{768}<\frac\pi2. \]

The internal-edge gap is at most 5/768. These strict bounds retain each listed branch, rather than inferring chamber membership from two winding values alone. The joined graph has cycle-space dimension 53-20+1=34; the complete branch vector also retains the mixed and fine-member cycles. Branch changes induced solely by integer changes of vertex lifts describe the same circular chamber and do not enlarge this cycle class.

3. Acute phase uniqueness and stationary pair synchronization

Let C denote the open chamber from Section 1 and let C_bar denote its closed version, obtained by replacing every strict acute inequality by -pi/2<=delta_e<=pi/2, with the same branches. The closure extension is proved separately here; no trajectory or numerical response is used to change the frozen question.

Both sets are convex on real lifts. For theta,phi in C_bar, orient each edge as in the declaration and take their gradient difference. Exact summation by parts gives

\[ (\theta-\phi)^T[\nabla V_U(\theta)-\nabla V_U(\phi)] =\sum_{e\in U} (\delta_e(\theta)-\delta_e(\phi)) [\sin\delta_e(\theta)-\sin\delta_e(\phi)]. \]

Sine is strictly increasing on the whole closed interval [-pi/2,pi/2]. Every summand is nonnegative, and it is positive whenever the two edge gaps differ. Connectedness implies that all gaps agree only when theta-phi is a common constant. Hence there is at most one stationary phase configuration in C_bar modulo common phase. This conclusion does not assume that a stationary point exists.

In the open chamber the Hessian is the positive cosine Laplacian and is strictly positive on relative phase directions. At an acute boundary some edge cosines may vanish. The preceding strict monotonicity argument still applies there and does not infer a positive Hessian from those zero values.

Each independent interchange of the two members of one pair preserves bare support, all the declared branch constraints and the potential. On old or cross blocks it permutes equal-branch edges. At the internal A_0 edge it reverses the zero-branch gap. Thus a stationary phase point would be carried to another stationary point of the same closed chamber. Choose the representative with sum theta=sum theta_*. Interchange preserves this sum, so the uniqueness result forces it to fix that point. Every stationary pair therefore has equal lifted member phases:

\[ \theta_{A,(k,+)}=\theta_{A,(k,-)}=A_k,\qquad \theta_{B,(k,+)}=\theta_{B,(k,-)}=B_k. \]

This is a consequence of the full fine stationary equations. The actual conductances need not have pair symmetry, and an evolving state with equal member phases need not remain on that stratum. No fixed-weight pair closure is used in the argument.

4. Branch-preserving reflection of a possible stationary phase

The bare graph also preserves the reversal k -> -k mod 5 in both components. Combine it with phase conjugation and the fixed lift correction

\[ (\mathcal R\theta)_{C,(k,s)}= \begin{cases} -\theta_{C,(0,s)}, & k=0,\\ 2\pi-\theta_{C,(5-k,s)}, & k=1,2,3,4, \end{cases} \qquad C\in\{A,B\}. \]

Old forward gaps are carried to reversed-index forward gaps with the same acute bounds, including the declared closing-edge correction. The A_0 -> B_0 gaps change sign, the A_0 -> B_1 and A_0 -> B_4 gaps exchange with a minus sign, and the internal A_0 gap changes sign. Thus R preserves the full closed chamber and V_U; it also maps S_U=0 to S_U=0. It need not preserve the actual weighted dynamical field, which is not required for this stationary phase argument.

After the derived pair equality, choose the common phase representative A_0=0. This is possible without altering any branch gap. Reflection preserves that representative. Uniqueness therefore forces R theta=theta, giving

\[ B_0=0,\qquad B_4=2\pi-B_1,\qquad B_3=2\pi-B_2. \]

Write a=B_1, so B_4=2*pi-a. The two contact branches give -pi/2<=a<=pi/2. These relations have been derived without fixing the receiver's origin or internal shape in advance.

5. Exact obstruction in the complete acute chamber and its closure

The fine members of B pairs two and three have only their four old neighbors. With the stationary member equality just proved, the current equation at B pair two is

\[ 0=2\sin(B_1-B_2)+2\sin(B_3-B_2). \]

Both consecutive forward gaps lie in [-pi/2,pi/2]. Strict injectivity of sine on that interval makes them equal. The same argument at B pair three gives a common gap q on the whole three-edge chain from B pair one through pairs two and three to pair four:

\[ B_2-B_1=B_3-B_2=B_4-B_3=q,\qquad -\frac\pi2\le q\le\frac\pi2. \]

Together with the derived reflection this gives

\[ 2a+3q=2\pi,\qquad a\ge\frac\pi4. \]

The endpoint B pair one has two neighbors in each of B_0,B_2,A_0. Its full fine current equation, with no changed edge multiplicities, is

\[ 0=2[-\sin a+\sin q-\sin a],\qquad \sin q=2\sin a. \]

But pi/4<=a<=pi/2, so the right side is at least sqrt(2), while the left side is at most one. The strict contradiction proves

\[ \boxed{\text{There is no stationary state in the declared complete acute chamber, or in its closed acute boundary.}} \]

The conclusion holds for every actual positive conductance assignment and positive held capacity produced by the frozen contact family, not only the nominal split control. Those quantities enter the full evolution and its storage but cancel from the necessary phase stationary problem after the full form equation has forced constant form. The proof changes neither the UM fan nor the pressure, phase law or clock.

An independent interior obstruction avoids reflection and helps identify the conflicting currents. Normalize A_0=0 after stationary member equality, and put a=B_1, b=B_4-2*pi, c=B_0. The uncontacted B chain gives 3*q=2*pi+b-a, hence d=a-b=2*pi-3*q>pi/2. Both contact gaps are acute, so a,-b>0. The remaining cycle gaps r=a-c,t=c-b are acute and have r+t=d, so they too are positive. The two endpoint balances require

\[ 2\sin q=(\sin r+\sin t)+(\sin a-\sin b). \]

Each parenthesis is strictly greater than one: for positive acute u,v with u+v>pi/2, one has sin v>cos u and sin u+cos u>1. The displayed right side is therefore greater than two, while its left side is less than two. This second argument is confined to the open chamber; the preceding reflection proof supplies the boundary result.

6. What the obstruction establishes

The actual UM contact fan cannot support an equilibrium retaining all of the original fine and cross-edge acute branches. In particular, there is no stationary relative-storage minimum of the complete law there. A minimum constrained to the closed chamber can have nonzero boundary forces; it is not a stationary solution of the unconstrained nodal rows. The result is an exact feasibility obstruction, rather than a failed optimizer or an uncertified small residual.

The earlier result required the receiver to leave its original small radius-1/16 identity tube. The present result is stronger about possible stationary targets: merely enlarging that tube within this complete acute chamber cannot produce a stationary continuation. It does not assert a time at which the actual trajectory leaves the chamber, or that it ever does. The conservative law can support motion without an equilibrium at the moving state; excluding equilibrium does not exclude bounded nonstationary organization.

An edge reaching |delta|=pi/2 also does not imply a winding change. It is a boundary of the acute certificate, whereas a principal phase cut lies at |delta|=pi. The actual winding observations can therefore survive loss of acuteness. Stationary states with nonacute edges, other complete cycle classes, nonstationary identities and acquisition from the actual contacted source retain their separate obligations. No statement about physical particles, a unique fundamental law, or autonomous contact occurrence follows from this graph-specific theorem.

The principal-class curvature analysis below resolves the larger stationary-minimum question: nonacute saddles exist, but no stable equilibrium is available in the same complete open class. Actual nonstationary motion remains a separate question.

7. Independent static controls and comparison scope

The static controls check all 53 edge orientations and branches, incidence rank nineteen and 34 independent cycle directions. Exact turns check the complete reference class. Independent fine edge sums check the monotonicity, reflected current map and receiver-current equations without calling a root finder. The fixed algebra probes use 100-digit arithmetic; graph-capture comparisons retain their represented phases and absolute tolerance 2^-46 separately.

The auxiliary positive conductances cycle through 3/8,...,7/8. They test the complete weighted field and stationary reduction; they are not new UM outputs or replacements for the preceding contact's frozen conductances. The theorem itself applies to every admitted actual positive weight list. No event or frozen response producer is replayed by these controls.

Two comparisons keep the exclusion's domain explicit. The disconnected original twists have zero phase current before the fan is added. On the same connected contact support, constant form and constant phase give an equilibrium in the distinct zero-winding class: its relative storage Hessian consists of positive weighted-form and bare-phase Laplacians. This does not place that equilibrium in the excluded original chamber or show that the contacted source reaches it. Nonconstant form at constant phase also checks that a zero sine current alone does not satisfy both complete stationary rows.

These finite checks support implementation and algebraic consistency. They neither prove the chamber-wide theorem nor certify a new organized source.

8. Frozen nonacute minimum question and static controls

This declaration precedes the new static evaluations. Retain the complete model, actual conductances, capacities, clock, support, orientations and branches of Section 1. Enlarge only the admitted phase domain to -pi<delta_e<pi on every edge. The question is whether a stationary local minimum of the full relative storage exists anywhere in this open marked class. A nonstrict minimum is included. The complete fine Hessian must be positive semidefinite, but that necessary condition alone is not sufficient.

Derive any pair or receiver restriction from full stationarity and local minimality. Preserve zero neighbor resultants and zero curvature until an argument on this actual graph excludes them as minima. The old acute uniqueness/reflection proof and the odd-degree bound for another contact graph are unavailable here. Holding the donor's relative phases fixed while varying its common phase is an allowed variation, not a dynamical closure.

The bounded evaluation is analytical. Independent static controls may check the zero-resultant fiber identity, adjacent-twin splitting curvature, the exact receiver restriction, all eight cosine-sign cases of the three-edge chain, and the final endpoint inequalities. Use exact rational matrices and turns where available and 100-digit arithmetic for fixed trigonometric identities. Auxiliary probes use the quadrature C4, adjacent-twin half-split pi/3, a rotation increment 1/64, and receiver phases (0, pi/8, pi/3, 3*pi/4, 5*pi/4, 7*pi/4) in order (O,B_0,B_1,B_2,B_3,B_4). These are algebra controls, not proposed equilibria or source responses. Retain the earlier represented-phase tolerance for shared-reader comparisons. No optimizer, root search, trajectory, event or frozen producer is replayed. A minimum obstruction would leave stationary saddles, nonstationary organization and principal-cut crossings separate.

9. Full stationary curvature in the principal marked chamber

Let P be the open principal chamber declared in Section 8. The full stationary reduction from Section 2 remains valid there: x is constant and S_U(theta)=0. The full storage Hessian is block diagonal,

\[ \nabla^2 H=\begin{pmatrix}L_W&0\\0&K\end{pmatrix},\qquad K=L_{\cos\delta}. \]

The actual positive weighted form block is positive on relative form directions. A stationary relative-storage minimum therefore requires the signed phase Hessian K to be positive semidefinite on all relative fine directions. Since K*1=0, this is equivalent to K>=0 on the whole phase space; changing a test vector by a common constant leaves its curvature unchanged. Fixing either conserved common mean does not discard a negative relative direction.

The following proof excludes every stationary point satisfying this necessary PSD condition in P, including singular Hessians. It thus excludes nonstrict as well as strict local minima. It does not apply the acute uniqueness or reflection argument outside its domain.

10. Stationary PSD conditions force fine twin equality

For each fine node define its bare neighbor resultant and Hessian diagonal

\[ R_i=\sum_{j\sim i}e^{i\theta_j},\qquad r_i=\Re(e^{-i\theta_i}R_i)=K_{ii}. \]

Stationarity makes Im(exp(-i*theta_i)*R_i)=0; PSD gives r_i>=0. If R_i is nonzero, these two conditions force exp(i*theta_i)=R_i/|R_i|. If r_i=0, PSD forces the whole row and column of K to vanish, so every incident edge has zero cosine. This zero case cannot be dropped merely because another graph had odd degrees.

Every pair except A_0 consists of nonadjacent twins with exactly the same external neighbors. Both members therefore share one resultant R. If R is nonzero, the alignment just derived gives equal circular phases. For any one common neighbor, the two incident edges have the same branch when oriented outward from the twin pair. Their strict principal bounds give |theta_+-theta_-|<2*pi. Equal circular phases must consequently have equal lifts in this chamber.

Now retain the possibly zero resultant R=0. Both PSD rows vanish, so every common neighbor is perpendicular to both twins. Unequal circular twins must then be antipodal. The same strict branch bound makes their lifted separation exactly pi in absolute value. Write them, after possibly interchanging their names, as Theta+pi/2,Theta-pi/2.

For a common neighbor j, let eta_j be its lift adjusted by the common outward edge branch, so both principal gaps are eta_j-theta_+ and eta_j-theta_-. Their two strict cut inequalities imply

\[ -\frac\pi2<\eta_j-\Theta<\frac\pi2. \]

The zero incident cosines require sin(eta_j-Theta)=0, hence eta_j=Theta. All common neighbors have the same circular phase, and there is at least one of them. Their resultant is therefore degree*exp(i*Theta), contradicting R=0. Unequal zero-resultant twins are excluded. Equal twins with zero resultant have not been excluded by this lemma; they remain covered by the subsequent argument.

The internal A_0 edge makes that pair adjacent twins. A separate local curvature calculation applies. Put theta_+=Theta+epsilon,theta_-=Theta-epsilon; the internal principal cut gives |epsilon|<pi/2. With every external phase fixed, define C=Re(exp(-i*Theta)*R) for their common external resultant. The part of the potential depending on epsilon is

\[ f(\epsilon)=-2C\cos\epsilon-\cos(2\epsilon)+\text{constant}, \qquad f'(\epsilon)=2\sin\epsilon(C+2\cos\epsilon). \]

If a stationary split has epsilon!=0, then C=-2*cos(epsilon) and

\[ f''(\epsilon)=-4\sin^2\epsilon<0. \]

That is a full fine antisymmetric phase direction and contradicts PSD. Thus the adjacent pair also has equal lifted phases. This argument retains zero external resultants and does not divide by them.

Every hypothetical stationary PSD state in P therefore has equal lifted members in all ten pairs. This is a necessary stationary restriction, not a weighted dynamical pair-closure claim. It used the actual complete branch class to handle the degenerate nonadjacent twins.

11. Exact receiver variation and a second twin restriction

Keep the derived fine-member equality. Vary each B pair through its common phase, and vary all ten A phases by one common rotation, holding every A relative phase fixed. This is a six-dimensional family of configurations, not a proposed invariant dynamical subsystem. Every internal A edge retains its phase gap exactly. With O=A_0 and the five B pair phases as its coordinates, the restricted phase potential is

\[ V_{\rm restricted}=\text{constant}+4V_G, \]

where G is the six-vertex graph consisting of the B cycle and the three edges O--B_0,O--B_1,O--B_4. The factor four is the actual fine edge multiplicity of every complete pair block. No actual conductance is averaged or altered: this is a restriction of the bare phase potential.

Full stationarity makes every derivative of this restricted potential zero. In particular, the O derivative sums all A node currents, whose internal contributions cancel. Full PSD makes its restricted Hessian PSD. No separate classification of the A cycle is required.

In G, the vertices O and B_0 are adjacent twins: both have the same two external neighbors B_1,B_4. Their internal contact edge has branch zero and strict principal gap. The adjacent-twin calculation in Section 10 therefore forces B_0=O. It is valid even if their common external resultant vanishes. This equality has been derived without reflection, acute uniqueness or a supplied relative origin.

Use the common phase freedom to put O=B_0=0, and write

\[ a=B_1,\qquad b=B_4-2\pi. \]

The actual contact branches give -pi<a,b<pi. The retained B cycle has the three interior chain gaps

\[ q_1=B_2-B_1,\quad q_2=B_3-B_2,\quad q_3=B_4-B_3, \qquad q_1+q_2+q_3=2\pi+b-a. \]

All three gaps lie strictly between -pi and pi by the original branch declaration. The identity on their sum retains the complete marked class; it is not obtained by wrapping a sum after the calculation.

12. The uncontacted chain retains no negative-cosine alternative

Stationarity at B pairs two and three gives sin(q_1)=sin(q_2)=sin(q_3). Thus their cosine magnitudes are equal: write cos(q_j)=epsilon_j*c, with c>=0 and signs epsilon_j when c>0. Restrict the six-vertex Hessian further to the two coordinates B_2,B_3, holding its other coordinates fixed. Apart from the positive factor four, its matrix is

\[ T=\begin{pmatrix} \cos q_1+\cos q_2&-\cos q_2\\ -\cos q_2&\cos q_2+\cos q_3 \end{pmatrix}\succeq0. \]

If c>0 and the middle cosine is -c, the two nonnegative diagonals force both endpoint cosines to be +c. The resulting determinant is -c^2, a contradiction. If the middle cosine is +c but an endpoint cosine is -c, one diagonal is zero while the off-diagonal is nonzero, again contradicting PSD. The only remaining case for c>0 is that all three cosines are positive. Their strict principal domains and equal sines then make all three gaps equal in (-pi/2,pi/2).

The singular case c=0 is retained explicitly. Each principal gap is +pi/2 or -pi/2; equality of their sines again makes all three equal. Consequently every hypothetical PSD state satisfies

\[ q_1=q_2=q_3=q,\qquad -\frac\pi2\le q\le\frac\pi2,\qquad 3q=2\pi+b-a. \]

Because a,b are strict principal contact gaps, 2*pi+b-a>0. Thus 0<q<=pi/2, including the possible zero-cosine endpoint. It follows that

\[ a-b=2\pi-3q\ge\frac\pi2. \]

No inverse cosine or square-root division has removed a singular case.

13. Endpoint currents and curvature contradict the marked winding

The B pair-one and pair-four equations in the restricted graph are

\[ \sin q=2\sin a=-2\sin b. \]

The two terms at each endpoint come from its distinct edges to O and B_0, whose phases have been proved equal. Their Hessian diagonals must also be nonnegative:

\[ 2\cos a+\cos q\ge0,\qquad 2\cos b+\cos q\ge0. \]

Since 0<q<=pi/2, stationarity gives 0<sin(a)=-sin(b)<=1/2. On the strict principal interval this puts a in either (0,pi/6] or [5*pi/6,pi), and -b in the same two intervals. The distant alternative is impossible for either endpoint: its cosine is at most -sqrt(3)/2, so its Hessian diagonal is at most 1-sqrt(3)<0. Therefore

\[ 0<a\le\frac\pi6,\qquad 0<-b\le\frac\pi6,\qquad a-b\le\frac\pi3. \]

This contradicts the necessary lower bound a-b>=pi/2 from the same retained chain. Hence

\[ \boxed{\text{No stationary phase in the original open principal marked chamber has a positive-semidefinite full fine Hessian.}} \]

Every stationary state in that chamber, if any exists, has a strictly negative relative phase-curvature direction. The full storage therefore has no stationary local minimum there, including nonstrict or degenerate minima. The conclusion holds for every actual positive conductance and positive held capacity in the frozen contact family: those actual values remain in the complete law and the positive form block, while stationary phase and its curvature use bare support.

14. Scope beyond the acute obstruction

This result closes the nonacute stationary-minimum question in the same complete cycle class. It derives all reductions from full stationarity, fine curvature and strict principal branch admission. Even-degree zero resultants, zero chain cosines and singular PSD tests have been retained until the graph-specific contradiction excludes the whole candidate set. No odd-degree resultant estimate, acute convexity, reflection ansatz, optimizer outcome or numerical Hessian eigenvalue is a premise.

The result excludes a stationary local-storage minimum, rather than every possible stationary state in the enlarged chamber. It does not extend to principal-cut boundary points, where the strict twin branch intersection changes, or to other complete cycle classes. The synchronized equilibrium from Section 7 remains a valid minimum in its distinct zero-winding class.

Nor does the theorem show that the actual contacted trajectory changes winding, approaches any equilibrium or acquires a different identity. Nonstationary organization, finite retention, full-time confinement and source entry retain separate dynamical obligations. The absence of a stationary minimum is a geometric restriction on one candidate maintenance mechanism under the supplied law, not a failure of the nodal identity or an identification with fundamental physical behavior.

15. Fixed algebraic stationary saddle control

15.1. Prospective declaration

Before evaluating this additional static control, fix

\[ c=\frac{1+\sqrt{17}}8,\qquad t=\arccos c,\qquad q=2t,\qquad a=\pi-3t. \]

Use synchronized members in each pair, the original A winding A_k=2*pi*k/5, and the B lifted phases

\[ (B_0,B_1,B_2,B_3,B_4) =(0,a,a+q,a+2q,2\pi-a). \]

Retain the same 53-edge support and every original branch, positive actual conductance, positive held capacity and complete law. Put x_i=m constant at any admitted common form origin. The fixed predictions are full stationarity, strict principal-chamber membership and negative fine phase curvature K_ii=4*cos(q)<0 at either B pair-two member. The algebraic constant is not fitted or retuned. Any static numerical check uses 100-digit arithmetic and the already declared controls; no root finder, trajectory, source re-preparation or actual event is performed.

15.2. Exact stationary and negative-curvature verification

The fixed algebraic constant satisfies

\[ 4c^2-c-1=0,\qquad \frac12<c<\frac1{\sqrt2}, \qquad \frac\pi4<t<\frac\pi3. \]

Consequently 0<a<pi/4, pi/2<q<2*pi/3 and 2*a+3*q=2*pi. The B cycle's forward gaps are exactly (a,q,q,q,a). They are all strict principal gaps with winding one. The cross gaps are zero, a and -a; the internal gap is zero. Together with the old A twist these are precisely the original complete branch assignments, although the three B chain gaps are nonacute.

The algebraic relation for c gives an exact current identity:

\[ \sin q=2\sin t\cos t =2\sin t(4\cos^2t-1) =2\sin(3t)=2\sin a. \]

Thus the two receiver endpoints satisfy their full current balances. The three identical chain gaps cancel at B pairs two and three, and the symmetric +a,-a contacts cancel at B pair zero and A pair zero. All remaining A currents are those of the original twist and vanish. Fine member synchronization supplies the exact factors two in each node current. Hence every fine sine current is zero. Constant form also gives L_W*x=0 for every retained actual conductance assignment, so both complete stationary rows vanish.

At either member of B pair two, all four incident edge cosines equal cos q. Its fine Hessian diagonal is

\[ K_{ii}=4\cos q=4(2c^2-1)=2(c-1)<0. \]

The corresponding one-node variation, recentered by any common phase constant to retain the declared mean, has strictly negative curvature. This is an explicit stationary saddle in the retained principal chamber. It distinguishes the no-minimum theorem from a claim that no stationary configuration exists there. The construction does not describe a state acquired by the actual contacted source.

16. Instability of every stationary point under the actual weighted law

The negative phase direction also has a dynamical implication for this specific conservative two-row model. Keep the actual mobility M=diag(nu_i/(pi*d_i)), positive weighted Laplacian L_W and stationary phase Hessian K. The tangent equations are

\[ \dot\xi=-MK\eta,\qquad \dot\eta=ML_W\xi, \qquad \ddot\eta=-AK\eta,\qquad A=ML_WM. \]

Let rho_i=d_i/nu_i and let E be the tangent space rho^T*v=0 of either conserved weighted-mean leaf. The symmetric matrix A has kernel span(rho) and is strictly positive on E; denote that positive restriction by A_E. For the orthogonal projection P_E, put K_E=P_E*K|_E. The complete second-order phase equation on that leaf is eta_ddot=-A_E*K_E*eta, because A annihilates the omitted rho component.

A negative fine curvature direction from Section 13 can be shifted by a common constant to lie in E, without changing its quadratic form since K*1=0. Hence K_E has a negative direction. The product A_E*K_E is similar to the symmetric matrix A_E^(1/2)*K_E*A_E^(1/2), which also has a negative eigenvalue. It follows that -A_E*K_E has a strictly positive eigenvalue sigma^2 and a real nonzero eigenvector v in E.

The growing solution of the full first-order tangent system is recovered without dropping the form row:

\[ \eta(t)=e^{\sigma t}v,\qquad \xi(t)=-\frac{e^{\sigma t}}\sigma MKv. \]

The form perturbation also lies in E, because rho^T*M=(1/pi)*1^T and 1^T*K=0. Both tangent equations therefore hold on the actual two conserved-mean leaves. Every stationary state in the retained open principal chamber has a real growing tangent mode under the same actual weighted law. Smooth finite-dimensional linearized instability then gives instability of that equilibrium on its relative leaf.

This result supplies no uniform growth rate over the chamber or conductance family, no departure time for the contacted preparation, and no claim that the preparation approaches one of these stationary states. A growing mode near an equilibrium does not itself select a principal-cut crossing or exclude every nonstationary organized motion.

The static controls check the full fine restriction, every chain cosine-sign case, a zero-resultant counterexample outside the retained branch class, and the fixed saddle's negative direction on its exact weighted-mean leaf. The rounded saddle is checked with an independent captured-state residual and enclosing rate bounds; neither an exact represented root nor a bound on the entire interval width is inferred from the declared midpoint tolerance. These finite checks support the shared arithmetic and field consumers, not a numerical substitute for the analytical exclusions.

17. Frozen conservative confinement question

Fix each exact actual post-word source from the contact protocol, its actual 53-edge support, generated positive conductances, held capacities, original structural clock and conservative weighted-form/bare-phase law. Retain every marked branch from Section 1 and the open principal chamber from Section 8. No new event, forcing or loss is inserted.

Let rho_i=d_i/nu_i, m_rho=sum(rho*x_0)/sum(rho), Q_0=(1/2)*x_0^T*L_W*x_0, V_0=V(theta_0) and h=Q_0+V_0. The question is whether storage and the two conserved lifted means can isolate the actual preparation from every principal cut of this complete marked class. The analytical prediction is a continuous configuration path starting at that source and ending on a first principal boundary, with H=h, both source means unchanged, and every edge strictly inside its original branch until the endpoint. Such a path is not claimed to satisfy either evolution row or select the cut reached by the actual source.

Use the full fine state, fixed-mean compactness, the no-minimum result and strict sublevel components. Preserve the actual positive form storage as a coordinate of H; its variation along a comparison path does not add an event reserve or an external reservoir. Prove source-specific positivity before dividing by Q_0. No optimizer, root finder, trajectory, frozen producer or new precision campaign belongs to this declaration.

Before static evaluation, fix these independent algebra controls. Reuse the existing auxiliary positive conductance list, capacities and 100-digit arithmetic; those weights are not regenerated UM outputs. Check the exact post-AL nominal forms m=5/16, increment b=1/1024 on A pair zero only, the component mean difference, the support diameter and weighted centering. For radial form lifting use the original doubled twists and the phase path that decreases B pair one by s and increases B pair four by s, recentered to its original weighted phase mean, at s=0,1/128,1/64. Check storage and means with both this nonconstant nominal form and a separately declared uniform-form comparison; the latter is not an actual post-word source.

For the separate cycle-information control, rigidly rotate all B phases by s=0,pi/2,3*pi/5,4*pi/5, with common weighted recentering. Predict unchanged old constituent cycle gaps and a first cross-edge cut at 3*pi/5; do not attribute the energy-path theorem to this rigid rotation. Exact turn and matrix checks precede represented shared-reader comparisons, whose residual, enclosure and existing 2^-46 midpoint tolerance remain distinct.

18. The actual source has strictly positive form storage

Fix one exact source from Section 17, including its actual W,nu and post-word forms and lifts (x_0,theta_0). All following means and paths refer to this one source; differing sources retain their own invariant values and generated conductances. Put

\[ Q(x)=\frac12x^TL_Wx,\quad V(\theta)=V_U(\theta),\quad H=Q+V,\qquad m_\rho=\frac{\rho^Tx_0}{\rho^T1},\quad \theta_\rho=\frac{\rho^T\theta_0}{\rho^T1},\quad \rho_i=\frac{d_i}{\nu_i}>0. \]

These are the actual weighted form mean and the declared lifted phase mean, not circular or separate constituent means. Both remain conserved under the selected complete law because rho^T*M=(1/pi)*1^T and both the sine current and weighted form gradient sum to zero.

Before the word, each constituent's arithmetic form mean is the same m. AL increases only the two selected A forms by b=1/1024; UM and SHA leave all forms unchanged. Thus the actual endpoint has

\[ \overline x_A=m+\frac b5,\qquad \overline x_B=m. \]

These arithmetic means certify an initial difference only; they are not claimed to be conserved after contact. The difference proves that x_0 is nonconstant. Since the actual positive weighted graph is connected, Q_0=Q(x_0)>0. This remains true for the synchronized-member control and does not require a nonzero selected internal split.

A uniform lower bound follows directly from the admitted source geometry. Some A node i has x_i>=m+b/5, and some B node j has x_j<=m. A path from i to the selected A pair has at most two edges. A contact edge followed, when necessary, by one B edge reaches j, so a joined path has length ell<=4. Every actual conductance is greater than 3/4. Cauchy's inequality on that path gives

\[ Q_0\ge\frac12\sum_{e\in\text{path}}W_e(\Delta_ex_0)^2 >\frac{3}{8\ell}(x_i-x_j)^2 \ge\frac{3b^2}{800} =\frac3{838860800}>0. \]

The strict inequality uses at least one nonzero path difference. It is a bound on the existing positive form term in H, not an additional reservoir or an event-work allowance. Write V_0=V(theta_0) and h=Q_0+V_0.

19. Compactness on the two fixed-mean leaves

Let A_theta be the affine phase leaf rho^T*theta=rho^T*theta_0, and write Omega=P intersect A_theta for the original strict principal chamber on that leaf. The actual source lies in Omega by Section 2. Its closed chamber replaces all strict gap inequalities by -pi<=delta_e<=pi. It is the closure of Omega: interpolation from any closed-chamber point to the strict source makes all edge inequalities strict before the closed endpoint.

This phase closure is compact. The joined graph has diameter at most four: any A node is within two edges of either selected A member, any B node is within two edges of that member, and the within-component distances are at most two. Along each edge, the retained branches b_e in {-1,0,1} give

\[ |\theta_j-\theta_i|\le\pi+2\pi|b_e|\le3\pi. \]

Consequently the full lifted phase range is at most 12*pi. Positivity of rho puts its mean inside that range, so every coordinate obeys |theta_i-theta_rho|<=12*pi. The closed edge inequalities and mean constraint make this bounded set closed.

The full closed storage sublevels on both source mean leaves are compact as well. For H<=E, nonnegativity of V gives Q<=E. With w_min=min_e W_e>0, Cauchy's inequality along a path of at most four edges yields

\[ |x_i-x_j|^2\le\frac{8E}{w_{\min}}. \]

The fixed positive weighted mean then bounds every |x_i-m_rho|<=sqrt(8*E/w_min). Together with the phase bounds, continuity of H proves compactness of each nonempty closed sublevel. This argument retains the actual conductances and means; it does not substitute an arithmetic mean or a unit-weight Laplacian.

There is no local minimum of V in Omega. Indeed a constrained critical point there has grad V=lambda*rho; translation invariance gives 1^T*grad V=0, hence lambda=0. At a local minimum its relative phase Hessian would be PSD. Since K*1=0, every phase test direction can be shifted by a common constant onto the weighted-mean tangent leaf without changing its curvature. This would make the full fine Hessian PSD, contradicting Section 13. Thus the no-minimum result applies on this actual phase leaf, including all possible singular candidates.

20. A strict phase sublevel must reach a principal boundary

Choose the source-specific threshold

\[ E_* = V_0+\frac{Q_0}{2}<h, \]

and let D be the connected component containing theta_0 of the strict phase sublevel {theta in Omega: V(theta)<E_*}. It is nonempty because Q_0>0. It is open in the finite-dimensional affine phase leaf. Its closure is compact by Section 19, and every closure point has V<=E_* by continuity.

Suppose that this closure avoided all principal boundary faces. It would then lie inside Omega, and V would attain a minimum at some point theta_min of the closure. Its value is at most V_0<E_*. A sufficiently small ball about that point lies in Omega and in {V<E_*}. Since the point is in the closure of D, the ball intersects D. The ball is connected, so it belongs to the same component D, including its center. This makes theta_min an interior local minimum of V on the phase leaf, contrary to Section 19.

Therefore there exists a point theta_b satisfying

\[ \theta_b\in\overline D\cap\partial\Omega,\qquad V(\theta_b)\le E_*<h. \]

At least one original edge gap equals +pi or -pi there; all other closed gap constraints still hold. The statement identifies neither that edge nor a unique boundary point.

This component argument supplies more than an arbitrary small excess-energy comparison. The actual positive form term leaves a strict margin between the boundary potential and the conserved total h. That margin will give an exact-energy path to an attained boundary endpoint.

21. A comparison path on the exact invariant shell

By continuity at theta_b, choose a convex ball in A_theta about that point on which

\[ V<\frac{E_*+h}{2}<h. \]

Choose y in D inside this ball, which is possible because theta_b in closure(D). An open connected subset of a finite-dimensional affine space is polygonally connected: the points reachable from a fixed one by finitely many segments inside the open set form a relatively open and relatively closed subset of each connected component. Hence a finite polygonal phase path from theta_0 to y lies entirely in D.

Append the straight segment from y to theta_b. It stays inside the chosen convex ball and inside the closed phase chamber. For every point before its endpoint, each edge gap is a strict convex combination of a strict interior gap at y and a closed gap at theta_b, so all edge gaps remain strict principal gaps. The appended endpoint is the first principal boundary reached by the whole path.

Write this finite piecewise-linear phase path as theta(s), 0<=s<=1. It retains the source phase mean and satisfies throughout

\[ V(\theta(s))<h,\qquad h-V(\theta(s))>\frac{Q_0}{4}. \]

The first polygonal portion even has h-V>Q_0/2; the weaker displayed bound also covers the final segment and its boundary endpoint. Now use the actual initial relative form direction to define

\[ \boxed{ x(s)=m_\rho1+ \sqrt{\frac{h-V(\theta(s))}{Q_0}}\,(x_0-m_\rho1). } \]

The square root is well-defined and strictly greater than 1/2 on the whole path. Since L_W*1=0, the actual quadratic form scales exactly:

\[ \rho^Tx(s)=\rho^Tx_0,\qquad Q(x(s))=h-V(\theta(s)),\qquad H(x(s),\theta(s))=h. \]

At s=0 the multiplier is one, so this path begins at the actual full source (x_0,theta_0). It ends at a principal-cut configuration, preserves both source means and total storage exactly, and keeps every original edge gap strictly within its branch before the endpoint. The forms and phases remain continuous and piecewise smooth; all weights, capacities and support stay at their actual source values. No pair synchronization or changing weighted boundary association is imposed along the path.

The parameter s is only a comparison-path parameter. It has no declared conversion to the original structural time. In particular, the radial form formula is not a pressure reconstruction or a solution of either nodal row. It redistributes the two existing terms of H among configurations without asserting that the complete law performs that redistribution along this path.

22. Geometric continuation across a cut and the barrier obstruction

The strict potential margin also permits a short continuation of the final straight phase segment beyond theta_b. If an edge gap equals +pi at the endpoint, its slope along the segment is positive because the starting gap at y was strictly smaller; the analogous slope is negative at a -pi face. Thus every face reached there is crossed transversely by a sufficiently short extension. The continuous sine potential stays below h in the same small ball, so the radial form lift continues with the same storage and means. This is an additional configuration comparison beyond the original chamber, not a prediction of the actual vector field.

In particular, the connected component of the actual invariant shell H=h on both mean leaves that contains the source reaches a principal cut and has configurations across it. Conservation of precisely these three quantities cannot separate that source component from every cut. There is no strictly positive storage barrier protecting the whole original complete principal chamber for any source in the frozen family.

This excludes a trapping certificate based solely on separation of that storage-and-means level component from the cuts. It does not exclude an additional dynamical invariant, an invariant subset of the shell, or a nonstationary organization maintained by another proved mechanism. The smooth sine law itself remains defined at a principal cut; the cut is a boundary of the retained phase identity, not an inserted support event or a singularity of the complete field.

23. Complete marked cuts and constituent winding are different readouts

The theorem does not select a cut or say that either original constituent winding changes. A separate exact kinematic comparison makes this distinction explicit. Start from the original doubled twists, rigidly rotate all B phases by s, and subtract the common phase increment s*sum_(i in B)(rho_i)/sum_i(rho_i) from every node to retain the same weighted phase mean. All old A and B edge gaps and the internal A pair gap are unchanged.

The contact gaps at B pairs zero, one and four are respectively s,alpha+s,-alpha+s, where alpha=2*pi/5. On 0<=s<3*pi/5 they remain strict principal gaps. At s=3*pi/5 the four contacts to B pair one reach +pi, while every old constituent cycle gap is still alpha. Both original winding observations remain one. Passing that cut changes the branch inventory of those cross edges; it does not change the old constituent cycle gaps.

This rigid comparison is not the exact-shell path constructed above. With the nominal forms held fixed, its contact potential changes by

\[ \Delta V_C(s)=4(1+2\cos\alpha)(1-\cos s),\qquad \Delta V_C(3\pi/5)=4+2\sqrt5>0. \]

The common recentering changes no edge difference and cannot remove that cost. These observations establish that a complete marked cut need not be a constituent-winding cut; they do not identify which kind is reachable along the invariant-shell path or which kind the actual flow might reach.

24. Consequence for maintenance under the supplied complete law

Every exact contacted source in the frozen family has a same-storage, same-two-means configuration path to a first principal cut, with its actual support, conductances and capacities retained. This rules out using the existing total-storage barrier mechanism alone to certify perpetual maintenance of the original complete cycle class for that source family. The result needed no trajectory, candidate optimization or increase in response precision.

Neither this path nor the absence of stationary minima proves that the actual trajectory crosses a cut, changes a constituent winding, dissolves, or acquires a new common identity. The analytical finite-retention and old-receiver-tube exit results remain valid with their original intervals. No selected boundary point, path-length bound, structural-time deadline, autonomous event selector or physical identification is supplied here. Actual confinement or passage needs information about the complete dynamics beyond these three conserved scalar values.

The static controls check exact weighted centering, the support diameter, positive restricted form curvature and radial storage scaling. A fixed short phase deformation tests the constant-energy lifting identities, including a separately labeled uniform-form comparison outside the actual post-word family. The rigid relative-rotation control checks old and mixed cycle information and treats the mixed winding as unavailable at its cut. These are algebraic and reader controls, not evaluations of the unknown source trajectory or constructions of the unspecified global path from Section 21.