Principal-sector minima: exact reduction and full exclusion¶
The proposed UM support's full original principal chamber contains no stationary storage minimum. This extends the closed-acute obstruction through an exact reduction from twenty relative fine phases to four real coordinates and complete analytical sector exclusions. The proof retains all 53 edge branches and the singular cases needed to justify the reduction. The source-specific corollary also shows that its storage and two means alone do not separate it from principal cuts.
This is a restriction of the existing phase potential for stationary classification. It is not a new pressure law, a closed temporal model, a selected equilibrium or a trajectory. The research program remains the sole active queue.
1. Actual law and principal domain¶
Keep the same re-admitted retained THOL input and hypothetical default UM reset, with 21 nodes, 53 edges, positive predicted conductances and capacities, structural clock and zero forcing. The complete law and branch inventory remain unchanged. Set
The integers k_ij are precisely the eight one-turn branches and 45 zero
branches already admitted. P is an open convex chamber of real lifts,
modulo common translation. It is larger than the acute chamber; its admission
does not prove that the hypothetical source stays there.
Write V=sum_U(1-cos delta) and K=Hess V=L_cos(delta). The actual
storage is H=Q_W(x)+V(theta). At a stationary state, connected positive
form conductances imply common form, and the phase equations are grad V=0.
For the actual positive weights rho_i=d_i/nu_i, a constrained critical
point obeys grad V=lambda*rho; translation invariance forces lambda=0.
For any phase direction z, replacing it by
z-1*(rho^T z)/(rho^T 1) preserves z^T K z because K*1=0.
Thus full K being positive semidefinite is necessary for a local minimum
on the actual weighted-mean leaf. Arithmetic-mean centering is not substituted.
Positive semidefiniteness alone is not a minimum theorem. A negative edge cosine alone is not a negative direction of the full Hessian. Both distinctions are retained below and in the independent controls.
2. Fine twins, leaf and zero resultants¶
The exact bare support has adjacent twins (0,1) and nonadjacent twins
(4,5), (6,7), (8,9), (10,11), (12,13),
(14,15), (16,17), (18,19).
Nodes 2 and 3 have different external neighbors and are not twins. The previous contact theorem's graph-specific receiver identification therefore does not apply.
At a stationary PSD candidate in P, each listed pair has equal lifted phases. The general twin argument applies because these pairs have identical outward branches to every common neighbor. For a nonzero common neighbor resultant, stationarity and a nonnegative diagonal select alignment with it. If that resultant is zero, PSD forces every entry of the corresponding zero-diagonal row to vanish. Unequal twins would be antipodal; their same strict principal branches then force every common neighbor to the same midpoint phase, contradicting a zero resultant. Equal twins with zero resultant are not discarded at this stage.
For the adjacent pair, a split Theta+epsilon,Theta-epsilon has restricted
potential -2*C*cos epsilon-cos(2*epsilon)+constant. A nonzero stationary
split inside the internal principal cut has second derivative
-4*sin(epsilon)^2<0, so PSD excludes it even when the external resultant
vanishes.
The child 20 is a leaf. Stationarity gives sin(theta_20-theta_2)=0;
its strict zero-branch principal gap forces equality. These restrictions
are consequences at candidate minima, not an invariant fine dynamical
pair reduction. Nonuniform capacities and conductances still govern motion.
3. Return chains and a singular case excluded by higher order¶
Use the common pair phases O=A0, R=A2, B0, B1, B2, with
P_node=theta_2 and Q_node=theta_3 distinct. Each three-block return
chain has two uncontacted interior pairs:
A2 -> A3 -> A4 -> A0,
B2 -> B3 -> B4 -> B0.
Stationarity equates the three edge sines. The principal two-interior-node Hessian restriction, apart from factor four, is
Its diagonals and determinant exclude every nonzero cosine sign pattern except all positive. If all cosines are zero, equal sines still force all three principal gaps equal. The full marked branches and the two-edge paths through the parent imply
The zero-cosine endpoint is PSD-compatible but cannot be a local minimum.
At t=pi/2, move the two common interior pair phases by u and v, keeping
endpoints and every other phase fixed. Only the three complete blocks change:
With v=2u, this is 8*sin u*(1-cos u), which has both signs arbitrarily
near zero. Thus every actual local minimum satisfies strictly
0<t_A,t_B<pi/2. The argument retains the singular Hessian and uses a
cubic variation; it does not invent a negative quadratic direction there.
4. Six independent core coordinates and exact curvature bookkeeping¶
Fix O=0 and use six coordinates (p,q,r,b,v,c) for the parent, node 3,
A2, B0, B1 and B2. Put t_A=(2*pi-r)/3 and
t_B=(2*pi+b-c)/3. Reconstruct the fine lifts as
(0,0, p,q, r,r, r+t_A,r+t_A, r+2*t_A,r+2*t_A,
b,b, v,v, c,c, c+t_B,c+t_B, c+2*t_B,c+2*t_B, p).
Check all original principal edge inequalities after reconstruction. On this affine chain restriction the full bare potential is
Each coefficient is the actual number of bare fine edges. The synchronized internal A0 edge and child edge contribute zero; the other multiplicities sum to 51. Actual conductances are not averaged into this phase potential.
Full fine stationarity on this restriction is equivalent to grad V_6=0:
the interior chain and child currents cancel identically, and the remaining
fine currents are recovered from the six derivatives and their zero sum.
At a stationary candidate with nonnegative chain cosines, the full fine
PSD test is equivalent to the six-core PSD test. To retain the singular case,
use an exact quadratic decomposition instead of dividing by a zero cosine.
For one chain of common cosine h and endpoint increments x0,x3, write
z1,z2 for interior deviations from linear interpolation. Its form is
Add the positive child discrepancy square and the fine twin antisymmetric
blocks. Complete pair blocks remove cross terms. Each chain-interior split
has eigenvalue 4h; endpoint splits equal half the corresponding ungrounded
core diagonal plus 4h/3, with an additional 2 for the adjacent A0 pair.
The B1 split is half its core diagonal. These identities show that core PSD
implies nonnegative omitted blocks; restriction gives the converse. Common
phase translation can then be fixed without changing PSD. This equivalence
is a stationary curvature statement, not complete nonlinear dynamics closure.
5. Eliminate the two middle nodes without losing zero-resultant cases¶
Node 3 and either member of B1 have odd degree five. At stationary PSD,
a zero resultant would give a zero Hessian diagonal and hence a zero full
row. All five incident cosines would vanish, leaving five sine values
+1 or -1 whose sum cannot be zero. Thus their resultants are strictly
nonzero at any local minimum.
Now fix the parent's phase to zero and describe the two endpoint pairs by
Define
The aligned middle lifts are Q_node=Arg Z_A and B1=Arg Z_B, on the
unique branches admitted by the parent and endpoint principal gaps.
The necessary regular domain is
Also require |Arg Z_A|,|Arg Z_B|<pi,
|Arg Z_A-(m-a)|,|Arg Z_A-(m+a)|<pi, and the two corresponding B endpoint
inequalities. Retain the original full fine branch check as the authoritative
admission. These conditions cannot be replaced by a midpoint angle or just
two winding numbers. A zero or unresolved resultant gives no regular chart.
The remaining fine coordinates follow from the equal return-chain gaps, the twins and the child. A common translation converts this parent-zero chart to the core's O-zero chart; it changes no edge branch. Neither chart substitutes an arithmetic phase mean for the actual conserved mean.
6. Four-coordinate potential and what it decides¶
With t_A=(2*pi-2*a)/3, t_B=(2*pi-2*b)/3, the exactly restricted fine
potential is V=51+F, where
The factors 1 and 2 on the resultants retain the unequal fine multiplicities. They are a material difference from the previous contact model.
Every full principal-sector local minimum maps to a local minimum of F
inside this admitted domain. Conversely, a local minimum of F reconstructs
a full relative phase minimum. The reason is local elimination of positive
directions, not an assumed symmetry of every trajectory. The middle-node
curvatures are R_A and 2*R_B; both chain interior forms are positive.
A minimum of F gives PSD reconstructed core curvature. The omitted twin
blocks are then strictly positive: chain endpoints inherit positive chain
terms, B1 inherits R_B, the adjacent A0 split has its extra 2, and the
child discrepancy has cosine one. Symmetry fixes the transverse minimum
at equal members. These positive directions admit unique local minimizing
continuations, leaving precisely F as the local-minimum question.
A certified critical point with positive-definite four-dimensional Hessian would therefore give a strict relative storage minimum for the actual positive-weight complete law. Common form is fixed by the actual form mean; common phase is translated to the actual phase mean. This would establish local conservative maintenance, not attraction or entry of the proposed source. If the four-dimensional Hessian is singular, PSD remains only a necessary condition and higher-order terms must be examined.
7. Executable interval calculations and their evidence boundary¶
The candidate owner
rebuilds the source/reset/chamber primitives for every public call.
evaluate_successor_core returns the reconstructed fine state, full branch
margins, fine currents and Hessian, and their six-coordinate restriction.
evaluate_successor_reduction additionally admits the nonzero resultants,
reconstructs the aligned middle nodes and returns the full potential
V=51+F, with the gradient and Hessian shared by V and F.
It uses whole interval enclosures; unresolved cuts, domains or denominators
reject. It does not install edges, execute a word or declare a root from a
small residual.
For the four-variable derivatives, eliminate the two middle coordinates of
the core Hessian by their strictly positive diagonal block, then pull back
the four endpoint coordinates. The external core order is (p,r,b,c) and
its linear derivative with respect to (a,m,b,n) is
Thus grad F=L^T*g_external and
Hess F=L^T*(K_external-K_cross*diag(1/R_A,1/(2*R_B))*K_cross^T)*L.
The aligned middle gradients vanish identically; no approximate response
gradient is substituted into this identity.
The full classification follows from the analytical arguments in Sections 8-12, including singular cases and approaches to the open domain boundary. The evaluator remains an observation API: failure of a solver, a finite set of candidates or an interval box to certify a root would not prove absence. The acute theorem is reused only after its actual hypotheses are established.
Verification uses exact reduction controls, candidate admission controls and independent fine-state comparisons. These include the singular chain's cubic variation and independent derivatives of F. A synchronized minimum in a different zero-winding class prevents a scope expansion to all graph states. No UM, trajectory or frozen producer is replayed, and the retained evidence remains unchanged.
8. Exclusion when both endpoint half-spans have positive cosine¶
The four-coordinate domain admits a useful additional restriction before stationarity is considered:
To prove the upper bound for either motif, write its half-span and midpoint
as u,v, and its aligned middle phase as Q. The two strict middle-to-endpoint
branches imply |Q-v|<pi-u. If u>=pi/2, then cos(Q-v)>0. Multiplying
Z=1+4*cos(u)*exp(i*v) by exp(-i*v) gives
Consequently cos u>-1/4. For u<pi/2 the bound already holds. This is a
branch and alignment restriction, not evidence that the whole domain is acute.
Conditional exclusion. There is no local minimum in the full admitted
principal chamber whose reduced coordinates satisfy
pi/4<a,b<pi/2. This excludes a region containing nonacute individual edges;
it does not assume those edges acute at the start of the proof.
Put d=m-a-n+b, c_A=cos a and c_B=cos b. The two midpoint stationarity
equations obtained from the actual potential are
All prefactors are positive in the sector under consideration, and the
principal branches give |m|,|n|,|d|<pi. Thus either m=n=d=0 and a=b,
or the midpoint signs are opposite. More precisely,
These conclusions use the sign of the admitted lifted d, not an arbitrary wrapped representative. Also, each aligned middle argument lies strictly between its nonzero midpoint and zero: its resultant is a positive combination of the two rays at those angles, separated by less than pi. Each of the three cases is excluded below.
8.1. Larger A half-span¶
Suppose a>b. Let z=a-pi/4, t=t_A=pi/2-2*z/3, h=cos t,
and q=Q_node/2. For fixed r=4*cos a, differentiation of the aligned
middle argument gives
The first inequality follows from cos(m)^2<=1; the second uses
r<2*sqrt(2) and cos m>1/sqrt(2). The argument is zero at m=0, hence
0<q<2*z/5. Since 0<2*z/3<pi/6, concavity of sine above its chord
on [0,pi/6] gives h=sin(2*z/3)>=2*z/pi. It follows that
Write y=a+m-q. The fine A2 current equation, with its actual two edges
to the parent and two to node 3, is
Here 0<y<pi. If y>=pi/2, the diagonal for the A2 endpoint after
eliminating the positive return-chain interiors is
because the square root exceeds h/sqrt(2)>h/3. This contradicts necessary
PSD curvature. Therefore y<pi/2. The stronger inequality
sin(q)^2<1-sin t then gives sin y<cos q, and hence
y+q=a+m<pi/2. Both upper A gaps, a+m and a+m-Q_node, are acute.
Every other original edge is also acute: the lower A gaps are bounded by a;
the B endpoint and middle gaps are bounded by b+|n|<a; the middle-to-parent
gaps are smaller than the corresponding midpoint magnitudes; the cross gap
has |d|<a-b; the return chains are already strictly acute. The internal A0
and child gaps vanish. Thus this hypothetical stationary PSD candidate lies
in the full acute chamber, where the existing
closed-acute obstruction
excludes stationarity.
8.2. Larger B half-span¶
Suppose b>a. Positive cosine places each aligned middle phase strictly
between its midpoint and zero. In A, the two upper gaps therefore lie in
(0,a): |m|<b-a<a and m<Q_node<0. Their current equation forces
sin t_A<sin a. Both angles lie in (0,pi/2), so
In B, 0<B1<n and n<b-a. Its two upper gaps, b+n and b+n-B1,
lie in (b,2*b-a), which is contained in (a,pi-a). Each sine therefore
exceeds sin a>sin(2*pi/5)>2/3. But the B2 current equation, retaining
two parent edges and four middle edges, requires
Its left side is strictly greater than 2, a contradiction. This case does not require a curvature test.
8.3. Equal half-spans and remaining scope¶
If a=b, midpoint stationarity gives m=n=0; positive resultants then
give both middle phases zero. The A2 and B2 equations respectively require
2*sin a=2*sin t_A and 3*sin a=2*sin t_A. They are incompatible because
sin a>0.
The three cases exclude every minimum with both half-spans strictly below
pi/2. The boundaries and negative-cosine sectors require the separate
arguments below. The static sector controls
check the fine current multiplicities, chain-eliminated curvature, argument
derivative bound and interval margins used here. These controls run neither
an event nor a trajectory, and finite probes are not the exclusion proof.
9. Wide endpoints, zero half-span cosines and the double-wide sector¶
Call a half-span wide when it is at least pi/2. By the branch bound in
Section 8 its aligned middle resultant has the form 1-r*exp(i*v), with
0<=r<1. Its real part is positive and its argument has the opposite sign
to v, or is zero when r=0. Thus a nonnegative midpoint makes both upper
endpoint gaps lie in [pi/2,pi).
For A, write these two gaps as alpha,beta, and put s=sin t_A,
h=cos t_A>0. The endpoint equation and the chain-eliminated endpoint
diagonal are
Here 1/2<s<1. For u=sin alpha, v=sin beta in [0,1] with
u+v=2s>1, concavity of the square root on this line gives
The minimum occurs at either endpoint u=1 or v=1. The final strict
inequality reduces to (1-s)*(8*s-1)>0. Since both cosines are nonpositive,
D_A<0, contradicting necessary PSD curvature. A wide A half-span therefore
requires m<0 at a candidate minimum.
For B, the corresponding equations retain the different middle multiplicity:
The first equation gives sin beta<=sin t_B. Because
beta in [pi/2,pi) and 0<t_B<pi/2, this implies
cos beta<=-cos t_B. Together with cos alpha<=0, it gives D_B<0.
Thus a wide B half-span requires n<0. These diagonals follow by eliminating
only the positive return-chain interiors while retaining the other endpoints;
they are necessary restrictions of full PSD, not independent edge tests.
If both half-spans are strictly wide, cos a,cos b<0 and m,n<0 make
both terms in
strictly positive. Hence that sector is impossible. If exactly one half-span
equals pi/2, its angular equation forces d=0. A strictly acute other
half-span then has midpoint zero, and d=0 makes the wide midpoint positive,
contradicting the endpoint restriction just proved. A strictly wide other
half-span has a strictly positive angular term, also impossible.
Finally, if a=b=pi/2, then d=0 gives m=n. Both aligned middle phases
are zero. The two endpoint equations would require
2*cos m=sqrt(3) and 3*cos m=sqrt(3) simultaneously. Thus the zero-cosine
boundaries also contain no candidate minimum. Only the two strict mixed
sectors remain after Sections 8 and 9.
10. Exclusion of the positive-A, negative-B mixed sector¶
Assume pi/4<a<pi/2<b<acos(-1/4). Section 9 gives n<0; the angular
equations then give m<0 and d>0. Write m=-u, n=-v, with u,v>0,
and write the A middle phase as Q_node=-z_A, where 0<z_A<u. Its upper
endpoint equation is
If u>=a, the first sine is nonpositive and the second at most one,
contradicting 2*sin t_A>sqrt(3)>1. Consequently 0<u<a<pi/2, and both
upper angles are acute. Their average gamma=a-u+z_A/2 satisfies
sin gamma*cos(z_A/2)=sin t_A, so gamma>=t_A. It follows that
a>t_A and a>2*pi/5.
10.1. A bounds the left contact and its current¶
Put a=pi/2-epsilon, so 0<epsilon<pi/10, and
D=a-t_A=pi/6-5*epsilon/3. For k=4*cos a=4*sin epsilon and
0<u<pi/2,
Indeed the denominator minus k+cos u is
1+k^2-k+(2*k-1)*cos u>0: use cos u<=1 when k<=1/2, and
cos u>=0 when k>=1/2. Integration gives z_A<=k*u; hence
The denominator is positive. The A left-endpoint magnitude A_L=a+u
satisfies A_L<=2*pi/3, because
The exact resultant identity sin(u-z_A)=sin u/R_A also gives the contact
current
The last inequality is equivalent to
20*epsilon^2-(2*pi+2)*epsilon+1>0. Its leading coefficient is positive
and its discriminant (2*pi+2)^2-80 is negative; pi<10/3 already suffices.
Let L=b+v be the B left-endpoint magnitude. Its parent branch gives L<pi,
and d=L-A_L<pi-a<3*pi/5. Since d>0 and sin d<1/6, the interval
[pi/2,3*pi/5] is impossible. Thus 0<d<pi/2. The certified scalar bound
then gives d<34/15-2*pi/3 and therefore L<34/15. The bound comes
from stationarity and the retained branches, without an observed trajectory.
10.2. The B endpoint cannot balance this contact¶
Write z=B1>0, R=b-v, beta=R-z, r=-4*cos b>0, and
t=(2*pi-2*b)/3. The original middle branches give
The exact middle and upper endpoint currents are
Set L_0=34/15 and split at b_0=pi/2+1/8.
For pi/2<b<=b_0, put delta=b-pi/2. Then 0<delta<=1/8,
r=4*sin delta<1 and z<=arcsin(4*sin delta). As R=2*b-L,
The function 2*delta-arcsin(4*sin delta) decreases on [0,1/8]:
its derivative is 2-4*cos(delta)/sqrt(1-16*sin(delta)^2)<0, using
delta<=1/8<pi/3. The certified inequality
sin(21/40)>4*sin(1/8) therefore gives beta>pi-61/24.
Since R,beta<b_0<min(L_0,61/24), the symmetric sine bounds yield
This contradicts the B endpoint equation throughout this interval.
For b>=b_0, the original middle equation can also be written
On 0<z<pi-L, the right side is a strictly concave function of z:
both sine terms are positive. Its endpoint values are 2*sin L and
sin L, so it exceeds sin L. Since 0<L-b<L_0-b<pi/2,
Define G(b)=sin b*cos(L_0-b)-sin((2*pi-2*b)/3). Its derivative is
because 0<2*b-L_0<7*pi/6-L_0<pi/2 and 0<t<pi/2. Its lower endpoint
satisfies the certified scalar inequality
Thus the B current again exceeds 2*sin t, a contradiction. Together the
two intervals exclude the entire positive-A, negative-B mixed sector.
All scalar inequalities above are enclosed with the shared rational outward
interval kernel in the static sector controls; rounded decimal estimates
are not their premises.
11. Exclusion of the negative-A, positive-B mixed sector¶
Assume pi/2<a<acos(-1/4) and pi/4<b<pi/2. Section 9 gives m<0.
Write a=pi/2+e, m=-u, r=-4*cos a=4*sin e, and denote the A middle
phase and resultant magnitude by Q and R_A. Its middle-to-endpoint branches
put the argument of exp(i*u)-r inside (-(pi-a),pi-a), an acute interval.
Thus cos u>r, and the parent branch also gives e+u<pi/2. In particular,
The A angular equation gives sin d=-T, where
T=cos a*sin m*(1+1/R_A)>0; the B angular equation then gives n<0.
Since sin Q=r*sin u/R_A,
Set delta=arcsin T, so 0<delta<Q<pi/2. The negative principal contact
gap is either d=-delta or d=-pi+delta. The second case is impossible:
the middle branch Q-O<pi gives O>Q-pi, hence
B0=O-d>Q-delta>0, whereas B0=n-b<0. Therefore
d=-delta and B0=O+delta.
In fact B0<-pi/2. Since R_A>=sin u,
T<=sin e*(1+sin u), while
Here cos e>sqrt(15)/4, sin e<1/4, and tan(u/2)<1. Since
e+u<pi/2, this proves delta<e+u and the claimed B0 bound.
11.1. Upper current and lower cosine restrictions¶
Put q=Q/2, w=a-u-q, t=t_A, s=sin t, and h=cos t. The upper
A endpoint equation is
Necessarily w<pi/2. Otherwise u+q<=e, so q<e and
sin w>=sin a=cos e; their product exceeds cos(e)^2>15/16, whereas
s<=sqrt(3)/2<15/16. The principal branches and the row give w>0.
Also sin(w-q)=2s-sin(a-u)>0, because t>pi/6. With w<pi/2,
Define X=-cos O>0 and Y=-cos(O-Q)>0. The exact projection of the
middle resultant onto its aligned phase is
The parent contribution is cos Q, not one. Moreover,
cos Q-R_A=r*(cos u-r)/R_A>0; consequently
11.2. A full fine direction moving both lower endpoints¶
Move O and B0 by the same infinitesimal amount, hold the parent, A2 and
B2 fixed, minimize each middle-node quadratic fiber, and use linear
interpolation along each return chain. The cross-contact variation vanishes
exactly. The resulting full fine-phase second variation is D_A+D_B, with
These are the actual Schur factors of the middle curvatures R_A and
2*R_B. This direction gives the same quadratic form after recentering to
the actual weighted phase mean. The fine twin and child coordinates remain
admitted; no collective temporal law is assumed.
Because B0<-pi/2, its first cosine is negative. Completing the square gives
4*z-8*z^2/R_B<=R_B/2, and the positive-B resultant satisfies
R_B<=1+4*cos b. Therefore
The middle expression decreases on pi/4<b<pi/2, since its derivative
is -2*sin b+(8/9)*sin t_B<0. It remains to prove D_A<-2.
For 0<e<=1/5, put C=cos(q)^2. The resultant geometry gives
sin Q<=r, hence
The second inequality follows from r^2<=16/25<3/4 and
sqrt(1-x)>=1-(2/3)*x on [0,3/4]. Concavity of sine gives
s<=sqrt(3)/2-e/3. Thus
where the strict comparison uses
1+1/sqrt(3)>3/2>58/45>=(58/9)*e. The exact lower cosine sum is
Elementary Taylor bounds give
For the latter, use sqrt(3)/2>6/7, cos x>=1-x^2/2 and
sin x<=x: (6/7)*(1-2/225)-1/15=137/175. Their product exceeds
(3/2)*e, because 20002/13125>3/2. Together with
cos(2e)>=1-2*e^2, this gives
As R_A<1 and Y>h>=1/2, it follows that
D_A<-2+(4/3)*h-4*h^2<=-7/3<-2.
For e>=1/5, the Taylor lower bound
r>=4*sin(1/5)>=4*(1/5-(1/5)^3/6)>3/4 gives
R_A<sqrt(7)/4<2/3. Since e>1/6,
Here cos(1/9)>=161/162, sin(1/9)>=485/4374 and sqrt(3)>5/3
justify every bound. Using just X>0 and Y>h,
Both ranges give D_A+D_B<0, a contradiction to full fine PSD. This
excludes the entire negative-A, positive-B mixed sector.
12. Full principal-chamber minimum exclusion and its scope¶
For the actual selected 21-node, 53-edge UM support, the phase potential
has no local minimum anywhere in its retained open principal chamber,
modulo common phase translation. Consequently the complete positive-weight
storage H=Q_W+V has no stationary local minimum on its actual fixed form
and phase mean leaf within that chamber.
The proof covers every possible minimum: twin and leaf restrictions retain the singular resultant cases; the zero-cosine return-chain endpoint fails by a cubic variation; every remaining minimum belongs to the regular four-coordinate reduction. Sections 8-11 exclude its two positive cosines, zero-cosine boundaries, two negative cosines and both mixed sectors. No compact truncation, finite candidate list or failed root search replaces coverage near the open domain boundaries.
This does not exclude stationary points, prove linear instability of every stationary point, or exclude nonstationary invariant sets. In particular, the zero-chain cubic argument does not supply a negative Hessian direction. The synchronized minimum in a different branch class remains valid. The result concerns the declared law and selected support; it neither executes UM nor predicts that the retained hypothetical endpoint crosses a branch cut, and it makes no claim of physical identification or operator selection.
13. This source's storage and means do not separate it from principal cuts¶
Fix the exact ideal hypothetical endpoint, or one admitted endpoint in its
conditional numerical envelope. Keep that endpoint's own positive W and
capacities, rho_i=d_i/nu_i, forms x_0, lifted phases theta_0, and
weighted means. The existing
source-specific storage admission
gives
This independently justifies division by Q_0. It is a conditional source
bound, not a new certificate of an unevaluated runtime reset error.
Let Omega be the open principal chamber intersected with this source's exact
phase-mean leaf. Its closed chamber is compact: a spanning-tree path in the
connected finite support bounds each lift difference by sums of the finite
edge bounds pi+2*pi*|k_e|; the positive weighted mean then bounds every
centered coordinate. Closed branch and mean constraints make the bounded set
closed. Interpolation from the strict source shows that this closed chamber
is exactly the closure of Omega.
Set E_*=V_0+Q_0/2<h, and let D be the component containing theta_0 of
{theta in Omega: V(theta)<E_*}. Its closure is compact. If that closure
avoided every principal boundary face, V would attain a minimum there with
value at most V_0<E_*. A small connected ball in the strict sublevel around
this point intersects D and therefore belongs to D. The point would be an
interior local minimum, contradicting Section 12. Consequently there is
theta_b in both the closure of D and the boundary of Omega, with
V(theta_b)<=E_*.
Choose a small convex neighborhood of theta_b on which
V<(E_*+h)/2=h-Q_0/4, and choose y in D inside it. Open connected subsets
of the finite-dimensional mean leaf are polygonally connected: join
theta_0 to y inside D, then append the straight segment to theta_b.
The entire phase path retains the source mean and satisfies h-V>Q_0/4;
all original principal gaps remain strict before its final endpoint.
Writing m_rho=(rho^T*x_0)/(rho^T*1), lift this path to full state by
The multiplier starts at one and remains greater than one half. Since
L_W*1=0, the path retains the source form mean, phase mean and exactly
H=h. At least one original gap reaches +pi or -pi at theta_b.
The final segment crosses every reached face transversely upon a short
extension. Continuity and the positive Q_0/4 margin permit the same radial
form lift beyond that boundary.
Thus the component of this source's storage-and-means shell reaches and crosses a principal cut. Those three conserved values alone cannot isolate the source from every boundary of its full branch identity. The construction applies separately to each admitted endpoint, without transferring another endpoint's weights, means or energy. It reuses the existing source storage and radial-lift controls, and changes no source, support, law or clock.
The parameter s is not structural time. This does not select a cut, crossing time or trajectory, prove loss of either old constituent winding, or exclude nonstationary confinement. A principal cut is an identity boundary rather than a singularity of the smooth sine law. The separate complete-state certificate retains all original branches through a fixed half-unit window and proves a relative-velocity reversal. That finite result does not extend this stationary classification into an all-time confinement or escape theorem.