Full-fine stationary contact in a declared winding chamber¶

Prospective declaration¶

This is a bounded, source-free stationary search under the complete law in continued contact. It does not evolve the acquired family or change the supplied interface. The reflection obstruction leaves possible stable targets outside its sectors unresolved.

The following declaration and its executable instrument are committed before generating any candidate. Their commit is recorded in the exclusive attempt record. The first outcome is retained; no budget extension or replacement is admitted by this declaration.

Item	Fixed rule
Complete law	54 nodes `(c,j)`, `c=0,...,5`, `j=0,...,8`; the existing 144 unit edges of `C9 square W6`; held unit capacity; `x'=(-Lx+gammaS)/4`, `theta'=gammaLx/4`, `gamma=1/(1023pi)`, prime in `tau=(1023/1024)t`; no input or event
Stationary reduction	Connectedness and the two conserved means give `x=0`, `S(theta)=0` on the zero-form-mean leaf. Ground phase node 0 only to choose coordinates. All other 53 coordinates, including relative donor origin, remain free; recenter a certified target to the zero lifted-phase mean
Observed identity	Six oriented internal C9 windings one and nine oriented old outer C5 windings one; all 99 marked edges remain away from principal cuts. No spoke winding is prescribed
Search domain	One lifted chamber: internal forward edges have branch integer 0 except the closing edge in each C9, which has -1; outer forward edges have 0 except those from child 4 to child 0, which have -1. Constrain each marked gap `theta_b-theta_a-2pik` to `[-pi+1/1024,pi-1/1024]`. Nonacute gaps are allowed. Spokes are unconstrained
Domain limitation	These branch integers fix the complete marked-graph cycle class, including mixed cycles not specified by the fifteen windings. No reflection, carrier or internal-shape condition is imposed. Other cycle classes and the omitted margin strips are not searched
Starts	In order, four donor origins `beta=kpi/2`, `k=0,1,2,3`, each with amplitudes `0,1/8,1/4,1/2,1,3/2`. Base phase is `2pij/9+2pic/5` on hosts and `2pij/9+beta` on donor. One NumPy PCG64 stream with seed 20261010 draws 53 uniform `[-1,1)` values for every attempt, including amplitude zero; node 0 perturbation is zero. Multiply by amplitude, then by `t=min(1,min_e slack_e/(2abs(q_b-q_a)))`, ignoring zero denominators, where `slack_e=pi-1/1024-abs(base_gap_e)`. This retains at least half the base slack
Candidate generator	Python 3.13.6, NumPy 2.5.3, SciPy 1.18.1, binary64, BLAS/OpenMP thread environment 1. One SLSQP minimization of `V=sum_all_144_edges(1-cos(delta))` per start, analytic gradient `-S`, the 99 linear constraints, `ftol=1e-12`, at most 500 iterations and 5000 value/gradient oracle calls. Every attempt and endpoint is retained, including failure and budget exhaustion
Screening and polish	Only endpoints with gradient infinity norm below `1e-7` and tightened-chart slack above `2e-7` receive one `root(method='hybr')` call, analytic grounded Hessian, `xtol=1e-10`, at most 600 residual and 600 Jacobian calls. A polished endpoint must satisfy the same two screens and have numerical grounded-Hessian minimum eigenvalue above `1e-7` before rigorous certification. Numerical eigenvalues and solver success labels are screens only
Certification budget	One box of exact radius `1/10^7` in all 53 grounded coordinates, existing dyadic128 outward arithmetic. Admit binary64 centers as their exact represented rationals. One binary64 inverse rounded to rational multiples of `2^-40` by Python nearest/ties-to-even `round(value2*40)`; verify it by the contraction inequalities below. Round edge cosine lower bounds downward to multiples of `2^-32` and use shared exact strict LDL. No radius, precision, inverse or root retry
Acceptance	Strict root inclusion, strict full relative-Hessian positivity and all marked branch intervals inside the tightened chart. These imply a stationary target locally asymptotically stable on the fixed-mean leaf for the complete two-row law. An optimizer flag or small residual alone never passes
Stopping	Stop at the first certified target; otherwise finish the 24 attempts and stop. An exception is retained as an incomplete attempt, with no automatic rerun. A boundary-constrained minimum is not an equilibrium of the unforced law. A finite nonfinding is neither nonexistence nor actual winding loss
Evidence	Retain the full Git source revision, source blob associations, runtime and numeric backend, exact binary64 start/endpoint coordinates, support order and branches, oracle counts, optimizer messages, residuals, boundary slack and numerical curvature. Write exclusive attempt/outcome records with the shared artifact writer

No trajectory horizon is introduced: optimizer iterations are not physical time. No acquisition, source capture, spontaneous symmetry breaking or fundamental physical identification is inferred from this search.

Root and stability acceptance¶

For a grounded phase center q, let F=-S on its 53 free nodes and let J be the grounded cosine Laplacian. The absent row follows from the exact zero sum of all currents. Every primitive coordinate and edge is re-admitted; reported residuals or Hessians are not accepted as premises.

Let R be the supplied rational approximate inverse. Outward trigonometric arithmetic at q gives bounds on F(q) and J(q). With exact midpoint matrix J0, interval radius row sums e_k, and radius r, define

\[ \eta\geq\|RF(q)\|_\infty,\qquad q_0=\max_i\left[\sum_j|\delta_{ij}-(RJ_0)_{ij}| +\sum_k|R_{ik}|e_k\right], \qquad q_B=q_0+4d_{\max}r\|R\|_\infty. \]

Here d_max=7. The cosine is 1-Lipschitz and each edge difference changes by at most 2r; the diagonal and off-diagonal row changes together are at most 4*d_max*r. Thus q_B<1 and eta+q_B*r<r make y -> y-RF(y) a strict self-map and contraction on the closed box. Its fixed point is unique there. Also q0<1 makes R*J(q) invertible, so R is invertible and the fixed point is a zero of F, not merely of a singular projection.

For each edge, bound its cosine throughout the box and round the lower bound down to the declared dyadic grid. The exact grounded Laplacian assembled from these lower weights is a Loewner lower bound for every J in the box: the difference is a sum of nonnegative multiples of edge outer products. This remains valid with negative individual edge weights. Strict exact LDL therefore certifies positive definiteness of the entire 53-dimensional phase quotient. Grounding and restricting to mean zero are congruent descriptions; their numerical eigenvalues need not coincide. The complete two-row tangent pencil from the contact theorem then has 106 stable relative directions and the two retained neutral means. A nonpassing LDL check alone proves neither instability nor nonexistence; extra zero directions remain unresolved.

For completeness, positivity suffices without simultaneous diagonalization of L and M. Linearization gives theta''=-L*theta'/4-gamma^2*L*M*theta/16. On the mean-zero subspace, any complex eigenpair satisfies (s^2*L^-1+s*I/4+gamma^2*M/16)v=0. Multiplying by the conjugate transpose of v gives a*s^2+b*s+c=0 with strictly positive real a=v*L^-1*v, b=||v||^2/4 and c=gamma^2*v*M*v/16. Both real roots are negative, and nonreal roots have real part -b/(2*a)<0. The smooth complete law therefore has local exponential stability on each fixed-mean leaf; neither common mean is damped.

For every marked edge, its interval gap minus 2*pi*k must lie strictly between -pi+1/1024 and pi-1/1024. The integer branch sum is -1 around each of the fifteen cycles, so telescoping gives winding one exactly. Interval overlap of a sum with 2*pi is not used as a winding proof.

Marked-cycle information and connected chambers¶

The fifteen prescribed winding numbers do not specify every winding on the marked support. Remove the 45 unmarked spokes. The resulting marked graph has 99 edges, 54 vertices and two connected components: the 45-node host product C9 square C5 and the donor C9. Its real cycle-space dimension is therefore 99-54+2=47. The six internal cycles and nine outer cycles are independent: they are pairwise edge-disjoint, and every one has an edge with nonzero coefficient that appears in no other prescribed cycle. Their span has dimension 15, leaving 47-15=32 cycle-space directions not specified by those observations. This dimension count does not assert that arbitrary integer assignments to the remaining directions are simultaneously realizable by phases with principal gaps.

For an oriented marked edge a -> b, choose real phase lifts and write

\[ \delta_{ab}=\theta_b-\theta_a-2\pi k_{ab}, \qquad -\pi<\delta_{ab}<\pi,\qquad k_{ab}\in\mathbb Z. \]

Changing a vertex lift by 2*pi*m_a changes k_ab by m_b-m_a, an integer coboundary. Its circulation around every cycle is unchanged, and the principal winding of an oriented cycle is minus that circulation. Along a continuous path avoiding all marked cuts, each principal gap is continuous; every integer cycle winding is therefore constant. Thus a marked-cycle class is invariant under such paths, even when only fifteen of its values are retained as the declared identity.

Conversely, suppose two phase configurations have equal windings on every marked cycle. Their branch difference has zero circulation, so integrating it along a spanning tree in each marked component supplies integer vertex shifts that make their branch vectors equal. Linear interpolation of these adjusted lifts keeps every marked gap strictly between its two cuts, because each gap interpolates between two points in the same open interval. Hence the complete realizable cycle class, modulo integer vertex shifts, specifies a connected component of the cut-free phase domain. This is a statement about continuous paths in the configuration space, not reachability under the supplied contact law.

There are two independent common marked-component angles. After quotienting the common phase of the whole connected contact graph, the relative donor angle remains a free circle coordinate; no marked edge constrains it. Spokes enter the stationary equations and storage but introduce no prescribed cut or winding here. Fixing the global lifted mean only chooses a common representative and does not remove this relative angle. The search above retains it as a free optimization coordinate. Its four supplied initial angles are starts, not four fixed-origin subproblems.

The declared branch chamber has zero winding on every elementary marked host square. It is strictly narrower than the class defined by the fifteen observations alone. An exact counterexample uses phases in turns, u=theta/(2*pi). For the five hosts set

\[ u_{c,j}=j/9+c/5+(9/25)s_c t_j,\qquad s=(1,-1,1,-1,1),\qquad t=(1,-1,1,0,0,0,0,0,0), \tag{1} \]

and set u_(5,j)=j/9 for the donor. All marked principal gaps lie strictly between -1/2 and 1/2 turns. On each perturbed internal cycle, the two changes of size 18/25 cross opposite cuts, so their winding changes cancel. The other changed gaps have principal representatives 1/9 +/- 9/25, both inside the interval. Every internal winding remains one. At each of the three perturbed outer slots, two positive and two negative 18/25 changes likewise give cancelling branch crossings. The remaining outer slots are uniform. Thus all nine outer windings also remain one.

Nevertheless, the oriented square (0,2) -> (1,2) -> (1,3) -> (0,3) -> (0,2) has principal gaps

\[ (12/25,\ 106/225,\ -1/5,\ 56/225) \]

in turns, whose sum is one. The unperturbed product state has sum zero on this square. These states therefore share the fifteen prescribed windings but lie in different marked-cut-free components, even after arbitrary integer vertex shifts. Both may be recentered to the same lifted mean without changing this conclusion.

This counterexample concerns retained information and chamber scope only. It is not a stationary or stable target: at node (0,3), the wheel currents cancel, while the internal current is sin(2*pi/9)+sin(112*pi/225)>0. No candidate search or trajectory is needed for these exact identities. The tightened margin and finite start budget further restrict the numerical search; neither expands it to the other 32 unspecified cycle-space directions or proves exclusion there.

For uninterrupted maintenance from the original product chamber, the complete marked class is a necessary target condition: every marked edge belongs to one of the fifteen observed cycles, and the all-times contract excludes its cut. Simultaneous compensating crossings can restore sampled winding values but fail that cut-free contract at the crossing. This is a kinematic restriction on a possible source-to-target connection, not a claim that the actual source reaches any stationary state.

Membership of the actual preparation in this chamber follows from the retained source bounds, rather than from those fifteen labels. The host is psi_j+Phi_c+error with coordinate error at most Z+m, Z<7*10^-70, m=2*g*B_o*T, T=1/5000, g<1/3000 and B_o<1/50; thus m<10^-8. Its carrier outer gaps are bounded by 7/5. The donor's corresponding error is at most 10^-70; its supplied common origin does not affect a marked gap. Scaling these errors to zero avoids every marked cut. The carrier's winding-one outer gaps can then interpolate to 2*pi/5, while internal gaps remain 2*pi/9. This proves membership using the existing admitted source contract, without rerunning its producers or implying that this interpolation is a trajectory of the law. The chosen zero-mean target representative can be translated to any retained common phase mean.

Retained finite outcome¶

The declaration and instrument were frozen in 75497035c5e0840b164897119ce5a50dcf6bda91 before their single execution. The attempt record retains source revision/blobs, numeric backend, the ordered support and branch vector. The same directory preserves all 24 exact binary64 starts and endpoints. The outcome is finite_search_without_certificate.

All 24 SLSQP calls returned a success label for their constrained optimization problem. None passed the independent stationary/interior screen, so there was no root polish or candidate certificate call. They used 925 iterations and 1,581 value/gradient calls in total; per attempt these ranged from 7 to 63 iterations and from 7 to 106 calls. There was no budget extension, exception, source evolution or subsequent retry.

Endpoint observation	Range over the retained attempts
Grounded gradient infinity norm	`0.0007324218215945957` to `2.4513699598554446`
Tightened-chart slack	`-3.4488e-12` to `6.4838e-14`
Numerical minimum grounded-Hessian eigenvalue	`-3.7261` to `-1.5304`
Phase potential	`62.4010` to `66.6337`

Every endpoint reached an imposed margin face to solver precision. The tiny negative tightened slacks are numerical constraint violations, not exact admission of that tightened closed domain. The actual principal cuts are still approximately 1/1024 radians away. Read-only reconstruction from the stored primitive coordinates checks all fifteen windings and a strictly nonzero current at every endpoint; it invokes neither initialization nor optimization. Numerical negative curvature here is a local storage diagnostic at nonstationary points, not an equilibrium-instability theorem.

The artificial constraints can stop a descent with nonzero gradient. Consequently an optimizer success label does not produce a stationary organization under the unforced complete law. These 24 endpoints supply no stable target, no global nonexistence proof, and no evidence that the actual source crosses a winding cut. Other minima in this chamber, the omitted margin strips, weakly stable or degenerate targets, and other realizable marked classes are not excluded. The new exact certificate remains a reusable acceptance mechanism; its analytic controls do not turn this nonfinding into a successful 54-node candidate.

The mathematical advance is the complete marked-class restriction and its explicit same-fifteen-windings counterexample. A target compatible with uninterrupted maintenance from the original source must lie in its product marked class. Existence and full-state stability within that class remain the missing analytical premise; increasing the number of optimizer starts would not supply it. The execution plan owns the reassessment and next admission, independently of this frozen result.