Two-port compatibility of unequal sine winding classes¶
Prospective claim and frozen protocol¶
The central-port composition retains all contacts at one node of each C9 component. Its compatible target has aligned contact phases and unchanged internal twists. A second contact at a different node creates an additional cycle and need not preserve that geometry. This gate asks whether the already specified winding classes one and two admit a joint acute equilibrium on one fixed two-port support.
The general methods are already owned: the acute circulation and uniqueness result and the full-law stability argument below apply to connected sine supports. The native C5 return-path example also has deformed geometry and nonzero interface current at zero mixed period, under its different complete law. The present obligation is the concrete C9 compatibility admission, not the first occurrence of either phenomenon.
Freeze this protocol before the new reserved assessment:
- Use eighteen nodes: donor cycle
0->1->...->8->0and receiver cycle9->10->...->17->9, with the two unit contacts(0,9)and(1,10). The graph has twenty edges. Nodes0,1,9,10have degree three; all others have degree two. Both evolution rows use those full degrees. - Retain the complete reciprocal normalized-sine law, unit held capacities,
beta=1,e=1023/1024,w=1/1024, scaled clocktau=e*t, andgamma=1/(1023*pi). No forcing, support change, state reset or separately installed contact law occurs in the assessed model. - The primary winding pair is
(2,1). The oriented interface cycle is0->9->10->1->0, with period zero. Use the acute edge branch on every edge. Form has degree-weighted mean zero, and the one global lifted phase gauge also has degree-weighted mean zero. Do not fix separate component means or supply an independently selected relative phase offset. - Predict one acute critical geometry modulo the common phase, with
nonzero opposed contact sine currents and deformed internal twists.
Predict that no independent component rotation can make both unequal
isolated twists an acute equilibrium on this support. The matched
(1,1)control must retain the undeformed twists and zero contact current. - Enclose the implicit critical geometry by the monotone scalar construction below. Use exact rational turn coordinates and the shared outward trigonometric intervals. Use thirty-two outer bisection refinements and sixty-four inner refinements. An unresolved required sign is unavailable; do not infer it from a floating midpoint, change the refinement budget after inspecting the result, or substitute a small residual for existence.
- Require the retained primary bracket to enclose the proved root, its outward contact-current lower bound to be strictly positive, every outward acute margin to be strictly positive, and the full-node balance, cycle periods, weighted gauge and matched-class control to agree with the exact construction. The full-law local attraction conclusion must follow from a positive relative Hessian, not a numerical eigenvalue sign.
- Preserve the protocol, producing source and proof before the first assessment. There is no finite response horizon or sensor-error budget: the observation is a static compatibility certificate for an implicit equilibrium. No trajectory, parameter search or previous producer is run.
This is an equilibrium and local-recovery admission. It does not transfer the earlier prepared sources into the new basin, prove attraction from arbitrary relative origins, select the support, fund an attachment event or identify a physical bond. The existing thirty-coordinate central-port surrogate is not assumed valid for this different interface.
Complete rows, means and the acute cycle cell¶
Let \(L\) be this graph's unit Laplacian, \(\mathsf D\) its degree matrix, \(K=\mathsf D^{-1}\), and \(S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i)\). In the declared clock the actual rows are
Their conserved charges are \(\mathbf1^T\mathsf D x\) and \(\mathbf1^T\mathsf D\theta\), where the latter uses continuously chosen real lifts. The degree mass is forty. Only one common form and one common phase are removed. Individual component means generally evolve. Changing a lift representative by nodewise integer multiples of \(2\pi\) does not change the circular state or (1); the declared weighted gauge fixes one representative after the edge branches have been selected.
Orient the two rings as above and each contact donor-to-receiver. Denote
their principal acute edge gaps by \(q_e\in(-\pi/2,\pi/2)\).
The two ring cycles and the four-edge interface cycle form an integer cycle
basis. Removing the chords (0,8), (9,17) and (1,10) leaves a spanning
tree; the three displayed cycles are its fundamental cycles, up to orientation.
The connected graph has cycle rank \(20-18+1=3\). Their required
periods are \((2,1,0)\). In fact an acute four-edge cycle has increment
sum strictly between \(-2\pi\) and \(2\pi\), so its integer period
must be zero.
On this fixed period cell, admissible edge gaps are the intersection of an affine cycle-constraint space with the open acute cube. That set is convex. The phase potential
is strictly convex in those edge coordinates. Equivalently, its Hessian is
for an edge-difference incidence matrix \(B\); it is positive definite on relative nodal variations. Consequently there is at most one acute critical geometry in the cell, modulo common phase. Existence still needs proof; convexity alone does not prevent a minimum on the acute boundary.
This formulation fixes cycle periods rather than treating arbitrary corners of independent edge intervals as realizable states. Two nodal lift choices reconstructing the same edge gaps differ only by integer vertex shifts and a common phase, and do not supply distinct relative circular geometries.
Exact reduction of the critical equations¶
At any equilibrium, the phase row and positive capacities imply \(Lx=0\). The zero form-mean leaf therefore has \(x=0\), and its form row then requires \(S(\theta)=0\). Every nonport ring node has degree two. Its sine balance equates its two oriented ring currents; injectivity of sine on the acute branch equates their gaps.
Write \(a\) for the donor gap on 0->1 and \(b\) for each of the
other eight donor gaps. Write \(c\) for 9->10 and \(d\) for the
other eight receiver gaps. The ring periods give
Let \(\delta_0,\delta_1\) be the contact gaps on 0->9 and
1->10. Summing nodal balance over one ring gives
\(\sin\delta_0+\sin\delta_1=0\). Both are acute, so
\(\delta_1=-\delta_0\). The interface period gives
\(\delta_0+c-\delta_1-a=0\). Hence, writing \(\delta=\delta_0\),
Conversely, any acute solution of (3)--(4) supplies every fine-node balance. For example the donor node-zero row is \(\sin a-\sin b+\sin\delta=0\), and the receiver node-nine row is \(\sin c-\sin d-\sin\delta=0\). The opposite port rows are their negatives; all fourteen nonport rows cancel along their arcs. Degree three multiplies a zero port sum and is retained in both full rows.
An explicit real lift is
The closing donor and receiver increments differ from their principal gaps by \(-4\pi\) and \(-2\pi\), respectively. Both contact gaps are as in (4). Subtracting the single affine weighted mean \(\sum_i d_i\theta_i/40\) imposes the declared gauge without changing any gap. This is a correlated implicit target, not a target obtained by rounding each phase independently.
A strictly monotone compatibility equation proves existence¶
Put \(\alpha=2\pi/9\), and define
On all intervals used below,
For \(a\in[\alpha,2\alpha]\), put \(g=h_2(a)\ge0\). There is a unique \(c=c(a)\in[\alpha,2\alpha]\) satisfying \(h_1(c)+g=0\). The lower endpoint has \(h_1(\alpha)=0\). For the upper endpoint, the exact inequality
provides strict bracketing whenever \(g>0\). At \(a=2\alpha\), \(g=0\) and \(c=\alpha\) exactly. Implicit differentiation gives \(c'(a)=-h_2'(a)/h_1'(c)<0\).
Define
Here \(|a-c(a)|/2\le\pi/9\), so (6) implies
At \(a=\alpha\), one has \(g>0\) and \(c>\alpha\), hence \(F(\alpha)>0\). At \(a=2\alpha\), one has \(F(2\alpha)=-\sin(\pi/9)<0\). The intermediate value theorem and strict monotonicity give exactly one root in the open interval.
At this root \(g>0\), so (8) forces \(a>c\). All gaps satisfy
Thus the constructed target is acute, with every margin strictly larger than \(\pi/36\). Equations (3)--(5) prove existence in the declared period cell. Its uniqueness among all acute geometries in that cell follows from (2), not just from restricting the search to arc-constant candidates.
The matched (1,1) control has
\(a=b=c=d=\alpha\) and \(\delta=0\). It satisfies every full row
and the same interface period. Strict convexity again gives uniqueness in
its own acute cell. The corresponding statement for matched (2,2) uses
\(2\alpha\); swapping the two rings maps (2,1) to (1,2) and
reverses the oriented contact current without changing its magnitude.
What incompatibility of the isolated twists means¶
Keep the two isolated twists undeformed, while allowing any independent common rotation of either component. Each internal sine sum is then zero. At every port, the acute full-node balance forces its single contact gap to be zero. But two zero contact gaps would give \(a=c\) around the interface cycle, contradicting the unequal winding slopes \(a=2\alpha\), \(c=\alpha\). Therefore no supplied relative origin can make both undeformed unequal twists critical on this support.
The compatible state instead changes both short and long internal arc gaps and derives its relative port phases from (4). Its nonzero opposed contact currents are a stationary sine circulation. They are not a sustained transfer of structural storage, a temporal oscillation or a measured energy current: at the critical state every form and phase velocity is zero.
There is also no energetic binding conclusion from this compatibility. Strict convexity of \(1-\cos s\) on the acute interval gives
Each isolated ring's fixed-period minimum is its uniform twist. The target deforms those rings and adds the positive contact potential \(2[1-\cos\delta]\). Thus its storage exceeds the sum of the two isolated minima. A law for removing or creating edges and its work accounting would be an additional model. Local attraction at fixed support does not select that support or make its formation passive.
Full-law local recovery without operator commutation¶
The exact fixed-support storage is
The mixed terms cancel between both rows of (1). At the target, (9) gives \(\mathcal H_*\succeq\sin(\pi/36)L\), positive on the relative space. On the fixed weighted-mean leaf the form energy and local phase excess are both positive definite. Choose a sufficiently small closed relative ball entirely inside the acute cell and a positive storage sublevel strictly below its boundary. Equation (11) makes that sublevel invariant. Its largest invariant zero-loss set has \(Lx=0\), hence \(x=0\), then \(S(\theta)=0\), hence the unique target. LaSalle's argument proves local asymptotic attraction for the full thirty-six-coordinate law, after removing the two conserved common modes.
For the stronger local exponential statement, use the full reciprocal stability argument. On \((K^{-1/2}\mathbf1)^\perp\), set \(P=K^{1/2}LK^{1/2}>0\) and \(Q=K^{1/2}\mathcal H_*K^{1/2}>0\). The phase variation can be transformed to the quadratic pencil
For an eigenvector, its scalar inner product has strictly positive real mass, damping and stiffness coefficients. A nonreal root therefore has strictly negative real part; a real root cannot be nonnegative. All relative linear modes decay, so smoothness yields local exponential attraction. Neither \(P\) and \(Q\) nor their eigenvectors are assumed to commute.
These are local statements around the new implicit target. They provide no uniform basin for the old formation sources, no finite acquisition time, no global attraction throughout the acute cell and no theorem spanning a support event. Those would require their own complete-state admission.
Certified scalar enclosure and implementation boundary¶
The implementation uses turns \(A=a/(2\pi)\), \(C=c/(2\pi)\) and
The outer bracket is \([1/9,2/9]\). For each admitted outer argument, the inner decreasing equation is \(\widehat h_1(C)+\widehat h_2(A)=0\) on the same bracket. The upper outer endpoint uses the exact identity \(C=1/9\). The decreasing outer residual is
The shared strict-sign root enclosure bisects only when its outward interval proves a sign. The inner enclosures are propagated through (13), retaining their uncertainty and operator order. The fixed sixty-four inner refinements are independent of the thirty-two outer refinements. A completed outer bracket has width at most \((1/9)2^{-32}\) turns. No unproved sign or adaptively enlarged numerical budget is used to force completion.
Endpoint monotonicity then encloses \(C(A)\), \(b,d,\delta\), the full affine lift (5), the gauge, currents, storage and acute margins. Exact reconstruction can retain the gauge as the affine turn expression
For the two root residuals \(F=\widehat F(A)\) and \(J=\widehat h_1(C)+\widehat h_2(A)\), the only nonzero symbolic nodal rows are \(S_0=-F\), \(S_1=F\), \(S_9=F-J\) and \(S_{10}=J-F\). This factorization vanishes at the implicit root. The exact periods and fine-row cancellations belong to the implicit correlated root; they are not asserted for every corner of the reported marginal intervals. Intervals containing zero in the computed residuals are implementation consistency checks, not substitutes for the analytic existence proof. The matched control uses its exact twist formulas and requires no root solve.
An unresolved numerical sign or margin leaves the represented certificate unavailable while preserving the analytic theorem. Conversely, a successful static certificate does not establish formation, an autonomous event selector, physical time units, a sensor model or an energy-conjugate contact readout.
The shared
assess_sine_two_port_compatibility
returns SineTwoPortCompatibility from the mandatory primitives classes,
outer_refinements and inner_refinements. It admits the class labels one
and two and ordinary integer refinement budgets from one through sixty-four;
the reserved protocol above fixes their selected values. It rebuilds the
actual graph, affine phase correlations and balance factorization rather than
accepting an incoming equilibrium report. The class-swap and matched-class
extensions retain the same support, complete law and fixed weighted means.
Status at archival¶
At source archival the protocol and proof above were complete, and the reserved assessment had not been performed. The following section records the subsequent first assessment of that unchanged declaration.
Retained two-port compatibility result¶
Both the primary (2,1) assessment and the matched (1,1) control returned
certified_compatible, with no unavailable reasons. All six frozen stopping
conditions passed: both implicit equilibria, the positive-versus-zero contact
current distinction, exclusion of the undeformed unequal twists, and strict
local attraction on the declared full-network mean leaf.
For the primary geometry, the completed donor short-arc bracket in turns is
The stored outer residual intervals have strictly positive and negative signs at the lower and upper endpoints, respectively. The fixed inner budget remains sixty-four refinements. The following exact outward enclosures use the common denominator \(2^{128}\):
| Quantity | Lower numerator | Upper numerator |
|---|---|---|
| First contact gap \(\delta/(2\pi)\) | 6220678650586981850326865582201125091 |
6220678657852373978098316676520651436 |
| First contact sine current \(\sin\delta\) | 38999787950219209243187699610900432965 |
38999787995869014311299513956505139888 |
| Minimum acute edge margin, in turns | 7233486662907790580349073231255408184 |
7233486664008181726380522364499074162 |
These correspond approximately to a contact gap between \(0.01828093153011452\) and \(0.01828093155146559\) turns, a sine current between \(0.1146100760468745\) and \(0.11461007618102723\), and an acute-margin lower bound \(0.021257306772491172\) turns. The second contact has the opposite gap and current. The matched control has exactly zero on both contacts and retains its uniform class-one twists.
The complete support has diameter nine and certified combinatorial Laplacian gap lower bound \(2/81\). Its primary relative phase-Hessian lower bound is
The actual degrees, zero weighted gauge, three integer cycle periods and full-node incidence factorization agree with the exact construction. The reported local attraction follows from this positive Hessian and (11)--(12). Residual intervals containing zero are retained only as consistency checks. No rounded point from the marginal phase intervals replaces the correlated implicit equilibrium.
The retained protocol,
source archive,
primary and matched reports
and evidence manifest
preserve the first assessment. The archive SHA-256 is
6ce65e3155dc41f4c0156f103cdc04fdcc33ddbd294929db050b75e244782eb2.
The archive retains the pre-evaluation status, while the protocol and
mathematical derivation preceding it are unchanged in this owner.
The contract, guide and independent controls cover exact reconstruction, integer cycle admission, fine-node balance, independent coupled-equation checks, the full Hessian, class exchange and explicit abstention when a required sign is unresolved.
This result admits a deformed, locally attracting geometry for two supplied winding identities on the new fixed support. It does not establish a source family that reaches it, a contact event or its work budget, global attraction, stationary energy transport or a physical binding mechanism. Those distinctions are unchanged by the successful static certificate.
An origin-independent obstruction to a storage-only handoff¶
This separate analytic result asks whether the already proved target-free sector capture theorem can admit the undeformed class pair directly on the two-port support. It does not revise the archived compatibility protocol or reevaluate its producer. Under F4 the result is an exact obstruction to a specified proof mechanism; no new trajectory, root solve or numerical campaign is involved.
Retain the fixed support, complete rows (1), positive loss, capacities and clock above. Supply the two isolated uniform twists of winding two and one, allowing any independent component phase origins. Form may be any admitted finite signed vector. The relevant capture theorem would require the complete initial state to be strictly acute and its total storage to be strictly below a common lower bound on phase storage on every nonempty signed boundary face of the same acute period cell. There are forty such signed face candidates on the twenty-edge support.
The conclusion below excludes that scalar full-boundary storage criterion for every relative origin. It also excludes the criterion for a specified neighborhood of the undeformed phase family. It does not exclude convergence under (1), a differently shaped invariant region, or a later same-law entry into a capture sublevel after dissipation.
Minimize over every component origin before constructing a witness¶
Write
The donor ring has all nine principal increments equal to \(q\), and the receiver has all nine equal to \(c\). If the first contact gap is \(r\), the second is \(r+c-q=r-c\), modulo full turns. Therefore
This is an exact minimization over all relative origins, including those whose contacts are not acute. Equality is attained by opposite contact gaps \(\delta,-\delta\). Thus the comparison uses the most favorable origin for total storage; it does not select an origin after a trajectory response.
One feasible boundary state lies below that minimum¶
For \(0\le s\le s_b:=\pi/144\), assign donor edge (1,2) the gap
\(q+8s\), every other donor ring edge the gap \(q-s\), every receiver
ring edge the gap \(c\), and the two contacts the gaps
\(\delta-s/2,-\delta+s/2\). These are correlated gaps of an actual
nodal state. Explicit real lifts are
The closing donor and receiver lift differences require the same additions
\(4\pi\) and \(2\pi\) to recover their principal ring gaps. The
ring sums are exactly \(4\pi\) and \(2\pi\). The interface cycle
(0,9,10,1) has sum
\(2(\delta-s/2)+c-(q-s)=0\). Consequently the periods remain
\((2,1,0)\) throughout this path. These are the same integer basis cycles
used in the compatibility proof. Subtracting
\(\sum_i d_i\theta_i(s)/40\) gives the zero weighted phase gauge
without changing any edge gap. A common shift then places the witness on
any source's conserved phase-mean leaf. Choose its uniform form equal to
the source's conserved weighted form mean; its form storage is zero.
Thus the comparison does not rely on moving to a different conserved-mean
leaf. The zero-mean representative has witness form zero.
Every gap is strictly acute for \(s<s_b\). At \(s_b\), only edge
(1,2) reaches the positive acute face. Its gap is \(\pi/2\); the
other eight donor gaps are \(7\pi/16\), the receiver gaps are
\(2\pi/9\), and the contact gaps are \(\pm31\pi/288\). Hence this
is a feasible point on one of the actual full-cell boundary faces, with
phase storage
The construction is an auxiliary geometric path, not a proposed solution of (1) or evidence that the dynamics visits that boundary.
Let \(D(s)=U(\theta(s))-U_{\min}\). Then \(D(0)=0\), and
Throughout the displayed interval, \(\sin(q+8s)\le1\), \(\sin(q-s)\ge\cos(\pi/16)\), and \(\sin(\delta-s/2)\ge\sin(31\pi/288)\). The elementary inequalities \(1-\cos z\le z^2/2\), \(\pi^2<10\), and \(\sin z\ge2z/\pi\) for \(0\le z\le\pi/2\) give
Integrating this strict inequality and using \(\pi>3\) proves
No approximate energy comparison or unproved minimizer is needed.
If \(B\) is any valid common lower bound on phase storage on every nonempty acute boundary face, the exhibited face forces \(B\le U_b\). For any nominal origin and any form state,
Thus no improvement to the lower-bound solver can make \(H(x,\theta_{\rm nom})<B\) true. This statement concerns a common bound on the full boundary; it does not assert that every individual face has low storage. If the initial contacts are not strictly acute, the capture theorem already fails its initial-domain premise. If they are acute, their four-edge cycle has period zero and (20) excludes its required storage inequality in the correct \((2,1,0)\) cell.
The obstruction includes an explicit phase-error family¶
Allow each node's continuous phase lift to differ from its nominal twist by at most \(\eta\ge0\) radians, after retaining the independent component origins. The errors may be correlated or independent; no restriction on the admitted form vector is needed. The global inequality \(|f(v)-f(u)|\le|v-u|\) gives, on all twenty edges,
Consequently every state in this full phase family satisfies
Whenever \(M(\eta)>0\), the internal phase errors are also too small to change the two acute ring periods: \(2\eta<\pi/18\), the smaller nominal ring margin. An actual state with acute contacts therefore lies in the same \((2,1,0)\) cell and fails its full-boundary storage criterion. A state with nonacute contacts fails the criterion's domain instead. Large form differences cannot repair this obstruction because the form storage is nonnegative.
Choose the explicit rational phase-error budget
It defines the admitted family without using the implicit equilibrium coordinates. The assessor's sufficient rule requires a strictly positive displayed margin. It abstains when that margin is zero or negative; such abstention does not claim that the sharper strict analytic inequality has failed. In particular, the threshold \(\eta=17/552960\) is not promoted to a positive certificate by rounding or by silently changing the rule.
The shared
assess_sine_two_port_handoff_obstruction
returns SineTwoPortHandoffObstruction from the mandatory primitive
phase_error_radius. It admits a nonnegative exact or represented real,
reconstructs the actual support, rational witness lifts, weighted gauge,
integer periods and boundary edge, and computes (21) by exact rational
arithmetic. The report retains both positive and nonpositive margins; its
certificate never substitutes a cached verdict, rounded storage difference
or source trajectory for those premises. This family theorem does not
admit an observed node array or authenticate a supplied endpoint report.
The previous uninterrupted formation-and-dwell proof already supplies \(10^{-32}\) Euclidean phase-error bounds per component under its own preparation, isolated support and declared dwell. Each coordinate then satisfies the larger bound in (22), so these endpoint families inherit the obstruction if inspected on the present two-port support without resetting their state. The comparison holds for every relative origin, including the former preparation's supplied origin. This implication uses the proved endpoint error, not a transfer of the old single-port capture verdict or its event-work allowance. It installs no new support, changes no prior preparation and makes no two-port event passive.
The static compatible equilibrium and its local attraction remain valid. The new result identifies a limitation of the proposed direct handoff: total storage versus every acute face cannot certify these undeformed source families. A convergence proof for them needs additional dynamical or geometric information beyond this scalar criterion. The result supplies no nonconvergence theorem, autonomous contact selector or physical identification.