Finite directional transit of the two-port C9 pair

Complete-state question and supplied preparation

The two-port compatibility theorem admits a deformed acute equilibrium for winding classes two and one. The subsequent handoff obstruction shows that undeformed pairs cannot pass a direct full-boundary total-storage capture test, even after minimizing over their relative origins. Neither statement decides the direction of their actual motion.

This result proves finite motion toward smaller donor short gaps and larger receiver short gaps under the same complete positive-loss law. It uses an auxiliary gradient flow only inside a proved comparison estimate. No trajectory solver, fitted endpoint, equilibrium reset or pressure change is part of the certificate. The mathematical result is an F4 conditional theorem; its rational certificate checks the explicit sufficient bounds below.

Supply two unit C9 cycles with nodes 0,...,8 and 9,...,17, and unit contacts (0,9) and (1,10). The contact ports have degree three; the other nodes have degree two. With the actual graph Laplacian \(L\), degree matrix \(M\), and \(K=M^{-1}\), retain held capacity one, beta one, \(e=1023/1024\), \(w=1/1024\), no forcing and no events. Set

\[ \gamma=\frac1{1023\pi},\quad \eta=\gamma^2,\quad \tau=et,\quad \sigma=\eta\tau,\quad S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i). \]

The actual complete rows in \(\tau\) are

\[ x'=-KLx+\gamma KS(\theta),\qquad \theta'=\gamma KLx. \tag{1} \]

All eighteen form and eighteen phase coordinates remain in this model.

The nominal initial form is zero. Declare the continuous phase lifts

\[ \widetilde\theta^0_j=(j-\tfrac12)\frac{4\pi}{9},\qquad \widetilde\theta^0_{9+j}=(j-\tfrac12)\frac{2\pi}{9}, \qquad 0\le j\le8. \tag{2} \]

Their full-network weighted mean is \(21\pi/20\). Use the zero-mean nominal representative \(\theta^0=\widetilde\theta^0-(21\pi/20)\mathbf1\). The implementation equivalently starts from turns \(2j/9\) and \(1/18+j/9\), whose weighted mean is \(229/360\), and subtracts that single full-network mean. The two constructions differ before centering only by one common shift and give exactly the same centered phases. Neither construction adjusts the components' relative origin or resets actual errors.

Both contact midpoints share a phase origin. Their principal gaps are \(\delta,-\delta\), with \(\delta=\pi/9\); the ring gaps are \(4\pi/9\) and \(2\pi/9\). This preparation is specified directly from the two undeformed twists, without the implicit compatible target. Relative to the earlier (j-4) isolated lifts, it supplies receiver-minus- donor origin \(-7\pi/9\). This is a new preparation premise, not a late rotation of an already evaluated response.

Admit every initial state of the form

\[ x_i(0)=\epsilon^x_i,\qquad \theta_i(0)=\theta_i^0+\epsilon^\theta_i,\qquad |\epsilon^x_i|\le r_x,\quad |\epsilon^\theta_i|\le r_\theta. \tag{3} \]

The phase radius is in radians and the form radius in the model's signed form units. They are separate nonnegative primitives. No reflection symmetry, zero error sum or correlation is required of these thirty-six errors. Common form and phase means are retained, not reset to nominal values. The support is supplied from this initial time. Any preceding formation, support-attachment event or associated work accounting needs its own premises.

Fix the endpoint at

\[ \Sigma=\tfrac14\quad\hbox{in slow time},\qquad \tau_* =\frac{1023^2\pi^2}{4},\qquad t_*=261888\pi^2. \tag{4} \]

These are declared structural clocks, without a laboratory-unit bridge.

A complete-law comparison using the dissipated storage

The following estimate improves the long-horizon exponential envelope of the existing finite slow-phase comparison when a common acute chart can be certified. It adds that chart premise and does not replace the earlier globally valid estimate outside it.

Write \(A=KL\), \(f(\theta)=KS(\theta)\), and let \(P_M\) remove the weighted common mode. Put

\[ z=\gamma P_Mx,\qquad y=\theta+z, \qquad \|v\|_M^2=v^TMv. \]

The exact mixed-coordinate equations are

\[ z_\tau=-Az+\eta f(\theta),\qquad y_\sigma=f(\theta). \tag{5} \]

These are the same-information coordinates of the form-phase reduction owner. Form is recovered as \(P_Mx=z/\gamma\); independently wrapping both mixed phases would discard this information.

Let \(\psi_\sigma=f(\psi)\) be a declared reference. Match its common phase origin to the actual state's conserved weighted phase mean. Suppose that \(A\) has weighted quotient gap at least \(\lambda>0\), and \(f\) is \(\Lambda\)-Lipschitz in this norm. On a common acute period cell its derivative is \(-K\mathcal H\), where the cosine-weighted Hessian \(\mathcal H\) is positive semidefinite. Therefore

\[ \langle u-v,f(u)-f(v)\rangle_M\le0 \tag{6} \]

whenever the segment from \(u\) to \(v\) stays in that cell. This is a gradient-flow comparison property, not an assertion that the actual phase alone obeys a gradient law.

Assume initial bounds \(\|z(0)\|_M\le Z_0\) and \(\|y(0)-\psi(0)\|_M\le v_0\). Let \(U_{\rm floor}\) be a proved lower bound on phase potential in this cell and choose \(E_0\ge H(0)-U_{\rm floor}\), where

\[ H=\tfrac12x^TLx+U(\theta),\qquad U=\sum_{\{i,j\}\in E}[1-\cos(\theta_j-\theta_i)]. \]

The complete-law identity is \(H_\tau=-\|Ax\|_M^2\). Consequently

\[ H_\sigma=-\frac{\|Az\|_M^2}{\eta^2},\qquad \int_0^\Sigma\|z\|_M^2\,d\sigma \le\frac{\eta^2E_0}{\lambda^2}. \tag{7} \]

The factor is \(\eta^2\): both the slow clock and the retained form scaling contribute. The inequality applies before any first exit of the actual phase from the cell. Positivity of the form storage and the phase floor bound the available storage decrease.

For \(v=y-\psi\), split its derivative as \(f(y)-f(\psi)+f(\theta)-f(y)\). Equation (6) and Lipschitz continuity give the upper norm derivative bound \(d\|v\|_M/d\sigma\le\Lambda\|z\|_M\), including zero norm by the upper Dini derivative. Cauchy-Schwarz and (7) yield

\[ V:=v_0+\frac{\Lambda\eta\sqrt{\Sigma E_0}}{\lambda}, \qquad \sup_{0\le\sigma\le\Sigma}\|y-\psi\|_M\le V. \tag{8} \]

On the acute reference trajectory,

\[ \frac d{d\sigma}\|f(\psi)\|_M^2 =-2f(\psi)^T\mathcal H(\psi)f(\psi)\le0. \]

Thus any \(F_0\ge\|f(\psi(0))\|_M\) bounds the reference speed. Variation of constants in the first row of (5), on the zero-weighted-mean space, now gives the closed bounds

\[ Z:=\frac{Z_0+(\eta/\lambda)(F_0+\Lambda V)} {1-\eta\Lambda/\lambda},\qquad \sup\|z\|_M\le Z,\qquad \sup\|\theta-\psi\|_M\le Q:=V+Z, \tag{9} \]

provided \(\eta\Lambda/\lambda<1\). Indeed, on any finite prefix the form supremum is at most \(Z_0+(\eta/\lambda)[F_0+\Lambda(V+\sup\|z\|_M)]\). Solving that inequality proves (9); no derivative of an unknown forcing or commutation of graph operators is used.

Equations (7)--(9) initially hold up to the first possible chart exit. If the proved reference edge margins exceed the edge projections of \(Q\), they preclude that exit for both \(\theta\) and \(y\). The reference cell is convex, so the segment premise in (6) is retained. This strict first-exit argument closes the comparison without assuming the conclusion. The estimate controls a finite horizon; its square-root growth in (8) is not an all-time capture or convergence theorem.

Explicit bounds for the fixed two-port preparation

The graph has weighted mass forty and diameter nine. For any real vector, the weighted variance is at most one quarter of its squared range, while the squared range is at most nine times \(x^TLx\), by Cauchy-Schwarz on a path between its extreme nodes. Hence

\[ \|P_Mx\|_M^2\le\frac{40\cdot9}{4}x^TLx, \qquad \lambda=\frac1{90}. \tag{10} \]

The weighted variance bound follows from \((x_i-\min x)(\max x-x_i)\ge0\) and averaging with positive weights. Equivalently, \(L-\lambda[M-dd^T/40]\) is positive semidefinite, with \(d=M\mathbf1\). The normalized Laplacian has norm at most two, and \(-L\preceq\mathcal H(\theta)\preceq L\); therefore \(\Lambda=2\) is a global Lipschitz bound.

Initialize the reference at the centered representative of (2). Every internal sine sum vanishes. Its four port rates have magnitudes \(\sin\delta/3\), with opposite signs at the two ports of each component. Consequently

\[ \|f(\psi(0))\|_M^2=\frac43\sin^2\delta <\frac{40}{243}<\left(\frac5{12}\right)^2, \qquad F_0=\frac5{12}. \tag{11} \]

Before any reference exit, its speed norm does not increase. Each edge functional has weighted dual norm \(\sqrt{1/d_i+1/d_j}\le1\). Every edge thus changes by at most \(F_0\Sigma=5/48\). The smaller initial acute margin is \(\pi/18>1/6\), so every reference edge retains margin strictly greater than \(1/6-5/48=1/16\) throughout \([0,\Sigma]\). A first-exit argument proves that this reference estimate holds on the entire interval.

The two actual ring periods remain two and one on this cell. Jensen's inequality for \(1-\cos s\) on the acute interval supplies

\[ U_{\rm floor}=9[1-\cos(4\pi/9)]+9[1-\cos(2\pi/9)]. \tag{12} \]

The contact terms are nonnegative. Initially, their nominal sum is at most \(\delta^2<10/81\). The global phase-potential Lipschitz estimate and the twenty-edge form quadratic give

\[ H(0)-U_{\rm floor} <\frac{10}{81}+40r_\theta+40r_x^2. \]

Require the exact rational admission

\[ \frac{10}{81}+40r_\theta+40r_x^2<\frac18. \tag{13} \]

This retains all form uncertainty, rather than substituting a nominal zero-form energy. It supplies \(E_0=1/8\) in (7)--(8).

The elementary bounds on pi imply

\[ \frac1{3216}<\gamma<\frac1{3069},\qquad \eta<\overline\eta:=\frac1{9000000},\qquad \sqrt{\Sigma E_0}<\frac3{16}. \]

For each member, center the form and shift the nominal reference by that member's conserved phase-mean error. Weighted centering is contractive. Since \(\sqrt{40}<7\), take the following entirely rational quantities:

\[ \begin{aligned} Z_0&=7r_x/3069,&v_0&=7r_\theta+Z_0,\\ V&=v_0+2\overline\eta(3/16)90,\\ Z&=\frac{Z_0+90\overline\eta(5/12+2V)} {1-180\overline\eta},& Q&=V+Z. \end{aligned} \tag{14} \]

All substitutions enlarge the earlier upper bounds. Require \(Q<1/16\). The reference margin from (11) then keeps both actual phase and mixed coordinate strictly acute throughout the finite interval. This also preserves the ring periods and the zero period of the four-edge cycle (0,9,10,1); the three cycles are the integer basis used in the compatibility owner. The proof retains all complete-state coordinates and memberwise means. In particular,

\[ \sup\|P_Mx\|_M<3216Z,\qquad |\overline x_M|\le r_x. \tag{15} \]

These are bounds on original form, not merely on its scaled surrogate.

The actual short gaps move by more than the declared amount

Let \(a(\sigma)=\theta_1-\theta_0\) and \(c(\sigma)=\theta_{10}-\theta_9\) in the retained acute lifts. The nominal reference has initial derivatives

\[ a_\psi'(0)=-\tfrac23\sin\delta,\qquad c_\psi'(0)=\tfrac23\sin\delta. \]

Both short-edge functionals have weighted dual norm \(\sqrt{2/3}<g:=5/6\). Since \(\|\psi''\|_M\le\Lambda F_0\) on the interval, Taylor's integral remainder bounds either signed reference displacement from below by

\[ \tfrac23\sin\delta\,\Sigma -\tfrac12g\Lambda F_0\Sigma^2. \]

Here \(\sin(\pi/9)>\sin(1/3)>1/3-(1/3)^3/6=53/162\). At \(\Sigma=1/4\), the displayed lower bound therefore exceeds \(53/972-25/1152\). The actual endpoint gap differs from its mean-matched reference by at most \(gQ\); each actual initial gap differs from its nominal value by at most \(2r_\theta\). Define

\[ J=\frac{53}{972}-\frac{25}{1152}-\frac56Q-2r_\theta. \tag{16} \]

Every admitted complete-state member satisfies

\[ a(0)-a(\Sigma)>J,\qquad c(\Sigma)-c(0)>J. \tag{17} \]

These are changes relative to each member's own actual initial gaps, not relative to idealized or reset starting values. They need not be monotone at every intermediate instant; for nominal zero form the actual phase velocity initially vanishes. The statement is a strict finite endpoint change under the full form-phase law.

For the explicit choice \(r_x=r_\theta=1/65536\), (13) holds and rational evaluation of (14) gives \(Q<1/8192\). In particular the entire finite acute margin is positive, and

\[ J>\frac1{32}+\frac{11491}{7962624}>\frac1{32}. \tag{18} \]

Thus at the declared time the donor short gap has decreased by more than \(1/32\) radian and the receiver short gap has increased by more than \(1/32\) radian, for every state in this full error family. Both ring identities and the interface period persist throughout the interval.

The implemented sufficient certificate requires (13), a strictly positive bootstrap margin \(1/16-Q\), and a strictly positive directional margin \(J-1/32\). A nonpositive reported margin leaves the requested certificate unavailable; it is not an observed reversed motion or an instability claim. The rational rules are declared explicitly even when a sharper strict inequality could resolve an equality case.

The shared assess_sine_two_port_transit returns SineTwoPortTransit from the mandatory nonnegative primitives form_error_radius and phase_error_radius. It rebuilds the actual support, centered rational source turns, integer cycle periods and normalized spectral slack matrices before applying the exact rational bounds. Candidate quantities remain separate from certified actual errors. If the energy or chart premise fails, actual error and directional fields remain unavailable. A certified error envelope and the stricter directional threshold retain separate flags. The SDK projection preserves these distinctions; no supplied endpoint, cached equilibrium report or prior capture verdict is consumed.

The result supplies the first bounded dynamical step beyond static compatibility and the failed direct storage handoff. It does not prove entry into the new equilibrium's basin, eventual convergence, all-time acuteness, formation of support, passive contact or physical binding. The prescribed origin, support, law, clock and uncertainty family remain supplied premises.