Capture after a certified two-port C9 transit

Prospective question and complete-state scope

The finite directional transit proves actual short-arc deformation through slow time \(1/4\). Its endpoint is not an admission to the compatible equilibrium's basin. This separate gate asks whether the same prepared full-state family can reach a certified basin after a longer, explicitly declared transit.

Retain exactly that owner's two unit C9 cycles, contacts (0,9) and (1,10), held unit capacities, beta one, \(e=1023/1024\), \(w=1/1024\), no forcing and no events. Let \(L\) be the actual unit Laplacian, \(M\) the actual degree matrix, \(K=M^{-1}\), and

\[ \gamma=\frac1{1023\pi},\quad\eta=\gamma^2,\quad \tau=et,\quad\sigma=\eta\tau. \]

The complete actual rows are unchanged:

\[ x_\tau=-KLx+\gamma KS(\theta),\qquad \theta_\tau=\gamma KLx, \quad S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i). \tag{1} \]

Keep nominal initial form zero and the same centered, midpoint-aligned undeformed winding pair. In local indices \(j=0,\ldots,8\), its raw phase turns are

\[ \widetilde\Theta_{{\rm D},j}=(j-\tfrac12)\frac29,\qquad \widetilde\Theta_{{\rm R},j}=(j-\tfrac12)\frac19. \]

Subtract the single full-network weighted mean \(21/40\) turns to obtain \(\Theta^0\). Equivalently, use raw turns \(2j/9\) and \(1/18+j/9\) and subtract their mean \(229/360\). These constructions give exactly the same eighteen centered phases. Supply the unchanged family

\[ |x_i(0)|\le r_x=\frac1{65536},\qquad |\theta_i(0)-2\pi\Theta_i^0|\le r_\theta=\frac1{65536}. \tag{2} \]

Form and phase errors are independent nodewise budgets in form units and radians respectively. No zero error sum or reflection constraint is imposed on the thirty-six actual coordinates. Each member retains its own conserved weighted form and continuous phase means.

The longer horizon below belongs to this new declaration. It neither lengthens the earlier evaluated quarter-unit result nor replaces its source by an implicit target or a state chosen near that target. The target used in the proof is the unique correlated root of the two-port compatibility equations. Its criticality and uniqueness are analytic premises already proved for the same support, law and \((2,1,0)\) period cell.

An exact eight-coordinate reference for the nominal source

Use the auxiliary slow gradient flow \(\psi_\sigma=KS(\psi)\), initialized at the nominal phase in (2). It is a proof reference, not the actual phase law. The full initial form and all actual errors enter the separate complete-law comparison below.

The nominal reference is invariant under simultaneous ring reflection \(j\mapsto1-j\pmod9\) and circular phase negation about the common contact midpoint. In the raw midpoint frame the fixed nodes have phases \(\pi k\) for class \(k\). Their two sine terms cancel, so their reference rates vanish. The support and both contact edges respect the reflection. Uniqueness of the smooth gradient flow therefore preserves this symmetry for the exact nominal reference.

For a ring of class \(k\), retain four turn coordinates \(u_{k,0},\ldots,u_{k,3}\) at local nodes 1,...,4. Its nine raw turns are exactly

\[ (-u_{k,0},\ u_{k,0},\ u_{k,1},\ u_{k,2},\ u_{k,3},\ k/2, \ k-u_{k,3},\ k-u_{k,2},\ k-u_{k,1}). \tag{3} \]

The full-network weighted mean of this reconstructed vector is always \(7(k_{\rm D}+k_{\rm R})/40=21/40\) turns, independently of \(u\). Subtract that mean to obtain the same centered full reference as above. The exact rational initial coordinates are

\[ u_{k,i}(0)=(i+\tfrac12)k/9,\qquad i=0,\ldots,3. \tag{4} \]

For either component let \(v_0\) denote the other component's port coordinate and set \(u_4=k/2\). In slow time the restricted equations are

\[ \begin{aligned} (u_0)_\sigma &=\frac{-\sin(4\pi u_0)+\sin[2\pi(u_1-u_0)] +\sin[2\pi(v_0-u_0)]}{6\pi},\\ (u_i)_\sigma &=\frac{\sin[2\pi(u_{i-1}-u_i)] +\sin[2\pi(u_{i+1}-u_i)]}{4\pi}, &&i=1,2,3. \end{aligned} \tag{5} \]

Reconstruction (3) verifies every one of the eighteen reference rows, including both degree-three contacts and the two fixed nodes. This is an exact invariant restriction of that reference, rather than a phase closure assumed for the actual complete system.

If two folded states differ by \(h\), their reconstructed physical phase difference has squared full weighted norm

\[ \|\Delta\psi\|_M^2=(2\pi)^2h^TWh,\qquad W=\operatorname{diag}(6,4,4,4,6,4,4,4). \tag{6} \]

The factors six and four count both mirrored nodes with their actual degrees. The fixed nodes contribute no deviation. On any convex tube whose reconstructed edges remain in the same acute branches, the full gradient field is dissipative in the \(M\)-norm. Equation (6) proves that the folded field has logarithmic growth rate at most zero in the \(W\)-norm. No numerical eigenvalue or absolute-Jacobian growth estimate is substituted for this exact metric statement.

The correlated equilibrium also has a folded representative. For its root turns \(A=a/(2\pi)\), \(C=c/(2\pi)\), these coordinates are

\[ u_{{\rm D},i}^*=A/2+i(2-A)/8,\qquad u_{{\rm R},i}^*=C/2+i(1-C)/8. \tag{7} \]

Reconstruction and the same mean subtraction agree with the compatibility owner's exact implicit target. Independent rounded phase midpoints are not declared to be critical.

For numerical evaluation use an equivalent affine coordinate chart: eight radian deviations \(\xi\) from the fixed nominal phase, at full-node representatives (0,2,3,4,9,11,12,13). For each component the conversion from the preceding absolute turns is

\[ \xi_{k,0}=-2\pi[u_{k,0}-u_{k,0}(0)],\qquad \xi_{k,i}=2\pi[u_{k,i}-u_{k,i}(0)],\quad i=1,2,3. \tag{7a} \]

The minus sign selects local node zero rather than its mirror, node one. Write \(p\) for the reflection permutation and let the reconstruction matrix have entries \(E_{ir}=1\) when \(i\) is representative \(r\), \(E_{ir}=-1\) when \(i=p(r)\), and zero otherwise. Then

\[ \psi=2\pi\Theta^0+E\xi,\qquad \xi(0)=0,\qquad E^TME=W,\qquad \mathbf1^TME=0. \tag{7b} \]

Thus the retained metric radius is directly in the full physical phase norm: \(\|E\Delta\xi\|_M=\|\Delta\xi\|_W\), with no additional factor of \(2\pi\). The reference rate of coordinate \(\xi_r\) is the full gradient rate at its representative node. Each edge angle is its nominal angle plus the corresponding row difference of \(E\xi\). This gives exactly (5) under (7a); it does not supply a second reference law. The constant nominal edge angles retain mathematical pi, while the initial numerical center and metric radius are exactly zero. Dissipativity and growth rate zero are unchanged by this scaled sign transformation.

Fixed reference-enclosure obligations

The shared validated_metric_taylor_step retains a full positive-metric uncertainty ball while using projected boxes only for Picard inclusion, derivatives and domain checks. Apply it to the affine radian field (7a)--(7b), with metric \(W\), declared growth rate zero and shared outward rational trigonometry. Pi remains the same mathematical constant in every row. Its outward enclosure covers arithmetic evaluation; it does not introduce independent physical constants in different coefficients.

For each step, reconstruct the twenty principal full-reference edge gaps with their fixed cycle branches. Require strict Picard inclusion and a whole-tube acute margin greater than

\[ \rho:=1/2048\quad\hbox{radian} \tag{8} \]

on every edge. This is stronger than merely checking midpoint or endpoint acuteness. The integer ring periods and the zero four-edge interface period remain those of the exact reconstruction on all admitted tubes.

At the fixed reference endpoint \(T=1024\), require

\[ \|\psi(T)-\theta_*\|_M\le\rho. \tag{9} \]

Here \(\theta_*\) has the same zero nominal phase mean. The endpoint metric radius and the uncertainty in the correlated implicit root must both be included in this upper bound. For example, if \(\widehat\xi\) is the retained radian center, \(r\) its radius in metric \(W\), and \(D_*\) encloses \(\|\widehat\xi-\xi_*\|_W\), then \(r+D_*\) is an admissible physical-norm upper bound. At each representative, \(\xi_{*,r}=\theta_{*,r}-2\pi\Theta_r^0\), using the same full-network zero gauge and the admitted implicit-root enclosure. A small center residual alone proves neither (9) nor an exact critical target.

Rebuild the compatibility root using its declared thirty-two outer and sixty-four inner strict-sign refinements. Require its admitted target to have minimum acute edge margin strictly greater than \(1/8\) radian. The old static proof supplies existence and criticality; the fresh enclosure checks this quantitative margin without replacing either proof by a verdict field.

If a step, target margin or endpoint obligation is unavailable, preserve the completed prefix and the failed condition. Neither increasing precision, extending the horizon, changing the source nor retrying a smaller step is part of this protocol. The reference's acuteness throughout its transit is an obligation of the enclosure, not a consequence of having acute endpoints.

One analytic continuation unit controls the original form

Assume the reference obligations above have passed. The ball in (9) lies strictly inside the target's acute chart: every edge functional has weighted dual norm at most one, while \(\rho<1/8\). The gradient-flow identity

\[ \frac12\frac d{d\sigma}\|\psi-\theta_*\|_M^2 =\langle\psi-\theta_*,f(\psi)-f(\theta_*)\rangle_M\le0 \]

prevents a first exit from this ball. Therefore (9) holds for all later reference times. During the additional declared unit \([T,T+1]\),

\[ \|f(\psi)\|_M\le2\rho=1/1024. \tag{10} \]

This continuation does not require additional numerical reference steps.

Apply the energy-informed complete-law comparison on the new total horizon \(\Sigma=T+1=1025\). Its weighted normalized gap \(\lambda=1/90\), Lipschitz bound two and phase floor remain valid on the same support and period cell. The fixed source satisfies

\[ H(0)-U_{\rm floor} <10/81+40r_\theta+40r_x^2<1/8. \]

Keep the same bounds \(1/3216<\gamma<1/3069\) and \(\eta<\overline\eta=1/9000000\). Since \(\sqrt{1025/8}<12\), define the rational quantities

\[ \begin{aligned} Z_0&=7r_x/3069,&v_0&=7r_\theta+Z_0,\\ V&=v_0+2\overline\eta\,12\,90,\\ Z&=\frac{Z_0+90\overline\eta(5/12+2V)} {1-180\overline\eta},&Q&=V+Z. \end{aligned} \tag{11} \]

For (2), exact rational arithmetic gives

\[ Q<1/2048=\rho,\qquad Z<1/200000. \tag{12} \]

Match the reference's common phase to each actual member's conserved phase mean. Its geometry and all enclosure margins are unchanged. Before any possible actual or mixed-coordinate exit, the comparison gives \(\|\theta-\psi\|_M\le Q\) and \(\|z\|_M\le Z\), with \(z=\gamma P_Mx\). The reference tube margins on \([0,T]\) exceed \(\rho>Q\); the reference target ball provides even larger margins on \([T,T+1]\). The strict first-exit argument therefore validates these bounds throughout the whole new interval, for the full thirty-six-coordinate family. Symmetry of its actual errors was never required.

The global form envelope \(Z/\gamma\) alone is insufficient for a sharp endpoint handoff. Retain the moving reference on the last unit and use (10) instead of its larger initial forcing bound. Variation of constants for \(z_\tau=-Az+\eta f(\theta)\) gives

\[ \|z(T+1)\|_M \le e^{-\lambda/\eta}Z +\frac{\overline\eta}{\lambda}(2\rho+2Q) <2^{-512}Z+\frac1{51200000}=:Z_{\rm end}. \tag{13} \]

The exponential estimate is analytic: \(\lambda/\eta>100000>512\) and \(\exp(1)>2\), so \(e^{-\lambda/\eta}<2^{-512}\). No unknown forcing is differentiated, and the original form is not replaced by a quasistatic algebraic row. Using (12) and the lower bound on \(\gamma\) yields

\[ \|P_Mx(T+1)\|_M<3216Z_{\rm end}<1/8192. \tag{14} \]

The actual phase bound at the same time is

\[ \|\theta(T+1)-\theta_*^{\,m}\|_M \le Q+\rho<1/1024, \tag{15} \]

where \(\theta_*^{\,m}\) is the exact compatible target shifted to that member's conserved phase mean. Its uniform limiting form will likewise equal the member's conserved form mean, not necessarily zero.

A strict full-state basin proves subsequent convergence

On that fixed-mean leaf use

\[ \mathcal R^2=\|P_Mx\|_M^2+\|\theta-\theta_*^{\,m}\|_M^2, \qquad R=1/12. \]

Within \(\mathcal R\le R\), each edge differs from its target by at most \(R\). The target margin exceeds \(1/8\), so every segment from the target has cosine at least

\[ \sin(1/8-1/12)=\sin(1/24)>1/25. \]

Criticality cancels the linear phase-potential term. The graph gap and Taylor's integral formula then give the lower bound

\[ H-H_* \ge\frac{\lambda}{2} [\|P_Mx\|_M^2+(1/25)\|\theta-\theta_*^{\,m}\|_M^2] \ge\frac{\mathcal R^2}{4500}. \tag{16} \]

Here \(H_*\) is the exact target potential; common form shifts have zero form storage. On the radius boundary the storage excess is therefore at least

\[ B_R=R^2/4500=1/648000. \tag{17} \]

Conversely, the normalized Laplacian upper bound two and \(\mathcal H(\theta)\preceq L\) give \(H-H_*\le\mathcal R^2\) for the endpoint, using the same exact criticality. Equations (14)--(15) prove that endpoint is inside the radius ball and has

\[ H(T+1)-H_*<\frac1{8192^2}+\frac1{1024^2} =\frac{65}{67108864}<\frac1{648000}. \tag{18} \]

Thus the complete state, including its original form, enters a strict forward-trapped sublevel. The continuous identity \(H_\tau=-(Lx)^TK(Lx)\le0\) precludes exit through the radius boundary. This is an excess-storage barrier around the new target after a proved transit; it does not contradict the earlier obstruction to a direct initial full-sector storage certificate.

On the fixed-mean leaf the sublevel is compact. Its largest invariant zero-loss subset has \(Lx=0\), hence uniform form and zero phase velocity. Preserving it further requires \(S=0\); summing the sine rows removes any putative common nonzero form rate. The strict acute period-cell uniqueness proved by the compatibility owner then leaves only \(\theta_*^{\,m}\). LaSalle's invariance argument proves convergence of every member to this compatible geometry and its own uniform form mean. All later uninterrupted flow retains the periods. The existing local exponential stability result applies near the limiting equilibrium, without supplying a new uniform acquisition or decay rate.

Frozen protocol and stopping rule

Freeze the following before the first new reference-transit assessment:

• The support, complete rows, clocks, source and independent radii are exactly (1)--(2). There is no source, origin, coefficient or error-budget search, support event, forcing or reset.
• Rebuild the primary (2,1) implicit target with thirty-two outer and sixty-four inner strict-sign refinements. Require its analytic equilibrium admission and outward minimum acute margin strictly above \(1/8\) radian.
• Propagate only the exact nominal eight-coordinate reference, using the affine radian deviations (7a)--(7b), zero initial center and radius, and representatives (0,2,3,4,9,11,12,13). Use metric \(W\), growth bound zero, shared dyadic128 interval trigonometry, Taylor order eight and fixed step \(1/4\) in slow time. The reference horizon is \(T=1024\): exactly 4096 steps are required. This is a declared finite work policy for this experiment, not a change to a shared kernel's mathematical hypotheses or dimension limits.
• Require strict Picard inclusion and all twenty reconstructed whole-tube acute margins above \(1/2048\) radian at every step. Retain the metric radius; projected endpoint boxes do not replace it.
• Require the final reference metric distance, including target-enclosure uncertainty, to be at most \(1/2048\) radian. Do not stop successfully at an earlier favorable state or extend the final time after inspecting the endpoint.
• Use the analytic last slow-time unit, not additional trajectory steps, to obtain (13)--(15). The complete-state capture time is \(\sigma_*=1025\), \(\tau_*=1025\cdot1023^2\pi^2\), and \(t_*=1025\cdot1023\cdot1024\pi^2\).
• Certify capture only if target, complete reference prefix, endpoint, comparison bootstrap, original-form tail and strict full-state basin obligations all pass. Preserve an unavailable result and its completed prefix on any failure. Do not retry, subdivide adaptively or silently enlarge the budget.
• Archive the complete producing source, this prospective proof and the protocol before evaluation. Retain the response and verdict separately. Do not regenerate earlier frozen producers or revise their predictions.

At archival this proof establishes a conditional capture mechanism; the fixed new reference assessment has not yet been performed. The outcome must be recorded separately from the declaration above. In particular, the proved local equilibrium and the earlier directional result do not count as passing the new transit or endpoint obligations.

Even a successful result concerns relaxation of two supplied winding patterns on supplied support. It does not derive their preparation, choose contact occurrence, establish passive attachment, prove autonomous formation or identify a physical composite. No laboratory clock, sensor model or physical binding energy follows from this conditional mathematical capture result.

Retained first capture result

The first assessment of the unchanged archived protocol returned certified_capture, with no unavailable reasons. All ten frozen stopping conditions passed. Every one of the 4096 reference steps was admitted at the declared quarter-unit step and Taylor order eight, reaching reference slow time \(T=1024\). The additional unit to complete-state time \(\sigma=1025\) uses the analytic continuation and original-form estimate above; it is not an additional numerical trajectory or an equilibrium reset.

The saved record retains exact rational bounds. The following decimal values are approximate displays of those bounds, not measured errors or replacement certificate thresholds:

Retained quantity	Approximate bound	Declared requirement
Minimum whole-reference-tube acute margin	0.04572542278762468 rad	Strictly above 1/2048 rad
Reference endpoint metric radius	5.418710124146611e-9 rad	Retained in the endpoint distance
Reference distance to the implicit target at \(T\)	5.684776349788719e-9 rad	At most 1/2048 rad
Full-family phase distance at \(\sigma=1025\)	3.510605027044329e-4 rad	Below 1/1024 rad
Original relative-form norm at \(\sigma=1025\)	5.39860958891343e-5	Below 1/8192
Full endpoint excess-storage upper bound	1.261579751084399e-7	Below 1/648000
Positive capture-storage margin	1.41705190143477e-6	Strictly positive

The reference endpoint distance includes its retained numerical radius and the implicit-target enclosure. The actual phase and form bounds instead include every member of the unchanged error family (2), through the complete-law comparison. A small nominal-reference distance alone would not have established either full-state bound.

The strict endpoint storage margin now admits the complete family into the forward-trapped neighborhood (16)--(18). Consequently, under uninterrupted evolution by (1), every member converges to the same compatible phase geometry with its own conserved common phase origin, and to uniform form equal to its own conserved form mean. Its winding periods remain \((2,1,0)\). This is a source-to-basin result for the stated finite uncertainty family, beyond the earlier local stability of an equilibrium supplied without a source.

The earlier initial-storage obstruction remains valid. It excluded a direct full-sector energy certificate for these undeformed sources. The present result supplies the previously missing transit information and applies a local full-state barrier only after that transit. It does not reinterpret the obstruction as a dynamical failure or replace the old result.

The retained protocol, producing source archive, exact response and evidence manifest preserve this first evaluation. The archive SHA-256 is b8df26578b59a32dae82b9cad58450b0d208d6c45dc234769575702830f0fb3f, based on revision 29e4c02339f6246d003a7fbfbdf7f37c8a2c1277 with the two declared runtime overlays. The prospective proof above remains byte-identical to the archived prefix. Hashes establish content consistency and recoverability, not independent chronology authentication or physical acquisition.

The contract and inspection guide retain the distinction between the validated nominal reference and the full-law convergence conclusion. The independent field controls and read-only evidence audit cover affine reconstruction, metric and gauge, all-node field agreement, step-chain consistency, admitted edge margins and the exact handoff bounds. The compact step record does not independently reproduce every Taylor remainder or authenticate the producing execution.

The supplied support, complete law, preparation and structural clocks remain premises. Convergence to joint geometry does not derive contact occurrence, passive attachment, autonomous formation or physical binding.