Coherent pattern contact and formation boundaries

This note owns the conditional two-ring contact result. It reuses the supplied phase law, cycle-period geometry and isolated-ring relaxation. The execution plan alone owns the next research task.

Its formation boundary separates this prepared survival from formation under the same phase law: isolated gated C5 cannot enter an acute winding sector from winding zero. The one-neighbor extension derives the actual contact budget and proves that this entrance obstruction persists with one fixed external phase neighbor. The distributed interface admits recovery from an existing winding. The gated potential then constrains formation from winding zero, including an exact exclusion for an initially uniform recipient.

Model boundary. These are primitive phase patterns under an explicitly supplied averaged-sine law. They are not the derived form angles of the passive collective model. The result concerns prepared identity during interaction. It does not derive the phase law, create support, establish spontaneous pattern formation or identify particles.

The later relational region interaction reuses this geometry and regional accounting with a different, fully coupled phase/form law. This note's supplied sine evolution and numerical contact results must not be transferred to that model.

The phase-premise review separates this supplied law from operator admission and derived form phase. Changing the cutoff changes the preparation bounds; the exclusions below cannot be transferred unchanged to another phase law. The remaining gate-itinerary question is open and deferred in the sole execution plan.

Model and prospective control

Prepare two unit undirected cycles, A=(0,1,2,3,4) and B=(5,6,7,8,9). For local index k=0,...,4, set

\[ q=2\pi/5,\qquad \theta_{A,k}(0)=qk,\qquad \theta_{B,k}(0)=qk+\beta,\qquad x_i(0)=1/2. \]

Capacity is held at one. Add a reciprocal unit bridge (0,5) for 0<=t<=T=1/2, then remove it without changing the nodal state. All transport weights and diagnostic lengths are one. The support schedule is a supplied intervention, not a generated operator sequence. On the contact graph, d_0=d_5=3 and all other degrees are two.

Choose equal normalized EPI and phase pressure coefficients, with capacity and topology coefficients zero. The latter must be selected explicitly: the degree-three ports make the joined graph irregular. No Gamma, capacity adaptation or glyph event is present. On the fully admitted acute domain,

\[ \dot\theta_i=1+\frac1{d_i}\sum_{j\sim i}\sin(\theta_j-\theta_i), \qquad \dot x=-\tfrac12L_{rw}x+\tfrac12g(\theta), \]

where L_rw=I-D^{-1}W and

\[ g_i=\frac1\pi\operatorname{wrap}\left( \operatorname{Arg}\sum_{j\sim i}e^{i\theta_j}-\theta_i\right). \]

The executable phase law has a U3 gate of pi/2. The proof below ensures that every edge remains strictly inside it; it does not replace the law outside admission by an ungated continuation. The structural clock and the identification of held capacity with free angular rate are supplied model choices, not consequences of dx/dt=nu_f*DeltaNFR.

Frozen before production execution, 2026-09-26: use only beta=pi/24 and beta=pi/12, with h=1/128, 64 same-snapshot Euler steps, and clipping rails [0,1]. Initial mathematical turns are k/5 and k/5+1/48 or k/5+1/24. Record their binary64 materialization separately. Use the existing execute_joint_step test adapter: NumPy pressure assembly, the shared U3 phase proposal, scalar DefaultIntegrator EPI execution and refreshed endpoint pressure. The scalar integrator path is selected locally in the test; this does not relabel the vectorized pressure path. There is no random preparation, fit, tolerance-selected identity or sweep.

Identity means each original oriented cycle retains winding one. Phase deformation, EPI response, resultant availability and acute admission are separate observations. Winding at isolated endpoints alone does not prove path preservation. The control must retain consistent phase lifts and check all eleven edges, including the bridge.

A sufficient energy barrier and its limitation

In the co-rotating frame, the cosine phase energy satisfies

\[ V=\sum_{\{i,j\}}(1-\cos(\theta_j-\theta_i)),\qquad \dot V=-\sum_i d_i(\dot\theta_i-1)^2\le0. \]

Unequal port degrees change the mobility, not this identity. One acute winding-one C5 has minimum V_*=5(1-cos(q)). At its first acute boundary, the offending oriented gap must be +pi/2: a gap of -pi/2, with the other four at most pi/2, cannot have total 2*pi. Convexity of 1-cos on the acute interval gives

\[ V_{\partial}\ge1+4(1-\cos(3\pi/8)),\qquad \Delta=V_{\partial}-V_*=5\cos(2\pi/5)-4\cos(3\pi/8). \]

The other ring still costs at least V_*; a bridge acute boundary costs one. Thus 1-cos(beta)<min(Delta,1) is sufficient to prevent exit. The low offset satisfies it and the high offset does not. For example, elementary squared-rational bounds give 0.0142<Delta<0.01825, whereas 1-cos(pi/24)<0.00858 and 1-cos(pi/12)>0.03375. Failure of this sufficient test is inconclusive, not a prediction of loss. The stronger comparison below decides both frozen cases without changing their model or preparation after execution.

A shared rotating reference protects both identities

The same joined graph has the exact reference

\[ \theta^*_{A,k}(t)=\theta^*_{B,k}(t)=t+qk. \]

The two ring sine currents cancel at every node, including the ports; the reference bridge current is zero. Use continuous lifts and set a_i=theta_i-theta_i^*. Initially a_A=0, a_B=beta. Reference edge representatives are delta^*=+q,-q,0. Their fixed integer branch shifts do not change the sine. Subtraction gives

\[ \dot a_i=\frac1{d_i}\sum_{j\sim i}c_{ij}(a)(a_j-a_i),\qquad c_{ij}(a)=\int_0^1\cos\bigl(\delta^*_{ij}+s(a_j-a_i)\bigr)\,ds. \]

If 0<beta<pi/10, every state in the box 0<=a_i<=beta has

\[ 0<\cos(q+\beta)\le c_{ij}\le1. \]

At a maximal offset the derivative is nonpositive; at a minimal offset it is nonnegative. The maximum principle preserves the box. Since q+beta<pi/2, the box is strictly inside the acute domain and closes the bootstrap for all times on this fixed contact graph. Smoothness there and bounded rates provide continuation of the specified solution.

Both predeclared offsets are below pi/10. Every original oriented cycle gap remains in [q-beta,q+beta]; the bridge gap stays in [-beta,beta]. Their lifts therefore cannot encounter an antipodal branch crossing, and each original winding remains one. This predicts persistence of a geometric identity while allowing its shape to change. It is not a claim that arbitrary preparations, capacities, interfaces or primitive laws have this property.

Phase geometry bounds the form response

In the reference frame at a ring node, the two ring-neighbor phasors sum to

\[ 2\cos\left(q+\frac{a_{\rm next}-a_{\rm prev}}2\right) \exp\left(i\frac{a_{\rm next}+a_{\rm prev}}2\right). \]

Its real amplitude is positive and its direction lies in [0,beta]. At a port, adding the bridge phasor with direction in the same cone preserves nonzero resultant and that direction bound. Consequently

\[ |g_i|\le\beta/\pi,\qquad \|x(t)-\tfrac12\mathbf1\|_\infty\le\frac{t\beta}{2\pi}. \]

The second inequality follows from positivity and maximum-norm contraction of the EPI diffusion semigroup, or its extremum principle. At T=1/2, the two radii are exactly 1/96 and 1/48. Both envelopes are strictly inside the frozen rails, so ideal clipping is inactive. Phase geometry controls a genuine form response; it is not an independent sustaining input.

The interaction is already nonzero at the initial state. Only the two ports have nonzero co-rotating phase rates, respectively +sin(beta)/3 and its negative. Their initial form rates are

\[ \dot x_0(0)=-\dot x_5(0)= \frac1{2\pi}\operatorname{atan2}\bigl( \sin\beta,\,2\cos q+\cos\beta\bigr)>0. \]

Every other initial form rate is zero. This cancellation uses the declared preparation symmetry; the general phase source is not transport-conservative.

Discrete and represented execution have separate obligations

In exact arithmetic, phase Euler has the offset update

\[ a_i^+=\left(1-\frac h{d_i}\sum_jc_{ij}\right)a_i +\frac h{d_i}\sum_jc_{ij}a_j. \]

For h<=1, its coefficients form a convex row. It preserves the same box, and so does straight interpolation of the consistently lifted offsets. For EPI Euler, I-(h/2)L_rw is stochastic for h<=2; the source estimate gives the same finite form envelope. These are properties of the ideal discrete method, separately from the continuous theorem.

An actual binary64 trace has materialization, phase proposal, pressure and integration defects. Retain these separately. An initial offset error r_0 and per-step phase proposal error bounds epsilon_k would inflate the box to [-r_k,beta+r_k], with r_k=r_0+sum epsilon_j. The sufficient admission condition becomes q+beta+2*r_k<pi/2, and the corresponding source bound is (beta+2*r_k)/pi. A high-precision recomputation is a numerical residual estimate, not automatically an outward-rounded interval certificate.

The finite control checks the actual lifted segment endpoints on consistent branches; an affine gap between acute endpoints remains acute throughout that segment. Small lifted nodal increments and explicit edge consistency checks rule out an unobserved whole-turn jump in the chosen interpolation. This establishes the retained discrete path property, not an error bound between that interpolation and the continuous solution. Source assembly and EPI Euler residuals reuse the existing forcing/transport observations.

Finite production evidence

The bounded control passes five contact tests, sharing the two frozen trajectories through one module fixture. On both traces, both original windings remain +1, every edge and each retained lifted segment stays acute, and every EPI proposal is strictly inside the rails. Initial port responses agree with the independent degree-three formulas above. Removal leaves the entire nodal state unchanged.

Readout	beta=pi/24	beta=pi/12
Predicted endpoint form radius	1/96	1/48
Observed maximum endpoint form deviation	0.00444678019	0.00891375558
Maximum endpoint phase deformation, relative to initial co-rotating phase	0.0177181698	0.0352466810
Minimum acute margin over all edges and stored states	0.297115913	0.280229412
Observed cosine-energy change	-0.00382244056	-0.01512052456
Endpoint regional form means	(0.501148432, 0.498851568)	(0.502299569, 0.497700431)

Across the two controls the largest local lifted-phase residual estimate is approximately 4.44e-16; the largest accumulated phase-plus-initial estimate is 2.23e-14. The source residual estimate is at most 3.95e-17. The exact represented pressure-assembly and EPI Euler state defects have displayed maxima 1.16e-18 and 5.55e-17, respectively. Euler accounting has zero identity residual. Tests use an explicit absolute comparison tolerance 1e-12, with no default relative-tolerance substitution. Numerical energy comparisons are finite observations, separately from the exact ODE dissipation identity; no complete-runtime stability conclusion follows.

Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1 and mpmath 1.3.0, with 80-digit residual recomputation against the represented inputs/lifts. The pressure path, scalar integrator, initialization and step schedule are fixed above. The JUnit receipt records those versions and per-case metrics from the same fixture, without additional trajectories:

python -m pytest tests/physics/test_coherent_pattern_contact.py -k "not formation_control and not contact_budget_control" -q --junitxml=artifacts/research/coherent_pattern_contact_2026_09_26.xml

The receipt is a local ignored artifact; this owner and the executable control retain the result and reproduction recipe. This verification covers the selected contact model and its five tests, not the full research inventory or empirical physics.

Removal and scope of recovery

Removing the bridge changes support and subsequent rates, without a jump in EPI, capacity or phase. The following recovery statement concerns the unclipped continuous model; the finite runtime control stops at removal and does not certify all-time clipping inactivity afterwards. Each endpoint is inside its ring's acute winding-one basin. The existing isolated-ring result then applies: gaps relax to q, with each ring retaining its own common phase offset, and EPI relaxes to that ring's post-contact mean. Degree-two acute ring phase sources have zero sum, so that form mean is conserved after removal. Contact does not generally preserve each regional mean; recovery to the original EPI value 1/2 is not asserted. No additional numerical long-time continuation is required by this control.

The result supplies conditional interaction, bounded deformation and recovery of an already prepared identity. Formation from preparations without that identity, independent justification of the primitive phase/capacity laws and physical correspondence remain open. The energy-test failure in the second case is resolved by shared nodal comparison, not by post-evaluation tuning.

Formation boundary under the same gated law

Prepared contact survival does not imply that the same law can form the identity on an isolated ring. In fact, the fixed unit C5 with common capacity, unit coupling and the literal pi/2 gate has a stronger entrance obstruction than the existing common-semicircle theorem. This result concerns phase geometry independently of the form row; pressure still reads full support.

An exact obstruction to acute winding formation

Let delta_i=wrap(theta_(i+1)-theta_i) be principal oriented gaps on an interval without antipodal edges. For winding one, sum delta_i=2*pi. Write b=pi/2 and define nonnegative gap excess and negative-gap mass

\[ E=\sum_i(\delta_i-b)_+,\qquad N=\sum_i(-\delta_i)_+. \]

If N>0, at most four of the five gaps are positive. Hence

\[ 2\pi+N=\sum_i(\delta_i)_+\le4b+E=2\pi+E, \qquad N\le E. \]

This count is specific to C5. It must not be generalized to every cycle, network or TNFR model.

Define admission a_i=1 when abs(delta_i)<=b, and zero otherwise. The co-rotating phase rate is

\[ f_i=\frac{a_i\sin\delta_i-a_{i-1}\sin\delta_{i-1}}{a_{i-1}+a_i}, \]

with zero coupling rate when the denominator is zero. In particular, if delta_i>b, the edge itself is excluded. Its endpoints have at most one other admitted neighbor, so the actual admitted-count normalization gives

\[ \dot\delta_i=a_{i-1}\sin\delta_{i-1}+a_{i+1}\sin\delta_{i+1}. \]

Each admitted negative gap appears at most twice when summing these rates over positive-excess edges. Positive contributions help, and sin(delta)>=delta for negative principal gaps. The positive-part chain rule therefore gives, almost everywhere,

\[ \dot E\ge-2N\ge-2E, \qquad E(t)\ge e^{-2(t-s)}E(s). \]

This covers simultaneous gate crossings and the inclusive threshold at equality. The statement applies to every absolutely continuous solution satisfying the literal gated equation almost everywhere. It does not assert global existence or uniqueness for a discontinuous vector field or silently replace it by a differential inclusion.

Suppose there were an entirely acute winding-one endpoint. On the maximal preceding antipodal-free interval, that endpoint has E=0; the estimate forces E=0 throughout the interval. Then N=0 and all gaps belong to [0,b]. This compact interval cannot approach an antipodal boundary, so the interval extends to the initial time, which must already have winding one. Reflection gives the winding-minus-one result.

Conclusion: a well-defined winding-zero preparation cannot enter, or converge to, a strictly acute winding-one or winding-minus-one lock under this fixed C5 law. Convergence to a strictly interior lock would imply finite entrance. The result permits winding changes, fragmented states and boundary accumulation; it does not exclude other supports, unequal capacities, external interactions or different constitutive phase laws. The isolated positive-formation search is closed under these premises.

A separate exact-Euler boundary

For ideal simultaneous Euler with 0<h<1/2, a step without an antipodal wrap on a winding-one branch obeys

\[ E_{n+1}\ge\sum_{\delta_i>b}(\delta_i+h\dot\delta_i-b) \ge(1-2h)E_n. \]

If an edge instead wraps at an antipode, its endpoint magnitude is at least pi-2*h>pi/2: the common free rate cancels and each coupling rate has magnitude at most one. Such an endpoint cannot be fully acute. Thus this step bound preserves the same no-entrance property for ideal Euler, separately from the continuous theorem. Binary64 residuals require their own accounting. Neither an excessively large step nor rounding-induced admission establishes a continuous formation mechanism.

A winding change with no coherent capture

The following family gives an explicit negative control beyond the common semicircle, using q=2*pi/5 and eta in [pi/40,pi/20]:

\[ \theta(0)=(0,q,4q+\eta,2q,3q),\qquad x(0)=\tfrac12\mathbf1. \]

The initial winding is zero. Only edges (0,1) and (3,4) are phase admitted; node 2 has no admitted phase neighbor. With

\[ d(t)=2\arctan\!\left(\tan(q/2)e^{-2t}\right), \]

the exact co-rotating solution is

\[ \theta(t)-t\mathbf1= (q/2-d/2,\ q/2+d/2,\ 4q+\eta,\ 5q/2-d/2,\ 5q/2+d/2). \]

It follows from the independent pair equation d_dot=-2*sin(d). The other three edges remain excluded for all time: their principal-gap magnitudes have lower bounds 0.55*pi, 0.625*pi and 0.8*pi. The raw closing gap is -2*q-d. It passes through -pi at t_*=log(5)/4, when d=q/2, changing the support-cycle winding from zero to minus one. At the crossing itself the antipodal branch observation requires its explicit boundary treatment. The active phase support remains two pairs and an isolated node; no fully acute winding cycle forms.

All local pressure resultants remain nonzero. The positive eta also keeps the phase pressure away from its antipodal displacement boundary. Full-support pressure, unlike phase evolution, continues to consume all five edges. Its exact phase source is

\[ \begin{aligned} g_0&=(q+d)/\pi,&g_4&=-g_0,\\ g_1&=-3(q+d)/(4\pi)+\eta/(2\pi),& g_3&=3(q+d)/(4\pi)+\eta/(2\pi),\\ g_2&=1-\eta/\pi. \end{aligned} \]

Thus sum g_i=1, and, while unclipped, the arithmetic form mean is mean(x)=1/2+t/10. The nonzero winding accompanies fragmented phase coupling and persistent uniform form drive. It is not evidence of formation of the admitted coherent cycle or bounded stationary form.

Frozen before production execution, 2026-09-26: use only eta=pi/40, T=1/2, h=1/128 and 64 steps. Retain the same pressure coefficients, unit capacities, rails [0,1], absent Gamma, absence of events and NumPy-pressure/scalar-EPI execution adapter as the contact control. Use exact initial turns (0,1/5,13/16,2/5,3/5) before binary64 materialization. The bound abs(g_i)<=1 gives norm(x-1/2,inf)<=t/2, keeping the finite trace inside [1/4,3/4]. The independent endpoint mean prediction is 0.55.

The finite check must retain the actual admitted-neighbor counts, full pressure support, a lifted closing-edge branch passage, source and nodal Euler defects, and the absence of coherent capture. The observed discrete crossing time must not be called the exact ODE crossing time. This one control checks the implementation's distinction between winding and formation; the theorem, not a sweep or simulation, closes the isolated formation question.

Finite formation-control evidence

Three additional tests in the shared control module passed, with the earlier two contact trajectories deselected. A separate module fixture executes only the newly frozen trajectory. Independent pressure evaluations at the two analytic endpoints are snapshots, not additional trajectories.

The phase-admitted neighbors remain ((1,),(0,),(),(4,),(3,)), while every pressure row consumes both support neighbors. Winding changes from zero to minus one, but no retained state is fully acute. The lifted Euler closing-edge passage occurs in [51/128,13/32]; the separate exact ODE event time is log(5)/4, approximately 0.4023594781.

Endpoint observation	Prediction or error budget	Measured value
Arithmetic EPI mean	0.55	0.5499999999999999
Mean forecast error	Exact represented accumulated defect budget 6.6691e-16	1.2213e-16
Maximum form deviation	At most 1/4	0.21738514
Pair-gap error against the continuous solution	At most T*h=0.00390625, plus arithmetic	0.00252501

For the last row, d_dot=-2*sin(d) has abs(d_ddot)<=2, giving local Euler error at most h^2. Its Euler map is nonexpansive on [0,q] for the frozen step, hence the accumulated exact-arithmetic error is at most t*h. The independently propagated scalar pair recurrence is checked against both the analytic solution and the actual retained phase lifts; it does not replace the production phase owner.

The mean budget accumulates the actual rate-mean defect from 1/10 and the exact represented Euler endpoint defects. All proposals stay strictly inside the rails. Maximum local estimated phase and source-sum defects are approximately 2.22e-16 and 5.00e-16; the largest represented EPI Euler defect is 5.51e-17. Source-sum arithmetic is not a proof of a general conservation law: the nonzero sum here follows from this preparation's explicit phasor branches.

Versions, path, clock, mean budget and both kinds of crossing evidence are retained in the local JUnit receipt, using the same runtime versions and 80-digit residual arithmetic as the contact control:

python -m pytest tests/physics/test_coherent_pattern_contact.py -k formation_control -q --junitxml=artifacts/research/coherent_pattern_formation_2026_09_26.xml

This resolves a potential false formation claim without altering the phase gate or pressure channels. A coupled environment changes the port equations used in the isolated proof; any induced-formation claim must retain those extra rates and the supplied environment explicitly.

One external phase neighbor: budget and entrance obstruction

Keep the same recipient unit C5, common capacity one and literal gated averaged-sine law. Connect recipient node 0 to one evolving external node 5. The donor can be the other C5 used above. No donor trajectory is imposed: its reciprocal evolution remains part of the full graph. This section asks whether the extra port can induce an acute winding identity in a recipient that initially has winding zero.

Exact regional contact balance

Use the preceding winding-one branch variables delta, b, E, N and admitted ring-edge indicators a_i. Set

\[ \gamma=\operatorname{wrap}(\theta_5-\theta_0),\quad c=\mathbf1_{|\gamma|\le b},\quad n=a_4+a_0,\quad S=a_0\sin\delta_0-a_4\sin\delta_4. \]

The co-rotating rate with only the two internal ring neighbors is f_0^0=S/n when n>0, and zero otherwise. The actual full-graph rate is f_0=(S+c*sin(gamma))/(n+c), with the same empty-neighborhood convention. This is a same-snapshot algebraic comparison, not evolution of an isolated replacement graph. The contact correction is

\[ u=f_0-f_0^0= \begin{cases} 0,&c=0,\\ (\sin\gamma-f_0^0)/(n+1),&c=1. \end{cases} \]

It changes only the recipient gap rates delta_4_dot by +u and delta_0_dot by -u. With P={i:delta_i>b}, the exact excess balance is

\[ \dot E=R+u\sigma,\qquad R=\sum_{i\in P}(a_{i-1}\sin\delta_{i-1}+a_{i+1}\sin\delta_{i+1}), \qquad \sigma=\mathbf1_{4\in P}-\mathbf1_{0\in P}. \]

As before, R>=-2*N>=-2*E. If both port-adjacent gaps have positive excess, their contact contributions cancel in the sum. The contact sign depends on the actual donor phase and internal port rate, not on a coherence score. All these quantities are derived observations; none is a new driver.

On any such branch interval, finite arrival at E(T)=0 would require

\[ \int_s^T e^{2(r-s)}[-u(r)\sigma(r)]\,dr\ge E(s). \]

This is necessary, not sufficient. Since abs(u)<=1, it also requires T-s>=log(1+2*E(s))/2. The inequalities retain actual admitted counts; using the full topological degree in the phase denominator would change the model. Pressure continues to use its separate full-support contract.

A local improvement does not establish capture

Let epsilon lie in (0,b), take any positive w_0,...,w_3 with sum one, and prepare recipient gaps

\[ \delta_i=b+w_i\epsilon\ (i=0,1,2,3),\qquad \delta_4=-\epsilon. \]

The recipient already has winding one, with excess E=epsilon; it is not an initially winding-zero formation trial. A donor with uniform winding one and acute bridge gap gamma has actual co-rotating port rates

\[ f_0=(\sin\epsilon+\sin\gamma)/2,\qquad f_5=-\sin\gamma/3. \]

Recipient rates f_1=f_2=f_3=0, f_4=-sin(epsilon) give

\[ R=-2\sin\epsilon,\qquad u=(\sin\gamma-\sin\epsilon)/2,\qquad \dot E=-\tfrac32\sin\epsilon-\tfrac12\sin\gamma. \]

For fixed positive gamma, this stays negative as epsilon tends to zero. Thus a uniform global estimate E_dot>=-C*E is false with the contact. Nevertheless, the middle excluded gaps have zero instantaneous rate. A decreasing sum can coexist with individual gaps that prevent capture.

Why a single external neighbor still cannot create the acute identity

The following argument covers every evolving donor through abs(sin(gamma))<=1; it does not freeze the donor or fit a source. On a winding-one branch with 0<E<b, every gap outside P is admitted, because N<=E<b excludes a negative gate-off gap.

Project the gaps onto bar_delta in [0,b]^5 with sum 4*b, keeping every P gap at b. Raise negative non-P gaps to zero, then distribute the remaining E-N among the non-P gaps without exceeding b. Such a filling exists, and its total increase on non-P gaps is exactly E. This is an algebraic projection: keep the original excluded set P when evaluating its expression at the projected point, even though the literal inclusive gate would admit the boundary gaps there. It is not a new trajectory or a reassigned gate policy.

When the bridge is excluded or sigma=0, the isolated bound applies. Otherwise suppose 0 in P, 4 not in P; the other case has reflected coefficients and the opposite bridge term. Let m_j count the two adjacent edge indices that belong to P. For the admitted bridge, the excess derivative becomes

\[ \dot E=\sum_{j\notin P}v_j\sin\delta_j-\tfrac12\sin\gamma, \qquad v_j=m_j-\tfrac12\mathbf1_{j=4},\quad 0\le v_j\le2. \]

The sine function is 1-Lipschitz, so replacing non-P gaps by their projection costs at most 2*E in this expression. On the projected polytope, deficits b-bar_delta_j sum to b. Concavity of sine on [0,b] puts a minimum at a vertex with one non-P gap zero and all others equal to b. If B counts the cyclic P/non-P interfaces, the worst value, including the bridge, is B-1-max(v_j).

For C5, B is two or four. If B is four, this is nonnegative. If B is two and the complement of P contains at least two edges, its endpoint weights are at most one, again giving nonnegativity. The sole exception has P={0,1,2,3}; reflection gives P={1,2,3,4}. Consequently E_dot>=-2*E holds near zero except in those two patterns.

Suppose a positive-excess interval ended at finite time with E=0. Unless the endpoint is (b,b,b,b,0) or (0,b,b,b,b), continuity excludes both exceptional patterns nearby and the local exponential bound forbids arrival. Near the first corner, gaps 0 through 3 remain positive. For the non-port edge 1, whenever its gap exceeds b its rate is a sum of nonnegative admitted neighboring sines. Therefore (delta_1-b)_+ is nondecreasing almost everywhere. Its zero endpoint value forces it to be zero throughout a preceding neighborhood, excluding the first exceptional pattern. Continuity excludes the other pattern there. At the second corner use edge 2 with positive neighboring gaps 1 and 3. The same exponential argument then rules out arrival at either corner as well.

Thus no positive-excess interval can end at zero. An acute winding-one endpoint would force E=0 throughout its preceding antipodal-free interval; the resulting gaps in [0,b] cannot encounter an antipodal endpoint. The initial recipient must already have winding one. Reflection proves the winding-minus-one result, and convergence to a strict acute lock would entail finite entry.

Conclusion: the fixed single-edge interface does not allow an initially winding-zero recipient to reach a strict acute winding identity under this common-capacity phase law. This remains true with an actual evolving donor, bridge admission/removal and simultaneous gate boundaries, provided there are no phase jumps and the equation holds almost everywhere along an absolutely continuous solution. It does not establish existence/uniqueness at every discontinuity. Multiple external neighbors, moving attachment ports, capacity changes, other supports or other phase laws are outside this theorem. No claim about arbitrary TNFR identities follows.

Frozen local balance controls

No induced-formation trajectory is justified after that obstruction. Instead, three finite, non-mutating snapshot checks test the new balance against the production owner. Frozen before production execution, 2026-09-26: use epsilon=pi/40, weights (1,2,3,4)/10, recipient turns (0,201/800,403/800,606/800,1/80), and donor turns gamma/(2*pi)+k/5, reduced modulo one. Use gamma=pi/6,-pi/6,3*pi/4 on the same two-C5 graph with bridge (0,5). All capacities and edge weights/lengths are one; EPI is 1/2. Keep the previous explicit EPI/phase pressure weights, both U3 limits pi/2, absent Gamma and no glyph events.

Materialize exact turns with 80-digit arithmetic into binary64. Capture detached forcing/support observations, actual phase-admitted neighbors and one non-mutating shared phase proposal with h=1/128 per snapshot. Do not write that proposal, integrate EPI or execute a trajectory. Check independent represented-input rates and proposal residuals separately from the ideal formulas above. The two acute bridge signs must give opposite contact effects; the excluded bridge has zero phase correction while still contributing to full-support pressure. The unequal gap excesses avoid zero-resultant and antipodal pressure-displacement ambiguities. Both prepared cycles have winding one; these checks cannot be called formation.

Finite local-budget evidence

The two contact_budget_control tests in the shared control module pass with the eight earlier tests deselected. They reuse one fixture containing the three frozen observations. No proposal is applied, no EPI integrator is invoked, and no trajectory is generated.

The common isolated rate is approximately R=-0.1569181915. The actual contact effect changes sign as predicted:

Bridge gap	Port correction u	Predicted excess rate	Rate inferred from the finite phase proposal
pi/6	+0.2107704521	-0.3676886436	-0.3676886436
-pi/6	-0.2892295479	+0.1323113564	+0.1323113564
3*pi/4, excluded	0	-0.1569181915	-0.1569181915

The comparison retains initial materialization, represented-input rates and proposal-increment residuals separately. The maximum estimated phase proposal residual is 3.47e-16, and the largest full-support pressure-source residual is 9.02e-17. The minimum resultant modulus and pressure branch margin agree with 2*sin(3*pi/800) and pi/800, respectively, within the declared absolute numerical tolerance. These are finite numerical comparisons, not validated interval certificates.

Pressure retains three support neighbors at both ports in every case. In particular, exclusion of the bridge from phase motion does not remove its pressure contribution. Detached support rows are sorted, while live phase admission preserves neighbor insertion order; the control retains both conventions instead of assuming identical array ordering. Nodes, edges and declared configuration remain unchanged after observation and proposal. This check makes no equality assertion about private pressure caches.

The local receipt records exact preparation turns, node/order conventions, versions, path and residual estimates from those same observations:

python -m pytest tests/physics/test_coherent_pattern_contact.py -k contact_budget_control -q --junitxml=artifacts/research/one_port_phase_budget_2026_09_26.xml

Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1 and mpmath 1.3.0, with binary64 production and 80-digit residual arithmetic. The continuous entrance theorem remains a separate analytical result. This control verifies the new balance and its observation path, not the creation or loss of an identity.

Distributed two-port contact: a realizable entrance witness

The one-port obstruction does not extend to the fixed interface (0,5),(1,6) between the same two C5 rings. Keep the supplied gated phase law, held unit capacities, unit conductances, equal EPI/phase pressure weights and absence of Gamma/events. Only support and preparation change. The second contact creates a third independent cycle; both external phase differences must come from one nodal state. They are not independent controls.

Two-port budget and exact preparation

For recipient port k=0,1, let n_k count its admitted ring neighbors, f_k^iso be their mean sine (zero if none), and gamma_k the phase of its external neighbor relative to that port. Its actual correction is

u_k = 0                                      if the bridge is excluded,
u_k = (sin(gamma_k)-f_k^iso)/(n_k+1)           if the bridge is admitted.

With the earlier excess set P and isolated excess rate R, the regional balance becomes

E_dot = R + u_0*(1_{4 in P}-1_{0 in P})
          + u_1*(1_{0 in P}-1_{1 in P}).

Both corrections act on gap 0, with opposite signs. Their allowed values are constrained by the square cycle and the evolving donor.

Set b=pi/2, q=2*pi/5, and prescribe

d = b+epsilon,        r = (2*pi-d)/4,       gamma = (d+q)/2,
recipient lifts = (0, d, d+r, d+2*r, d+3*r),
donor lifts     = (gamma, gamma-q, gamma-2*q, gamma-3*q, gamma-4*q).

For sufficiently small positive epsilon, the recipient has winding +1, one excluded gap d and four acute gaps r. The donor has acute winding -1; the two bridge gaps are gamma,-gamma. The new square (0,1,6,5) has period zero, since d-gamma+q-gamma=0. Thus the joint preparation is realizable. This is not a winding-zero initial recipient.

Subtract the common angular rate 1 and denote the remaining nodal rates by f. Initially, while gap 0 is excluded,

f_0 = (-sin(r)+sin(gamma))/2,       f_1 = -f_0,
f_5 = -sin(gamma)/3,               f_6 = -f_5,
all other f_i = 0,
d_dot = sin(r)-sin(gamma).

At epsilon=0 the excluded-side limit is -m, where m=sin(9*pi/20)-sin(3*pi/8)>1/16. The literal inclusive gate instead uses three neighbors at each recipient port, giving d_dot=(2/3)*(sin(r)-sin(d)-sin(gamma))<0. Both sides point inward. For comparison, removing bridge (1,6) at the same preparation gives d_dot=(3*sin(r)-sin(gamma))/2>0. This comparison is algebraic, not a second executed preparation. The donor rates above are nonzero: admission does not assume a stationary environment or imposed port torques.

The centered bridge phases maximize the available initial inward contrast within this fixed uniform-donor family. Joint square realizability requires gamma_0-gamma_1=d+q, so

sin(gamma_0)-sin(gamma_1)
    = 2*cos((gamma_0+gamma_1)/2)*sin((d+q)/2)
    <= 2*sin((d+q)/2).

Equality occurs at the chosen gamma_0=-gamma_1=(d+q)/2, with both bridges acute. A uniform donor of the same winding instead fixes gamma_0-gamma_1=d-q; at d=b its maximum contrast is 2*sin(pi/20)<2*sin(3*pi/8), insufficient for this inward direction. This comparison concerns the declared uniform donor families. It does not make opposite winding a universal requirement for induced formation.

A short continuous entrance theorem

Fix epsilon=pi/8000 and T=1/64. In the co-rotating frame, abs(f_i)<=1, so every support gap changes by at most 2*t. The smallest initial noncritical acute margin is pi/20-epsilon/2>2*T. Consequently only recipient edge (0,1) can change gate before T; no support gap can reach an antipodal branch during this interval.

On the excluded chamber, the recipient gap derivative differs from its initial value by at most 4*t: each nodal mean sine is 2-Lipschitz in the supremum norm of co-rotating phases. Here r<=3*pi/8 and gamma>=9*pi/20, both below pi/2, hence

d_dot(t) <= -m+4*t < 0                       for 0 <= t <= T,
d(T)-b <= epsilon-m*T+2*T^2 < 0              if the edge stayed excluded.

The last inequality follows from m>1/16 and epsilon<1/2048=T/16-2*T^2; thus exclusion through T is impossible. One direct bound for m follows from cos(z)>=1-z^2/2 and cos(z)<=1-z^2/2+z^4/24 at these arguments:

m >= pi^2*(1/128-1/800)-pi^4/98304 > 1/16.

The final strict inequality already follows from 157/50<pi<22/7. On the admitted chamber, extend its smooth field back to the initial preparation. Its gap derivative there is (2/3)*(sin(r)-sin(d)-sin(gamma))<-1/2; the same 4*t bound keeps it negative through T. There can be no immediate return to the excluded side. Smooth existence on each chamber and transverse inward crossing therefore construct a local absolutely continuous solution of the literal law, with a strictly acute recipient at T and the acute donor retained. The value at the single crossing instant does not change that a.e. law. This is local existence/admission, not a global well-posedness claim.

Full-support pressure is also regular. Initially its relative resultants at recipient ports are conjugates of exp(i*d)+exp(-i*r)+exp(i*gamma); the other recipient rows are 2*cos(r). Donor port resultants are 2*cos(q)+exp(+-i*gamma) and other donor rows are 2*cos(q). Every real part exceeds 1/2: at the most restrictive recipient port use cos(3*pi/8)+cos(9*pi/20)-3*epsilon/2>1/2. At most three terms each change real part by 2*t, giving Re(z_i(t))>1/2-6*T=13/32. No resultant vanishes or reaches its relative Arg cut. The phase pressure source satisfies abs(g_i)<1/2; the more general abs(g_i)<=1 bound suffices below.

With initial EPI 1/2, the shared form equation is x_dot=(-L_rw*x+g)/2. Its maximum principle gives abs(x_i(t)-1/2)<=t/2<=1/128. Thus clipping to [0,1] is inactive for this continuous witness. These are conditional phase and form results for the supplied laws, not consequences of the nodal product alone.

Frozen shared-engine control

Declared before execution, 2026-09-26: execute this single preparation for 32 joint steps with h=1/2048, T=1/64. Store exact rational nodal turns and materialize with 80-digit arithmetic into binary64. Reuse the topological chord-extension owner to check cycle rank 2 -> 3 and the additional square. Compute exact periods from these declared nodal turns; do not weaken the strict-acute reconstruction API to admit the initial excluded edge.

Reuse the existing U3 phase proposal, NumPy full-support pressure and scalar nodal Euler through tests/joint_phase_helpers.py. Both U3 limits are pi/2; capacity/topology pressure weights are zero, capacities and all conductances/lengths are one, initial EPI is 1/2, extended dynamics is disabled and there are no Gamma/glyph events. Record node/support order, gate membership, both ring windings, the new square period, full-support resultants, pressure branches, clipping margins and actual donor motion. Compare initial rates with the formulas above, and retain represented-input phase/source/held-Euler residuals throughout the trace.

This control tests local entrance by the actual shared executor. The continuous theorem and binary64 trace have separate scopes; a discrete gate bracket is not a continuous crossing-time enclosure. No topology, offset, horizon or step-size sweep is admitted. Success would demonstrate recovery into an acute sector from an already wound state, not formation from winding zero or indefinite maintenance under contact.

Finite distributed-contact evidence

The three focused tests in test_two_port_coherent_contact.py pass using one shared fixture. The fixed control required no parameter adjustment. Exact fundamental cycle periods are (1,0,-1) in the shared owner's basis; the explicitly oriented recipient, donor and square periods are (1,-1,0). These periods persist at all 33 retained states.

Observation	Finite result
Initial recipient excess rate	-0.06387707919015755
First admitted endpoint for edge (0,1)	Step 13
Discrete gate bracket	[3/512,13/2048]
Endpoint minimum all-support acute margin	0.006582062193005909
Endpoint maximum form displacement from 1/2	0.002714587714422123
Endpoint maximum donor deformation after subtracting common rotation	0.005129347493660036
Minimum retained full-support resultant modulus	0.6131536621655245
Minimum retained pressure branch margin	2.039457036751049

The recipient starts with only edge (0,1) excluded, enters the acute region and remains there for the rest of the retained trace. The donor remains acute and moves relative to common rotation. Full pressure keeps all three neighbors at both recipient ports even before edge (0,1) is phase-admitted; using the gated neighbor set would give a different source. Form stays within the predicted radius 1/128, and every held Euler proposal is strictly inside the clipping interval.

Maximum estimated phase proposal, phase-source and held nodal state residuals are 4.41e-16, 9.73e-17 and 5.55e-17, respectively. These numerical comparisons are not validated interval bounds or an error enclosure for the continuous trajectory. In particular, the discrete gate bracket above does not enclose a proven ODE crossing time. The continuous entrance theorem independently supplies its own horizon.

python -m pytest tests/physics/test_two_port_coherent_contact.py -k "not formation_budget and not gate_barrier_control" -q --junitxml=artifacts/research/two_port_entrance_2026_09_26.xml

The receipt retains exact preparations, configuration, gate rows, cycle periods, execution path and arithmetic estimates. Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1 and mpmath 1.3.0; production values are binary64 and reference residual arithmetic uses 80 decimal digits. This closes local two-port entrance admission. A path from winding zero to a new acute regional identity remains a separate open obligation.

Winding-zero formation: a dissipation and preparation boundary

Keep the same fixed two-contact graph, common unit capacity and supplied phase/form laws. A recipient winding change and the later existence of two acute wound rings are different conditions. This section derives a necessary preparation budget before attempting the latter. It does not exclude every nonuniform winding-zero preparation.

A potential valid across phase gates and winding changes

The earlier cosine energy assumes all support edges are phase-admitted. For the literal gated law its global continuation is instead

psi(z) = 1-max(cos(z),0),
V_g(theta) = sum_{undirected support edges {i,j}} psi(theta_j-theta_i).

It equals 1-cos(z) on acute separations and stays at one outside them. It is periodic, nonnegative and Lipschitz. It is a potential of this specified phase law, not physical energy, the tetrad energy or a new decision rule for selecting operators. Its cap follows by integrating the existing gated sine torque; it does not modify the gate or phase dynamics.

Let a_ij be the symmetric inclusive gate, n_i=sum_j a_ij, and f_i=sum_j a_ij*sin(theta_j-theta_i)/n_i, with f_i=0 when n_i=0. The literal row is theta_dot_i=1+f_i. For any absolutely continuous solution satisfying that equation almost everywhere,

dV_g/dt = sum_edges a_ij*sin(theta_j-theta_i)*(f_j-f_i)
        = -sum_i n_i*f_i^2 <= 0,
V_g(t) + integral_0^t sum_i n_i*f_i^2 ds = V_g(0).

Away from cos(z)=0, this follows by the chain rule and pairing the two ends of every edge. On a gate boundary level set, the phase difference has derivative zero almost everywhere on that set, so the same formula holds with the inclusive gate. At an antipodal crossing psi is locally constant and has no jump. This proves the integrated identity through gate changes without asserting global existence or uniqueness of the discontinuous ODE. Common angular drift cancels. Unequal free rates, directed interactions, support events or another phase law require their own balance; Euler/binary64 nonincrease does not follow automatically.

The EPI row remains driven by full-support pressure. It supplies no return term to this phase row, so changing initial form or its pressure does not replenish V_g under this declared closure. The ordinary runtime's different phase map cannot be substituted into this theorem.

What two acute identities cost in this model

For a strictly acute C5 of winding +1, each of its five oriented gaps is positive: the other four sum to less than 2*pi. Reflection gives the negative-winding case. With q=2*pi/5, convexity and the strict chord bound 1-cos(z)<2*z/pi for 0<z<pi/2 give

5*(1-cos(q)) <= V_ring < 4.

Both strict-acute wound rings, with either relative winding sign, thus require

V_g >= Q = 10*(1-cos(q)) = (25-5*sqrt(5))/2 > 6.9.

Bridge contributions are nonnegative, so this is a necessary lower bound even when the final bridges are excluded. If the recipient initially has uniform phase, its ring potential is zero. An acute wound donor contributes less than four and the two contacts at most two. Therefore V_g(0)<6<Q: an initially uniform recipient cannot form an acute winding identity while an acute wound donor is present at the endpoint. The exclusion even allows the donor to leave and re-enter its acute sector between endpoints. No contact duration can fix the missing budget.

For a general nonuniform winding-zero recipient, the total-potential necessary filter is V_g(0)>=Q. The donor/contact bounds also imply the weaker necessary condition

V_recipient(0) > Q-6 = (13-5*sqrt(5))/2 = approximately 0.90983.

Neither inequality is sufficient. A state can have enough potential yet be blocked by gates, joint cycle geometry, donor loss or pressure singularities. These exclusions concern this supplied dissipative phase closure; they are not a no-emergence theorem for every TNFR model.

A branch-crossing candidate that still cannot form the target

Use a same-winding uniform donor and a nonuniform winding-zero recipient:

epsilon = pi/8000,  d = pi+epsilon,  r = (2*pi-d)/4,
gamma = (d-q)/2,
recipient lifts = (0,d,d+r,d+2*r,d+3*r),
donor lifts     = (gamma,gamma+q,gamma+2*q,gamma+3*q,gamma+4*q).

The recipient, donor and square (0,1,6,5) periods are (0,+1,-1). Only recipient edge (0,1) is phase-excluded; its principal gap is -pi+epsilon. Both bridges are acute, with gaps gamma,-gamma. The unwrapped recipient port separation obeys d_dot=sin(r)-sin(gamma)<0. Thus a winding change is locally feasible.

More precisely, let m=sin(3*pi/10)-sin(pi/4)>1/10 and T=1/64. The strict speed bound follows from sqrt(5)>223/100 and sqrt(2)<99/70, which give m>281/2800>1/10. The common bound abs(f_i)<=1 retains all other gates and gives d_dot(t)<=-m+4*t. Since epsilon-m*T+2*T^2<0, d crosses pi before T; edge (0,1) remains excluded on both sides of this branch crossing. The actual donor evolves and remains acute. Full-support pressure is regular: initial relative resultants have real parts above 1/4, and their change is at most 6*T, leaving real parts above 5/32. The continuous local path therefore changes recipient winding 0 -> +1 and square period -1 -> 0, with donor winding unchanged. The square already carried a supplied winding; this is a redistribution of cycle periods, not creation of the substrate.

Nevertheless, its initial potential is

V_g(0) = 1 + 4*(1-cos(r)) + 5*(1-cos(q)) + 2*(1-cos(gamma)) < Q.

At epsilon zero the deficit is exactly -2+2*sqrt(2)-5*cos(q)+2*cos(3*pi/10)>3/10. For example, use sqrt(2)>7/5, cos(q)<1/3, cos(3*pi/10)>7/12. The derivative of initial potential with respect to epsilon is -sin(r)+sin(gamma), of magnitude at most two; the frozen epsilon cannot exhaust that deficit. The dissipation identity excludes the two-acute-ring target at every future time, conditional on existence of the stated phase solution. An extended simulation of this preparation is therefore not a formation experiment worth pursuing.

Frozen non-mutating budget check

Declared before execution, 2026-09-26: check only the preparation above as one static snapshot. Use exact rational nodal turns, 80-digit materialization into binary64, and the same configuration and pressure owner as the two-port entrance control. Record exact cycle periods, actual gate membership, full-support pressure/resultant availability, V_g, Q and the initial potential derivative.

Request one shared phase proposal with h=1/2048 but do not apply it or integrate EPI. Compare its represented-input rate against the independent edge derivative and -sum_i n_i*f_i^2. The uncapped cosine sum exceeds Q in this preparation, whereas V_g is below Q: reusing the acute-only potential outside its domain would miss the exclusion. Check graph nodes, edges and declared configuration remain unchanged, excluding private pressure caches from that equality. Reuse one fixture for the two budget tests in the existing two-port control module. No formation trajectory, offset search or duration extension is authorized by this protocol. Arithmetic residuals are finite estimates, not an all-time numerical certificate for the continuous theorem.

Finite budget evidence

The two formation_budget tests pass, with the three earlier trajectory tests deselected. They reuse one non-mutating observation and one unapplied phase proposal. The exact oriented periods are (0,+1,-1); the shared fundamental basis reports (0,1,0). The initial gap rate is approximately -0.1020950324, consistent with motion toward the antipodal passage.

Quantity	Finite observation
Gated phase potential V_g	6.4509574551312285
Two-acute-ring lower bound Q	6.9098300562505255
Missing potential Q-V_g	0.45887260111929734
Uncapped cosine sum, inapplicable globally	7.450957378024945
Gated potential rate from independent admitted rates	-0.446886878829151
Rate inferred from the shared phase proposal	-0.4468868788288669

The independent edge derivative agrees with the degree-weighted squared rates within the 80-digit calculation. The maximum estimated proposal rate residual is 3.06e-13 (increment residual divided by h); the phase-source residual is below 8.61e-17. The minimum resultant modulus is 0.3118524913428369, and the pressure branch margin is 2.5505192562434775. Actual pressure still reads the excluded recipient edge. The retained nodes, edges and declared configuration are unchanged.

python -m pytest tests/physics/test_two_port_coherent_contact.py -k formation_budget -q --junitxml=artifacts/research/two_port_formation_budget_2026_09_26.xml

The receipt retains exact turns, configuration, actual neighbor counts, potential observations and arithmetic estimates. Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1 and mpmath 1.3.0; binary64 production was compared with 80-digit reference arithmetic. No trajectory was run for this preparation, and no sampled derivative is promoted to the all-time theorem. The excluded preparations are closed; general nonuniform winding-zero reachability remains open.

Joint port dynamics: sufficient potential does not give a formation path

The two port corrections are not independently adjustable. Their shared bridge terms cancel in a weighted combination of recipient and donor gap rates. This supplies a path obstruction beyond the potential filter. All statements retain fixed support, unit capacity, the same supplied phase law and an actually evolving donor.

An excluded-side barrier for a donor of the same winding

Write d=delta_0 for the positive recipient gap on edge (0,1) after a winding change, and e=eta_0 for the donor gap on (5,6). Consider b<=d<=pi, an acute donor of winding +1, the other four recipient edges admitted, and both bridges admitted. The donor's gaps are positive; the other recipient gaps may have either sign. With bridge separations gamma_0,gamma_1, define

A = sin(delta_1)+sin(delta_4),
B = sin(eta_1)+sin(eta_4),
C = sin(gamma_0)-sin(gamma_1).

As long as the recipient shared edge is excluded, the actual admitted neighbor counts are two at recipient ports and three at donor ports:

d_dot = (A-C)/2,
e_dot = (B-2*sin(e)+C)/3,
Z = 2*d+3*e,
Z_dot = A+B-2*sin(e).

The factors two and three are derived from those counts. They are not fitted coefficients. The cancellation retains both reciprocal donor responses; replacing the donor by fixed phases would lose it.

Recipient cycle closure gives delta_1+delta_4>=pi-d. For u,v in [-b,b] with s=u+v>=0, the minimum sine sum at that s is 1-cos(s), achieved when one argument is b. Thus A>=1+cos(d). Donor closure similarly gives eta_1+eta_4>=pi-e and B>=1+cos(e). At the surface Z=2*pi, where d=pi-3*e/2 and 0<e<=pi/3,

Z_dot >= 2-cos(3*e/2)+cos(e)-2*sin(e)
      >= 2-sqrt(3) > 0.

Here cos(e)>=cos(3*e/2). Therefore the side Z>=2*pi is protected throughout this chamber. A shared-edge antipodal passage starts with d=pi, hence Z>2*pi. At a proposed subsequent entrance d=b, the barrier forces e>=pi/3.

At that entrance the joint square has period zero: both bridges are admitted and d-e in [0,b], so gamma_0-gamma_1=d-e. Consequently

C <= 2*sin((b-e)/2) <= 2*sin(pi/12),
A >= 1,
d_dot >= (1-2*sin(pi/12))/2 > 0.

This is the incoming, excluded-side rate. It prevents arrival from d>b; the different inclusive-gate rate cannot bypass that obstruction. The argument closes the route from a shared-edge slip to acute recovery while the other recipient edges and both bridges remain admitted and the donor remains acute. It is a phase-law theorem, not an energy condition or an unrestricted statement about all switching itineraries.

An uninterrupted-admission obstruction for either donor sign

The result does not need a guessed single-slip trajectory. Suppose the recipient starts with well-defined winding zero and its four edges other than (0,1) remain admitted. Their principal gaps are continuous and define S=sum_{i=1}^4 delta_i and the continuous lift d=2*pi-S of the port separation. Initially d in (pi,3*pi); a strict-acute winding-one endpoint requires d in (0,b).

For a donor of winding +1, consider the final connected interval in b<d<3*b before the first arrival at b. It starts either at the initial time with d>pi, or at d=3*b. Thus Z>2*pi initially in this interval. The shared edge is excluded throughout the open strip, including any repeated antipodal crossings, so its admitted-count equations hold. On Z=2*pi within this strip, d=pi-3*e/2 automatically lies in [b,pi]; the preceding barrier calculation applies. The positive incoming rate at b rules out the proposed endpoint. Earlier admission of the shared edge or repeated changes of its winding cannot evade this argument while the other admission hypotheses hold.

For a donor of winding -1, its corresponding gap satisfies -b<e<0. At the unavoidable decreasing crossing d=pi, joint circular geometry with both bridges admitted requires

kappa = gamma_0-gamma_1 = -pi-e in (-pi,-b),
C = sin(gamma_0)-sin(gamma_1) <= -(1+cos(e)) < -1.

This follows by writing C=2*cos(M)*sin(kappa/2) and using abs(M)<=b-abs(kappa)/2. These phases are geometrically realizable; the square may carry nonzero winding. It is their direction of motion that blocks formation: recipient closure gives A>=0 at d=pi, and therefore d_dot>1/2. A decreasing passage is impossible. Reflection in co-rotating phases (equivalently theta -> 2*t-theta) covers a target recipient of winding -1 while preserving the common free drift.

In every excluded-side neighborhood used above, all other phase edges remain admitted, so the trajectory follows one fixed smooth sine field almost everywhere. Its integral representation is continuously differentiable there. The strict outward rates therefore prevent a downward arrival even for the stated absolutely continuous solution class; isolated gate or branch instants do not bypass the barrier.

Conclusion: a winding-zero recipient cannot reach a strict-acute nonzero winding while all four non-shared recipient edges and both bridges remain admitted and the donor remains strictly acute with nonzero winding. This allows arbitrary admitted asymmetry, adequate initial potential, and changes of the shared edge's gate or branch. It is conditional on absolutely continuous solutions of the supplied law, not a global existence theorem or an unrestricted numerical claim.

Any successful route preserving the donor must consequently involve an additional bridge or non-shared recipient edge being phase-excluded at some time. This can already occur in its preparation; the result does not force a unique first event. Gate switches and simultaneous corners remain open mechanisms. A counterexample to this uninterrupted route is not a proof that a route with those events succeeds. The support graph has not changed; it is the law's active phase-neighbor set that must change or start outside this class.

A high-potential winding-zero preparation

The obstruction is relevant even after the earlier budget test passes. Fix the following exact preparation:

epsilon = pi/8000, d = pi+epsilon,
a = pi/20, s = (2*pi-d-2*a)/2,
e = pi/8, u = (2*pi-e)/4 = 15*pi/32, gamma = (d-e)/2,
recipient lifts = (0,d,d+a,d+a+s,d+a+2*s),
donor lifts = (gamma,gamma+e,gamma+e+u,gamma+e+2*u,gamma+e+3*u).

The oriented recipient, donor and square periods are (0,+1,-1). Only recipient edge (0,1) is excluded. The exact initial potential is

V_g = 1+2*(1-cos(a))+2*(1-cos(s))
        +(1-cos(e))+4*(1-cos(u))+2*(1-cos(gamma))
    > 9-7*pi/20-pi/8000 > 7 > Q.

For this elementary bound, drop the positive a/e terms and use cos(s)<a+epsilon/2, cos(u)<pi/32 and cos(gamma)<pi/16. The preparation passes the necessary potential test; it does not establish a successful trajectory.

Initially d_dot=sin(a)-sin(gamma)<-3/4. Over T=1/128, the bound abs(f_i)<=1 keeps all other gates unchanged and changes this rate by at most 4*t. Since epsilon-(3/4)*T+2*T^2<0, the recipient crosses its antipode before T. All initial relative resultants have real parts greater than 1/8; their change is at most 6*T, leaving real parts above 5/64. Full-support pressure remains defined and away from its relative Arg cut. This constructs a local continuous branch passage with an acute evolving donor and no clipping requirement beyond the earlier form maximum principle. It does not construct the later gate entrance.

The donor starts at e=pi/8 but has positive e rate; it is not held at that small gap. After the slip, the Z barrier prevents completion unless one of its admission hypotheses changes. Starting with sufficient potential and favorable local motion has not removed this dynamical obligation.

Frozen non-mutating port-balance checks

Declared before execution, 2026-09-26: inspect exactly two static preparations, with no applied proposal and no EPI integration:

1. The high-potential preparation above. Its exact nodal turns have common denominator 32000: recipient (0,16002,16802,24001,31200) and donor (7001,9001,16501,24001,31501).
2. A point on Z=2*pi: d=3*pi/4, e=pi/6, equal remaining recipient gaps 5*pi/16, equal remaining donor gaps 11*pi/24, and centered bridges gamma=7*pi/24. Exact nodal turns have denominator 96: recipient (0,36,51,66,81), donor (14,22,44,66,88). Its periods are (+1,+1,0) and only edge (0,1) is excluded.

Reuse the existing two-port control owner, configuration, full-support pressure capture and one non-mutating shared phase proposal per state with h=1/2048. Materialize exact turns with 80-digit arithmetic into binary64. Check exact cycle realizability, phase admission, independent initial sine rates and the identity 2*d_dot+3*e_dot=A+B-2*sin(e) against the actual proposal. At the second state require positive outward Z rate; at the first retain both the passed potential filter and the negative recipient gap rate. Record pressure availability and graph-state preservation, excluding private pressure caches from that equality. Keep proposal-increment and recovered-rate rounding estimates separate. These checks do not supply a formation trajectory, interval enclosure or an all-time binary64 barrier certificate.

Finite joint-balance evidence

The two gate_barrier_control tests pass with the previous five tests deselected. Both preparations pass the necessary potential filter, and the actual shared phase proposal agrees with the independent port rates. No proposal is applied and no trajectory is executed.

Observation	High-potential branch preparation	Z-boundary preparation
V_g	8.00562295359448	7.172066037239505
V_g-Q	1.0957928973439555	0.262235980988979
Recipient gap rate from proposal	-0.8243891023519354	0.03811627201135204
Donor gap rate from proposal	1.0622165744662195	0.8565321344433414
Z rate from proposal	1.5378715186947878	2.6458289473527286
Independent Z rate	1.5378715186946759	2.6458289473527112
Minimum relative resultant real part	0.18258615921863527	0.261052384440103

The boundary observation has Z_dot>2-sqrt(3) as predicted. The first observation confirms that a rapidly decreasing recipient gap coexists with a changing donor gap and increasing Z. It does not establish formation; this is precisely the coupled response omitted by a fixed donor approximation.

The largest estimated recovered-rate residual is 6.06e-13, below the declared finite comparison allowance 2*ulp(2*pi)/h, approximately 3.64e-12. This allowance addresses subtraction followed by division by h, not physical uncertainty or an all-time arithmetic bound. Full phase-source residual estimates are below 1.92e-16. Both snapshots retain regular full-support pressure and unchanged nodes, edges and declared configuration.

python -m pytest tests/physics/test_two_port_coherent_contact.py -k gate_barrier_control -q --junitxml=artifacts/research/two_port_gate_barrier_2026_09_26.xml

Execution used Python 3.13.6, NumPy 2.5.3, NetworkX 3.6.1 and mpmath 1.3.0, with binary64 production and 80-digit reference arithmetic. The receipt retains exact turns, cycles, admitted neighbors, configuration, execution paths and estimates. The analytical obstruction has broader scope than these two finite checks. Exclusion/re-admission of another phase edge remains an unproved route, not a demonstrated mechanism.