Native phase domains and capture¶

Regular-domain continuation, winding passage, reflected basins and the full-state acute-sector barrier, including capacity/storage-scale covariance and regularity margins.

Part of Native relational exchange admission. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

17. Regular-domain admission before a formation experiment¶

Four boundaries with different meanings¶

For the full continuous law of sections 1-3, write the local relative resultant and displacement as

\[ z_i=\sum_{j\sim i}e^{i(\theta_j-\theta_i)},\qquad \alpha_i=\operatorname{Arg}z_i\in(-\pi,\pi),\qquad g_i=\alpha_i/\pi,\quad H_i=\pi|z_i|\operatorname{sinc}\alpha_i. \]

The open regular domain is z_i outside the nonpositive real axis for every node. Here H_i>0 and the field is smooth. Its definition uses only the existing state and support. The separate execution admission certifies sufficient point and proposal-chord margins; a domain definition alone is not a numerical guarantee.

Boundary	Consequence for the stated continuous law
An ordinary edge reaches `abs(gap)=pi/2`	The acute executor stops; the resultant and complete field can remain regular
An ordinary edge reaches antipodal separation	Its ordinary cycle winding is undefined at that instant; the nodal field need not be singular
A local resultant vanishes	Its direction and native phase source are undefined; the phase metric tends to zero
A nonzero resultant lies on the negative real axis	The principal displacement has its `-pi/pi` branch; its two source limits differ and `sinc(alpha)` tends to zero

At zero resultant, different admitted approaches can give different limits for the native source. The phase velocity also generically diverges when (Bx)_i stays nonzero while H_i vanishes. Even if that numerator vanishes, the law does not select a value for the resulting ratio or a source direction on an open full-state neighborhood. Likewise the negative-real branch does not become regular merely because the resultant magnitude is positive. A special limiting trajectory would require its own existence and uniqueness argument; assigning a direction or clipping H^-1 would add a new law.

A derived cycle invariant gives a genuine pure-cycle obstruction¶

On a pure C_n, n>=5, every node has exactly its two cycle neighbors. Consequently

\[ |z_i|=|e^{i\theta_{i-1}}+e^{i\theta_{i+1}}|=0 \quad\Longleftrightarrow\quad \theta_{i+1}-\theta_{i-1}=\pi\pmod{2\pi}. \]

Consider the derived oriented edges i -> i+2 modulo n. They form one cycle when n is odd, and two parity cycles when n is even. Their non-antipodal winding integers are observations of the same phases, not added support or new primitive variables. Every such edge has a non-antipodal gap whenever the original local resultants are nonzero. The ordinary cycle-period argument from the existing classification therefore preserves these derived windings along any continuous path with nonzero resultants, independently of the dynamics or the Arg-branch admission.

For an acute original phase geometry, let its oriented gaps be delta_i and ordinary winding be ell. Each delta_i+delta_(i+1) is strictly inside (-pi,pi), so it is exactly the shortest two-hop gap. Summing gives

\[ \boxed{\quad W_{\mathrm{skip}}=2\ell\quad(n\text{ odd}),\qquad W_{\mathrm{even}}=W_{\mathrm{odd}}=\ell\quad(n\text{ even}). \quad} \]

For odd n the sum includes every original edge twice; each parity sum for even n includes every original edge once. In particular, C5 consensus has derived winding zero, while its acute winding-one twist has derived winding two on (0,2,4,1,3). Every continuous fixed-C5 path between them must encounter a zero local resultant. Allowing obtuse ordinary edges or removing the acute executor cutoff cannot remove this constitutive obstruction. The same reasoning separates different acute winding sectors on any pure cycle. It does not apply unchanged when extra neighbors contribute to a resultant.

Ordinary winding alone does not identify these regular components. For example, on C5 take

\[ \theta=(0,\pi,3\pi/4,\pi/2,\pi/4). \]

The edge (0,1) is antipodal, but the relative resultants are (-1+exp(i*pi/4), -1+exp(-i*pi/4), sqrt(2), sqrt(2), sqrt(2)). Every one is regular. Replacing theta_1 by pi+tau for sufficiently small positive/negative tau keeps all resultants regular while ordinary winding changes from zero to minus one. The derived winding remains zero on both sides. This is a regular geometric crossing, not yet an assertion that the prescribed path is a trajectory, nor a connection to the acute minus-one twist (whose derived winding would be minus two).

A sharp C5 zero-resultant storage barrier¶

Suppose a C5 resultant vanishes. Its two neighbors are antipodal, so the phase-storage contributions of the two edges incident at that node sum to exactly two, independently of the node's own phase. The remaining three-edge path connects antipodal endpoints. Fix them at 0,pi, with internal phases A,B. The sum of its three cosines is

\[ \cos A+\cos(B-A)-\cos B \le \cos A+2|\sin(A/2)| =\tfrac32-2(|\sin(A/2)|-\tfrac12)^2\le\tfrac32. \]

Thus that path costs at least 3/2, attained at A=pi/3,B=2*pi/3, and

\[ \boxed{\qquad V_\phi\ge\tfrac72 \quad\text{at every zero-resultant C5 configuration}.\qquad} \]

The bound is sharp: choose those three equal path gaps; the remaining node may have any phase. Nonnegative Dirichlet storage and the complete loss identity imply the necessary budget E(0)>=7*beta/2 for a trajectory to reach such a configuration. With positive capacities and e>0, any nonstationary phase evolution consumes positive integrated form loss, so a trajectory reaching this barrier after that evolution needs strictly more initial storage. Equality cannot pay both the barrier and that loss.

For initial phase consensus, the entire budget is initial form storage. The final acute winding-one phase storage is only 5*(1-cos(2*pi/5)), approximately 3.454915, below the crossing barrier 3.5. Comparing only initial and target energies therefore misses a necessary intermediate cost. Even sufficient stored energy would not define continuation through the singularity or establish a connecting solution of the current law. The regular ordinary-antipodal example above has V_phi=6-2*sqrt(2)<7/2; that lower cost does not contradict the distinct zero-resultant obstruction.

Added support gives a regular route to an admitted target¶

The pure-cycle obstruction is not a universal restriction on the full law. Prepare two unit cycles (0,1,2,3,4) and (5,6,7,8,9), with bridges (0,5) and (1,6). The second bridge is an explicit support change relative to section 16's one-bridge preparation; it is not generated by an operator or hidden in a pressure coefficient. On both rings choose the phase path

\[ \theta_{0:4}(\delta)=\theta_{5:9}(\delta) =(0,-4\delta,-3\delta,-2\delta,-\delta), \qquad 0\le\delta\le2\pi/5. \]

Each bridge joins equal phases. Interior degree-two nodes have z_i=2*cos(delta)>0; ports 0 and 5 have

\[ z_0=z_5=1+e^{-i\delta}+e^{-4i\delta},\qquad \operatorname{Re}z_0=1+\cos\delta+\cos4\delta \ge\cos\delta\ge\cos(2\pi/5)>0. \]

Ports 1 and 6 have the conjugate resultant. This is a compact regular path with a positive lower bound on every metric, from consensus to acute winding-one geometry in both rings. The additional four-cycle has winding zero wherever its ordinary winding is defined; it also encounters antipodal edges at delta=pi/4. At delta=pi/3, the two ring-neighbor phasors cancel at each port, but the aligned bridge neighbor contributes one. That is precisely a resultant-zero encounter on the isolated C5 which the supplied extra support prevents; no direction is assigned to an undefined resultant.

Each ring's oriented gaps are wrap(-4*delta),delta,delta,delta,delta. Its winding changes from zero to one when delta crosses pi/4. At the endpoint delta=2*pi/5, all ring gaps are acute and uniform, bridge gaps are zero, and native phase pressure vanishes. Uniform form there supplies an equilibrium covered by section 15's local-recovery theorem. This proves regular geometric connectivity between the preparations. It does not yet show that their connecting path obeys the joint equations.

The phase storage along this specified path is V_phi(delta)=2*[1-cos(4*delta)+4*(1-cos(delta))]; bridge contributions vanish. At delta=pi/3, all resultants equal one and g=0, but the phase geometry is nonacute. Here V_phi=7 and the second derivative along the path is -12, while changing only an interior node has positive phase curvature. It is a saddle of phase storage. The final acute target has V_phi=10*(1-cos(2*pi/5))<7. This locates a barrier on this particular regular path, not a proven minimum barrier over all paths or a connecting orbit of the joint ODE.

The same coupled law realizes a transversal crossing¶

A geometric path alone would be insufficient. Here the same coupled law can realize a transversal crossing. At any regular phase state with positive capacities, let M=N*H^-1 and prescribe an instantaneous relative phase velocity represented by v. Define

\[ \gamma=-\frac{\mathbf1^TM^{-1}v}{\mathbf1^TM^{-1}\mathbf1},\qquad Bx=\frac\beta w M^{-1}(v+\gamma\mathbf1). \]

The right side has zero sum. Connectedness of the graph supplies a finite real x, unique modulo a common offset. The actual phase row then satisfies theta_dot=v+gamma*1; the form row remains the unchanged native nodal law. This realizes any instantaneous relative phase direction through supplied form. It is not arbitrary control of a whole phase path, because subsequent form and phase must both follow the joint equations.

The two-ring state gives a concrete witness without a fitted tangent. At delta=pi/4, unit capacities, beta=1, e=w=1/2, take identical form on both rings:

\[ x_{0:4}=x_{5:9} =\frac{2\sqrt2}{5}(\pi+8,-\pi-8,-3\pi-4,0,3\pi+4). \]

The full-graph Bx on each ring is (8*sqrt(2),-8*sqrt(2),-2*pi*sqrt(2),0,2*pi*sqrt(2)). Bridge differences vanish. Port metrics are 2*sqrt(2) and interior metrics are pi*sqrt(2), so the actual phase row is

\[ \dot\theta_{0:4}=\dot\theta_{5:9}=(2,-2,-1,0,1). \]

This is the path tangent (0,-4,-3,-2,-1) plus the permitted representative offset gamma=2, obtained from the phase row, not externally imposed. Smooth local ODE existence and uniqueness give a solution on both time sides with theta_1(t)-theta_0(t)=-pi-4*t+O(t^2) and the same crossing on the second ring. The remaining ring edges stay non-antipodal locally. Thus an actual local solution changes ordinary winding from zero to one in both rings under the same law. No phase forcing, support event, telemetry selector or new evolution rule is inserted.

This supplied form carries finite Dirichlet storage 128*pi/5+48*pi^2/5+512/5; it is not free phase actuation or evidence that a small perturbation of consensus forms the target. Neither arrival from consensus at the crossing state nor later entry into the acute winding-one equilibrium's recovery basin has been proved. The regular path, local crossing solution and admitted stable target are distinct facts; concatenating them verbally would not establish one connecting trajectory.

Admission verdict and execution boundary¶

The same mathematics yields a negative and a positive result: pure-cycle acute sectors require an undefined resultant between them, whereas the specified higher-degree support admits a regular geometric route to a recoverable target and a local topological crossing under the complete law. Neither a universal formation obstruction nor a complete autonomous formation trajectory follows. The full regular domain is meaningful and can exceed the engine's default acute domain, but global continuation within it remains an independent obligation.

The default acute execution contract and every earlier frozen response remain unchanged. Section 18 supplies opt-in certified admission for positive resultants and the full regular domain. A prescribed geometric route does not choose the numerical trajectory or its outcome. No broader formation sweep, new state primitive or unsupported singular continuation is justified by this local existence result. The sole execution queue owns the next gate.

The exact cycle observer reuses the existing support geometry and rational scalar owner. It records ordinary/derived winding and both resultant boundary types from declared exact turns, leaving live radians and runtime admission separate. The admission controls check those observations, the sharp C5 budget and the two-port phasor, tangent and native-pressure identities. They contain no time integration or formation parameter search. API scope owns inputs and availability; the symbolic proof above owns the conditional continuous conclusions.

18. Certified phase-domain execution and a retained crossing control¶

A sufficient regular chamber for the unchanged joint law¶

The production model accepts explicit phase_domain="positive_resultant"; "acute" remains the default. This option requires Re(z_i)>0 at every node. This right-half-plane chamber is strictly narrower than the full regular domain in section 17, which also admits left-half-plane resultants away from the negative-real branch. It is nevertheless broad enough for the two-port witness and its prescribed geometric route. Support, capacities, storage scale and phase/form equations remain the same. This extends execution admission, not constitutive selection.

The shared rational helper reuses the existing Machin enclosure of mathematical pi. For each exact rational difference of materialized phase coordinates it encloses cosine: integer true-turn reduction, a rational midpoint Taylor enclosure, and a Lipschitz allowance for the reduction interval. Outward dyadic rounding retains a rational lower bound. Near zero an elementary quadratic/quartic enclosure suffices. Bounded numerical work can return an inconclusive bound; neither that outcome nor a nonpositive lower bound proves an actual singularity. The ordinary edge's antipodal branch requires no special cosine continuation.

For the nodewise lower bounds and actual represented endpoint increments, write

\[ L_i\le\sum_{j\sim i}\cos(\theta_j-\theta_i),\qquad d_i=\theta_i^{\rm after}-\theta_i^{\rm before}. \]

Cosine is globally one-Lipschitz. Every point of the straight Euler proposal chord therefore satisfies

\[ \operatorname{Re}z_i(s)\ge L_i-\sum_{j\sim i}|d_j-d_i|, \qquad 0\le s\le1. \]

Strict positivity of this exact rational right side certifies that the whole represented phase chord remains in the chamber. On that chord, H_i>=pi*Re(z_i)>0: for |delta_i|<pi/2, sin(delta_i)/delta_i>=cos(delta_i). The engine additionally evaluates materialized positive cosine sums and metrics and re-admits the endpoint. The exact margins are retained as resultant_real_lower_bounds and segment_resultant_real_lower_bounds. Rejection precedes live graph commit.

Using only endpoint admission would be insufficient: two regular P2 endpoints can differ by a full relative turn whose interior crosses the excluded chart. Conversely, a common phase increment contributes zero to the exact drift bound. The increments used here are the actual rounded endpoint differences, not an uncommitted dt*phase_rate approximation. The execution controls exercise these distinctions, atomic rejection, zero capacity, the analytic crossing tangent, and admission independent of libm cosine signs.

This certificate does not enclose the exact continuous ODE trajectory, bound its global numerical error, or prove Euler storage decrease. Native pressure and joint rates still use their existing materialized arithmetic; pressure split, work and step defects remain measured. Raw relative angles are used for trigonometric evaluation in this chamber; the default acute path retains its previous represented wrapping. Earlier frozen producers and records are untouched. Their historical source fingerprints are not rewritten to claim compatibility with this source revision.

Certified admission on the full regular phase domain¶

The separate phase_domain="regular" option admits the same joint equations when each relative resultant avoids the entire nonpositive real axis. A negative real component is allowed if the imaginary component is nonzero. No extra phase coordinate, pressure channel, operator or material property is introduced. The domain follows from the existing source and metric; selecting the complete joint law still retains the independent storage, capacity and support premises of sections 1-2.

Write z_i=C_i+i*S_i and alpha_i=Arg(z_i) in (-pi,pi). The equivalent metric expression

\[ H_i=\pi S_i/\alpha_i\quad(\alpha_i\ne0),\qquad H_i=\pi C_i\quad(\alpha_i=0,\ C_i>0) \]

is positive throughout this open domain: S_i and alpha_i have the same sign. In particular, C_i<0 does not make the metric negative. This is the same H_i=pi*|z_i|*sinc(alpha_i) identity, with its regular positive-real extension. Therefore H*g=-grad(V_phi), unchanged native pressure, nodewise phase/form exchange cancellation and total storage loss all retain their proofs. Nothing here continues the law through the excluded zero or negative-real resultant.

Numerical admission reuses the rational trigonometric enclosure owner. For exact represented raw phase differences it produces a rectangle

\[ C_i\in[C_i^-,C_i^+],\qquad S_i\in[S_i^-,S_i^+]. \]

Its sufficient separation margin is

\[ m_i=\max(0,C_i^-,S_i^-,-S_i^+). \]

Let K={r+i*0:r<=0} be the excluded closed ray. Each positive term in this maximum is a lower bound for dist(z_i,K), hence so is their maximum. Strict m_i>0 certifies a regular point even in the left half-plane. A zero margin is inconclusive or outside the domain; finite enclosure failure does not prove that the mathematical resultant actually lies on the ray. The helper's bounded work is part of numerical admission, not an additional constitutive boundary.

The field retains the component rectangle as resultant_bounds and its margin as resultant_regular_margin_lower_bounds. Computed component signs must agree with the separation certificate. A materialized principal argument at either represented branch endpoint is rejected, even if the exact point has a smaller unresolved separation. The regular numerical metric uses the displayed pi*S/Arg identity and positive-axis limit. Native pressure consumes the same captured phase source, avoiding a second branch selection; pressure split, rate and work residuals are still reported from their actual arithmetic.

For actual represented endpoint increments d_i, the straight proposal is theta_i(s)=theta_i(0)+s*d_i, 0<=s<=1. The elementary chord inequality |exp(i*a)-exp(i*b)|<=|a-b| yields

\[ |z_i(s)-z_i(0)|\le\sum_{j\sim i}|d_j-d_i|. \]

Distance to a closed set is one-Lipschitz, so

\[ \operatorname{dist}(z_i(s),K) \ge m_i-\sum_{j\sim i}|d_j-d_i|. \]

Strict positivity at every node certifies the whole represented chord. The step retains segment_resultant_regular_margin_lower_bounds. It may reject a longer regular chord whose actual geometry is safer than this conservative bound. Common phase increments cost zero margin. Checking only the two endpoints would miss a branch crossing in the interior. Materialized endpoint rates and metrics are additionally admitted before atomic commit; their arithmetic residuals remain observations rather than being replaced by their ideal zero values. This certificate neither encloses the continuous solution nor guarantees future regularity or a storage decrease during a finite Euler step.

Regular-domain continuation and boundary access¶

All statements here retain the fixed connected simple unit support, nonnegative held capacities, e>=0,w,beta>0, no input and no events. They concern the ideal continuous law. Neither finite Euler steps nor the narrower acute execution chamber inherit a new invariance guarantee.

Derived resultant dynamics. Put v_i=theta_dot_i, delta_ij=theta_j-theta_i, and z_i=C_i+i*S_i. Direct differentiation gives

\[ \dot z_i=i\sum_{j\sim i}e^{i\delta_{ij}}(v_j-v_i),\qquad \dot C_i=-\sum_j\sin\delta_{ij}(v_j-v_i),\quad \dot S_i=\sum_j\cos\delta_{ij}(v_j-v_i). \]

The actual law supplies v_i=(w/beta)*nu_i*q_i/H_i. These identities do not close a law in resultants alone: form, capacities and neighboring rates remain necessary. A useful cancellation is

\[ \frac{d}{dt}|z_i|^2 =2\sum_j(S_i\cos\delta_{ij}-C_i\sin\delta_{ij})v_j. \]

The node's own phase rate rotates its relative resultant without directly changing its magnitude. Its effect on magnitude returns through neighboring rates. For nonzero z_i, Arg(z_i)'=(C_i*S_dot_i-S_i*C_dot_i)/|z_i|^2. The kinematic derivative of z_i can exist at zero even though this quotient and the full nodal law are undefined.

For the excluded ray K, the same chord estimate used by the step admission gives the continuous conditional bound

\[ \operatorname{dist}(z_i(t),K)\ge \operatorname{dist}(z_i(0),K) -\int_0^t\sum_j|v_j(s)-v_i(s)|\,ds. \]

A whole-time independent rate bound can make this a continuation certificate. An instantaneous rate is insufficient for that conclusion.

A sufficient storage theorem. Let d_min be minimum degree and v=E(0)/beta. If

\[ \boxed{\mathcal E(0)<\beta\max(2,d_{\min}),} \]

the ideal regular law continues for all finite times. To prove this, count each undirected edge once in the nonnegative storage. At a node of degree d_i>=2,

\[ C_i=d_i-\sum_{j\sim i}(1-\cos\delta_{ij})\ge d_i-v>0,\qquad H_i\ge\pi C_i\ge\pi(d_i-v). \]

\[ H_i\ge\frac{\pi}{2}(2-v)>0\quad\text{at a leaf}. \]

All resultants are uniformly separated from the excluded ray. With leaves a common conservative lower distance is min(1,2-v): when a leaf has negative real part its distance is |S_i|, and S_i^2=(1+C_i)(1-C_i)>=2-v; otherwise its distance is one. Without leaves, d_min-v is a common lower bound. Storage bounds centered form on connected support. The source is bounded and the phase metrics stay positive, so form and phase rates are bounded on every finite interval, including the common form velocity. Smooth local existence therefore extends globally.

This proves continuation, not convergence, a physical identification or stability of a numerical step. Zero held capacities are permitted; no inverse capacity enters the bound. The universal 2*beta boundary-storage scale cannot be increased using only such graph-independent energy information: with uniform form, P2 at antipodal phase has a nonzero branch resultant with storage 2*beta, and P3 at phases (-pi/2,0,pi/2) has a zero central resultant at that storage. Graph-specific sharper bounds remain possible.

A nonzero branch can be reached in finite time. On P2 take e=w=1/2, beta=nu_0=nu_1=1, x=(p,-p), theta=(a,-a) and initially p=3,a=pi/4. For pi/4<=a<pi/2,

\[ H=\pi\frac{\sin(2a)}{2a},\qquad \dot p=-p-a/\pi,\qquad \dot a=p/H. \]

While p>=1, 0<H<=2 and dp/da=-H-sin(2a)/(2p)>=-5/2. Thus p(a)>=3-5*pi/8>1, closing that bootstrap up to a=pi/2. The smooth scalar equation for p(a) extends to this endpoint and t(a)=integral(H/p) da is strictly increasing in the interior with finite terminal value T<=pi/2. Its inverse is an actual solution of the original law until T. Both resultants approach -1, while a_dot diverges. Positive dissipation and nonzero resultant magnitude do not prevent this branch collision. Unwrapping the argument would change the prescribed pressure and is not a harmless chart continuation.

The two-ring zero-resultant construction separately proves finite-time access to zero from acute states with E<7*beta in the central reflected component. Its full-state vector field has no continuous extension at the endpoint. Neither counterexample determines the future of the frozen source/receiver preparation.

Shared implementation. observe_relational_regularity captures one admitted field, encloses its initial storage with mathematical trigonometry and applies the strict sufficient bound. It reports explicit unavailability when that bound cannot certify continuation. Its resultant derivative enclosure uses captured numerical rates as a supplied tangent; it does not certify that tangent as the ideal ODE rate. The shared relative_resultant_rate_bounds owns that kinematic calculation, including common-rate cancellation. No observer installs a boundary law or executes a trajectory.

Which earlier conclusions retain their scope¶

The wider execution option removes a numerical restriction; it does not erase theorem hypotheses or historical evidence:

Existing result	Scope after full-regular admission
Metric, native-pressure alignment and ideal storage/work balances	Already valid throughout the regular domain, including left-half-plane resultants
Acute equilibrium classification, local recovery and explicit capture basins	Their acute or basin hypotheses remain required; they do not classify every nonacute regular equilibrium
Ordinary cycle winding preservation	Holds while relevant edge gaps stay nonantipodal; full resultant regularity alone does not guarantee this
Pure-cycle derived-winding obstruction	Still holds on every continuous path with nonzero resultants, even outside the positive-resultant chamber
Uniform-receiver storage deficit	Its initial positive-resultant restriction remains essential; the full-regular two-port counterexample already prevents promoting it to a universal obstruction
Retained crossing, formation and memory evidence	Keeps its declared preparation, phase domain, coefficients, clock, numerical budget and source identity; broader admission is not a new response evaluation

The full-regular source/receiver preparation is consequently a justified candidate for further analysis, not a demonstrated formation trajectory. Enough initial storage is necessary for its proposed acute target, but is not sufficient for reachability, regular continuation or capture.

Fixed preparation and independently stated questions¶

The bounded producer uses the supplied two-ring, two-bridge support of section 17, unit capacities, beta=1, and e=w=1/2. Each ring starts at theta=(0,-4*delta_0,-3*delta_0,-2*delta_0,-delta_0), with delta_0=pi/4-1/256, and the explicit transverse form state derived there. Thus both ordinary windings initially equal zero, but this is already a structured, finite-storage preparation close to a crossing; it is not phase consensus, a random perturbation or substrate birth.

Before evolution, the protocol fixes one horizon T=16, three Euler grids dt=(1/64,1/128,1/256), and common checkpoints. The short positive hypothesis is winding (1,1) by t=1/64, retained at t=1/32. Later acute-domain entrance, distance to the prepared acute winding-one equilibrium and a local basin indicator are separate questions, without a predeclared positive capture claim. The complete quotient state includes both rings together, so relative regional offsets are not discarded.

For the basin indicator, the target edge margin is m=pi/10. Choose r=pi/(20*sqrt(2)), giving c_r=sin(pi/20)>1/10. Every connected ten-node unit graph has lambda_2>=2/81>1/45: summing the path bound (y_i-y_j)^2<=9*y^T*B*y over all pairs gives this conservative estimate. Section 15 then yields k_r>1/900, r^2>9/800, and k_r*r^2>1/80000. In the declared beta-one quotient normalization,

\[ \|y\|^2<9/800,\qquad 0\le E_{\rm rel}<1/100000 \]

are sufficient ideal-law basin conditions. The retained finite-state norm and excess storage use materialized phases and trigonometry relative to the supplied target. Their threshold verdict is therefore a numerical indicator, not a rigorous transcendental energy enclosure or an exact-ODE capture certificate. Negative estimated excess is retained, not clipped.

Every grid stops at its first shared-owner rejection, retaining its last admitted full state, attempted step and unchanged-graph check. It neither invents a rejected proposal report nor retries with a smaller step or broader chamber. Refinement compares only checkpoints reached by all grids. No later form tuning, prescribed phase path or new pressure term is allowed.

Retained response: crossing succeeded; final capture is unavailable¶

The prediction was frozen before the single evaluation. Its SHA-256 is 4ed42b2a94138e5a5ed7318c39aa9fdb485234f38cbdcab97ab8a2da75959956; the response SHA-256 is 651cf5bcdeec81d64c4289c56110d052ea23d48a0ecdd08af3c76e65ab258d25. The records retain the complete package/producer fingerprints, declared binary64 runtime, exact arithmetic defects and all saved full states. No random initialization, retry, altered pressure or subsequent recalibration was used.

Euler step	Completed steps	Last admitted time	First observed winding `(1,1)`	First observed acute edges	Last minimum real-resultant bound
`1/64`	37	`0.578125`	`0.015625`	`0.46875`	`0.00289508`
`1/128`	73	`0.5703125`	`0.0078125`	`0.46875`	`0.0278649`
`1/256`	147	`0.57421875`	`0.0078125`	`0.46875`	`0.0121522`

Both rings start with winding zero, and all three grids satisfy the frozen winding-one prediction at both 1/64 and 1/32. The common checkpoint state discrepancies decrease under refinement: at t=1/4, the combined form/phase infinity differences are approximately 1.26349e-3 and 6.27253e-4 for successive grids. This is finite refinement evidence, not a validated bound against the exact ODE. The maximum measured work residual is below 2.64e-14; the declared 1e-10 numerical policy passes. Euler energy step defects remain nonzero and recorded separately.

Each run subsequently enters the acute phase region but stops when the next whole-chord positive-resultant bound cannot be certified. Every last state is still regular, winding-one and acute. Rejection leaves its live graph unchanged. The limiting rows are nodes 3 and 8, whose phase metrics approach zero as their two neighbors approach antipodal separation. The endpoint evidence and the symmetry calculation below locate the relevant boundary; the stopped Euler runs do not prove that the exact ODE reaches it in finite time or lacks every other continuation.

None reaches the declared horizon 16. Initial joint storage is about 283.894; last-state values are about 203.45--204.46, while the target phase storage is about 6.90983. The final saved quotient squared norms are about 276.25--277.64, well outside the conservative 9/800 basin indicator. Thus acute entrance alone is plainly insufficient for capture. Later common checkpoints and final-horizon basin verdicts are unavailable, not failed completed predictions or inferred equilibria.

The retained-response controls reconstruct winding/availability/acute geometry from every saved phase state, check the original preparation and time accounting, and replay detached last-state observations. They do not rerun the trajectories. The result establishes an executed, same-law change of geometric sector under supplied initial structure. It does not establish autonomous formation and maintenance of the target, spontaneous support creation, or physical particles.

Reflection reduction explains the next mathematical obligation¶

The fixed preparation has more structure than two arbitrary five-node rings. It is invariant under copying the rings and under simultaneous form-sign and phase-reflection about a constant phase c. Each ring can be written

\[ x=(A,-A,-B,0,B),\qquad \theta=c\mathbf1+(a,-a,-b,0,b). \]

The graph automorphism exchanges ports 0/1, nodes 2/4 and the corresponding second-ring nodes, while fixing nodes 3 and 8. Unit capacities and the unforced joint law respect these symmetries on its regular domain. Uniqueness therefore preserves this four-coordinate subspace of the ideal ODE; it is not imposed on the numerical states to alter their evolution. In the component containing the preparation, the central form and phase rows vanish because q_3=g_3=0, so c remains constant.

Writing q=3*A-B and r=2*B-A, the nonredundant relative resultants are

\[ z_0=1+e^{-2ia}+e^{i(b-a)},\qquad z_4=2\cos(a/2)e^{i(a/2-b)},\qquad z_3=2\cos b. \]

The other port/interior rows are conjugates. For |a|<pi, |b|<pi/2 and the retained regular branch, define g_0=Arg(z_0)/pi, H_0=pi*|z_0|*sinc(Arg(z_0)), g_4=(a/2-b)/pi, H_4=2*pi*cos(a/2)*sinc(a/2-b). The unchanged complete joint equations reduce exactly to

\[ \begin{aligned} \dot A&=-e\,q/3+w\,g_0,& \dot B&=-e\,r/2+w\,g_4,\\ \dot a&=(w/\beta)\,q/H_0,& \dot b&=(w/\beta)\,r/H_4. \end{aligned} \]

These are derived coordinates of the existing law, not a new pressure, controller or independent reduced solver. In particular, positive r still drives b upward after the edges become acute. The retained last states have r near 7.5, so their remaining form contrast has not disappeared merely because their winding and acute status now match the target's labels.

The full two-ring storage in this subspace is

\[ \begin{aligned} E_D&=6A^2-4AB+4B^2,\\ V(a,b)&=10-2\cos(2a)-4\cos(a-b)-4\cos b,\\ \dot E&=-e\left[\tfrac43(3A-B)^2+2(2B-A)^2\right]\le0, \qquad E=E_D+\beta V. \end{aligned} \]

The invertible proof coordinates A=(2*q+r)/5, B=(q+3*r)/5 give E_D=(4*q^2+4*q*r+6*r^2)/5 and diagonal loss -e*(4*q^2/3+2*r^2). They connect the existing first-exit proof to a potential interval storage bound without projecting the runtime state. The full central-resultant admission 2*cos(b)>0 remains necessary even though its exact symmetric dynamical row vanishes.

At the observed central cancellation face,

\[ V(a,\pi/2)=7+4(\sin a-1/2)^2\ge7. \]

This is the same 7/2 cycle barrier from section 17 applied to both reflected rings, with zero bridge storage. An ideal trajectory with E<7*beta cannot reach this face while the nonincrease law remains admitted. The acute target (a,b)=(4*pi/5,2*pi/5) has V=10*(1-cos(2*pi/5))<7; the nonacute saddle (2*pi/3,pi/3) has V=7. These facts connect storage loss to a potentially protected region around the target. They do not prove that a winding-zero preparation can enter that region, nor that E<7*beta alone identifies its basin rather than another allowed region.

As b approaches pi/2, the full central resultant vanishes. The exactly symmetric central row may look removable because its numerator is identically zero, but arbitrary nearby full states do not share that cancellation. A continuation of four formal coordinate equations through the face would not by itself extend the full phase pressure and inverse-metric law. Beyond the face, the central resultant is negative real: assigning it zero phase source would change the native law, while either Arg-side limit breaks the imposed central zero-form row. Conversely, the finite guard stops before that face and is not a singular-hitting theorem.

Section 19 uses these reduced equations and the existing local recovery theorem to prove protected capture and distinguish its possible limits. Entry from winding zero still requires a crossing and loss budget. The supplied high-storage response remains the first retained test; it is not retuned after observing its stop.

Static formation-admission controls independently reconstruct the reduced Laplacian, resultants and storage from the full support, check a nonacute regular field against the four equations, and verify both signs of the sharp central-cancellation barrier. They add neither a reduced execution path nor another temporal campaign.

19. Protected capture and conditional basin selection¶

The reflection reduction supplies a global capture result within an explicit region, without another integration. Retain the same two-ring, two-bridge support, unit held capacities and fixed e,w,beta>0. Work in the exact ideal reflection subspace of section 18. Arbitrary common form and phase offsets may be added; both are constant here because the central rows vanish.

A protected region around the winding-one target¶

Define the positive-target phase rectangle

\[ \mathcal R=(2\pi/3,\pi)\times(0,\pi/2). \]

Every relative resultant has positive real part there. In particular, Re(z_0)=1+cos(2a)+cos(a-b)>cos(a)*(2*cos(a)+1)>0; z_4=2*cos(a/2)*exp(i*(a/2-b)) has positive modulus and argument in (-pi/6,pi/2); and z_3=2*cos(b)>0. Thus the full joint law is smooth, not merely its formal four-coordinate expression. The four boundary costs are

\[ \begin{aligned} V(2\pi/3,b)&=11-4\cos(b-\pi/3)\ge7,\qquad V(\pi,b)=8,\\ V(a,0)&=9-4(\cos a+1/2)^2\ge8,\\ V(a,\pi/2)&=7+4(\sin a-1/2)^2\ge7. \end{aligned} \]

Conditional capture theorem. Any exact reflected initial state with (a,b) in R and E=E_D+beta*V<7*beta has a global forward solution and converges to (a,b,A,B)=(4*pi/5,2*pi/5,0,0).

Indeed, E_D=6*A^2-4*A*B+4*B^2 is positive definite, so a fixed sublevel bounds form. Storage nonincrease and the boundary costs prevent a first exit from the rectangle. The phase sublevel stays a positive distance from its boundary; together these give a compact subset of the smooth full-law domain. Continuation is therefore global. Zero loss forces q=r=0, hence A=B=0. To remain in that set requires g_0=g_4=0. The latter gives b=a/2, and the former reduces to

\[ \sin(2a)+\sin(a/2)=2\sin(5a/4)\cos(3a/4)=0. \]

Its unique solution in this rectangle is a=4*pi/5. LaSalle's invariance principle then gives convergence to that target; the local-recovery theorem supplies an eventual exponential rate, without a graph-independent bound. The limiting storage is beta*10*(1-cos(2*pi/5)), not zero.

Every phase point in R already has ordinary winding one on both rings: the positively wrapped ring gaps are 2*pi-2*a, a-b, b, b, a-b, whose sum is 2*pi. This theorem establishes capture of a supplied sector; it does not establish its formation from winding zero. The strict storage test concerns the ideal law and exact trigonometry, not a rounded energy estimate or successful admission of an arbitrary Euler step.

Simultaneous sign reversal (a,b,A,B) -> (-a,-b,-A,-B) preserves the same law and storage. Hence the mirror rectangle R_minus=(-pi,-2*pi/3) x (-pi/2,0) has the identical E<7*beta capture theorem, with target (-4*pi/5,-2*pi/5,0,0) and both windings equal to minus one. These are distinct protected regions of the same completion, not different operator policies or extra dynamical rules.

The saddle has a target-directed unstable branch¶

At the nonacute regular equilibrium s=(2*pi/3,pi/3,0,0), both active metrics equal pi, storage equals 7*beta, and the phase-storage Hessian is [[-2,-2],[-2,4]]. Put

\[ K=\begin{pmatrix}3&-1\\-1&2\end{pmatrix},\quad G=\begin{pmatrix}1/2&1/2\\1/2&-1\end{pmatrix},\quad D=\operatorname{diag}(3,2),\quad \omega^2=\frac{w^2}{\beta\pi^2}. \]

In phase/form coordinates (u,p)=(delta(a,b),delta(A,B)), the Jacobian is

\[ J_s=\begin{pmatrix}0&(w/(\beta\pi))K\\ (w/\pi)G&-eD^{-1}K\end{pmatrix}. \]

Eliminating form gives the symmetric quadratic eigenvalue pencil P(lambda)=lambda^2*K^(-1)+e*lambda*D^(-1)-omega^2*G. At zero it has one negative and one positive eigenvalue; for real lambda>=0, its derivative is positive definite and its large-lambda limit is positive definite. Hence it has exactly one simple positive root. Zero is excluded because G is invertible. For a nonreal root and nonzero phase vector u, the imaginary part of u^*P(lambda)u=0 gives

\[ \operatorname{Re}\lambda =-\frac{e\,u^*D^{-1}u}{2u^*K^{-1}u}<0. \]

Thus the equilibrium is hyperbolic with a one-dimensional unstable manifold in this reflected subspace. Its unstable phase vector has nonzero first component, since P_22=3*lambda^2/5+e*lambda/2+omega^2>0. Orient it by u_a>0; its slope is

\[ \frac{u_b}{u_a} =\frac{\omega^2/2-\lambda^2/5} {3\lambda^2/5+e\lambda/2+\omega^2}\in(0,1/2). \]

For the lower bound, P is already positive definite when lambda^2=5*omega^2/2, where its off-diagonal entry vanishes, so the unstable root lies below that value. The displayed ratio then gives both bounds. These are properties of the local linearization, not a simulated connection or a finite-time error estimate.

Every nontrivial unstable-manifold orbit approaches the saddle as t -> -infinity. Its strict energy deficit follows from the actual balance:

\[ E(t)=7\beta-e\int_{-\infty}^{t} \left[\tfrac43q(s)^2+2r(s)^2\right]ds<7\beta. \]

The integral converges by the local exponential unstable asymptotics and is positive because the unstable form component is nonzero. The branch oriented by increasing a enters R. The capture theorem therefore continues that branch globally forward to the acute winding-one target: a conditional same-law heteroclinic orbit from the supplied nonacute saddle.

The opposite branch approaches consensus¶

The other local branch enters S=(-2*pi/3,2*pi/3) x (-pi/2,pi/2) with the same strict storage deficit. Here the boundary values are V(±2*pi/3,b)=11-4*cos(b∓pi/3)>=7 and V(a,±pi/2)=7+4*(sin(a)∓1/2)^2>=7. Within a sublevel V<7,

\[ V\ge8-4\operatorname{Re}z_0,\qquad V\ge8-4\operatorname{Re}z_4, \]

so both real parts exceed 1/4; z_3=2*cos(b)>0 as well. Consequently this sublevel is regular in the same positive-resultant chamber, despite S itself not supplying that assertion at arbitrary energy. Compactness and LaSalle apply again. Now g_4=0 gives b=a/2, and the displayed critical equation has only a=0 inside S. Thus every exact reflected state in S with E<7*beta converges to consensus, including the opposite unstable branch. It starts with winding one near the saddle and eventually has winding zero near consensus. This is a forward loss of winding, not a reversal of the target-directed orbit. In particular it reaches a<pi/2 forward in time. The mirrored saddle has the corresponding two branches toward the negative target and consensus. This gives three sufficient capture regions; it is not a classification of every initial state or a proof that no other behavior exists outside them.

The exact saddle itself never departs. These are two distinct trajectories with the same backward limit, not pieces that can be concatenated into a winding-zero-to-target solution of the autonomous law. Capture and consensus selection here remain conditional on supplied support, constitutive premises and the prepared state; no spontaneous substrate or physical identity follows.

The exact reflection restriction is sufficient, not an assertion that every slightly asymmetric state fails. From each state in any of these three protected regions, a finite compact regular segment reaches the corresponding full-state local-recovery neighborhood. Continuous dependence therefore gives an open neighborhood of that state in the full form/phase state, with support, capacity and model held, whose trajectories also converge modulo common offsets. This is an existence statement: no radius has been computed, and no rounded preparation, retained earlier trajectory or numerical capture indicator is thereby certified.

Shared admission of the sufficient ideal-law basins¶

certify_relational_capture and Network.relational_capture(model, cycles=...) reuse one fresh detached field and the shared rational cosine/pi enclosures. They check the exact support, copied/reflected represented state, unit capacities, positive coefficients, rectangle and strict energy bound, without projecting the state or integrating a second solver. The output retains all symmetry defects, candidate rectangle margins, exact energy bounds and unavailable reasons. A target sector is supplied only on complete admission, independently of current winding. The certificate concerns the ideal continuous law from that represented state; Euler trajectories retain their separate obligations. The API contract owns implementation details, and SDK usage supplies a nonacute zero-form preparation.

Static production controls exercise both twist sectors, consensus from a winding-one state, strict energy equality, exact symmetry, common offsets, unsupported domains and exact report export. The symbolic controls derive boundary costs, resultant lower bounds, critical geometry and the saddle pencil from the full support. No new temporal campaign is used to establish these conclusions, and no old frozen outcome is rewritten.

The same owner also exposes certify_relational_local_capture and Network.relational_local_capture. These apply section 15's full-state local theorem on this support, with beta=nu=1, to a declared target sector. They use all centered form/phase coordinates, exact affine-pi reference errors and outward cosine energy bounds. The conservative conditions are norm_squared<9/800 and excess_storage<1/100000 from section 17; exact reflection is unnecessary. Their proof is independent of any observed trajectory. If applied to a numerical endpoint, the certificate concerns the ideal continuation restarted at that precise represented state. It does not supply an error tube from the original initial-value problem.

A distinct winding-zero preparation with a bounded crossing theorem¶

The previous high-storage preparation is not the only way to drive the same law. Retain unit capacities, e=w=1/2, beta=1, both supplied rings and both bridges, and the exact reflection coordinates. A separate preparation is

\[ a_0=b_0=\pi/2-\varepsilon,\qquad A_0=2/5,\quad B_0=1/5,\qquad \varepsilon=1/64. \]

It has q_0=1, r_0=0 and winding zero on each ring. All initial relative resultants have positive real part:

\[ z_0=2-\cos(2\varepsilon)-i\sin(2\varepsilon),\quad z_4=1+\sin\varepsilon-i\cos\varepsilon,\quad z_3=2\sin\varepsilon>0. \]

The form storage is exactly 4/5; the full storage is

\[ E_0=44/5-4\sin\varepsilon-4\sin^2\varepsilon, \qquad 44/5-65/1024<E_0<44/5. \]

This state lies above the protected 7*beta threshold. It is not admitted by the capture theorem simply because its storage is smaller than that of the earlier frozen response.

Put chi=atan(sin(2*epsilon)/(2-cos(2*epsilon))). Then 0<chi<2*epsilon, g_0=-chi/pi, g_4=-1/4+epsilon/(2*pi), H_0=pi*sin(2*epsilon)/chi and H_4=2*pi*cos(epsilon)/a_0. The initial derivatives obey

\[ \begin{aligned} \dot q_0&=-3/8-(6\chi+\varepsilon)/(4\pi)<0,\\ \dot r_0&=-1/12+(\varepsilon+\chi)/(2\pi)<-29/384<0,\\ \dot a_0&=1/(2H_0)>0,\qquad \dot b_0=0,\qquad \ddot b_0=\dot r_0/(2H_4)<0. \end{aligned} \]

Thus the form imbalance initially moves a toward the winding boundary while moving b away from the central-resultant cancellation face. As epsilon -> 0+, the derivatives tend to qdot=-3/8, rdot=-1/12, adot=1/(2*pi) and bddot=-1/96. The limiting corner itself is singular and is not an admissible initial state; the selected positive epsilon remains part of the preparation.

Bounded crossing, not capture. For this fixed ideal preparation, the solution remains regular through t=1/4 and both windings change from zero to one by t=7/48. This follows without numerical integration. Use the box

\[ \begin{gathered} \pi/2-1/64\le a\le\pi/2+1/16,\qquad \pi/2-1/32\le b\le\pi/2-1/64,\\ 3/4\le q\le1,\qquad -1/8\le r\le0. \end{gathered} \]

Here 0<=a-b<=3/32, abs(a+b-pi)<=3/64, and Re(z_0),Re(z_4)>2039/2048 (the non-strict lower bound also suffices). The elementary bounds sin(t)<=t, cos(t)>=1-t^2/2 and 3<pi<22/7 give

\[ |g_0|<1/60,\quad -1/4<g_4<-11/48,\quad 5/2<H_0<7/2,\quad H_4>5/2. \]

The reduced equations therefore imply -1/2<qdot<0, rdot>-1/6, and, on the face r=0, rdot<-13/240. Consequently, before any first exit,

\[ \begin{aligned} q(t)&\ge1-t/2,& r(t)&\ge-t/6,\\ 3/28&<\dot a(t)<1/5,& b_0-t^2/60&\le b(t)\le b_0. \end{aligned} \]

For 0<=t<=1/4 these inequalities prevent exit through every face; r<0 for positive time prevents exit through the upper b face. The central resultant stays at least 2*sin(1/64)>0. Smooth continuation on this compact box establishes the interval, and the lower bound on adot gives a(7/48)>pi/2. Since a increases, while b remains positive and 0<=a-b<pi, the ordinary ring winding crosses once at the antipodal edge 0--1 (and its copied edge). The full resultants remain regular at that crossing. The winding itself is undefined at the crossing instant.

The signs and corridor motivate a separate finite prediction through the shared executor. They are an ideal-law theorem, not an Euler-error bound or a result for the binary representation of the supplied phases. In particular, this argument does not show entry into R, enough subsequent storage loss for E<7, or convergence to the winding-one target. The static controls check the initial identities and the rational corridor estimates without executing a trajectory.

Frozen upper-corner response: regular sector acquisition and a remaining certificate gap¶

relational_capture_response.py froze one materialized preparation, e=w=1/2, beta=nu=1, no Gamma, the supplied ten-node/twelve-edge support, horizon 64 and three Euler steps 1/64,1/128,1/256 before evaluation. Form and phase offsets were zero. The binary values corresponding to (2/5,1/5) have exact r=0 but q=1+2^-54; the preceding ideal-input proof is not silently identified with those represented values. Python 3.13 and dependency/platform versions, all 599 package/producer source fingerprints, exact inputs and numerical settings are retained in the prediction.

The protocol stopped each grid at its first admitted target certificate, shared-owner rejection, or horizon. No projection, retry, phase forcing, parameter sweep or horizon extension was performed. All three grids reached 64 without rejection; the target certificate was unavailable on every grid. The compound finite_prediction_passed is therefore false, while the numerical work-residual policy passed.

Euler step	Executed steps	Final storage upper bound, rounded	Final squared quotient norm, rounded	Final target excess upper bound, rounded
`1/64`	4096	6.919042007	0.03467713	0.009211950
`1/128`	8192	6.919040986	0.03467310	0.009210930
`1/256`	16384	6.919040485	0.03467112	0.009210429

Every grid starts with windings (0,0) and has (1,1) at all retained checkpoints from t=1/4 through 64. This is sampled numerical evidence, separate from the ideal short-crossing theorem. At the retained times 32 and 64, all first-ring (a,b) coordinates lie in the positive-twist rectangle and the full-state certified storage is below 7. For the finest grid at 32, (a,b) is approximately (2.267268628,1.109711608) and the storage upper bound is approximately 6.970551621.

However, the represented states have small nonzero reflection defects. At 64 the largest form/phase defects across the grids are below 2e-18, while ring-copy defects are zero. The reflected theorem rejects those states solely on its exact-symmetry premise. The full-state local certificate also rejects them: both distance and excess storage remain above its conservative thresholds. Neither certificate can be promoted by rounding those defects away or by treating energy below 7 as a support-independent basin test.

The maximum native work residual across the three traces is below 2.176e-16; clock defects are zero. Maximum absolute energy-step defects decrease from about 2.503e-5 to 6.257e-6 to 1.564e-6. All retained states and represented Euler chords satisfy the positive- resultant guard. At 64 the coarse/middle and middle/fine state infinity differences are about 4.391e-6 and 2.154e-6; differences decrease under refinement at every common checkpoint. These are numerical comparisons, not rigorous global trajectory-error bounds or asymptotic convergence.

Immutable record	SHA-256
Prediction	`4045ab698013f2bb071c5083a99e8aea760c7ce42e568e1be5a230af9629db7c`
Response	`7085c782f0b85c465c530d2e84ff2990e71dfcc89d950f1f993025a952310ed9`

The retained-response controls recompute certificate decisions from saved states without re-executing the campaign. Earlier frozen responses remain unchanged. This result supports a regular same-law route toward the maintained geometry on supplied support; it does not yet certify the original continuous trajectory's entry into the protected basin, autonomous support creation, or a physical NFR identity.

20. A full-state acute-sector barrier without reflection¶

The retained response exposes a gap between the dynamics and the available sufficient certificates: its tiny symmetry defects invalidate the reflected theorem, while its distance from the target exceeds the deliberately small local bound. Neither restriction is intrinsic to every capture proof. The same support admits a wider full-state theorem based on its cycle periods and cosine storage, with no symmetry projection or changed evolution law.

Retain the two supplied C5 rings and the two bridges between matching positions 0 and 1. Let e,w,beta>0 and all held capacities be strictly positive. Suppose every true wrapped support-edge phase difference is strictly acute and both supplied oriented rings have winding s, where s=+1 or s=-1. Define

\[ V_* =5\bigl(1-\cos(2\pi/5)\bigr),\qquad V_{\rm face}=5-4\cos(3\pi/8),\qquad \mathcal B=V_*+V_{\rm face}. \]

Conditional full-state capture. If the actual full-state storage satisfies

\[ E_D+\beta V<\beta\mathcal B, \]

the ideal trajectory remains in its acute component for all forward time and converges to uniform form and the aligned winding-s twist, modulo common offsets. Exact ring copying, form/phase reflection and a prescribed small Euclidean radius are not premises. The theorem concerns circular phase geometry: the final raw phase lift can additionally contain fixed nodewise integer multiples of 2*pi, determined by its initial acute component.

The cycle geometry supplies the barrier¶

For s=+1, the five oriented acute differences of either ring sum to 2*pi. Convexity of 1-cos(delta) on [-pi/2,pi/2] and Jensen's inequality give ring storage at least V_*. At a first acute-boundary encounter on that ring, a difference of -pi/2 is impossible: the other four would need sum 5*pi/2, exceeding their maximum 2*pi. Thus a boundary edge has difference pi/2, and the other four have sum 3*pi/2. Another application of Jensen gives

\[ V_{\rm ring}\ge1+4\bigl(1-\cos(3\pi/8)\bigr)=V_{\rm face}. \]

The other ring still costs at least V_*; bridge costs are nonnegative. Consequently every first boundary through a ring edge has total phase storage at least mathcal B. At a bridge boundary, its own cost is 1, so the total is at least 1+2*V_*, which is larger: since cos(3*pi/8)>cos(2*pi/5),

\[ 1+V_*-V_{\rm face} =1-5\cos(2\pi/5)+4\cos(3\pi/8) >1-\cos(2\pi/5)>0. \]

Sign reversal proves the same statements for s=-1. The one-ring Jensen bound is sharp, but mathcal B is only a sufficient whole-support boundary bound; simultaneous equality need not respect both bridges. No claim of an optimal basin or a complete classification of the regular domain is made.

The barrier has an exact radical expression and a simple rational lower bound:

\[ \mathcal B=\frac{45-5\sqrt5}{4}-2\sqrt{2-\sqrt2} >\frac{6924179}{1000000},\qquad \mathcal B\simeq6.924181298665. \]

For example, the certified inequalities cos(2*pi/5)<309017/1000000 and cos(3*pi/8)<382684/1000000 imply the displayed lower bound. They follow by squaring positive rational bounds in the exact radical formulas, independently of binary trigonometric evaluation. An executable certificate can instead reuse the shared rational pi/cosine enclosures.

Compactness, the target and the zero-loss set¶

The two rings and the four-cycle through both bridges form a cycle basis. Four strictly acute differences cannot sum to a nonzero integer multiple of 2*pi, so the four-cycle has period zero. The three periods therefore agree with the aligned twist: both rings have phase step s*2*pi/5 and both bridges have zero difference.

Choose compatible node lifts of that reference and the initial phases. Equality of all cycle periods makes their edge-difference discrepancy an exact graph gradient. The common acute component is therefore a convex polytope in lifted phase coordinates modulo their common offset, with each edge difference restricted to its selected interval of length pi. The aligned twist belongs to this component. The phase-storage Hessian is the graph Laplacian with edge weights cos(delta)>0, hence is positive definite on the common-phase quotient. Its sole critical geometry in the component is consequently the aligned twist.

Storage nonincrease and the strict boundary gap prevent a first exit. Connected support bounds phase differences modulo the common offset; Dirichlet storage bounds form modulo its common offset. Thus a fixed joint sublevel below beta*mathcal B is compact in this quotient and stays a positive distance from the acute boundary. All H entries stay strictly positive. Quotient continuation is global, and bounded common-offset rates also exclude finite-time failure of the full state.

For the invariant zero-loss set, positive capacity and e>0 force Bx=0. Uniform form alone does not yet imply zero common-form drift. To preserve that condition requires N*g=c*1 for a scalar c. But the undirected phase potential satisfies

\[ \sum_i H_i g_i=0,\qquad c\sum_i H_i/\nu_i=0. \]

Every H_i/nu_i is positive, so c=0 and g=0. LaSalle's invariance principle on the compact quotient now gives convergence to the unique twist geometry and uniform form. The existing local theorem supplies eventual exponential recovery and finite limiting common offsets. Those offsets need not equal their initial means in an asymmetric state.

Capture shared by a declared strictly dissipative law class¶

The sector barrier also separates the maintained geometry from a uniquely selected transient law. Keep the supplied support, fixed positive capacities, positive e,w,beta, native form row and chosen storage:

\[ \dot x=N(-eD^{-1}Bx+w g),\qquad E=\tfrac12x^TBx+\beta V,\qquad L=e\,q^TND^{-1}q,\quad q=Bx. \]

Consider a phase law theta_dot=F(x,theta) with the following additional law-level premises:

The law is autonomous and C1, well-defined on circular phase states, and invariant under both common form shifts and common phase rotations. Its domain contains an open regular neighborhood of the entire protected storage sublevel, including all sufficiently small perturbations of the target. The fixed support, capacities and constitutive coefficients do not change during the flow; there is no input or event.
It has rest at every state with q=g=0: F=0 there, including its common phase component.
For some fixed c>0, its actual full vector field satisfies the pointwise inequality

[ \dot E\le-cL ]

throughout that neighborhood. A sampled decrease, a condition only along one trajectory, or merely E_dot<=0 does not supply this premise.

Under these premises, every preparation admitted by the strict acute-sector barrier above remains in that sector and converges to the same aligned twist and uniform form. Convergence is eventually exponential on the joint quotient, and the full trajectory has finite limiting common form and phase offsets. This is a conditional theorem for the declared class; the storage and dissipation assumptions are not derived from the nodal identity.

Retention and continuation use the unchanged geometry. Nonincrease of the same storage gives exactly the previous first-exit contradiction and compact quotient sublevel. The geometric acute-edge, resultant and metric lower bounds in the quantitative consolidation therefore apply with the same initial storage deficit. They do not depend on a shared transient trajectory. The C1 field is bounded on the compact quotient; shift invariance bounds the common-offset velocities there as well. These facts give global continuation without requiring bounded absolute offsets before convergence has been proved.

The native form row fixes the invariant zero-loss set. Since c,e,nu_i are positive, zero loss requires q=0. A trajectory remaining in that set must satisfy BNg=0, hence Ng=a*1. The identity sum(H_i*g_i)=0 and positivity of H_i/nu_i give a=0 and g=0, exactly as above. The phase rest premise makes this set invariant. The phase potential has only the aligned-twist critical geometry in the acute component, so LaSalle's argument gives quotient convergence to that geometry and uniform form for every law in the class.

Local exponential recovery also follows from the declared conditions. An extra assumption about a named law's Jacobian is unnecessary here. At the target, write K=Hess(V) and let the columns of R be an orthonormal basis of the complement of the constant vector. Define

\[ B_q=R^TBR,\qquad K_q=R^TKR,\qquad M_D=R^TND^{-1}R,\qquad M_H=R^TNH^{-1}R. \]

All four matrices are symmetric positive definite: the support is connected, the target is acute, and N,D,H are positive diagonal matrices. In quotient coordinates z=(u,v), the Hessian of storage is

\[ P=\operatorname{diag}(B_q,\beta K_q)>0. \]

The identity grad(V)=-H*g gives Dg=-H^{-1}K at g=0. Consequently the first block row of the quotient Jacobian A is fixed, independently of the choice of F:

\[ (Az)_x=-eM_DB_qu-wM_HK_qv. \]

Here BR=RB_q and KR=RK_q justify the projected products. The product M_H*K_q is invertible; it need not be symmetric, and the proof does not treat it as a symmetric positive definite matrix.

Expanding the pointwise loss inequality to quadratic order at rest gives

\[ \frac{PA+A^TP}{2} \preceq-ce \begin{pmatrix} B_qM_DB_q&0\\ 0&0 \end{pmatrix}. \]

This expansion requires the inequality on a full neighborhood, not only on the selected orbit. For a nonzero complex eigenvector z=(u,v) of A,

\[ \operatorname{Re}(\lambda)\,z^*Pz \le-ce\,u^*B_qM_DB_qu. \]

Thus no eigenvalue has positive real part. Equality of its real part to zero would force u=0; the first block row would then force w*M_H*K_q*v=0, hence v=0, a contradiction. The quotient Jacobian is therefore Hurwitz for every admitted law. The C1 nonlinear stability theorem gives local exponential recovery, and the previously proved quotient convergence supplies eventual entry into that neighborhood. Shift invariance and rest make the common-offset velocities vanish at the target and bound them by a constant times the quotient deviation nearby. Their exponential decay is integrable, giving finite limiting common offsets. These limits need not be the preparation's means.

The named comparison laws satisfy the conditions separately. The reference law has E_dot=-L, so it admits c=1. The phase-damping completion F_rho=F_rel+rho*N*g, with fixed rho>=0, satisfies

\[ \dot E=-L-\beta\rho\sum_i\nu_iH_i g_i^2\le-L, \]

and also admits c=1. The nonlinear signed-form completion, with fixed 0<eta<2, satisfies

\[ \dot E=-L+\mathcal R\le-(1-\eta/2)L, \]

so it admits c=1-eta/2. Each law is smooth on the regular neighborhood, shift invariant and at rest when q=g=0. Their previous independent calculations establish these premises; sector geometry alone does not. The nonlinear completion retains the reference Jacobian, whereas positive phase damping changes it. Both nevertheless share this capture conclusion.

Geometry evidence is not law admission. A strict snapshot certificate can establish the support, winding, acute-margin and storage premises. It cannot establish an arbitrary callback's pointwise dissipation, regularity, shift invariance or rest. Those require a separately identified law and proof. The reference certificate keeps its declared reference-model meaning; reuse of its geometry does not silently install an alternative law.

The theorem supplies neither a rate uniform over all admitted laws nor the reference law's previously calculated rates, validated transient tubes, formation-entry times or frozen responses for a changed law. In particular, it does not establish that an alternative law carries the original winding-zero preparation into this sector. That requires a separate transit argument. Zero capacities, w=0, a nonacute target, nonsmooth laws, inputs and support events also fall outside this proof. The conclusion is robust maintenance on supplied support under stated constitutive premises, not autonomous support birth or physical identification of the pattern.

Evidence and execution boundary¶

The hypotheses can be checked from a saved full-state snapshot using strict pi-enclosed edge gaps, exact integer cycle periods and certified cosine storage bounds. Numerical winding labels, rounded symmetry or an estimated distance to the target cannot replace those checks. The shared sector certificate admits the theorem's arbitrary held positive capacities and positive beta; the separate reflected and local owners retain their own more restrictive premises.

This theorem was derived after the reserved response in section 19. Applying it to a retained endpoint is a separately identified mathematical reanalysis. It cannot turn the failed original frozen prediction into a successful preregistered capture test, change its earlier certificate decisions, or justify rerunning it with another gate. An admitted endpoint certifies the ideal law restarted at that precise represented state. It still does not bound the discrepancy between the numerical trace and the original continuous trajectory from its winding-zero preparation. No longer trajectory, support event, additional pressure, controller or physical-identity claim is introduced by this argument.

The static controls check the support's cycle basis, target periods, one-ring face costs and the strict rational barrier bound without executing any evolution.

Shared certificate and retained endpoint reanalysis¶

certify_relational_sector_capture and Network.relational_sector_capture in the shared capture owner implement this full-state theorem. The first retained endpoint audit used unit capacity and unit beta; extending the shared admission to the theorem's positive held capacities and beta does not change that record. The existing exact topology, phase-storage and pi/cosine owners are reused. Each candidate wrapped edge gap is represented as a rational plus an integer multiple of mathematical pi; two strict enclosed inequalities must certify its acute interval. Summing the admitted integer turns yields both ring periods and the bridge square's period. Floating winding telemetry remains separate. The full storage upper bound must lie below a certified lower bound on the barrier. The API contract owns report fields, unsupported domains and restart-only scope.

The separate read-only auditor binds the immutable section-19 records and reconstructs their three final states without modifying any coordinate. All three states pass the new theorem. Their lower bounds on the energy margin mathcal B-E are about 0.00513929210, 0.00514031242, 0.00514081319 in coarse-to-fine order; every strict acute margin exceeds 0.16215 radians. Both exact ring periods are one and the bridge-square period is zero. Consequently the ideal law restarted at each actual represented endpoint remains regular and converges to the aligned twist without a reflection premise or further forcing.

Separate immutable evidence	SHA-256
Post-evaluation endpoint audit	`8d5dee62f7514118d69a19173599a6a536f1c516d2ee486c98abf3ea4e22352a`
All 599 original fingerprinted source files	`eb0600da8e479e8b7f1bfede31dbf802ef420ed8419fad982ca49e3e7ba8d049`
Four-file source overlay for the endpoint audit	`a0c9075f1514ab0b8fdac5d54a1c7e8c79e7c535d48348b9f5c1ce72730d23b2`

The source archive was verified against every original prediction digest before post-evaluation integration changed the engine/SDK source. It does not bundle dependencies; their versions remain in the original prediction. The new audit retains its own source/runtime manifest and explicitly records both all_endpoints_admitted=true and the unchanged original finite_prediction_passed=false. Tests check archive byte identity, detached report data and the old/new decisions without re-executing any trajectory. The small overlay plus the base source archive reproduces every source hash in the endpoint audit; it preserves the original unit-restricted owner before the generalization below without duplicating the complete package archive.

The endpoint audit alone leaves an original-IVP obligation: a validated finite-time transit enclosure, now supplied below. The exact represented preparation is initially reflected, so its ideal trajectory stays in the existing four-coordinate slice even though finite Euler arithmetic breaks that symmetry. Enclosing that same flow to an admitted reflected or acute-sector basin would join formation and maintenance for the original continuous state. It requires no new constitutive law, tuned preparation or longer finite observation.

Consolidation: capacity, storage scale and a quantitative regularity margin¶

The sector theorem already uses the same independent-capacity law as the heterogeneous-capacity balance and local recovery theorems. No equality of capacities appears in its barrier or zero-loss argument. All capacities must remain strictly positive and held; e,w,beta>0, the supplied support, the strict acute component and its exact cycle periods remain premises. Multiplying every capacity by the same a>0 multiplies both differential rows by a, so it changes only their common clock. Heterogeneous positive capacities can change transients and limiting offsets while preserving this sufficient geometric basin. None of these statements admits capacity events or extends the separate reflected/local production certificates.

Dividing storage by beta gives E_D/beta+V_phi. Thus, on exact real form charts, replacing beta by beta_new and multiplying all form contrasts by sqrt(beta_new/beta) preserves the sector energy test. It need not preserve the trajectory under a time change: with y=x/sqrt(beta), form damping has coefficient e while the two exchange terms have coefficient w/sqrt(beta). Beta therefore controls relative storage and exchange as well as admission; it is not generally a pure clock scale.

An admitted energy deficit supplies an explicit lower bound on future regularity. It is a derived quantity, not a new primitive or policy threshold. Put

\[ \eta=\mathcal B-\mathcal E(0)/\beta>0,\qquad \kappa=2\pi/5,\qquad F(t)=1-\cos t+4\left[1-\cos\left((2\pi-t)/4\right)\right], \]

and d=F(pi/2)-F(kappa)=mathcal B-2*V_*. Every oriented signed gap y=s*delta on either acute ring is positive: the four other gaps are each strictly below pi/2 and all five sum to 2*pi. Jensen's inequality on those other four gaps gives ring storage at least F(y). On kappa<=t<=pi/2,

\[ 0\le F'(t)=\sin t-\sin((2\pi-t)/4) \le M=1-\sin(3\pi/8)<1/13. \]

The final strict rational inequality has an exact static check: sin(3*pi/8)=sqrt(2+sqrt(2))/2>12/13, because 238^2=56644<57122=2*169^2. The factor 13 is a conservative rational choice below 1/M, not a distinguished TNFR constant or an evolution parameter. Also 0<eta<=d<M*pi/10<pi/130, using the two-ring Jensen minimum and integrating the displayed derivative. Storage nonincrease makes these same initial-budget estimates valid at every future time. For a ring gap y>=kappa, they imply

\[ \eta\le F(\pi/2)-F(y)\le M(\pi/2-y),\qquad \pi/2-y>13\eta. \]

For y<=kappa, its margin is at least pi/10>13*eta. A bridge instead has cost 1-cos(delta)<=d-eta<pi/130<1/2, hence abs(delta)<pi/3 and margin greater than pi/6>13*eta. Every support edge therefore obeys the uniform, all-future strict acute margin

\[ \boxed{\quad \pi/2-|\delta_{ij}(t)|>13\eta>0.\quad} \]

Let z_i=sum_{j~i} exp(i*(theta_j-theta_i)) be the relative neighbor resultant and d_i the support degree. Sine concavity on [0,pi/2] gives cos(delta_ij)>=sin(13*eta)>=26*eta/pi. In the resulting positive-real chamber, alpha_i=Arg(z_i) lies in (-pi/2,pi/2) and H_i=pi*Re(z_i)*tan(alpha_i)/alpha_i, with ratio one at zero. Thus

\[ \boxed{\quad \operatorname{Re}z_i(t)\ge\frac{26d_i\eta}{\pi},\qquad |z_i(t)|\ge\frac{26d_i\eta}{\pi},\qquad H_i(t)\ge26d_i\eta>0. \quad} \]

The phase Hessian likewise satisfies Hess(V_phi)>= (26*eta/pi)*B as quadratic forms. These bounds quantify the compact regular domain used by the proof; they do not assert a global exponential rate or introduce additional phase damping.

The shared certificate can retain a rational eta_lower=B_lower-E_upper/beta, where B_lower encloses the geometric barrier and E_upper encloses the actual full-state storage. Only complete admission of all sector hypotheses makes the future bounds available. Replacing eta by this positive lower bound and pi by its certified upper bound on the right-hand side preserves every non-strict lower bound above. Unit-capacity and unit-beta flags remain descriptive evidence, not admission requirements. These are exact ideal-law implications from the captured represented snapshot. They neither bound the preceding numerical error nor by themselves close the finite-time transit obligation for the original initial value problem; the validated proof below supplies that separate link. The frozen response and its later endpoint audit remain separate.