Native validated transit and preparation controls

Validated continuous transit, bounded constitutive robustness and zero/reversed/consensus-form controls; frozen responses retain their original numerical verdicts.

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

Validated continuous transit on the exact reflected subsystem

The read-only owner physics/relational_transit.py checks the original continuous IVP through the existing invariant reduction. It uses the supplied two-ring support, exact copied/reflected initial state, unit held capacities and the same positive-coefficient, source-free relational law. This is a proof computation; it does not advance a live graph or replace the shared production integrator. The SDK delegate and exact report exporter retain the full initial admission and every accepted interval step.

With q=3A-B, r=2B-A, let d=a/2-b,

\[ C_0=1+\cos(2a)+\cos(a-b),\quad S_0=-\sin(2a)-\sin(a-b), \quad u=S_0/C_0,\quad R(u)=\operatorname{atan}(u)/u, \]

where R(0)=1. On the admitted positive-real chamber,

\[ g_0=uR(u)/\pi,\quad H_0^{-1}=R(u)/(\pi C_0),\quad g_4=d/\pi,\quad H_4^{-1}=[2\pi\cos(a/2)\operatorname{sinc}(d)]^{-1}. \]

The original proof used the zero-safe analytic series on abs(u)<=1/2 and the sinc series on abs(d)<=1. These are sufficient numerical domains, not new physical limits. The current arctangent-ratio owner retains that same series near zero and, outside it, uses atan(u)'=u'/(1+u^2) followed by formal division when the entire expansion interval avoids zero. A wide interval crossing zero outside the first domain is still unavailable. This analytic extension was tested before evaluating the zero-form control; for example, the already proved consensus-basin state q=r=0,a=b=1 has u about -0.5741, outside the old numerical domain. It is not a new law. Positive 2*cos(a/2)*cos(d) together with abs(d)<=1 selects exactly the displayed branch for g_4. The zero-safe analytic ratios avoid dividing by a pressure source that crosses zero. The reduced rows are

\[ \begin{aligned} \dot q&=-eq+(e/2)r+w(3g_0-g_4),\\ \dot r&=(e/3)q-er+w(2g_4-g_0),\\ \dot a&=(w/\beta)qH_0^{-1},\qquad \dot b=(w/\beta)rH_4^{-1}. \end{aligned} \]

The actual frozen form values are binary64 0.4 and 0.2, so the exact initial values are q=1+2^(-54), r=0, not (1,0). The phase value is the exact represented rational 875483625981347/562949953421312 at both coordinates. The earlier ideal-pi/rational-fifths short-crossing theorem is not silently substituted for this different IVP.

Whole-time enclosure and error propagation

All arithmetic bounds use rational endpoints rounded outwards to a fixed 128-bit dyadic grid. Mathematical pi and cosine bounds reuse the existing Machin-series and cosine owners. Their optional higher-precision path leaves the production cosine defaults unchanged. Sinc and arctangent-ratio series bound every normalized derivative through the requested order; a scalar series remainder alone would not validate time derivatives.

For a current box Y, an attempted compact convex tube B must satisfy

\[ Y+[0,h]F(B)\subset\operatorname{int}B, \qquad C_0>0,\quad 2\cos(a/2)\cos(d)>0,\quad 2\cos b>0 \quad\text{throughout }B. \]

The last condition retains the full central-node resultant even though its row vanishes under reflection. A first-exit argument and smoothness on an open neighborhood of the tube prove existence and containment throughout the step. This does not require a Picard contraction or identify an Euler chord with a continuous trajectory. Inflation constructs a candidate tube; only the strict displayed inclusion admits it.

Let c be the exact midpoint of Y. Generate normalized flow coefficients a_j recursively by j*a_j = [s^(j-1)]F(sum_l a_l*s^l). The center solution at the endpoint is enclosed by

\[ P_h(c)+h^{p+1}a_{p+1}(B),\qquad P_h(c)=\sum_{j=0}^{p}h^j a_j(c). \]

The last coefficient is evaluated over the entire tube, bounding the normalized derivative at every possible remainder point. Outward arithmetic also encloses the represented midpoint and all polynomial operations.

To avoid discarding the linear damping while propagating earlier uncertainty, form a Metzler comparison matrix from a first-order interval jet of the same vector field:

\[ M_{ii}\ge\sup_B\partial_iF_i,\qquad M_{ij}\ge\sup_B|\partial_jF_i|\quad(i\ne j). \]

The diagonal keeps its sign. Both the center solution and every solution starting in Y remain in the convex tube, so the upper-Dini comparison inequality gives componentwise separation at most exp(h*M)*rad(Y). The shared comparison kernel shifts the diagonal to a nonnegative matrix, uses a positive rational series with a norm-bounded tail, and encloses the compensating scalar exponential. Adding this propagated radius to the center Taylor enclosure gives Y_next. Its intersection with B remains valid because both sets already enclose the endpoint. No midpoint projection of the actual state is performed.

Joining transit to maintenance

The endpoint test applies to the whole box, not to a synthetic midpoint graph. By default it requires a>2*pi/3, a<pi, b>0, b<pi/2 and

\[ \mathcal E=\frac45(q+r/2)^2+r^2 +\beta[10-2\cos(2a)-4\cos(a-b)-4\cos b]<7\beta. \]

Exact initial symmetry and uniqueness preserve the invariant slice. These inequalities therefore join directly to the protected-capture theorem: the ideal continuation is regular for all future time and converges to the positive aligned twist modulo its constant common offsets. The proof of initial winding zero additionally requires exact symmetry and each initial raw cycle gap strictly inside (-pi,pi). At a positive-rectangle endpoint only the edge -2a gains 2*pi under principal wrapping; the ring period is one. No separate numerical estimate of the crossing instant is needed.

The current certificate also accepts requested_sector=0 or -1, and None classifies any admitted one of the three disjoint protected rectangles. The point and interval certificates evaluate one shared affine-margin ledger; their precision and input scopes remain distinct. The actual sector, selected rectangle and all candidate bounds are retained. positive_rectangle_margins remains a compatibility field, not the only gate. Consensus is sector zero, not an unavailable or falsey target. The original positive-only audit and its exact archived source remain unchanged.

The caller must declare the horizon, step and order. Unsupported analytic domains, unresolved tube inclusion or inconclusive endpoint margins return unavailable evidence, retaining the accepted prefix and first failed tube. They do not prove that the true trajectory fails. A proof audit of an already evaluated preparation is explicitly post-evaluation verification; neither a successful proof nor an improved enclosure rewrites the original frozen finite-executor prediction.

Retained original-IVP proof audit

The producer froze its inputs, numerical policy, runtime and all 604 source files before this post-evaluation proof computation. It reused the immutable represented upper-corner seed, e=w=1/2, beta=nu_i=1, fixed horizon 32, step 1/8, Taylor order 12, 128-bit outward intervals, at most 16 Picard inflations per step, and no retry.

All 256 whole-time steps pass. The complete endpoint box lies in R+, and its storage is enclosed in the conservative decimal interval [6.97054580874334, 6.97054580874345], strictly below 7. Every endpoint coordinate width is below 7e-15. Across all tubes, the port, interior and central relative-resultant real lower bounds exceed 0.98293, 0.84597 and 0.03114543, respectively; every strict Picard inclusion margin exceeds 1.4389e-7. Exact rational inequalities and every tube are in the report; the displayed decimals summarize them. No tube failed and no bound was repaired by changing the initial state, law, step, order or horizon.

Consequently this exact represented winding-zero initial state generates a winding-one pattern and converges to the maintained aligned twist under the declared continuous law. This joins formation to maintenance for the original IVP, rather than restarting the law at a numerical endpoint.

Separate immutable proof evidence	SHA-256
Frozen proof protocol	d8913d6af0d40fd1c903eaf24be29fb8ebeb6d3be28d6415d4e8dafb9974db99
Validated transit report	ac3104c42cf9054968eac770fee5c1b1e174547a408875edac0b635fe04ca781
Executed source archive	381212374968f32fe59751c752b4dc812844a3309e00f771704a72b31a154c93

An independent retained-record audit recomputes the strict self-inclusions, all resultant bounds, time chain, exact initial seed and whole-endpoint conditions without re-integrating the trajectory. Tests also bind every archived source byte, check independent analytic references for the interval kernels, and preserve the original finite-executor false verdict. This is a computer-assisted conditional mathematical result. The proof is not a formally verified implementation and does not validate the constitutive law physically. Initial support, capacity and geometric/storage preparation remain supplied; neither substrate creation nor generic pattern selection has been established.

Synergy: qualitative robustness beyond exact reflection

Whenever the validated transit above succeeds from a strictly winding-zero preparation, it also implies a qualitative open-set formation result for the full state. Exact reflection is needed by this proof's reduced enclosure; it is not thereby necessary for every nearby trajectory to form the pattern.

The reflected capture theorem gives convergence to the aligned positive twist. Its limiting storage is E_*=beta*[10-10*cos(2*pi/5)], strictly below the full-state acute-sector barrier B_*=beta*[10-5*cos(2*pi/5)-4*cos(3*pi/8)]. Positivity of their gap is exact: 5*cos(2*pi/5)>4*cos(3*pi/8) reduces, by squaring positive sides, to 176*sqrt(2)>239, whose squared comparison is 61952>57121. The limiting edge gaps are strictly acute. The reference solution therefore enters the open full-state protected sector at some finite time T_*. This argument does not locate T_*; an endpoint in the reflected basin need not already be in the smaller acute-energy basin.

The reference trajectory is compact and regular on [0,T_*]. Smooth dependence on the initial state pulls that open target basin back to an open neighborhood of the original preparation. All sufficiently nearby full states remain regular through T_* and then converge to the same positive-twist orbit. For the frozen upper-corner preparation, the smallest initial antipodal margin is pi-2*a_0>1/32, so a sufficiently small phase perturbation also preserves initial ring winding zero. Intersecting the two open neighborhoods proves winding-zero formation followed by maintenance without requiring exact copy or reflection of the perturbed state.

The same conclusion permits small unequal held positive capacities: augment the ODE by nu_dot=0, use smooth dependence on this parameter, and apply the full-state sector theorem, which already admits heterogeneous positive capacities. Keep the graph, unit conductances, positive e,w,beta and unforced constitutive law fixed. Common form/phase offsets remain neutral and their limiting values may differ between preparations.

This consequence supplies neither a quantified perturbation radius nor a uniform entry time. It does not cover arbitrary positive capacities, topology changes, capacity events, forcing or finite binary64 execution of neighboring states. It is a conditional dynamical robustness result, not an identification of the pattern with a physical constituent.

Formation robustness under small admitted changes of phase law

The retained transit also supplies a quantitative comparison with the two admitted passive completions. This uses the same exact represented initial state, supplied support, unit held capacities, e=w=1/2, beta=1, clock and horizon T=32. Only the phase law changes. No reference producer is rerun, and no changed-law trajectory is sampled.

The target used in this comparison is the reflected rectangle R+ with storage below 7. The retained endpoint storage is about 6.9705458087434, above the full-state acute-sector barrier of about 6.924181298665. It would therefore be incorrect to call that endpoint an admitted shared acute-sector basin. The following symmetry-preserving extension of the reflected theorem closes the actual endpoint obligation.

The reflected barrier also permits a strict-loss class

Keep the native form row and chosen storage, and require a smooth autonomous phase law preserving the exact ring-copy and simultaneous form/phase reflection symmetries. On an open regular neighborhood of the relevant reflected sublevel, suppose E_dot<=-c*L for a fixed c>0, and require rest at q=g=0. These are whole-law premises, not snapshot observations. The four boundary costs in the reflected theorem depend only on geometry and storage. They still prevent exit from R+ when E<7, and give a compact regular sublevel. Zero loss forces q=r=0; the unchanged two form rows require g_0=g_4=0 to remain there. Their unique solution in R+ is the aligned twist. LaSalle's argument therefore gives convergence for this reflected class too. For laws also satisfying the full-neighborhood premises of the law-class theorem, its local Hurwitz argument gives eventual exponential recovery at the acute target. No full-state convexity is being asserted throughout R+.

Both named completions meet these conditions. The rho*N*g correction is odd under phase reversal. In the eta correction, h(q/(d*sqrt(beta))) is odd under form reversal and g^2 is even under phase reversal. Both respect node relabeling, ring copying and common shifts, and preserve the simultaneous reflection. The central rows have q_3=g_3=0 and remain zero for both laws. Thus their exact full states retain the derived four-coordinate form, with the actual common form and phase offsets constant on this preparation. No mean equation is discarded or externally frozen. The rho law admits c=1; the eta law admits c=1-eta/2>0.

A corridor around the retained reference tubes

Let y=(q,r,a,b) and let y_0(t) be the exact reference solution enclosed by the original 256 whole-time tubes. Inflate each retained tube in every coordinate by

\[ \delta=\frac1{1024}. \]

Exact rational comparisons with the retained tube bounds give, throughout these enlarged boxes,

\[ |q|<\frac{101}{100},\qquad |r|<\frac14,\qquad C_0>\frac{97}{100},\qquad C_4>\frac{21}{25},\qquad C_3>\frac{29}{1000}. \]

Here C_i=Re(z_i) for the port, interior and central resultants. Their phase-coordinate Lipschitz constants are respectively 4,3,2. Consequently the retained lower bounds minus 4*delta, 3*delta and 2*delta establish the displayed strictly positive corridor margins. The enlarged boxes also satisfy 0<a<pi, |a/2-b|<pi/2, so the same branch has g_4=(a/2-b)/pi. Every complete nodal resultant is regular there, including the central row absent from the reduced equations.

All comparison norms below are the infinity norm of these four exact coordinates. Their scaling is fixed by the retained reference preparation; it is not an inferred physical unit convention. The inverse relations A=(2*q+r)/5, B=(q+3*r)/5, together with the retained common offsets, reconstruct every nodal form and phase. An error bound in these four coordinates therefore also bounds the full nodal infinity error; this comparison discards no changing mean or hidden state. The reference field f_0 has a Lipschitz constant

\[ L_0=\frac{25}{8} \]

on each convex enlarged tube. To obtain an independent bound, write c_0=97/100, c_4=21/25, Q=101/100 and R=1/4. Then

\[ \begin{aligned} \|\nabla g_0\|_1&\le\frac4{\pi c_0},& \|\nabla g_4\|_1&=\frac3{2\pi},\\ H_0^{-1}&\le\frac1{\pi c_0},& \|\nabla H_0^{-1}\|_1&\le\frac4{\pi c_0^2},\\ H_4^{-1}&\le\frac1{\pi c_4},& \|\nabla H_4^{-1}\|_1&\le\frac3{\pi c_4^2}. \end{aligned} \]

For completeness, when z=C+iS and C>0, the metric has the zero-safe representation

\[ H^{-1}=\frac1\pi\int_0^1\frac{C}{C^2+t^2S^2}\,dt. \]

The integrand's Euclidean gradient in (C,S) has norm at most 1/(C^2+t^2*S^2)<=1/C^2. The summed phase-direction norms of z_0 and z_4 are at most 4 and 3, proving the inverse-metric derivative bounds without a division by a possibly zero imaginary part. Substitution into the four unchanged reference rows, using pi>3, bounds their absolute Jacobian row sums by

\[ \frac{297}{97},\qquad \frac{1079}{582},\qquad \frac{8350}{9409},\qquad \frac{1325}{3528}, \]

respectively. Each is strictly below 25/8. This controls the derivative on the whole enlarged tubes, not merely the saved centers or endpoints.

Bounded discrepancy and a first-exit argument

The named phase corrections alter only the reduced a_dot,b_dot rows:

\[ \begin{aligned} f_\rho-f_0&=(0,0,\rho g_0,\rho g_4),\\ f_\eta-f_0&=\left(0,0, \frac{\eta e}{\pi}h(q/3)g_0^2, \frac{\eta e}{\pi}h(r/2)g_4^2\right),\qquad h(s)=\frac{s^3}{1+s^2}. \end{aligned} \]

These are their full reduced differences at the same state. The subsequent form evolution changes through the native phase-to-form feedback. On the corridor, positive real resultants give |g_i|<1/2, and |h(s)|<=|s|. For nonnegative parameters this yields

\[ \|f_\rho-f_0\|_\infty\le\frac\rho2,\qquad \|f_\eta-f_0\|_\infty\le\frac\eta{64}. \]

For example, the eta port bound per unit eta is at most 101/7200<1/64; the interior bound is at most 1/192<1/64.

As long as the changed solution remains within delta of y_0(t), both points and their connecting segment belong to the relevant enlarged tube. If b_lambda bounds the displayed whole-field discrepancy, the upper-Dini comparison with the same exact initial state gives

\[ \|y_\lambda(t)-y_0(t)\|_\infty \le b_\lambda\frac{e^{L_0t}-1}{L_0} \le b_\lambda t e^{L_0t}. \]

The estimate carries continuously across the retained tube boundaries; it does not restart at a numerical center. Since L_0*T=100 and exp(100)<2^150, the following separate sufficient parameter ranges keep the error strictly below delta through the entire horizon:

\[ \boxed{\quad 0\le\rho\le2^{-164},\qquad 0\le\eta\le2^{-159}\quad} \]

for the rho and eta laws respectively. These are independent analytic fallback bounds, before the sharper static comparison below. The exponential inequality follows, for instance, from log(2)>2/3: the integral of 1/t on [1,2] strictly exceeds its midpoint bound. The worst bounds are rho*16*exp(100)<2^-10 and eta*exp(100)/2<2^-10. At a putative first exit they contradict equality with delta, so the assumed corridor containment proves itself. Smoothness and the strict regular margin give existence and continuation to T=32. This is a new comparison enclosure around the exact reference solution, not a claim that the changed solution lies inside the old, uninflated tubes.

A static logarithmic-norm refinement

The existing interval Jacobian owner also retains the negative diagonal derivatives rather than replacing them by absolute values. Applying it to the same enlarged boxes, without integrating either law, gives a Metzler matrix for each retained time interval:

\[ M^{(k)}_{ii}\ge\sup\partial_i(f_0)_i,\qquad M^{(k)}_{ij}\ge\sup|\partial_j(f_0)_i|\quad(i\ne j). \]

Define its nonnegative logarithmic-norm bound

\[ \mu_k=\max\left(0,\max_i\left[M^{(k)}_{ii} +\sum_{j\ne i}M^{(k)}_{ij}\right]\right). \]

The first-order interval jets evaluate derivatives on every point of each inflated tube. Exact rational evaluation over all 256 boxes verifies

\[ \sum_k h_k\mu_k<9. \]

The detached audit_relational_formation_robustness reader binds the three original files to their declared hashes, verifies the archived source membership and checks the reference formulas before performing this new static calculation. It retains every comparison matrix, its logarithmic-norm bound, corridor and endpoint margins, and the separate law bounds. The focused controls check the exact comparison arithmetic, endpoint distinction, export and unavailable/inconsistent evidence paths while forbidding trajectory execution.

The mean-value Jacobian along the segment between the two states obeys the same bound. At a coordinate attaining the infinity norm of their difference, the signed diagonal term can be retained in the upper-Dini inequality; thus D+error<=mu_k*error+b_lambda. Nonnegative mu_k makes the cumulative bound apply also at every earlier time. The same exact initial state and first-exit construction therefore give

\[ \|y_\lambda(t)-y_0(t)\|_\infty \le b_\lambda t\exp\left(\int_0^t\mu(s)\,ds\right) \le32b_\lambda\exp(9),\qquad 0\le t\le32. \]

Using exp(9)<(11/4)^9<2^14 gives the larger, still conservative, certified sufficient ranges

\[ \boxed{\quad 0\le\rho\le2^{-29},\qquad 0\le\eta\le2^{-24}\quad} \]

for the separate rho and eta comparisons. Each has uniform error less than 1/2048, half the admitted corridor inflation. The elementary estimate exp(1)<11/4 follows from its series: the terms through order 3 sum to 8/3, and the tail from order 4 is at most (1/24)/(1-1/5)=5/96. The inequality (11/4)^9<2^14 is an integer comparison. No optimized logarithmic norm, fitted response or parameter search is needed. These stronger radii depend on the separately checked static Jacobian sum; the preceding smaller bounds remain an analytic fallback if that check is unavailable.

Entry and the resulting formation conclusion

Every retained endpoint rectangle margin exceeds 17/100, and its exact upper storage bound is below 6971/1000. The first-exit error is less than 1/1024, so the changed endpoint remains in R+. On the connecting endpoint segment, the reduced storage

\[ \mathcal E=\frac45(q+r/2)^2+r^2 +10-2\cos(2a)-4\cos(a-b)-4\cos b \]

has gradient one-norm at most

\[ \frac{12}{5}Q+\frac{16}{5}R+16 =\frac{2403}{125}<20. \]

Consequently its changed value is strictly below 6971/1000+20/1024<7. The reflected strict-loss theorem just proved now applies. Each named law in its displayed parameter range takes the unchanged winding-zero preparation to winding one and then converges to the same aligned twist. Once close to that acute target, the full-state local stability argument also applies. Formation followed by maintenance is thus not isolated to the exact reference phase row within these two admitted families.

The radii are deliberately small, conservative sufficient bounds from a uniform Gronwall estimate. They are neither optimal thresholds nor evidence of robustness over a practically useful parameter range, physical scales or selection of either law by observation. The theorem evaluates no new trajectory and changes no frozen source, response, endpoint or verdict. The exact preparation, coefficients and clock matter; no conclusion for arbitrary preparations, nonunit capacities, new support, combined corrections or the production finite-step executor is being supplied here.

For a broader family of full-state phase laws, the existing reference convergence still gives qualitative openness: at some finite T_* it enters the strict full-state acute basin, and sufficiently small uniform field discrepancies on a compact regular corridor preserve that entry by continuous dependence. A changed law must independently meet the shared capture premises after entry. This requires no reflection preservation, but the present retained horizon-32 endpoint does not locate that acute entry or quantify its corridor. The explicit radii above therefore belong only to the two symmetry-preserving comparisons just proved.

Zero initial form contrast: a prospective phase-to-form discriminator

The prospectively frozen zero-form control changes only the initial form of the successful reference: x_i=0 for every node. Its represented phases, supplied two-ring support, unit capacities, e=w=1/2 and beta=1 remain identical. The common producer relational_transit_proof.py --zero-form freezes a distinct protocol and source archive. No subsequent response is used to choose another preparation, horizon or outcome gate.

What is predicted before evolving the control

For arbitrary uniform initial form x(0)=m*1 under this unit-capacity law,

\[ \dot x(0)=w g(\theta_0),\qquad \dot\theta(0)=0,\qquad \ddot\theta(0)=\frac{w^2}{\beta}H(\theta_0)^{-1}B g(\theta_0). \]

The acceleration is a derivative of the existing two coupled first-order rows, not an added inertial equation. Therefore zero form contrast need not be an equilibrium: nonuniform phase pressure can create form contrast and then change phase through the same feedback. In the selected preparation, rigorous initial derivative intervals give

\[ \begin{aligned} 0.10885&<\dot q(0)<0.10886,& -0.24255&<\dot r(0)<-0.24254,\\ \dot a(0)&=\dot b(0)=0,& 0.01730&<\ddot a(0)<0.01732,\\ &&-0.03003&<\ddot b(0)<-0.03001. \end{aligned} \]

Although q initially grows, the original form coordinates have A_dot=w*g0<0 and B_dot=w*g4<0: this dynamically generated direction is different from the supplied positive reference seed. Initial local signs alone cannot select a terminal basin.

Form Dirichlet storage grows at quadratic order, E_D''(0)=w^2*g^T*B*g>0, while phase storage loses the same leading amount. For the joint storage,

\[ \dot{\mathcal E}(0)=\ddot{\mathcal E}(0)=0,\qquad \mathcal E^{(3)}(0)=-2e\left[\frac43\dot q(0)^2+2\dot r(0)^2\right]<0. \]

Its initial value is about 7.93652606007>7; dissipation starts at cubic order. These are static mathematical consequences evaluated before any control trajectory. They do not imply capture from the initial energy test.

The protocol fixes horizon 32, step 1/8, order 12 and 128-bit outward arithmetic, with no retry. Terminal positive twist, consensus, negative twist and unresolved admission are predeclared alternatives. An admitted basin settles a conditional asymptotic limit; merely failing to enter one by the fixed horizon does not. A numerical proof-domain failure remains distinct from a zero resultant of the actual law. Prepared phase geometry supplies storage and pressure: none of these claims describes creation from nothing or emergence of the initial graph.

Retained result: transient winding followed by proved consensus

The first frozen control completes all 256 validated steps to time 32, with no unresolved tube or retry. Its full endpoint box lies in the consensus rectangle S, with joint storage conservatively enclosed in [2.11675502876186, 2.11675502876328], strictly below 7. Maximum endpoint coordinate width is below 1.5e-13. Every tube retains positive port, interior and central resultants; their respective global lower bounds exceed 0.96412, 1.01502 and 0.03098482. Thus this original continuous initial state converges to consensus under the same conditional law.

Preparation, with identical phase/support/capacity/law	Proved limiting basin	Whole endpoint storage upper bound at time 32
Retained supplied form contrast	Positive aligned twist	6.97054580874345 < 7
Zero initial form contrast	Consensus	2.11675502876328 < 7

The source manifest and every saved Picard inclusion, resultant bound, initial input and terminal inequality were independently checked without re-integrating either trajectory. The three protected rectangles share one definition in the point and interval owners, so consensus is explicitly target_sector=0 with terminal_basin_admitted=true. The positive-pattern flag remains false for this control; it is not an unavailable outcome.

Immutable control evidence	SHA-256
Frozen prospective protocol	920413db8d67dc46c13914bdd550bb097b416ade8dadd0417883b5b5eeca49d1
Validated response	bb30b7b2ac8812871b7f7667c4a898d79293a4e25e7b96bff13ec29fdf5996c4
604 fingerprinted source files	dcd7b49d4f596c9703ea191b74fadb0d28c6952d529363ce559b245984dbe8ee

A post-evaluation inspection of retained endpoint enclosures also proves both ring windings zero through time 15/8, one at every saved endpoint from 2 through 23/4, and zero from 47/8 through 32. Raw cycle gaps telescope; only -2a acquires an additional 2*pi wrap in the middle group. Consequently continuity gives at least one crossing in each of (15/8,2) and (23/4,47/8). These are retrospective crossing brackets, not a frozen crossing-time prediction, exact event times or proof of an uninterrupted winding-one lifetime between every sampled point.

This control therefore generates form contrast and a transient winding-one configuration, but does not maintain the reference pattern. The selected initial form preparation changes the asymptotic basin. That does not establish a universal need for initial form contrast: other phase preparations remain unclassified. Nor does it prove that scalar storage alone chooses a basin; removing form changes its signed direction as well as its magnitude and energy. The equal-storage control below separates those two effects.

General mechanism and its exact stationary boundary

The initial exchange does not depend on the special two-ring graph. On any connected fixed unit support admitted by the relational law, let held N=diag(nu_i)>0, regular H>0, w,beta>0 and x_0=m*1. Then

\[ \dot x_0=wNg,\qquad \dot\theta_0=0,\qquad \ddot\theta_0=\frac{w^2}{\beta}H^{-1}NBNg. \]

Reciprocity gives 1^T H g=0 by cancellation of edge sine contributions. If Ng=c*1, this identity gives c*sum_i(H_i/nu_i)=0, hence g=0. For any nonzero phase source, Ng is therefore nonconstant, BNg is nonzero, and form contrast and relative phase acceleration arise. An acceleration proportional to 1 would similarly imply BNg=c*H*N^-1*1; summing its entries forces c=0, which excludes nonzero acceleration without relative phase change. Furthermore,

\[ \ddot E_D(0)=w^2g^TNBNg>0,\qquad \beta\ddot V(0)=-\ddot E_D(0), \]

and, for e>0,

\[ \mathcal E^{(3)}(0) =-2ew^2(BNg)^TND^{-1}(BNg)<0. \]

Conversely, uniform form with g=0 is exactly stationary under the held unforced law. The model does not depart spontaneously from a completely balanced equilibrium. This theorem identifies a local conversion of supplied nonequilibrium phase structure into form, not the eventual identity or lifetime of the configuration it generates. The maintained acute geometries already have their circulation classification; that result should be reused, not counted as a new discovery from this control.

Common ideal shifts x -> x+m*1 and theta -> theta+c*1 leave both rows, contrast storage and winding invariant (B*1=L*1=0). The zero-form control therefore represents the entire common-uniform-form family and all common phase rotations. Its zero origin is not a privileged physical value. Arbitrary rounded additions to stored binary64 phases need fresh admission because their exact represented differences may no longer coincide.

Equal-storage form reversal: separating energy from direction

The prospectively frozen control exactly negates the successful reference's stored initial EPI values and changes no phase, support, capacity or coefficient. In the reflected coordinates, (q,r) -> (-q,-r) with (a,b) unchanged. The represented q_0=1+2^(-54) becomes its exact negative; signed zero in the raw JSON is retained and represents the same mathematical zero. This is neither simultaneous form/phase reflection nor time reversal.

Exact initial match and local discriminator

On the admitted fixed graph with held positive capacity, write y=Bx and K=N*D^-1. A common-offset reflection x^- = 2m*1-x^+ has y^-=-y^+. Identical phase geometry keeps g,H,V fixed. Consequently,

\[ E_D^-=E_D^+,\quad V^-=V^+,\quad \mathcal E^- =\mathcal E^+,\quad \dot{\mathcal E}^- =\dot{\mathcal E}^+=-e\,y^TKy, \qquad \dot\theta^-=-\dot\theta^+. \]

The signed exchange term

\[ J=w\,y^TNg,\qquad \dot E_D=-e\,y^TKy+J,\qquad \beta\dot V=-J \]

reverses sign. It is a read-out of work already present in the joint law, not a new pressure channel, state primitive or control policy. Equal total loss therefore allows opposite initial transfer between phase and form. For the actual reference, certified intervals give J_ref about -0.0198749634744: form initially supplies phase storage. The reversed preparation has opposite transfer while losing total storage at exactly the same instantaneous rate.

Since y_dot=-e*B*K*y+w*B*N*g,

\[ \ddot{\mathcal E} =2e^2y^TKBKy-2ew\,y^TKBNg,\qquad \ddot{\mathcal E}^- -\ddot{\mathcal E}^+ =4ew\,y^TKBNg. \]

For the selected two-ring state r_0=0, put h=w*(3*g0-g4)>0. This reduces to

\[ \ddot{\mathcal E}(-q_0)-\ddot{\mathcal E}(q_0) =\frac{16}{3}e q_0 h>0. \]

Static exact-input calculations, performed before evolution, retain rigorous intervals around E_0=8.736526060070739, equal ideal E_dot(0)=-(2/3)*(1+2^(-54))^2, and an acceleration difference between 0.29026 and 0.29028. Initial a_dot changes from about +0.159026 to -0.159026; b_dot=0 in both. Native engine fields independently check represented form-storage equality, loss equality and phase-rate negation; their floating work residual is not substituted for the ideal identities.

Thus (E,E_dot) already fails as an autonomous state description: two identical summary states have different derivatives of E_dot. Before evolution this did not decide the limiting basin. The frozen criterion was that different admitted terminal sectors would also exclude a basin selector based only on those initial summaries, even with identical phase geometry and model parameters. The same limit would not establish storage sufficiency; numerical unavailability would remain inconclusive.

The shared producer's --reverse-form mode freezes one control with horizon 32, step 1/8, order 12 and 128-bit outward arithmetic. Its initial-match gates must pass before a protocol can be prepared. The whole endpoint may admit any existing protected basin; no preferred sector, new phase law, retry or changed horizon is introduced. The reference and zero-form records remain immutable and are not re-executed by this control.

Retained result: equal initial energy, different limiting identity

The first evaluation validates all 256 whole-time steps to time 32, without retry or unresolved interval. Its entire endpoint box lies in the consensus rectangle S, with joint storage conservatively enclosed in [1.37994484500693, 1.37994484500711] < 7. The smallest rectangle margin is greater than 1.05417; the maximum endpoint coordinate width is below 1.889e-14. All resultants remain positive, with a whole-run lower bound greater than 0.03080034. The smallest strict Picard inclusion margin exceeds 3.0648e-5. Conditional continuation therefore converges to consensus.

Initial form, with identical phase/support/capacity/law	Initial joint storage	Initial loss	Proved limiting basin
Reference contrast	About 8.73652606007	(2/3)*(1+2^(-54))^2	Positive aligned twist
Zero contrast	About 7.93652606007	0	Consensus
Negated reference contrast	Exactly equal to reference	Exactly equal to reference	Consensus

The reference/reversal pair now disproves a deterministic limiting-basin selector using only initial (E,E_dot), even with the phase geometry and other model inputs supplied. It does not disprove every possible scalar encoding, a classifier using additional state, or a model using the subsequent energy history. The initial acceleration of energy already differs, and no claim of identical losses throughout the trajectory was made.

The retained report has terminal_basin_admitted=true, target_sector=0 and initial_storage_selector_refuted=true. Its historical positive-pattern flag is false because consensus is a different resolved basin. The earlier finite reference verdict also remains false; the later continuous proof and these controls do not rewrite that experiment's acceptance criterion.

Immutable control evidence	SHA-256
Frozen prospective protocol	050e265497f9bd7c483c20ab77bbf58baf763bb13e42575d8227bca9801bd9a8
Validated response	f3de2b74e9f0ab9ba3b621dfc471266e22f23a391bb1f4f76153538d924e2963
604 fingerprinted source files	9816d178479941e1d9ac007452b06c41d75e2fe7eb5d20d83c81bb1c42494e20

An independent retained-record audit checks every source hash, the sign-only intervention, initial identities, all strict Picard inclusions, chained endpoints, resultant bounds and terminal inequalities, without rerunning the trajectory. A retrospective consequence of the same whole-time tubes is that every raw oriented ring gap (-2a,a-b,b,b,a-b) stays strictly inside (-pi,pi), with margin greater than 0.02616569. Their sum telescopes to zero, so both ring windings remain zero throughout [0,32]. This is stronger than endpoint sampling but was not a frozen winding-time prediction. The zero-form control's separately reported transient winding remains a different case.

What the three preparations establish together

The uniform-form result shows how nonbalanced phase geometry first produces form contrast. Form contrast in turn moves relative phase; reversing it changes both that direction and the signed exchange J while preserving initial scalar storage and loss. Dissipation and the existing circulation classification then allow distinct final phase geometries with uniform form. Thus transient form can affect the maintained full-state identity even when the final scalar EPI profile is uniform. The earlier regional interaction result supplies transmission through the same declared law and supplied connections. These mechanisms share one model; they do not require a new operator, telemetry-based selector or primitive state variable.

This closes the named energy-versus-direction discriminator. It does not close arbitrary graph formation, autonomous support/capacity evolution, physical identification or selection of this constitutive law by nature. The shared work integration now exposes signed exchange and regional boundary accounting without extending this preparation into another basin sweep. The composition owner retains the tangent reduction, nonlinear closure obstructions and conditional passive bridge-relocation result. The sole execution plan selects any further formation or preparation question; the controls here remain reusable evidence rather than a separate task queue.

Phase consensus with a bounded reflected form preparation cannot form the target

The preceding successful formation proof supplies both nonuniform phase and form. Removing the initial phase structure is a different initial-value question from either of its retained form controls. The following analytic result excludes an entire bounded class under the same native law; it does not depend on a sampled response or on the numerical transit solver's domain.

The exact preparation class and conclusion

Keep the same two unit C5 rings with the two corresponding adjacent bridges, unit held capacities, e=w=1/2, beta=1, structural clock, and no inputs, events or changing support. After removing a common phase origin, set every initial phase to zero. On each ring supply the copied reflected form

\[ x(0)=(A_0,-A_0,-B_0,0,B_0),\qquad A_0,B_0\in\mathbb R. \]

Common uniform form offsets are immaterial to this relative statement. Write the existing sufficient reflected state as p=(A,B)^T, z=(a,b)^T, and define

\[ K=\begin{pmatrix}3&-1\\-1&2\end{pmatrix},\qquad D=\operatorname{diag}(3,2),\qquad \binom q r=Kp. \]

Its initial joint storage is exactly its form storage,

\[ F_0=2p(0)^TKp(0)=6A_0^2-4A_0B_0+4B_0^2. \]

Conditional obstruction. If F_0<=9, the ideal full native solution exists through structural time one, remains in |a|,|b|<1/2 throughout that interval, and satisfies

\[ \boxed{E(1)\le\frac23F_0\le6<7.} \]

It then belongs to the already proved consensus capture basin, so its continuation is regular for all forward time and converges to consensus. In particular it cannot converge to either maintained aligned unit-winding target. The number 9 is an analytic preparation ceiling, not a selected amplitude, a changed laboratory unit or a candidate obtained by trial.

If F_0=0, positive definiteness forces A_0=B_0=0; the complete state is stationary. For 0<F_0<=7, the previous strict sublevel argument already settles the result: the exact loss is initially strictly positive, so an initial equality F_0=7 immediately enters E<7 while the phases remain inside the consensus rectangle. The proof below additionally covers the whole upper part 7<F_0<=9, rather than treating the necessary energy barrier as a sufficient formation condition.

A regular small-phase interval is guaranteed before any basin test

Reuse the exact four-coordinate rows from the reflection reduction:

\[ \dot p=-eD^{-1}Kp+w g(z),\qquad \dot z=w H(z)^{-1}Kp, \qquad H=\operatorname{diag}(H_0,H_4). \]

The phase metric is the actual native metric, not its consensus linearization. While |a|,|b|<=1/2, every incident fine-edge phase difference lies in [-1,1]: the nonzero ring gaps are (-2a,a-b,b,b,a-b) and the bridge gaps vanish by copying. Since cos(1)>1/2 and 1<pi/2, every relative resultant has strictly positive real part, and its argument alpha_i lies in (-pi/2,pi/2). The native identity

\[ H_i=\pi\operatorname{Re}(z_i) \frac{\tan\alpha_i}{\alpha_i}, \qquad \frac{\tan0}{0}:=1, \]

therefore gives H_i>pi*d_i/2, including the central rows of the full graph. In the two retained phase rows this implies H>(pi/2)D as positive diagonal matrices. No zero-resultant continuation or special treatment of a formally vanishing central numerator is used.

The existing work identity gives E(t)<=F_0 while the law is regular, and phase storage is nonnegative. The exact form quadratic in the gradient coordinates is

\[ F_D=\frac{4q^2+4qr+6r^2}{5} =\frac23q^2+\frac65(r+q/3)^2 =r^2+\frac45(q+r/2)^2. \]

Thus q^2<=3F_0/2 and r^2<=F_0, yielding the complete-row speed bounds

\[ |\dot a|\le\frac{\sqrt{3F_0/2}}{3\pi}<\frac12,\qquad |\dot b|\le\frac{\sqrt{F_0}}{2\pi}<\frac12 \qquad(F_0\le9). \]

Starting from a=b=0, neither coordinate can first reach the box boundary by time one: integration up to such a first exit would still give absolute coordinates strictly below 1/2. Form is bounded by the positive quadratic, and the entire closed phase box is a compact subset of the full regular domain. Smooth continuation therefore supplies the claimed whole interval. More precisely,

\[ |a(t)|\le\frac{t\sqrt{3F_0/2}}{3\pi},\qquad |b(t)|\le\frac{t\sqrt{F_0}}{2\pi} \qquad(0\le t\le1). \]

These are analytic bounds for the actual nonlinear trajectory. They neither hold form fixed nor identify a tangent or Euler extrapolation with the solution.

Form loses storage faster than this interval can build phase storage

Introduce only proof norms, not new dynamical coordinates,

\[ Y(t)^2=2p(t)^TKp(t)=F_D(t),\qquad Z(t)^2=2z(t)^TKz(t). \]

The symmetric matrix K^(1/2) D^(-1) K^(1/2) has eigenvalues 1-1/sqrt(6) and 1+1/sqrt(6). The latter is less than 3/2. The phase-metric bound and the induced K norm therefore give

\[ \sqrt{2\dot z^TK\dot z} \le w\lambda_{\max}(K^{1/2}H^{-1}K^{1/2})Y \le\frac{3}{2\pi}Y\le\frac12Y. \]

Since Y(t)<=sqrt(F_0) and z(0)=0, the integral triangle inequality implies Z(t)<=sqrt(F_0)*t/2. The global elementary bound 1-cos(s)<=s^2/2 applied to the actual reflected phase storage then gives

\[ \begin{aligned} V(a,b)&=2[1-\cos(2a)]+4[1-\cos(a-b)]+4[1-\cos b]\\ &\le6a^2-4ab+4b^2=Z^2 \le\frac{F_0t^2}{4}. \end{aligned} \]

Independently, the exact loss satisfies

\[ \mathcal L=4e\,(Kp)^TD^{-1}(Kp) \ge2e\left(1-\frac1{\sqrt6}\right)Y^2 \ge\frac12Y^2. \]

All quantities are evaluated along the same native solution. With beta=1, E=Y^2+V, so throughout the guaranteed interval

\[ \dot E=-\mathcal L \le-\frac12E+\frac{F_0t^2}{8}. \]

The explicit comparison function

\[ B(t)=F_0\left(1-\frac t2+\frac{t^2}{6}\right) \]

starts at B(0)=E(0)=F_0 and obeys

\[ B'(t)-\left[-\frac12B(t)+\frac{F_0t^2}{8}\right] =\frac{F_0t(2-t)}{24}\ge0\qquad(0\le t\le1). \]

Equivalently, the positive part of E-B cannot grow from zero under the displayed scalar differential inequality. Therefore E(t)<=B(t) on the entire interval, proving E(1)<=2F_0/3. This bound retains nonlinear phase feedback through its metric and potential bounds; it is not a calculation with that feedback removed.

At time one the phase box lies strictly inside (-2*pi/3,2*pi/3) x (-pi/2,pi/2), and joint storage is below the exact barrier 7. The existing consensus capture theorem now supplies global regular continuation and convergence to consensus. The proof did not construct a synthetic endpoint or restart a different law there: all inequalities concern the same continuous initial-value problem.

Application to the frozen successful reference's storage budget

The retained original protocol fixes the successful source's represented state and law independently of its response (SHA-256 d8913d6af0d40fd1c903eaf24be29fb8ebeb6d3be28d6415d4e8dafb9974db99). Reconstruct its graph, form, phases, capacities and model from that protocol, then use the shared native capture's rational storage_bounds, not the field's floating storage or a current producer's defaults. This encloses

\[ 8.7365260600707<E_{\rm ref}<8.7365260600729<9. \]

That total includes its initial phase storage as well as form storage; its former form amplitude is not substituted for the total budget. The decimal bounds above are outward summaries of the retained rational enclosure, not an exact decimal definition of equal energy.

Every phase-consensus preparation in the requested ellipse 6A_0^2-4A_0B_0+4B_0^2<=E_ref therefore belongs to the proved F_0<=9 class. Its storage by time one is at most 2E_ref/3, and it converges to consensus. The ceiling 9 is only a convenient rational outer bound for the analytic proof; the requested reference budget has not been raised, and no larger-amplitude response was selected or evaluated.

This is a class-level exclusion of maintained target formation for the supplied phase-consensus restriction. Nonuniform form can still generate phase motion: at the initial state, a_dot=w*q/(3*pi) and b_dot=w*r/(2*pi), so both vanish only when A_0=B_0=0. What fails is delivery of the maintained nonzero winding from this bounded initial class. The first-unit phase bounds keep its raw cycle gaps inside (-pi,pi), hence winding zero there. Convergence to consensus alone is not used to classify every possible intermediate winding after that interval.

Simultaneously negating (A,B,a,b) is a symmetry of the same law. At initial phase consensus this also represents form reversal, and both orientations have the same consensus conclusion. It is not dissipative time reversal. There is no contradiction with the earlier reference/reversal result, whose common initial phases were already nonuniform and were held fixed under that different intervention.

The result does not extend to arbitrary ten-node form states, broken reflection or ring-copy symmetry, larger budgets, changed capacities, other supports, or the sine and alternative-mobility laws. It does not establish that all formation requires an initial phase seed, or rule out other preparations under the native paradigm. It identifies a precise obstruction under fixed supplied structure and a declared budget, rather than a universal no-formation theorem.

Shared analytic admission and independent checks

certify_relational_consensus_capture and RelationalConsensusCaptureCertificate in the native capture owner reuse the existing support, copied-reflected state, coefficient and full-law admission. They check exact initial phase consensus and the exact form-storage ceiling before exposing the analytic time-one storage and phase bounds. An initial snapshot need not already pass the strict E<7 capture test for this new theorem to apply.

The certificate records a future bound, not an evaluated endpoint state. Refusal of one theorem hypothesis leaves this conclusion unavailable; it is not evidence of successful formation. The prior native finite-executor, continuous-transit and control verdicts remain unchanged. No producer was regenerated and no new trajectory or basin sweep was used to prove this exclusion.

The phase-consensus contract and formation-admission tests own the executable scope and independent controls. Static admission, the nonlinear class theorem and a selected numerical trajectory are different kinds of evidence. None identifies physical matter or selects the original microscopic law, initial form or support.

Full-form nonlinear phase-consensus obstruction

The copied-reflected restriction can be removed for target exclusion. Keep the same two C5 rings, the two bridges at matching positions zero and one, unit held capacities, beta=1, effective e=w=1/2, and the unforced native regular-domain law. Prepare all phases equal, but allow any real ten-node form modulo a common offset. Write L for the combinatorial Laplacian, D=diag(d_i), and

\[ F=\tfrac12x^TLx,\qquad P=V_\phi,\qquad E=F+P, \qquad \mathscr L=\tfrac12q^TD^{-1}q,\quad q=Lx. \]

Suppose F_0<=9. Then on every regular existence interval of the ideal nonlinear solution,

\[ \boxed{P(t)\le e^{-\pi/7}F_0\le\frac7{10}F_0\le\frac{63}{10}.} \]

The middle inequality is strict when F_0>0. The aligned unit-winding target in either orientation has phase storage greater than 55/8, so neither entry into its acute component nor convergence to that target is possible. This covers the reference budget F_0<=E_ref<9 without imposing copy or reflection. Unlike the preceding theorem, this result does not prove global regular continuation or convergence to consensus.

Argument pressure has a global storage bound on its regular domain

Let z_i=C_i+i*S_i, alpha_i=Arg(z_i) and g_i=alpha_i/pi. The incident phase cost is d_i-C_i. If C_i>=0, then |alpha_i|<=pi/2 and |z_i|<=d_i give

\[ d_i-C_i\ge d_i(1-\cos\alpha_i) \ge 2d_i\alpha_i^2/\pi^2\ge d_i g_i^2. \]

The trigonometric bound follows from concavity of sine on [0,pi/2]. If C_i<0, the cost exceeds d_i, whereas g_i^2<1 throughout the regular domain. Thus in either case

\[ d_i g_i^2\le\sum_{j\sim i}(1-\cos(\theta_j-\theta_i)), \qquad \sum_i d_i g_i^2\le2P. \]

This includes zero pressure and zero real resultant with nonzero imaginary part. It assigns no pressure at a zero resultant or on the excluded branch. The exact storage balance and weighted Cauchy--Schwarz now give

\[ \dot E=-\mathscr L,\qquad \dot P=-\tfrac12q^Tg,\qquad |\dot P|\le\tfrac12\sqrt{q^TD^{-1}q}\sqrt{g^TDg} \le\sqrt{\mathscr L P}. \tag{A} \]

This controls the actual nonlinear exchange, not a frozen tangent or a phase-metric approximation. The potentially large native phase velocity near a regular-domain boundary does not invalidate the work identity.

The actual full support supplies a uniform loss bound

The graph contains the spanning cycle (0,4,3,2,1,6,7,8,9,5,0) in the supplied ring labels, and its maximum degree is three. Let lambda_* be the nonzero spectral gap of D^(-1/2)L D^(-1/2). For any nonconstant form, its weighted Rayleigh denominator is min_c sum_i d_i(x_i-c)^2, at most three times its ordinary centered squared norm. The spanning C10 supplies a lower bound on the numerator. Consequently

\[ \lambda_*\ge\frac{\lambda_2(L_{C_{10}})}3 =\frac{3-\sqrt5}{6}>\frac19. \]

The last inequality follows from sqrt(5)<7/3. The same symmetric normalized Laplacian gives q^TD^-1q>=lambda_* x^TLx, hence

\[ \boxed{\mathscr L\ge\frac19 F.} \tag{B} \]

All bridge degrees and costs are retained. No isolated-ring rate, sampled eigenvalue or restriction to a few preparation modes enters this bound.

A nonlinear comparison includes all signs of the exchange

On the positive stratum F,P>0, put

\[ f=\sqrt F,\quad y=\sqrt P,\quad k=\mathscr L/F\ge1/9,\quad h=\dot P/(2fy). \]

Equations (A)--(B) imply

\[ \dot f=-\tfrac k2f-hy,\qquad \dot y=hf, \qquad 2h\le\sqrt k\le3k. \]

The sign of h is unrestricted: exchange may return storage to form. Let varphi=atan2(y,f) in [0,pi/2] and define the auxiliary comparison quantity

\[ \Psi(\varphi)=\frac27(\varphi+\sin\varphi\cos\varphi),\qquad W=(F+P)e^{\Psi(\varphi)}. \]

This is a derived bound, not a replacement storage or a new evolution law. Since Psi'=4*cos(varphi)^2/7 and varphi_dot=h+(k/2)sin(varphi)cos(varphi),

\[ \frac{\dot W}{W} =-k\cos^2\varphi+ \frac47\cos^2\varphi\left(h+\frac k2\sin\varphi\cos\varphi\right) \le k\cos^2\varphi \left[-1+\frac27\left(3+\sin\varphi\cos\varphi\right)\right] \le0. \]

The zero strata are included. On any compact time interval within the regular existence interval, x,theta are continuously differentiable. The quantities f and y are norms of, respectively, real edge-form differences and complex phase-chord differences, so they are locally Lipschitz in time. The displayed W(f,y) extends locally Lipschitz to the closed quadrant, with value zero at the origin. It is therefore absolutely continuous along the solution. A nonnegative Lipschitz function has derivative zero almost everywhere on its zero set. At f=0, q=0 makes both loss and phase work zero; at y=0, g=0 makes phase work zero and W_dot=-mathscr L almost everywhere on that stratum. Both zero means the stationary uniform state. Thus W_dot<=0 almost everywhere, including boundary visits, and absolute continuity gives W(t)<=W(0)=F_0.

Finally, P/W=sin(varphi)^2 exp[-Psi(varphi)] increases on [0,pi/2]: its logarithmic derivative in the interior is 2*cot(varphi)-4*cos(varphi)^2/7>0. Its largest value is exp(-pi/7). This proves the boxed bound. The rational relaxation follows from exp(pi/7)>1+pi/7>10/7, using pi>3.

The same comparison also gives a necessary preparation condition without assuming initial phase consensus. For any initial regular state of this same complete law, put varphi_0=atan2(sqrt(P_0),sqrt(F_0)). Then

\[ P(t)\le E_0\exp[\Psi(\varphi_0)-\pi/7]. \]

In particular, approaching the aligned target requires E_0*exp(Psi(varphi_0))>=P_*exp(pi/7). This is a necessary condition on the initial form/phase storage split, not a sufficient formation criterion or a selection of that preparation. It introduces no phase-threshold fit or additional response campaign.

What is excluded and what remains unresolved

The existing acute-sector Jensen bound gives phase storage at least

\[ P_*=10(1-\cos(2\pi/5))=\frac{25-5\sqrt5}{2} >\frac{55}{8}>\frac{63}{10} \]

whenever both supplied rings have the same unit winding and all edges are acute. Here sqrt(5)<9/4 proves the first rational inequality. The target itself has exactly P_*. The all-time strict separation therefore excludes entry into that acute target component and approach to the aligned target, even asymptotically. It is not merely a failed sufficient capture test or absence of formation in sampled trajectories.

The native maximal solution might instead approach another admissible state or reach a singular regular-domain boundary. The phase ceiling alone does not decide between those cases, exclude all transient winding, or define continuation through a missing native field. No general consensus claim follows. The earlier copied-reflected theorem retains its stronger continuation and limiting-consensus conclusions on its subset. The prepared positive formation, zero-form and reversal controls retain their nonuniform initial phases and are not contradicted.

The shared certify_relational_consensus_formation_obstruction and RelationalConsensusFormationObstruction in the capture owner admit one fresh full native source and the exact support. Their prospective bound is 7*F_0/10; the target cost reuses the existing mathematical-cosine owner. Failed phase, capacity, coefficient or budget premises withhold that prospective conclusion. The report explicitly leaves continuation uncertified. No reflection projection, solver, frozen producer or changed constitutive law is used. Independent algebra and actual-field admission controls belong to the existing formation and capture tests.

Supplied events have a separate comparison obligation. For a piecewise native path on this same support, capacity, storage scale and coefficient law, with regular reset endpoints and finitely many resets on the interval, W is nonincreasing on every regular continuous segment. A reset may change it. If Delta W_j=W_j^+-W_j^- denotes its actual signed jumps, entry into the acute target requires

\[ \sum_j\Delta W_j\ge P_*e^{\pi/7}-F_0, \qquad \sum_j(\Delta W_j)_+\ge P_*e^{\pi/7}-F_0. \]

This follows by telescoping the continuous decreases and jumps and using W>=P*exp(pi/7) at the endpoint. It is an accounting bound for this derived comparison quantity, not an energetic reservoir or an event-occurrence law. Structural-storage passivity alone does not enforce it: from the source x_2=3, all other forms zero and all phases zero, F_0=9. A supplied reset to uniform form and the aligned two-ring twist has E^+=P_*<9 but W^+=P_*exp(pi/7)>275/28>9=W^-. Both endpoints are native regular states. Thus an explicitly supplied passive reset can escape the continuous preparation obstruction. This comparison does not establish that any named operator executes that reset or explain when a node selects it. A support, capacity or coefficient change requires new constants and admission; it cannot silently reuse this same-model jump inequality.