Sine pattern geometry and dissipative capture

These results use the complete smooth reciprocal normalized-sine law. Supplied support, coefficients, held capacities, initial state, clock and phase-domain assumptions remain part of each theorem. The native neighbor-argument law and its event analysis have a separate composition owner. Shared variables and methods do not transfer conclusions between those complete laws.

Chapters and scope

Chapter	Responsibility
Exact and controlled form-phase reductions	Equivalent state/memory representations, slow-time comparison and full-state capture handoff.
Conservative regional formation and its barriers	Finite winding entry, channel balance, source geometry, barriers and robust passage.
Collective phase families and live contact feedback	Exact partitions, moving windows, contact/storage exchange and receiver rigidity.
Conservative source preparation and finite retention	Retention bounds, saddle/corridor construction and its operational and constitutive limitations.

A formation-to-retention claim must connect an identity-absent preparation to the full admitted retention set under the same law, including its live environment, conserved storage, clock and phase lifts. Local growth, available storage or a compatible invariant family alone does not supply that connection. Positive-loss capture and conservative finite retention keep distinct premises; exact hidden-state representations retain their independent initialization. None of these results derives support birth or physical identification.

Section numbers remain stable. The execution plan alone assigns research work.

24. Equilibrium geometry of components joined by bridges

The nine attracting geometries of the eleven-node sine model are an instance of a more general composition rule. This section concerns the smooth reciprocal sine law, not the native Arg law in the reflected-domain analyses. It combines component geometry under the existing complete dynamics without assuming independent component motion or installing a support law.

Complete law and decomposition premises

Let G be a finite connected simple undirected unit graph with n>=2 nodes. Partition its nodes into m>=1 nonempty sets G_a, each inducing a connected graph; a singleton is allowed. Require exactly m-1 intercomponent edges, with the quotient graph a tree. These edges are then bridges of the full graph. In particular, a second connection between the same components or a cycle in the quotient is outside this decomposition. Internal component edges need not themselves be bridges.

Retain strictly positive held capacities, fixed support, no forcing or events, and e,w,beta>0. Define full-graph degrees d_i, K=diag(nu_i/d_i), the unit Laplacian L, S_i(theta)=sum_{j~i} sin(theta_j-theta_i), a=w/pi and b=w/(beta*pi). The consumed rows are

\[ \dot x=-eKLx+aKS(\theta),\qquad \dot\theta=bKLx. \]

Bridge degrees and all nodes remain in these rows. No isolated-component degree normalization replaces K; a singleton's full degree is positive because the full graph is connected. The total one-node graph is excluded from this row-normalized statement.

At equilibrium, positivity of K gives Lx=0, hence uniform form, and then S(theta)=0. Choose an exact critical phase geometry, not a state declared critical merely because its residual is small. Let its phase Hessian be

\[ v^\mathsf THv= \sum_{\{i,j\}\in E(G)} \cos(\theta_j^*-\theta_i^*)(v_j-v_i)^2. \]

A bridge cannot carry a stationary sine current

Delete a bridge {u,v} and let A be the resulting component containing u. Summing S_i(theta*)=0 over i in A cancels every internal sine current against its reversed edge. The single remaining term is sin(theta_v*-theta_u*). Consequently every bridge has gap zero or pi modulo 2*pi, and its cosine is respectively +1 or -1.

Removing these zero bridge currents from a node's balance leaves exactly the internal component balance. The restricted phase geometry is therefore critical on every G_a. Conversely, choose a critical phase geometry on each component and independently prescribe zero or pi on each bridge. Starting at one root component, shift each child's common phase to satisfy its joining bridge. The quotient tree makes this consistent and unique modulo the common phase of the whole graph. Thus these choices produce all full critical geometries on this decomposition.

This argument uses the conservative pairwise sine current S. The native neighbor-resultant direction is not generally such a current, so the same cut sum cannot silently be applied to it.

Relative Hessian inertia adds across the bridge tree

For each component let (p_a,q_a,z_a) be the positive, negative and zero inertia of its internal Hessian after removing that component's common phase direction. Their sum is |G_a|-1; a singleton contributes (0,0,0). Let b_+ and b_- count zero-gap and pi-gap bridges, respectively. Then the full relative inertia is

\[ \boxed{\quad \operatorname{inertia}(H_{\rm relative})= \left(\sum_a p_a+b_+,\ \sum_a q_a+b_-,\ \sum_a z_a\right). \quad} \]

To prove the direct sum, coordinatize a nodal perturbation modulo its one common origin by every component's internal relative perturbation and the m-1 actual bridge endpoint differences. Given these coordinates, solve the component common offsets successively along the quotient tree. This gives a linear isomorphism: the dimensions are sum_a(|G_a|-1)+(m-1)=n-1, and only the global common offset is discarded. The Hessian quadratic form in these coordinates is the sum of the internal component forms and the independent scalar bridge forms +v_e^2 or -v_e^2. Sylvester's law of inertia proves the formula. The unrestricted Hessian adds one common-phase zero direction.

The coordinate transformation is a congruence, not a dynamical decoupling. It does not diagonalize the full generator, remove mediated interaction or identify component degrees with full degrees.

Full nodal stability follows under positive loss

Write r=n-1, j=sum_a q_a+b_- and z=sum_a z_a. Reuse the full reciprocal Jacobian argument, which depends on connected support and the Hessian inertia rather than the number eleven. On the conserved weighted-mean leaf, put B=K^(1/2)LK^(1/2)>0 and C=K^(1/2)HK^(1/2), restricted to (K^(-1/2)1)^perp. The latter has the relative inertia just derived. The exact two-row variation reduces to the quadratic pencil

\[ P(s)=s^2I+esB+abB^{1/2}CB^{1/2}. \]

For a nonreal characteristic root, its imaginary-part identity forces strictly negative real part. On s>=0, P'(s)=2sI+eB is positive definite. Each initially negative stiffness direction therefore crosses zero once at positive s, counted with multiplicity. There are exactly j positive real characteristic roots.

If z=0, no relative zero root remains. The full relative equilibrium is hyperbolic, with j unstable directions and 2r-j stable directions. In particular it is locally exponentially attracting precisely when every component has positive-definite relative Hessian and every bridge is aligned. A negative component direction or an antipodal bridge makes the full equilibrium nonlinearly unstable.

If z>0, a negative direction still proves instability, but the relative zero modes require their own nonlinear analysis. A positive-semidefinite Hessian with additional kernel is not certified as an attractor by this test. The common form and phase origins are separate symmetry freedoms; they are removed before counting these relative zero modes.

Strictly positive changes of the held capacities or coefficients preserve this classification while changing rates and potentially basins. The claim does not extend to zero capacity, zero loss, another phase law, time-dependent support or forcing.

Trees and assemblies of C5 components

For a tree, take every component to be one node. All critical phase gaps are zero or pi, and the negative index equals the number of antipodal edges. Hence consensus is the only locally attracting relative equilibrium; every other critical geometry is unstable. This says nothing about convergence of each preparation or finite-time transients.

For m disjoint unit C5 components, with any additional singleton intermediaries, suppose the joining edges form a tree on these components. The exact C5 classification supplies three positive-definite relative critical geometries on each cycle: consensus and the two uniform unit windings. Every other C5 critical geometry is nondegenerate and has a negative direction. All bridges must therefore align for attraction, and independent cycle choices yield exactly 3^m locally attracting relative geometries. The count keeps the supplied node labels and the two handedness choices; it removes one global phase origin, not graph automorphisms or reflection. The two-C5/intermediary graph gives nine. This is neither a bound on TNFR geometries nor a claim that all these states are accessible from a particular preparation.

Adding a connection that creates a cycle between components removes the independent-bridge premise. Nonzero stationary currents and coupled cycle constraints can then appear. The bridge formula is unavailable; the actual full Hessian or an appropriate cycle-space argument must be used instead of extrapolating 3^m.

Shared geometry reader and evidence boundary

compose_bridge_tree_hessian_inertia centralizes the direct-sum calculation, retaining the declared component dimensions, relative inertias and bridge signs. Its result is a conditional geometric certificate: component inertia inputs require independent derivation, and arithmetic composition alone does not verify a supplied phase state, select a complete law or execute its dynamics. The specialized C5 report can supply those component premises from its exact catalog. The independent bridge controls compare the shared calculation with full Hessians and invalid decomposition cases rather than treating a sum of supplied labels as a formation test.

These results explain how restoring geometry can survive composition while the full system remains coupled. They do not create the support, choose its partition or occurrence time, prove entry into a basin, generate a persistent pulse or identify a physical constituent. Formation, maintenance and autonomous hierarchical organization remain distinct questions under their stated complete laws.

25. Cycle periods constrain joint composition

Retain Section 24's finite connected simple unit support, positive held capacities, e,w,beta>0, complete two-row normalized-sine law and absence of forcing or events. Remove only the bridge-tree decomposition premise. All internal nodes, full degrees and edge currents remain in the model. The result concerns strictly acute phase gaps; nonacute equilibria and formation from a specified preparation require separate arguments.

Existing circulation criterion on the complete nodal law

Use D for the oriented incidence matrix and let C contain the integer fundamental-cycle columns supplied by PhaseCycleGeometry (the transpose of its cycle_rows). Write b_1=|E|-|V|+1 for the cycle rank. With delta_e=wrap(theta_head-theta_tail) and f_e=sin(delta_e), the nodal sine sum is S=-Df. At a full equilibrium, the phase row and positive K force Lx=0, hence uniform form; the form row then forces Df=0. Conversely these two conditions make both rows vanish.

The circulation and reconstruction theorem therefore supplies the exact necessary and sufficient equations

\[ f=Cz,\qquad |(Cz)_e|<1,\qquad C^{\mathsf T}\arcsin(Cz)=2\pi\mathbf{k},\qquad \mathbf{k}\in\mathbb Z^{b_1}. \]

Its integer-lattice qualification, principal acute arcsine and at-most-one geometry per sector all apply unchanged. This transfers its phase geometry, not its separately supplied common-capacity phase dynamics: here the full equilibrium is stationary and capacities may be unequal. Existence still requires a solution of these coupled equations. A zero mixed period need not give zero connecting circulation, as the return-path equilibrium already demonstrates. Component winding labels omit these currents and the internal phase deformation needed to satisfy the full nodal balances.

At any such equilibrium, every edge cosine is positive. Consequently H=D diag(cos(delta)) D^T is positive definite modulo the one common phase direction. Section 24's full-Jacobian quadratic-pencil argument then gives local exponential recovery on the conserved weighted-mean leaf, with 2*(|V|-1) stable relative directions. There are no additional relative zero modes in this strict acute domain. Uniform form and common phase remain the two independent origin freedoms. This conditional recovery conclusion does not establish existence, an acute invariant domain for arbitrary preparations or entry into the local basin.

A general exact obstruction from combined cycle periods

Set t=delta/(2*pi) in turns. Circular reconstruction requires C^T t=k, while strict acuity requires |t_e|<1/4. For any nonzero integer cycle-coordinate vector r, form the actual signed edge chain c=Cr. Cancellation of shared edges occurs before taking its norm. Every strictly acute realization must satisfy

\[ \boxed{\quad |r^{\mathsf T}\mathbf{k}| < \frac{\|Cr\|_1}{4}. \quad} \]

Indeed, r^T k=c^T t and |c^T t|<=sum_e |c_e||t_e|<||c||_1/4; full column rank makes c nonzero. Equality also excludes a strictly acute realization. The bound uses the combined chain, not the sum of separate cycle lengths. It therefore detects conflicts that each basis-cycle bound can miss. It is an obstruction to phase geometry itself, hence also to an acute equilibrium under the complete law, independently of capacities or positive coefficient values.

There is also an exact geometry-only existence characterization. For b_1>0, define the open zonotope

\[ \mathcal Z=C^{\mathsf T}(-1/4,1/4)^{|E|}. \]

An integer sector k has a strictly acute circular phase realization if and only if k belongs to Z. Equivalently, the boxed inequality holds for every nonzero real r. To see the converse, Z is the interior of the full-dimensional compact zonotope C^T[-1/4,1/4]^|E|; its support function in direction r is ||Cr||_1/4. Separation excludes every point outside that interior by one failed strict support inequality. Since this polytope is rational, its facets have rational normals, so testing all nonzero integer r is equivalent as well. The original integral-period reconstruction then supplies circular phases. For a tree the cycle-period vector is empty and this period condition is vacuous; its equilibrium condition still forces zero edge currents.

This equivalence supplies phase compatibility only. It does not impose D sin(2*pi*t)=0; even a phase realization is not a full equilibrium certificate. Without a completeness proof, a finite collection of successful inequalities does not certify membership in Z. One failed inequality suffices for exclusion.

The difference is substantive. Take three internally disjoint paths from A to B of lengths (4,4,1), all oriented from A to B. Prescribe period one on each long path followed by the reversed single edge. Giving every long-path edge turn 5/24 and the single edge turn -1/6 supplies a strictly acute circular realization of both periods. Nevertheless this sector has no acute sine equilibrium. At such an equilibrium, every internal degree-two node would force consecutive sines equal, hence consecutive acute gaps equal. If the single-edge gap is phi, each long-path gap must then be (2*pi+phi)/4. Acuity requires -pi/2<phi<0, so the outgoing sine sum at A would obey

\[ 2\sin((2\pi+\phi)/4)+\sin\phi >2\sin(3\pi/8)-1>0. \]

This contradicts nodal balance. Both five-edge cycles separately admit a stable unit twist, and the joint sector passes every geometric support inequality, but circulating-current compatibility still fails.

For a simple triangle or square, the integer period and strict length bound force its period to zero. This retains the earlier short-cycle consensus obstruction but also constrains individual sectors when those short cycles span only part of the cycle space. For example, let three internally disjoint paths between two nodes have lengths (3,2,2). The cycles consisting of the first path followed by either reversed short path each have length five. Proposed periods (1,-1) pass both individual bounds 1<5/4. Their difference is the four-edge cycle between the short paths, however, and has proposed period two, violating 2<4/4. Independent admissible cycle labels cannot be composed here.

Independent components: a compatible and an excluded composition

Take two disjoint oriented C5 components with nodes L_i,R_i, indices modulo five, and add all five matching edges L_i--R_i. Each supplied component admits either uniform unit winding and has positive relative phase Hessian. Any one matching edge can separately be aligned by a common rotation of one component. These facts do not decide whether all five connections can coexist in a strictly acute phase state.

Orient each square as Q_i=(L_i,L_(i+1),R_(i+1),R_i). Strict acuity gives every square period zero, while the exact integer-chain identity is

\[ \sum_{i=0}^{4}Q_i=c_L-c_R. \]

Thus any strictly acute state on the complete graph must have equal inherited ring periods k_L=k_R. Opposite unit windings cannot compose in this domain, even with arbitrary internal phase deformations. This excludes any strictly acute full equilibrium preserving those two periods; it does not exclude nonacute states or a history that changes the supplied sectors.

On the same support, equal windings have the exact positive control theta_(L_i)=theta_(R_i)=2*pi*i/5. Ring gaps are 2*pi/5, all matching gaps vanish, and opposite ring currents cancel at every node. Uniform form completes a stationary state. Its full phase Hessian is positive on relative coordinates, so the preceding full-law recovery result applies for every choice of strictly positive held capacities. The negative-winding counterpart and consensus give further controls. The short-cycle span here has rank five while the full cycle rank is six: the old consensus-only topology flag cannot express the exclusion of opposite sectors while retaining these nonzero compatible geometries.

Shared certificate and retained formation boundary

assess_acute_cycle_periods returns AcuteCyclePeriodAssessment for a supplied integer fundamental sector and one nonzero rational cycle combination. The bound is homogeneous, so rational rescaling preserves its verdict; supported represented-real coefficients use the shared exact-materialization boundary. It reconstructs the support owner, computes the exact combined chain, period, strict bound and margin, and marks an obstruction only when that inequality fails. A passing witness remains unresolved: the function neither searches all separating directions nor solves the sine-current equations. The certificate is unchanged by consistent changes of integer cycle basis, though its displayed coordinates change. It reads no live nodal state, installs no edge and does not replace execution admission.

The composition mechanism is therefore constrained circulation with integral phase periods, with a full nodal restoring test once a compatible equilibrium exists. Neither compatible geometry nor local recovery selects the support, its creation event, its winding preparation or its attainment from an initial state. No lower-dimensional autonomous component law or physical identification follows from the certificate.

26. Storage below every sector face forces capture

Retain Section 25's complete positive-loss sine model: finite connected simple unit support, held nu_i>0, e,w,beta>0, the declared clock, and no forcing or events. Put K=diag(nu_i/d_i), a=w/pi, b=w/(beta*pi). The actual two rows and their storage balance are

\[ \dot x=-eKLx+aKS,\qquad \dot\theta=bKLx,\qquad E=\tfrac12x^{\mathsf T}Lx+\beta V(t),\qquad \dot E=-e(Lx)^{\mathsf T}K(Lx), \]

where S=-D sin(2*pi*t) and V(t)=sum_e[1-cos(2*pi*t_e)]. These are the existing complete-law identities, not a new phase-only dynamics. In particular, all form coordinates remain in the initial storage and subsequent evolution.

The entire acute cell, without an equilibrium target

For the full integer fundamental-cycle matrix C and a supplied integer sector k, define

\[ \mathcal A_k=\{t:C^{\mathsf T}t=k,\ |t_e|<1/4\},\qquad \overline{\mathcal A}_k= \{t:C^{\mathsf T}t=k,\ |t_e|\le1/4\}. \]

Require an initial state in A_k. This establishes nonemptiness, so the displayed closed polytope really is the closure: mix any closed-cell point with the strict initial point. Integral periods reconstruct circular phases; on this cell, edge turns determine the complete phase geometry modulo one common rotation. Its tangent space is ker(C^T)=im(D^T). The relative boundary is the union of all signed edge faces

\[ F_{e,\sigma}=\{t\in\overline{\mathcal A}_k: t_e=\sigma/4\},\qquad \sigma\in\{-1,1\}. \]

Some faces can be empty. Neither a subset of basis cycles nor selected boundary samples can replace this full union. For a tree there are no cycle-period constraints: the acute cell is the nonempty full open edge cube, its closure is the closed edge cube, and the minimum boundary phase storage is one.

Target-free capture theorem. Let B be any proved lower bound on V on every nonempty face. If the full initial storage satisfies

\[ \boxed{\qquad E(x_0,t_0)<\beta B,\qquad t_0\in\mathcal A_k,\qquad} \]

then the sector has exactly one sine-critical phase geometry. The complete solution remains strictly inside this sector for all forward time and converges to that geometry and uniform form, modulo the common origins. No critical coordinates are an input to this criterion.

To prove existence, minimize V on the compact closed cell. Since beta*V(t_0)<=E(x_0,t_0)<beta*B, no minimizer lies on its boundary. At an interior minimizer, stationarity in im(D^T) gives D sin(2*pi*t)=0, hence S=0. Uniform form completes an equilibrium of both rows. The Hessian of V is diagonal with strictly positive entries (2*pi)^2*cos(2*pi*t_e) in the open cube. Thus V is strictly convex on the affine cell and has only this one critical geometry.

The smooth complete vector field is globally Lipschitz in real form and phase lifts, so it has unique global solutions. Before any putative first exit, storage is nonincreasing. A boundary state would instead have E>=beta*B>E(x_0,t_0), a contradiction. The strict energy gap also keeps the entire trajectory away from the closed-cell boundary.

For compactness and convergence, use the two conserved lifted means

\[ \overline x_K= \frac{\sum_i(d_i/\nu_i)x_i}{\sum_i d_i/\nu_i},\qquad \overline\theta_K= \frac{\sum_i(d_i/\nu_i)\theta_i}{\sum_i d_i/\nu_i}. \]

The first fixes the eventual uniform form. The second is defined after choosing the cell's continuous lift and fixes its common phase origin; it is not an additional circular observable. Their conservation follows from 1^T L=0 and 1^T S=0. Connectedness and the form energy bound make the fixed-form-mean leaf bounded, while the phase cell modulo its common origin is compact. Hence the trapped storage sublevel is compact.

In the set dot(E)=0, positivity of e and K forces Lx=0. Then dot(theta)=0. To remain in that set requires L dot(x)=a LKS=0, so KS=c*1; summing S and using positivity of 1^T K^-1 1 gives c=0. Its largest invariant subset therefore has Lx=0,S=0. The invariance principle and sector uniqueness imply convergence to the single equilibrium on the fixed-mean leaf. Section 25's local exponential result applies once the trajectory is near it; the certificate does not supply a uniform global convergence rate.

A computable lower bound covering every face

The shared recovery owner uses convex supporting planes and exact rational linear algebra, without solving for a critical target or sampling a trajectory. Write R=C^T. On face (e,sigma), remove column e to obtain B_e, and put d=k-sigma*R[:,e]/4. For any auxiliary cube vector u on the remaining edges, let f(u_j)=1-cos(2*pi*u_j) and g_j=2*pi*sin(2*pi*u_j). For any rational multiplier vector lambda, convexity and B_e*t=d imply

\[ \begin{aligned} V(t)&\ge L_{e,\sigma}(u,\lambda),\\ L_{e,\sigma} &=1+\lambda^{\mathsf T}d+ \sum_{j\ne e}\bigl[f(u_j)-g_ju_j\bigr] -\frac14\sum_{j\ne e} \left|g_j-(B_e^{\mathsf T}\lambda)_j\right|. \end{aligned} \]

Indeed, each supporting line bounds its convex edge potential from below. Subtracting the equality multiplier leaves a linear residual, whose minimum over the remaining closed cube is minus one quarter of its l1 norm. The fixed edge contributes exactly one. The auxiliary u need not satisfy the sector or face constraints and is not an equilibrium candidate. An empty face also satisfies this bound vacuously.

_sine_sector_boundary.py chooses u by clipping the exact least-norm solution B_e^T (B_e B_e^T)^-1 d to the closed cube. It chooses rational lambda=(B_e B_e^T)^-1 B_e g_mid, where g_mid contains interval derivative midpoints. Deleting a single edge leaves the cycle rows independent: a nonzero cycle-space vector supported on one nonloop edge would have nonzero incidence divergence. Thus the reduced Gram inverse exists; a rank-one update reuses the full Gram inverse. The tree case needs no inverse.

Outward rational sine, cosine and arithmetic bound the displayed dual expression from below. Taking the maximum with the elementary bound one on each face, then the minimum over all 2*|E| faces, gives B. The fixed arithmetic budget and deterministic supporting planes can make this conservative; failure to obtain a strict storage margin is unavailable, not nonexistence, instability or a failed physical law. No exact minimization or optimal barrier is claimed.

For an independent nonzero-sector control, a unit-winding C5 has phase storage V_5=5*(1-cos(2*pi/5)). On a feasible boundary face the remaining four oriented turns sum to 3/4. Convexity gives B_5=1+4*(1-cos(3*pi/8)); the opposite signed face is infeasible. The kernel's supporting planes recover this lower bound, with the strict outward-certified gap B_5-V_5>1/100. This comparison remains strict for small nonuniform form and phase perturbations, so it is not limited to a stationary initial state. For example, perturb one node's phase by 1/1000 turn and its form by 1/1000 with beta=1: the form cost is 10^-6 and the phase increase is below 4*10^-5. Two such cycles joined through a supplied intermediary retain the full-support boundary bound B_5+V_5, including the intermediary-edge faces; this follows from the disjoint cycle potentials and is also obtained by the same kernel.

The feasible (4,4,1) no-equilibrium sector of Section 25 cannot pass this theorem. Its closed-cell minimum must occur on the boundary; otherwise the interior-minimizer argument would produce the excluded equilibrium. Therefore every true boundary lower bound is at most the initial phase storage for any admitted state in that sector, and the strict full-storage margin is impossible. A conservative unavailable report preserves that obstruction without pretending to decide all other sectors.

This is capture inside a supplied sector under the complete sine law. It selects no support event or preparation, proves no entry from another sector, and does not transfer the native-law continuation certificate, an operator-runtime claim, a closed coarse dynamics or physical identity.

27. Prepared form can acquire a nonzero phase sector

The preceding theorem retains a sector already present at its initial instant. A separate global estimate now supplies a sufficient route into such a sector from equal initial phases, under the same complete sine law throughout the transit. The preparation includes a supplied signed form profile on the full fixed graph. It is structured initial information, not an unstructured substrate or a derived source of support. The full-state uncertainty extension below quantifies an open neighborhood of this mechanism, including nonflat initial phase, under the same held capacities and complete law.

A global form-to-phase estimate with a finite declared horizon

Keep Section 26's finite connected simple unit support, positive held capacities, e,w,beta>0, and absence of inputs or events. Set

\[ M=K^{-1},\quad A=KL,\quad r=w/e,\quad \tau=et,\quad z=(b/e)(x-\overline x_K\mathbf1),\quad \eta=ab/e^2=\frac{r^2}{\beta\pi^2}. \]

Here tau is a rescaled variable used in the proof; the declared original-clock endpoint is reported in the model clock t. Choose the common initial phase origin so that theta(0)=0, and write v=z(0). The conserved form mean gives 1^T M z=0. Rewriting both complete rows, without deleting their nonlinear sine term, gives

\[ z'=-Az+\eta KS(\theta),\qquad \theta'=Az,\qquad (\theta+z)'=\eta KS(\theta), \]

where primes denote d/dtau. These real phase lifts remain legitimate through nonacute and antipodal configurations because the sine field is globally smooth; no native-Arg continuation theorem is being imported. More precisely, its linear form rows and globally bounded sine derivatives make the full vector field globally Lipschitz on all real form and phase lifts. Every finite declared preparation therefore has a unique global solution, so the endpoint being enclosed exists and the enclosure is nonempty.

Use ||q||_M^2=q^T Mq. The operator A is self-adjoint in this inner product, annihilates the common mode and has a positive gap on 1^T Mq=0. Let lambda>0 be any certified lower bound on that gap. One reusable choice is

\[ \lambda=k_{\min}\lambda_{L},\qquad k_{\min}=\min_i\nu_i/d_i, \]

where lambda_L is the shared exact rational lower bound on the combinatorial Laplacian's positive gap. Indeed, on the weighted-mean-zero leaf,

\[ \|q\|_M^2=\min_c\sum_i\frac{(q_i-c)^2}{k_i} \le\frac1{k_{\min}}\min_c\sum_i(q_i-c)^2, \qquad q^{\mathsf T}Lq\ge\lambda\|q\|_M^2. \]

The forcing remains in this leaf because 1^T S=0. The graph alone supplies the global bound

\[ \|KS(\theta)\|_M^2=\sum_i k_iS_i^2 \le\sum_i\nu_i d_i=:F^2, \]

using |S_i|<=d_i; this does not assume that phase gaps stay small. Fix a finite supplied horizon tau>0 and any proved rho>=exp(-lambda*tau). In particular the fixed rational expression rho=(1+lambda*tau/16)^(-16) is valid, since (1+s/16)^16<=exp(s) for s>=0. Put

\[ \boxed{\begin{aligned} R_z&=\rho\|v\|_M+\eta F\min(\tau,1/\lambda),\\ R_\theta&=R_z+\eta F\tau. \end{aligned}} \]

Then every solution of the declared preparation satisfies

\[ \|z(\tau)\|_M\le R_z,\qquad \|\theta(\tau)-v\|_M\le R_\theta. \]

For the first inequality, variation of constants and the self-adjoint semigroup estimate give ||z(tau)||_M<=exp(-lambda*tau)||v||_M+ eta*F*(1-exp(-lambda*tau))/lambda. The integral factor is at most min(tau,1/lambda). Integrating the exact third row gives theta(tau)-v=-z(tau)+eta*integral_0^tau KS ds, proving the second inequality. Thus the complete nonlinear response is bounded without evaluating or fitting that response.

Convert the estimate into a full-state capture certificate

At the original-clock time T=tau/e, the entire endpoint belongs to the explicit coordinate box

\[ \begin{aligned} |x_i(T)-\overline x_K|&\le(e/b)\sqrt{k_i}\,R_z,\\ |\theta_i(T)-v_i|&\le\sqrt{k_i}\,R_\theta. \end{aligned} \]

This is an analytic endpoint enclosure, not a measured snapshot or a validated Taylor trajectory. Independent coordinate intervals may discard correlations, but still contain the complete true endpoint. If retained, the dual norm gives sharper edge errors |(z_j-z_i)(T)|<=sqrt(k_i+k_j)*R_z and |(theta_j-theta_i)(T)-(v_j-v_i)|<=sqrt(k_i+k_j)*R_theta. When these sharper bounds are consumed, the admitted endpoint set is the intersection of the node box, the correlated edge constraints and the conserved weighted-mean leaf. Its actual endpoint is in that intersection; arbitrary corners of the full rectangular box need not satisfy the sharper constraints. A claim about the entire product box instead requires bounds valid on that larger set, as in the independent control below. Also Lambda=2*max_i nu_i bounds the largest eigenvalue of A, since q^T Lq<=2*sum_i d_i*q_i^2<=Lambda*||q||_M^2; hence

\[ E_{\rm form}(T)\le\frac{\Lambda}{2}(e/b)^2R_z^2. \]

certify_sine_prepared_entry retains the captured complete-law source, the supplied scaled horizon, both full endpoint coordinate bounds and their sharper edge bounds. Its SinePreparedEntry report uses the shared exact weighted diffusion gap and outward exponential/trigonometric arithmetic, then consumes the same private capture core as certify_sine_sector_capture. A tighter proved gap or exponential bound can improve this estimate without changing the theorem. Missing law premises or unresolved strict margins remain explicit; an unavailable enclosure is not failed formation. The prepared-entry tests exercise the preparation, full-law hypotheses, endpoint provenance and capture handoff.

Supply integer edge offsets for this admitted endpoint set. If every resulting edge-turn interval is strictly acute, all its states lie in the same sector k=C^T h. If the whole endpoint storage upper bound is below the full-face barrier of Section 26, apply that theorem at time T. This proves subsequent retention and convergence to the sector's unique equilibrium, without requiring its phase coordinates in advance. When k!=0, the initially phase-flat trajectory has acquired a nonzero sector. Continuity then also implies at least one intervening antipodal edge crossing; no event or discontinuity is added to the sine law.

The inequalities are finite conditions on the supplied preparation, coefficients and horizon. A small ratio alone at a fixed horizon is not a certificate: the remaining homogeneous form term is multiplied by e/b=beta*pi/r when converted back to the original form coordinate. For an asymptotic family, fix the support, K, beta and a weighted-mean-zero profile v while r tends to zero. Then one sufficient choice is tau=1/r. With the displayed fixed rational bound, rho=O(r^16), R_z=O(r^2) and R_theta=O(r), so the endpoint form vanishes and its phase approaches v. Consequently any v with strictly acute edges and phase storage strictly below a proved sector boundary has such prepared entrants for sufficiently small positive r. This corollary follows from the quantitative bounds; it does not replace their evaluation at a selected finite preparation.

That family carries the explicit initial storage

\[ E(0)=\frac12(e/b)^2v^{\mathsf T}Lv =\frac{\beta^2\pi^2}{2r^2}v^{\mathsf T}Lv. \]

It diverges as r^(-2) for fixed nonconstant v. The phase information was encoded in the supplied form profile. Varying r=w/e changes the constitutive ratio, whereas a common clock rescaling preserves it; neither a bounded source budget nor a universal coefficient ratio is established by this construction.

One exact prepared control and its zero-form comparison

Take the complete unit C5 with node order (0,1,2,3,4), unit held capacities, beta=1, and the exactly representable effective weights

\[ e=1023/1024,\qquad w=1/1024,\qquad x(0)=4092(-2,-1,0,1,2),\qquad\theta(0)=0. \]

Its form mean is zero and v=(-8,-4,0,4,8)/pi. This is the phase profile predicted by the preparation, not an equilibrium supplied to the capture theorem. Four oriented cycle gaps are 4/pi; the closing gap is 2*pi-16/pi. They are all strictly acute and sum to 2*pi, so the prospective sector has unit winding. The unequal closing gap also shows that this profile need not itself be sine-critical.

Choose tau=100, so T=102400/1023 in the declared original clock. An independent elementary gap bound suffices: shortest paths between the ten unordered C5 node pairs have equal edge load five in the path-counting inequality of the whole-support recovery owner. Thus lambda_2(L)>=5/5=1, and lambda>=1/2 because K=I/2. Here F=sqrt(10), ||v||_M=sqrt(320)/pi and outward rational evaluation of the displayed finite formulas gives

\[ R_z<6.132\,10^{-7},\qquad R_\theta<3.123\,10^{-5}. \]

The independent node intervals have form radius below 0.001394 and phase radius below 0.000022083 radians. On their complete product box, all five wrapped edges remain strictly acute, the winding is one, and full storage is less than 3.456051. The independently derived C5 full-face bound in Section 26 exceeds 3.469266. The strict margin therefore proves finite-time entry followed by permanent sector retention and convergence. No trajectory samples or search over preparations are needed for this control. The initial form storage is exactly 160*1023^2=167444640; the large transient energy budget is part of the preparation, not omitted phase-only information.

With the same graph, capacities, coefficients and equal phases but uniform initial form, both complete rows vanish identically. That control remains in consensus and cannot acquire nonzero winding. The positive result is thus a conditional conversion of supplied nonuniform form into a maintained phase sector. It neither reopens the separately bounded eleven-node receiver source class nor derives support birth, an autonomous preparation, a unique fundamental law or physical matter.

Quantitative robustness for independent form and phase uncertainty

The exact witness's strict margins and continuous dependence already imply some open neighborhood of successful preparations. The additional result here is a computable bound for a declared full-state family. It uses the existing SineRelativePattern observation set, with every node retained and capacities exact:

\[ \begin{aligned} x_i(0)&=x_i^{\rm nom}+c_x+\xi_i, &|\xi_i|&\le\epsilon_i^x,\\ \theta_i(0)&=\theta_i^{\rm nom}+c_\theta+\zeta_i, &|\zeta_i|&\le\epsilon_i^\theta. \end{aligned} \]

The two common offsets are arbitrary and independent. The residual errors in form and phase are independently bounded; phase centers and errors refer to the owner's supplied consistent real lifts. No radius is a fitted confidence level, and nominal capacities do not replace an uncertain capacity law. The theorem retains the original common-offset-plus-residual set, not the larger rectangular relative box that forgets reference-error correlations.

Write m_i=M_ii, W=sum_i m_i, and P_M q=q-1*(sum_i m_i*q_i)/W. This is the orthogonal centering projector for ||.||_M. For each actual initial state put v=alpha*P_M x(0), where alpha=b/e>0. The previous form row and global forcing bound are unchanged, while the exact integrated phase identity is now

\[ \theta(T)=\theta(0)+v-z(\tau) +\eta\int_0^\tau KS(\theta(s))\,ds. \]

The nonflat initial phase is an essential term. Its omission would replace the declared preparation by a different one. Since P_M is a contraction, the triangle inequality gives the uniform initial norm bound

\[ \|P_Mx(0)\|_M\le N_x:= \|P_Mx^{\rm nom}\|_M+ \left(\sum_i m_i(\epsilon_i^x)^2\right)^{1/2}. \]

Use alpha*N_x in place of ||v||_M in the same finite formulas for R_z and R_theta. They then bound every preparation in the original observation set, including its nonlinear phase feedback during nonacute passage. No nominal trajectory or new approximation to the sine law is introduced. With zero residual radii, this estimate recovers the exact initial norm used above.

For a directed edge i--j, let X_ij and Theta_ij be the initial form and phase difference intervals obtained directly from the original residuals. Their radii are respectively epsilon_i^x+epsilon_j^x and epsilon_i^theta+epsilon_j^theta; the common offsets cancel before any interval construction. The same actual endpoint therefore satisfies

\[ \begin{aligned} x_j(T)-x_i(T)&\in \left[-\alpha^{-1}\sqrt{k_i+k_j}R_z, \alpha^{-1}\sqrt{k_i+k_j}R_z\right],\\ \theta_j(T)-\theta_i(T)&\in \Theta_{ij}+\alpha X_{ij}+ \left[-\sqrt{k_i+k_j}R_\theta, \sqrt{k_i+k_j}R_\theta\right]. \end{aligned} \]

These direct edge bounds retain the common-mean cancellation needed by the shared sector capture calculation. They are not obtained by differencing independent outer node intervals. If every initial raw phase-edge interval is strictly acute, the supplied real lifts telescope around every cycle without wrapping, proving initial sector zero for the whole family. Apply the supplied final integer edge offsets to the endpoint intervals and require the same strict acute and total-storage margins as before. A nonzero captured sector then proves acquisition and subsequent convergence for every state in the family; acuteness is not required during transit.

The uncertainty report also distinguishes centered and absolute endpoint coordinates. Define

\[ d_i^x=(1-m_i/W)\epsilon_i^x+ \sum_{j\ne i}(m_j/W)\epsilon_j^x \]

and define d_i^theta in the same way. These bound the respective centered residual coordinates. Centered endpoint form has radius alpha^(-1)*sqrt(k_i)*R_z about zero; centered phase has radius d_i^theta+alpha*d_i^x+sqrt(k_i)*R_theta about (P_M theta^nom)_i+alpha*(P_M x^nom)_i. Arbitrary common offsets leave absolute endpoint coordinates and weighted means unavailable for a relative source. Each actual state has its own conserved weighted means, so this family ranges over a union of conserved leaves, not one common leaf. For an exact captured source, its known means separately reconstruct the absolute endpoint bounds. The correlated edge constraints remain distinct from the larger product of centered node intervals in either case.

One fixed preparation neighborhood

Keep the published C5 nominal preparation, support, held capacities, e=1023/1024, w=1/1024, beta=1 and tau=100. Declare, before evaluating the bound, the same independent residual radii at every node:

\[ \epsilon_i^x=1/16,\qquad \epsilon_i^\theta=1/65536. \]

There is no coefficient, horizon or radius search. Every initial phase gap has magnitude at most 1/32768 radian, so all preparations have zero winding. The shared weighted estimate and direct edge enclosures give endpoint storage below 3.456318, against the complete boundary lower bound above 3.469266; the computed strict margin exceeds 0.0129487. Consequently the complete family acquires unit winding and converges to the captured sector's unique phase geometry, with each preparation's own conserved common origins. Its positive radii contain an open neighborhood in all initial form and phase coordinates. Arbitrary common translations and rotations do not change the conclusion.

The storage budget remains part of that family. The initial form storage is bounded above by 21433437701/128=167448732.0390625, and the initial phase potential is at most 5/(2*32768^2) by 1-cos(u)<=u^2/2. The shared source retains its own outward storage enclosure; the exact nominal storage cannot replace the uncertain preparation's budget.

For the same uncertainty radii but uniform nominal form and phase, the whole initial set instead has zero-sector storage at most 5/128+5/(2*32768^2)<1. The elementary full-face bound one therefore certifies zero-sector capture and convergence to consensus directly. This uncertain control family is not stationary: its nonuniform residuals can evolve. Exact uniform form and phase remain the stationary control proved above. These two controls distinguish robust conversion of the supplied structured preparation from generic claims about small noise, autonomous preparation or the separately deferred receiver source class.

31. A fixed original preparation budget forces consensus at small ratio

The successful preparation in Sections 27 and 30 retains nonzero scaled form while its original form budget increases as the exchange/loss ratio decreases. A different question fixes that original budget. For exactly phase-flat preparations, the exact mixed-coordinate identity from Section 28 gives an all-time obstruction to winding acquisition, stronger than a finite slow-time comparison.

The preparation class and sufficient condition

Fix a finite connected simple unit graph with at least two nodes, positive held capacities, beta>0 and complete-law coefficients e,w>0. There are no external inputs or support events. Retain K=diag(nu_i/d_i), M=K^-1, A=KL, alpha=b/e and eta=ab/e^2=beta*alpha^2. Let lambda>0 be a certified weighted quotient gap, so that

\[ x^{\mathsf T}Lx\ge\lambda\|P_Mx\|_M^2. \]

The preparation class consists of all signed form vectors and exactly equal initial circular phases satisfying

\[ \frac12x(0)^{\mathsf T}Lx(0)\le B, \qquad B\ge0. \]

Choose equal initial real phase lifts. Common form and phase origins are arbitrary and may differ between members. Independent nonzero phase residuals do not in general belong to this exactly flat class.

Write k_i=K_ii, kappa=max_edges(k_i+k_j) and

\[ \overline W=\frac{2\eta B}{\beta\lambda}. \]

For B>0, a sufficient condition is

\[ \boxed{\qquad 0<\eta\le1,\qquad 2\sqrt{k_i+k_j}\sqrt{\overline W}<\pi \quad\text{on every edge}. \qquad} \]

Equivalently, the strict edge condition is 8*kappa*eta*B<beta*lambda*pi^2. Under these hypotheses every member retains zero principal cycle periods for all future time and converges to full consensus: form tends to its conserved weighted mean and phase to its conserved common phase lift. The result has no selected slow horizon. If B=0, connectedness already forces constant initial form; the entire class is exactly stationary for every positive e,w,beta, without the restriction eta<=1.

An exact auxiliary Lyapunov function

For each member, center the phase on its conserved weighted mean and set

\[ \vartheta=P_M\theta,\qquad z=\alpha P_Mx, \qquad y=\vartheta+z,\qquad \tau=e t. \]

The exact rows are vartheta'=Az, z'=-Az+eta*KS(vartheta) and y'=eta*KS(vartheta). Both centered coordinates have zero weighted mean. Define the auxiliary function

\[ W=\frac12\|\vartheta+z\|_M^2+\frac12\|z\|_M^2 =\left\|z+\frac{\vartheta}{2}\right\|_M^2 +\frac14\|\vartheta\|_M^2. \]

This is a positive definite function of the retained joint state on the centered leaf. It is not the original storage E, a new evolution law, or a physical energy identification. Since vartheta(0)=0, the preparation budget implies

\[ W(0)=\|z(0)\|_M^2 \le\frac{2\alpha^2B}{\lambda}=\overline W. \]

Orient each edge ij arbitrarily, and write its raw real-lift differences as delta_ij=vartheta_j-vartheta_i and zeta_ij=z_j-z_i. The exact edge summation identity u^T S(vartheta)=-sum_edges (u_j-u_i)*sin(delta_ij) gives

\[ \begin{aligned} W' &=-z^{\mathsf T}Lz +\eta(\vartheta+2z)^{\mathsf T}S(\vartheta)\\ &=-\sum_{ij}\left(\zeta_{ij}+\eta\sin\delta_{ij}\right)^2 -\eta\sum_{ij} \left(\delta_{ij}\sin\delta_{ij} -\eta\sin^2\delta_{ij}\right). \end{aligned} \]

For |delta|<pi, the real numbers delta and sin(delta) have the same sign and |sin(delta)|<=|delta|. Hence delta*sin(delta)>=sin(delta)^2. If 0<eta<=1, every term in the last sum is nonnegative. It is strictly positive whenever delta!=0 in this open strip, including when eta=1. Thus W'<=0 throughout the strip. This calculation uses the weighted inner product and the edge identity directly; it requires no commutation of K and L.

First-exit exclusion and full-state convergence

Before any possible first exit from the raw strip |delta_ij|<pi, monotonicity gives W<=overline W, and weighted duality gives

\[ |\delta_{ij}| \le\sqrt{k_i+k_j}\|\vartheta\|_M \le2\sqrt{k_i+k_j}\sqrt{\overline W}<\pi. \]

The final margin is uniform in time, so continuity excludes that first exit. Global existence follows from the complete field's global Lipschitz property. Consequently the entire future remains in the same strip. Every principal phase difference equals its raw lift difference, and raw differences telescope around every cycle; all principal cycle periods therefore stay zero. No acuteness assumption is needed for this conclusion. The stronger optional condition 32*kappa*eta*B<beta*lambda*pi^2 keeps every edge strictly acute.

The sublevel W<=overline W is compact on the finite-dimensional centered leaf and, by the strict margin, lies entirely inside the raw strip. It is positively invariant. Within it, W'=0 requires delta_ij=0 on every edge, followed by zeta_ij=0. Connectedness and the two zero-mean constraints give vartheta=z=0. LaSalle's invariance principle therefore proves vartheta(t),z(t)->0. Since the fixed coefficient alpha is strictly positive, P_Mx(t)=z(t)/alpha->0 as well. Each member retains its own conserved origins throughout this argument; no common unknown origin is assigned a value.

The same invariant sublevel also supplies uniform full-state bounds ||z(t)||_M<=sqrt(2*overline W) and ||P_Mx(t)||_M<=sqrt(4*B/lambda). The latter follows by cancelling the exact positive alpha algebraically, so its evaluation need not divide by a rounded interval for a very small form-to-phase scale.

This proves consensus directly from a different invariant sublevel. It does not assert that the initial original storage lies below the ordinary acute-sector boundary barrier. In particular, the allowed budget can exceed that barrier while the sufficient ratio condition still holds.

Budget dependence, shared certificate and fixed controls

At fixed support, capacities, beta and finite B, the strict condition holds for all sufficiently small r=w/e, since eta=r^2/(beta*pi^2). This is an all-time statement for each admitted positive ratio, not an exchange of a finite-time approximation with an infinite-time limit. Its contrapositive supplies a conservative necessary cost: if an exactly phase-flat preparation reaches nonzero principal cycle period while 0<eta<=1, its original initial form storage must satisfy

\[ \frac12x(0)^{\mathsf T}Lx(0) \ge\frac{\beta\lambda\pi^2}{8\kappa\eta} =\frac{\beta^2\lambda\pi^4}{8\kappa r^2}. \]

This lower bound is not sufficient for formation and is not claimed sharp. It derives the inverse-square budget requirement for the whole phase-flat class, rather than only evaluating the cost of one successful preparation.

certify_sine_budget_consensus implements this sufficient budget condition using admitted support, held capacities, the shared weighted gap and outward arithmetic. It does not integrate a trajectory or install a phase reference. A failed sufficient margin remains unavailable, not an instability or formation verdict.

The budget tests retain the published C5 law e=1023/1024, w=1/1024, beta=nu_i=1 and fix the whole preparation class B=160. The independent gap lambda>=1/2 from Section 27 and kappa=1 give

\[ \overline W\le\frac{640}{1023^2\pi^2},\qquad |\delta_{ij}(t)| \le\frac{2\sqrt{640}}{1023\pi}<\frac1{50} \quad\text{for all }t\ge0. \]

For example, x_i(0)=4*(i-2) has exactly this form budget, but the certificate covers every phase-flat form vector within the ceiling and every common origin. The earlier nonzero-sector source x_i(0)=4092*(i-2) has budget 160*1023^2 and does not meet this sufficient strip condition. Its retained acquisition certificate and the present bounded-budget exclusion concern distinct preparation classes under the same law. No ratio, support or budget is searched to alter either control.

The theorem does not extend to arbitrary initial phase records, select the origin of a supplied preparation or support, or establish physical identity. It distinguishes what this complete law can do with a fixed original form budget from its previously certified large-preparation mechanism.

32. Equal original budgets do not determine the acquired geometry

Section 31 supplies a necessary resource bound for winding acquisition. That scalar budget does not specify the organization reached by the full dynamics. The following fixed comparison separates the budget from the geometric information in the preparation. Its symmetry mechanism reuses the sine orientation argument and the reflection obstruction to receiver formation; it is not a new symmetry law.

A capacity-compatible automorphism restricts cycle periods

Consider the complete unforced sine rows on admitted fixed unit support, with held capacities nu_i>=0, beta>0, w>0 and e>=0. Let p be a graph automorphism preserving the capacities, and let P be its permutation matrix. Degrees are preserved as well, so for K=diag(nu_i/d_i) one has PK=KP, PL=LP and S(Ptheta)=P*S(theta). Thus the full field is equivariant:

\[ F(Px,P\theta)=P F(x,\theta), \]

where the same permutation acts on each row. This argument requires neither positive loss nor strictly positive capacity; it does not use K^-1. There are no inputs or events that break the supplied symmetry.

Suppose the actual initial form and phase lifts are exactly fixed: Px(0)=x(0) and Ptheta(0)=theta(0). Global uniqueness for the smooth complete field then gives Px(t)=x(t) and Ptheta(t)=theta(t) for all t>=0. These are raw equalities on consistent real lifts, without a floating tolerance or an assumed nodewise phase rewrapping. Symmetric error intervals alone would not establish them for every independently uncertain member. Independent common form and phase shifts preserve the equalities.

For an oriented cycle chain c, let p_*c be the chain obtained by mapping each ordered edge through the automorphism. At a time when the relevant edges are not antipodal, their principal angle increments Delta_e are uniquely defined in (-pi,pi) and reverse sign when an edge orientation reverses. Define

\[ k(c)=\frac1{2\pi}\sum_e c_e\Delta_e. \]

Raw state invariance implies k(p_*c)=k(c). Consequently,

\[ \boxed{\qquad p_*c=-c\quad\Longrightarrow\quad k(c)=0.\qquad} \]

More generally the cycle-period functional is invariant under the automorphism's induced action on cycle chains. The reversal test can therefore use a supplied simple cycle on a larger support; it need not be a particular fundamental cycle or the entire graph.

At an antipodal edge, a half-open wrapping convention need not satisfy wrap(-delta)=-wrap(delta). No zero-period conclusion is assigned at that boundary by this argument. The raw symmetry itself remains exact, and the smooth sine evolution continues through the boundary. In particular, an invariant source can never enter a strict acute state with nonzero winding on a reversed cycle. This is an exclusion, not a proof of consensus or of avoidance of antipodal edges.

Any circular limiting configuration with no antipodal edge on this cycle has a neighborhood where its winding is constant. A nonzero-winding limit would therefore contradict the zero-winding restriction at all sufficiently late times. This also excludes nonacute limits with nonzero winding when their cycle edges stay strictly away from antipodes; no convergence is assumed.

The fixed C5 preparations have exactly equal storage

Use nodes 0,...,4, the oriented cycle 0,1,2,3,4,0, unit capacities, beta=1, e=1023/1024, w=1/1024 and initially zero phase. Retain Section 27's acquisition source and fix the comparison source as

\[ \begin{aligned} x_+(0)&=4092(-2,-1,0,1,2) =1023(-8,-4,0,4,8),\\ x_R(0)&=1023(8,4,-8,-8,4). \end{aligned} \]

Both sums vanish, so both conserved weighted form means are zero. Initial phase potential is zero. Directly from the five support edges,

\[ \begin{aligned} \frac{E_+(0)}{1023^2} &=\frac12(4^2+4^2+4^2+4^2+16^2)=160,\\ \frac{E_R(0)}{1023^2} &=\frac12(4^2+12^2+0+12^2+4^2)=160. \end{aligned} \]

Thus the full initial storage is exactly 160*1023^2=167444640 in both cases, under the identical support, capacities, law and clock. The equality does not assume equality of their other state coordinates or observations.

The reflection p(i)=-i mod 5 fixes x_R(0) and its flat phase, preserves the capacities, and reverses the declared cycle. Its full trajectory therefore has the raw phase form (a,b,c,c,b) at every time. Away from antipodal edges the cycle increments occur in opposite pairs, with the edge 2--3 having zero increment, so the winding is zero. The control cannot enter either strict acute nonzero winding sector or converge to either corresponding C5 twist: those phase geometries are outside the closed reflection-fixed set, even after a common phase rotation.

This control is not stationary. Initially KLx_R=1023*(4,4,-6,-6,4), so its full phase row is nonzero. The reflection obstruction neither identifies its eventual equilibrium nor proves convergence or high-budget consensus.

In contrast, Section 27 already certifies that x_+(0) reaches the strict winding-one capture sector at tau=100, or original time t=102400/1023, and subsequently converges within that sector. That certificate is reused without another trajectory or horizon search. Reflecting this successful full preparation while holding the declared cycle orientation fixed reverses the certified winding sign. Relabeling both the state and the oriented observation cycle instead leaves the observation unchanged. There is no orientation-independent absolute sign selected by this comparison.

The successful ramp still has a combined symmetry. The reflection p(i)=4-i mod 5 sends x_+(0) to -x_+(0), and the complete law also commutes with the global sign reversal (x,theta)->(-x,-theta). Their composition fixes the flat-phase preparation and is preserved by uniqueness. Each operation separately reverses the oriented winding; their composition preserves it, so this stabilizer is compatible with the certified nonzero winding. The preparation breaks the pure reflection that would force zero winding; it does not lack all symmetries. The reader below checks pure node permutations only.

What the comparison establishes

assess_sine_cycle_symmetry checks an admitted exact comparison, a supplied node permutation and a supplied simple cycle. It validates the support automorphism, capacity compatibility, raw full-state equalities and reversed cycle chain. The symmetry tests combine this reusable restriction with the retained entry certificate and the independently checked equal-budget preparations. An unsupported symmetry does not imply acquisition, and the reader makes no conclusion about a winding value on an antipodal boundary.

The budget supplies a necessary resource; the initial form supplies information that breaks or preserves geometric symmetries. Both sources start without phase winding. The complete reciprocal rows convert the successful form preparation into actual phase winding, while the capture theorem establishes its later retention and convergence. This does not identify form with an initially present primitive phase winding.

Scalar budget alone therefore cannot predict the organization, and the successful preparation is not evidence that the model autonomously selects that preparation or a handedness from a reflection-fixed state. The comparison supplies no preparation-occurrence law, support origin, new constitutive selector or physical identification.

33. Retained relative origins permit a full-state composition handoff

This result connects the existing relative-state observation to Section 26's capture theorem under the same complete positive-loss sine law. It does not select that law. The common-origin result explains why eliminating one global origin differs from independently eliminating two origins before an interaction. Here the missing relative information is declared explicitly and kept in a reusable state handoff.

The admitted family and the exact offset map

Let A and B have disjoint ordered node sets, connected simple unit supports, positive held capacities, identical admitted e,w,beta>0 and one declared structural clock. At a declared common observation/event time, each relative source supplies nominal form and real phase lifts with per-node residuals:

\[ x_i=\bar x_i+C^x_A+\epsilon_i,\qquad \theta_i=\bar\theta_i+C^\theta_A+\eta_i\quad(i\in A), \]

and similarly on B, with |epsilon_i|<=r_i, |eta_i|<=s_i. The common offsets can be unknown. These are admitted preparation/observation premises, not an assertion that separate measurements were synchronized. Declare exact relative common-origin differences

\[ \Delta_x=C^x_B-C^x_A,\qquad \Delta_\theta=C^\theta_B-C^\theta_A. \]

They are not actual reference-node differences: the latter also contain the nominal reference difference and both reference residuals. A phase difference uses a supplied consistent real lift; no unwrap or favorable branch is inferred. Define the joint nominal rows by retaining A and adding Delta_x,Delta_theta to every B node, then concatenate the original residual radii. Every member of the two-source family is represented exactly by these rows plus one common form and phase origin. Conversely each member of that joint residual family splits into the two declared source families with those offsets. No Cartesian relative-coordinate box or independently chosen nominal member is substituted. This exact map relies on scalar known origin differences; uncertain origin differences require a separately justified correlated or outer-enclosure contract.

Rebasing a source's nominal form by s_A or s_B requires Delta_x -> Delta_x+s_A-s_B, and likewise for phase. The joined family then changes by just the common rebase s_A. Holding the offset fixed while rebasing one component describes a different proposed contact.

A supplied bridge changes weights and adds storage

Admit one unit bridge a in A, b in B, with no nodal reset and unchanged capacities. Put d_i^+=d_i^-+1_{i=a,b}, K_i^+=nu_i/d_i^+ and rho_i^+=d_i^+/nu_i. The pressure and phase rows after the event must use these new degrees. Reusing either isolated mobility gives the wrong field at the bridge endpoints. For the weights and charges in a common frame,

\[ W^+=W_A+W_B+\frac1{\nu_a}+\frac1{\nu_b},\qquad Q^+=Q_A+Q_B+\frac{x_a}{\nu_a}+\frac{x_b}{\nu_b}. \]

This is event reweighting, not a continuous flux or violation of a held-support invariant. Only the postevent weighted mean determines the eventual uniform form if full-state capture holds. An unknown common form origin leaves the absolute mean unknown; an interval for the representative with A's common origin zero must be labelled as such. Lifted phase means have the analogous frame dependence and are not additional circular observables.

The exact storage jump for each member is

\[ \boxed{\Delta E=\frac12(x_b-x_a)^2+ \beta[1-\cos(\theta_b-\theta_a)]\ge0.} \]

Internal edge costs are unchanged. The bridge form-gap interval is centered at bar(x_b)+Delta_x-bar(x_a) with radius r_a+r_b; the phase-gap interval has the corresponding phase center and radius s_a+s_b. Outward interval square and cosine give a conservative jump bound. Rebuilding all internal and bridge bounds gives a bound for the whole postevent storage. Bounds are per-member statements about one family; subtracting two independently rounded storage intervals is not needed to identify the event work.

A nonnegative supplied event-work allowance can be compared against this jump. A certified within-allowance result does not establish capture or event occurrence. Conversely, a postevent capture theorem does not supply the work required to perform the event. Continuous storage dissipation is not an undeclared reserve from which this jump can be paid.

Missing relative information is a mathematical obstruction

If Delta_x is unspecified, a common shift of B leaves all of its internal observations and isolated certificates unchanged while making the bridge form cost arbitrarily large. If Delta_theta is unspecified, a relative rotation can make the bridge antipodal. There is then no whole-family strict acute admission. These obstructions concern what the sources determine; they do not say that every particular contact fails. Missing offsets therefore remain explicitly unavailable and never default to alignment.

Even known offsets and two isolated capture certificates are insufficient. Take two P2 components with unit capacities, beta=1, zero phase and forms (-1,0) and (0,1). Each isolated energy is 1/2, below its tree boundary barrier 1. Join the two zero-form endpoints. The bridge jump is zero but total postevent storage is 1, so the strict whole-tree energy criterion is unavailable. Its failure is not a proof that the joined trajectory cannot converge. It proves why isolated verdicts or bridge cost alone cannot replace Section 26's full-state premises.

Shared implementation and fixed controls

assess_sine_pattern_composition re-admits both SineRelativePattern sources and their authoritative model coefficients, then rebuilds their fields from exact primitive rows. It shifts only the right nominal coordinates, preserves every residual radius, joins one supplied edge and rebuilds all joint fields through the shared relative pattern owner. No graph-to-binary64 roundtrip changes an admitted rational state. The existing sector owner receives this entire joined source and explicit integer edge offsets; it checks every signed boundary face. Its absolute weighted means remain unavailable, even when the frame-relative representative means have finite bounds.

The composition controls fix two P2 sources with nominal zero form/phase, capacities (1,2) and (3,4), e=w=1/2,beta=1, bridge the inner endpoints and declare Delta_x=Delta_theta=1/4. In the resulting P4 order,

\[ d^-=(1,1,1,1),\quad d^+=(1,2,2,1),\quad W^-=25/12,\quad W^+=35/12, \]

\[ Q^-_A=7/48,\quad Q^+_A=11/48,\quad M^-_A=7/100,\quad M^+_A=11/140. \]

Here the subscript A denotes the representative with A's common form origin zero, not an absolute charge measurement. The jump is 1/32+1-cos(1/4)<1, so the full tree is captured without supplying an equilibrium target. Missing-frame, large-phase-gap, full-energy and per-node uncertainty controls test separate obligations. The implementation neither installs a live bridge nor evolves a trajectory; it supplies the previously missing preparation-to-capture handoff under a declared event and law.

A second fixed control joins two C5 sources with zero form, unit capacities, phases 5*j/4 for j=0,...,4, zero relative common origins and a bridge between their first nodes. A -1 edge turn on each canonically oriented closing edge retains unit winding in both cycles. The same whole-set reader certifies capture against all 22 signed boundary faces of the joined support. This checks composition of nonzero winding patterns; their preparation and the occurrence of the bridge are still supplied premises.

34. Analytic acquisition can hand off to maintained composition

Sections 27 and 33 can be connected without replacing an analytic endpoint by an independently supplied observation. Each component evolves on its own declared support up to one common event time; a supplied unit bridge then changes the support without resetting any nodal coordinate. The isolated entry theorem proves the finite prefix. Only a new full-support certificate proves maintenance after the event. This construction still assumes the preparations, complete sine law and bridge event.

Preserve the actual endpoint family and its clock

For component c retain its original relative preparation, positive held capacities and the complete law of Section 27. Let rho_ci=d_ci/nu_ci, W_c=sum_i rho_ci, and let its nominal weighted means be mbar_c^x and mbar_c^theta. The memberwise conserved means, excluding the unknown common origins, are

\[ m_c^x=\bar m_c^x+\frac{\sum_i\rho_{ci}\epsilon_{ci}}{W_c},\qquad m_c^\theta=\bar m_c^\theta+\frac{\sum_i\rho_{ci}\eta_{ci}}{W_c}. \]

Consequently the original residual radii imply intervals for these means; using only the nominal means would lose admitted states. Section 27 provides centered endpoint intervals U_ci^x,U_ci^theta and tighter internal edge intervals for the actual endpoint set. Every endpoint has the form

\[ x_{ci}(T)=C_c^x+m_c^x+u_{ci}^x(T),\qquad \theta_{ci}(T)=C_c^\theta+m_c^\theta+u_{ci}^\theta(T). \]

The centered intervals use the original component degrees and weighted norm. The common mean residual and centered errors need not be independent; interval sums give safe outer bounds without asserting such independence. The internal edge intervals retain their separate, stronger proof. They do not describe every corner of the product of endpoint node intervals.

With declared initial structural times t_A,t_B and scaled durations tau_A,tau_B, an instantaneous join requires

\[ T=t_A+\tau_A/e=t_B+\tau_B/e. \]

Equality is checked from admitted original preparations, coefficients and scaled durations, not cached horizon fields. Different starting times may give the same endpoint time. Unequal endpoint times cannot be repaired by relabelling an observation or silently holding the earlier state fixed. Clock agreement is a preparation premise, not authenticated measurement data.

Bound the bridge without discarding the isolated constraints

Declare exact right-minus-left common origins Delta_x,Delta_theta as in Section 33. For either coordinate q, the bridge from a in A to b in B has the interval bound

\[ q_b-q_a\in J_q:=\Delta_q+M_B^q-M_A^q+U_{Bb}^q-U_{Aa}^q, \]

where M_c^q encloses the memberwise mean above. Retain every original internal endpoint edge bound and add this one bridge bound. The actual joint set is the product of the isolated flow images with the declared frame relation, not a fabricated SineRelativePattern endpoint. The smooth held-law flow supplies a nonempty image for every admitted preparation member. These bounds therefore satisfy the shared sector core's producer contract.

The new degrees, mobility and exact event-work identity are those of Section 33. In particular, endpoint error bounds use d^-, whereas subsequent joint dynamics uses d^+. The bridge work is bounded by J_x^2/2 + beta*(1-cos(J_theta)). A supplied allowance is an independent comparison; positive width of a work enclosure does not establish positive actual work.

The post-event weighted mean also retains the old conserved inventory. In the representative with A's common origin zero, put M_B^q <- M_B^q+Delta_q. A valid bound is

\[ \boxed{ M_+^q\subseteq \frac{(W_A+1/\nu_a)M_A^q+(W_B+1/\nu_b)M_B^q +U_{Aa}^q/\nu_a+U_{Bb}^q/\nu_b} {W_A+W_B+1/\nu_a+1/\nu_b}.} \]

Here the inclusion means that every actual representative mean belongs to the interval on the right. Summing independent node-box corners would forget the conserved inventory. The absolute global origin remains unavailable; the lifted phase mean is not an additional circular observable.

If the entire joined endpoint set is acute and its total storage upper bound lies strictly below every full-support boundary face, Section 26 applies at T. The complete post-event trajectory then remains in that sector and converges on its new conserved-mean leaf. When both isolated preparations also pass Section 27's zero-to-nonzero winding test, this proves a conditional acquisition -> supplied join -> maintenance chain. Neither isolated capture flags nor bridge work alone imply this conclusion.

Shared reader and frozen analytic control

assess_sine_prepared_composition rebuilds both entry reports from their original SineRelativePattern sources, scaled times and declared integer sectors. Zero residual radii are supported. The report retains the initial times and derived common endpoint time, the analytic source provenance, original edge bounds, mean intervals and the new joint capture. It reuses the static composition's support/work accounting and the shared sector core; it neither runs a solver nor installs a live edge. Missing relative origins remain unavailable. Invalid declarations and unequal derived endpoint times reject. A failed sufficient bound remains inconclusive.

The handoff controls freeze two relabelled C5 copies of Section 27's preparation:

\[ x_j(0)=4092(j-2),\quad\theta_j(0)=0,\quad\nu_j=1, \quad e=1023/1024,\quad w=1/1024,\quad\beta=1. \]

Both begin at zero with tau=100, so T=102400/1023. Nodes are ordered 0,...,4 then 5,...,9; the bridge joins ports 0,5, and both relative common origins are zero. The canonical edge turns are (0,-1,0,0,0,0,0,-1,0,0,0). No coefficients, preparation or horizon are fitted to the returned result.

Outward rational bounds certify joint cycle periods (1,1) against all 22 signed boundary faces, with total storage below 6.912, boundary lower bound above 6.924 and a strict margin greater than 9/1000. The bridge work enclosure contains zero and has upper bound below 1/20000. An allowance is not inferred from those bounds. In fact the identical exact copies have equal port trajectories by symmetry and uniqueness; the general enclosure does not use that extra correlation, so its positive upper endpoint is conservative.

The total post-event weight is 22. Its representative form mean has a small interval around zero, while its phase mean lies near -8/(11*pi) because the new weights include the two port phases. No nodal reset causes that reweighting. The controls also retain source uncertainty and reject cached endpoint substitutions or mismatched times. This joins two existing theorems under one explicit hybrid protocol; it does not derive when contacts occur or identify the resulting patterns with physical matter.

Prepared acquisition with an inherited receiver response

Prospective claim and frozen protocol

This gate asks whether two explicitly prepared, initially phase-flat sources can acquire the same maintained winding geometry and retain distinguishable actual receiver form at acquisition, on their uninterrupted full-law trajectories. Prepared sector acquisition is already established above; the new obligation is its joint admission with a receiver signature of the initial internal allocation. The positive-loss law here is a supplied constitutive premise. The conservative pair-response coefficients and its backward-invariant trapped family are not used.

Freeze the following before evaluating the response:

• Nodes are 0,...,9, with pairs (2*a,2*a+1), a=0,...,4. Adjacent pairs on C5 have all four unit cross edges, no internal edges, and degree four. All capacities and beta are one, with fixed effective coefficients e=1023/1024, w=1/1024. No inputs, events or resets occur.
• In the original structural clock, x_dot=-e*L*x/4+(w/pi)*S(theta)/4 and theta_dot=(w/pi)*L*x/4, where S_i=sum_j sin(theta_j-theta_i). Declare tau=e*t, gamma=w/(pi*e), z=gamma*(x-mean(x)), and eta=gamma**2. Thus z'=-L*z/4+eta*S(theta)/4, theta'=L*z/4.
• Every nominal initial phase is zero. Nominal pair mean forms are 4039*(a-2). Preparation A adds (+96,-96) in pair 0; B adds the same internal form increment in pair 3. The nominal initial collective means and full storage agree. Each node independently admits initial form and phase errors of magnitude at most 1/10000000000, including their common origins. Perturbed members need not be energy matched.
• At the fixed scaled time T=100, or original time 102400/1023, read Y=(x_2+x_3)/2 with independent additive error at most 1/10000000000 per branch. Predict recorded Y_A-Y_B>1/100000000. This is a complete form response; a sine-current sign alone cannot pass.
• The target has zero centered form and phases (a-2)*2*pi/5, duplicated in each pair, modulo each member's conserved phase mean. Require the whole reached family to lie within full relative Euclidean radius 1/8 and strictly below its acute winding-one forward storage barrier. Initial winding must be zero throughout both source boxes. The endpoint family is the actual flow image; it is never reset to the target.
• Use exact rational primitives and the shared outward dyadic128/Machin interval arithmetic. The fixed analytic budget uses M=4*I, lambda=2/3, Lambda=2, F=sqrt(40), the shared negative-exponential enclosure, and exact rational Poisson inversion. Retain the full transient, nonlinear-history remainder and conserved form-mean error. No trajectory search, solver, seed or precision sweep is included.
• Close with both whole-family formation/retention and the strict recorded response margin, or with the precise failed obligation. Do not retune the source, law, clock, horizon, observation, errors or numerical budget after evaluation. Store the outward certificate endpoints separately from this preparation declaration.

The integer form profile deliberately supplies organized information. This is prepared acquisition and a finite inherited response, not spontaneous selection of the preparation, support, loss mechanism or physical constituent.

Complete-law semigroup and independently prepared source equality

Let \(L\) denote the fine combinatorial Laplacian, \(A=L/4\), \(M=4I\), and \(f(\theta)=S(\theta)/4\). Both form and continuously lifted phase means are conserved. Remove their common modes only in the proof: \(z=\gamma(x-\bar x\mathbf1)\), and \(\widehat\theta=\theta-\bar\theta\mathbf1\). The complete scaled rows are

\[ z'=-Az+\eta f(\widehat\theta),\qquad \widehat\theta'=Az,\qquad \eta=\gamma^2 . \]

The actual receiver form still includes \(\bar x\); it is not replaced by this centered coordinate. The prescribed initial phase-error lifts fix the conserved phase mean, and the current is unchanged by its removal.

The pair-constant and pair-antisymmetric subspaces are orthogonal and invariant for \(L\). On the former, \(L=2L_{C5}\); on the latter, \(L=4I\). Its nonzero eigenvalues are \(5-\sqrt5\), \(4\), and \(5+\sqrt5\). Thus the positive gap of \(A\) exceeds \(2/3\), since \(\sqrt5<7/3\). On the mean-free space \(A\) is self-adjoint in the \(M\) inner product and

\[ \|e^{-As}\|_M\le e^{-\lambda s},\qquad \|I-e^{-As}\|_M\le1,\qquad \lambda=2/3 . \]

These are fixed-support operator inequalities, rather than fitted relaxation rates. Odd edge currents make \(f\) mean free. The degree-four rows give \(\|f(\theta)\|_M\le F=\sqrt{40}\). Its derivative is a cosine-weighted Laplacian divided by four, whose operator norm is at most \(\Lambda=2\), globally in phase. Consequently \(\|f(\theta)-f(\psi)\|_M\le\Lambda\|\theta-\psi\|_M\).

The nominal sources have zero form mean. Their pair-constant profile contributes \(40\cdot4039^2\) to the form storage; either selected antisymmetric increment contributes \(4\cdot96^2\), with no cross term. Their exact nominal initial storage is therefore

\[ H_A(0)=H_B(0)=40\cdot4039^2+4\cdot96^2=652577704 . \]

Every nominal phase is zero, so no phase potential is hidden in this budget. The equality concerns the two supplied source centers, not all independently perturbed preparations.

For either nominal source define \(v=\gamma x^{\rm nom}(0)\). Put

\[ \ell_0=4039\gamma,\qquad d_0=96\gamma . \]

The pair means of \(v\) are \((a-2)\ell_0\); its selected pair also has the internal entries \(+d_0,-d_0\). In both cases,

\[ V^2:=\|v\|_M^2=80\ell_0^2+8d_0^2 . \]

This phase profile is predicted from the integer form preparation and the independently supplied law. It is not a measured endpoint.

A Poisson prediction for actual endpoint form

Let \(P=I-\mathbf1\mathbf1^T/10\), and define the exact rational matrix

\[ G=\left(L+\frac{\mathbf1\mathbf1^T}{10}\right)^{-1} -\frac{\mathbf1\mathbf1^T}{10}. \]

Connectivity gives \(LG=GL=P\) and \(G\mathbf1=0\). The mean-free vector

\[ q(v)=G S(v) \]

therefore satisfies \(Aq(v)=f(v)\) and \(\|q(v)\|_M\le F/\lambda\). Edge-current antisymmetry supplies its zero-mean premise; an interval rectangle around the currents need not have zero sum at every arbitrary corner.

Write the actual initial centered errors as \(\epsilon_v=\gamma P\xi_x\) and \(\epsilon_\theta=P\xi_\theta\). Then

\[ z(0)=v+\epsilon_v,\qquad \widehat\theta(0)=\epsilon_\theta,\qquad \|\epsilon_v\|_M\le E_v:=\sqrt{40}\gamma\rho_x,\qquad \|\epsilon_\theta\|_M\le E_\theta:=\sqrt{40}\rho_\theta . \]

Orthogonal centering cannot increase these norms. Variation of constants and the exact sum row \((z+\widehat\theta)'=\eta f(\widehat\theta)\) give

\[ \|\widehat\theta(s)-v\|_M \le V e^{-\lambda s}+E_v+E_\theta +\eta F\bigl(s+\min(s,1/\lambda)\bigr). \]

In deriving this bound, the homogeneous initial-form perturbation enters through \((I-e^{-As})\epsilon_v\), whose norm is at most \(E_v\). It is not counted twice as an independently varying endpoint error.

Subtract the constant Poisson response inside the full Duhamel formula:

\[ \begin{aligned} z(T)={}&\eta q(v) +e^{-AT}\bigl(v+\epsilon_v-\eta q(v)\bigr)\\ &+\eta\int_0^T e^{-A(T-s)} [f(\widehat\theta(s))-f(v)]\,ds . \end{aligned} \]

This retains the form-loss operator and the entire nonlinear phase history. It is not a stationary replacement of the actual dynamics. For a certified upper bound \(\rho\ge e^{-\lambda T}\), define

\[ \boxed{ R_z=\rho(V+E_v+\eta F/\lambda) +\eta\Lambda\left[ VT\rho+\frac{E_v+E_\theta}{\lambda} +\frac{\eta F(T+1/\lambda)}{\lambda} \right].} \]

Then \(\|z(T)-\eta q(v)\|_M\le R_z\). Indeed the decaying nominal term in the history bound contributes exactly \(VT e^{-\lambda T}\); the constant initial errors contribute at most \((E_v+E_\theta)/\lambda\), and the remaining term is bounded by \(\eta F(T+1/\lambda)/\lambda\). The shared negative-exponential enclosure supplies \(\rho\); no decay rate is inferred from an evaluated response.

The simultaneous phase bound used for geometric admission is

\[ R_\theta=\rho V+E_v+E_\theta+\eta F(T+1/\lambda),\qquad \|\widehat\theta(T)-v\|_M\le R_\theta . \]

All these estimates are global through the transit, including nonacute and antipodal phase configurations. The complete field is globally Lipschitz on real form and lifted-phase coordinates, so its finite endpoint exists uniquely.

Receiver form, loss compensation and the allocation signal

Let \(l=(e_2+e_3)/2\) be the mean-receiver row. On the mean-free space its \(M\)-dual norm is

\[ \|l-\mathbf1/10\|_{M^{-1}}=\frac1{\sqrt{10}} . \]

The actual receiver and its independently predicted center are

\[ Y(T)=\bar x+\gamma^{-1}l^Tz(T),\qquad Y^{\rm qs}(v)=\gamma\,l^Tq(v). \]

Here \(\eta/\gamma=\gamma\) in the frozen unit-beta law. Since \(|\bar x|\le\rho_x\), every recorded scalar differs from this center by at most

\[ \boxed{ B_Y=\rho_x+\frac{R_z}{\gamma\sqrt{10}}+\rho_y , } \]

where \(\rho_y\) denotes its declared readout-error radius. The two independent conserved form-mean errors cannot be canceled between branches. The Poisson center includes form diffusion; the sign of the sine-current term alone would not bound \(Y(T)\).

There is also a direct exact check of the allocation signal. Let \(p_0=1-\cos d_0\). Taking pair means of \(Lq=S(v)\) gives the five-cycle Poisson equation. Its mean-free inverse has receiver row

\[ (0,\ 2/5,\ 0,\ -1/5,\ -1/5). \]

The coarse Poisson forcing difference between the two placements is

\[ p_0(-\sin\ell_0-\sin4\ell_0,\; \sin\ell_0,\ \sin\ell_0,\ 0,\ \sin4\ell_0-\sin\ell_0). \]

Consequently

\[ \boxed{ Y_A^{\rm qs}-Y_B^{\rm qs} =\frac{\gamma(1-\cos d_0)}5 [3\sin\ell_0-\sin4\ell_0].} \]

The nonuniform closing gap of the rationally prepared profile is retained. In particular \(\ell_0\) is not replaced by \(2\pi/5\). The displayed center difference is positive when \(\pi/4<\ell_0<\pi/2\) and \(0<d_0<\pi\), as here. The finite recorded claim still requires subtracting both complete error bounds \(B_Y\); the sign of this center is not the stopping rule.

Although the source centers have equal storage, their predicted phase-potential increments are respectively \(4p_0(\cos\ell_0+\cos4\ell_0)\) and \(8p_0\cos\ell_0\). Neither these endpoint values nor the two accumulated losses are assumed equal.

The reached full family enters a forward-invariant winding chart

Set \(\alpha=2\pi/5\), \(c=\cos\alpha\) and \(\theta_*=(a-2)\alpha\) in both members of pair \(a\). The target storage is \(H_*=20(1-c)\). For each nominal profile, define its phase potential \(U(v)=\sum_{\{i,j\}}[1-\cos(v_j-v_i)]\), and put

\[ \begin{aligned} R_x&=\frac{\eta\|q(v)\|_M+R_z}{2\gamma},& E&=R_\theta/2,\\ d_*&=\|v-\theta_*\|_2,& N&=R_x^2+(d_*+E)^2 . \end{aligned} \]

Because \(M=4I\), these imply \(\|Px(T)\|_2\le R_x\) and \(\|\widehat\theta(T)-v\|_2\le E\). Both nominal profiles have \(d_*^2=20(\ell_0-\alpha)^2+2d_0^2\), but their Poisson norms and potential bounds are retained separately. Thus the actual endpoint's full relative squared distance to the target is at most \(N\).

The fine Laplacian and the absolute phase-Hessian norm are bounded by eight, while \(\nabla U(v)=-S(v)\). Taylor's inequality therefore gives the complete endpoint excess-storage upper bound

\[ \boxed{ G_H=U(v)-H_*+\|S(v)\|_2 E+4E^2+4R_x^2,\qquad H(T)-H_*\le G_H .} \]

The last term bounds the actual form storage; it is not discarded because the predicted phase pattern is acute. Each actual preparation retains its own mean and storage history.

For the declared radius, put

\[ \mu=\frac\pi2-\alpha-\sqrt2r,\qquad \kappa=\frac{5-\sqrt5}{2}\cos(\alpha+\sqrt2r). \]

If \(\mu>0\), the same full-state target Hessian argument used in the joint persistence theorem gives \(H-H_*\ge\kappa Z_f^2\) throughout the closed relative radius ball. Here \(\kappa\) is geometric; the law during transit and retention is the positive-loss law declared for this gate. In the original clock its exact balance is

\[ \boxed{\dot H=-\frac e4\|Lx\|_2^2\le0 .} \]

The form and phase exchange terms cancel in this balance. Loss is neither fitted from the response nor omitted from the endpoint readout.

For either reached family the strict conditions

\[ \boxed{r^2-N>0,\qquad \kappa r^2-G_H>0,\qquad \mu>0} \]

exclude any subsequent first radius exit. All target-compatible fine edge gaps remain acute, so all target cycle periods are retained, including winding one around the base C5. This is forward retention of the actual reached state, without a reset, law switch or fresh preparation at \(T\). The conservative backward-invariance obstruction does not apply: storage increases in reverse time for this different supplied law.

Initially all fine phases lie in one lifted interval of width \(2\rho_\theta<\pi/2\). Every principal edge increment is therefore its ordinary lifted difference, and every cycle sum telescopes to zero. Both complete source boxes start at zero winding. Their positive widths admit open neighborhoods in all twenty fine form and phase coordinates; endpoint membership and retention refer to their actual flow images, not independently chosen target corners.

Evaluated certificate and the joint stopping rule

The declared analytic evaluation uses scaled time \(T=100\), or original time \(102400/1023\). Both source centers have initial storage \(652577704\). The shared interval calculation gives the following strict outward lower endpoints. Each numerator in the table is divided by \(2^{128}\); these are evaluated bounds, separate from the frozen preparation above.

Obligation	Certified lower-endpoint numerator
Initial zero-winding margin, \(\pi/2-2\rho_\theta\)	534514291964426900545652493888260470163
Acute target-radius margin, \(\mu\)	46748866114495546899709594196780814092
Actual endpoint radius margin, A	4708693058859986293445299107047880637
Actual endpoint radius margin, B	4708693058875486025435602082713862883
Actual endpoint storage-barrier margin, A	630219352267490091696073985417643471
Actual endpoint storage-barrier margin, B	630541569722428563034335359212869245
Recorded receiver difference, A minus B	23314612080077889112572910327611

The complete recorded-difference enclosure is

\[ \boxed{ Y_A^{\rm rec}-Y_B^{\rm rec}\in \frac{[23314612080077889112572910327611, 48558337973699762375085869740872]}{2^{128}} .} \]

Its lower endpoint exceeds \(10^{-8}\), the prospective response threshold. For scale, it is approximately \(6.85\times10^{-8}\); the exact rational comparison supplies the verdict. The weighted scaled-form remainder is at most approximately \(1.806\times10^{-11}\), and each recorded receiver error is less than \(1.855\times10^{-8}\). These decimal summaries do not replace the retained interval evidence.

Thus every admitted source member begins with zero cycle periods, reaches the target's full acute winding chart at the declared time, and remains in that chart under its uninterrupted supplied law. Every admitted A record exceeds every admitted B record by the declared finite margin at that same acquisition time. This combines formation, forward geometric retention and a form response inherited from the initial internal allocation; no independent endpoint preparation is used to combine those conclusions.

Shared assessor and the boundary of the result

The primitive-only assess_sine_formation_response returns SineFormationResponse. It rebuilds the fixed support, complete reference law, source preparations, exact Poisson identities and all consumed bounds before issuing a verdict. Its report retains both source-specific norms and storage bounds, the actual receiver enclosures, and the strict initial, geometric and response margins. The reusable status tests strict positive separation; the frozen gate also requires the stronger \(10^{-8}\) threshold checked above. The contract, usage guide and independent controls retain the execution and admission boundaries. This is an analytic certificate; no numerical trajectory or physical measurement was observed in evaluating it.

The positive-loss law, integer form preparation and support remain supplied premises. Equal nominal storage does not make all perturbed members energy matched, nor does it imply equal accumulated loss. The result does not derive how an unstructured source selects this preparation, support or loss law. Forward retention concerns the target's geometry and cycle periods. The two reached families have a distinguishable finite signature at the declared readout; their internal activity, signature and mutual separation need not persist indefinitely. No autonomous constituent selection, universal law selection or physical identification follows.

Two prepared nonzero winding classes under one complete law

Prospective class-admission claim and frozen protocol

This gate asks whether one fixed positive-loss sine law admits two symmetry-inequivalent, nonzero winding attracting geometries reached from explicit nominally phase-flat form preparations. It tests formation and class admission before any future common-probe experiment. Both entire source families must reach their respective recovery neighborhoods at one declared time and subsequently converge under the unchanged law.

Freeze the following before evaluating the certificate:

• The support is the simple unit cycle C9, with node order 0,...,8 and edges j--(j+1 mod 9). All capacities and beta are one; e=1023/1024 and w=1/1024 remain fixed. No forcing, support events, capacity interventions, state resets or coefficient changes occur.
• In the original structural clock, the complete rows are x_dot=-e*L*x/2+(w/pi)*S(theta)/2 and theta_dot=(w/pi)*L*x/2, with S_i=sum_j sin(theta_j-theta_i). Declare tau=e*t, gamma=1/(1023*pi), eta=gamma**2, and z=gamma*x on the zero-mean leaf. Then z'=-L*z/2+eta*S(theta)/2 and theta'=L*z/2.
• Put alpha=2*pi/9 and the exact rational m=(2046/9)*(355/113)**2. Classes are ordered by k=(1,2). Their nominal initial forms are x_j=k*m*(j-4) and all nominal phases are zero. Their respective target phases are theta_j^*=k*alpha*(j-4), with target form zero. Target phase coordinates are compared modulo circular wrapping and the declared common-phase quotient, not substituted for reached states.
• At every node, admit initial form and continuous phase-lift residuals of magnitude at most 1/10000000000. In each class separately, require the exact constraints sum_j xi_j^x=0 and sum_j xi_j^theta=0. All eighteen source coordinates remain uncertain subject to these two constraints. The families contain sixteen-dimensional relative neighborhoods, not open sets in the full eighteen-dimensional state space. They have the same actual conserved means, both zero. The residuals are otherwise unrestricted; no response-dependent correlations are admitted.
• The common scaled acquisition time is T=100, corresponding to original time 102400/1023. For each complete reached family, require full relative Euclidean radius strictly less than r=1/12 about its own target and storage excess strictly below the shared acute-neighborhood barrier proved below. Require initial winding zero and attained winding k for every admitted source member.
• Initial storage uses the same law and is at most the common budget 1000000000 for both whole source families. The nominal costs are allowed to differ and must be reported separately. Equal source costs are not part of this claim.
• Use exact rational primitives and the shared outward dyadic128/Machin interval arithmetic. The declared semigroup bounds use M=2*I, lambda=1/5, F=sqrt(18) and lambda_max(L)<=4, with the shared negative-exponential enclosure. Retain every initial residual and the complete nonlinear forcing bound. No trajectory integration, search over sources or horizons, or precision sweep is included.
• Pass only if both full source budgets, both geometric and storage margins, the strict acute margin and exact inequivalence checks pass. Otherwise retain the failed obligation without adjusting this preparation, law, time, uncertainty, budget or arithmetic protocol. Record evaluated bounds separately from this declaration.

The requested inequivalence removes graph automorphisms, common phase rotations, common form translations and simultaneous form/phase sign reversal. It is a claim about two prepared relative geometries under a supplied model. There is no probe, readout discrimination, autonomous source selection, support birth or physical identification in this gate.

Why the earlier allocation signature cannot define two lasting classes

The doubled-cycle formation-response result retains both prepared allocations in one acute neighborhood of the same critical geometry. Its compact forward trapping, positive loss and strict phase convexity give a direct application of the convergence argument in Section 26. On each conserved-mean leaf, zero storage dissipation forces uniform form. The phase row then vanishes; remaining in this zero-dissipation set requires zero sine pressure. Strict convexity leaves only the target phase geometry. LaSalle's argument therefore gives convergence to that geometry and the member's conserved uniform form.

The two nominal preparations have identical conserved means, so their full relative states converge to the same target and their nominal receiver-form contrast tends to zero. The earlier independent source errors can give different conserved common origins; such offsets are not a retained identity of their internal allocation. Neither assertion equates distinct fine states at a finite time. In particular, the earlier finite receiver certificate remains valid. A uniform positive late-time allocation margin cannot be inferred from it. The new gate instead requires different relative attracting geometries from the outset.

Complete C9 law, source costs and two exact critical geometries

Write \(P=I-\mathbf1\mathbf1^{\mathsf T}/9\), \(A=L/2\), \(M=2I\) and \(f(\theta)=S(\theta)/2\). The declared source constraints fix both conserved means to zero. The scaled rows, including their nonlinear sine term, are exactly

\[ z'=-Az+\eta f(\theta),\qquad \theta'=Az,\qquad (\theta+z)'=\eta f(\theta). \]

The globally smooth periodic phase field and linear form rows give a unique global solution on continuous phase lifts. These lifts are fixed initially by the small residuals around phase zero and are continued without resetting them through any winding changes. The error constraints do not impose an additional equation of motion.

For a cycle, the eigenvalues of \(A\) are \(1-\cos(2\pi j/9)\). Its positive gap is \(1-\cos(2\pi/9)\). The elementary bounds

\[ \cos(2\pi/9)<\cos(2/3) \le 1-\frac{(2/3)^2}{2}+\frac{(2/3)^4}{24} =\frac{191}{243}<\frac45 \]

use \(\pi>3\), monotonicity of cosine on \([0,\pi]\), and the alternating cosine-series bound at \(2/3\). Hence the declared \(\lambda=1/5\) is a strict lower bound for the semigroup gap, and \(\lambda_2(L)>2/5\). Also \(\lambda_{\max}(L)\le4\), by \(\sum_{i--j}(u_i-u_j)^2\le2\sum_i d_i u_i^2\). No computed small-eigenvalue approximation is required. The full nonlinear forcing satisfies

\[ \|f(\theta)\|_M^2 =\frac12\sum_i S_i(\theta)^2\le18=F^2 \]

at every phase state, including nonacute configurations during acquisition.

For source \(k\), put \(a_k=km\). Its eight nonclosing nominal form differences are \(a_k\), and its closing difference is \(-8a_k\). Consequently its exact nominal storage is

\[ H_k(0)=\frac12(8+64)a_k^2=36k^2m^2. \]

The phase potential initially vanishes for the nominal source. For the entire residual family, \(|\Delta\xi^x|\le2\rho_x\) and \(|\Delta\xi^\theta|\le2\rho_\theta\) imply the sufficient common-clock storage bound

\[ B_k=36k^2m^2+32km\rho_x +18\rho_x^2+18\rho_\theta^2 . \]

The linear term uses the sum \(16km\) of the absolute nominal edge differences; each quadratic term uses the nine edges and \(1-\cos u\le u^2/2\). This bound is valid on the full source family and need not saturate. The two nominal costs differ by a factor four, with both whole-family bounds compared to the same declared budget. No equality of accumulated loss is assumed. Since \(2\rho_\theta<\pi\), every initial principal phase difference equals its supplied small lift difference. Their sum around the cycle telescopes to zero, proving initial winding zero for every member. The assessor uses the stronger sufficient initial acute condition \(\pi/2-2\rho_\theta>0\); failure of that sufficient test alone would not prove nonzero initial winding.

At target \(k\), all nine oriented principal phase increments are \(k\alpha\): the closing edge includes its integer offset \(2\pi k\). The two sine contributions cancel at every node, so \(S(\theta_k^*)=0\) exactly. Both targets have acute edges because \(0<\alpha<2\alpha<\pi/2\). Their phase Hessians are

\[ \nabla^2U(\theta_k^*)=\cos(k\alpha)L, \]

strictly positive on the mean-zero space. The positive-loss full-state stability result above therefore supplies local exponential attraction, including perturbations of form and phase together. The finite source admission below does not replace this complete law by its tangent.

Full-family transit to the two target neighborhoods

Set \(\rho\ge e^{-\lambda T}\), using the declared outward exponential enclosure, and define

\[ \begin{aligned} V_k&=\sqrt{120}\,k\gamma m, &E_v&=\sqrt{18}\,\gamma\rho_x, &E_\theta&=\sqrt{18}\,\rho_\theta,\\ R_{z,k}&=\rho(V_k+E_v)+\eta F/\lambda, &R_{x,k}&=\frac{R_{z,k}}{\gamma\sqrt2},\\ E_k&=\frac{E_v+E_\theta+R_{z,k}+\eta FT}{\sqrt2}, &D_k&=\sqrt{60}\,k\,|\gamma m-\alpha|,\\ Z_k^2&=R_{x,k}^2+(D_k+E_k)^2. \end{aligned} \]

Here \(V_k\) is the nominal initial scaled-form norm, since \(\sum_{j=0}^8(j-4)^2=60\), and \(E_v,E_\theta\) bound the two complete initial residual vectors in the \(M\) norm. No node is removed from either bound. The reusable implementation can replace \(1/\lambda\) in the forcing part of \(R_{z,k}\) by \(\min(T,1/\lambda)\); this is the same bound at the frozen time and a valid improvement at shorter times.

The operator \(A\) is self-adjoint for \(\|\cdot\|_M\). Variation of constants and the global forcing bound give

\[ \|z(T)\|_M \le e^{-\lambda T}(V_k+E_v) +\eta F\frac{1-e^{-\lambda T}}{\lambda} \le R_{z,k}. \]

Writing \(v_k=\gamma x_k^{\rm nom}\), the integrated complete rows also give

\[ \theta(T)-v_k =\xi^\theta+\gamma\xi^x-z(T) +\eta\int_0^T f(\theta(s))\,ds . \]

Thus \(\|\theta(T)-v_k\|_2\le E_k\), using \(\|u\|_M=\sqrt2\|u\|_2\), and \(\|v_k-\theta_k^*\|_2=D_k\). The actual endpoint therefore satisfies

\[ \|x(T)\|_2^2+\|\theta(T)-\theta_k^*\|_2^2\le Z_k^2. \]

This statement concerns every actual flow image of the original constrained source family. It neither installs an endpoint reset nor replaces the family by independently selected endpoint coordinates.

Let \(H=\tfrac12x^{\mathsf T}Lx+U(\theta)\), where \(U(\theta)=\sum_{i--j}[1-\cos(\theta_j-\theta_i)]\), and \(H_k^*=9[1-\cos(k\alpha)]\). Stationarity at the target cancels the linear phase term. Because \(\nabla^2U\preceq L\preceq4I\) everywhere, the endpoint storage excess obeys

\[ H(T)-H_k^*\le2\|x(T)\|_2^2 +2\|\theta(T)-\theta_k^*\|_2^2 \le2Z_k^2. \]

This is an upper bound derived from the complete state, not from the phase coordinate alone.

One strict barrier gives retention, recovery and inequivalence

Put

\[ \mu=\frac\pi2-2\alpha-\sqrt2r,\qquad c_r=\cos(2\alpha+\sqrt2r),\qquad \kappa=\frac{c_r}{5}. \]

If \(\mu>0\), every phase edge in either relative radius-\(r\) ball stays strictly acute: its deviation from the target edge is at most \(\sqrt2r\). Throughout such a ball, the phase Hessian is bounded below by \(c_rL\). Taylor's integral remainder, the zero means and \(\lambda_2(L)>2/5\) give

\[ H-H_k^*\ge\kappa \bigl(\|x\|_2^2+\|\theta-\theta_k^*\|_2^2\bigr). \]

In the original clock the complete law gives exactly

\[ \frac{dH}{dt}=-\frac e2\|Lx\|_2^2\le0. \]

Therefore the sufficient strict inequalities

\[ Z_k^2<r^2,\qquad 2Z_k^2<\kappa r^2 \]

place every reached member inside a compact forward-trapped acute neighborhood. A first exit through its radius boundary would require storage at least \(\kappa r^2\), contradicting the admitted endpoint bound and subsequent nonincrease. The attained winding \(k\) is retained for all later times. Since it began at zero, winding acquisition requires an intervening antipodal crossing, permitted by the unchanged smooth sine law.

Within this compact neighborhood, zero dissipation forces \(Lx=0\), hence \(x=0\) on the fixed mean-zero leaf. For a trajectory to remain there, its form row requires \(S(\theta)=0\). Strict phase convexity in the ball makes \(\theta_k^*\) its only such state. LaSalle's invariance argument then gives convergence to the corresponding target. Together with its local exponential stability, this identifies a recoverable attracting geometry. No quantitative uniform recovery time or driven recovery protocol is asserted here.

The automorphisms of C9 are rotations and reflections. They preserve the absolute winding magnitude; simultaneous form/phase sign reversal can only reverse its sign. Common translations and rotations do not alter winding. Thus magnitudes one and two are inequivalent under all the declared symmetries. Independently, their target storage values \(9[1-\cos\alpha]\) and \(9[1-\cos2\alpha]\) are different, whereas every declared symmetry preserves this storage. The distinction is not a relabeling of opposite orientations of one pattern.

Retained class-admission result and execution boundary

The complete saved report is retained byte-for-byte alongside an evidence manifest. The manifest distinguishes preserved evaluation output from subsequently versioned implementation; it does not authenticate evaluation-time source or chronology.

The declared source families pass at scaled time \(100\), or original time \(102400/1023\), without changing the preparation or method. Their exact nominal initial costs, in class order, are

\[ \left( \frac{29548956783610000}{163047361},\quad \frac{118195827134440000}{163047361} \right). \]

The full-family source upper bounds are respectively

\[ \frac{2216171758770837798673556004402278747} {12228552075000000000000000000},\qquad \frac{8864687035083175597347112004402278747} {12228552075000000000000000000}, \]

both below the common \(10^9\) budget. Let \(D=2^{128}\). The retained outward margin intervals below are \([N,N+d]/D\). Endpoint norm and excess-storage margins use the implementation's upward-rounded full-state bounds and downward-rounded barrier; they do not assume that the nominal endpoint is the exact target.

Strict margin	Lower numerator \(N\)	Width numerator \(d\)
Initial acute margin, both classes	534514291964426900545652493888260470163	2
Target-radius acute margin, both classes	19287815364497400734698836429388645131	17
Radius-squared margin, winding one	2355544828433730297092109439281976691	1
Radius-squared margin, winding two	2355429713531756461723392506835889501	1
Storage-barrier margin, winding one	11719962073973842397261359842045276	1
Storage-barrier margin, winding two	11489732270026171659827494949870897	1
Source-budget margin, winding one	278613236597683921307924110315898488355996937937	1
Source-budget margin, winding two	93605845627925181166669313907300163947479107149	1

In particular, both storage margins exceed \(3\times10^{-5}\). Each entire sixteen-dimensional relative source neighborhood acquires its own nonzero winding by the declared time, retains it thereafter, and converges to its own symmetry-inequivalent attracting geometry. This is an analytic certificate for complete nonlinear trajectories; no trajectory samples or physical measurements were evaluated.

The primitive-only assess_sine_formed_class_pair returns SineFormedClassPair. It rebuilds the fixed complete support, law and both source rows through one freshly admitted shared preparation domain. It also checks the rational inequality \(L/2\succeq P/5\) directly. All consumed norms, source budgets, endpoint bounds and strict margins are rebuilt; no incoming report or cached verdict is accepted as a premise. An unavailable sufficient certificate is not evidence of failed formation. The contract, guide and independent controls retain its execution and admission boundaries.

These two classes differ in phase geometry while their limiting form is the same zero mean. Their different target Hessians do not by themselves certify a finite measurement or a response to an unspecified probe. A subsequent probe must retain the actual reached states, its own forcing or event law, observation errors and recovery obligations. The present construction supplies organized form information and a positive-loss law; it derives neither that preparation nor the loss mechanism, the support, an autonomous constituent selection or a physical interpretation of the two geometries.

A common supplied probe distinguishes the two formed classes

Prospective formed-state probe and frozen protocol

This gate continues the two full source families admitted in the preceding class-formation result. It asks whether one declared probe gives different finite receiver-form responses while both actual post-probe families recover their own attracting geometry. The reached states are not reset to their targets.

Freeze the following before evaluating the response:

• Retain exactly the preceding simple unit C9 support, node order, capacities, complete positive-loss sine law, clock, rational source slope, ordered winding classes (1,2), zero-mean source residual constraints, per-coordinate form and phase budgets 1/10000000000, initial common storage budget 1000000000, and radius r=1/12. Their already declared acquisition time remains tau=100.
• Continue each actual complete-law trajectory without a reset to the fixed probe time P=200 in scaled clock tau=e*t. This is original time 204800/1023. No forcing, event or coefficient change occurs before that time.
• Supply the exact simultaneous phase jump theta(P+)=theta(P-)+delta*q, with delta=1/100 and q=e_0-(1/9)*1; form is unchanged. The jump, its support, amplitude and time are part of the admitted experiment. Its zero mean preserves the common phase origin. There is no support or capacity change, and no inverse jump is subsequently applied. This supplied event is not an autonomous event-selection law.
• Resume the same unforced complete continuous law immediately after the jump. At the single elapsed scaled time h=1, read actual form Y_k=x_0(P+h), with independent additive scalar readout error at most 1/10000000000 for each class. The readout is at original time 205824/1023. Predict recorded Y_2-Y_1>1/10000000. Event amplitude, timing, capacity and clock are exact premises; the initial state and final readout have the declared errors.
• Require both actual families immediately after the jump to remain strictly within their own full relative radius 1/12 and below the corresponding acute-neighborhood storage barrier. Subsequent winding retention and recovery use the unchanged positive-loss law. Account separately for the exact phase-potential jump; do not assume that the supplied probe is passive or free of work.
• Use the same exact rational primitives and outward dyadic128/Machin interval arithmetic. The fixed warmup method is the complete Duhamel bound below with lambda=1/5, Lambda=2, M=2*I, F=sqrt(18) and the shared negative-exponential enclosure. The reference is a constant-forcing comparison at each exact target plus the same jump; bound its error against the actual unreset trajectory. The receiver heat factor uses the fixed spectral bounds below, not a fit or a trajectory sample. Retain the original source uncertainty, nonlinear phase drift and final readout error.
• Pass only if the unchanged source-formation obligations, both post-jump recovery admissions and the strict recorded-response threshold all pass. Otherwise retain the failed obligation without changing the source, law, probe, horizon, observation or budgets. Record evaluated response bounds separately from this declaration.

The prediction concerns two prepared geometries under one supplied model and one externally specified intervention. It neither identifies physical constituents nor selects a unique constitutive law, an autonomous preparation mechanism or the origin of the probe.

Complete-law warmup retains both original source families

Keep the preceding notation \(A=L/2\), \(M=2I\), \(f=S/2\), \(\gamma=1/(1023\pi)\), \(\eta=\gamma^2\), \(\lambda=1/5\), \(\Lambda=2\) and \(F=\sqrt{18}\). The map \(f\) is globally \(\Lambda\)-Lipschitz in both the Euclidean and \(M\) norms. Indeed, its derivative is minus one half of a Laplacian with cosine edge weights. Its absolute quadratic form is bounded by \(u^{\mathsf T}Lu/2\le2\|u\|_2^2\). In particular, \(\|f\|_2\le3\) globally.

Let \(v_k=\gamma x_k^{\rm nom}\), and retain the earlier exact quantities \(V_k,E_v,E_\theta,D_k\). On the mean-zero space define the Poisson vector

\[ \chi_k=A^{-1}f(v_k). \]

Since \(f(\theta_k^*)=0\), its two sufficient bounds are

\[ \|\chi_k\|_M\le F/\lambda,\qquad \|\chi_k\|_M\le\Lambda\sqrt2D_k/\lambda. \]

The inverse here is only on the mean-zero space, where the positive gap has already been proved. No new forcing law is defined by this vector.

Write \(\sigma\ge e^{-\lambda P}\). The exact variation-of-constants identity gives

\[ z(P)=\eta\chi_k+e^{-AP}[z(0)-\eta\chi_k] +\eta\int_0^P e^{-A(P-s)}[f(\theta(s))-f(v_k)]\,ds . \]

For every \(s\ge0\), the two complete scaled rows also give

\[ \theta(s)=\theta(0)+(I-e^{-As})z(0) +\eta\int_0^s(I-e^{-A(s-u)})f(\theta(u))\,du . \]

On the mean-zero space, \(\|I-e^{-As}\|_M\le1\). Consequently the following conservative history bound includes the original form and phase residuals without treating them as independent endpoint errors:

\[ \|\theta(s)-v_k\|_M \le e^{-\lambda s}V_k+E_v+E_\theta +\eta F(s+1/\lambda). \]

The displayed identity even permits \(s\) in the last term; the larger expression is the retained method used for this gate.

Convolving this bound with \(e^{-\lambda(P-s)}\), using \(\int_0^P e^{-\lambda(P-s)}e^{-\lambda s}ds =Pe^{-\lambda P}\), gives

\[ \begin{aligned} R_k={}&\sigma(V_k+E_v+\eta F/\lambda)\\ &+\eta\Lambda\left[ V_kP\sigma+ \frac{E_v+E_\theta+\eta F(P+1/\lambda)}{\lambda} \right],\\ \|z(P)-\eta\chi_k\|_M&\le R_k . \end{aligned} \]

Thus valid Euclidean pre-probe bounds are

\[ \begin{aligned} b_{x,k}&= \frac{\eta\Lambda\sqrt2D_k/\lambda+R_k}{\gamma\sqrt2},\\ b_{\theta,k}&=D_k+ \frac{\sigma V_k+E_v+E_\theta+\eta F(P+1/\lambda)}{\sqrt2},\\ \|x(P^-)\|_2&\le b_{x,k},\qquad \|\theta(P^-)-\theta_k^*\|_2\le b_{\theta,k}. \end{aligned} \]

The refinement is needed because a global forcing bound alone need not make the remaining form sufficiently small for this response comparison. It uses the near-critical nominal phase proxy and the entire prior history, without assuming that the actual warmup endpoint is stationary.

One common phase jump and a receiver comparison with explicit error

For \(q=e_0-\mathbf1/9\), \(\mathbf1^{\mathsf T}q=0\), \(\|q\|_2^2=8/9\), and \(q^{\mathsf T}Aq=1\). In particular its norm is \(\sqrt{8/9}\). Write \(c_k=\cos(k\alpha)\) and \(s_k=\sin(k\alpha)\). Directly summing the two incident currents at each affected node yields

\[ f(\theta_k^*+\delta q) =-c_k\sin\delta\,Aq +\frac{s_k(1-\cos\delta)}2(e_1-e_8). \]

The reflection \(j\mapsto-j\pmod9\) commutes with \(A\), fixes node zero and makes \(e_1-e_8\) odd. Its heat response therefore vanishes identically at the observed node.

Define an analytic comparison \(\widetilde x_k\), with initial form zero, by holding only this comparison's forcing constant:

\[ \widetilde x_k'=-A\widetilde x_k +\gamma f(\theta_k^*+\delta q),\qquad \widetilde x_k(0)=0 . \]

This auxiliary linear equation is not installed on the actual graph. Its node-zero value is exactly

\[ \widetilde Y_k(h)=-\gamma c_k\sin\delta\,g(h),\qquad g(h)=q^{\mathsf T}(I-e^{-Ah})q. \]

For \(0\le\ell\le2\), \(1-e^{-\ell h}\ge\ell(1-e^{-2h})/2\), by concavity in \(\ell\). The spectral theorem and \(q^{\mathsf T}Aq=1\) give the fixed sufficient enclosure

\[ \frac{1-e^{-2h}}2\le g(h)\le\min(h,8/9). \]

The upper bounds use respectively \(1-e^{-\ell h}\le\ell h\) and \(I-e^{-Ah}\preceq I\). This heat factor is exactly the same for both classes because their support, capacities and elapsed time are identical.

In the scaled clock the actual continuous rows after the jump are

\[ x'=-Ax+\gamma f(\theta),\qquad \theta'=\gamma Ax. \]

Semigroup contraction gives \(\|x(s)\|_2\le b_{x,k}+3\gamma s\), and therefore

\[ \|\theta(P+s)-\theta(P^+)\|_2 \le\gamma\Lambda(s b_{x,k}+3\gamma s^2/2). \]

The actual post-jump phase and the comparison phase initially differ by at most \(b_{\theta,k}\). A second variation-of-constants bound, with the global Lipschitz constant \(\Lambda\), consequently gives

\[ \begin{aligned} |Y_k(h)-\widetilde Y_k(h)|&\le B_k,\\ B_k&=b_{x,k}+\gamma\Lambda h b_{\theta,k} +\frac{\gamma^2\Lambda^2h^2b_{x,k}}2 +\frac{3\gamma^3\Lambda^2h^3}{6}. \end{aligned} \]

Each term concerns the same actual trajectory: residual initial form, residual initial phase, phase drift driven by that form, and the further drift from nonlinear forcing. With independent readout errors bounded by \(\varepsilon_Y\), every admitted pair of recorded responses obeys

\[ \begin{aligned} Y_2^{\rm rec}-Y_1^{\rm rec} \ge{}&\gamma(c_1-c_2)\sin\delta\, \frac{1-e^{-2h}}2-B_1-B_2-2\varepsilon_Y,\\ Y_2^{\rm rec}-Y_1^{\rm rec} \le{}&\gamma(c_1-c_2)\sin\delta\, \min(h,8/9)+B_1+B_2+2\varepsilon_Y. \end{aligned} \]

For the declared \(\delta\), the coefficient is positive. Retaining the common heat factor in this contrast is an exact structural correlation, not a presumed correlation of the independent preparation or readout errors. Separately rounded marginal response intervals may discard it. The assertion concerns the recorded two-class contrast; it is not a general inverse identification of an unknown fine state.

Probe work and recovery of each actual reached family

The phase jump changes no form and preserves the common phase mean. Its exact storage jump on any actual pre-probe state is

\[ W_k=U(\theta(P^-)+\delta q)-U(\theta(P^-)). \]

At the exact target this is \(2c_k(1-\cos\delta)\). Globally \(\|\nabla^2U\|_2\le4\); applying the fundamental theorem of calculus to the gradient of this potential difference gives

\[ \left|W_k-2c_k(1-\cos\delta)\right| \le4\delta\sqrt{8/9}\,b_{\theta,k}. \]

This accounts for a supplied hybrid event. Subsequent continuous loss is still \(-e\|Lx\|_2^2/2\). The event work is not counted as continuous dissipation, assumed passive, or attributed to an autonomously selected pressure source.

Let

\[ \begin{aligned} b_{\theta,k}^+&=b_{\theta,k}+\delta\sqrt{8/9},\\ N_k^+&=b_{x,k}^2+(b_{\theta,k}^+)^2,\\ C_k^+&=\cos\!\left(\max(0,k\alpha-\sqrt2b_{\theta,k}^+)\right),\\ E_k^+&=2b_{x,k}^2+2C_k^+(b_{\theta,k}^+)^2,\\ \kappa_k&=\frac15\cos(k\alpha+\sqrt2r). \end{aligned} \]

The first two expressions bound the full post-jump displacement. For the energy upper bound, every edge on the phase interpolation from its target deviates by at most \(\sqrt2b_{\theta,k}^+\). Its cosine is at most \(C_k^+\): if the lower angular endpoint is positive, the whole interval lies inside \((0,\pi)\); otherwise the global upper bound one applies. Target stationarity cancels the linear term, so \(H(P^+)-H_k^*\le E_k^+\).

Whenever the class-specific acute margin is positive and

\[ N_k^+<r^2,\qquad E_k^+<\kappa_k r^2, \]

the earlier compact trapping and LaSalle argument applies immediately after the actual jump. Each full family retains its own winding and converges to its own original target, without an inverse intervention or a replacement of the reached state. This proves response followed by recovery under the declared experiment; it does not make the transient receiver-form contrast persist indefinitely.

The earlier uninterrupted formation certificate applies to the actual prefix up to \(P^-\). Its no-event forward conclusion cannot by itself be continued across the phase jump. The separate post-jump radius and storage admissions above provide that handoff to the resumed continuous law.

Retained common-probe response and recovery certificate

The complete saved report is retained byte-for-byte under the same evidence manifest. This consolidation copies the existing evaluated output without rerunning its assessor or changing the frozen protocol.

The unchanged frozen experiment passes. The probe and readout occur at original times \(204800/1023\) and \(205824/1023\), respectively. With \(D=2^{128}\), the retained recorded contrast is

\[ Y_2^{\rm rec}-Y_1^{\rm rec}\in \frac1D\left[ 116252861521748717635664206406824,\; 712441838942872580413879874471662 \right]. \]

Its lower endpoint exceeds \(3.4\times10^{-7}\), and therefore the prospective \(10^{-7}\) threshold. The comparison includes both original preparation families, the complete warmup, the nonlinear post-probe phase evolution and both independent readout errors.

The retained recovery margins below have intervals \([N,N+d]/D\). All are strict; the full state immediately after the actual jump, rather than the constant-forcing comparison, is the admitted state.

Post-probe margin	Lower numerator \(N\)	Width numerator \(d\)
Radius-squared margin, winding one	2332435485037970454562726074935752860	1
Radius-squared margin, winding two	2332429554268500590392795835718422695	1
Storage-barrier margin, winding one	276350974143488465480789243144019653	1
Storage-barrier margin, winding two	15323309165369098245604323637474595	1
Acute-radius margin, winding one	256849722934490011370183363798693263646	10
Acute-radius margin, winding two	19287815364497400734698836429388645131	17

The storage margins exceed \(8\times10^{-4}\) and \(4.5\times10^{-5}\), respectively. Both actual post-probe families thus retain their different winding identities and recover their corresponding targets under the resumed positive-loss law. Their actual event-work enclosures are

\[ \begin{aligned} W_1&\in\frac1D[ 25291040304426712266496923605374385,\; 26842808498892619294493488281813055],\\ W_2&\in\frac1D[ 5121222566067361708584972223263012,\; 6696561533508206534511574126693058]. \end{aligned} \]

Both are positive: this supplied phase probe injects storage in the admitted experiment. Its work is neither omitted nor identified with the ensuing continuous loss.

For comparison, subtracting only the separately enclosed marginal readouts gives a lower bound of approximately \(9.4957531\times10^{-8}\). That weaker bound is positive but does not meet the declared \(10^{-7}\) threshold. The common deterministic heat factor retained in the direct contrast is needed for this certificate's stronger margin. No cancellation of unknown preparation or measurement errors is used.

The primitive-only assess_sine_formed_class_response returns SineFormedClassResponse. It rebuilds the formation prerequisite from the supplied primitive budgets, requires the probe time not to precede that checkpoint, and reconstructs the warmup bounds through the shared exact preparation and Duhamel kernels. It accepts no incoming certificate as evidence. The report retains the ideal comparison, actual response-error bounds, direct recorded contrast, event work and separate post-jump recovery admissions. Its reusable response status requires strict positivity; this frozen experiment additionally passes the stronger threshold declared above. The contract, guide and independent controls retain the usage and admission boundaries.

Together with the preceding formation result, this supplies two symmetry-inequivalent, prepared nonzero winding classes with different finite responses to one common probe and recovery afterward. It is an analytic complete-model result, not an executed trajectory or measured physical response. Both limiting forms are still zero; the transient readout is not an indefinitely retained form record. The structured preparation, support, positive loss, probe and measurement map remain supplied premises, so no autonomous constituent selection or physical identification follows.

Central contact admission for the formed C9 classes

This result audits an interface before proposing a finite interaction experiment. It uses the actual nominal preparations in the two-class formation result, continued without the subsequent supplied phase probes. It proves an exact obstruction for one contact, with no new numerical response evaluation. The relative-origin handoff and prepared composition result retain the separate full-family admission obligations.

Complete law, source histories and the supplied contact

Take two disjoint copies of the unit cycle C9, each with local node order \(j=0,\ldots,8\), and retain capacities and beta one, \(e=1023/1024\), \(w=1/1024\). In the original clock and on any fixed admitted support, the complete rows are

\[ \dot x=-eKLx+\frac{w}{\pi}KS(\theta),\qquad \dot\theta=\frac{w}{\pi}KLx,\qquad K=\operatorname{diag}(1/d_i), \]

where \(L\) and \(d_i\) belong to that support and \(S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i)\). Thus in \(\tau=et\), with \(\gamma=1/(1023\pi)\),

\[ x'=-KLx+\gamma KS(\theta),\qquad \theta'=\gamma KLx. \tag{C1} \]

Both rows use the same declared clock. On an isolated C9, \(K=I/2\).

Choose a donor class \(k\in\{1,2\}\) and a fixed receiver class \(\ell\in\{1,2\}\). Their nominal initial conditions are respectively

\[ x_j(0)=km(j-4),\quad \theta_j(0)=0,\qquad x_j(0)=\ell m(j-4),\quad \theta_j(0)=0,\qquad m=\frac{2046}{9}\left(\frac{355}{113}\right)^2. \tag{C2} \]

Use these real initial phase lifts and their continuous evolutions. Both form origins and both phase origins are zero in one shared frame. Continue the two actual isolated solutions to one declared common contact time \(\tau_c\). For \(\tau_c\ge100\), the preceding formation certificate already places these nominal members in their respective formed classes. The obstruction itself holds at every \(\tau_c\ge0\).

At that time supply one unit bridge between the two local nodes \(4\), without changing any nodal value, capacity or coefficient. Hold the joined support afterward, with no forcing, additional event or probe. The two port degrees become three; all other degrees remain two. Equation (C1) therefore uses \(K_{44}^{+}=1/3\) at each port after contact. This is an explicitly supplied support event, not an autonomous contact-selection law.

Reflection and continuous lifts give an exact invisible interface

Let \(R\) act within a cycle by \(j\mapsto8-j\). It preserves its edges and fixes node \(4\). Reflection commutes with the isolated Laplacian, while

\[ S(-R\theta)=-R S(\theta). \]

Consequently the transformation \((x,\theta)\mapsto(-Rx,-R\theta)\) is an exact symmetry of the complete isolated law. Each initial state in (C2) is fixed by this transformation. The lifted vector field is globally Lipschitz: its form dependence is linear and all sine derivatives are bounded. Global existence and uniqueness therefore imply, at every time,

\[ x_j=-x_{8-j},\qquad \theta_j=-\theta_{8-j},\qquad x_4=\theta_4=0. \tag{C3} \]

The phase conclusion is an equality of the continued real lifts. A circular fixed-point argument alone would also allow phase pi; it would not justify replacing (C3) by the selected zero lift. No wrapping, resetting or favorable branch choice is made during formation.

Concatenate the two isolated solutions and continue them past \(\tau_c\). This candidate solves the joined complete law exactly. Indeed, its bridge has zero form difference and zero phase difference. At either port the internal contributions also vanish separately:

\[ (L_{\mathrm{cycle}}x)_4=2x_4-x_3-x_5=0,\qquad S_{\mathrm{cycle},4} =\sin(\theta_3-\theta_4)+\sin(\theta_5-\theta_4)=0. \tag{C4} \]

Adding the bridge contributes zero to each expression. Multiplying these zero sums by the correct new mobility \(1/3\) leaves both port rows zero. Every nonport row and its mobility are unchanged. Thus the candidate satisfies every joined row and the actual postevent initial state. Uniqueness of the joined lifted law proves that it is the actual joined solution for all later times.

In particular, zero instantaneous bridge current has here been extended to an all-time statement by an invariant-state and uniqueness proof. Zero current at one arbitrary contact state would not suffice. The result concerns nonstationary prepared trajectories, not only the two limiting critical geometries.

The nominal support-event storage jump is exactly

\[ \Delta H=\frac12(x_{4,\mathrm R}-x_{4,\mathrm D})^2+ 1-\cos(\theta_{4,\mathrm R}-\theta_{4,\mathrm D})=0. \tag{C5} \]

The postevent weights still change. Their total is \(38\), and their form charge is

\[ Q_x^+=2\sum_{\mathrm D\cup\mathrm R}x_i+ x_{4,\mathrm D}+x_{4,\mathrm R}=0, \]

with the analogous identity for the lifted phase charge. These equalities use the nominal reflection property. They do not transfer unchanged to arbitrary uncertain source members.

Consequence for a uniform class-response claim

Fix the receiver preparation, contact time and interface convention, and compare donor \(k=1\) with donor \(k=2\). In both cases the receiver's entire nominal state history after contact is exactly its disconnected isolated history. Hence every common observation depending only on that receiver history is identical between the two donor cases, at any horizon.

Both admitted initial uncertainty families in the formation result include their zero-residual nominal members. The corresponding sets of possible receiver records therefore overlap. This remains true with bounded readout errors that admit zero: select the nominal members and the same zero readout error. No strictly positive uniform donor-class contrast or disjoint receiver-record certificate can hold over these whole source families for this zero-origin central contact. The obstruction also applies to an observation of the receiver's complete history; it is not caused by selecting one unfortunate scalar readout.

This does not assert that every uncertain member has zero bridge current. General allowed errors can break the reflection. Their responses cannot remove a nominal counterexample to a uniform claim. Nor does the argument cover the later common phase probe or repeated supplied probes: the declared jump \(e_0-\mathbf1/9\) generally breaks (C3). This contact is a separate continuation of the original uninterrupted preparation flow, not a revision of either frozen probe protocol.

What a different interface would have to establish

A declared relative common phase origin \(\varphi\) is additional preparation information. For example, prepare the donor with phase origin zero and the receiver with phase origin \(\varphi\), using the same \(\varphi\) in both donor comparisons. Isolated phase-shift symmetry then gives port phases zero and \(\varphi\), respectively, while both port forms remain zero. It does not permit a silent rotation of just one already prepared component at contact; such a rotation would be a distinct supplied state event.

At contact, the correctly normalized additional form rates are

\[ x'_{4,\mathrm D}=\frac{\gamma}{3}\sin\varphi,\qquad x'_{4,\mathrm R}=-\frac{\gamma}{3}\sin\varphi, \tag{C6} \]

and both initial port phase rates are zero. These rates are independent of donor \(k\). An offset with \(\sin\varphi\ne0\) breaks the preceding zero-current condition, but neither proves nor excludes a later class-dependent response through internal feedback. A merely nonzero circular offset is insufficient: at \(\varphi=\pi\), the product trajectory again solves the joined law with zero bridge current, although the nominal bridge adds storage \(2\). Event work and transmitted current are different observations.

Before any positive interaction evaluation, a new protocol must retain:

1. The actual two source histories, a common clock, and the declared relative form and lifted phase origins, including their uncertainty.
2. The full joined rows with their changed degrees and weighted means, the exact support-event storage jump, and any independently supplied event-work allowance.
3. A whole-family postcontact identity-retention argument on the joined support, and recovery if claimed. Isolated capture radii, spectral gaps and maintenance return constants cannot be reused without proving the new hypotheses.
4. A prospective receiver comparison with the same interface premise for both donor classes, its disconnected control, and fixed observation, horizon and error budgets. Equal initial interface rates cannot substitute for this finite response proof.

No offset, response horizon or numerical budget is selected here. An admitted bridge remains supplied support. A finite response or postcontact trapping theorem would not by itself derive a binding force, support formation or autonomous composite selection.

The separate finite contact certificate meets these obligations for one declared nonzero relative origin and its actual prepared families. The aligned central-contact obstruction remains valid under its original zero-origin premise.

The contact contract retains these admission boundaries. Static complete-row controls exercise the shared evaluator, reflection, changed endpoint degrees and source-class comparisons with exact inputs and outward enclosures. They do not propagate trajectories; the all-time conclusion follows from (C3)--(C4) and uniqueness.

Section link directory

These aliases route existing citations to their substantive owner.

• Smooth-sine pattern dynamics

• 28. Exact form, phase and memory representations

• sine-form-phase-memory-equivalence

• The sum coordinate retains the full joint state

• Exact elimination gives second-order phase and retained memory

• form-reconstruction-scope

• What an emergent-form hypothesis must distinguish

• The nonlinear storage representation contains the existing resonance pencil

• A phase-flat preparation distinguishes the full law from phase descent

• 29. A controlled fast-form and slow-phase comparison

• sine-controlled-slow-phase

• Complete state, reference initialization and clocks

• Explicit composite, phase and form bounds

• Uniform finite-time meaning and the retained storage budget

• Means, uncertain sources and phase potential

• Shared certificate and bounded numerical controls

• 30. Full-state capture from controlled phase geometry

• sine-slow-phase-capture-handoff

• A proved reference neighborhood from the preparation

• The actual endpoint and its complete storage

• Handoff to the whole-sector theorem

• Fixed analytic controls and scope

• 35. Conservative exchange can produce finite regional winding

• conservative-regional-winding-entry

• A global full-law phase enclosure

• One frozen two-port preparation

• sine-regional-storage-balance

• Regional work retains the cross-boundary storage

• conservative-regional-phase-transport

• Acute winding requires internal phase redistribution

• The complete rows already transmit phase motion to the inner path

• conservative-regional-work-retention

• A finite regional work budget can retain an admitted acute state

• conservative-regional-organization-control

• A bounded full-state organization control

• Controls, implementation and limits

• sine-regional-channel-accessibility

• 36. Regional channel balance and phase accessibility

• Separate internal conversion from boundary input

• Equal storage does not determine the initial phase response

• A running phase barrier for any acute unit-winding entry

• Shared implementation and evidence boundary

• Retrospective channel diagnosis of the frozen control

• 37. Distributed source geometry can produce finite acute winding

• sine-conservative-source-geometry

• The environmental source image uses full degrees

• A source image plus a global remainder gives actual finite passage

• One supplied distributed preparation

• The same storage and mean do not select the acquired winding

• Shared admission and scope

• 38. Low regional storage does not close the retention handoff

• sine-conservative-handoff-obstruction

• Admit the actual full-support velocity ramp

• Acquired entry near the storage minimum

• Strict finite exit and a necessary positive work transfer

• Zero instantaneous work does not remove the missing information

• Relative rhythm means compatible phase rates, not identical phases

• Shared admission and boundary of the result

• 39. Compatible internal geometry and collective phase dynamics

• sine-phase-offset-partition

• A declared partition with fixed internal phase offsets

• Necessity and sufficiency use both consumed rows

• Reciprocity, inherited storage and conserved means

• One moving realization on the retained private-leaf support

• Exact compatibility is not finite-time formation into the family

• Shared admission distinguishes proof from undecided equality

• 40. A finite-width window around compatible moving geometry

• sine-moving-pattern-window

• Reference motion, error chart and declared duration

• The moving-reference relative-storage identity

• Internal identity and actual work remain compatible

• Conserved storage and backward time restrict acquisition claims

• 41. Actual contact motion can exchange storage with internal structure

• sine-collective-pulse-transfer

• Mean-contact rows from the complete state

• The storage split retains its phase-lift dependence

• A phase-flat preparation has a discriminating fourth derivative

• One phase-flat preparation above the necessary phase-path budget

• Retained full-law control excludes acute winding on its declared horizon

• A structural alternative separates storage release from acquisition

• 42. A larger cycle sector gives a global conservative barrier

• sine-cycle-sector-barrier

• The boundary minimum is exactly 7/2

• A conserved full budget separates the sector in both directions

• Negative collective feedback can still lie below this barrier

• Compatible conserved quantities still do not determine reachability

• 43. Fast contact motion can exclude finite receiver organization

• sine-contact-averaging

• The full rows bound the accumulated contact currents

• An exact change of variables retains the nonlinear receiver

• A prospective finite exclusion despite a large total budget

• 44. A distributed preparation has a robust full-state passage

• robust-conservative-passage

• Direct full-row bounds compose into an independent target box

• An exact distributed source and a fixed short window

• What this passage establishes and what it does not

• 45. Complete contact feedback distinguishes cancellation from rigidity

• sine-receiver-rigidity

• sine-relative-phase-feedback

• The complete centered rows and their derivatives

• Vanishing first and second rates do not imply a fixed geometry

• Exact rigidity has no additional moving compensation family

• A retained second-order description exposes the same feedback

• Consequence for the longer retention question

• 46. A conserved budget bounds finite relative-phase travel

• sine-energy-speed-retention

• An edge-speed bound from the actual full form storage

• First-exit certificate, including full-state uncertainty

• An explicit target with uncertainty in all twenty coordinates

• What this closes, and what it leaves open

• A directed preparation for a possible reverse-time check

• 47. A reversible enclosure can certify an independent formation source

• sine-reversible-preparation

• Complete state, clock and global error bound

• From a reverse endpoint to a forward source ball

• Numerical admission is part of the composition

• Frozen bounded evaluation and its scope

• Retained response: the directed checkpoint does not supply a zero-winding source

• 48. An exact sign/reflection reduction retains the sector saddle

• sine-involution-saddle-reduction

• Exact invariant family and reconstruction

• Inherited storage and oriented winding

• The full saddle's unique hyperbolic pair lies in this family

• A finite nonlinear envelope for the tangent comparison

• A nonlinear local sector passage with an explicit error budget

• What the reduction does not remove

• 49. A directed nonlinear corridor reaches the winding seam

• sine-directed-saddle-corridor

• The principal seam has a lower minimum than the sector saddle

• Exact transverse storage and a directed momentum

• A uniform force bound throughout the corridor

• A directional exit condition keeps the lower face closed

• Why this exit changes winding, and what it leaves open

• 50. Conservative formation and retention on the same full-state orbit

• sine-conservative-formation-retention

• One exact preparation supplies both corridor admissions

• Outer and inner nonlinear connections

• An explicit acute band supplies a unit of retained evolution

• Positive widths include all twenty form and phase coordinates

• 51. Rational preparation and retained-metric propagation

• sine-operational-saddle-preparation

• A rational preparation must retain the same finite gates

• Why independent coordinate boxes lose essential cancellations

• An exact rational metric retains all relative coordinates

• A finite-neighborhood nonlinear bound

• Conversion to primitive coordinates and execution boundary

• A validated step retains the metric radius

• Sine execution with retained full-state uncertainty

• Certifying growth on a full Picard tube beyond the saddle neighborhood

• Retained finite connection and its preparation boundary

• An inverse-inclusion test separates an image from a preparation

• sine-constitutive-robustness

• 52. A constitutive change preserves equilibria but blocks the same source's formation

• The complete changed law and fixed source

• The exact changed acquisition barrier

• Applying the barrier to the retained source

• The complete critical geometry and its inertia remain unchanged

• The sufficient error-transport method has a separate role