Exact and controlled form-phase reductions

Equivalent form/phase/memory representations, controlled slow-time comparison and its full-state capture handoff; hidden initialization and transient errors remain explicit.

Part of Sine pattern geometry and dissipative capture. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

28. Exact form, phase and memory representations

The completed composition, capture and prepared-entry results share more than their graph. They use the same reciprocal exchange, conserved means and phase potential. The following identities connect those results to the existing memory and resonance owners without introducing a new primitive coordinate, deleting the initial form information or assuming a slow phase approximation.

Retain finite connected simple unit support with at least two nodes, positive held capacities, e,w,beta>0, the declared clock and no forcing or events. Use A=KL, M=K^-1, W=1^T M1, and the weighted centering projector P_M=I-1*(1^T M)/W. The means mu_x and mu_theta are those of the actual state; the latter belongs to a chosen continuous phase lift. Uncertain relative sources retain one conserved pair of means per member, with their absolute common origins unobserved. Neither mean is replaced by a nominal value in the identities below.

The sum coordinate retains the full joint state

In Section 27's variables tau=e*t, z=(b/e)P_Mx and eta=ab/e^2, define the mixed coordinate y=theta+z. The exact rows become

\[ \boxed{\qquad \theta'=A(y-\theta),\qquad y'=\eta KS(\theta). \qquad} \]

This change is invertible on consistent real lifts: z=y-theta, x=mu_x*1+(e/b)z, with 1^T Mz=0. Thus it retains both consumed coordinates and the conserved form mean. A change of circular representative sends (theta,y) to (theta+2*pi*m,y+2*pi*m) for the same integer vector m; their real difference is unchanged. Equivalently, retain real z and circular exp(i*y), from which exp(i*theta)=exp(i*y)*exp(-i*z) is reconstructed.

Treating theta and y as two independently wrapped phase vectors loses information. On the unit P2 with unit capacities, K=I. The preparations theta=0,z=0 and theta=0,z=(2*pi,-2*pi) give the same two separately wrapped phase vectors, but their scaled phase rates are respectively 0 and A z=(4*pi,-4*pi). The real form contrast has been discarded by that wrapping, although it is consumed by the phase row.

Nor does y alone close the dynamics. On the same P2, y=0 can arise from theta=z=0, giving y'=0, or from theta=(delta,-delta),z=(-delta,delta), giving y'=eta*(-sin(2*delta),sin(2*delta)). For 0<delta<pi/4 these rates differ even on the same conserved-mean leaf. A first-order law depending only on y therefore needs an additional approximation or restricted invariant family.

The cancellation behind this coordinate is already used operationally in the hidden-state inverse: dot(x)+(e/b)*dot(theta)=aKS removes the form-gradient contribution from paired rate observations. Its time derivative supplies the paired acceleration channel there. The present transformation and the entry estimate reorganize that same complete-law identity; they do not select a new pressure law from those observations.

Exact elimination gives second-order phase and retained memory

In the original clock, differentiate the complete phase row and use the form row. The following second-order identity and velocity reconstruction also hold at e=0, with positive w,beta and held positive capacities; they do not use the preceding change of variables dividing by e. With the actual ordered matrix products this gives

\[ \boxed{\ddot\theta+eA\dot\theta=abAKS(\theta),\qquad \dot\theta(0)=bAP_Mx(0).} \]

The constraint 1^T M dot(theta)=0 must remain. An unconstrained second-order equation on all phase lifts would also permit a constant common phase velocity, which the original positive-capacity law does not supply. On the weighted-mean-zero space, A is invertible and the phase velocity reconstructs the full centered form as P_Mx=b^(-1)A^(-1)dot(theta). The constant form mean must be retained separately if the original absolute form is needed. This is a same-information representation, not a removal of a state variable.

What an emergent-form hypothesis must distinguish

Formation of an organized pattern from joint evolution is different from an all-state instantaneous identity x=F(theta,nu,support,p). The latter does not hold for the admitted conservative sine law. On unit P2, take e=0,w=beta=nu_1=nu_2=1 and theta=(0,0). The states x=(0,0) and x=(q,-q), with nonzero real q, have the same phase, capacity, support, form mean and instantaneous pressure p=(0,0). Their phase rates are respectively zero and (2q/pi,-2q/pi). The common observed coordinates therefore do not determine form or the ensuing response.

Phase velocity, together with the original form mean, supplies the missing information through the inverse above. This uses the independently specified phase law, not the nodal identity alone. At zero loss its exact history representation is

\[ \theta(t)=\theta(0)+btAP_Mx(0) +ab\int_0^t(t-s)AKS(\theta(s))\,ds. \]

The initial form remains an independent source, or equivalently the initial phase velocity. This equation does not derive it from phase alone. At positive loss, pressure computed from the full state can encode form differences; that does not make such pressure an independent measurement or remove the state information needed to compute it. A special invariant or controlled slow family could restrict form in terms of other variables, but its preparation, domain and error require separate proof. Neither global reconstruction nor impossibility of every restricted family follows.

Thus the supported synergy is a same-information description of relative form as phase-motion information. The stronger claim that primitive form emerges from the other instantaneous parameters remains unestablished. The channel-geometry analysis keeps this state question separate from unequal response of the two channels.

Alternatively, variation of constants and integration give the exact nonlinear Volterra equation for e>0

\[ \boxed{\begin{aligned} \theta(t)={}&\theta(0) +\frac be\left(I-e^{-eAt}\right)P_Mx(0)\\ &+\frac{ab}{e}\int_0^t \left(I-e^{-eA(t-s)}\right)KS(\theta(s))\,ds. \end{aligned}} \]

The source term is the retained initial form, not an external input. The phase current in the integral uses the actual earlier phase state; this is not a fixed linear convolution in theta. Differentiating the identity with its supplied initial state recovers the second-order row, and reconstructing form recovers the original joint system. Global Lipschitz continuity of the full sine field supplies uniqueness. The projector could be omitted only from the first source term because I-exp(-eAt) annihilates the common form mode; displaying it makes the mean accounting explicit.

No commutation of K with L is used. In particular AK=KLK, whereas KA=K^2L is generally different. The corresponding phase-velocity memory operator is

\[ \mathcal R_e(s)=A e^{-eAs}K =K^{1/2}B e^{-eBs}K^{1/2},\qquad B=K^{1/2}LK^{1/2}. \]

It is symmetric positive semidefinite for each nonnegative lag, but its individual entries need not be nonnegative. Spectral integration gives

\[ \int_0^\infty\mathcal R_e(s)\,ds =\frac1e\left(K-\frac{\mathbf1\mathbf1^{\mathsf T}}W\right). \]

Acting on a sine-current vector, whose ordinary sum is zero, this integrated operator gives KS/e. This explains the frozen-history gain behind a candidate slow phase law; it does not bound the error made by replacing the moving history by its current value. It also does not remove the initial source or justify exchanging an infinite-time limit with a constitutive limit.

The single-intermediary sine owner already derives a nonlinear second-order representation and derivative-free Volterra memory, including the hidden initial state and moving visible boundary. The general elimination owner owns the variation-of-constants principle and its source obligation. The identities here specialize that principle to eliminating the entire form coordinate of this complete nonlinear sine model. They neither turn the existing signed intermediary kernel into a positive-entry kernel nor transfer a pure-diffusion or conservative-memory conclusion between models.

The nonlinear storage representation contains the existing resonance pencil

Remove the conserved origins, and put

\[ h=K^{-1/2}\mathbf1,\quad\mathcal Q=h^\perp,\quad \xi=K^{-1/2}P_Mx,\quad \vartheta=K^{-1/2}P_M\theta,\quad \Phi(\vartheta)=V(K^{1/2}\vartheta). \]

The symmetric matrix B above is positive definite on Q; all inverse matrices in this paragraph act only there. Since grad(Phi)=-K^(1/2)S, the exact nonlinear transformed rows and their equivalent second-order equation are

\[ \dot\xi=-eB\xi-a\nabla\Phi(\vartheta),\qquad \dot\vartheta=bB\xi,\qquad B^{-1}\ddot\vartheta+e\dot\vartheta +ab\nabla\Phi(\vartheta)=0. \]

Their storage and loss are still the original quantities:

\[ b^2E=\frac12\dot\vartheta^{\mathsf T}B^{-1}\dot\vartheta +ab\Phi(\vartheta),\qquad \frac{d}{dt}(b^2E)=-e\|\dot\vartheta\|^2. \]

The velocity contribution is the original form storage in different coordinates. The derived quotient matrix B^-1 depends on the graph and held capacities; this supplies no identification of capacity with inverse physical mass. Initial velocity, nonlinear phase history and the conserved origins remain part of the model.

At a critical phase geometry with Hessian H, the Hessian of Phi is C=K^(1/2)HK^(1/2). Linearizing the displayed equation gives precisely the existing full-support resonance pencil, B^-1*ddot(vartheta)+e*dot(vartheta)+ab*C*vartheta=0 for perturbations. The matrices B and C need not commute, so scalar cycle discriminants still cannot classify a general attachment. Strict acute-sector convexity, equilibrium recovery and the full storage capture barrier use the same phase curvature; the nonnegative velocity/form term is why a phase-only storage check cannot replace the capture certificate.

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

Take equal initial phases and any nonconstant initial form on the retained connected graph. Then S(0)=0, H(0)=L and

\[ V(0)=\dot V(0)=0,\qquad \ddot V(0)=b^2(Ax(0))^{\mathsf T}L(Ax(0))>0, \qquad \dot E(0)=-e(Lx(0))^{\mathsf T}K(Lx(0))<0. \]

For strict positivity, Ax(0) has zero weighted mean. If it were a constant vector, it would vanish; positivity of K and connectedness would then force x(0) constant, contrary to preparation. Thus phase potential initially increases while total storage decreases. The form reservoir and reciprocal phase row account for both signs under one law.

In contrast, the candidate phase-gradient flow dpsi/dsigma=KS(psi) initialized at that same uniform theta(0) remains there by uniqueness. It would discard the prepared form information. Initializing a proposed comparison from theta(0)+z(0) retains that information, but the exact identities alone do not prove closeness to this comparison flow. Section 29 derives the finite-horizon error bound while retaining the initial transient. A later capture handoff, support creation and physical identification remain separate obligations.

The full-law equivalence controls differentiate the shared field with interval jets, check the two opposite storage signs and the failure of a sum-coordinate-only closure, and retain noncommuting capacity/Laplacian products. Independent numerical quadrature checks the nonlinear memory identity along a complete trajectory; it is an equivalence check, not a reserved-response prediction or the analytic proof.

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

The exact representations in Section 28 permit a quantitative comparison with a first-order phase flow. This is an approximation theorem for the same complete sine law, with an explicit transient and error. It does not install that reference flow as a replacement pressure or remove the prepared form information.

Complete state, reference initialization and clocks

Fix the finite connected unit support, positive held K, positive beta and positive complete-law coefficients e,w. Retain Section 27's M=K^-1, A=KL, alpha=b/e, z=alpha*P_Mx, tau=e*t and eta=ab/e^2=beta*alpha^2. For the actual initial state write z_0=z(0) and take a proved bound Z>=||z_0||_M. Define the slow time and reference on consistent phase lifts by

\[ \sigma=\eta\tau=\frac{ab}{e}t,\qquad \frac{d\psi}{d\sigma}=f(\psi),\qquad f(\theta)=KS(\theta),\qquad \psi(0)=\theta(0)+z_0. \]

Both the complete and reference fields are globally Lipschitz, so their solutions exist uniquely for all finite times. The reference is initialized with the actual retained form contribution, not only the initial phase. As in Section 28, the mixed coordinate uses real z_0 and consistent lifts; wrapping its two summands independently would discard information.

Let lambda>0 be a certified weighted quotient gap for A, and set

\[ F=\left(\sum_i\nu_i d_i\right)^{1/2},\qquad \ell=2\max_i\nu_i. \]

The earlier forcing bound gives ||f(theta)||_M<=F globally. Its derivative is -KH(theta), where H is the cosine-weighted phase Hessian. The edge formula gives -B<=K^(1/2)H(theta)K^(1/2)<=B for B=K^(1/2)LK^(1/2). The similar matrix KL has Gershgorin intervals [0,2*nu_i], so the symmetric matrix B has norm at most ell. Integrating the derivative along a segment therefore proves the global Lipschitz estimate ||f(u)-f(v)||_M<=ell*||u-v||_M. This bound needs no acute phase domain for either path or the segment between them.

Explicit composite, phase and form bounds

For a supplied finite sigma>=0, put tau=sigma/eta and D(tau)=exp(-A*tau). Define

\[ \boxed{\begin{aligned} R(\tau)&=\frac{\eta F}{\lambda} \left(1-e^{-\lambda\tau}\right),\\ C(\sigma)&= \frac{\eta(\ell Z+F)}{\lambda+\eta\ell} \left(e^{\ell\sigma}-e^{-\lambda\tau}\right). \end{aligned}} \]

Then the exact complete solution and its reference satisfy

\[ \boxed{\begin{aligned} \|z(\tau)-D(\tau)z_0\|_M&\le R(\tau),\\ \|\theta(\tau)+D(\tau)z_0-\psi(\sigma)\|_M&\le C(\sigma),\\ \|\theta(\tau)-\psi(\sigma)\|_M &\le Ze^{-\lambda\tau}+C(\sigma). \end{aligned}} \]

The middle quantity is the composite correction. It retains the fast initial form contribution instead of calling the corrected angle the actual phase. For y=theta+z, the same estimates also give ||y(tau)-psi(sigma)||_M<=C(sigma)+R(tau).

The form estimate follows from the exact variation-of-constants identity

\[ z(\tau)=D(\tau)z_0+ \eta\int_0^\tau D(\tau-s)f(\theta(s))\,ds. \]

All integrand vectors have zero weighted mean. Thus the quotient semigroup bound ||D(s)||_M<=exp(-lambda*s) applies to them and gives R. Section 28's exact phase memory, now in scaled time, gives

\[ \theta(\tau)+D(\tau)z_0 =\theta(0)+z_0+ \eta\int_0^\tau\left[I-D(\tau-s)\right]f(\theta(s))\,ds. \]

Subtract the reference integral equation. If q(tau)=theta(tau)+D(tau)z_0-psi(eta*tau), the forcing and Lipschitz bounds imply

\[ \|q(\tau)\|_M\le \eta\ell\int_0^\tau\|q(s)\|_M\,ds+ \frac{\eta(\ell Z+F)}{\lambda} \left(1-e^{-\lambda\tau}\right). \]

Here the ell*Z term retains the initial transient inside theta-psi=q-D(s)z_0; the F term bounds the remaining memory integral. The scalar comparison function with zero initial value solves c'=eta*ell*c+eta*(ell*Z+F)*exp(-lambda*tau). Its explicit solution is the displayed C, proving the composite bound by Gronwall's inequality. The uncorrected phase bound then follows by the triangle inequality. No phase linearization, commuting K,L assumption or sampled response enters this argument.

Conversion back to the original form coordinate is also explicit:

\[ \left\|P_Mx(t)-e^{-eAt}P_Mx(0)\right\|_M \le\frac ae\frac F\lambda \left(1-e^{-\lambda\tau}\right) =\sqrt{\beta\eta}\frac F\lambda \left(1-e^{-\lambda\tau}\right). \]

Thus the original form remainder after its homogeneous transient is O(sqrt(eta)) at fixed support, K and beta. A small scaled z remainder alone would not establish this original-coordinate statement. The homogeneous form term remains until its own transient has decayed.

Uniform finite-time meaning and the retained storage budget

Fix a finite slow-time horizon Sigma, the support, K, beta and a uniform bound Z on the scaled initial form. For 0<=sigma<=Sigma,

\[ C(\sigma)\le \eta\frac{\ell Z+F}{\lambda}e^{\ell\Sigma}. \]

This is a uniform O(eta) composite comparison including the initial instant. It is not uniform O(eta) proximity of the actual phase from that instant: exactly theta(0)-psi(0)=-z_0. For any fixed sigma_0>0, the uncorrected estimate on [sigma_0,Sigma] instead has the additional transient Z*exp(-lambda*sigma_0/eta). More generally, the finite displayed inequalities decide whether a chosen transient is sufficiently small, without retuning the requested horizon.

The original-clock comparison time is

\[ t=\frac{\sigma}{e\eta} =\beta\pi^2\frac e{w^2}\sigma. \]

As eta changes, the constitutive ratio changes; this is not just a common clock rescaling. Keeping a nonzero scaled preparation fixed also retains the original initial storage

\[ E(0)=\frac{\beta}{2\eta}z_0^{\mathsf T}Lz_0 +\beta V(\theta(0)). \]

Its form term grows as 1/eta. For uncertain preparations the expression and its enclosing budget apply to every actual member, rather than to its nominal source alone.

Means, uncertain sources and phase potential

Since 1^T M f=1^T S=0, the reference conserves its weighted phase mean. Its initialization has the same weighted phase mean as the complete trajectory because z_0 is centered. Common form shifts cancel from z_0; common phase shifts move the actual and reference solutions together. All comparison differences and the composite correction have zero weighted mean on the selected lift.

For a SineRelativePattern, reuse Section 27's original residual set and its weighted initial norm bound. Each member is compared with its own reference initialized at that member's theta(0)+z_0. A uniform Z makes the displayed error uniform over the family but does not replace these references by a single nominal reference trajectory. Doing that would require a separate bound on the initialization differences. Unknown absolute origins and memberwise conserved means remain explicit.

The potential satisfies the global bound |V(u)-V(v)|<=F*||u-v||_M, since ||grad(V)||_(M^-1)=||KS||_M<=F. Hence

\[ |V(\theta(\tau))-V(\psi(\sigma))| \le F\left[Ze^{-\lambda\tau}+C(\sigma)\right]. \]

The reference itself obeys dV(psi)/dsigma=-S(psi)^T K S(psi)<=0. This does not make the actual phase potential monotone: Section 28's phase-flat, nonconstant-form preparation has strictly positive initial second phase-potential derivative while total storage decreases. Initializing the reference only at that uniform phase would leave it at consensus and discard the preparation mechanism; the shifted initialization and explicit transient resolve that mismatch in the comparison theorem.

Shared certificate and bounded numerical controls

bound_sine_slow_phase accepts an admitted exact or relative source and a supplied rational slow time. It reuses the complete-law preparation admission, weighted norm and exact reversible gap owners, and returns outward analytic comparison bounds. The scaled and original times are enclosed directly from their mathematical-pi formulas; these are enclosures of the declared comparison instant, not a new class of independently uncertain times. The reader runs neither the full trajectory nor the reference phase solver. It does not issue an endpoint capture or final-basin verdict.

The reduction tests freeze a small heterogeneous support with edges (0,1),(1,2),(2,3),(0,2), capacities (1,3/2,2,5/4) and beta=2. The primary constructor weight ratio is 16:1; the stored normalized coefficients remain authoritative, with exact ratio w/e=1/16. The declared preparation is x=(beta*e/w)*(1/2,-3/4,5/4,-1) and theta=(1/8,-1/4,1/2,-3/8), with slow horizon 1/16. A separately fixed 8:1 comparison retains the same scaled initial form and checks the feedback-parameter dependence. Independent SciPy integrations of the complete and reference rows cross-check the inequalities under the test's explicit numerical settings; they are not validated trajectory enclosures or a search over sources, ratios or horizons. The analytic certificate does not depend on their sampled responses.

The error grows with the declared slow horizon and is not an infinite-time theorem. It proves neither a final basin match nor an exchange of the limits eta->0 and t->infinity. Section 30 supplies a separate capture criterion that admits the actual full-state endpoint, including the form remainder and all sector margins. The result controls a mechanism within the supplied law and preparation; it does not select that law, support, source budget or physical interpretation.

A conditional comparison using acute geometry and dissipation

The two-port transit proof derives a different finite-horizon estimate for its unit-capacity, beta-one model. On a common acute chart, the slow phase field is dissipative in the degree metric. Integrating the complete law's storage loss bounds the squared scaled-form norm over time, yielding a comparison error that grows with the square root of the slow horizon. A strict first-exit argument must establish that both the actual and mixed phases stay in that chart. The nominal reference also retains its explicit initialization mismatch against every actual source member.

That conditional argument supplies a finite deformation result with the original form coordinate retained. It does not replace the global, possibly nonacute comparison above or establish eventual capture. Its model-specific coefficients and prepared phase origins do not transfer without admission of the new complete model.

30. Full-state capture from controlled phase geometry

Section 29 bounds the difference from a reference flow but does not by itself locate that reference at the requested endpoint. A sufficient handoff needs both a justified reference neighborhood and the remaining original form storage. The following construction supplies these from the admitted preparation and an exactly verified stationary phase geometry. It requires neither a supplied response endpoint nor a reference solver.

A proved reference neighborhood from the preparation

Retain all hypotheses and notation of Section 29. Supply exact rational node turns q_i, in the source node order, and put phi_*=2*pi*q on the selected real lift. Require every principal target edge angle delta_*e to lie strictly between -pi/2 and pi/2, and require S(phi_*)=0. The shared target owner verifies these hypotheses using the exact acute geometry and sine cancellation; a floating residual close to zero does not prove criticality. This implemented algebraic admission is sufficient, not a classification of all possible sine equilibria.

For each actual preparation member, shift phi_* by a common constant so that its weighted mean equals that of theta(0). Write the resulting lift as phi_*^m. Its phase geometry and potential V_* are unchanged. The reference psi(0)=theta(0)+alpha*P_Mx(0) has this same weighted mean. Choose a proved bound

\[ D_0\ge \left\|P_M\left(\theta(0)+\alpha P_Mx(0)-\phi_*\right)\right\|_M. \]

Since f(phi_*^m)=0, the constant curve phi_*^m solves the reference equation. The global Lipschitz bound from Section 29 and Gronwall give

\[ \|\psi(\sigma)-\phi_*^m\|_M \le D_{\rm ref}(\sigma):=e^{\ell\sigma}D_0. \]

This estimate assumes no contraction or acute evolution of the reference. The supplied stationary geometry is a proof reference; it is not inserted into the complete law or assigned to the evolving state.

For a relative source, retain exactly Section 27's original residual family. If its componentwise form and phase error radii are epsilon_xi,epsilon_thetai, a uniform choice is

\[ \begin{aligned} D_0={}& \left\|P_M\left(\theta_{\rm nom} +\alpha P_Mx_{\rm nom}-\phi_*\right)\right\|_M\\ &+\left(\sum_i m_i\epsilon_{\theta i}^2\right)^{1/2} +\alpha\left(\sum_i m_i\epsilon_{x i}^2\right)^{1/2}, \qquad m_i=M_{ii}. \end{aligned} \]

This follows from the triangle inequality and the contractivity of the weighted orthogonal projection P_M. Every member keeps its own reference, mean-matched target and conserved means. Unknown common origins cancel before bounding the mismatch; no absolute phase or form origin is inferred. The estimate uses the selected consistent phase lifts, with the chart and quotient distinctions from Section 28.

The actual endpoint and its complete storage

At the supplied slow time sigma, set tau=sigma/eta and define

\[ \begin{aligned} Q(\sigma)&=Ze^{-\lambda\tau}+C(\sigma),\\ \rho(\sigma)&=D_{\rm ref}(\sigma)+Q(\sigma),\\ X(\sigma)&=e^{-\lambda\tau}N_x+ \frac ae\frac F\lambda\left(1-e^{-\lambda\tau}\right), \qquad N_x\ge\|P_Mx(0)\|_M. \end{aligned} \]

Section 29 proves that the actual endpoint satisfies ||theta(t)-phi_*^m||_M<=rho and ||P_Mx(t)||_M<=X, where t=sigma/(e*eta). The phase bound includes the initial transient; substituting only the composite error C would not prove this statement. The form bound is in the original coordinate, with its conversion factor retained.

For an edge ij, write k_i=K_ii and g_ij=sqrt(k_i+k_j). Weighted duality gives

\[ \left|(\theta_j-\theta_i) -(\phi_{*j}^m-\phi_{*i}^m)\right|\le g_{ij}\rho, \qquad |x_j-x_i|\le g_{ij}X. \]

Thus a sufficient strict acute margin on every edge is

\[ \boxed{\quad |\delta_{*ij}|+g_{ij}\rho<\frac\pi2 \quad\text{for every retained edge}.\quad} \]

Use the integer edge offsets that reduce the supplied target lift to its principal acute angles. They place every enclosed actual endpoint in the target's cycle-period sector, without inferring a future unwrapping from samples.

A useful storage bound retains the correlations lost by independent edge intervals. Let h=theta(t)-phi_*^m. The phase Hessian satisfies H<=L globally, since every edge cosine is at most one. Taylor's integral formula and the exact criticality of the target therefore give

\[ V(\phi_*^m+h) =V_*+\int_0^1(1-s)h^{\mathsf T}H(\phi_*^m+sh)h\,ds \le V_*+\frac12h^{\mathsf T}Lh \le V_*+\frac\ell2\rho^2. \]

The linear term vanishes because grad V(phi_*^m)=0. The segment need not remain acute for this upper bound. Likewise x^T Lx<=ell*||P_Mx||_M^2, so the complete storage obeys

\[ \boxed{\qquad E(t)\le E_{\rm handoff}:= \frac\ell2X^2+\beta\left(V_*+\frac\ell2\rho^2\right). \qquad} \]

This uses the same cancellation principle as local recovery, now with a preparation-derived phase neighborhood. Bounding every cosine separately can lose that cancellation and fail to establish a margin that the correlated bound proves. The admitted endpoint set retains the norm and storage constraints together with the edge intervals and memberwise mean leaf; arbitrary corners of the independent outer intervals need not obey the correlated storage bound. Global existence of the complete law makes the set nonempty for every admitted preparation member.

Handoff to the whole-sector theorem

Let B_k be the shared certified lower bound for the phase potential on every boundary face of the target sector, as in Section 26. If all strict edge margins above hold and

\[ \boxed{\qquad E_{\rm handoff}<\beta B_k,\qquad} \]

then Section 26 applies to the actual full-state endpoint of every source member. Subsequent complete evolution stays in that sector and converges on its conserved-mean leaf to its unique acute equilibrium. The admitted target already supplies the stationary geometry in that sector, so uniqueness identifies the limiting phase geometry with it. The theorem does not assume that geometry was the initial winding or that the reference and actual paths shared their earlier sectors.

certify_sine_slow_capture recomputes the preparation and slow comparison, admits the exact target, and passes producer-proved edge and correlated storage bounds to the shared sector-capture owner. The outer report retains the enclosed original endpoint time; its nested capture is neither a timestamped observation nor a sampled forecast. Outward rational arithmetic encloses the pi, trigonometric, norm and exponential quantities. Malformed inputs are rejected; insufficient strict margins return unavailable. An unavailable sufficient condition proves neither instability nor impossibility.

Fixed analytic controls and scope

The handoff tests reuse Section 27's published C5 preparation unchanged: nu_i=beta=1, e=1023/1024, w=1/1024, x_i(0)=4092*(i-2) and theta_i(0)=0 for i=0,...,4. The supplied target is q_i=(i-2)/5 and the fixed slow time is 1/16. Its initial form budget remains 160*1023^2; neither that budget nor the ratio is retuned to obtain the handoff. The analytic bounds give

\[ \begin{aligned} D_0&<0.074249,&\rho&<0.084137,\\ \frac\ell2X^2&<0.000002028,& E_{\rm handoff}&<3.461996083,\\ B_k&>3.469266270,& \beta B_k-E_{\rm handoff}&>0.00727018. \end{aligned} \]

All edge margins are positive. The same previously fixed independent radii epsilon_x=1/16 and epsilon_theta=1/65536 also pass, with total storage below 3.462017026 and margin above 0.00724924. These are analytic endpoint certificates, with no integrated reference or complete trajectory used as their producer. Initial zero winding is separately known for these preparations, so their captured nonzero sector also establishes acquisition. The general API does not assume zero initial winding and therefore does not label every capture an acquisition.

Here the declared original time is approximately 646182.7393 structural units, obtained from t=beta*pi^2*e*sigma/w^2. This fixed handoff is not an optimized entry time or a replacement for the earlier tau=100 entry certificate. Controls with resolved phase but excessive remaining form storage, and with unresolved phase margins, remain unavailable under their respective sufficient conditions.

The result connects one justified reference geometry to complete-state capture at a finite declared instant. It supplies neither autonomous support creation, selection of the preparation or target, a unique constitutive law, nor a physical identification. It also does not turn the finite comparison into a uniform infinite-time approximation.