Quantitative maintenance of formed classes under repeated probes

Prospective return-map claim and frozen protocol

The formed-class experiment admits two actual C9 preparation families, one common supplied phase probe, a distinct finite form response and recovery afterward. This gate asks for a quantitative uniform return bound that permits repetition. Qualitative existence of sufficiently long relaxation is already implied by the earlier compact capture and asymptotic stability; the new obligation is a fixed finite dwell, a strict invariant return inclusion and per-cycle response and work bounds.

Freeze the following before evaluating this certificate:

• Retain exactly the preceding simple unit C9 support, node order 0,...,8, unit capacities, beta one, e=1023/1024, w=1/1024, and complete law. In the scaled clock tau=e*t, put A=L/2, gamma=1/(1023*pi) and f(theta)=S(theta)/2. Then x'=-A*x+gamma*f(theta) and theta'=gamma*A*x.
• Retain the two original nominally phase-flat sources x_j=k*m*(j-4), with k=(1,2) and m=(2046/9)*(355/113)**2. At every node both initial residual bounds are 1/10000000000; their form and continuous phase-lift sums vanish separately in each class. The actual conserved means remain zero. Their common initial storage budget is 1000000000; their unequal nominal costs are unchanged.
• Preserve the acquisition checkpoint tau=100, actual first probe time P=200, phase increment delta=1/100, probe vector q=e_0-(1/9)*1, readout offset h=1, and independent per-readout additive error 1/10000000000. Each event is exactly theta_plus=theta_minus+delta*q, with form unchanged. No support, capacity, coefficient or clock change, state reset, adaptive controller, additional disturbance or inverse jump is admitted.
• Define each compact pre-probe set K_k on its actual zero-mean leaf by ||x||_2<=b_x,k and ||theta-theta_k^*||_2<=b_theta,k in the retained local lift chart. Its radii are exactly the upper bounds warmup_form_norm_upper_bounds and warmup_target_phase_radius_upper_bounds from the preceding frozen complete response report. The target lifts remain theta_k,j^*=k*(2*pi/9)*(j-4). Rebuild these radii from the original primitive protocol when evaluating; an incoming report or cached verdict is not an admission premise. Each set has nonempty sixteen-dimensional relative interior and contains its entire actual first pre-probe family. Put K_k^+=J(K_k) for the exact supplied phase jump J.
• Fix the common dwell D=1000000000000 in scaled clock, strictly longer than h. Apply the same jump at tau=P+n*D for every integer n>=0, and read actual node-zero form at tau=P+n*D+h. The dwell in the original clock is 1024*D/1023. This deliberately conservative mathematical dwell carries no practical-speed or laboratory-time claim.
• Require Phi_D(J(K_k)) to lie strictly inside the half-radius pre-probe set for each class: returned form and phase-error norms are respectively less than b_x,k/2 and b_theta,k/2. Require the complete intervening state to retain winding k, and recorded class contrast Y_2-Y_1>1/10000000 at every declared readout. Account for cumulative signed event work and continuous loss separately. Indefinite operation is an externally supplied repeated-intervention protocol, not finite-budget autonomous maintenance.
• Use the shared exact rational and outward dyadic128/Machin arithmetic. The strict Lyapunov method below uses lambda=1/5, Lambda=2, epsilon=lambda/4, the common worst-class acute cosine, and the shared negative-exponential enclosure with exponent at most 4096. The analytic selection of D uses the sufficient rational envelopes stated below; no dwell, rate, source, amplitude or precision sweep is admitted. Outward rounding must retain strict return margins.
• Pass only if the rebuilt formation and one-probe admissions, both uniform half-radius returns, whole-window identity, per-cycle contrast and work accounting pass. Otherwise retain the insufficient bound or separately proved obstruction without changing this protocol. Record evaluated bounds separately from this declaration.

Finite-time recovery here means return to the admitted neighborhood. Asymptotic convergence to an exact target applies after interventions stop. An invariant return set does not establish a unique periodic orbit, autonomous event selection, a fundamental law or physical constituent identity.

A strict Lyapunov bound on each retained acute chart

All coordinates below are on the mean-zero space. For class \(k\), write \(y=\theta-\theta_k^*\). Let \(\underline\eta>0\) and \(\overline\eta\) be the freshly reconstructed rational lower and upper bounds for \(\eta=\gamma^2\). Choose a rational \(\underline c>0\) no greater than either \(\cos(k\alpha+\sqrt2r)\), where \(\alpha=2\pi/9\). The preceding post-jump certificates place all of \(J(K_k)\) inside the corresponding compact, forward-trapped acute neighborhood. They apply uniformly to these sets because their proof consumes only the two pre-probe norm bounds defining \(K_k\).

The mean-free operator \(A=L/2\) is symmetric positive definite with \(\lambda I\preceq A\preceq\Lambda I\). Define auxiliary proof coordinates and a potential by

\[ \xi=A^{-1/2}y,\qquad v=\xi'=\gamma A^{1/2}x,\qquad W_k(\xi)=\frac{\gamma^2}{2} [U(\theta_k^*+A^{1/2}\xi)-U(\theta_k^*)]. \]

They introduce no new physical coordinates or evolution law. Since \(f=-\nabla U/2\), differentiating the complete original scaled rows gives exactly

\[ \xi'=v,\qquad v'=-Av-\nabla W_k(\xi). \]

On the trapped chart, \(2\underline c A\preceq\nabla^2U\preceq2A\). Consequently the following rational constants bound the transformed Hessian:

\[ \mu=\underline\eta\,\underline c\,\lambda^2>0,\qquad M=\overline\eta\,\Lambda^2,\qquad \mu I\preceq\nabla^2W_k\preceq MI. \]

The interpolation from the target stays in the same convex chart, so

\[ \frac\mu2\|\xi\|^2\le W_k(\xi)\le\frac M2\|\xi\|^2, \qquad \xi^{\mathsf T}\nabla W_k(\xi)\ge\mu\|\xi\|^2. \]

These inequalities concern the actual nonlinear phase potential. Positive tangent eigenvalues alone are not substituted for them.

Set \(\varepsilon=\lambda/4\) and use the modified energy

\[ \mathcal V_k(\xi,v) =\frac12\|v\|^2+W_k(\xi) +\varepsilon\xi^{\mathsf T}v +\frac\varepsilon2\xi^{\mathsf T}A\xi. \]

Young's inequality \(\varepsilon|\xi^{\mathsf T}v| \le\|v\|^2/4+\varepsilon^2\|\xi\|^2\) gives

\[ \begin{aligned} a_-&=\frac\mu2+\frac{\varepsilon\lambda}{2}-\varepsilon^2 =\frac\mu2+\frac{\lambda^2}{16}>0,\\ a_+&=\frac M2+\frac{\varepsilon\Lambda}{2}+\varepsilon^2,\\ \frac14\|v\|^2+a_-\|\xi\|^2 &\le\mathcal V_k \le\frac34\|v\|^2+a_+\|\xi\|^2. \end{aligned} \]

The two mixed damping terms cancel on differentiation:

\[ \begin{aligned} \mathcal V_k' &=-v^{\mathsf T}(A-\varepsilon I)v -\varepsilon\xi^{\mathsf T}\nabla W_k(\xi)\\ &\le-(\lambda-\varepsilon)\|v\|^2 -\varepsilon\mu\|\xi\|^2 \le-\kappa\mathcal V_k,\\ \kappa&=\min\left( \frac{4(\lambda-\varepsilon)}3, \frac{\varepsilon\mu}{a_+}\right)>0. \end{aligned} \]

Both classes therefore have the same proved rate bound \(\mathcal V_k(s)\le e^{-\kappa s}\mathcal V_k(0)\) through any uninterrupted post-probe relaxation interval. The chart assumption is supplied by the independently retained trapping barriers; it is not inferred from this rate inequality after leaving the chart.

A computable half-radius return and the chosen dwell

Write \(b_{x,k},b_{\theta,k}\) for the exact rational radii defining \(K_k\), and let \(b_{\theta,k}^+\) be their retained upper bound after the jump. Thus \(b_{\theta,k}^+\ge b_{\theta,k}+\delta\sqrt{8/9}\). For every state in \(J(K_k)\),

\[ \|v(0)\|^2\le\overline\eta\Lambda b_{x,k}^2, \qquad \|\xi(0)\|^2\le\frac{(b_{\theta,k}^+)^2}{\lambda}. \]

It follows that

\[ V_{0,k}=\frac34\overline\eta\Lambda b_{x,k}^2 +\frac{a_+}{\lambda}(b_{\theta,k}^+)^2 \]

is a uniform upper bound for \(\mathcal V_k(0)\). Let \(\rho_D\ge e^{-\kappa D}\) be the outward exponential upper bound actually used in the calculation. Since \(\|v\|^2\ge\underline\eta\lambda\|x\|^2\) and \(\|y\|^2\le\Lambda\|\xi\|^2\), valid returned squared-norm bounds are

\[ X_k^2=\frac{4V_{0,k}\rho_D}{\underline\eta\lambda},\qquad Y_k^2=\frac{\Lambda V_{0,k}\rho_D}{a_-}. \]

The required strict half-radius admission is exactly

\[ \frac{b_{x,k}^2}{4}-X_k^2>0,\qquad \frac{b_{\theta,k}^2}{4}-Y_k^2>0. \]

Equivalently, the following energy target is sufficient before the outward endpoint checks:

\[ Q_k=\min\left( \frac{\underline\eta\lambda b_{x,k}^2}{16}, \frac{a_-b_{\theta,k}^2}{4\Lambda}\right), \qquad V_{0,k}\rho_D<Q_k. \]

The strict margins use the full coordinates. A small phase displacement alone cannot supply the form return or repeat the response certificate.

The following coarse rational envelopes of the preceding frozen one-probe bounds motivate the fixed dwell without a horizon search: [ \begin{gathered} 9\times10^{-8}<\underline\eta\le\overline\eta<10^{-7},\qquad \underline c>1/20,\ 10^{-7}<b_{x,k}<10^{-6},\qquad 10^{-5}<b_{\theta,k}<10^{-4},\qquad |q|<1. \end{gathered} With \(\lambda=1/5\), \(\Lambda=2\) and \(\varepsilon=1/20\), these give

\[ \mu>1.8\times10^{-10},\quad a_->1/400,\quad a_+<53/1000,\quad \kappa>10^{-10},\quad V_{0,k}<10^{-3},\quad Q_k>10^{-23}. \]

For example, the form part of \(Q_k\) exceeds \(9\times10^{-22}/80=1.125\times10^{-23}\); its phase part exceeds \(10^{-10}/3200\). These are sufficient design bounds, not fitted response values. They are independently checked from the freshly reconstructed premises when certifying this frozen instance.

At \(D=10^{12}\), the exact exponential is less than \(e^{-100}\), which is less than \(10^{-40}\). One elementary verification is \(e^5>\sum_{j=0}^6 5^j/j!>100\), followed by the twentieth power. Thus the exact returned modified energy is below \(10^{-43}\), well inside both sufficient targets. Also \(\kappa D<80000/21<4096\), using \(\underline c\le1\), \(\overline\eta<10^{-7}\) and \(a_+>21/400\), so the shared exponential work limit is respected. The interval implementation still checks its own actual outward return margins: rounding an extremely small decay to a dyadic128 upper endpoint does not preserve every decimal bound on the exact exponential. No such rounded upper endpoint is silently replaced by the exact exponential.

Induction over the actual repeated experiment

The original full-law warmup places every first pre-probe state in \(K_k\). The existing probe certificate applies uniformly to all of \(K_k\), because its response-error, work and post-jump bounds use only \(b_{x,k}\) and \(b_{\theta,k}\). It does not require a particular warmup history once this state-set admission has been proved.

The post-jump barrier first places every state of \(J(K_k)\) in its class's forward-trapped acute region. The strict return margins then give

\[ \Phi_D(J(K_k))\subset\tfrac12K_k\subset\operatorname{int}_{\rm rel}K_k, \]

where \(\tfrac12K_k\) means halving both radii about the same target and zero form. This yields the equivalent post-probe inclusion \(J\circ\Phi_D(K_k^+)\subset K_k^+\), with \(K_k^+=J(K_k)\). Induction applies the identical jump, uninterrupted flow interval and readout bound at every cycle. Winding is retained throughout each interval and across every admitted jump. No exact target reset, state-dependent waiting rule or new source error is inserted.

Each cycle uses the same two-class receiver comparison and deterministic heat factor at offset \(h=1\). The original preparation uncertainty is propagated by the invariant sets; readout errors remain independent bounded observations and do not drive the state. The common contrast bound therefore applies at every cycle without assuming error cancellation or accumulating a new independent state box at each step. After a last intervention, the unchanged positive-loss law gives asymptotic recovery to the corresponding target. During indefinite repetition only neighborhood return and geometric identity are asserted; no exact equilibrium or unique periodic orbit is claimed.

Cumulative work and the boundary of conditional maintenance

For each actual event the supplied work remains exactly

\[ W_{k,n}=U(\theta_{k,n}^-+\delta q)-U(\theta_{k,n}^-). \]

The one-probe work enclosure is uniform on \(K_k\), so it bounds every \(W_{k,n}\). With original event times \(t_n=(P+nD)/e\), the exact balance across \(N\) complete cycles is

\[ H_k(t_N^-)-H_k(t_0^-) =\sum_{n=0}^{N-1}W_{k,n} -\frac e2\sum_{n=0}^{N-1} \int_{t_n}^{t_{n+1}}\|Lx_k(t)\|_2^2\,dt. \]

The values at the finitely many jumps do not affect the continuous integrals. If \([w_k^-,w_k^+]\) is the retained per-probe work interval, then supplied work for \(N\) probes lies in \([Nw_k^-,Nw_k^+]\). Its positive lower endpoint in the frozen protocol implies unbounded cumulative supplied work for indefinitely many repetitions. State storage remains bounded in the trapped charts; continuous loss accounts for the difference, not an invented reservoir.

This is quantitative maintenance under an explicitly powered schedule. The original \(10^9\) budget concerns preparation, not an unlimited probe resource. A finite external work allowance would require its own finite-cycle admission. No practical operating speed, autonomous event selection, physical identification or independently justified clock bridge follows from this return-map certificate.

Retained quantitative return certificate

The frozen protocol passes without changing its dwell or premises. The complete evaluated report retains the exact rate, pre/post-probe bounds, prerequisite evidence and return margins. Its protocol record and source archive were captured before evaluation, including the proof derivation above. The evidence manifest records the committed base and producing source overlay separately from subsequent result documentation.

The common rate lower bound is approximately \(2.0894361086775972\times10^{-10}\) per scaled time unit. At the fixed \(D=10^{12}\), the retained outward decay interval is exactly \([0,2^{-128}]\). Its zero lower endpoint is an enclosure artifact, not exact finite-time equilibration. With \(B=2^{128}\), the following retained margin intervals are \([N,N+1]/B\):

Strict return margin	Lower numerator \(N\)
Modified-energy target, winding one	14574548981967538
Modified-energy target, winding two	15020686919729160
Quarter form-radius squared, winding one	3010759966127372195420062
Quarter form-radius squared, winding two	3102921599673989472552128
Quarter phase-radius squared, winding one	310975950929306447248182251052
Quarter phase-radius squared, winding two	320494915300103012563382764566

Both form and phase half-radius returns are strict. Therefore every admitted source member can undergo every declared repetition while retaining its class winding. The rebuilt per-cycle recorded contrast keeps the preceding lower bound above \(3.4\times10^{-7}\), exceeding the frozen \(10^{-7}\) requirement at every cycle. The separately retained positive work intervals apply to each intervention and scale linearly as bounds on cumulative supplied work.

The primitive-only assess_sine_formed_class_maintenance returns SineFormedClassMaintenance. It rebuilds formation, the first probe, the pre-probe radii and the post-jump trapping premises from the original scalar inputs. Return bounds require those trapping premises; an acute cosine by itself cannot certify an out-of-chart trajectory. The exponential work limit consumes the proved slow rate times the dwell, rather than the initial fast semigroup gap. The contract, guide and independent controls retain the execution boundary and unavailable cases.

This certificate proves a uniform return inclusion for the full nonlinear state sets. It was not obtained by sampling a long pulse train. It supplies no practical-speed estimate, finite total work budget, autonomous intervention mechanism or physical identification.