Effective memory of an acquired sine-class mediator¶
The acquired-class response establishes a finite donor-induced receiver contrast on the three-C9 path. This owner derives a causal donor-receiver description after hiding the middle component. It retains that component's class and full initialization, and bounds the nonlinear reduction error without resetting reached states. The supplied law, capacities, interface and structural clock remain fixed. No new response campaign or physical identification is implied.
The component port kernel, linear coordinate-memory method and nonlinear environmental memory already establish their own elimination identities. The additional result here concerns the actual two-contact acquired C9 mediator, its matched nonlinear source histories and a finite error small enough to preserve the certified class contrast. A generic convolution identity alone would not meet that obligation. Task status remains in the execution plan.
Full state and declared visible observation¶
Use the same ordered 27-node graph, unit capacities and class tuples \((1,k,1)\), \(k=1,2\), as the response owner. Components are donor, mediator and receiver, each with local nodes \(0,\ldots,8\); both contacts join their central nodes 4. Actual port degrees are \((3,4,3)\). With \(e=1023/1024\), \(\tau=et\), and \(\gamma=1/(1023\pi)\), the full rows are
where \(K\) is inverse actual degree and \(L\) is the unit graph Laplacian. The phases are circular; deviations use the continuous lifts \(y=\theta-\Theta_k\) around the aligned class target. Form and phase are in the same normalized structural units as the response owner, with angles in radians and primes denoting the declared structural time.
The full deviation \(z=(x,y)\) has 54 coordinates. Retain all donor and receiver forms and phase deviations as the 36-coordinate visible state \(v\), and hide all 18 mediator coordinates as \(h\). Each block orders forms before phases and retains the original local node order. This is a coordinate partition, not a replacement of either visible component by its mean or central port. Hidden coordinates remain parts of the complete autonomous state. Keeping all eighteen hidden coordinates is not a minimality claim. Eight mediator reflection-odd tangent coordinates are invisible through its central contacts, while nonlinear residuals can still couple to those coordinates. Their contribution is retained in the full-state error bound.
Let \(C_k=KL_{c,k}\), where cycle phase Hessian weights are \(\cos\alpha,\cos(k\alpha),\cos\alpha\), the two contact weights are one, and \(\alpha=2\pi/9\). In the original unscaled coordinates,
This decomposition is exact; the residual is not discarded. It obeys \(\|n_k(z)\|_\infty\le2\gamma\|y\|_\infty^2\), from the scalar sine Taylor remainder and actual degree normalization.
Exact tangent memory and the nonlinear residual¶
Partition (2) as
The matrices \(E,B,C\) are class independent. All class dependence in the tangent generator is confined to \(D_k\), because only the mediator's internal phase Hessian changes. No changed contact coefficient or hidden clock is supplied.
Variation of constants gives the exact tangent law
The full tangent initial state is still \((v(0),h(0))\). A class label does not determine its hidden initial source. Supplying a visible history externally would describe a driven subsystem, not the autonomous composed solution without its feedback equation.
For the complete nonlinear law, the exact right-hand side of (4) also contains
Its hidden state satisfies the corresponding variation-of-constants row from (3). Equations (3)--(5) therefore retain the nonlinear history and initialization. Omitting (5) is a controlled tangent approximation only after the error argument below; it is not an exact autonomous nonlinear convolution with a fixed kernel.
The cross-port kernel exposes where the observed class information enters. For receiver and donor central form selectors \(e_{R_x},e_{D_x}\),
This is the same actual-degree walk as in the response theorem: \(1/4\) at mediator incidence, \(1/2\) for its internal central diagonal and \(1/3\) at receiver incidence. The common cross entry at zero lag is \((1-\gamma^2)/12\). Thus the class-dependent kernel derivative gives the cubic paired contrast \(a\gamma^2[\cos\alpha-\cos(2\alpha)]t^3/144\), while lower orders are class blind. Neither this onset order nor causal memory defines a positive waiting time before influence begins.
Finite nonlinear error with the reached initialization retained¶
Suppose an actual reached state has maximum-norm form and target-phase errors at most \(\epsilon\). Continue it with and without the same donor central form jump \(a\ge0\); both continuations retain their complete visible and hidden initial states. This hypothesis is an endpoint ball for the following bound. Proving that the original preparation reaches it is the separate matched formation/handoff obligation, already owned by the response theorem. The implementation retains that owner's sufficient per-component Euclidean norm budgets, which imply the maximum-norm premise used here.
For \(0\le t\le H\le1/4\), let \(g\) be an outward upper bound for \(\gamma\), and put
Diffusion contraction and the global sine Lipschitz bound imply
The numerator retains both original channels; no phase uncertainty is divided by \(\gamma\). Compare the actual full trajectory with the tangent trajectory initialized at that same complete state. Their initial difference is zero, \(\|J_k\|_\infty<3\), and integration of the quadratic residual in (2) gives the whole-window error
Setting \(a=0\) gives \(Q_0,E_0\) for the actual unprobed baseline.
Within each class, tangent linearity cancels the same complete initial state exactly in the probe-minus-baseline response, including its hidden source term. That cancellation does not erase the initialization of either individual trajectory or cancel the nonlinear residual. The error between one actual paired receiver response and its tangent memory prediction is at most \(E_a+E_0\). For both independently prepared classes and four scalar recording errors of magnitude at most \(\delta\),
where \(P\) and the correlated tangent tail \(B_T\) are the independently derived response coefficients. Equation (10) adds no spatial truncation: (4) is exact for the full tangent system. It also adds no separate \(4\epsilon\) ideal-initialization penalty, because both tangent continuations match their actual source.
At the previously frozen values \(a=10^{-3}\), \(H=10^{-4}\), \(\epsilon=10^{-32}\), \(\delta=10^{-30}\), elementary bounds give \(g<1/3000\), \(Q_a<101a/100\) and \(Q_0<2\epsilon\). Substitution into (9), with \(1-3H=9997/10000\), yields
The previous elementary trigonometric inequalities give \(P>3\,10^{-25}\) and \(B_T/P<13/100\). Therefore (10) has recorded lower bound greater than \(2.6\,10^{-25}\), whereas the nonlinear stationary-comparator allowance below is less than \(5\,10^{-30}\). The independent budget controls check these rational inequalities without generating a response. This analytic transfer shows the memory description retains the already certified finite effect. It supplies no new reserved assessment or replacement of the archived outcome.
A declared instantaneous comparator loses the class contrast¶
Freeze the visible tangent state and solve the hidden stationary rows. Their unique solution is
For form, the hidden Dirichlet Laplacian is the internal cycle Laplacian plus two central contact terms. Its quadratic form is positive definite, and the midpoint constant solves every row. The phase Dirichlet Laplacian has the same contacts and a positive multiplier \(\cos(k\alpha)\) on internal cycle edges, so the same argument proves its unique midpoint. The hidden tangent generator is therefore invertible.
Consequently the Schur comparator \(E-BD_k^{-1}C\) is identical for both classes. This comparison keeps the inherited visible mobility, including degree three at its ports; it does not renormalize a graph after deleting the mediator. Its within-class paired receiver response, and hence its cross-class contrast, is class independent. It cannot reproduce the positive contrast retained by (10).
Each comparator starts from the visible projection of the actual postevent state. Its midpoint reconstruction generally replaces the hidden state: even the ideal donor-only kick has actual hidden form zero while (11) assigns the value \(a/2\). It therefore does not share the complete initialization of the true memory law. The reduced visible mobility remains the inherited one, with central denominator three rather than a newly normalized degree \(5/2\). Moving the hidden midpoint can change the reconstructed full degree-weighted charge; full-law storage, charge and identity statements must not be transferred to this comparator.
The stationary hidden section is not asserted invariant when the visible boundary moves. This comparator result concerns precisely (11), not every possible instantaneous model or every finite-error approximation. Storage minimization, a zero-frequency coefficient and a causal response law are different constructions.
There is also an exact nonlinear stationary section on the corresponding acute lift chart: retain each hidden reference twist and set its phase deviation and form to the two visible central midpoints in (11). Internal sine currents cancel, and the two central bridge gaps are opposite, so all hidden nonlinear rates vanish. The grounded phase Hessian is positive definite on this chart, making that stationary hidden phase unique there. This is a local branch statement, not a global minimizer across windings. The midpoint section generally moves when the visible state moves; its instantaneously zero hidden rates do not track that motion.
Substituting this section into the visible nonlinear rows gives a second declared instantaneous comparator. Its field is class independent, with the same inherited visible mobility. Its paired response is identical between classes only if the visible initial states also coincide. For independently uncertain visible source states, its recorded class contrast is instead contained in
This follows by comparing its four histories with the common ideal visible initialization under the same Lipschitz bound three. For the linear Schur comparator, exact within-class source cancellation gives the sharper recorded null interval \([-4\delta,4\delta]\). The bound in (10) separates the prior finite design from both nulls. Neither comparator is installed into the engine or supplies a new formation, work or identity proof.
An acquired-family obstruction to visible-only closure¶
An obstruction for arbitrary endpoint perturbations would not by itself establish an obstruction on actual formation images. Here the original source has a relative interior in the sixteen-dimensional plane of zero-sum mediator form and phase residuals. The isolated complete sine field is globally Lipschitz on continuous phase lifts and preserves both component means. Its finite-time flow is a smooth invertible map of that plane onto itself. Thus the image of any source interior is open in the same relative plane, even after the long but finite formation/dwell time. This argument asserts existence of a sufficiently small perturbation; it supplies no numerically resolved radius for that open image.
Choose one reached interior state of either fixed mediator class. Perturb only its hidden local form 3 by \(+s\) and local form 2 by \(-s\), with sufficiently small \(s>0\), leaving all other forms and all phases unchanged. Openness gives another actual admitted source whose unprobed formation trajectory reaches this perturbed endpoint. The donor and receiver histories before contact are unchanged. The hidden perturbation has zero component sum; its two nodes both have degree two, so the joined degree-weighted form charge is also unchanged. The phase charge and mediator central form/phase are identical. Supply the same contact and optional donor probe to both states, preserving these equalities.
All 36 visible coordinates and their initial first derivatives agree. At the hidden central node, the changed form derivatives are
Consequently the receiver central accelerations differ by
The actual bridge phase enters this equality; it need not equal its ideal target. Positivity follows from \(\gamma<1\). Contact and donor-probe work are equal for this pair because every participating central value is unchanged. The separate original source costs remain within their admitted budgets and are not asserted equal.
No single regular autonomous first-order law of the visible coordinates, the class label and conserved charges can reproduce both future visible trajectories on this acquired family: the identical initial observations would have one locally unique continuation, whereas (14) separates them. Even retaining the mediator central pair does not close its evolution without further hidden information, since that same pair has the different derivatives (13). This is not an obstruction to memory, a larger retained state, separately constrained preparations or a controlled finite-error approximation.
Admission and inherited obligations¶
The conditional endpoint-ball bound supplies no formation, contact work, probe work or identity certificate. Applying it to the acquired family requires the same original source, finite formation and dwell, actual handoff, complete law, support, capacity and event premises as the retained response owner. Those proofs keep their own source costs and conserved-mean leaves. They are not consequences of a kernel or of an available matrix decomposition.
Matrix coefficients involving \(\gamma\) and the class cosines must retain their ideal algebraic definition and outward parameter enclosures. An exact identity for a binary64-materialized rational generator certifies that supplied generator, not automatically this transcendental one. The shared coordinate-memory machinery can be reused within that declared coefficient boundary; no fitted kernel or hidden initial state is supplied by an export.
The memory reader and bound calculator
separate those responsibilities. derive_sine_class_mediated_memory
rebuilds the fixed rational spatial blocks, named scalar factors, kernel
moments and stationary map without evaluating an exponential.
bound_sine_class_mediated_memory accepts only a probe amplitude, horizon,
conditional endpoint radius and scalar readout error. Its finite bounds
and any positive conditional contrast do not admit an actual preparation.
The contract
and guide
retain that distinction and the two comparator scopes.
The theorem derives an interaction description of organized nodal patterns and preserves a class-dependent finite observable under explicit errors. It does not select the substrate, law, supplied contact occurrence, physical measurement map or a fundamental-particle interpretation. All earlier frozen proofs, protocols and assessed responses remain unchanged.