Reduced ports for actually formed sine classes¶
Prospective transfer claim and frozen protocol¶
The formed-class contact result establishes one finite donor-class response against receiver class one. This gate asks whether separately derived component models predict an additional contact against receiver class two, without refitting their parameters. The reduction retains five reflection-even spatial coordinates in each of two channels per component, an exact nonlinear bridge, and explicit bounds for all discarded fine-state effects. It is neither an exact nonlinear symmetry quotient nor a replacement of each component by one scalar form/phase pair.
Freeze the following before evaluating the held-out response:
- Retain the complete positive-loss sine law, structural clock \(\tau=et\), \(e=1023/1024\), \(w=1/1024\), \(\gamma=1/(1023\pi)\), unit capacities and beta one. Two simple unit C9 cycles have local node order \(0,\ldots,8\). A single supplied unit bridge joins their central nodes \(4\); afterward the two port degrees are three and all other degrees two.
- The donor cases remain \(k=1,2\); the held-out receiver is class \(\ell=2\) in both cases. For either component of class \(j\), retain source form \(jm(i-4)\), with \(m=(2046/9)(355/113)^2\). Donor nominal phases start at zero; receiver nominal phases start at the supplied common origin \(\varphi=1/1000\). This offset is a preparation premise, not a rotation or reset performed at contact.
- Each component retains initial form and continuous phase-lift residual bounds \(10^{-10}\), with each residual row summing to zero separately. All allowed even and odd perturbations remain. Retain each component's original common storage ceiling \(10^9\); the nominal preparation costs need not agree. Both full source families, including the receiver-class-two source, must be freshly admitted under the original formation theorem.
- Retain acquisition time \(100\), radius \(r=1/12\), and the uninterrupted unprobed relaxation duration \(D=10^{13}\). Contact occurs at \(\tau_c=100+D\). Retain the endpoint Euclidean bounds \(\epsilon=10^{-32}\) separately for each component's form and phase error, proved with the guarded exact decay bound \(2^{-512}\) and \(512\le\kappa D\le4096\). No target reset or previous maintenance probe is admitted.
- Derive one component matrix for each class directly from its C9 geometry and the declared one-contact mobility. Retain the radial order \((4),(3,5),(2,6),(1,7),(0,8)\), in form and in real phase deviations from that class's fixed twist. The class dependence is \(\cos(2\pi j/9)\), derived from the internal sine field. Use these same two component matrices and the same two-channel port convention in every composition; no receiver-specific parameter or response fit is admitted.
- The predictive system has twenty retained coordinates, ten per component. Its internal phase field is the derived linear tangent, while its bridge sine current remains exact and nonlinear. Retain the initial-state source term in the port representation. For the nominal prediction, use the limiting comparison coordinates and enclose the actual source deviations through the proved error terms; never reset the actual fine state.
- At elapsed scaled time \(h=1/100\) after contact, read actual receiver-port form with independent additive error at most \(d=10^{-30}\) per case. Predict recorded donor-two minus donor-one contrast greater than \(10^{-25}\). Derive its fourth-order leading term from the composed reduced matrices, retaining their first three derivative cancellations. Do not substitute a coefficient or verdict from the earlier full-state contact certificate.
- Bound the finite reduced prediction by its correlated semigroup tail and reduced nonlinear remainder. Add the separately proved lifted-surrogate discrepancy, complete actual preparation error and independent readout error. Let \(E_{\mathrm{total}}\) be the sum of these nonnegative error bounds and let \(L_{\mathrm{full}}\) be the lower endpoint of the final outward recorded-contrast enclosure. Require \(L_{\mathrm{full}}>10^{-25}\) and the strict error-fraction condition \(E_{\mathrm{total}}<(1/4)L_{\mathrm{full}}\). The denominator is a certified full-law recorded gap, not its formal leading amplitude.
- Retain the supplied contact-work allowance \(10^{-6}\), both actual cycle identities and full joined-family recovery. Rebuild the contact storage, quotient radius, weighted means and barrier for donor \(k\) and receiver two. Removing the bridge must leave receiver evolution independent of donor data; retain the preparation/readout control band.
- Use exact rational primitives and outward dyadic128 trigonometric intervals, keeping the tiny warmup squared bounds as exact rationals. Preserve the protocol and producing source before the first reserved evaluation. There is no trajectory integration, parameter search, new horizon sweep or precision search. If any prerequisite, error fraction, identity, work or contrast condition fails, retain that outcome without changing this protocol.
The earlier receiver-one result is a retained reference, not a calibration dataset for these component matrices. The new question is the controlled transfer of a derived reduced model to one declared receiver change. The structural dwell, initial source control and tiny readout budget have no independent laboratory realization here.
The retained component state and one-contact normalization¶
Use local reflection \(R:i\mapsto8-i\), fixing the central port \(p=4\). For each channel, let \(S\) take the values or averages on the five layers
Let \(T\) lift a five-vector by assigning its value to each node in its layer. Then \(ST=I_5\) and \(TS=(I+R)/2\). Both maps are contractions in the maximum norm. The reduced state is \(z_k=(\bar x_k,\bar y_k)\in\mathbb R^{10}\), where \(\bar x_k=Sx_k\) and \(\bar y_k=S(\theta_k-\Theta_k-o_k\mathbf1)\). Here \(\Theta_{k,i}=k\alpha(i-4)\), \(\alpha=2\pi/9\), and the declared common origin \(o_k\) is zero for the donor and \(\varphi\) for the receiver. This projection defines retained coordinates of a fine state. Evolving equation (2) instead defines a surrogate trajectory, which generally differs from the projection of the evolving nonlinear fine state.
The phase deviations are real retained lifts. Wrapping them before applying the internal linear operator changes this model. The internal reference twist and known origins must be restored when forming a circular bridge difference.
The component has one admitted unit contact. Its internal mobility is \(K_*=\operatorname{diag}(1/2,\ldots,1/3\text{ at }p,\ldots,1/2)\), even though \(L_{\mathrm{C9}}\) contains only its internal edges. Its reduced internal operator is
The edge joining nodes zero and eight lies inside the outermost layer and contributes zero on reflection-even vectors. The layer weights are \((3,4,4,4,4)\), of total nineteen. This is not the isolated-cycle normalization \(L_{\mathrm{C9}}/2\). A different number or arrangement of contacts would require its own mobility and reduction admission.
Every internal edge of the class-\(k\) twist has cosine \(c_k=\cos(k\alpha)>0\). Linearizing only those internal sine currents gives the following component model:
The two input channels have the definite convention
They are not two unspecified forces. In the actual composition they are determined by the other component's retained port outputs.
For \(z_k=(\bar x_k,\bar y_k)\), define
The exact input/output identity for this component model is
Thus \(G_k(t)=Ce^{J_kt}B\) is its derived two-channel port kernel. It depends on class, supplied law and contact normalization, not on a fitted receiver response. The initial-state term is essential: the kernel does not determine the initial internal shape.
The twenty-coordinate realization retains both component means. Writing \(m_*=(3,4,4,4,4)^{\mathsf T}\), equation (1) gives
Opposite inputs on the two components preserve the combined degree-weighted charges, while the separate component means can change. Holding them independently fixed after contact would introduce a different law.
This is a strict reduction from thirty-six fine coordinates to twenty retained coordinates, or an equivalent port history with its initial source. No minimality theorem is asserted. In particular, writing a kernel is not evidence that all hidden initialization can be discarded or that one scalar state per formed class closes the dynamics.
Autonomous composition keeps the bridge nonlinear¶
Compose the two component models using
The resulting twenty-coordinate system is autonomous after the supplied support event. No observed response is installed as an input, and no port is held to an externally selected future history. Only the choice \(\ell=2\) of the receiver's already defined component matrix changes in the held-out experiment.
Put
In ten-component vectors \(\bar x,\bar y\), (2)--(4) become
The first row is linear in the internal deviations but not in the bridge phase difference. Replacing that sine by a fitted gain or by its initial value would be another approximation requiring an additional error term.
Why the discarded nonlinear modes remain controlled¶
The exact fine sine law does not preserve this even subspace in general. Around a winding-\(k\) twist, an internal edge contributes
The terms proportional to \(\sin(k\alpha)\delta^2\) can generate reflection-odd deviations under an even drive. Linear reflection symmetry alone cannot prove an exact nonlinear quotient.
For the error proof, extend (5) to all nine coordinates of each component: use the linearized internal \(c_kK_*L_{\mathrm{C9}}\) and the exact bridge sine, without projecting either channel. This thirty-six-coordinate surrogate is a proof device. Its matrices commute with each component reflection and its bridge depends only on the fixed ports. Its ideal initial deviations are zero, so its ideal flow stays even and is exactly the lift of (5). The actual predictive realization remains twenty-dimensional.
Compare this lifted ideal surrogate to the ideal full sine flow with the same zero form and target twists plus receiver origin \(\varphi\). Let \(y_{\mathrm{fine}}(t)\) be phase displacement from those initial comparison phases and put \(s=|\sin\varphi|\). The complete joined field has initial normalized sine forcing \((\sin\varphi/3)(e_a-e_b)\), irrespective of \(k,\ell\). The contraction/Volterra argument in the contact remainder proof uses only this forcing, the normalized row bounds and \(\eta=\gamma^2\). It therefore applies with receiver class two:
At the actual fine phase state, the omitted internal Taylor terms have normalized magnitude at most \(2\|y_{\mathrm{fine}}\|_\infty^2\). They enter the physical form row multiplied by \(\gamma\) and do not directly enter the phase row. Both the full field and the extended surrogate have maximum-norm Lipschitz constant at most \(2+2\gamma<3\). Thus for \(0<h<1/3\), their full-state discrepancy obeys
Taking the supremum in the integral difference inequality proves the denominator: \(E_h\le3hE_h+\int_0^h\|\mathrm{defect}\|\,dt\). The same argument applies to every prefix of the window. Generated odd modes and their subsequent feedback are included through the full fine trajectory in (6), not set to zero.
For each actual formation image, arbitrary initial even and odd errors remain within the endpoint bound \(\epsilon\). Comparison of the actual full flow to the ideal full flow gives the separate maximum-norm bound
This is precisely the complete-field argument in the actual-state contact handoff. Combining (7) and (8) bounds the actual receiver form against the nominal reduced flow without resetting any actual coordinate. Unknown initial shape is handled by a proved family enclosure, not inferred from winding, means or port values alone.
A finite prediction computed from the reduced matrices¶
The analytic evaluator of (5) starts with \(\bar x=\bar y=0\). Put
Direct differentiation of the reduced rows gives
The receiver is reduced coordinate five. Because its row of \(\bar D\) vanishes, the two donor cases have identical receiver derivatives through order three. Their fourth derivative is obtained from the rational reduced matrices through
For example, (1) gives \((\bar A^+q)_0=4/3\), \((\bar A^+q)_1=-1/2\), and \((\bar D\bar A^+q)_0=11/9\). The receiver bridge entry is \(-1/3\), so \(\chi_4=11/81\). This value is derived by composing the reduced operators. It is not a copied parameter from the earlier contact report. The changed receiver class also changes the individual derivative rows in (9), even though it cancels from this donor contrast.
Let \(d_c=c_1-c_2>0\). The exact reduced leading expression is
Its finite response is not identified with this derivative alone. The normalized joined matrix \(\bar A^+\) has a contracting nonnegative semigroup, and \(\|\bar A^+\|_\infty,\|\bar D\|_\infty\le2\). The correlated first-\(\gamma^2\) semigroup expansion has the receiver terms
All \(p=0\) terms vanish. Bounding the \(n=p+q\ge2\) tail uses \(n\) terms and successive ratio at most \(h/2\). Restoring the physical factors gives
This derives the tail from the reduced rows and their shared forcing, rather than subtracting two unrelated marginal estimates.
The reduced bridge remainder has normalized Taylor bound at most \(2\|\bar y\|_\infty^2\); this deliberately conservative bound also covers the sharper bridge-only constant. The same Volterra bootstrap used for (6), now on the reduced rows, bounds the paired nonlinear evaluation remainder by
Equations (12)--(13) bound the finite reduced-flow contrast around (11). Equation (7) has a different responsibility: it bounds the reduced-flow approximation to the original nonlinear fine law. Neither error can stand in for the other.
Recorded transfer criterion and actual joined identities¶
Use independently admitted errors in both donor comparisons. An upper bound for the total error around the exact expression \(P\) is
The five terms retain finite reduced evaluation, omitted nonlinear fine-state effects, arbitrary initial preparation error and independent observation error. No unknown error correlation is used to reduce this sum.
The implementation encloses \(P\) and forms the outward interval for \(P+[-E_{\mathrm{total}},E_{\mathrm{total}}]\), using outward upper bounds for every error term. Let \(L_{\mathrm{full}}\) be its final lower endpoint. The strict frozen rule is
An outward enclosure of the fraction margin must also retain a strictly positive lower endpoint. Consequently rounding a positive exact margin to zero yields an unavailable certificate. The relevant denominator is the final recorded gap; neither a plotted midpoint nor the formal leading term can replace it.
The fixed analytic design has room for this extra obligation. Before interval rounding, at \(h=1/100\),
The bounds \(1/4000<\gamma<1/3000\), \(d_c>1/2\), \(1/2000<s<1/1000\), with the frozen \(\epsilon,d\), bound the other terms in (14) by less than \(P/1000\). Therefore
The same coarse bounds give \(P>(1.5625)\,10^{-25}\), so \((5/6)P>10^{-25}\). These sufficient analytic inequalities select no new horizon, offset or precision. The reserved evaluation must still verify its actual outward endpoints and all prerequisite admissions.
The original formation theorem covers both classes separately. Receiver two uses its own freshly derived class-two source bounds; the common worst-class Lyapunov estimate gives the same \(\epsilon\) handoff after the declared unprobed dwell. The support and port normalization are unchanged, and both internal twists have edge magnitude at most \(4\pi/9\). Thus the full joined proof applies after explicitly replacing the receiver-one twist by receiver two. In particular it again gives
against the barrier \(r^2/1620\), and work bound \(2\epsilon^2+(\varphi+2\epsilon)^2/2\). The complete actual families retain winding pairs \((1,2)\) and \((2,2)\), respectively, with recovery on their actual postevent weighted-mean leaves. These are full-law claims with matched hypotheses; they do not follow merely from the reduced output error.
The joined weighted means remain those of the actual nodal values, with total weight thirty-eight. The class-two receiver's internal twist still has zero weighted mean by reflection. The degree reweighting, positive supplied event work and disconnected control bound \(2\epsilon/(1-3h)+2d\) therefore retain their original formulas. No report from the earlier receiver-one experiment supplies a cached passing verdict for this new instance.
Interpretation and limits of the component law¶
The new claim is conditional predictive reuse: the same independently derived class matrices and exact bridge convention are composed with a different receiver, with a uniform finite error for actual formed families. The model does not discard memory by definition. Equation (3) retains its initial-state source and the dynamics of its internal even coordinates; equations (7)--(8) retain the effects of omitted odd and uncertain initial coordinates.
The finite derivative expansion is one certified evaluation of this reduced dynamical model. It is not a substitute for defining the model's retained state, port feedback and initialization. An insufficient error bound would not itself disprove that a useful reduction exists; an actual response outside a claimed enclosure would instead challenge its premises or proof.
The positive-loss microscopic law, source preparation, contact normalization and work remain supplied. The very long structural warmup and small readout budget are still mathematical premises. No minimal realization, universal constituent law, autonomous contact selection or physical sensor interpretation follows.
The shared
reduced-port owner
implements evaluate_sine_reduced_class_ports as a detached instantaneous
evaluation of the twenty-coordinate field, and
assess_sine_reduced_class_ports as its primitive-only finite certificate.
The latter rebuilds the same matrices, computes the individual derivative
rows and contrast coefficient, and consumes freshly derived formation
and contact bounds. It accepts no incoming report as evidence and
does not integrate a reduced or full trajectory.
The contract
and independent controls
retain admission, error-budget and usage boundaries.
Retained evaluation of the unchanged transfer protocol¶
The complete saved report returns certified_reduced_class_ports with no unavailable reasons. The frozen stopping rule passes: the held-out receiver-two contrast exceeds \(10^{-25}\), and the total error is strictly below one quarter of the final outward full-law recorded gap. Both unprobed handoffs, the two actual joined identities and the supplied contact-work allowance also pass.
Every endpoint in the following table is the displayed integer divided by \(2^{128}\). The fraction margin is \((1/4)L_{\mathrm{full}}-E_{\mathrm{total}}\), using the final outward lower endpoint as declared.
| Retained quantity | Lower numerator | Upper numerator |
|---|---|---|
| Recorded receiver-two contrast, donor two minus donor one | 289363867316106 | 397865303363279 |
| Disconnected recorded contrast | -687580866 | 687580866 |
| Strict error-fraction margin | 18090248805440 | 18090248805441 |
The exact recorded lower endpoint is therefore approximately \(8.503639784054314\,10^{-25}>10^{-25}\). The saved exact rational error bounds have the decimal summaries
These summaries do not replace the exact retained fractions. The last line isolates the paired surrogate-to-full discrepancy; it is only one of the five terms in the total error. The disconnected record band remains below \(10^{-29}\) in absolute value.
The reduced matrices produce the geometric coefficient \(11/81\) and retain the changed individual receiver-two derivative rows. Their parameter values were not recalibrated from this response. The complete actual winding pairs \((1,2)\) and \((2,2)\) retain their full-law identity and recovery certificates; those certificates do not follow from the reduced output interval alone.
The frozen protocol, producing source archive and evidence manifest retain the source and pre-evaluation declaration. The archive contains this owner's protocol and complete derivation before the first reserved evaluation. That prefix remains byte-identical; only this result section replaces the former unevaluated-status paragraph. The endpoints above were read through the shared JSON reader from the saved report, without reassessment or producer regeneration.
This certifies one controlled transfer of the smaller dynamical model under the supplied complete law and retained full preparation families. It is not an exact nonlinear quotient, a minimal realization or an empirical validation. The uninterrupted structural dwell \(10^{13}\), readout error \(10^{-30}\), supplied bridge and contact work remain unchanged mathematical premises, with no laboratory clock or sensor bridge established.