Two-probe superposition on an acquired sine composite¶
The class-mediated response and effective-memory theorem derive a class-dependent interaction from the complete three-C9 law. This owner asks what two successive supplied form inputs add beyond that linear memory. It derives a nonlinear onset, then proves a finite observation obstruction at a declared error budget. No trajectory or reserved response is evaluated, and earlier frozen evidence is unchanged.
The obstruction concerns four final scalar observations. It does not make the nonlinear and tangent laws identical, or identify their complete continuous histories. The execution plan owns subsequent admissions.
F1-F2: complete state, events and matched histories¶
Use the response owner's ordered donor, mediator and receiver C9 rings, with central contacts at local node 4 and classes \((1,k,1)\), \(k\in\{1,2\}\). Fix one class throughout the four continuations. Actual central degrees are \((3,4,3)\), the total degree mass is 58, and all capacities are one. In the inherited structural clock \(\tau=et\), \(e=1023/1024\), \(\gamma=1/(1023\pi)\), the complete 54-coordinate law is
Here \(L\) is the unit-support Laplacian, \(K\) is inverse actual degree, and \(S_i(\theta)=\sum_{j\sim i}\sin(\theta_j-\theta_i)\). Form is signed and phases are circular. Calculations use continuous deviations \(y=\theta-\Theta_k\) around the aligned acquired target; angles are radians and form uses the previous normalized structural units. Neither a laboratory clock nor a physical measurement map follows.
Start all four histories from the same complete reached state \(z_0\), including every hidden mediator coordinate. Its form and phase errors in each component have Euclidean norms at most \(\epsilon\), with each component's unweighted residual sums zero. These are conditional endpoint premises; the calculator below does not establish acquisition. Reusing the actual formation family requires the unchanged original preparation, formation, dwell and handoff proofs. No ideal target is substituted for the reached state.
Let \(a,b\in\mathbb R\), \(0\le s\le T\le1/4\). A supplied event of amplitude \(q\) changes only \(x_D\) by \(q\), where \(D=(0,4)\). Declare these four continuations:
| History | Event at 0 | Event at \(s\) |
|---|---|---|
| \(00\) | none | none |
| \(10\) | \(a e_D\) | none |
| \(01\) | none | \(b e_D\) |
| \(11\) | \(a e_D\) | \(b e_D\) |
All intervening rows follow (1). In particular, history \(01\) reaches its second event by unprobed evolution from \(z_0\), while \(11\) retains the entire first-probe state. When \(s=0\), the two declared form jumps compose without a flow interval. No state, mean or hidden initialization is reset.
At the common final time, read receiver central form \(R=x_{(2,4)}\). The true and recorded mixed statistics are
There are four final readings, not three successive readings of one trajectory. Their total scalar error is at most \(4\delta\). Every fixed linear tangent law, including its exact causal memory with hidden initialization retained, gives \(M=0\) for any common full initial state. This follows directly by linearity of both flow and jumps.
F3: symmetry and the first allowed mixed response¶
Reflect every ring by \(\mathcal R:j\mapsto8-j\). It fixes both contacts and the two observed central nodes. The centered twist satisfies \(\mathcal R\Theta_k=-\Theta_k\). The complete law in deviations is equivariant under
The ideal target and the nominal formation source are fixed by this transformation; the latter has form proportional to \(j-4\) and flat phase. Thus their unforced nominal histories remain fixed. A central form kick reverses sign under (3), as does the central form readout. Consequently the nominal response is odd under simultaneous reversal \((a,b)\mapsto(-a,-b)\). All even total amplitude degrees vanish, and inclusion-exclusion removes pure single-input terms. A quadratic mixed term \(ab\) cannot survive this observation symmetry.
This does not make the complete state reflection even. Quadratic reflection-odd internal corrections are allowed and enter later output. Nor is an arbitrary admitted source residual fixed by (3): its transformed source can be a different member of the family. Actual residuals are bounded below without imposing this symmetry on them.
An exact local identity exposes the nonlinear response. Just before the second event, let \(d_a\) and \(d_0\) be the actual lifted bridge gaps \(\theta_{M,4}-\theta_{R,4}\) on the first-probe and unprobed histories. The mixed full state and its first derivative vanish at \(s^+\), whereas receiver mixed curvature is
Indeed a donor form jump \(b\) changes mediator form rate by \(b/4\) and phase rate by \(-\gamma b/4\). Its change in receiver form curvature is therefore \(b(1-\gamma^2\cos d)/12\); subtract the two carried histories. This equality uses actual bridge gaps, not fitted or inferred coefficients.
The sign can also be certified on an actual uncertain source family. Choose \(0<\gamma_-\le\gamma\le g<1/3000\), and define
Then \(|d_0|\le B_0\) and \(m\le|d_a|\le v\). To see this, the ideal first-probe history obeys \(\|x'(t)\|_\infty\le2Q(1+2g^2t)\). The difference of the two bridge phase-rate rows has norm at most \(4g\). Integration around the ideal initial slope \(-\gamma a/4\) bounds its gap error by \(4gQs^2+(8/3)g^3Qs^3\), below the first term in \(E_d\). The original-state comparison supplies the additional \(B_0\), without requiring symmetric residuals.
If \(v\le1\) and \(m^2/3-B_0^2/2>0\), elementary cosine bounds yield the strict oriented certificate
If \(a=0\), \(b=0\) or \(s=0\), this instantaneous mixed curvature is exactly zero. A failed sufficient sign bound is unavailable, not zero. This right derivative is a mathematical feature of the carried flow, not a measured derivative or a finite endpoint separation.
For the ideal target only, the first-time-order receiver bridge gap is \(-\gamma[a t+b(t-s)_+]/4\). Its cubic sine contribution to the receiver form row is \(\gamma^4[a t+b(t-s)_+]^3/1152\). Subtracting the two single-input histories and integrating gives the first joint-time term, with \(u=T-s\),
For fixed amplitudes, scaling both \(s,T\) by \(\lambda\) makes this term order \(\lambda^4\), with higher-time-order remainder. It is cubic in amplitudes and class blind at this order. At a fixed positive delay, the immediate postevent onset is generally quadratic in \(u\), as (4) shows. These are different expansions. Neither a formal leading term nor its positive sign at positive amplitudes supplies a finite lower bound or observable separation by itself.
F4: a finite symmetry-aware bound¶
Use a tight outward bound \(\gamma\le g<1/3000\), and set
These denominators are positive on the admitted time domain. Consider an ideal-target history with absolute pulse budget \(A_*=|a|+|b|\), or the corresponding smaller single-input budget. Diffusion contraction and the global sine bound give
The full tangent generator \(J_k\) commutes with \(\mathcal R\) and has maximum row norm less than three. Its exact nonlinear form residual \(n(y)\) obeys \(\|n(y)\|_\infty\le2gY^2\). All jumps are reflection even, so integration of its odd part gives
At reflection-even phase deviation, the even nonlinear residual contains only the cosine-weighted \(\sin d-d\) remainder: the base-sine quadratic part is odd and cancels in that projection. Thus
The second term follows from \(\|Dn\|_\infty\le4gY\) between the even projection and the full phase. Integrating (9), using (7)-(8) and the tangent bound \(1/\ell_T\), proves
This derivation retains the odd internal modes rather than incorrectly declaring an even nonlinear invariant subspace.
The full nonlinear and tangent fields each have maximum-norm Lipschitz bound three in the original \((x,y)\) coordinates. Their respective actual-to-ideal comparisons are at most \(\epsilon/\ell_T\), with the same jumps; no phase error is divided by \(\gamma\). Two consequences are useful:
and, separately for each history \(j\),
The tangent in (12) starts from the same actual full source and retains its hidden initialization. The ideal references are proof comparisons, not reset preparations. Equation (11) uses exact tangent superposition; equation (12) instead compares each of the four individual endpoints.
Scalar cancellation and four-endpoint record overlap¶
If \(U_M\le4\delta\), every admitted true mixed value can be recorded as zero. In the order \((11,10,01,00)\), choose errors \((-M/4,M/4,M/4,-M/4)\). Each obeys the stated scalar budget, and their signed sum cancels \(M\) exactly. Therefore this design cannot guarantee a nonzero recorded mixed statistic under every allowed error. Favorable realized errors can still reveal a nonzero value.
The stronger sufficient condition
proves intersection of the complete four-endpoint record sets. Choose the common record vector to be the four same-source tangent endpoints. The tangent model uses zero recording errors, while the nonlinear model uses the endpoint differences bounded in (12). This is an existential common record for every admitted actual source. It does not assert that every allowed record is shared, or that the continuous trajectories, intermediate measurements, phase observations or another protocol are indistinguishable.
Failure of either sufficient test alone is not a proof of discrimination. A useful positive protocol needs a finite lower bound exceeding its complete preparation and observation errors. The model's nonlinear coefficient (5), the scalar cancellation test and the stronger endpoint overlap test have separate logical roles.
Carried event work and collective identity¶
Retain the original storage \(\mathcal H=\tfrac12x^{\mathsf T}Lx+\sum_{ij}(1-\cos(\theta_j-\theta_i))\), with continuous loss \(\mathcal H'=-(Lx)^{\mathsf T}K(Lx)\). A donor form jump \(q\) at its actual preevent state has exact work
Thus the second work in \(11\) uses \(x_{10}(s^-)\), while the second work in \(01\) uses \(x_{00}(s^-)\). These are not interchangeable single-probe certificates.
For \(Q_a(s)=(|a|+\epsilon+2gs\epsilon)/(1-2g^2s^2)\), the respective upper bounds are
The unprobed control uses \(Q_0(s)\le Q_a(s)\). Contact work remains in \([0,8\epsilon^2]\). Starting from the original componentwise zero-sum endpoint family, a whole-history excess ceiling is
Continuous loss cannot increase that ceiling. For radius \(r\le1/12\), the matched full-law local capture barrier is strictly greater than \(r^2/2700\). Entry also needs the quotient radius bound, not energy alone. With \(A_*=|a|+|b|\),
is a sufficient global Euclidean representative bound after either event. Together with (16) strictly below the barrier, it retains each cycle identity and the unforced recovery conclusion after the last event. The identity conclusion requires both strict inequalities; the separate work policy requires all declared allowances.
The joined form mean changes by \(3a/58\) and then \(3b/58\) in the histories where those events occur; the phase mean is unchanged. The initial joined means remain bounded by \(4\epsilon/58\). Identity is therefore on each history's own conserved-mean leaf. The work, mean and identity statements concern the complete nonlinear law, not an assumed invariant property of its tangent observation comparator.
A finite design with unresolved nonlinear observations¶
Consider the single analytic design
Use contact allowance \(10^{-12}\) and each probe-work allowance \(2\,10^{-6}\). The elementary bound \(g<1/3000\), substituted in (6)-(17), gives
and the squared-radius ceiling is less than \(28\,10^{-6}<1/144\). Both probe-work ceilings are below \(2\,10^{-6}\); contact work is below \(10^{-12}\). Thus both observation-obstruction conditions coexist with the work, entry and storage guards. Using also \(\gamma>1/3300\), (4b) gives actual right mixed curvature greater than \(10^{-36}\) for this positive second impulse. The leading expression (5) is nonzero as well, while the admitted endpoint records still overlap the same-source tangent records. These bounds follow by exact rational substitution and require no selected trajectory.
This is a response-free theorem and arithmetic control, not a simulated response or another reserved assessment. It identifies a limit of this readout/error regime without changing source precision, breaking the symmetry, discarding a baseline or fitting a response after observing it.
Implementation and scientific boundary¶
The conditional calculator returns finite bounds and separate observation, work and identity guards. It consumes no source report or trajectory and establishes no formation or physical error budget. The contract and guide own its normalized inputs and usage. Independent tests rebuild the full-support symmetry, local derivative identities and rational budgets.
The result addresses nonlinear interaction of organized patterns under one supplied complete law. It does not derive autonomous probe selection, a physical scattering law, particle identity, or a uniquely selected fundamental pressure law. The existing two-phase-input inverse concerns different state, support, events and observations and supplies no coefficients or verdicts for this four-history test.