Amplitude feasibility for mediator-organization discrimination¶
The central cubic response has a resolved organization-dependent sign, while its original recording allowance admits scalar cancellation. Combining it with the assessed spatial statistic does not remove that joint recording limitation. This owner changes only a common positive scale of the two existing donor form impulses. It proves a nonempty scale interval satisfying response, noise, work and identity requirements simultaneously.
The argument uses the retained coefficient through exact homogeneity and a derived delayed-work estimate. It evaluates no new coefficient, source, nonlinear trajectory or response sweep. The execution plan owns selection of a later intervention or evaluation.
Fixed model and one changed input¶
Retain the complete three-C9 law
Here \(D\) is the actual degree matrix, \(K=D^{-1}\), the contacts are \((4,13)\), \((13,22)\), and every capacity is one. All 54 coordinates, both classes \((1,1,1)\), \((1,2,1)\), their acquired full source families and independent residuals remain as in the source owner. In particular, each original component has separate zero-sum form and target-phase deviations with Euclidean norms at most \(\epsilon=10^{-32}\). The four histories within each class share one actual source; the two classes need not share their residuals.
The only variable is a common positive amplitude scale:
The form impulses retain their original donor node \(4\) and event times. All phase, mediator and receiver coordinates carry; no state is reset. Retain the eight final receiver-center readings, their separate errors \(\delta=10^{-30}\), radius \(r=1/12\), contact allowance \(10^{-12}\) and each probe-work allowance \(2\,10^{-6}\).
Let \(D_{\mathrm c}(\lambda)\) be the class-one-minus-class-two difference of the two four-history mixed central responses. Its eight scalar errors give
The separately noisy eight-record zero-contrast alternative has its own \(8\delta\) allowance. Consequently a true oriented contrast strictly greater than \(16\delta\) separates these statistic images; the weaker \(8\delta\) condition guarantees only the recorded sign. Neither threshold is changed by scaling the input.
Homogeneous coefficient and complete finite error¶
The complete amplitude hierarchy is triangular: its first level is linear in the supplied impulses, its second forcing is quadratic in level one, and its third forcing consists of a level-one/level-two product and a level-one cube. The second event changes only level one and carries every previous coefficient. Induction first on amplitude degree and then across that event proves
This holds separately for all four histories and both classes. It retains full phase feedback and the quadratic internal recoupling. It is not a homogeneity assumption on the nonlinear trajectory or on its arbitrary actual residuals.
The previously enclosed complete cubic contrast has the conservative retained outer interval
It includes the complete coefficient, its time truncation and outward arithmetic. Therefore \(\lambda^3J_0\) encloses the scaled complete cubic contrast without a new coefficient calculation. Scaling this whole enclosure also scales its retained coefficient error; it does not assert that a fresh numerical execution would have identical rounding widths.
Use the fixed rational upper bound \(g=1/3000>\gamma\). The complex-amplitude theorem gives
when \(16g^2T^2<1\) and \(2gA_*<1\). The zero history has zero nominal remainder. Define
The first factor of two accounts for the two classes; the inner factor accounts for the two separate single-impulse histories. The source allowance is unchanged by \(\lambda\): real sine is globally Lipschitz in phase, and heat contraction compares each actual history with its matching nominal history under identical scaled jumps. Thus
The coefficient-generating execution remains a premise of the retained evidence. Admitting an arbitrary supplied numerical pair as a calculator input cannot prove that it encloses this coefficient.
For an interval \(0<\lambda_-\le\lambda\le\lambda_+\) in the domain of (6), \(E\) is increasing. If \(J_0=[j_-,j_+]\) with \(j_+<0\), a uniform enclosure is
In particular, \(-\lambda_-^3j_+-E(\lambda_+)>16\delta\) is a sufficient whole-interval separation condition. It is stronger than testing a few scales or scaling the earlier full-error interval as though every error were cubic.
A sharper bound on the actual delayed work¶
The complete storage is \(\mathcal H=\tfrac12x^{\mathsf T}Lx+ \sum_{ij}(1-\cos(\theta_j-\theta_i))\). Its continuous derivative is nonpositive. At a donor form jump \(q\), however, the exact work is
The first term is the full preevent donor Laplacian, not \(3qx_4^-\), and the delayed event must use the actual carried history.
A spectral estimate bounds its heat contribution without evaluating a heat trajectory. Set \(P(t)=\exp(-At)\). The similarity \(D^{1/2}AD^{-1/2}=D^{-1/2}LD^{-1/2}\) is symmetric positive semidefinite with spectrum in \([0,2]\). For each normalized coordinate vector, the diagonal of \(A\exp(-A)\) is therefore a probability-weighted mean of \(\mu e^{-\mu}\), which lies in \([0,1/\exp(1)]\). Since the donor degree is three,
The last strict inequality follows from \(\exp(1)>1+1+1/2+1/6=8/3\). The exponential constant here is unrelated to the model's weight \(e=1023/1024\).
Suppose the first impulse is \(u\ge0\), including \(u=0\) for a control. Over \(0\le t\le1\), define
Diffusion contraction and the complete phase row give \(\|x(t)\|_\infty\le Q_u\) and \(\|y(t)\|_\infty\le\epsilon+2gtQ_u\). Relative to the ideal positive heat impulse \(uP(t)e_4\), the full actual form at time one satisfies
The donor row of \(L\) has absolute row sum six. Combining (11) and (13) proves the complete carried-pressure enclosure
This explicitly retains source errors and sine feedback. It is not a pure-diffusion replacement for the original law.
For a positive delayed impulse \(q\), (10) now gives
The second-only history uses \(u=0\), not the probed preevent state. These upper bounds are monotone in \(u,q\ge0\), so the both-probe history at \(\lambda_+\) dominates all history/event upper bounds throughout the claimed interval.
Storage, identity and conserved means¶
Contact work remains in \([0,8\epsilon^2]\), and the initial joined excess storage is at most \(22\epsilon^2\). Without crediting any amount of continuous dissipation, every history has the excess ceiling
This bounds the first event as well as the second because the used work upper bounds are nonnegative. Storage can only decrease between jumps; no amount of continuous loss is credited as an event reserve.
For \(A_*=u+q\), use the original full-history representative bounds
The quotient radius is no larger than this global Euclidean representative bound. The matched capture theorem has barrier strictly greater than \(r^2/2700\). Its sufficient guards remain
Both must hold; a work allowance or small total amplitude alone does not certify identity.
The graph's degree mass is 58. Initially the separate component zero sums give joined form and phase means bounded in magnitude by \(4\epsilon/58\). A donor jump \(u\) changes the weighted form mean by \(3u/58\), and a later jump \(q\) changes it by \(3q/58\). Phase means are unchanged. The continuous law preserves both weighted means between events. Each history consequently retains its own mean leaf; none is reset to a common postevent mean for comparison. Under (18), the cycle identities and unforced recovery after the last event retain their original conditional meaning.
A certified nonempty scale interval¶
Take the complete interval
Use \(\lambda_+\) in every increasing error, work and radius majorant, and \(\lambda_-\) in the oriented cubic lower bound. This is a uniform proof, not interpolation between sampled responses. The largest per-event amplitude is \(7/10000\), and the largest history amplitude is \(7/5000\). Hence the complex-amplitude and real comparison denominators are strictly positive throughout.
Direct exact rational substitution in (6)-(9) gives
Thus the whole recorded interval lies below \(-8.537\,10^{-30}\), whereas the separately noisy zero-contrast alternative has statistic in \([-8,8]\,10^{-30}\). Their separation margin is greater than \(5.37\,10^{-31}\), uniformly on (19). In particular the recorded sign is guaranteed under every allowed eight-error vector. This excludes that named alternative's record set through the statistic; it does not select a unique law or identify a physical interaction.
The same exact substitution in (12)-(18) gives, for every history and both classes,
The contact ceiling \(8\epsilon^2=8\,10^{-64}\) is below its unchanged allowance \(10^{-12}\). The delayed work improvement uses the full donor Laplacian and the source/full-law correction. No storage loss, initial precision, observation accuracy or work ceiling has been changed to obtain these inequalities.
The result admits an interval of interventions mathematically. It does not choose or execute a new reserved response, claim that all amplitudes preserve identity outside this interval, or establish laboratory preparation, clock or noise feasibility. The original unit-scale cancellation results remain valid for their unchanged protocols.
Implementation and evidence boundary¶
The conditional calculator admits a positive scale interval and a supplied base cubic interval. It reconstructs homogeneous response bounds, higher-amplitude and source allowances, carried work, identity guards and mean intervals. The upper-scale work ledger is explicitly an endpoint majorant; its individual mean shifts must not be substituted for the separately reported mean intervals covering all scales.
The supplied coefficient endpoints are a mathematical premise, not an authenticated observation or a certificate generated by this API. For (5), that premise is supplied by the retained cubic calculation and its read-only evidence audit. The independent amplitude controls check homogeneity, spectral pressure, complete error propagation and the whole interval with exact arithmetic. Failed sufficient guards report that the method has not certified the requested regime; they do not prove dynamical infeasibility.
No new coefficient or response archive is needed for these deductions. The existing coefficient evidence is reused without modification or duplication, and its generation remains an explicit execution premise. Any later changed-input observation requires its own fixed source, event, numerical and recording protocol before evaluation.