Complete cubic-amplitude response of an acquired mediator

The organization contrast isolates a finite mediator contribution but leaves its sign unresolved by an independent-class heat remainder. This owner retains the complete cubic-amplitude response instead. Linear phase feedback and quadratic internal corrections are part of the coefficient, not discarded into that remainder. A complex-parameter estimate controls all higher amplitude orders without changing the real nodal model.

This is a conditional coefficient calculation and finite-error theorem. It executes no formation assessment or nonlinear reserved trajectory and does not reuse the retained class-two response as new cross-class data. The execution plan owns its research status.

Complete source, events and observation

Retain both actual class families \((1,k,1)\), \(k=1,2\), on the same three-C9 graph, with contacts \((4,13)\), \((13,22)\), unit capacities and all 54 original coordinates. Use the supplied law

\[ x'=-Ax+\gamma KS(\theta),\qquad \theta'=\gamma Ax, \qquad A=KL,\quad\gamma=1/(1023\pi),\quad \tau=et, \quad e=1023/1024. \tag{1} \]

Each class's four histories share its complete actual acquired source; the two classes may have independent residuals. The original source and cost obligations are unchanged, including separate zero component sums and form and target-phase Euclidean error bounds \(\epsilon\). No phase, form, mean or hidden coordinate is reset at either event.

Donor central form impulses occur at times zero and \(s\). The fixed design remains

\[ a=b=1/2000,\quad s=1,\quad T=2,\quad \epsilon=10^{-32},\quad\delta=10^{-30},\quad r=1/12. \tag{2} \]

Let \(M_k=R_{k,11}-R_{k,10}-R_{k,01}+R_{k,00}\), where \(R\) is receiver central form at \(T\), and let \(D=M_1-M_2\). The eight scalar recording errors satisfy \(|e_j|\le\delta\). Consequently \(|\widehat D-D|\le8\delta\); no probability law or laboratory error calibration is assumed.

The nominal equilibrium is only a proof reference for the already admitted full source family. For one history scale all its supplied form impulses by an auxiliary parameter \(\zeta\), leaving times and every law unchanged. Its nominal trajectory has ordinary Taylor coefficients

\[ x(\zeta,t)=\sum_{n\ge1}\zeta^n x_n(t),\qquad y(\zeta,t)=\theta-\Theta_k=\sum_{n\ge1}\zeta^n y_n(t). \tag{3} \]

Thus the coefficients include the usual Taylor factorial normalization. Amplitude derivatives are derived from the law, not measured extra channels or new initial preparations.

The complete triangular variation equations

Write \(C_k\) for the degree-normalized phase Laplacian at \(\Theta_k\), and put \(d_{ij}=\Theta_{k,j}-\Theta_{k,i}\). Define normalized multilinear maps

\[ Q_k(v,w)_i=-\frac1{2d_i}\sum_{j\sim i} \sin d_{ij}\,(v_j-v_i)(w_j-w_i), \qquad T_k(v,w,z)_i=-\frac1{6d_i}\sum_{j\sim i} \cos d_{ij}\,(v_j-v_i)(w_j-w_i)(z_j-z_i). \tag{4} \]

Their signs and factors follow by expanding the original sine current. In original coordinates, with \(J_k=\left(\begin{smallmatrix}-A&-\gamma C_k\\\gamma A&0\end{smallmatrix}\right)\), the first three levels satisfy

\[ \begin{aligned} (x_1,y_1)'&=J_k(x_1,y_1),\\ (x_2,y_2)'&=J_k(x_2,y_2)+(\gamma Q_k(y_1,y_1),0),\\ (x_3,y_3)'&=J_k(x_3,y_3) +(2\gamma Q_k(y_1,y_2)+\gamma T_k(y_1,y_1,y_1),0). \end{aligned} \tag{5} \]

Every present impulse adds its amplitude to \(x_{1,D}\). Levels two and three have no jump, and all phase coefficients are continuous. The second event uses every carried coefficient from the first window.

For conditioned coefficient arithmetic use

\[ x_1=u_1,\ y_1=\gamma w_1,\qquad x_2=\gamma^3u_2,\ y_2=\gamma^4w_2,\qquad x_3=\gamma^4u_3,\ y_3=\gamma^5w_3, \quad \eta=\gamma^2. \tag{6} \]

All three pairs then have the same linear block \(\mathcal J_k=\left(\begin{smallmatrix}-A&-\eta C_k\\A&0\end{smallmatrix}\right)\). The second form forcing is \(Q_k(w_1,w_1)\); the third is \(2\eta Q_k(w_1,w_2)+T_k(w_1,w_1,w_1)\). The factor \(\eta\) in the quadratic recoupling must remain. These scaled coordinates are used for the nominal coefficient calculation only; actual phase uncertainty is never divided by \(\gamma\).

The port-fixing reflection gives \(\mathcal R(x_n,y_n)=(-1)^{n+1}(x_n,y_n)\). Hence the nominal central coefficient vanishes at every even amplitude degree. Quadratic internal coefficients generally do not vanish: their reflection-odd phase component feeds the third level through (5). The complete tangent mixed response cancels within each class, so the nominal cubic mixed coefficient is

\[ C_k^{(3)}=\gamma^4 [u_{3,k,11,R}(T)-u_{3,k,10,R}(T)-u_{3,k,01,R}(T)], \qquad D^{(3)}=C_1^{(3)}-C_2^{(3)}. \tag{7} \]

The nominal unprobed coefficient is exactly zero. Equations (5)-(7) retain full tangent feedback and all intermediate nodes, unlike the earlier heat-cubic approximation.

A rigorous bound beyond cubic amplitude

Fix \(g=1/3000>\gamma\). For a history with total absolute impulse budget \(A_*>0\), require \(2gA_*<1\), and choose

\[ \rho=\frac1{2gA_*}>1,\qquad D_T=1-2g^2T^2>0,\qquad 16g^2T^2<1. \tag{8} \]

Complex values of \(\zeta\) below are auxiliary analytic variables; physical form and phase are still real. The heat reference on \(|\zeta|\le\rho\) has phase

\[ y_{\rm heat}(t)=\gamma\zeta\sum_{t_j\le t} q_j[I-P(t-t_j)]e_D,\qquad P(t)=e^{-At}. \tag{9} \]

Markov contraction and nonnegativity imply \(\|[I-P(v)]e_D\|_\infty\le1\), so \(\|y_{\rm heat}\|_\infty\le\gamma\rho A_*\le1/2\). This bound retains both events; replacing it by a linear-in-time phase estimate would needlessly shrink the analytic disk.

For real \(d\) and complex \(z\), derivatives of sine have modulus at most \(\cosh|\operatorname{Im}z|\). On the bootstrap \(\|y\|_\infty\le1\), all edge increments have modulus at most two and \(\cosh2<4\). Thus the normalized sine difference is at most \(8\|y\|_\infty\). Comparing full and heat form, then integrating the complete phase row, gives

\[ Y(t)\le\frac12+16g^2\int_0^t(t-v)Y(v)\,dv, \qquad \sup_{v\le t}Y(v)\le\frac12+8g^2t^2\sup_{v\le t}Y(v). \tag{10} \]

At a first point with \(Y=1\), (8) makes the right side strictly less than one. This closes the bound on the disk. The form remains bounded by its heat value and a finite forcing integral. The entire sine field, linear jumps and strict bound give holomorphic parameter dependence on a neighborhood of the closed disk throughout the complete finite history. This is a mathematical extension for the error proof, not a complex-state replacement of the model.

The normalized sine defect after its linear part is at most \(8\|y\|_\infty^2\). Therefore its form forcing is at most \(8g\) on this disk. Diffusion contraction and the phase integral bound the complete nonlinear-minus-tangent form difference by

\[ \sup_{|\zeta|\le\rho,t\le T} |x_R(\zeta,t)-\zeta x_{1,R}(t)|\le\frac{8gT}{D_T}. \tag{11} \]

Indeed the linear phase feedback contributes at most \(2g^2T^2\) times the form-error supremum. No exponential bound for the full block generator is needed here.

This central difference is odd in \(\zeta\) and starts at degree three. Cauchy's coefficient estimate, summed over degrees five, seven and higher at \(\zeta=1\), proves

\[ |x_R(1,T)-x_{1,R}(T)-x_{3,R}(T)| \le R_5(A_*,T) :=\frac{256g^6A_*^5T} {(1-2g^2T^2)(1-4g^2A_*^2)}. \tag{12} \]

If \(A_*=0\), the nominal history is stationary and this remainder is exactly zero; no division by zero in (8) is required. Actual residuals need not have the reflection symmetry used in this proof.

Fixed finite evaluation of the triangular coefficient

The coefficient policy fixes dyadic128 outward arithmetic, time order \(N=64\), and \(L=201/100\) before evaluating any new coefficient. The two time pieces are \([0,1]\) and \([1,2]\). For each class, one first-probe prefix supplies the first-only and both-probe suffixes; the delayed-only suffix starts from the exact nominal unprobed state. This uses eight analytic segments in total. Every level carries at the second event. There is no adaptive order, subdivision or parameter search. These are coefficients of (5), not new nonlinear full-state observations.

The full block obeys \(\|\mathcal J_k\|_\infty\le2+2\eta<L\). The normalized forcing norms in scaled coordinates are bounded by \(2\|w_1\|^2\) and \(4\eta\|w_1\|\|w_2\|+(4/3)\|w_1\|^3\). Let \(v_1,v_2,v_3\) bound the complete scaled pairs at a segment's initial point. The following positive series dominate their absolute time coefficients coefficient by coefficient:

\[ \begin{aligned} V_1(t)&=v_1e^{Lt},\\ V_2(t)&=(v_2+2v_1^2t)e^{2Lt},\\ V_3(t)&=[v_3+(4\eta v_1v_2+4v_1^3/3)t +4\eta v_1^3t^2]e^{3Lt}. \end{aligned} \tag{13} \]

One proof first uses the coefficientwise comparison system \(V_1'=LV_1\), \(V_2'=LV_2+2V_1^2\), \(V_3'=LV_3+4\eta V_1V_2+(4/3)V_1^3\). Its convolution coefficients are bounded by the positive exponential-polynomial series in (13): substitution leaves nonnegative coefficients in each differential supersolution residual, so induction on the Taylor degree gives the comparison. Pointwise growth bounds alone would not justify a Taylor tail. Numerical majorants replace \(\eta\) by its admitted outward upper bound; coefficient intervals retain the original parameter enclosure.

For \(z\ge0\) and \(z<m+1\), define

\[ E_m(z)=\frac{z^m}{m!\,[1-z/(m+1)]}. \tag{14} \]

It bounds the exponential series starting at power \(m\), by its first term and the largest subsequent ratio. For segment length \(h\), the tails after time degree \(N\) are bounded by

\[ \begin{aligned} T_1&=v_1E_{N+1}(Lh),\\ T_2&=v_2E_{N+1}(2Lh)+2v_1^2hE_N(2Lh),\\ T_3&=v_3E_{N+1}(3Lh) +(4\eta v_1v_2+4v_1^3/3)hE_N(3Lh) +4\eta v_1^3h^2E_{N-1}(3Lh). \end{aligned} \tag{15} \]

Thus order 64 uses the starts 65, 64 and 63 for the constant, linear and quadratic prefactors. Each outward endpoint includes its own tail before being carried into the next segment. Source, sine/cosine, coefficient arithmetic, time truncation and amplitude truncation are separate obligations; the ordinary full-state solver's dimension or time-order cap is not changed by this triangular coefficient calculation.

Transfer to the actual families and recording limits

Let \(J^{(3)}\) be an outward interval for (7), including the complete triangular coefficient and time tails. The true contrast satisfies

\[ D\in J^{(3)}+[-F,F],\qquad F=2\{R_5(|a|+|b|,T)+R_5(|a|,T)+R_5(|b|,T)\} +\frac{8\epsilon}{1-2\bar\gamma T}. \tag{16} \]

Here \(\bar\gamma\) is an admitted outward upper bound for \(\gamma\), with \(\bar\gamma\le g=1/3000\). The source comparison may use this sharper bound while the Cauchy disk and (12) retain the fixed \(g\). The last term compares each actual history with its own nominal class reference in the original coordinates. It grants no cancellation of independent cross-class residuals. At (2), exact rational arithmetic gives

\[ 2\{R_5(1/1000,2)+2R_5(1/2000,2)\}<1.6\,10^{-33},\qquad \frac{8\epsilon}{1-2\bar\gamma T}<8.02\,10^{-32}. \tag{17} \]

This bounds approximation and source allowances independently of the coefficient. It proves neither sign nor separation without the coefficient enclosure and is not a new measured error.

Recorded-sign certification requires the true interval (16) to lie strictly beyond \(\pm8\delta\). Excluding a separately noisy eight-record null model through this statistic retains the different \(16\delta\) condition. Neither threshold is inferred from the coefficient's sign.

There is also a precise scalar observation obstruction. If (16) instead proves \(|D|\le8\delta\) throughout the actual family, write \(\sigma_j\in\{-1,1\}\) for the eight coefficients defining \(D\). For each actual pair of class sources choose

\[ e_j=-\sigma_jD/8. \tag{18} \]

Every \(|e_j|\le\delta\), and the recorded statistic is exactly zero. Thus every such actual source pair admits at least one allowed error vector erasing this scalar contrast. It does not follow that all records are uninformative, that full eight-record sets overlap, or that the underlying organization-dependent response vanishes. Favorable errors or different functions of those records can have different discrimination properties.

The source costs and event/identity proof are unchanged by improving the approximation. They retain different class preparations, independent residuals, the full delayed state, both work allowances and each resulting conserved-mean leaf. Their validity is not inferred from a signed coefficient or a noise obstruction.

Evaluation boundary

The fixed policy above precedes evaluation of the refined coefficient. Its finite outcome, whether a resolved sign, a scalar-resolution obstruction or an unresolved interval, must be stated separately from the formulas. No refined coefficient or complete-law observation is asserted by the admission alone. Earlier frozen responses and the valid limitation of the independent heat-remainder method remain unchanged.

Finite coefficient result and scalar recording obstruction

The first calculation under the fixed policy completed all eight analytic segments at order 64, without subdivision or a repeated coefficient call. Its complete cubic contrast, including interval arithmetic and the carried time tails, satisfies the outward bound

\[ J^{(3)}\subset [-7.013764940694,-7.013764940692]\,10^{-30}. \tag{19} \]

The exact retained interval is much narrower than these displayed endpoints: its width is less than \(1.182\,10^{-49}\). The higher-amplitude allowance is less than \(1.493\,10^{-33}\), and the independent-source allowance is less than \(8.010\,10^{-32}\). Reconstructing (16) from those exact rational quantities gives

\[ -7.096\,10^{-30}<D<-6.932\,10^{-30}<0. \tag{20} \]

Both complete class families therefore have a strictly ordered finite nonlinear mixed response in the declared class-one-minus-class-two orientation. This sign includes the quadratic internal recoupling, every higher amplitude order through its bound, and the independently admitted actual residuals. The shared source/work ledger also satisfies the original contact and probe allowances and both identity guards; the coefficient calculation does not replace those premises.

At the unchanged \(\delta=10^{-30}\), however, the whole true interval lies strictly inside \((-8\delta,8\delta)\). The recorded interval obeys only

\[ \widehat D\in[-15.096,1.068]\,10^{-30}, \tag{21} \]

and (18) supplies an allowed error vector making this statistic zero for every admitted actual pair. The exact cancellation margin exceeds \(9.046\,10^{-31}\). Thus the result proves both an organization-dependent true response and a worst-case scalar recording limitation at the fixed noise budget. It certifies neither a recorded sign nor exclusion of a noisy null alternative through this statistic. It does not assert equality or overlap of the complete eight-record vectors, and it does not identify a physical property.

This calculation evaluates amplitude coefficients and rigorous remainders, not a new complete nonlinear response or formation run. The earlier independent-class heat estimate remains a valid, weaker unresolved enclosure. Its limitation has been removed here by retaining the omitted dynamics in the coefficient, without changing the law, source, events, horizon or recording budget.

The maintained coefficient implementation, implementation controls, independent algebra controls and shared contrast decisions keep coefficient calculation, source admission and observation decisions separate.

The byte-preserving evidence bundle retains the declared policy, first attempt ledger, exact outcome, evaluated source and driver. A source-packaging operation initially used an incorrect dependency path and stopped before creating the attempt ledger or importing/calling the coefficient calculation. Its partial archive and error record are preserved alongside the separately named complete archive. Correcting that path changed no policy, inputs, numerical budget or runtime bytes; there was one coefficient evaluation. The bundle stores the original 33,439,119-byte outcome losslessly.

The relevant SHA-256 identities are:

Object	SHA-256
Policy	f958646ea96ca8d850563e12b18e72ce0d51b8605967de208f29ffd9e1281245
Complete evaluated-source archive	6f559fdc6d1fb69739915723e78ac9690701236328a34c23542ef35d935d5209
Exact outcome	a56348c90862d0acecbedb610c6599b2de956dba11b7524f005989eb655dbe12
Evidence bundle	991d771464d84f65a25c0a82c9b08f7e5a579e450885141b32fef1a1ed8239cb

The read-only evidence audit rebuilds consumed downstream interval arithmetic and decisions from retained primitives. Generating the stored coefficient enclosures remains an explicit execution premise, supported by the derivation and independent controls; reading or hashing the bundle is not a replay of that calculation or a proof of physical identification.