Spatial observation of an acquired mediator's nonlinear response

The complete cubic central response has a resolved organization-dependent true sign, but the fixed recording budget permits cancellation of its scalar contrast. This owner admits a different receiver observation by deriving its symmetry, locality and complete error budget before assessing its coefficient. A lower amplitude degree alone does not guarantee greater organization sensitivity.

The law, acquired families, support, clock, events and recording budget remain unchanged. This is a conditional mathematical observation study, not a physical identification or a new formation or nonlinear-response campaign. The execution plan owns its status.

Complete model and scalar observation

Use all 54 original coordinates on three C9 rings, contacts \((4,13)\) and \((13,22)\), unit capacities and nominal classes \((1,k,1)\), \(k=1,2\). In the previously declared structural clock,

\[ x'=-Ax+\gamma f_k(y),\qquad y'=\gamma Ax,\qquad A=K L,\quad f_k(y)=K S(\Theta_k+y),\quad \gamma=\frac1{1023\pi},\quad \tau=et,\quad e=\frac{1023}{1024}. \tag{1} \]

Here \(y=\theta-\Theta_k\) is a continuous lift relative to that class's equilibrium, \(K\) is the inverse actual degree matrix and \(\Theta_{k,c,j}=k_c(2\pi/9)(j-4)\). Thus \(f_k(0)=0\). The central donor, mediator and receiver have degrees \(3,4,3\). All original phase and form residuals remain in the source.

The same four histories within each class share one complete actual acquired source, with separate component zero sums and Euclidean form and target-phase errors at most \(\epsilon\). The two classes may have independent source residuals. Their original preparation, event work and identity obligations remain premises; selecting an observation installs no event or reset. Keep

\[ a=b=\frac1{2000},\qquad s=1,\qquad T=2,\qquad \epsilon=10^{-32},\qquad\delta=10^{-30},\qquad r=\frac1{12}. \tag{2} \]

The donor form jumps are at zero and \(s\), with the appropriate absent jump in each control history. Every coordinate carries across \(s\).

Let \(\ell x=x_{23}-x_{21}\), the difference between the two receiver neighbors of its center \(22\). Define

\[ N_k=\ell x_{k,11}(T)-\ell x_{k,10}(T) -\ell x_{k,01}(T)+\ell x_{k,00}(T), \qquad D_{\mathrm{sp}}=N_1-N_2. \tag{3} \]

This observation uses two nodal readings per history, hence sixteen readings in total. Each retains its separate bound \(\delta\). Without assuming correlations or a separately calibrated direct difference sensor,

\[ |\widehat D_{\mathrm{sp}}-D_{\mathrm{sp}}|\le16\delta. \tag{4} \]

The observation is at the final time; no temporal derivative or hidden state is measured as an extra channel.

Parity and the central-contact restriction

Let \(\mathcal R\) reflect each ring by \(j\mapsto8-j\). It fixes all ports, the graph and both supplied donor impulses. On a nominal source, scaling all impulses of a history by an auxiliary parameter \(\zeta\) gives the exact relation

\[ x_k(-\zeta,t)=-\mathcal R x_k(\zeta,t),\qquad y_k(-\zeta,t)=-\mathcal R y_k(\zeta,t). \tag{5} \]

It follows from the sine law and uniqueness, with the reflected target phases interpreted through their circular differences. Since \(\ell\mathcal R=-\ell\), this spatial response is even in \(\zeta\). Its nominal constant term is zero and every odd amplitude coefficient vanishes. Actual source residuals need not obey (5); they are handled separately below.

Let \(P_R^-\) be the projection onto the receiver-supported reflection-odd subspace: take the receiver restriction, antisymmetrize it, and set the other components to zero. In the maximum norm, \(\|P_R^-\|\le1\), \(\ell P_R^-=\ell\), and

\[ P_R^- A=A P_R^-. \tag{6} \]

Receiver-odd vectors have zero receiver-center coordinate; the two internal neighbor contributions cancel at that center. Thus no bridge term enters or leaves this subspace through its single central contact. Equation (6) concerns the heat/tangent operator and projected equations. It does not assert a decoupled nonlinear receiver law: local nonlinear forcing can depend on receiver-even phases delivered through the port.

For example, in the complete quadratic amplitude hierarchy below, mediator-odd internal forcing cannot itself diffuse into receiver-odd coordinates. At \(\eta=\gamma^2=0\) in the scaled hierarchy, the first-level heat and phase fields are class blind. The receiver's quadratic forcing then uses only its own fixed target sine, and its projected propagation is also class blind. Consequently the leading heat-only quadratic spatial response is identical for \(k=1,2\) at every finite time. Exposing an odd quadratic mode is therefore insufficient to demonstrate organization sensitivity.

Complete quadratic response and a nonzero locality check

Use the ordinary amplitude coefficients and maps from the complete triangular variation law. Only its first two levels are needed by the new observation:

\[ \begin{aligned} (u_1,w_1)'&=\mathcal J_k(u_1,w_1),\\ (u_2,w_2)'&=\mathcal J_k(u_2,w_2)+(Q_k(w_1,w_1),0),\\ \mathcal J_k&= \begin{pmatrix}-A&-\eta C_k\\A&0\end{pmatrix},\qquad x_1=u_1,\ y_1=\gamma w_1,\quad x_2=\gamma^3u_2,\ y_2=\gamma^4w_2, \end{aligned} \tag{7} \]

where

\[ Q_k(v,w)_i=-\frac1{2d_i}\sum_{j\sim i} \sin(\Theta_{k,j}-\Theta_{k,i})(v_j-v_i)(w_j-w_i). \tag{8} \]

Each present form jump changes only \(u_{1,4}\). All second-level coordinates and all phase coefficients carry continuously at the second event. The complete quadratic contrast is

\[ D_{\mathrm{sp}}^{(2)} =\gamma^3\ell\left[ (u_{2,1,11}-u_{2,1,10}-u_{2,1,01}) -(u_{2,2,11}-u_{2,2,10}-u_{2,2,01})\right](T). \tag{9} \]

No tangent phase feedback is dropped. The nominal unprobed coefficient is exactly zero.

The quadratic contrast is not identically class blind once this feedback is retained. A local formal check makes the first surviving mechanism explicit. Put \(\alpha=2\pi/9\), \(d_c=\cos\alpha-\cos2\alpha>0\), and \(u=T-s\). Under joint time scaling \(s\mapsto\lambda s\), \(T\mapsto\lambda T\), the degree-six term of (9) is

\[ \lambda^6\, \frac{\gamma^5 d_c\sin\alpha\,ab}{3456} \left(\frac{s^3u^3}{3}+s^2u^4+su^5+\frac{u^6}{3}\right). \tag{10} \]

For one impulse \(a\), the receiver-center first phase coefficient starts as \(w_{1,22}=-at^2/24+\cdots\); its class-one-minus-class-two difference starts as \(-\eta d_c at^3/144+\cdots\). Receiver neighbors start one time degree later. The center-to-neighbor gap is therefore \(at^2/24+\cdots\), with class difference \(\eta d_c at^3/144+\cdots\). The leading projected receiver forcing in (8) is \(\sin\alpha\) times half the squared gap. After cancelling the single-impulse histories, its class difference is

\[ \frac{\eta d_c\sin\alpha\,ab}{3456} [t^2(t-s)^3+t^3(t-s)^2], \tag{11} \]

whose integral yields (10). Other receiver edges and subsequent heat propagation contribute at later joint time degrees.

Equation (10) is a check on the complete quadratic amplitude coefficient, including its \(\gamma^5\) suppression. It is neither a finite-horizon sign prediction nor a claim about the leading time degree of every amplitude order of the full nonlinear response.

A correlated bound on all even orders at least four

A separate remainder for each class would lose the useful locality in (6). Instead compare their nominal full histories directly on a common complex amplitude disk. This is only an analytic proof device; the supplied nodal model and its source remain real.

For a history with total absolute impulse budget \(A_*>0\), fix \(g=1/3000>\gamma\) and

\[ \rho=\frac1{2gA_*}>1,\qquad 16g^2T^2<1. \tag{12} \]

The closed-disk bound already proves \(\|y_k\|_\infty<1\) for \(|\zeta|\le\rho\), both classes and every time up to \(T\). It retains all event amplitudes and gives holomorphic dependence on a neighborhood of the closed disk. Normalized sine forcing there obeys \(\|f_k(y_k)\|_\infty\le8\), since \(\cosh2<4\).

The two class histories have identical nominal form impulses and zero initial deviations. Write \(X_\Delta(t)=\|x_1(t)-x_2(t)\|_\infty\) and \(Y_\Delta(t)=\|y_1(t)-y_2(t)\|_\infty\). The heat semigroup is a contraction, so directly

\[ X_\Delta(t)\le16gt,\qquad Y_\Delta(t)\le2g\int_0^t X_\Delta(v)\,dv \le16g^2t^2. \tag{13} \]

These bounds use the already closed individual phase disks. They need no further comparison denominator or hypothesis that the two nonlinear class fields coincide.

The receiver base phase gaps, including its bridge, are identical in both classes. Along the straight segment between the two complex phase states their edge imaginary parts remain bounded by two. Hence the receiver sine difference is at most \(8Y_\Delta\). Projecting the form difference onto (6) eliminates incoming form and bridge forcing. Another heat-contraction estimate gives

\[ \|P_R^-(x_1-x_2)(T)\|_\infty \le8g\int_0^T Y_\Delta(t)\,dt \le\frac{128}{3}g^3T^3,\qquad |\ell(x_1-x_2)(T)|\le\frac{256}{3}g^3T^3. \tag{14} \]

Thus the receiver difference benefits from locality even though the global class difference does not.

The scalar in (14) is holomorphic and even in \(\zeta\), with zero constant term. Cauchy's estimate summed over amplitude degrees four, six and higher proves the cross-class per-history bound

\[ R_{4,\Delta}(A_*,T) =\frac{4096\,g^7 A_*^4T^3} {3(1-4g^2A_*^2)}. \tag{15} \]

For \(A_*=0\) the nominal histories are stationary and this remainder is exactly zero. This is already a difference between classes. The mixed contrast therefore uses \(R_{4,\Delta}(|a|+|b|,T)+R_{4,\Delta}(|a|,T)+R_{4,\Delta}(|b|,T)\), without another factor of two.

Actual sources and recording criteria

Let \(J_{\mathrm{sp}}^{(2)}\) enclose (9), including all coefficient arithmetic and finite time tails. Real full-law heat comparison bounds each source perturbation in the original coordinates by \(\epsilon/(1-2\bar\gamma T)\), where \(\gamma\le\bar\gamma\le g\). Applying the two-node observation to eight histories gives the uniform true contrast enclosure

\[ D_{\mathrm{sp}}\in J_{\mathrm{sp}}^{(2)}+[-E,E],\qquad E=\sum_{h\in\{11,10,01\}}R_{4,\Delta}(A_h,T) +\frac{16\epsilon}{1-2\bar\gamma T}. \tag{16} \]

Here \(A_{11}=|a|+|b|\), \(A_{10}=|a|\) and \(A_{01}=|b|\); the indexed histories retain both single-impulse contributions when \(a=b\). This retains arbitrary actual source residuals, with no imposed reflection symmetry or cross-class correlation. At (2), exact rational arithmetic independently of any coefficient gives

\[ \sum R_{4,\Delta}<5.619\,10^{-33},\qquad \frac{16\epsilon}{1-2\bar\gamma T}<1.603\,10^{-31},\qquad E<1.66\,10^{-31}. \tag{17} \]

The bound is sufficiently small to assess an organization contrast without the former independent-class remainder. It does not establish the size or sign of the as-yet-unassessed spatial coefficient.

If the true interval in (16) lies strictly beyond \(\pm16\delta\), the recorded statistic has the same sign for every admitted error vector. Excluding a separately noisy sixteen-record null model through this statistic requires the different sufficient condition beyond \(\pm32\delta\). Neither conclusion follows merely from (10).

If instead (16) proves \(|D_{\mathrm{sp}}|\le16\delta\), each actual source pair admits errors \(e_j=-\sigma_jD_{\mathrm{sp}}/16\), where \(\sigma_j\) are the sixteen signs in (3), making the recorded statistic exactly zero. This is a scalar cancellation witness, not equality or overlap of the whole record vectors and not blindness of every other observation. Failure of either test is an unresolved sufficient bound, not an exact nonidentifiability theorem.

Fixed analytic projection policy

Before assessing this observation, retain exactly the prior numerical policy: order 64, dyadic128 outward arithmetic, full 54-coordinate amplitude levels one through three, eight carried analytic segments and their proved time tails. No new solver, subdivision, precision, source, pulse, horizon or noise choice is introduced. Use the fixed weights \((23,+1),(21,-1)\), project level two, subtract both class mixed coefficients and only then multiply by the shared \(\gamma^3\) enclosure.

The previous coefficient evaluation retained every coordinate at all three amplitude levels. Those records are prior information. Reusing their justified enclosures for this separately declared projection is a conditional analytic assessment, not an unseen reserved full-response experiment. Its finite outcome must be retained separately; this admission evaluates neither those off-center entries nor a new response.

The event work and identity ledger is inherited without alteration. There is no extra preparation or stimulus cost from a passive mathematical observation, while realizing or calibrating such an observation physically remains a separate obligation.

Finite analytic result at the unchanged design

The first declared assessment completed the fixed calculation and projection. Both retained full-coordinate parameter and coefficient arrays matched the previous analytic evidence exactly. The new complete quadratic contrast obeys the outward bound

\[ J_{\mathrm{sp}}^{(2)} \subset[7.95100431108890,7.95100431108891]\,10^{-30}. \tag{18} \]

Its exact rational width is less than \(7.396\,10^{-46}\). The higher-amplitude allowance is less than \(5.619\,10^{-33}\) and the actual-source allowance is less than \(1.602\,10^{-31}\). Reconstructing (16) from the exact retained quantities proves

\[ 0<7.785\,10^{-30}<D_{\mathrm{sp}}<8.117\,10^{-30}. \tag{19} \]

Thus the true spatial mixed response depends on mediator organization through the complete nonlinear law on the admitted actual source families. Its sign is positive in the declared class-one-minus-class-two orientation. The original work allowances and identity guards also remain satisfied; this sign supplies no replacement for their premises.

The recording allowance is \(16\delta=16\,10^{-30}\), however. The retained recorded interval has the outward enclosure

\[ \widehat D_{\mathrm{sp}}\in[-8.215,24.117]\,10^{-30}. \tag{20} \]

Equation (19) lies wholly inside \((-16\delta,16\delta)\), so the scalar cancellation witness following (17) applies to every admitted actual pair. Its exact margin exceeds \(7.883\,10^{-30}\). This observation resolves a true sign without guaranteeing a recorded sign or excluding the separately noisy null alternative. Selecting a lower amplitude degree has not overcome the inherited observation limit: locality suppresses the organization-sensitive part, and the new statistic also consumes twice as many nodal readings.

A joint limitation of the two admitted statistics

The central statistic and (3) are defined on the same complete class histories, events and source family. Their bounded-error witnesses can hold simultaneously. Write \(D_{\mathrm c}\) for the central statistic. For each actual source pair choose

\[ e^{\mathrm c}_j=-\sigma^{\mathrm c}_jD_{\mathrm c}/8 \quad\text{on the eight node-22 readings},\qquad e^{\mathrm{sp}}_j=-\sigma^{\mathrm{sp}}_jD_{\mathrm{sp}}/16 \quad\text{on the sixteen node-21/23 readings}. \tag{21} \]

The two sets of error coordinates are disjoint. The proved bounds \(|D_{\mathrm c}|<8\delta\) and \(|D_{\mathrm{sp}}|<16\delta\) make all 24 errors admissible under the same per-reading budget. Thus

\[ (\widehat D_{\mathrm c},\widehat D_{\mathrm{sp}})=(0,0) \tag{22} \]

is an allowed pair of statistics for every actual source pair. No homogeneous linear combination of these two statistics can guarantee a nonzero class contrast under that error model. This is stronger than merely observing that two separate sufficient tests fail, but remains precisely scoped: it does not prove overlap of the full 24-record vectors, cancellation under correlated or differently calibrated errors, blindness of other receiver observations, or vanishing of the true organization-dependent dynamics.

Implementation and retained evidence

The spatial report implementation reuses the complete amplitude hierarchy and shared observation decisions. Its contract controls and independent algebra controls check full-state parity, receiver isolation, the class-blind heat limit, the first surviving time coefficient and the correlated error bound.

The compact evidence bundle retains the prospective projection policy, first attempt, evaluated source, driver and exact 45,004-byte outcome. It references the earlier full-coordinate coefficient evidence by exact member and bundle hashes rather than duplicating the large series. Both coefficient arrays were verified identical; the record does not treat their reuse as independent or previously unseen data. No new nonlinear trajectory or formation assessment was evaluated. The read-only evidence audit reuses the original carried-coefficient reconstruction and rebuilds the spatial projection, source/remainder allowances and recording decisions.

Object	SHA-256
Projection policy	3ac58f780e7d65db29f38367c9e22414af50cdad78948ddfaae05b44bc873e69
Evaluated-source archive	d3db4cce394f8bc5e470a5d69f02bcf2d42a634a3ac13da07a0316bdf9dd7632
Compact outcome	64e770071f83fad454607d243feec434dc9615ccf783dc40adff14df921ae148
Evidence bundle	6dd3633441a21004f5f7299a0806a4336bc8cca4befbe0164a32dd465215975b

The coefficient-generation and analytic-tail execution premises remain explicit. Rebuilding projection arithmetic, source allowances and recording decisions from their retained primitives does not regenerate the original coefficient calculation or establish a physical measurement bridge.