Constitutive freedom in the interaction of matched organized patterns

The same organized targets, target storage values, complete tangent and quadratic dynamics can support opposite local nonlinear interactions. This owner proves that statement for the existing positive-loss three-C9 support. It supplies an explicit analytic alternative to the sine potential; it neither selects that alternative nor repeats a finite response calculation. An own-law acquisition argument below also places the original isolated-C9 source families in their capture neighborhoods for every member of this potential family, at the original common time and error budgets.

The separate source-path robustness theorem also carries the original uncertain interior Cayley corridor through its phase passage into winding-one capture for every member of this family. Its own-law current and storage bounds make that formation outcome non-discriminating. The nominal interaction distinction below retains its scope; neither formation result transfers it to an acquired residual state or a finite response.

The carried-state observation below closes that separate obligation for the two endpoint laws and the original class-one ramp sources. Own-law recovery, prescribed attachment work and a real nonlinear remainder give opposite finite measured signs under an explicit mathematical error allowance. No laboratory resolution or physical clock is inferred from that allowance. The scalar calibration theorem then separates preservation of those signs from separation of complete recorded observations under a common independently specified sensor.

The earlier mobility counterfamily and smooth local continuation already establish other nonselection results. Here mobility stays constant, loss stays positive, and a target-matched analytic potential changes the distinct-neighbor interaction of the present joined objects. The observation and supplied interventions retain their original coordinates.

Complete state, potential and reciprocal law

Keep all 27 nodes, the 27 internal cycle edges and the two bridges (4,13) and (13,22). All edges have unit weight. Node order is 9*c+j, with component index c=0,1,2 and local index j=0,...,8. Let \(L\) be the unnormalized graph Laplacian,

\[ D=\operatorname{diag}(d_i),\quad A=D^{-1}L,\quad (d_4,d_{13},d_{22})=(3,4,3),\quad \sum_i d_i=58. \tag{1} \]

The complete state is signed real form \(x\in\mathbb R^{27}\) and 27 primitive circular phases, represented by continuous lifts when taking derivatives and means. Every capacity is held at one. Support, loss, mobility and clock are held; no controller, support selector or capacity evolution is inferred.

For \(0\le\lambda\le1\), define

\[ \begin{split} W(\delta)&=\frac{(1-\cos9\delta)^2}{6\,9^4},\\ V_\lambda(\delta)&=1-\cos\delta+\lambda W(\delta),\\ j_\lambda(\delta)=V_\lambda'(\delta) &=\sin\delta+\frac{\lambda}{2187} \left(\sin9\delta-\frac12\sin18\delta\right). \end{split}\tag{2} \]

The potential is even, nonnegative, real analytic and \(2\pi\)-periodic. It vanishes at zero. Its current is odd and periodic. The parameter and the ninth harmonic are declared constitutive choices, constructed to test this support's target constraints; they are not inferred universal constants.

Set \(U_\lambda(\theta)=\sum_{\{i,j\}}V_\lambda(\theta_j-\theta_i)\),

\[ q=Lx,\qquad S_{\lambda,i}=\sum_{j\sim i}j_\lambda(\theta_j-\theta_i),\qquad \nabla U_\lambda=-S_\lambda. \tag{3} \]

Use the same structural normalization

\[ e=\frac{1023}{1024},\quad w=\frac1{1024},\quad \tau=e t,\quad \gamma=\frac{w}{e\pi}=\frac1{1023\pi}. \]

In the original clock and the structural clock respectively, the complete rows are

\[ \begin{array}{ll} \displaystyle\frac{dx}{dt}=-eD^{-1}q+\frac w\pi D^{-1}S_\lambda, &\displaystyle\frac{d\theta}{dt}=\frac w\pi D^{-1}q,\\[3pt] x'=-Ax+\gamma D^{-1}S_\lambda, &\theta'=\gamma Ax. \end{array}\tag{4} \]

Thus the pressure for the held unit-capacity nodal row is specified by the first original-time expression, independently of the evaluated response. Both evolution rows transform under the clock change. All coefficients use the adopted structural units: \([p]=[x]/([\nu_f][t])\). No laboratory time, energy unit, inverse-mass interpretation or physical identity is supplied.

For

\[ H_\lambda=\frac12x^TLx+U_\lambda(\theta), \]

the constant reciprocal exchange block is

\[ \begin{pmatrix}x'\\\theta'\end{pmatrix} =\left[ \begin{pmatrix}0&-\gamma D^{-1}\\\gamma D^{-1}&0\end{pmatrix} -\begin{pmatrix}D^{-1}&0\\0&0\end{pmatrix} \right]\nabla H_\lambda. \tag{5} \]

The skew part cancels between the two rows. Consequently, on the actual full state,

\[ H_\lambda'=-(Lx)^TD^{-1}(Lx)\le0,\qquad \frac{dH_\lambda}{dt}=-e(Lx)^TD^{-1}(Lx). \tag{6} \]

The same positive loss law is retained, not merely its value on a target. Odd edge currents and Laplacian row sums also give

\[ (\mathbf1^TDx)'=0,\qquad (\mathbf1^TD\theta)'=0. \tag{7} \]

These are lifted means in the declared coordinates, not circular-resultant angles. Adding the same constant to all forms or to all phases preserves the dynamics and storage differences.

Global admission and constitutive derivative bounds

For every real gap and integer \(r\ge0\), differentiating (2) gives

\[ |j_\lambda^{(r)}(\delta)| \le1+\frac{\lambda}{2187} \left(9^r+\frac{18^r}{2}\right). \tag{8} \]

In particular, \(|j_\lambda'|\le1+2\lambda/243\). Thus the complete real-lift field (4) is globally Lipschitz: its form-row difference is bounded by \(2\|\Delta x\|_\infty+ 2\gamma(1+2\lambda/243)\|\Delta\theta\|_\infty\), and its phase-row difference by \(2\gamma\|\Delta x\|_\infty\). It has a unique global solution from every finite full state, in either time direction, between the explicitly supplied events. This does not assert global convergence to one winding target.

Equation (8) supplies real derivative bounds for this law. Although (2) also extends analytically to complex phases, its ninth and eighteenth harmonics have different complex growth from sine. The earlier sine Cauchy tails, source bounds, finite-time intervals and error budgets do not transfer without a new estimate.

Target storage and full local dynamics agree

Let \(\alpha=2\pi/9\), choose mediator class \(k\in\{1,2\}\), and use the nominal joined target

\[ x=0,\qquad \Theta_{k,9c+j}=k_c\alpha(j-4),\qquad (k_0,k_1,k_2)=(1,k,1). \tag{9} \]

Each cycle's principal edge gaps are \(\pm k_c\alpha\); the bridges have gap zero. The closing-edge lifts differ by known full turns, which do not change the periodic current or its derivatives. Direct differentiation gives

\[ \begin{split} W''(\delta)&=\frac{\cos9\delta-\cos18\delta}{243},\\ W'''(\delta)&=\frac{-\sin9\delta+2\sin18\delta}{27},\\ W''''(\delta)&=\frac{-\cos9\delta+4\cos18\delta}{3}. \end{split}\tag{10} \]

At every \(\delta=2\pi m/9\), \(m\in\mathbb Z\),

\[ W=W'=W''=W'''=0,\qquad W''''=1. \tag{11} \]

Therefore both classes have the same equilibrium, target storage value, gradient and full phase Hessian for every \(\lambda\). In particular,

\[ H_{*,k}=18(1-\cos\alpha)+9(1-\cos k\alpha),\qquad L_{c,k}=\nabla^2U_\lambda(\Theta_k) \]

are independent of \(\lambda\). Opposite internal currents cancel at every cycle node, and both bridge currents vanish. The full 54-coordinate tangent is

\[ J_k=\begin{pmatrix}-A&-\gamma D^{-1}L_{c,k}\\ \gamma A&0\end{pmatrix}. \tag{12} \]

The entire quadratic field is also identical: it consumes \(j_\lambda''=V_\lambda'''\), whose target values do not change. The cubic field consumes \(j_\lambda'''=V_\lambda''''\), and does change. No hidden quadratic mode is removed by this matching statement.

A locally re-admitted identity and exact event work

Let \(y=\theta-\Theta_k\), \(P_0=I-\mathbf1\mathbf1^T/27\), and \(Z^2=\|P_0x\|_2^2+\|P_0y\|_2^2\). This is a quotient radius, not an operation that resets the state or its weighted means. Within \(Z\le r=1/12\), every edge residual has magnitude at most \(\sqrt2r\). The graph diameter is ten: four cycle edges, the two bridges, then four cycle edges suffice between any pair. For any real vector \(v\), its range and the variance bound give \(\|P_0v\|_2^2\le27\operatorname{range}(v)^2/4 \le(135/2)v^TLv\), by Cauchy-Schwarz along a shortest path joining its extreme values. Thus \(\lambda_2(L)\ge2/135\), independently of the phase law. The existing angular bound is \(\cos(4\pi/9+\sqrt2r)>1/20\). It also follows from \(\pi>31/10\), \(\sqrt2<17/12\) and \(\sin(13/240)>13/240-(13/240)^3/6>1/20\).

Writing \(c=\cos9\delta\), equation (10) gives \(W''=(1+c-2c^2)/243\ge-2/243\). Hence throughout this chart,

\[ V_\lambda''>\frac1{20}-\frac2{243} =\frac{203}{4860}>\frac1{25}. \tag{13} \]

Integrating the Hessian along the segment from the target yields

\[ H_\lambda-H_{*,k} \ge\frac12x^TLx+\frac1{50}y^TLy \ge\frac{Z^2}{3375}. \tag{14} \]

An initial state inside the chart with relative storage below \(r^2/3375=1/486000\) cannot reach its boundary under unforced flow, by (6) and (14). The target is the unique phase critical orbit there. On a fixed weighted-mean leaf the trapped sublevel is compact. The largest invariant subset with zero loss has \(Lx=0\); invariance then requires \(D^{-1}\nabla U_\lambda\) to be common across nodes. Its degree-weighted sum is zero, so that gradient vanishes. Thus LaSalle's argument gives local convergence to the corresponding target orbit. No common form or phase is forced back to zero.

For any prescribed form jump \(f\), phases do not jump, and each law has the exact work

\[ \Delta H_\lambda=f^TLx^-+\frac12f^TLf. \tag{15} \]

For simultaneous donor and receiver jumps \(f=ae_4+be_{22}\), \(L_{4,22}=0\), so (15) is \(a(Lx^-)_4+b(Lx^-)_{22}+3(a^2+b^2)/2\). It is the same functional in each model, but later pre-event states need not coincide. The weighted form mean changes by \(3(a+b)/58\), with the phase mean unchanged.

From the prescribed nominal source (9), the existing amplitudes \(a=b=m=7/10000\) give work \(3m^2<1/486000\) and \(Z_+^2=50m^2/27<r^2\). The single-input and zero-input branches also satisfy these bounds. All four branches consequently retain the joined quotient identity for both laws, without borrowing an acquisition theorem. Any support change needs its own edge-storage work; continuous loss is not an undeclared event reserve.

A changed interaction of the same prescribed organized states

Use the same complete nominal pre-input state (9) in four histories with form jumps \(0,ae_4,be_{22},ae_4+be_{22}\). Observe the mediator form and define, as in the distinct-neighbor owner,

\[ M_{\lambda,k}(a,b,T)=x_{13}^{11}(T)-x_{13}^{10}(T) -x_{13}^{01}(T)+x_{13}^{00}(T). \tag{16} \]

Immediately after the joint jump, the phase slopes are \(\theta'_4=\gamma a\), \(\theta'_{22}=\gamma b\), \(\theta'_{13}=-\gamma(a+b)/4\), and \(\theta'_{12}=\theta'_{14}=0\). Thus the four incident gap slopes into node 13 are \(\gamma(5a+b)/4\), \(\gamma(a+5b)/4\), and twice \(\gamma(a+b)/4\). The first two edges are bridges; the other two are internal mediator edges.

The form derivatives through order two are linear in the impulses. The third derivative is at most quadratic. In the fourth derivative, the only cubic contribution is the current's third derivative applied to three initial gap slopes; every other term is at most quadratic. Port-fixing ring reflection with simultaneous form/phase-deviation sign reversal is a symmetry of every even potential (2). At the nominal source, the central output is odd in \((a,b)\), so its quadratic contributions vanish. Hidden quadratic modes generally remain nonzero.

The mixed bridge cubes sum to \(90ab(a+b)\), and the two mixed internal cubes sum to \(6ab(a+b)\). Including the degree-four normalization, gap denominators and \(4!\) gives the complete nonlinear local onset

\[ \begin{split} M_{\lambda,k}(a,b,T) &=\frac{\gamma^4ab(a+b)}{1024} [15j_\lambda'''(0)+j_\lambda'''(k\alpha)]T^4+O(T^5)\\ &=\boxed{\frac{16\lambda-15-\cos(k\alpha)}{1024} \gamma^4ab(a+b)T^4+O(T^5)}. \end{split}\tag{17} \]

For fixed positive \(a,b\), the sine member \(\lambda=0\) has negative onset, whereas \(\lambda=1\) has positive onset, since \(1-\cos(k\alpha)>0\) for both classes. Analyticity supplies a positive local time neighborhood in which the corresponding exact signs hold; (17) supplies no declared numerical horizon or measurement resolution.

The leading coefficient vanishes at \(\lambda_{\mathrm{crit},k}=(15+\cos k\alpha)/16\). Since \(\cos2\alpha<\cos\alpha\), \(\lambda_{\mathrm{crit},2}<\lambda_{\mathrm{crit},1}\). Between these two values the classes have opposite local orientations under one law. For example, the fixed rational \(\lambda=31/32\) gives the factor \(1/2-\cos(k\alpha)\): it is negative for class one and positive for class two because \(\alpha<\pi/3<2\alpha\). At \(\lambda=1\), the bridge contribution vanishes and the entire positive onset comes from the internal mediator edges. Omitting those edges would therefore give a false null. Within this particular family, the normalized class-one-minus-class-two coefficient remains \((\cos2\alpha-\cos\alpha)/1024\); that robustness is not asserted for every matched-target potential. At a critical value only the order-four coefficient is known to vanish; exact additivity does not follow. Zero input in either channel cancels (16) for every common source. At the nominal source, \(a=-b\) is also an exact null by outer-ring exchange and the sign symmetry.

Own-law acquisition from the original isolated C9 sources

This section changes support explicitly to one isolated unit C9, with nodes \(0,\ldots,8\), edges \(j--(j+1\bmod9)\), degree matrix \(D=2I\), capacities one and the same \(e,w,\gamma\) and clocks as (4). It does not apply the joined graph's spectral bound to an isolated component or identify the two supports' reached states. The original two-class preparation supplies the following unchanged experiment for each law:

• Classes \(k=1,2\), rational slope \(m=(2046/9)(355/113)^2\), initial form \(x_j(0)=km(j-4)+\xi_j^x\), and initial phase lifts \(\theta_j(0)=\xi_j^\theta\).
• Bounds \(|\xi_j^x|\le\rho_x=10^{-10}\) and \(|\xi_j^\theta|\le\rho_\theta=10^{-10}\), with separate exact zero sums \(\sum_j\xi_j^x=\sum_j\xi_j^\theta=0\). Each class has a complete sixteen-dimensional relative source neighborhood, not an independently selectable eighteen-dimensional ambient box.
• Acquisition time \(T=100\) in \(\tau\), original time \(102400/1023\), radius \(r=1/12\), and common whole-source storage budget \(10^9\). No input, support event, state reset or coefficient change occurs.
• Target \(x^*=0\), \(\theta_{k,j}^*=k\alpha(j-4)\), with \(\alpha=2\pi/9\). Targets label the endpoint comparison, not replacement endpoint states.

Write \(A=L/2\), \(z=\gamma x\), \(\eta=\gamma^2\), and \(f_\lambda=S_\lambda/2\). The complete scaled law is

\[ z'=-Az+\eta f_\lambda(\theta),\qquad \theta'=Az,\qquad (\theta+z)'=\eta f_\lambda(\theta). \tag{18} \]

Odd currents preserve both zero means. The full real-lift field is globally Lipschitz by (8), including nonacute states during acquisition. Continuous lifts are therefore retained through any winding changes.

For \(P_9=I-\mathbf1\mathbf1^T/9\) and the norm \(\|u\|_{2I}=\sqrt2\|u\|_2\), the C9 graph gives the same law-independent semigroup gap \(g=1/5\): \(\|e^{-A\tau}P_9\|_{2I}\le e^{-g\tau}\). Indeed \(1-\cos(2\pi/9)>1/5\), so \(\lambda_2(L)>2/5\), while \(\lambda_{\max}(L)\le4\). The forcing and Hessian constants must instead be re-admitted:

\[ \begin{aligned} J_\lambda&=1+\lambda/1458,& \|f_\lambda(\theta)\|_{2I}&\le F_\lambda=\sqrt{18}\,J_\lambda,\\ u_\lambda&=1+2\lambda/243,& \nabla^2U_\lambda(\theta)&\preceq u_\lambda L . \end{aligned}\tag{19} \]

The first bound uses \(|\sin9\delta-\tfrac12\sin18\delta|\le3/2\); the second uses (10) and \(\cos\delta\le1\). They hold globally, not only near a target. The raw initial metric norms remain \(\sqrt{18}\rho_\theta\) and \(\sqrt{18}\gamma\rho_x\); multiplying those errors by the enlarged forcing constant would confuse distinct premises.

The nominal phase storage initially vanishes. Since \(1-\cos(9v)\le81v^2/2\), the new potential term obeys \(0\le W(v)\le v^4/24\). The entire source family therefore has

\[ B_{\lambda,k} =36k^2m^2+32km\rho_x+18(\rho_x^2+\rho_\theta^2) +6\lambda\rho_\theta^4 \quad\hbox{as a storage upper bound}. \tag{20} \]

The linear form term uses the sum \(16km\) of the absolute nominal edge differences; the last term uses all nine edges and \(|\Delta\xi^\theta|\le2\rho_\theta\). The exact nominal costs \(36k^2m^2\) are unchanged and differ by a factor four. The quartic source correction is retained even though at these budgets it is below one dyadic128 grid spacing. No equality of later losses or phase histories follows.

Let \(\rho\) be an outward upper bound for \(e^{-g T}\). Define

\[ \begin{aligned} V_k&=\sqrt{120}\,k\gamma m,& E_x&=\sqrt{18}\gamma\rho_x,& E_\theta&=\sqrt{18}\rho_\theta,\\ R_{z,\lambda,k}&=\rho(V_k+E_x)+\eta F_\lambda/g,& R_{x,\lambda,k}&=R_{z,\lambda,k}/(\gamma\sqrt2),\\ D_k&=\sqrt{60}\,k|\gamma m-\alpha|,& Y_{\lambda,k}&=D_k+ (E_x+E_\theta+R_{z,\lambda,k}+\eta F_\lambda T)/\sqrt2,\\ Z_{\lambda,k}^2&=R_{x,\lambda,k}^2+Y_{\lambda,k}^2 . \end{aligned}\tag{21} \]

Variation of constants bounds \(\|z(T)\|_{2I}\) by \(R_z\). Integrating the last row of (18) gives

\[ \theta(T)-\gamma x^{\rm nom}(0) =\xi^\theta+\gamma\xi^x-z(T) +\eta\int_0^T f_\lambda(\theta(s))\,ds . \]

Consequently every actual endpoint satisfies \(\|x(T)\|_2\le R_x\), \(\|\theta(T)-\theta_k^*\|_2\le Y\) and the full radius bound \(Z^2\). This uses the entire nonlinear forcing and all initial residuals; it is not a tangent trajectory or a Cartesian reset of the reached family.

Target stationarity and the global Hessian upper bound yield the split endpoint estimate

\[ H_\lambda(T)-H_k^* \le E_{\lambda,k}:= 2R_{x,\lambda,k}^2+2u_\lambda Y_{\lambda,k}^2,\qquad H_k^*=9(1-\cos k\alpha). \tag{22} \]

The form and phase terms retain their different coefficients. For the common capture ball set

\[ \mu=\pi/2-4\pi/9-\sqrt2r,\quad c_r=\cos(4\pi/9+\sqrt2r),\quad c_\lambda=c_r-2\lambda/243,\quad b_\lambda=c_\lambda r^2/5 . \tag{23} \]

When \(\mu>0\) and \(c_\lambda>0\), the own-law phase Hessian throughout the ball is bounded below by \(c_\lambda L\). With the zero means and the isolated graph gap, this gives \(H_\lambda-H_k^*\ge(c_\lambda/5) (\|x\|_2^2+\|\theta-\theta_k^*\|_2^2)\). Thus \(Z_{\lambda,k}^2<r^2\) and \(E_{\lambda,k}<b_\lambda\) prove forward trapping under the own-law identity \(H_\lambda'=-\|Lx\|_2^2/2\).

Uniform fixed-budget result and its provenance

The shared outward dyadic128 square-root, Machin-pi, cosine and negative-exponential arithmetic applied to (19)--(23) at \(\lambda=1\) gives the following strict rational bounds for both classes:

\[ \begin{gathered} \mu>56/1000,\qquad c_1>48/1000,\qquad Z_{1,k}^2<23/10^6<r^2,\\ E_{1,k}<45/10^6,\qquad b_1>672/10^7,\qquad b_1-E_{1,k}>222/10^7,\\ B_{1,1}<181230000,\qquad B_{1,2}<724918000<10^9 . \end{gathered}\tag{24} \]

Here the subscript one on \(c_1,b_1\) means deformation one, not winding class one. The bounds are rounded outward from direct scalar interval evaluation of the displayed formulas, not from a sampled trajectory or the earlier sine report. In particular the computed storage-margin lower bounds exceed \(2.29504274\,10^{-5}\) for winding one and \(2.22733831\,10^{-5}\) for winding two.

Only the endpoint \(\lambda=1\) is needed: \(F_\lambda,u_\lambda\), the source-cost bound and the endpoint bounds increase with \(\lambda\), whereas \(c_\lambda,b_\lambda\) decrease. Hence (24) proves acquisition and capture uniformly for all \(0\le\lambda\le1\), including both laws used in (17). This is an analytic family argument, not a parameter sweep.

Every initial phase edge has its small principal lift, so winding starts at zero. Endpoint membership in the target's acute ball fixes winding \(k\); it remains there thereafter. The change requires an intervening antipodal passage, allowed by (18); an acute constraint during the whole acquisition would be a different model. On the compact trapped zero-mean leaf, zero loss forces \(x=0\), and invariance forces \(S_\lambda=0\). Strict phase convexity leaves only \(\theta_k^*\). LaSalle therefore gives convergence; the unchanged positive target Hessian also retains local exponential stability of the complete form/phase tangent. For each positive eigenvalue \(\ell\) of \(L/2\), its characteristic polynomial is \(s^2+\ell s+\gamma^2\cos(k\alpha)\ell^2\), whose roots have negative real parts. Winding magnitudes one and two remain inequivalent under cycle automorphisms, common origins and joint sign reversal.

This establishes acquisition from the same prescribed source family, with the same time, uncertainty and cost limits. It does not identify the reached fine states between laws, reproduce later very small residual budgets, or transfer a dwell, support-contact, joined-source or finite-response certificate. Those require the new law's own bounds.

Carrying actual acquired states through recovery and contact

Use three copies of the original class-one ramp family from the acquisition section, each with its full separate zero-sum form and phase residuals bounded by \(10^{-10}\). Use \(\lambda=0\) or \(\lambda=1\) consistently throughout each complete history. At \(\tau=100\), each actual state is in the forward-trapped radius-\(r=1/12\) neighborhood of its isolated winding-one target. The following dwell changes no source, mean, state coordinate or law.

Re-admit the modified-energy argument using the actual nonlinear potential. Put \(g=1/5\), \(\Lambda=2\), \(c_\lambda=\cos(4\pi/9+\sqrt2r)-2\lambda/243\), and \(u_\lambda=1+2\lambda/243\). The common lower curvature is conservative for class one. On the retained mean-zero chart, with \(y=\theta-\Theta_1\), the proof coordinates \(\xi=A^{-1/2}y\), \(v=\gamma A^{1/2}x\) obey

\[ \xi'=v,\qquad v'=-Av-\nabla\mathcal W_\lambda(\xi),\qquad \mathcal W_\lambda=\frac{\eta}{2} [U_\lambda(\Theta_1+A^{1/2}\xi)-U_\lambda(\Theta_1)]. \]

With freshly reconstructed \(\underline\eta,\overline\eta\), its Hessian lies between \(\mu I\) and \(MI\), where

\[ \mu=\underline\eta c_\lambda g^2,\quad M=\overline\eta u_\lambda\Lambda^2,\quad \epsilon_L=g/4, \quad a_-=\mu/2+g^2/16,\quad a_+=M/2+\epsilon_L\Lambda/2+\epsilon_L^2. \]

The upper coefficient includes this law's \(u_\lambda\). Target tangent matching does not justify setting it to one. The modified energy

\[ \mathcal V=\tfrac12\|v\|^2+\mathcal W_\lambda +\epsilon_L\xi^Tv+\tfrac{\epsilon_L}{2}\xi^TA\xi \]

satisfies the same algebraic sandwich and derivative inequality,

\[ \tfrac14\|v\|^2+a_-\|\xi\|^2\le\mathcal V \le\tfrac34\|v\|^2+a_+\|\xi\|^2,\qquad \mathcal V'\le-\kappa\mathcal V, \quad \kappa=\min\{4(g-\epsilon_L)/3,\epsilon_L\mu/a_+\}. \]

This controls the actual nonlinear evolution inside its already admitted trap. It is not a tangent-only convergence claim.

Fix a common unforced dwell \(D=10^{13}\) in \(\tau\). The elementary own-law bounds \(0.048<c_\lambda<0.06\), \(9\,10^{-8}<\eta<10^{-7}\), \(u_\lambda<1.01\) give

\[ a_->1/400,\quad 21/400<a_+<53/1000,\quad 10^{-10}<\kappa<8/(35\,10^9),\quad 1000<\kappa D<16000/7<4096. \]

The initial modified-energy bound is \(V_0=3\overline\eta\Lambda b_x^2/4+a_+b_\theta^2/g<1/500\), using the freshly proved acquisition radii. Thus \(e^{-\kappa D}<2^{-512}\), with an exponent inside the shared arithmetic work limit. Exact rational return estimates yield

\[ \|x\|_2^2\le\frac{4V_0\,2^{-512}}{\underline\eta g} <\frac{4\,10^6}{9}2^{-512}<10^{-148},\qquad \|y\|_2^2\le\frac{\Lambda V_0\,2^{-512}}{a_-} <\frac85\,2^{-512}<10^{-148}. \]

Both norms are consequently below \(\varepsilon=10^{-70}\) at the common actual handoff \(\tau_h=100+10^{13}\), or \(t_h=(1024/1023)(100+10^{13})\). These conservative structural budgets are not laboratory times. The tiny bounds are retained as exact rationals.

Now prescribe the two bridges \((4,13)\), \((13,22)\), keeping the reached state unchanged. Each new edge has form and phase differences at most \(2\varepsilon\), so its actual work and the total joined excess obey

\[ 0\le\Delta H_{\rm attach}\le4(1+u_\lambda)\varepsilon^2, \qquad H_\lambda^- -H_{*,1}\le10(1+u_\lambda)\varepsilon^2. \]

The second estimate combines three isolated excesses bounded by \(2(1+u_\lambda)\varepsilon^2\) with both bridge costs. It preserves the relative phase origins and includes form storage. No loss is spent as an undeclared event reserve.

For each prescribed input \(f=ae_4+be_{22}\), with \((a,b)\in\{0,m\}^2\), \(m=7/10000\), formula (15) gives

\[ \left|\Delta H_{\rm input}-\tfrac32(a^2+b^2)\right| \le6(a+b)\varepsilon, \]

and the actual quotient radius after that input satisfies

\[ Z_+^2\le6\varepsilon^2+2(a+b)\varepsilon+a^2+b^2 -(a+b)^2/27<r^2. \]

Every branch's complete excess is less than \(1/486000\), because the largest nominal input work is \(3m^2=1.47\,10^{-6}\) and the retained corrections fit strictly below that barrier. Equations (13)--(14) therefore preserve the joined identity throughout all four unforced observation histories and afterwards. Contact and input occurrence are supplied events, not consequences of passivity or an autonomous selector.

The new degree-weighted means are retained explicitly. Just before input,

\[ \bar x_D^-=(x_4+2x_{13}+x_{22})/58,\quad \bar y_D^-=(y_4+2y_{13}+y_{22})/58,\quad |\bar x_D^-|,|\bar y_D^-|\le2\varepsilon/29. \]

Input adds \(3(a+b)/58\) only to the form mean. The four-history mean combination cancels by common pre-input state and affine jumps; individual histories need not have zero means. Reached states and event work need not agree between laws. Within each law all four branches share the same actual pre-input state. The entire joined raw state remains within \(\varepsilon\) of its nominal target before those inputs, which supplies the next proof's actual-state premise without resetting it.

A finite observation with the actual law's nonlinear remainder

This section fixes the mediator to class one, both endpoint laws \(\lambda\in\{0,1\}\), and the same two form impulses \(a=b=m=7/10000\). It proves a finite nominal estimate and its transfer to a full pre-input state within a declared radius. The following bounds use the alternative law's real derivatives, including its extra harmonics. They do not reuse a sine complex-analytic tail or evaluate a time-coefficient producer, a trajectory or a reserved response.

In the estimates below, \(t\) denotes elapsed \(\tau\)-time, not the original clock; the observation duration is converted back explicitly.

For a nominal post-input branch write \(u=ae_4+be_{22}\), with each amplitude either zero or \(m\), and introduce the analytical coordinate \(v=(\theta-\Theta_1)/\gamma\). The pair \(X=(x,v)\) retains all 54 coordinates; this scaling is neither a reduced model nor a phase reset. Use the maximum norm on this pair. The target-normalized quadratic and cubic current maps, including their factorials, are denoted by \(Q,C\). Their static target jets in (11) give

\[ \|Q(v,w)\|_\infty\le2\|v\|_\infty\|w\|_\infty, \qquad \|C(v,w,z)\|_\infty\le\frac43 \|v\|_\infty\|w\|_\infty\|z\|_\infty. \]

Indeed \(|j''|\le1\) and \(|j'''|\le1\) at every target edge for these two laws and class one. The fourth-derivative bound is instead the own-law global value from (8), \(|j_\lambda^{(4)}|\le1+(9^4+18^4/2)/2187=28\). Taylor's real remainder on each complete edge therefore gives the exact field

\[ X'=\mathcal J X+ \bigl(\gamma^3Q(v,v)+\gamma^4C(v,v,v)+R_4(v),0\bigr), \qquad \|R_4(v)\|_\infty\le\frac{56}{3}\gamma^5\|v\|_\infty^4, \]

where \(\mathcal J\) is the full tangent in these scaled coordinates. The normalized Laplacian and target phase Hessian have maximum-norm operator bounds two. Thus \(\|\mathcal J\|_\infty<3\), and the complete scaled field is globally Lipschitz with constant below three because \(2+2\gamma^2(1+2/243)<3\). For \(0\le t\le1/100\), \(e^{3t}<11/10\); this last rational bound also follows from \(e^{3/100}\le(1-3/100)^{-1}<11/10\).

Define the first three full amplitude variations by variation of constants: \(X_1=e^{\mathcal Jt}(u,0)\), with the second and third forced by \((\gamma^3Q(v_1,v_1),0)\) and \((2\gamma^3Q(v_1,v_2)+\gamma^4C(v_1,v_1,v_1),0)\), respectively. These are definitions used in inequalities, not evaluated histories. The phase row is always \(v_i'=Ax_i\). Direct integration gives

\[ \begin{array}{lll} \|X_1\|_\infty\le11m/10,& \|v_1\|_\infty\le11mt/5,& \|v_1-Aut\|_\infty\le9mt^2/4,\\ \|X_2\|_\infty\le4\gamma^3m^2t^3,& \|v_2\|_\infty\le2\gamma^3m^2t^4,&\\ \|X_3\|_\infty\le4\gamma^4m^3t^4,& \|v_3\|_\infty\le2\gamma^4m^3t^5.& \end{array} \]

For example, \(\|x_1'\|_\infty<9m/4\) proves the first-row phase approximation. The second-state coefficient is at most \((11/10)2(11/5)^2/3=1331/375<4\); the third-state coefficient is at most

\[ \frac{11}{10}\left[ \frac{(4/3)(11/5)^3}{4} +\frac{(88/5)\gamma^2t^2}{6}\right]<4. \]

All these bounds use \(\gamma<1/3000\) and \(t\le1/100\). The quadratic history has not been deleted: it contributes to the third variation and can be nonzero at unobserved coordinates.

To control the entire remaining amplitude series without assuming its convergence, let \(X_{\rm app}=X_1+X_2+X_3\). Its phase coordinate satisfies \(\|v_{\rm app}\|_\infty<9mt/4\), and \(\|v_2+v_3\|_\infty\le3\gamma^3m^2t^4\). Substitution into the exact field leaves quadratic and cubic defects each bounded by \(\gamma^5m^4t^4\): their sharper coefficients relative to that quantity are at most \(18\gamma^2t^2+18\gamma^4t^4\) and \(243\gamma^2t^2/4\), respectively. The fourth-order remainder is bounded by \((56/3)(9/4)^4\gamma^5m^4t^4\). Global Lipschitz stability and time integration therefore give

\[ \|X-X_{\rm app}\|_\infty \le\frac{11}{50} \left[2+\frac{56}{3}\left(\frac94\right)^4\right] \gamma^5m^4t^5 <106\gamma^5m^4t^5. \]

It remains to compare the complete cubic history to its first nonzero time term. Since \(\|e^{\mathcal Js}-I\|_\infty\le33s/10\), transport of the leading direct force contributes at most \((44/25)\gamma^4m^3t^5\). Replacing \(v_1(s)\) by \(Au s\) in its cubic force contributes at most \((10923/1250)\gamma^4m^3t^5\): the force difference before integration is bounded by

\[ \frac43\frac94 \left[\left(\frac{11}{5}\right)^2+ 2\frac{11}{5}+4\right]m^3s^4 =\frac{993}{25}m^3s^4. \]

The retained quadratic recoupling contributes at most \((242/75)\gamma^6m^3t^6\). Consequently

\[ \left\|x_3(t)-\frac{\gamma^4}{4}C(Au,Au,Au)t^4\right\|_\infty \le\left[\frac{44}{25}+\frac{10923}{1250} +\frac{242}{75}\gamma^2t\right] \gamma^4m^3t^5 <11\gamma^4m^3t^5. \]

At the nominal source the exact reflection/sign symmetry makes the central second variation zero, while the first variation cancels in the mixed observation. Combining the last two bounds costs less than \(12\gamma^4m^3t^5\) in each nonzero-input branch, since \(106\gamma m<1\). The zero-input nominal branch stays at equilibrium. The complete onset calculation (17), with all internal and bridge edges, therefore yields the finite estimate

\[ \boxed{\left|M_{\lambda,1}(m,m,T;0) -\frac{16\lambda-15-\cos\alpha}{512} \gamma^4m^3T^4\right| \le36\gamma^4m^3T^5},\qquad 0\le T\le\frac1{100}. \]

Fix the common observation duration to \(T=10^{-6}\) in \(\tau\), or \(1024T/1023\) in the original clock. Since \(\alpha>2/3\), cosine monotonicity and its alternating Taylor bound give \(\cos\alpha<1-2/9+2/243=191/243<4/5\). This and \(\gamma>1/3300\), together with \(1/2560-36/10^6>1/3000\), give opposite nominal signs with the same explicit lower margin:

\[ M_{1,1}(m,m,T;0)>S,\qquad M_{0,1}(m,m,T;0)<-S, \qquad S=\frac{m^3T^4}{3000\,3300^4}>9\,10^{-52}. \]

Now retain an actual pre-input full state satisfying \(\|(x,\theta-\Theta_1)\|_\infty\le\varepsilon=10^{-70}\), with its actual continuous lifts. Each law's four branches start from that same reached state before their prescribed form inputs. In these unscaled coordinates the global field Lipschitz constant is also below three. Each actual-to-nominal branch discrepancy at the observation time is below \(2\varepsilon\), hence the mixed discrepancy is below \(8\varepsilon\). This estimate retains arbitrary symmetry-breaking residuals; it does not impose nominal reflection on a reached source.

Declare a supplied absolute error allowance \(\delta=10^{-54}\) for each of the four raw mediator readings. Their mixed reading has error at most \(4\delta\), and \(8\varepsilon+4\delta<10^{-53}\). Thus every admitted reached state and every admitted readout error preserve the signs, with measured mixed responses above \(8\,10^{-52}\) for \(\lambda=1\) and below \(-8\,10^{-52}\) for \(\lambda=0\).

This is a prospective mathematical observation allowance. Its very small scale is not established laboratory resolution, and the finite horizon is in structural time. Reaching the pre-input radius, preserving identity at contact, and accounting for support and impulse work are supplied by the own-law handoff above. The theorem supplies neither a physical identification nor an evaluated experimental response.

Independent scalar calibration and the recorded distinction

Retain both complete laws, the original acquired class-one sources, their own-law recovery, attachment work, four prescribed inputs and fixed \(T=10^{-6}\) in structural \(\tau\)-time from the preceding theorem. Every elapsed duration below uses that clock. This section changes only the prospective observation. For each law write its four actual mediator readings as \(Y_{ij}=x_{13}^{ij}(T)\), and supply one static scalar sensor

\[ r_{ij}=h(Y_{ij})+\eta_{ij},\qquad |\eta_{ij}|\le\delta_h. \]

The same \(h\) is used in all four histories and both candidate laws. Its domain, orientation, units and calibration are specified independently of those responses. No particular sensor is measured here. The error \(\delta_h\) is in sensor units, not a transferred raw-form error allowance. The map is a passive observation, not a state-chart change; it changes none of the laws, inputs, capacities, clocks or work balances.

Absolute operating range and single-input increments

Let \(\varepsilon=10^{-70}\), \(m=7/10000\), and

\[ q=Y_{00},\qquad u=Y_{10}-q,\qquad v=Y_{01}-q,\qquad M=Y_{11}-Y_{10}-Y_{01}+q. \]

These are actual readings, with no resetting or subtraction of a newly chosen source mean. The full unscaled vector field is globally Lipschitz with constant below three. Within one law, branches with and without one input have initial full-state maximum-norm distance \(m\), while their mediator form initially agrees. Hence

\[ |u|,\ |v|\le m(e^{3T}-1)<4mT=:B. \]

Indeed the mediator derivative difference is bounded by \(3me^{3t}\); integrating it and using \(e^{3T}\le(1-3T)^{-1}\) proves the estimate. The same argument bounds \(|Y_{11}-q|<B\), because the two simultaneous inputs still have maximum norm \(m\). The zero-input branch stays within \(2\varepsilon\) of its nominal equilibrium, so \(|q|<2\varepsilon\). The raw readings therefore lie in the smaller interval with radius \(R_{\rm raw}=2\varepsilon+B\). Every \(q+s+t\) with \(s\) between zero and \(u\) and \(t\) between zero and \(v\), and the segment from \(q+u+v\) to \(Y_{11}\), lie in

\[ [-R_*,R_*],\qquad R_*=2\varepsilon+2B<10^{-8}. \]

This derives an absolute operating range from the admitted source origins; common form shifts would require a correspondingly shifted calibrated domain. The individually nonzero post-input means are still retained.

Calibration for a mixed threshold and the opposite signs

Supply a convex interval \(I\) containing \([-R_*,R_*]\), a sensor \(h\in C^2(I)\), and independently justified constants

\[ h'(z)\ge g>0,\qquad |h''(z)|\le K \quad(z\in I),\qquad \delta_h\ge0. \]

Here \(g\) has sensor units per form unit and \(K\) has sensor units per squared form unit. An arbitrary common sensor offset cancels. No upper gain bound is needed for the following one-sided sign certificate. Two applications of the fundamental theorem of calculus give the exact identity

\[ \begin{split} h(Y_{11})-h(Y_{10})-h(Y_{01})+h(Y_{00}) &=a_h M+\int_0^u\int_0^v h''(q+s+t)\,dt\,ds,\\ a_h&=\int_0^1 h'(q+u+v+\sigma M)\,d\sigma\ge g. \end{split} \]

The integrals are oriented when an increment is negative; their absolute value is at most \(K|uv|\le KB^2\). Let \(s_0,s_1>0\) be the fresh carried raw-response margins, so every admitted source obeys \(M_0\le-s_0\) and \(M_1\ge s_1\). These margins include the full nonlinear tail and actual-state residual, but no raw-reading error: the sensor error is added separately. The recorded mixed scalars therefore obey

\[ M_{r,0}\le-\sigma_0,\qquad M_{r,1}\ge\sigma_1,\qquad \sigma_\lambda=g s_\lambda-KB^2-4\delta_h. \]

Strictly positive \(\sigma_0,\sigma_1\) certify opposite recorded signs. Strict separation by a mixed threshold needs only \(\Delta_h=\sigma_0+\sigma_1>0\): the fixed threshold \((\sigma_1-\sigma_0)/2\) lies strictly between the two admitted mixed ranges even when one sign margin is nonpositive. The resulting complete four-record distance is at least \(\Delta_h/4\) in maximum norm. The four errors may be correlated or adversarial; no cancellation or independence is assumed. A nonpositive sufficient margin is an unavailable certificate for its stated statistic, not proof of observational equivalence.

One explicit prospective contract, in the chosen structural and sensor units, is

\[ I=[-10^{-8},10^{-8}],\qquad g=\tfrac12,\qquad K=10^{-35},\qquad \delta_h=10^{-54}. \]

The preceding theorem gives \(s_\lambda>8\,10^{-52}\), whereas \(KB^2=7.84\,10^{-53}\). Thus \(\sigma_\lambda>3\,10^{-52}\) under both endpoint laws. This class is nonempty and includes nonlinear maps: for example \(h(z)=b+z+z^2/(2\,10^{35})\) satisfies the contract for any fixed offset \(b\). The small curvature allowance is a calibration premise, not a discovered property of an apparatus. No laboratory access to that allowance, the source accuracy, recovery dwell or observation clock follows.

Full records, sensor nuisance and a nonlinear sign counterexample

There is a separate consequence needing only positive gain. Fix one common \(h\) with \(h'\ge g\) throughout the interval containing all raw readings. For any pair of admitted raw four-vectors \(Y^0,Y^1\), possibly from different reached sources under their respective laws, the mixed functional has coefficient norm four, so

\[ \|Y^1-Y^0\|_\infty\ge\frac{s_0+s_1}{4}. \]

Applying the same increasing sensor to both vectors and then admitting their individual errors yields the complete-record bound

\[ \boxed{\ \|r^1-r^0\|_\infty \ge \frac{g(s_0+s_1)}4-2\delta_h\ }. \]

A positive right-hand side separates those four-reading record sets for that common sensor even without a curvature bound. With \(g=1/2\) and \(\delta_h=10^{-54}\) it exceeds \(10^{-52}\). This is a fixed-common-map statement, uniform over the admitted choices of that map. It does not establish separation after independently unioning the two laws over different nuisance maps \(h_0,h_1\). By contrast, whenever the curvature-based \(\Delta_h\) is positive, the same mixed threshold works for every admitted sensor, so even those separately unioned classes retain that separation. The explicit calibration above also preserves both signs. Neither argument fits an observation law to a response.

Monotonicity alone does not preserve the mixed sign, even for these actual TNFR histories. The mediator derivative difference at the input time is exactly \(m/4\) for either single input: only its adjacent form value changes, and the phases initially agree. Relative to the common target, the initial full-state norms are at most \(m+\varepsilon\) for that branch and \(\varepsilon\) for the zero-input branch. Since the field vanishes at the target, their displacements through time \(t\le T\) are bounded by \(4(m+\varepsilon)t\) and \(4\varepsilon t\). Lipschitz continuity and integration then give

\[ |u-mT/4|,\ |v-mT/4|\le6(m+2\varepsilon)T^2, \qquad u,v>mT/8. \]

The finite raw-response bound also implies \(|M_\lambda|<10^{-48}\): the absolute leading coefficient is at most \(\gamma^4m^3/32\), the additional time-tail coefficient is \(36T<1/32\), and \(\gamma<1/3000\), with the residual \(8\varepsilon\) retained. Now fix the single sensor \(h(z)=z+z^2\) on \(I=[-10^{-8},10^{-8}]\). It has \(h'(z)\ge1-2\,10^{-8}>1/2\), but its curvature is two. Its exact noiseless mixed scalar is

\[ [1+2(q+u+v)+M]M+2uv. \]

The bracket lies between \(1-2\,10^{-8}\) and \(1+2\,10^{-8}\), because it is the average derivative on the segment already admitted. Thus the recorded mixed scalar under either law exceeds

\[ \frac{m^2T^2}{32}-2\,10^{-48}-4\,10^{-54}>10^{-20}. \]

Both signs are positive although the raw signs are opposite. This example uses the actual full source families and declared error bounds, not a synthetic trajectory or an evaluated response. Nevertheless, the fixed-common-map complete-record separation above remains positive. Losing this particular mixed-sign statistic therefore does not erase the law distinction or establish an overlap of complete recorded observations.

What this comparison constrains, and what it leaves open

The prescribed source in (17) is an exact organized target. The acquisition section proves own-law capture from the original isolated source families; it does not replace that nominal source by a reached fine state or transfer the onset sign to its residuals. Arbitrary residuals need not have the nominal reflection symmetry. The finite theorem above retains those residuals under its explicit pre-input radius and readout allowance, with recovery and contact now admitted separately above. Acquisition alone does not establish that smaller radius or admit contact. No sign at the earlier \(T=1/8\) is asserted for the alternative law.

The interaction differs in the held structural chart with the same form impulses and mediator readout. A common clock rescaling cannot account for it while leaving the nonzero tangent (12) fixed. This does not exclude every nonlinear conjugacy accompanied by changed inputs and observations.

The supplied additive current consumes first, ninth and eighteenth phase harmonics. Its extra information is outside a law depending only on degree and the first neighbor phasor. The existing first-moment sufficiency theorem can constrain that freedom only if its information restriction is independently justified. Restating that restriction, imposing cosine storage globally, or assuming the desired current is not a new derivation.

These complete models retain signed form, primitive circular phase, whole-state memory, common-origin covariance, held capacities and the declared identity. Additive edge currents, fixed support, constant reciprocal mobility and the pressure completion remain supplied hypotheses. No registered operator's descent, autonomous event occurrence, inherited scale law, composite binding energy or fundamental constituent follows. The foundation compatibility audit owns those distinctions; the execution plan owns subsequent work.

The separate registered-composition comparison derives inherited complete laws for two different two-C9 interfaces, with their own support work and actual-source bounds. Those conditional maps do not supply an inherited law for the three-C9 support used here.

Implementation and independent controls

_sine_class_constitutive_comparison.py provides detached exact/static descriptions of this family, target jets, local interaction coefficients and whole-support balance observations. These static descriptions allow every \(\lambda\ge0\); the uniform local trapping proof above is restricted to \(0\le\lambda\le1\). Its independent controls rebuild target derivatives and mixed edge weights on all 27 nodes, check both evolution rows and event/loss balances, and retain invalid-domain and symmetry controls. This does not install an alternative live law or reuse a sine response as evidence for it. No trajectory, selected time-coefficient producer or frozen scientific response is regenerated.

The separate _sine_class_constitutive_acquisition.py rebuilds the isolated full-family transit, law-specific storage bounds and capture margins from primitive inputs, using shared semigroup arithmetic without treating the sine preparation's storage as the alternative law's. Its controls check the forcing versus initial-error norms, quartic source correction, split endpoint storage and strict capture margins. The uniform conclusion uses the proved monotonic dependence and the endpoint \(\lambda=1\); it does not require a sweep or a trajectory producer.

The actual-source handoff uses the shared modified-energy kernel with the actual potential's upper Hessian. Its controls retain raw source errors, uninterrupted recovery, support work, input work and joined identity. The finite discriminator then combines fresh handoff and static onset with the real nonlinear tail. Its independent controls rebuild the tail constants and response coefficients, retain exact tiny rational records and reject unsupported source or readout margins. The returned intervals admit a prospective comparison; they are not evaluated responses or calibration records.

The scalar-sensor assessment rebuilds that raw comparison with zero raw-reading error before admitting the independent operating-domain, gain, curvature and sensor-error bounds. It distinguishes uniform mixed-threshold separation, preservation of both signs and a fixed-common-map complete-record certificate. Its controls exercise strict calibration margins, the retained absolute range and the actual-source quadratic-sensor sign counterexample. No sensor callable, cached report or response-derived calibration substitutes for these primitive premises.