Nonlinear excess storage and loss of a joined class interaction

The distinct-neighbor response and its retained comparison distinguish joint form response from the sum of nonlinear single-neighbor responses. A form-reading sign does not determine an energetic sign. This owner derives a different conditional property: the same interaction has a nonzero rectangular storage difference after removing the complete tangent model's own quadratic interaction.

This is a response-free theorem under the supplied smooth-sine law. Static graph powers and rational inequalities are evaluated; no trajectory, finite-time variational coefficient or retained response is evaluated here. The execution plan owns the next gate. Structural storage is not identified with physical energy.

Complete models and the storage observation

Retain the three joined C9 patterns, all 29 unit edges, unit capacities, central degrees (3,4,3), and class (1,2,1). The proof also applies to the existing class (1,1,1) separately; it makes no cross-class distinction. Write \(D=\operatorname{diag}(d_i)\), \(A=D^{-1}L\), and \(\gamma=1/(1023\pi)\). Primes below denote the structural clock \(\tau=(1023/1024)t\). All forms and phase deviations use the original unscaled coordinates, with continuous lifts about the wound target \(\Theta\). Let \(U(\theta)=\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]\) and \(L_c=\nabla^2U(\Theta)\), the full target Hessian including bridges. The complete nonlinear and tangent laws are

\[ \begin{aligned} x'&=-Ax-\gamma D^{-1}\nabla U(\Theta+y),&y'&=\gamma Ax,\\ \xi'&=-A\xi-\gamma D^{-1}L_c\eta,&\eta'&=\gamma A\xi. \end{aligned}\tag{1} \]

Each model retains 27 form and 27 phase coordinates. In particular, \(\eta\) is not the scaled phase \(y/\gamma\) of the coefficient producer. Both laws have the same complete pre-event source \((x_0,y_0)\), with the original acquired-family component Euclidean bounds \(\epsilon=10^{-32}\), zero sums and declared origins. No actual-source reflection symmetry is assumed.

At time zero both models receive the corresponding form jump \(f_{00}=0,f_{10}=ae_4,f_{01}=be_{22},f_{11}=ae_4+be_{22}\). Phases do not jump. Keep the existing word

\[ a=b=m=7/10000,\qquad H=1/8.\tag{2} \]

Define target-relative full storage and tangent storage on their own states:

\[ \mathcal H_N=\tfrac12x^TLx+U(\Theta+y)-U(\Theta),\qquad \mathcal H_T=\tfrac12\xi^TL\xi+\tfrac12\eta^TL_c\eta. \tag{3} \]

For history weights \((1,-1,-1,1)\), write \(\Delta_4\) for their weighted sum and set

\[ E_{\rm ex}(H)=\Delta_4[\mathcal H_N(H)-\mathcal H_T(H)].\tag{4} \]

Evaluating the Hessian storage at the nonlinear endpoint would define a different observable. A linear system already has quadratic cross-energy; subtracting its complete response in (4) is essential. The functional uses the model's dimensionless structural normalization; its rate has units per \(\tau\). No joule, mass, force or laboratory-clock bridge is supplied.

Exact event and flow balance

The whole-support balance gives, by the two cancellations between form and phase rows,

\[ \mathcal H_N'=-\mathcal D(x),\qquad \mathcal H_T'=-\mathcal D(\xi),\qquad \mathcal D(v)=(Lv)^TD^{-1}(Lv).\tag{5} \]

No commutation of \(L\) and \(L_c\) is needed. Although each loss is nonnegative, their difference and its rectangular sum are signed. For a form jump \(f\), both stores change by \(f^TLx^-+\tfrac12f^TLf\). Here their pre-event forms coincide, so the work agrees in each history. Also \(L_{4,22}=0\), giving \(W=a(Lx_0)_4+b(Lx_0)_{22}+3(a^2+b^2)/2\) in the joint history. The common initial storage defect is the same in all four histories. Consequently

\[ E_{\rm ex}(H)=-\int_0^H \Delta_4[\mathcal D(x(s))-\mathcal D(\xi(s))],ds.\tag{6} \]

Delayed jumps after the two states separate would require their different work terms. Contact/acquisition costs retain their earlier ledger; no loss is reassigned as an event reserve. A negative (4) means positive integrated rectangular excess loss in (6), not greater loss in every individual branch.

Local onset and the role of hidden coordinates

For the nominal source, simultaneous port-fixing ring reflection with sign reversal takes input \((a,b)\) to \((-a,-b)\) and preserves either storage. The full-minus-tangent scalar is therefore even in an auxiliary common amplitude. Its quadratic term vanishes because both models share the complete first variation. The first possible excess is quartic, not cubic.

Use the complete amplitude hierarchy \((x_1,y_1),(x_2,y_2),(x_3,y_3)\). For \(B(u,v)=u^TLD^{-1}Lv\), the quartic loss difference is \(2B(x_1,x_3)+B(x_2,x_2)\); the cubic cross term vanishes by reflection. The hidden quadratic level is retained. At the earliest time order, \(y_1=\gamma Af\,s+O(s^2)\) and \(x_3=\gamma^4 T(Af,Af,Af)s^4/4+O(s^5)\), where \(T\) includes the sine factorial \(-1/6\). Oriented edge summation in (6) gives

\[ E_{\rm ex}(H;0)=-\frac{\gamma^4H^5}{60} \Delta_4\sum_e c_e\,[\Delta_e A^2f]\,[\Delta_e Af]^3+O(H^6). \tag{7} \]

Here \(\Delta_e v=v_j-v_i\), not the four-history difference. The exact coefficients of \((a^3b,a^2b^2,ab^3)\) in the edge sum are

Edge channel	Coefficients before its target cosine
Donor internal	\((9/16,0,0)\)
Mediator internal	\((5/64,15/128,5/64)\)
Receiver internal	\((0,0,9/16)\)
Both bridges	\((341/96,145/64,341/96)\)

For equal inputs the sum is \(m^4[9\cos(2\pi/9)/8+35\cos(2k\pi/9)/128+1799/192]\). The next sections, rather than this local onset alone, certify the finite word.

Finite direct quartic term from static edge cones

Put \(g=1/3000>\gamma\), \(D_H=1-2g^2H^2\), and \(P(s)=\exp(-As)\). The existing edge cones bound the complete tangent phases \(y_{1,p}=\gamma w_p\) by \(\Delta_e w_p(s)/s\in I_{p,e}\), for \(0<s\le H\). They include tangent phase feedback. Let \(U_e=I_{4,e}\), \(V_e=I_{22,e}\).

Self-adjointness of \(P\) in the \(D\)-inner product, and its commutation with \(A\), give \(B(P(t)f,P(t-s)r)=B(P(2t-s)f,r)\). The extra spatial columns needed are \(A^j e_p\), \(j=2,3,4,5\). For every \(0\le v\le2H\), Taylor's integral remainder and heat contraction give the following outward cone for \(\Delta_e A^2P(v)e_p\):

\[ J_{p,e}=\Delta_e A^2e_p -\Delta_e A^3e_p[0,2H] +\tfrac12\Delta_e A^4e_p[0,(2H)^2] +[-r_p,r_p],\quad r_p=\frac{2(2H)^3}{6}\|A^5e_p\|_\infty.\tag{8} \]

All these powers are static exact rational graph operations, not evaluated phase or form responses. Multiplication in (8) means interval multiplication.

For the direct cubic heat form \(z_3=\gamma^4\int_0^tP(t-s)T(w_1^3)(s)ds\), the rectangular loss pairing with the heat form has edge factor

\[ K_e=C_e\{J_{4,e}(3U_e^2V_e+3U_eV_e^2+V_e^3) +J_{22,e}(U_e^3+3U_e^2V_e+3U_eV_e^2)\}.\tag{9} \]

Here \(C_e=[1/6,1]\) on every internal edge and \(C_e=1\) on the bridges. These enclose both classes' exact target cosines. Independent interval products relax shared dependencies outward. Summing all 29 edges at (2) gives a lower endpoint greater than 0.50635907863 and an upper endpoint less than 16.07223944120; in particular \(1/2<\sum_e K_e<17\).

The double-integral weight is \(\int_0^H\int_0^t s^3dsdt=H^5/20\). The direct contribution consequently obeys the deliberately coarser rational certificate

\[ -\frac{17}{60}g^4m^4H^5<E_{\rm direct} <-\frac{m^4H^5}{120\,3300^4}.\tag{10} \]

The lower bound \(\gamma>1/3300\) is the same elementary coefficient bound used in the neighbor theorem. Neither selected output nor a favorable subset of edges is used in this sign proof.

Complete quartic, amplitude-tail and source errors

Since \(\sum_i d_i=58\) and \(\|A\|_\infty\le2\), \(|B(u,v)|\le232\|u\|_\infty\|v\|_\infty\). All three nonzero nominal histories satisfy \(\|f\|_\infty,\|Af\|_\infty\le m\), including the simultaneous one. The complete hierarchy bounds give

\[ \begin{aligned} \|x_1\|&\le m/D_H,&\|x_1-P(t)f\|&\le g^2mt^2/D_H,\\ \|z_3\|&\le g^4m^3t^4/(3D_H^3),& \|x_2\|&\le 2g^3m^2t^3/(3D_H^3),\\ \|x_3-z_3\|&\le 4g^6m^3t^6/(15D_H^4) +g^8m^3t^8/(63D_H^5). \end{aligned}\tag{11} \]

Norms here are maximum coordinate norms. Integrating \(2B(x_1-Pf,z_3)+2B(x_1,x_3-z_3)+B(x_2,x_2)\) and counting three nonzero histories gives the complete quartic correction

\[ E_4=3\cdot232m^4\left\{ g^6H^7\left[\frac2{21D_H^4}+\frac8{105D_H^5}+\frac4{63D_H^6}\right] +\frac{2g^8H^9}{567D_H^6}\right\}.\tag{12} \]

Index the histories by \(h\in\{00,10,01,11\}\), with \((A_{00},A_{10},A_{01},A_{11})=(0,m,m,2m)\); both single-input errors are counted. For each nonzero history variation \(A_*=A_h\), the existing complex-amplitude disk has radius \(\rho=(2gA_*)^{-1}\), full phase norm at most one, and nonlinear-minus-tangent form norm at most \(8gt/D_H\). On it \(\|x_N\|\le\rho A_*+8gt\) and \(\|x_T\|\le\rho A_*/D_H\). The exact loss balance bounds the analytic storage difference on that disk by

\[ C_H=232\left[\frac{4H^2}{D_H^2} +\frac{64g^2H^3}{3D_H}\right],\qquad R_6(A_*)=\frac{C_H(2gA_*)^6}{1-(2gA_*)^2}.\tag{13} \]

Indeed the loss integrand is at most \(232[8t/D_H^2+64g^2t^2/D_H]\). This analytic extension uses the bilinear form \(B\), without complex conjugation. Evenness and the vanishing quadratic excess allow the Cauchy tail to start at degree six. Set \(R_6(0)=0\); the disk is an auxiliary proof device, not complex physical form.

Actual-source transport requires a separate argument. Write \(\ell_H=1-2gH\), \(\sigma=\epsilon/\ell_H\), and let \(E_A=E_{\rm init}(A_*,H,\epsilon)\) be the shared nonlinear initialization defect. The nominal full-minus-tangent form has the bound \(d_A=2g^3A_*^2H/D_H^3\). Its change from adding the complete source is bounded by \(E_A\), whereas the tangent's source change is bounded by \(\sigma\). Expanding \(B(2x_T+d,d)\), with \(d=x_N-x_T\), proves

\[ E_s=232H\sum_{A_*\in(0,m,m,2m)} \left[2\sigma d_A+ (2A_*/D_H+2\sigma+2d_A+E_A)E_A\right].\tag{14} \]

This controls source effects in the quadratic loss functional; it does not reuse linear scalar-output cancellation. The zero-input source term is retained. Parity is used only for the nominal source.

At (2) and \(\epsilon=10^{-32}\), exact rational bound evaluation gives \(E_4<2.568\,10^{-38}\), \(\sum R_6<9.885\,10^{-36}\), and \(E_s<9.476\,10^{-44}\). The total is below \(9.911\,10^{-36}\). Adding these errors to (10) proves, for every admitted common source,

\[ -2.565\,10^{-32}<E_{\rm ex}(1/8) <-5.049\,10^{-34}<0.\tag{15} \]

Thus the same fixed interaction has a finite nonlinear rectangular loss excess beyond the complete tangent model. This is a conditional theorem, not an independently evaluated storage response. A zero input or zero horizon gives exact zero by identical-history cancellation, without requiring parity.

Observation requirements and the remaining comparison

The earlier allowance for four mediator form readings supplies no storage sensor. For example, at the wound phase target the forms 0 and \(s(e_0-e_1)\) have the same mediator reading, component form sum, weighted form charge and winding, but their stores differ by \(3s^2\) and their instantaneous losses by \(10s^2\). Arbitrarily small such perturbations fit the geometric source envelope. This is an instantaneous observation obstruction; it does not assert both states were acquired or prove a four-history inverse obstruction for the selected preparation.

A sufficient alternative observation map consumes original form coordinates and phase deviations of each full and tangent state. If their true coordinate bounds are \(X,Y\), with coordinate errors \(\delta_x,\delta_y\), then stationarity \(\nabla U(\Theta)=0\), edge differences and \(|\cos|\le1\) give, for either store,

\[ |\widehat{\mathcal H}-\mathcal H| \le116(X\delta_x+Y\delta_y)+58(\delta_x^2+\delta_y^2).\tag{16} \]

Eight such state observations suffice for (4), so eight times (16) is a total recording allowance \(r\) when common bounds apply. Alternatively, independently justified scalar storage allowances may be summed. A computed tangent state still needs its own initialization and numerical error budget; it is not assumed exactly observed.

For a true band \([L_E,U_E]\), the recorded band is \([L_E-r,U_E+r]\). Recorded negativity requires \(U_E+r<0\). Separation from an independently recorded zero-excess functional with the same allowance requires \(U_E+2r<0\). Equation (15) therefore supplies the conservative sufficient total allowance \(r\le2.524\,10^{-34}\) for that stronger comparison. This is a structural mathematical requirement, not demonstrated laboratory resolution. The previous form-reading budget cannot be inserted into (16) without a new observation model.

An eventual evaluation must propagate the full and tangent models from the same admitted complete source, retain their event/loss ledgers and produce independent numerical enclosures. Direct subtraction of large absolute cosine stores can destroy the required resolution; a validated excess/loss representation needs its own contract. The proof must not narrow its response interval. Existing endpoints would supply retrospective model-state evidence, not a new reserved storage observation. Binding, autonomous preparation, universal forces and physical energy identification remain open.

Shared implementation and checks

The private storage-excess owner reuses target coefficients, normalized support, neighbor phase cones and source fidelity. It admits primitive bounds before constructing coefficients, keeps direct/quartic/tail/source bounds separate and introduces no integrator or runtime law. The independent tests check edge algebra, complete balances, event work, finite rational bounds, observation errors and invalid domains under the shared no-regeneration guard.

Prepared passive-loss observer

The detached observer provides bound_sine_class_storage_readout with nine mandatory primitives: mediator_class, initial_form_bounds, initial_phase_deviation_bounds, donor_amplitude, receiver_amplitude, horizon, time_step, order and max_steps. It retains all 27 form and 27 phase coordinates, adding only a passive cumulative loss row \(\ell'=\mathcal D(x)\) to each model. The shared validated box Taylor kernel and its dimension limit are unchanged.

One complete pre-input source cover and the named exact target are shared associations. Full phases are constructed as \(\Theta+y_0\); tangent phases are \(y_0\) in original radians. The history order is (00,10,01,11), each full then tangent. Events change only forms at nodes 4 and 22. The loss accumulator starts at zero and carries without reset. The source, acquisition, same-source correlation and exact target remain mathematical premises; independent interval enclosures relax their correlations conservatively.

The complete reported observable is \(-\Delta_4(\ell_N-\ell_T)\), which equals (4) by the matched initial/event cancellation. Retained Taylor loss increments remove the accumulator baseline. This avoids subtracting large absolute cosine stores, but does not guarantee a sufficiently narrow full-minus-tangent interval. Raw accumulator endpoints and each individual loss remain available. A failure stops later flow calls and events and retains the completed prefix; incomplete sets provide no mixed verdict.

The arithmetic reader re-admits primitives and rebuilds target/law, event carry, every consumed 55-coordinate Taylor step and the signed observation before comparing cached fields. Derivative and strict Picard generation remain execution premises. Controls cover invalid domains, independent field algebra, synthetic carry/failure/corruption, and an unrelated small reference; they do not evaluate the selected word (2).

This is a prepared instrument. No storage source/protocol freeze, reserved attempt or selected storage response exists. Observation and numerical resolution policies still need admission before such a comparison. Its status and reopening conditions belong solely to the execution plan.