Prepared mobile encounter and a controlled interaction observable

Scope: two supplied localized preparations evolve on the same fixed cycle under the complete positive-loss law of prepared transport. The results below retain two band observations through their encounter and bound a prospective nonlinear response calculation. The separate retained endpoint assessment now certifies the nominal nonlinear defect; the original source-family test remains unresolved. No paired nonlinear trajectory was evaluated. Interaction, a new organization and physical identification remain separate claims. The research program owns the next task.

1. Common support, complete law and shared preparation

Keep signed real form x, real phase lifts theta about the uniform zero background, held unit capacities, unit cycle conductances and lengths, beta one, positive actual coefficients e,w, and no inputs or events. In the declared clock tau=w*t/pi, both rows remain

\[ x'=-\sigma Ax+\tfrac12S(\theta),\qquad \theta'=Ax, \qquad A=L/2,\quad \sigma=\pi e/w. \tag{1} \]

The notation z=x+i theta retains two real channels; its squared modulus is a localization diagnostic. It is not a complex form coordinate or probability. Let v(j,tau) be the norm-a Fejer reference in the transport owner. For an even integer 0<T<=N/2, set c=T/2 and prescribe

\[ v_A(j,\tau)=v(j-c,\tau),\qquad v_B(j,\tau)=v(-j-c,\tau). \tag{2} \]

Their centers are c-tau and -c+tau. Their disjoint Fourier bands are B_+={N/4+m:|m|<Q} and B_-=-B_+. Thus their joint reference has norm sqrt(2)*a at every time, including spatial overlap. Their relative internal registration is fixed by (2), with no later fit. In the fixed case c=128, the carrier factors at this integer translation are exactly unity.

Write Phi_tau for the actual complete flow. Choose two primitive states

\[ u_A=v_A(0)+r_A,\quad u_B=v_B(0)+r_B, \qquad \|r_A\|,\|r_B\|\le\epsilon. \tag{3} \]

The three histories are z_A=Phi(u_A), z_B=Phi(u_B) and z_AB=Phi(u_A+u_B). The same primitive errors are reused in the joint state; there is no third independent preparation error. This is a mathematical source association, not physical copying or source authentication. All fine coordinates and common means are allowed in each ball. The supplied support already connects every site; encounter means overlap of moving observations, not an attachment event.

2. Joint identity and band localization throughout the encounter

Reuse the transport owner's actual contractive tangent U(tau)=exp(tau J_sigma) and its conservative comparison only for the template error. Put

\[ d=2\sigma T+\frac T6\left(\frac{2\pi(Q-1)}N\right)^3, \qquad h=\epsilon/a+d+\tfrac{32}3Ta^2, \]

\[ E=2\epsilon+\sqrt2\,a d+\tfrac{32}3Ta^3. \tag{4} \]

The joint nominal tangent error is at most sqrt(2)*a*d, by orthogonality of the conservative bands and the common operator bound. Initial joint error is at most 2*epsilon. Inside ||z_AB||<2a, the cubic remainder contributes at most (32/3)*T*a^3. Consequently the strict condition sqrt(2)*a+E<2a closes that bootstrap and proves

\[ \|z_{AB}(\tau)-v_A(\tau)-v_B(\tau)\|\le E, \qquad 0\le\tau\le T. \tag{5} \]

The individual bounds require h<1 as before. These small-state bounds also retain the common phase-lift chart in the fixed protocol. General uses must retain their chart admission; a lift is not reconstructed from form.

Let P_+,P_- be the orthogonal complex Fourier projections. They are fixed linear observations of the two real channels. Equation (5) gives

\[ \|P_+z_{AB}-v_A\|\le E,\quad \|P_-z_{AB}-v_B\|\le E,\quad \|P_\pm z_{AB}\|\ge a-E. \tag{6} \]

The omitted bands have norm at most E. With the same reference tail bound H=N^3/[16 Q^3(R-1)^3], each projected component's fraction outside its own moving radius-R window is at most

\[ \left(\frac{\sqrt H+E/a}{1-E/a}\right)^2, \qquad E<a. \tag{7} \]

For N=1024,Q=16,a=2^-12,epsilon=a/1000,T=256,R=96, and the unchanged e=2^-20,w=1-2^-20, outward bounds give

\[ E/a<0.051378,\qquad E/(\sqrt2 a)<1/25,\qquad (7)<0.039954<1/25. \]

This strict concentration result can also be checked without decimal approximations: use pi<22/7, sqrt(2)<70711/50000 and sqrt(H)<432/3125 in (4) and (7). The old looser root bound 1/7 is insufficient for this particular paired certificate.

Both band observations therefore stay nonvanishing and more than 24/25 concentrated in their declared moving windows. The windows meet at tau=128 and are disjoint again at tau=256. The per-component full-template error is E, which is larger than the previous single-packet a/25 tolerance; that entire old contract is not transferred unchanged. Band projections are not closed nonlinear subsystems, separately formed NFRs, or an additive decomposition of the joint local squared density. During overlap, no unique spatial peak or assignment of sites to particles is required.

3. A nonlinear response with an exact linear null control

Define the full-state superposition defect and fixed endpoint observation

\[ I(\tau)=z_{AB}(\tau)-z_A(\tau)-z_B(\tau), \qquad q=\|I(T)\|_2. \tag{8} \]

Every linear flow on the same support gives I=0 for these shared primitive states, including their errors. In contrast, |z_A+z_B|^2-|z_A|^2-|z_B|^2=2 Re(z_A conj(z_B)) can be nonzero under perfectly linear evolution. Intensity interference is therefore not this interaction discriminator. With an independently perturbed joint source, even the linear defect is U(tau)[u_AB-u_A-u_B]; it cannot be discarded.

For an oriented cycle incidence D, the form-row remainder and its cubic part are

\[ R(\theta)=-\tfrac12D^T[\sin(D\theta)-D\theta],\qquad C_3(\theta)=\tfrac1{12}D^T(D\theta)^3. \tag{9} \]

All powers in edge coordinates are componentwise. With y_A=U(tau)u_A, y_B=U(tau)u_B, and their phase rows eta_A,eta_B, the prospective cubic approximation to (8) is

\[ I_3(T)=\int_0^T U(T-s) \binom{\tfrac14D^T[(D\eta_A)^2D\eta_B+ (D\eta_A)(D\eta_B)^2]}{0}\,ds. \tag{10} \]

This uses the actual damped tangent, not the rigid moving templates or the conservative dispersion law. It retains both rows and the entire history. A nonzero mixed instantaneous rate alone does not decide q: the propagated contributions in (10) can cancel.

A rigorous fifth-order remainder

Suppose an actual trajectory and its own tangent stay within full-state norm B and phase supnorm S. Elementary sine bounds and ||D||<=2 give

\[ \|R\|\le\tfrac43 S^2 B,\quad \|R-C_3\|\le\tfrac4{15}S^4 B,\quad \|DR\|,\|DC_3\|\le4S^2. \tag{11} \]

For example ||D theta||_infinity<=2S and ||D theta||_2<=2B prove the first two inequalities. The derivative bound holds along convex interpolations with the same supnorm cap. Duhamel contraction gives ||z-y||<=(4/3)*s*S^2*B. Substituting it into the cubic Taylor comparison and integrating proves the one-history remainder

\[ K(B,S)=\left(\tfrac4{15}T+\tfrac83T^2\right)S^4B. \tag{12} \]

Use B_s=9a/8 for each single and B_p=3a/2 for the pair. Their independent norm bootstraps close when

\[ a+\epsilon+\tfrac43T B_s^3<B_s,\qquad \sqrt2 a+2\epsilon+\tfrac43T B_p^3<B_p. \tag{13} \]

Since |F_Q|<=Q, the complete template bounds supply

\[ S_s=\frac{aQ}{\sqrt{NS_2}}+ah,\qquad S_p=\frac{2aQ}{\sqrt{NS_2}}+E. \tag{14} \]

These also bound the actual tangent histories: omitting the nonlinear template error only decreases their bound. Equations (13), the single and joint template bootstraps, and (14) apply to every preparation in (3). Summing three one-history remainders yields

\[ \|I(T)-I_3(T)\|\le K_5=K(B_p,S_p)+2K(B_s,S_s). \tag{15} \]

The simpler substitution S=B is valid but needlessly discards localization. Using the already proved Fejer height is a shared improvement to both nonlinear-response and source-sensitivity bounds, without any response fit.

Retaining the correlation of source errors

Let I^0 denote (8) for the exact nominal sources in (2). The variational equation and (11) bound the derivative of Phi_T-U(T) by exp(4S^2T)-1. Every straight interpolation from a nominal source to (3) remains in its original error ball, so the same bounds apply throughout. The linear terms cancel exactly in the shared three-history difference. For k_s=4S_s^2T<1 and k_p=4S_p^2T<1, this gives the rational bound

\[ \|I(T)-I^0(T)\|\le V=2\epsilon\left(\frac{k_p}{1-k_p}+\frac{k_s}{1-k_s}\right). \tag{16} \]

Here exp(k)-1<=k/(1-k) for 0<=k<1. This is of order epsilon*a^2, not an independent additive three-reading error. It is not a statement about physical sensor uncertainty.

For the fixed inputs, (13) holds and (14) gives S_s<19a/100 and S_p<9a/25. Even these convenient rational caps yield

\[ K_5<4.266\,10^{-15},\qquad V<4.939\,10^{-12}. \tag{17} \]

The calculator retains the tighter actual bounds from (14). These are prospective error allowances, not a measured or inferred interaction size.

4. Frozen endpoint test before response evaluation

Retain exactly the fixed inputs above and the source registration in (2). The reserved scalar is q at tau=256; no endpoint, carrier, amplitude or relative origin is selected from an evaluated response. Calculate (10) once for the nominal sources and seek a certified approximation with full Euclidean error at most eta_num=a^3/1000. This numerical error includes source materialization, elementary functions, propagators, integral approximation and norm reduction; a sampled integral alone is insufficient.

The two prospective levels are fixed in advance:

• Nominal source: a lower bound on ||I_3^0(T)|| strictly above K_5 proves q>0 for the specified exact preparations.
• Whole original source family: a lower bound strictly above K_5+V proves q>0 for every pair of primitive errors in (3), reused as declared.

For a point approximation whose norm is used instead of an outward norm interval, add eta_num to each threshold. Do not subtract numerical error twice. Failure at either level means the corresponding sufficient test does not resolve a response; it proves neither exact superposition nor impossibility of observing an interaction. These criteria were fixed before evaluating (10); the result owner records both verdicts without changing the original protocol.

The admitted calculation is a detached certificate for (10), using finite cycle Fourier blocks of the actual tangent and a validated integral. Its budget is the shared 128-bit outward rational arithmetic and at most 65,536 deterministically refined quadrature panels; refine only to meet the stated error target, not in response to the sign or size of the scientific result. If that budget cannot certify the target, retain numerical unavailability. The frozen tangent phase has only the two radius-15 carrier bands; their cubic products occupy radius-45 bands about the two carriers after modular folding, at most 182 modes. Each mode retains the actual two-by-two generator (1-cos(2*pi*k/N))*[[-sigma,-1],[1,0]]. This finite representation can reduce the integral's cost without a nonlinear spectral truncation. Any closed exponential-integral evaluation must handle coincident or nearly coincident frequencies by its continuous integral limit, not division by an interval that might contain zero. Any subsequent full nonlinear trajectory must first admit the shared integrator and complete-row residual evidence for this support size. The existing small-support flow wrapper's dimension cap is not waived by this analytical result, and held-pressure nodal RK4 is not substituted for (1).

5. Implementation and research boundary

The private paired budget reader re-admits primitive scalar inputs and actual model coefficients, reuses the transport owner, and computes only outward sufficient bounds. Failed bootstraps leave actual-state conclusions unavailable. A successful budget report does not claim a detected signal: its interaction observation remains unavailable. The static tests check geometry, exact linear cancellation, intensity interference, the mixed cubic source and remainder, correlated-source budgets and rejected domains without executing a response.

The hypothesis that interacting organizations can generate new organizations, which then interact at another level, motivates this study. Nonadditivity alone does not establish that hierarchy. A newly formed NFR would need an independently specified identity absent from the preparation, an occurrence and formation argument, retained constituent and boundary state, persistence or maintenance, and a subsequent response. Transient overlap and forced band projections supply none of those claims on their own. The present law need not increase complexity. Why particular forms and functions are selected remains a later question conditional on an actual formation result.