Certified endpoint observation of a prepared mobile encounter

Scope: evaluate the one reserved cubic-response certificate declared in the frozen protocol. The complete law, source balls, source registration, clock, horizon and identity contract are unchanged. This is a detached calculation of an analytic Duhamel integral with a proved transfer to the complete nonlinear flow, not numerical integration of a nonlinear trajectory. It installs no new native flow or support event. The research program alone owns the next task.

1. Exact finite Fourier representation

Use the unitary transform hat u_k=N^-1/2 sum_j u_j exp(-2*pi*i*k*j/N). Let c=T/2, k0=N/4, b_m=1-|m|/Q and S2=sum_{|m|<Q} b_m^2. The nominal complex bookkeeping variable of source A has Fourier coefficients

\[ \widehat z^A_{k_0+m}(0)=\frac{a b_m}{\sqrt{S_2}} e^{-2\pi i(k_0+m)c/N},\qquad |m|<Q, \]

and zero coefficients elsewhere. The real channels are separately retained: on that positive band hat x=hat z/2, hat theta=hat z/(2i); the opposite coefficients follow by conjugacy of each real channel. Source B is the exact spatial reflection of A, so hat x^B_k=hat x^A_{-k} and likewise for phase. No extra relative phase is selected from the response.

For lambda_k=1-cos(2*pi*k/N), the actual tangent block is

\[ B_k=\lambda_k\begin{pmatrix}-\sigma&-1\\1&0\end{pmatrix},\qquad \sigma=\pi e/w. \]

With omega=sqrt(1-sigma^2/4)>0, its exact propagator is

\[ U_k(s)=e^{-\sigma\lambda_k s/2} \left[\cos(\omega\lambda_k s)\,\mathrm{Id} +\frac{\sin(\omega\lambda_k s)}\omega \begin{pmatrix}-\sigma/2&-1\\1&\sigma/2\end{pmatrix}\right]. \tag{1} \]

The instrument requires sigma*T<=1, in addition to its positive-loss law admission. This keeps every decay argument within the shared unit-exponential domain and ensures the square root is separated from zero. It is a numerical instrument domain, not a new restriction on TNFR generally. The fixed model satisfies it. Form and phase must not be replaced by a single scalar complex propagator: positive form loss mixes the two bookkeeping carrier directions.

Take D theta_j=theta_j-theta_{j+1} and d_k=1-exp(2*pi*i*k/N). If alpha and beta are the Fourier coefficients of the two tangent edge-phase differences, the mixed form forcing is the cyclic convolution

\[ \widehat g_k(s)=\frac{\overline d_k}{4N} [\alpha*\beta*(\alpha+\beta)]_k. \tag{2} \]

The two unitary Fourier products each contribute N^-1/2; omitting this normalization would give an incorrect signal. The cubic integrand has separate form and phase channels U_k(T-s)*(hat g_k(s),0).

Both tangent phase histories occupy only the two radius-15 bands. Equation (2) occupies the two radius-45 bands around +/-N/4 after modular folding, 182 modes in total. No cubic output mode is discarded. This finite support statement concerns the cubic predictor, not the nonlinear full flow. The instrument requires 3(Q-1)<N/4, N<=4096 and Q<=32 for this representation.

Reflection exchanges A and B and preserves their mixed forcing, so g is reflection-even. Its Fourier coefficients are real and equal at opposite modes, as are those of the propagated integrand. The implementation intersects the two independently enclosed equal real coefficients and checks that the imaginary enclosures contain zero before retaining this exact symmetry. It retains both partners when applying Parseval's norm.

2. Outward arithmetic and the integration error

Elementary functions, square roots and final coefficient boxes use the shared 128-bit outward rational intervals. For polynomial convolution, their dyadic endpoints become exact integers on the same grid. All four endpoint products are enclosed, and integer sums are accumulated before a single outward projection per output coefficient. This is an optimization of a finite polynomial reduction, not an alternative interval arithmetic or a floating FFT. An opaque evaluation closure captures constants rebuilt from fresh primitive source and model inputs; externally supplied derived reports are not consumed. Trigonometric argument reduction subtracts an integer multiple of an outward pi/2 interval, then applies the exact quarter-turn identities and the shared sine/cosine owners to the residual. Choosing that integer from a midpoint is only a representation choice: the identity holds for every integer, including at uncertain quadrant boundaries. The residual interval retains pi uncertainty.

Let qmax=2*pi*(Q-1)/N. On the input and output Fourier supports, respectively,

\[ K_{\rm in}=(1+\sigma)(1+q_{\max}),\qquad K_{\rm out}=(1+\sigma)(1+3q_{\max}) \]

bound the tangent operator norm: its rotational part has norm one and its loss part norm sigma, while 1-cos(pi/2+q)<=1+|q|. Actual tangent propagators are contractive. Each differentiated input has norm at most a*K_in^j. The trilinear forcing has norm at most 8*a^3, using ||D||<=2. Differentiating its three inputs and the final propagator gives, by the multinomial theorem, the whole-window integrand derivative bound

\[ \|f^{(j)}(s)\|_2\le8a^3\Lambda^j, \qquad \Lambda=K_{\rm out}+3K_{\rm in} =(1+\sigma)(4+6q_{\max}). \tag{3} \]

This retains the actual damped generator in every derivative. It does not infer an error from the difference of two sampled numerical answers.

Use an n-point Gauss-Legendre rule on each of P equal panels. Its nodes are certified by exact rational sign bisection of the degree-n Legendre polynomial: n disjoint sign-changing intervals exhaust its roots. Outward weights are 2/[(1-r^2)*P_n'(r)^2]. The positive Peano kernel gives the same remainder constant for the Euclidean vector norm as for a scalar, with no dimension multiplier. Equations (3) and the Gauss remainder imply

\[ E_G\le 8a^3 T\left(\frac{\Lambda T}{P}\right)^{2n} \frac{(n!)^4}{(2n+1)((2n)!)^3}. \tag{4} \]

Choose n=12. Starting at one panel, double P until this prior bound is at most one tenth of eta_num=a^3/1000, or the frozen limit 65,536 prevents another doubling. This selection reads no integrand or response. For the fixed inputs, Lambda<23/5 and P=64 suffices; hence 768 integrand evaluations are prescribed before the response is known.

Node, weight, initial coefficient, propagator and convolution rounding are retained in the accumulated spectral boxes. The Euclidean norm of their component radii bounds the vector arithmetic error. The separate outward Parseval norm interval retains norm-reduction uncertainty; the numerical certificate uses E_G plus the larger of these two arithmetic allowances. All source materialization is included. The final cubic norm interval expands the quadrature norm interval by E_G. If the total error exceeds eta_num, numerical certification is unavailable even if a computed lower bound is positive. There is no automatic higher-precision retry.

3. Transfer to the actual nonlinear response

Rebuild the original analytical budgets from their primitive inputs. For an enclosed cubic norm [l,u], the actual nominal defect norm lies in

\[ [\max(0,l-K_5),\ u+K_5], \]

and every original shared-source preparation has defect norm within

\[ [\max(0,l-K_5-V),\ u+K_5+V]. \tag{5} \]

The two lower-bound tests remain separate. Their scientific certificates also require the declared identity bounds and successful numerical certification. A zero lower bound means unresolved sufficiency, not an observed zero. Bounds already include quadrature uncertainty; it is not subtracted a second time. Linear shared-source evolution remains the exact zero control. The result, if positive, concerns endpoint nonlinear deformation relative to separate evolution, not creation of a new NFR or a physical particle.

4. Implementation, controls and retained evaluation

The spectral owner computes the integrand, the shared quadrature owner encloses nodes, weights and truncation error, and the response reader performs the fixed endpoint reduction and separate verdicts. It does not modify the shared nonlinear integrator or relax existing support caps.

Independent controls check the spectral transform against direct real-site preparation, a matrix-exponential series and pointwise mixed cubic forcing; all Gauss polynomial moments through degree 2n-1 and the sharp degree-2n remainder; and exact interval-product corners. Response wiring uses explicitly mocked polynomials and a zero curve, not the reserved integrand. It also checks that a positive computed value cannot bypass a failed numerical budget.

The one-shot driver uses the shared exact artifact and strict JSON owners. A source archive and freeze are written before evaluation; an exclusive attempt precedes the scientific call. The first outcome, including failure, is retained. Archives associate declared bytes and configuration; they do not independently authenticate chronology or execution. The method was admitted before the reserved response below.

5. Retained endpoint result and its limits

The first assessment used the unchanged protocol, all 182 cubic output modes, 12 Gauss nodes on each of 64 panels, and 128-bit outward arithmetic. The following decimals are deliberately rounded outwards from the exact rational report:

Quantity	Certified enclosure or upper bound
Total numerical error	<4.639280e-16, below a^3/1000 > 1.45519e-14
Cubic endpoint norm	[1.2888499e-12, 1.2897778e-12]
Complete-flow nominal defect norm	[1.2847097e-12, 1.2939180e-12]
Complete-flow defect norm over the original shared-source family	[0, 6.1613890e-12]

Thus the exact nominal preparations have a strictly nonzero endpoint superposition defect under the declared complete nonlinear law. The linear shared-source control has exactly zero defect. Both prepared band identities retain their separate whole-window certificate. This is a finite mathematical interaction result, conditional on the supplied sources, support and law. It is not a full nonlinear trajectory calculation or laboratory observation.

The original source-family test remains unresolved. Its sensitivity allowance is about 4.86747e-12, whereas the nominal fifth-order allowance is about 4.14022e-15 and numerical error is below 4.64e-16. More quadrature precision alone therefore cannot make this same sufficient uniform bound positive. The zero lower endpoint is not a witness of zero interaction for an admissible preparation. Neither acceptance level, source tolerance nor other frozen input was changed after evaluation.

The protocol pins the scientific admission to revision 14cbad44f92194774c4cd66a8763f2069c95270c. The manifest, source archive, freeze and exclusive attempt retain the implementation and first outcome, based on source HEAD a8c02f0b. The read-only evidence controls re-admit primitive inputs, rebuild analytical and quadrature budgets, and reconstruct Parseval intervals and verdicts from the retained spectral boxes. They forbid producer replay. These reductions are conditional on those boxes; byte associations do not independently authenticate execution or prove each integrand evaluation.

The defect I is a difference between three histories, not an independently evolving object. Even additional Fourier modes can describe ordinary deformation or radiation. In this preparation T*a^2=1/65536; the certified nominal defect is less than 5.3e-9*a. No additional NFR identity, formation, post-separation lifetime or subsequent response is established. Investigating recursive organization requires an independently motivated actual-state identity and a complete-law mechanism that can sustain it. The formation-compatibility owner retains those premises. The execution plan owns reopening; this result authorizes no replay or parameter search.