Finite transport of a prepared localized sine-law pattern

Scope: a conditional full-state transport proof on supplied finite support. The law, localization observation and preparation are specified here before any trajectory evaluation. No trajectory, fitted response, formation source or physical measurement is used. The ontology contract separates prepared transport from formation, permanent motion and particle identification; the research program alone owns subsequent work.

1. Complete law, support and observation

Use the existing normalized-sine completion on the labeled cycle C_N, with N divisible by four, unit conductance, unit geometric edge length and held capacities nu_i=1. Form x is signed real, phase is circular, and beta is fixed at one. There is no input, clipping, support event or selected operator word. With actual positive coefficients e,w, the original structural clock t gives

\[ \dot x=-\frac e2Lx+\frac{w}{2\pi}S(\theta),\qquad \dot\theta=\frac{w}{2\pi}Lx,\qquad S_j=\sin(\theta_{j+1}-\theta_j)+\sin(\theta_{j-1}-\theta_j). \]

Both rows change under tau=w*t/pi. Put A=L/2 and sigma=pi*e/w:

\[ x'=-\sigma Ax+\tfrac12S(\theta),\qquad \theta'=Ax. \tag{1} \]

The supplied background is x=theta=0. A small real phase lift about that background is part of the preparation; the bound below keeps it in the same chart. z=x+i theta is notation for two real state channels with beta one, not a quantum amplitude or a new complex EPI. Its Euclidean norm measures full-state departure from this background. These structural coordinates have the fixed scale used by this law; no laboratory unit conversion is asserted.

Integer shifts and reflection preserve both complete rows, support and capacities. Define distance to a real position c on the supplied unit-length circle by d_N(j,c)=min_{k in Z}|j-c+kN|. This continuous position describes an observation window on fixed sites. Fractional shifts of an interpolated profile are not exact nonlinear graph symmetries.

The localization weights are rho_j=|z_j|^2=x_j^2+theta_j^2. They are nonnegative diagnostics, not local physical energy. A single complex Fourier mode has uniform rho, so motion of its form maxima cannot pass a concentrated packet criterion. Position is retained in the fixed site reference, rather than removed by a time-dependent relabeling.

2. One full-state reference and a prepared source class

Let Q>=2, Q-1<N/4, k0=pi/2 and q_m=2*pi*m/N. For m=-(Q-1),...,Q-1, set

\[ b_m=1-\frac{|m|}{Q},\qquad S_2=\sum_m b_m^2=\frac{2Q^2+1}{3Q},\qquad F_Q(u)=\sum_m b_m e^{iq_m u} =\frac1Q\left[\frac{\sin(Q\pi u/N)}{\sin(\pi u/N)}\right]^2. \tag{2} \]

Use the continuous limit at multiples of N. For amplitude a>0, prescribe the reference history

\[ z^*_j(\tau)=\frac{a}{\sqrt{NS_2}} e^{i(k_0j+\tau)}F_Q(j+\tau),\qquad 0\le\tau\le T. \tag{3} \]

Distinct discrete frequencies give ||z*(tau)||_2=a for every real tau. The envelope center is c(tau)=-tau mod N. The carrier band omits frequency zero, so the nominal real form and lifted phase means both vanish. A prepared state is any full real state satisfying

\[ \|z(0)-z^*(0)\|_2\le\epsilon. \tag{4} \]

All sites and both channels may be perturbed independently subject to this one joint bound. No restriction to an invariant Fourier subspace is imposed on the actual nonlinear trajectory. The source already contains a localized organization and a direction; neither is claimed to form autonomously.

The factor exp(i tau) mixes form and phase in the reference. It is a specified prediction of internal evolution, not common phase-origin gauge and not an exact symmetry of (1). Identity here means proximity to the complete history (3), including that rotation, its amplitude and its location. It does not mean a rigid full-state spatial translate at every time.

3. A whole-interval nonlinear error bound

The actual consensus tangent and the conservative comparison generator are

\[ J_\sigma=\begin{pmatrix}-\sigma A&-A\\A&0\end{pmatrix},\qquad J_0=\begin{pmatrix}0&-A\\A&0\end{pmatrix}. \]

J_sigma has a contractive Euclidean semigroup: its symmetric part contributes -sigma*x^T A x<=0. J_0 is skew, so its semigroup preserves norm. This comparison does not remove the actual law's positive loss. The coupled conservative comparison has dispersion omega(k)=1-cos k, not the square root of a Laplacian eigenvalue. The actual positive-loss tangent still has damping and its own eigenvalues. On the prepared band, the conservative comparison gives

\[ \omega(k_0+q)=1+\sin q,\qquad |\omega(k_0+q)-(1+q)|\le |q|^3/6. \]

Consequently Parseval's identity and |exp(iu)-exp(iv)|<=|u-v| bound the conservative tangent's difference from (3) by a*T*q_max^3/6, with q_max=2*pi*(Q-1)/N. The difference between the positive-loss and conservative tangents is at most 2*sigma*T*a, since ||A||<=2 and the positive-loss propagator is contractive.

For an oriented incidence matrix D, ||D||<=2 and the actual nonlinear remainder is

\[ R(\theta)=-\tfrac12D^T[\sin(D\theta)-D\theta],\qquad \|R(\theta)\|_2\le\tfrac43\|\theta\|_2^3. \tag{5} \]

Indeed |sin u-u|<=|u|^3/6, and bounding each edge difference by 2||theta||_2 gives (5). In the bootstrap region ||z||_2<=2a, the nonlinear Duhamel contribution is at most (32/3)*T*a^3. Initial error propagates by at most epsilon under the same contractive tangent.

Define the dimensionless sufficient error

\[ h=\frac\epsilon a+2\sigma T+ \frac{T}{6}\left(\frac{2\pi(Q-1)}N\right)^3 +\frac{32}{3}Ta^2. \tag{6} \]

If h<1, continuity closes the bootstrap: at a hypothetical first exit from ||z||<2a, the preceding estimates instead give ||z-z*||<=a*h and ||z||<=a*(1+h)<2a. Thus, throughout the entire interval,

\[ \|z(\tau)-z^*(\tau)\|_2\le ah,\qquad a(1-h)\le\|z(\tau)\|_2\le a(1+h). \tag{7} \]

The sine law is smooth, with linear form dependence and bounded sine currents, so finite-time continuation itself is not the bootstrap hypothesis. If 2a<pi/2, each retained phase lift and every edge phase difference stay away from the circular branch boundary. These sufficient estimates can fail without excluding transport; failure is unavailability, not a negative theorem. For the calculator's integer horizons T>=1, h<1 already implies 2a<sqrt(3/(8T))<pi/2, so it also supplies this small-lift condition.

4. Uniform localization and distinct endpoint windows

For 1<R<N/2, the elementary inequality sin(pi*d/N)>=2d/N for 0<=d<=N/2 bounds

\[ |F_Q(j-c)|\le\frac{N^2}{4Q\,d_N(j,c)^2}. \]

For any fractional center, a unit distance shell contains at most two cycle sites. Extending each tail to an infinite lattice bounds its fourth-power sum by 2/[3(R-1)^3]. As S_2>=2Q/3, the reference tail fraction satisfies

\[ \frac{\sum_{d_N(j,-\tau)>R}|z^*_j(\tau)|^2}{a^2} \le H=\frac{N^3}{16Q^3(R-1)^3}. \tag{8} \]

Combining (7) with the triangle inequality on the outside coordinates gives

\[ \frac{\sum_{d_N(j,-\tau)>R}\rho_j(\tau)} {\sum_j\rho_j(\tau)} \le\left(\frac{\sqrt H+h}{1-h}\right)^2. \tag{9} \]

This follows the same continuous full-state trajectory at all times, rather than matching two unrelated peaks. The nonvanishing norm floor in (7) prevents a vanishing trace from passing after normalization. A center is specified by the prospective reference, not selected afterward among tied maxima. The bound asserts concentration around that moving center; it does not reconstruct a unique center from arbitrary partial observations.

If T is an integer, T<=N/2 and 2R<T, the radius-R windows at zero and -T are disjoint. A stationary state cannot have over half its norm squared in each of these disjoint windows. Reflection of the full preparation reverses the direction under the same law. No support mutation or external actuation is needed during the admitted interval.

5. Fixed analytical admission

The first case fixes the following inputs, without a trajectory search:

Input	Declared value
Actual loss and exchange	e=2^-20, w=1-2^-20, beta one; these represented values sum exactly to one
Support and capacity	C_1024, unit lengths, conductances and held capacities
Carrier and envelope	k0=pi/2, Q=16, modes m=-15,...,15 in (2)
Nominal joint amplitude	a=2^-12
Full-state initial error	epsilon=a/1000
Structural horizon	T=256 in tau=w*t/pi
Localization radius	R=96 in supplied edge-length units
Required identity error	h<1/25 relative to the full-state reference norm
Required concentration	More than 24/25 of actual rho remains in the moving radius-R window

The initial spatial localization and spectral direction are supplied. This small positive loss is an explicit member of the existing family, not a value selected by ontology or physical data. Its eventual dissipative relaxation is compatible with a finite transport result. The initial uncertainty is a mathematical norm ball, not a certified laboratory preparation.

Equations (6) and (8), even using the rational upper bound pi<22/7, give h<1/25 and H<1/49. Therefore (9) is strictly below

\[ \left(\frac{1/7+1/25}{24/25}\right)^2 =\frac{16}{441}<\frac1{25}. \]

The signal norm stays above 24a/25, full-state template error stays below a/25, and initial/final radius-96 windows are disjoint after displacement 256. The conclusion applies to every state in (4) at every time in the declared interval. It is a prepared finite transport and robustness theorem, not an invariant moving family, indefinite stability or formation theorem.

6. Computation, reuse and remaining obligations

The detached budget reader re-admits primitive inputs and the actual normalized model coefficients. It evaluates only outward scalar bounds from this proof, not a trajectory, sampled response or numerical eigensystem. The actual field remains owned by the shared sine rate rows; this result installs no native execution law or additional solver. The controls check the complete rows, clock, translation direction, nonlocalized normal modes, analytic budgets and rejection domains separately.

The result retains a prepared envelope and internal evolution. Acquisition from a class lacking that organization, interactions between transported packets, a closed collective state, scaling under composition and physical observations each need their own admission. The positive-loss recurrence boundary still applies to the actual finite model. A carrier Fourier index is not primitive phase winding: this small-lift construction remains in winding zero. No mass, charge, spin, quantum detection rule or constituent identification is derived by these estimates.