Source resources for localized mobile identities

Scope: a conditional source-to-target obstruction under the same complete positive-loss sine law as the prepared transport theorem. No trajectory, response search, source reconstruction or physical measurement is evaluated. This result identifies a missing preparation resource for one class of mobile identities; it does not exclude every moving organization. The research program owns closure and subsequent admission.

1. Complete law and an independent source class

Keep the labeled unit cycle C_N, held unit capacities, beta one, positive coefficients e,w, no inputs or events, and clock tau=w*t/pi from the law owner. With A=L/2 and sigma=pi*e/w, both complete rows are

\[ x'=-\sigma Ax+\tfrac12 S(\theta),\qquad \theta'=Ax. \]

Retain real phase lifts relative to the supplied uniform zero background. As before, z=x+i theta denotes two real state channels. The source class is

\[ z(0)=(u,0)+r_0,\qquad u\in\mathbb R^N,\qquad \|u\|_2\le M,\qquad \|r_0\|_2\le\epsilon,\qquad M>0. \tag{1} \]

The nominal phase is uniform. Its zero lift fixes the background origin; it does not infer phase from form. The independent error covers both channels at every site, including common means. No zero-mean source is assumed.

The real profile u is otherwise arbitrary: it can already contain localization, spatial asymmetry, a spectral envelope or several peaks. Thus this class lacks the nominal form/phase correlation of a one-sided carrier, not necessarily all spatial organization. Its definition uses no desired final coordinates, backward evolution or fitted source. The ontology contract keeps these different formation claims separate.

For N divisible by four, Q>=2 and Q-1<N/4, define the disjoint Fourier bands, with indices interpreted modulo N,

\[ B_+=\{N/4+m:|m|<Q\},\qquad B_-=-B_+. \]

Use the unitary discrete Fourier transform and its orthogonal projectors P_+,P_-. A target v has norm a>0 and support entirely in either one of these bands. This includes every translated or internally rotated Fejer reference of the prepared theorem. Allowing all such vectors makes the obstruction independent of that particular envelope. Fourier orientation is not a new primitive phase, topological winding, charge or particle label.

2. The actual damped tangent preserves opposite-band balance

Linearize the actual law at the uniform background:

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

At wave number k, lambda(k)=1-cos(k)=lambda(-k). The two real modal functions f_k,g_k are defined by

\[ \binom{f_k(\tau)}{g_k(\tau)}= \exp\!\left[\tau\lambda(k) \begin{pmatrix}-\sigma&-1\\1&0\end{pmatrix}\right]\binom{1}{0}. \]

They are the same at k and -k, including overdamped and critically damped cases. Let z_lin be this actual tangent launched from (u,0). Since u is real, u_hat(-k)=conj(u_hat(k)), while

\[ \widehat z_{\rm lin}(k)=(f_k+i g_k)\widehat u(k),\qquad \widehat z_{\rm lin}(-k)=(f_k+i g_k)\overline{\widehat u(k)}. \]

Their magnitudes agree. Consequently

\[ \|P_+z_{\rm lin}(\tau)\|_2=\|P_-z_{\rm lin}(\tau)\|_2. \tag{2} \]

This statement uses actual positive loss; it needs no conservative comparison or dispersion-error allowance. It also needs no spatial reflection symmetry of u. Real form alone does not imply a reflection-even spatial profile.

For any state y satisfying (2), write the common band norm as r. If v has norm a and is supported in B_+, orthogonality and the reverse triangle inequality give

\[ \|y-v\|_2^2\ge(r-a)^2+r^2 =2(r-a/2)^2+a^2/2\ge a^2/2. \tag{3} \]

The other band is identical. Components outside both bands only increase the distance. Removing common form/phase origins does not remove this lower bound, since neither band contains the zero Fourier mode.

3. Transfer to the complete nonlinear trajectory

The shared contractive tangent and cubic remainder give ||R(theta)||_2<=(4/3)||theta||_2^3. In ||z||_2<=2M, Duhamel's formula yields, for every 0<=tau<=T,

\[ \|z(\tau)-z_{\rm lin}(\tau)\|_2 \le\delta=\epsilon+\frac{32}{3}TM^3. \tag{4} \]

If delta<M, continuity closes the bootstrap: contraction gives ||z_lin||<=M, so at a putative first exit the actual norm is instead bounded by M+delta<2M. This also admits the initial state because epsilon<M. The complete smooth sine law supplies finite-time continuation. Retain 2M<pi/2 for the small lifted-phase chart; for the calculator's integer T>=1, delta<M already implies this condition.

Combining (3) and (4) proves the uniform lower separation

\[ \boxed{\quad\|z(\tau)-v\|_2\ge a/\sqrt2-\delta\quad} \tag{5} \]

for every source in (1), every time in the interval and every norm-a target in either declared band. Entry into a target tube of radius eta*a is therefore excluded when a/sqrt(2)-delta>eta*a. A failed sufficient test does not prove entry, and an unclosed bootstrap supplies no nonlinear separation certificate. Balance (2) is not asserted as a nonlinear invariant.

Fixed source and target budgets

Keep the prepared theorem's actual law, N=1024, Q=16, a=2^-12 and T=256. Supply M=a, epsilon=M/1000 and target tolerance eta=1/25. Then

\[ \frac\delta a=\frac1{1000}+\frac1{6144}<\frac2{1000},\qquad \frac1{\sqrt2}>\frac{707}{1000}. \]

The source remains more than 141a/200 from every one-band norm-a target, well outside the a/25 identity tube, throughout the interval. This excludes even an instantaneous acquisition of that identity, hence also its later retention within the same interval. The result covers arbitrary nominal form profiles; it is stronger than testing one failed impulse or one guessed source.

A directional target has an appropriate correlation between both channels. For example, x_j=b*cos(kj) and theta_j=b*sin(kj) give z_j=b*exp(ikj), which occupies just one nonzero band. This is a boundary control outside the phase-flat class, not an acquisition proof. Supplying phase norm without the appropriate correlations is not sufficient either. Different envelopes, two-sided targets, larger source resources or longer nonlinear evolution retain separate questions.

4. An independent exact reflection restriction

For the reflection j -> -j on the same cycle, both full nonlinear rows are equivariant. If the complete actual initial state, including its retained real phase lifts, is reflection-even, uniqueness preserves that symmetry for all forward time. Thus rho_j=x_j^2+theta_j^2 satisfies rho_j=rho_{-j}. Whenever a window W and its reflection are disjoint, each contains at most half the total rho. For zero total rho the signal-floor requirement fails instead.

This excludes concentration above 24/25 in one such window at any amplitude for those declared continuous lifts. Circular phase symmetry alone does not imply equality of squared lifts chosen on different sheets. The frozen center c=-tau and radius 96 have disjoint reflected windows for 96<tau<=256 on C_1024. The restriction does not apply to arbitrary spatially asymmetric u in (1), nor to arbitrary errors that break reflection symmetry. It permits two counterpropagating concentrations.

The existing symmetry owner already requires complete support/capacity equivariance and actual-state symmetry; its involution controls retain these premises. No new event selector or symmetry-breaking rule is installed by this argument.

5. Computation and research interpretation

The private source calculator in _localized_identity_transport.py re-admits scalar source/target budgets and actual model coefficients. It shares the absolute cubic remainder and exact clock conversion with the prepared reader, while keeping the source verdict distinct. The source proof has no packet-dispersion term or conservative-loss penalty. Reported separations are available only after the actual nonlinear bootstrap closes. The static controls derive modal balance, the geometric distance, a correlated-phase boundary case and the finite budgets without evaluating a response.

The scientific result is a necessary preparation restriction for a specified family, not failure of the emergence hypothesis. A full-state directional organization is information that cannot be supplied by an arbitrary small phase-flat form preparation on this horizon. Localization itself can already be present in such a source, and bilateral motion remains possible. A new formation claim must identify which property is absent initially and allow the corresponding complete-law symmetries. The distinction also prevents an analyst-chosen waveform from becoming a definition of every possible NFR.