12 / C3-J: The Moving-Gauge Re-entry Audit, Absolute-Shell Hysteresis, and the Flux-Variation No-Go

The question left by C3-I is: if a distant defect re-enters the moving genealogical core later on, must it pay a cost? This round first corrects the very notion of "re-entry" itself, making it the sharpest methodological correction in the series so far. The Moving Spectral Balance Identity (Theorem 4.1) precisely splits the high-side energy change under a moving frequency frontier into $\mathcal G_\Lambda$ (gauge sweep — the apparent change caused merely by the frontier itself sweeping over existing content) and $\Phi_\Lambda$ (the genuine nonlinear spectral transfer); the Moving Spatial-Core Balance (Theorem 10.1) performs the same decomposition for a moving spatial core, yielding $\mathcal G_\chi$. Two no-gos (§6, §12) prove that pure coordinate relabeling (relative-high-frequency becoming relative-low-frequency, outside-the-core becoming inside-the-core) does not imply genuine physical transport — the very same absolute shell can be labeled UV → frontier → IR in sequence purely because the frontier itself moves, with no actual spectral transfer at all. The X-Integration must therefore preserve both the absolute shell $q$ and the relative shell $q-Q$ simultaneously; neither may be substituted for the other. After subtracting out the gauge component, this round proves two genuine finiteness results: the Absolute-Shell Direct-Reuse Bound (Theorem 15.1, purely combinatorial): a fixed absolute shell can serve as a direct local parent in at most $C_L$ frontier generations; and the Fixed-Shell Hysteretic Re-entry Bound (Theorem 21.1, a Lipschitz argument): the number of complete $\beta_0\to\beta_1$ threshold cycles for a fixed shell and chirality is necessarily finite within finite time. But both of these finiteness results are entirely compatible with $q_n\to\infty$ (infinitely many distinct shells), echoing C3-F's finite-branching reversal — "finite reuse of a single token" does not imply "a finite total genealogy." More seriously, two new no-gos emerge: the normalized time gap $\delta_q\gtrsim C\lambda_q^{-1}$ is only the weakest lower bound, and $\delta_q\to0$ still cannot be ruled out (the Energy-Only Time-Gap No-Go — C3-H's causal-collapse problem cannot be repaired by energy alone); and, most critically of all, the Signed-Flux Variation No-Go (Proposition 27.1): any argument of the form "$E(T)-E(0)+D=\int\Phi$" can only control the signed net flux $\int\Phi$, not the total variation $\int|\Phi|$ or the positive part $\int[\Phi]_+$ — an explicit counterexample, $\Phi_N(t)=N\sin(Nt)$, keeps its signed integral at $O(1)$ over a fixed interval, while the integral of its positive part grows linearly in $N$. So the "re-entry = finite additive cost" strategy fails a second time: many apparent re-entries are merely gauge sweeps, and once these are subtracted out, the genuine flux remains signed and can repeatedly reverse direction. The research frontier formally shifts to C3-K: rather than counting "how many times it appears to come back," the question becomes whether this is genuinely an infinite total variation of flux, or a moving frontier repeatedly sweeping over a phase-space field that is already highly congested.