Wave Dynamics from the Finite Real-J Address Ledger: The Centered Source Operator, Unique A6 Bounded Packet Space, Exact Orthogonal Propagator, d'Alembert Shadow, and Linear-Free Vacuum Falsifier

A finite-difference wave law with a unique bounded packet space A finite address update determines a wave operator, its admissible spectrum, and its exact propagation at every tick. The ordinary d'Alembert equation appears as the long-wavelength readout. \[\boxed{\delta_T^2\theta_J=\Delta_{\tilde\ell}^{(d)}\theta_J}\] In the declared radius-one, real, centered, isotropic source class, the operator and native unit coefficient are fixed. Its bounded free packet sector and one-tick rotor are \[\boxed{Q=-\frac14\Delta^{(d)},\quad P_{A6}=\mathbf1_{[0,1]}(Q),\quad C=I-2Q,\quad R=2\sqrt{Q(I-Q)},\quad U=\begin{pmatrix}C&R\\-R&C\end{pmatrix}.}\] The exact Chebyshev powers of \(U\) preserve the packet norm. The inherited directional phase correction is fixed before comparison with a laboratory row: \[\Delta\Phi_N=N(\Omega-K)=\frac{NK^3}{24}\left(1-\sum_a n_a^4\right)+O(NK^5).\] The one-rail correction vanishes exactly; the analytic linear channel is absent. A laboratory verdict requires a separately qualified source-to-field map and adequate sensitivity. Version 4.1 completes the infinite-address initial-data condition. Spectral support alone does not make every pair of initial fields a uniformly bounded history. With \(g=\theta_1-C\theta_0\), the exact criterion is \[\boxed{\sup_{n\in\mathbb Z}\|\theta_n\|<\infty\quad\Longleftrightarrow\quad g\in\operatorname{Ran}R.}\] Equivalently, the endpoint part of \(g\) vanishes and its spectral measure satisfies \[\int_{(0,1)}\frac{d\langle g,E_Q(q)g\rangle}{4q(1-q)}<\infty.\] The companion quadrature is then \(R^\dagger g\). The proof uses the spectral theorem and an averaged sine-square bound; an explicit infinite-domain counterexample shows why the additional condition is necessary. The release also constructs dynamics satisfying two distinct gates: bounded wave support and the global A6 coherent-mass postulate. For a physically recorded legal partition, block propagators and compatible boundary unitaries preserve both gates. Spectral stability alone does not certify a mass ceiling, and a source mass count is not identified with a wave norm. The nonlinear bandwidth counterexample is strengthened to normalizable packets. All earlier operator, continuum, exact-propagator, directional-residual, and falsifier results are retained. The package contains the 98 inherited checks, 12 inherited mutation tests, and 47 additional compatibility checks. Structural closure, the adopted physical postulate, and laboratory selection remain distinct. A6 hard-wall companion · Main Book v10.01 · A6 lexicon · Artian's Universe