Wave Dynamics from the Finite Real-J Address Ledger: A QTT Uniqueness Theorem for the Centered Source Operator, Its d'Alembert Shadow, Exact Dispersion, and Linear-Free Vacuum Falsifier

Why the wave equation has one finite-address source operator What must be true before the familiar continuum wave equation can appear? This paper starts from a finite tick, a finite address step, real linear propagation, time reversal, hypercubic symmetry, constant-phase preservation, and a nondegenerate second-order update. The complete radius-one class reduces to one operator shape, \[ LA1⁽ᵈ⁾ = δ_T^2-Δ_{}⁽ᵈ⁾, \] and A1 massless-cone saturation fixes its remaining coefficient through \[ cQTT=/ t. \] The resulting finite source law is \[ { δ_T^2θ_J = Δ_{}⁽ᵈ⁾θ_J } \] with the exact plane-mode spectrum \[ { sin^2\!(ω t/2) = ∑ₐ₌₁ᵈ sin^2\!(k_a/2) }. \] This exact relation contains two sharply different results. Along one native rail, \[ d=1 ω=cQTT|k| \] throughout the principal band: finite address dynamics is exactly nondispersive there. A simultaneous multi-rail source mode instead carries a derived quadratic, direction-dependent correction, \[ {ω}{cQTTk} = 1+ ^2/24 ( k^2-∑_a k_a^4/k^2 ) +O(k^4^4). \] Finite-range locality makes the source symbol analytic, while reflection and time-reversal symmetry make it even. Therefore the declared constructor class contains no source correction linear in energy: \[ { {ω}{cQTTk} = 1+O\!(E^2/E_*^2), O\!(E/E_*)=0 }. \] The A6 bounded-mode theorem also identifies the exact real-frequency sector. Writing \[ S( k)=∑_asin^2\!(k_a/2), \] a nonzero two-sided bounded free mode exists if and only if \[ {0≤ S( k)≤1}. \] Only after the long-wavelength access limit does the standard field equation appear: \[ { ∂_T^2θ_J -cQTT^2∇^2θ_J =0 }. \] The paper separates the finite source relation from laboratory exposure. Existing photon time-of-flight searches strongly constrain a linear channel and remain consistent with the linear-free theorem. Under the conventional Planck-endpoint audit, the derived quadratic coefficient is about 14.4 orders of magnitude below the conservative present reach. A laboratory verdict therefore requires an independently certified source-to-photon access map. Version: 2.0 Concept DOI: 10.5281/zenodo.20723081 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory: Main Book v10.01 Scientific status: GREEN: SIGMA-A1-RADIUS-ONE-OPERATOR-UNIQUENESS-CLOSED GREEN: SIGMA-A1-CAUSAL-SATURATION-COEFFICIENT-CLOSED GREEN: SIGMA-A6-BOUNDED-FREE-MODE-GATE-CLOSED GREEN: SIGMA-DALEMBERT-CONTINUUM-SHADOW-RECOVERED GREEN: SIGMA-LINEAR-DISPERSION-FORBIDDEN-BY-LOCAL-ANALYTICITY DERIVED / UNOBSERVED: SIGMA-QUADRATIC-SOURCE-DISPERSION Related public anchors: Action and stationary-phase framework Artian Lagrangian framework Artian Lorentz compliance audit Derivation Atlas QTT Lexicon