Quantum-Adaptive KS$varphi$: A Parameterized Three-Qubit Gate Family Embedding Toffoli with Measurement-Free Phase Kickback and Intrinsic Error Non-Amplification

We introduce Quantum-Adaptive KS$varphi$ $K$ = kickback, $S$ = sandwich, a parameterized three-qubit gate family that structurally embeds the Toffoli (CCX) gate within two additional components: (1)a palindromic Hadamard sandwich on the first control qubit q₀ that conjugates Z-type errors to X-type in the CCX frame, providing simultaneous sensitivity to both error types without ancilla overhead; and (2)a controlled-phase (CP) gate whose quantum phase kickback propagates post-CCX target-state information into the control-qubit phase without measurement. The term Quantum- Adaptive refers to amplitude steering conditioned by the compile-time parameter varphi via a Quantum Neural Cellular Automaton (QNCA) majority-inspired bias rule; the gate does not self-modify at runtime. Two QA-KS(π) gates chained on a shared control qubit q₀ produce outputs completely orthogonal to two sequential CCX gates on q₀=1 inputs output fidelity F=0.000, while agreeing exactly on q₀=0 inputs F=1.000. This subspace-dependent divergence is the direct computational signature of coherent phase retention across gate boundaries - impossible for CCX-only circuits. On the q₁ = 0 subspace the gate acts deterministically (up to a relative phase), providing intrinsic error non-amplification. On the q₁ = 1 subspace it produces four-component entangled superpositions, making it a strictly distinct quantum-native primitive from CCX. We present the complete 8 times 8 unitary matrix, confirmed exact to ||U^daggerU-I||_infty < 10^-15, and define two canonical variants: QA-KS_π/2 $varphi = π/2$, $S$ gate and QA-KS_π $varphi = π$, $Z$ gate. Qiskit depolarizing-noise simulation demonstrates near-unit fidelity at p leq 10^-2 with an honest depth cost at higher error rates. The gate preserves the three-qubit footprint of CCX with no qubit overhead.