Interpretive Note: Cosmic Acceleration as Bank Relaxation

**Abstract.** This note checks whether the Double-Gear mechanism can sharpen the logic of *Cosmic Acceleration as Bank Relaxation*. The answer is positive but structural rather than magical. The exact arithmetic module already forces a cyclic \( \mathbb Z_{48} \) clock through the simultaneous short exact sequences\[0\to \mathbb Z_{24}\to G\to \mathbb Z_2\to 0,\qquad0\to \mathbb Z_8\to G\to \mathbb Z_6\to 0,\]and the appendix-level affine scaffold provides an exact coarse descent\[1\to \mathbb Z_2\to \mathbb Z_{48}\to \mathbb Z_{24}\to 1\]together with the even projector \( (\Pi_{\rm even}f)(m)=\tfrac12(f(m)+f(m+24)) \). The internal synthetic audit performed here finds no hidden contradiction between these ingredients and the Bank-relaxation or glass-to-crystal notes. On the contrary, they can be organized into a coherent three-tier stack: exact affine \( \mathbb Z_{48} \) scaffold at the microscopic level, a mesoscopic synchronization/binding layer, and the late-time FRW projection at the infrared level. The key extra result of the audit is a useful repackaging of the fiber-product sector. Writing the Coxeter phase as \( j=b+6t \) with \( b\in\mathbb Z_6 \) and \( t\in\mathbb Z_5 \), while the coarse clock is \( a=b+6r\in\mathbb Z_{48} \), shows that the 240-state arithmetic sync sector is a five-sheet lift above the coarse 48-state clock. This removes the earlier temptation to talk about “incommensurable” 48 and 30: the correct statement is a rational mismatch with residual \( \mathbb Z_5 \) microphase structure. That residual lift suggests a natural coarse-grained order parameter\[C_{\rm DG}(z)=C_{\rm bind}(z)\,C_{\rm sync}(z),\]where \( C_{\rm bind} \) measures the emergence of the exact half-cycle pairing \( m\leftrightarrow m+24 \) and \( C_{\rm sync} \) measures coarse 48-clock synchronization. A corresponding Lyapunov/free-energy ansatz is proposed, with an explicit penalty against the “8-collapse” branch and an entropy term for unresolved fivefold microphases. The practical update is then immediate: the glass-to-crystal toy system can be re-read by replacing the original phenomenological \( C(z) \) with \( C_{\rm DG}(z) \), so that the same order parameter entering the transfer law and the growth equation now carries a Double-Gear meaning. The existing scan results are preserved under this reinterpretation: out of 9216 toy points, 7507 already have the correct sign pattern \( \Delta H_0>0 \), \( \Delta S_8