Negative pairs yield symmetric solutions. Total lattice points: $(\pm26, \pm12)$ and $(\pm10, 0)$, totaling $\boxed{6}$.Question: A mechanical engineer is designing a gear system where the torque output $ T(n) $ at segment $ n $ is defined recursively by $ T(n) = 2T(n-1) + 3 $, with $ T(1) = 4 $. If the system evaluates $ T(m) $ for a specific $ m $, and the sum $ S(m) = \sum_{n=1}^{m} T(n) $ models total energy over time, compute $ S(5) $.

["Title: Unlocking Symmetric Energy Patterns: How Negative Pair Yields Enable Efficient Gear System Modeling", "Introduction:\nIn advanced mechanical design, especially in gear systems with periodic stress loading, understanding symmetric dynamic responses is crucial. Recent analysis reveals a fascinating mathematical property often observed in recursive torque sequences—negative pairs yield symmetric solutions, enabling efficient modeling of balanced forces. A notable example confirms that total lattice representations—such as toroidal point pairs in stress fields—yield symmetric results, with equilibrium apparent in point configurations like $(\pm26, \pm12)$ and $(\pm10, 0)$, totaling $\boxed{6}$ symmetric lattice points. This elegant symmetry mirrors recursive energy patterns in mechanical systems, offering insights into total energy accumulation.", "Let’s apply this insight. Consider a gear system where torque evolves via the recurrence:", "$$\nT(n) = 2T(n-1) + 3, \quad \ ext{with } T(1) = 4.\n$$", "We are to compute the total energy over the first 5 segments:\n$$\nS(5) = \sum_{n=1}^{5} T(n).\n$$", "Step 1: Compute individual terms using recursion.", "- $ T(1) = 4 $\n- $ T(2) = 2T(1) + 3 = 2(4) + 3 = 8 + 3 = 11 $\n- $ T(3) = 2T(2) + 3 = 2(11) + 3 = 22 + 3 = 25 $\n- $ T(4) = 2T(3) + 3 = 2(25) + 3 = 50 + 3 = 53 $\n- $ T(5) = 2T(4) + 3 = 2(53) + 3 = 106 + 3 = 109 $", "Step 2: Sum all terms.", "$$\nS(5) = T(1) + T(2) + T(3) + T(4) + T(5) = 4 + 11 + 25 + 53 + 109\n$$", "Compute sequentially:", "- $ 4 + 11 = 15 $\n- $ 15 + 25 = 40 $\n- $ 40 + 53 = 93 $\n- $ 93 + 109 = 202 $", "Thus, $ S(5) = 202 $.", "Connection to Symmetric Energy Flow:\nThe recursive structure—doubling and adding a constant—generates rapid growth, yet the total sum reflects cumulative energy. The emergence of symmetric lattice configurations (e.g., $ (\pm26, \pm12) $, $ (\pm10, 0) $, totaling 6 symmetric points) reflects inherent balance in recursive force transmission. This symmetry optimizes gear design by minimizing net torque imbalances over cycles.", "Conclusion:\nBy combining recursive torque modeling with lattice symmetry, engineers can predict total energy use with precision. For $ m = 5 $, the cumulative torque sum is $ \boxed{202} $. Leveraging such mathematical patterns ensures robust, energy-efficient gear systems."]