From a window overlooking Central Park, a mathematician in New York City looks up — past the treetops, past the skyline — at the oldest light in the universe. 從一扇俯瞰中央公園的窗口，一位紐約的數學家抬起頭——越過樹梢，越過天際線——望向宇宙中最古老的光。
He found a simple formula that textbooks had overlooked — and it matches satellite data within 3%. 他找到了一個被教科書忽略的簡潔公式——與衛星數據的吻合度在 3% 以內。
The cosmic microwave background (CMB) is relic radiation from the early universe — a snapshot of the moment when light first traveled freely through space. 宇宙微波背景輻射（CMB）是早期宇宙的遺跡輻射——記錄了光第一次在空間中自由傳播的那個瞬間。
When we decompose CMB temperature variations by angular scale, a remarkable pattern of peaks and troughs emerges — acoustic oscillations frozen at the moment of decoupling. 當我們按角尺度分解 CMB 的溫度變化，一組引人注目的峰谷模式浮現——聲學振盪凍結在退耦的那一刻。
Hover over a peak to compare observed and predicted positions. 將滑鼠移到峰上，可比較觀測值與預測值。
Existing formulas predict peaks at ln ≈ nπ/θ*, but these systematically miss the observed positions. The first peak is predicted at ℓ = 309 — the actual value is 221. 現有公式預測峰值在 ln ≈ nπ/θ*，但系統性地偏離觀測位置。第一個峰被預測在 ℓ = 309——實際值是 221。
Wang's insight: the power spectrum Cℓ is an integral over wave numbers. The extremum of an integral need not coincide with the extremum of its integrand. This is first-year calculus — hiding in plain sight. 王教授的洞見：功率譜 Cℓ 是對波數的積分。積分的極值不一定與被積函數的極值重合。這是大一微積分——就藏在眾目睽睽之下。
The peaks of the integral (bottom) are shifted left by π/4 compared to the integrand (top).
This quarter-period shift is the origin of the n − ¼ correction.
積分的峰（下方）比被積函數的峰（上方）向左偏移了 π/4。
這個四分之一週期的偏移，就是 n − ¼ 修正的來源。
Replace n with n − ¼ in the peak formula, and use the angular acoustic scale θ* = 1/96.35 computed from published cosmological parameters: 在峰值公式中把 n 替換為 n − ¼，並使用從已發表的宇宙學參數計算出的角聲學尺度 θ* = 1/96.35：
| Peak n峰 n | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Planck 2018 | 220.6 | 538.1 | 809.8 | 1147.8 | 1446.8 | 1779 | 2075 |
| Formula公式 | 227.0 | 529.7 | 832.4 | 1135.1 | 1437.8 | 1740 | 2043 |
| Error誤差 | +2.9% | −1.6% | +2.8% | −1.1% | −0.6% | −2.2% | −1.5% |
From six cosmological parameters and the Friedmann equation — numerically integrated with Mathematica — the paper computes these characteristic distances: 從六個宇宙學參數和弗里德曼方程式——用 Mathematica 數值積分——論文計算出以下特徵距離：
Published in the European Journal of Physics 發表於歐洲物理期刊
Frank Wang · Department of Mathematics, LaGuardia Community College, CUNY
Numerical computations performed with Mathematica. 數值計算使用 Mathematica 完成。
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