Beyond Geometric Scaling: Toward a Compute-Centric Metric in the Post-Moore Era 後摩爾時代的度量革命：從幾何尺度走向 pC 算力指數

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寫下這篇隨筆的初衷，源於最近與幾位朋友聊到半導體技術的近況。看著產業媒體天天被「台積電 A16」、「英特爾 14A」或是各種「等效 1 奈米」的宣傳轟炸，高科技的行銷術語逐漸滑向了某種原子雕刻的文字遊戲。

清晨走在台北四獸山的山路上，迎著晨光，我突然意識到：當平面二維的線寬在微觀下被嚴酷的量子穿隧與漏電無情「拉平」時，這場半導體界的集體焦慮，與一百年前物理化學家所面臨的困境何其相似？

1930 年代，化學家發現不論多強的酸，一旦丟進水溶液裡，全都會被水分子無情地「拉平（Leveling Effect）」。在那碗水裡，pH 尺規徹底失去了對強酸的解析度。今天的半導體製程，正陷入屬於自己的幾何拉平效應中。當年，化學家路易斯（G. N. Lewis）拋棄「質子外殼」、直奔「電子對收發本質」完成了酸鹼定義的範式轉移；而今天的計算科學，也許到了必須扒光奈米與埃米外衣、直奔計算本質的時候。

這不是一門單純的電子工程課，而是一場關於「我們如何度量文明前進」的認識論反思。我嘗試借用化學尺度革命的歷史為參照，將這段時間在 AI 筆記本裡的隨筆凝練成一篇嚴謹的學術 Perspective。以下是一點不成熟的思辨，與諸位跨領域的同好切磋。

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Abstract / 摘要

As semiconductor scaling approaches fundamental physical limits, traditional geometry-based metrics such as feature size and node naming conventions increasingly fail to capture system-level performance. This Perspective argues that the current transition in computing resembles earlier epistemic shifts in physical chemistry, where measurement frameworks were replaced once their discriminatory power collapsed. We propose that future progress may be more naturally described in terms of system-level effective compute, expressed on a logarithmic scale, providing a robust framework to map multi-dimensional architectural trajectories onto a unified, dimensionally consistent scale.

隨著半導體微縮逐漸逼近根本的物理極限，傳統基於幾何的度量衡（如特徵尺寸與製程節點命名慣例）已日益無法精確捕捉系統級的實質性能。本評論指出，當前計算科學的範式轉移，極其相似於早期物理化學史上的認識論變革——當既有的測量框架失去解析度時，舊的尺規必然會被淘汰。我們提出，未來的技術進步應更自然地以系統級的「有效計算能力」來描述，並將其映射至對數尺度上，為現代計算系統中多維度的架構演進提供一個大一統且自洽的座標系。

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1. The breakdown of geometric interpretability / 幾何可解釋性的崩潰

For decades, semiconductor progress has been communicated through geometric scaling laws, most notably feature size reduction. However, as devices enter regimes dominated by quantum transport, parasitic effects, and three-dimensional integration, the correspondence between nominal node labels and physical dimensions has weakened significantly.

Recent technology nodes labeled in angstrom-scale terminology no longer correspond to a single, well-defined geometric quantity. Instead, they function as generational identifiers for increasingly heterogeneous system architectures. This shift suggests a deeper issue: geometry is no longer the primary determinant of computational progress.

過去數十年來，半導體的技術進步始終透過幾何微縮定律（最顯著的即特徵尺寸縮小）來傳傳播。然而，當電子元件進入由量子輸運、寄生效應與三維立體整合主導的微觀領域時，標稱的「製程節點標籤」與「物理實質尺寸」之間的對應關係已大幅削弱。近期被冠以「埃米級」術語的先進製程節點，在物理上早已不再對應任何單一、定義明確的幾何寬度。相反地，它們更像是某種世代代號，用來指代日益異質化的系統架構。這種轉變揭示了一個更深層的本質問題：幾何，已不再是衡量計算技術進步的主導變數。

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2. A historical analogy from physical chemistry / 來自物理化學史的歷史類比

A similar transition occurred in physical chemistry during the early development of acid–base theory. The pH scale, bound to aqueous activity, loses discriminatory power in strongly acidic regimes due to the solvent’s leveling effect, wherein structurally distinct superacids are indifferently converted to the hydronium ion (\(H_3O^+\)).

To overcome this epistemic saturation, alternative frameworks such as the Hammett acidity function (\(H_0\)) and later Lewis’s electron-pair definition shifted the analytical focus away from observable chemical proxies toward the fundamental quantum-mechanical nature of chemical reactivity. Importantly, these historical advances did not merely refine the existing coordinates; they redefined the relevant observable.

類似的範式轉移在物理化學早期的酸鹼理論發展中曾完美上演。受限於水溶液環境的 pH 尺度，在面對極強酸的領域時會因為溶劑的「拉平效應（Leveling Effect）」而徹底失去分辨力——在此環境下，結構與本質完全不同的各種超強酸，全都會被無情地拉平成相同濃度的氫離子（\(H_3O^+\)）。為了克服這種認識論上的飽和，化學家隨後發展出了哈米特酸度函數（\(H_0\)）以及最終的路易斯（G. N. Lewis）電子對理論，將分析焦點從宏觀的觀測代號，轉移至更具微觀本質的電子收發行為。至關重要的是，這些歷史性的跨越並非在舊的尺規上做精細修正，而是徹底重新定義了「何謂核心觀測量」。

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3. From components to systems: redefining computational progress / 從單元到系統：重新定義計算進步

Modern computing systems exhibit a similar collapse of single-dimensional metrics. Performance is no longer governed solely by transistor density or clock frequency, but by a nonlinear combination of:

• architectural parallelism (架構並行度)
• memory hierarchy efficiency (記憶體階層效率)
• interconnect topology (互連拓撲結構)
• energy constraints (功耗與散熱限制)
• system-level utilization (系統級實際利用率)

These factors interact nonlinearly, making isolated hardware parameters insufficient as predictive indicators of performance. This motivates a shift in perspective: from component-level descriptors to system-level effective compute.

現代計算系統同樣面臨著單一維度幾何指標失效的困境。晶片的最終性能不再僅由電晶體密度或時脈頻率決定，而是由上述諸多系統級因素非線性耦合的結果。這些因素相互交織，使得孤立的硬體幾何參數根本不足以作為預測系統效能的有效指標。這促使我們必須發生視角上的根本轉變：從傳統的「單元級描述符」，全面轉向「系統級的有效計算能力」。

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4. A logarithmic representation of effective compute / 有效計算能力的對數化表述

To capture multi-order-of-magnitude variation across computing systems, it is natural to consider a logarithmic transformation of effective compute:

\[pC = \log_{10}\left(\frac{C_{eff}}{C_0}\right) = \log_{10}\left(\frac{\text{FLOPS} \times \eta}{C_0}\right) \]

where \(C_{eff}\) denotes the effective computational throughput, \(C_0\) defines a reference baseline (\(1\text{ FLOPS}\)), and \(\eta \in (0, 1]\) is a spatiotemporal efficiency vector encapsulating memory bandwidth constraints, interconnect topologies, and thermal design power (TDP) utilization.

This representation does not introduce new physical assumptions; rather, it provides a compact coordinate system in which heterogeneous computing architectures can be compared on a unified, dimensionally consistent scale.

為了精準捕捉計算系統跨越數十個數量級的巨大跨度，引入有效計算能力的對數化轉換（即 \(pC\) 指數）是最自然且嚴謹的數學工具。在此公式中，\(C_{eff}\) 代表有效計算吞吐量，\(C_0\) 為基準算力（定為 \(1\text{ FLOPS}\)），而 \(\eta\) 則是一個時空效率向量，將記憶體頻寬瓶頸、互連拓撲損耗與熱設計功耗（TDP）利用率完美收納。此表述並未引入任何憑空捏造的物理假設，它僅提供了一個簡潔、無因次且自洽的座標系，讓全然不同的異質計算架構得以在同一個誠實的規矩下進行橫向對比。

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5. Interpretation: a coordinate system for architectural evolution / 詮釋：架構演進的時空座標系

Within this framework, historical progression in computing can be interpreted as approximately linear trajectories in logarithmic compute space, despite fundamentally different underlying mechanisms across eras:

• planar CMOS scaling (平面幾何微縮時代)
• three-dimensional integration (三維立體整合時代)
• heterogeneous accelerators (異質加速晶片時代)
• system-level co-design of hardware and software (軟硬體協同設計時代)

This suggests that what appears as discrete technological revolutions may, under an appropriate representation, correspond to continuous movement along a single abstract axis.

在此框架下，人類計算文明的歷史進程，可以被重新詮釋為在對數算力空間中一條穩定前進的「近似線性軌跡」——儘管在不同的技術時代，背後的物理推進機制截然不同。從早期的平面幾何微縮，到如今的三維立體堆疊、異質加速器與軟硬體協同優化，這些表面上看起來彼此割裂、躍遷的技術革命，在適當的數學表述（座標系）下，其實只是沿著同一個誠實的抽象軸線（\(pC\) 軸），優雅且連續地一格格向前跨步。

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Conclusion / 結語

As traditional geometric scaling loses descriptive power, computing may require a shift in its dominant descriptive variables. Rather than treating feature size as the primary axis of progress, future frameworks may benefit from system-level, efficiency-weighted measures of compute expressed on logarithmic scales.

Such a perspective does not replace existing metrics, but offers an alternative lens through which multi-scale architectural evolution can be interpreted. In this view, the challenge is not merely to build smaller devices, but to develop representations that remain meaningful as systems become increasingly complex.

當傳統的幾何微縮徹底失去其描述物理真相的能力時，計算科學必須迎來主導描述變數的根本移轉。未來的科技框架不應再將「特徵尺寸」視為衡量進步的唯一軸線，而應受益於這種基於系統級、經過效率加權並以對數尺度呈現的算力指標。這種視角並非為了取代現有的工程細節，而是提供了一盞嶄新的明燈，讓我們得以看清多尺度架構演進的真貌。在後摩爾時代的文明地平線上，真正的挑戰從來不只是如何把元件刻得更小，而是如何建立起一套當系統變得無窮複雜時，依然能保持物理誠實與科學理性的度量衡。

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歡迎來到 pC 算力級別的世界。身處後摩爾時代的我們，也許正在見證一場從「幾何文明」走向「計算文明」的偉大範式轉換。歡迎在下方留言，分享你對這場尺度革命的看法。