用离散循环群构建可计算的语义层次与上下文一致性机制
ModularPhaseNet: Finite-Cyclic Phase Geometry for Computable Semantic Hierarchy, Direction, and Context Consistency in Standard Transformers
- 将相位几何离散化为有限循环群，仅通过模运算实现
- 引入商滤链与群连接模块，提升语义层级与方向性建模能力
- 无需量子硬件，适合标准Transformer部署，理论完备
我们提出ModularPhaseNet，一种对量子相位网络中连续复数相位几何的类经典、整数可计算离散化方法。保留标准Transformer的实值隐藏状态，仅将辅助相位通道量化为域F_p乘法群的循环子群G = <g>，阶数q满足q | (p-1)。连续相位e^{i phi}由z = g^a mod p表示，相位合成变为群乘法，相对相位变为群除法，概念层次由循环商群的滤链诱导，语义方向由有向群元表示，上下文一致性通过规范不变的环路全息性度量。该方法在标准Transformer中引入三个组件：有限相位编码器、商滤链层次模块、群值连接模块，其输出以实值偏置形式进入自注意力。训练使用实群代数分布或直通式Gumbel-Softmax，推理采用精确模指数和预计算表。无需量子硬件、复数矩阵乘法或离散对数计算。证明了量化失真界、商诱导划分的嵌套性、规范不变性、平坦连接的离散可积性以及注意力输出的有界性。核心假设是这些精确离散不变量能在控制算力下提升层次恢复、话语对齐、矛盾检测及幻觉风险校准。本文给出理论框架与预注册评估计划；第14节描述的实验尚未执行，此处不宣称任何实证结果。
原文摘要 · Abstract (English)
We propose ModularPhaseNet, a classical and integer-computable discretization of the continuous complex phase geometry introduced in QuantumPhaseNet. The real-valued hidden states of a standard Transformer are retained, while only an auxiliary phase channel is quantized into a cyclic subgroup G = <g> of order q | (p-1) in the multiplicative group of F_p. A continuous phase e^{i phi} is represented by z = g^a mod p; phase composition becomes group multiplication, relative phase becomes group division, conceptual hierarchy is induced by a filtration of cyclic quotients, semantic direction is represented by oriented relative group elements, and contextual consistency is measured by gauge-invariant cycle holonomy. The method introduces three components into an otherwise standard Transformer: a finite-phase encoder, a quotient-filtration hierarchy module, and a group-valued connection module. Their outputs enter self-attention as real-valued bias terms. Training uses distributions in the real group algebra or straight-through Gumbel-Softmax, whereas inference uses exact modular exponentiation and precomputed tables. No quantum hardware, complex-valued matrix multiplication, or discrete-logarithm computation is required. We prove quantization-distortion bounds, nesting of quotient-induced partitions, gauge invariance, a discrete integrability result for flat connections, and boundedness of the resulting attention output. The central empirical hypothesis is that these exact discrete invariants improve hierarchy recovery, discourse alignment, contradiction detection, and calibrated hallucination-risk prediction under a controlled compute budget. This paper reports the theory together with a pre-registered evaluation plan; the experiments described in Section 14 have not yet been carried out, and no empirical result is claimed here.
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