ModularPhaseNet：标准Transformer中可计算的语义层级、方向与上下文一致性的有限循环相位几何
ModularPhaseNet: Finite-Cyclic Phase Geometry for Computable Semantic Hierarchy, Direction, and Context Consistency in Standard Transformers
AI总结：
ModularPhaseNet通过有限循环群离散化复相位几何，为标准Transformer引入可计算的语义层级、方向与一致性不变量，理论证明其性质并预注册评估计划。
AI中文摘要：
我们提出ModularPhaseNet，这是对QuantumPhaseNet中引入的连续复相位几何的一种经典且整数可计算的离散化。标准Transformer的实值隐藏状态被保留，仅一个辅助相位通道被量化到F_p的乘法群中阶数q | (p-1)的循环子群G = <g>。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，概念层级由循环商群的过滤诱导，语义方向由有向相对群元素表示，上下文一致性由规范不变的循环和乐（cycle holonomy）度量。该方法在原本标准的Transformer中引入了三个组件：有限相位编码器、商过滤层级模块和群值连接模块。它们的输出作为实值偏置项进入自注意力。训练使用实群代数中的分布或直通Gumbel-Softmax，而推理使用精确模幂运算和预计算表。不需要量子硬件、复值矩阵乘法或离散对数计算。我们证明了量化失真界、商诱导划分的嵌套性、规范不变性、平坦连接的离散可积性结果以及所得注意力输出的有界性。核心实证假设是，这些精确的离散不变量在受控计算预算下改善了层级恢复、话语对齐、矛盾检测和校准的幻觉风险预测。本文报告了理论及预注册的评估计划；第14节中描述的实验尚未进行，此处不声称任何实证结果。
英文摘要：
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.
