模参数 τ 的三体闭包选择机制:来自立体矩阵公理的推导与检验

摘要 本文以立体矩阵(SM)结构世界观为公理体系,对模块化 A₄ 味模型模参数 τ 的选择机制给出第一性推导、定量核定与可观测预言;SM 的公理有效性由理论自身的内部一致性、枚举可复算性与可证伪预言认证。推导侧:以三体闭包公理(三体、对称、闭合的唯一代数解为单位根集{1,ω,ω²})为锚点,建立模形式约束 Y₂²+2Y₁Y₃=0 与 1+ω+ω²=0 的同构,证明权重2三重态模形式在 τ=ω 处精确呈现单位根结构(精度10⁻¹⁶),并给出 Z₃ 剩余对称的本征实现与 i、i∞ 的结构排除。定量核定侧(预注册判据 A–E):判据D在最小 Feruglio A₄ seesaw 模型下给出唯一3σ混合角解区 ω 带(Re≈0.44–0.5、Im∈[0.826,0.946]),随机基线确认解区真实;判据C(选择效应)与判据E(质量谱)均为负结果,如实入册。解构侧:将文献中模量势能动力学在 τ≃ω 附近的极小还原为三体闭合的结构吸引子,而非独立输入。诊断侧:对 DESI 宇宙学-实验室张力、最小模型质量谱偏差与 NuFIT 小偏差给出统一诊断,并逐例给出可证伪判据。预言侧:P1(τ锁定 ω 带)具预注册效力,P2(Σmᵢ≪0.1 eV)与 P3(m_ββ 为 meV 量级)为方向性承诺。

Abstract Taking the Stereo Matrix (SM) structural worldview as its axiomatic system, this paper gives a first-principles derivation, quantitative determination and observable predictions for the selection mechanism of the modulus parameter τ in modular A₄ flavour models; the validity of the SM axioms is certified by the theory's own internal consistency, enumerative reproducibility and falsifiable predictions. On the derivation side: anchored on the three-body closure axiom (the unique algebraic solution of three-body, symmetric, closed being the root-of-unity set {1, ω, ω²}), we establish the isomorphism between the modular-form constraint Y₂²+2Y₁Y₃=0 and 1+ω+ω²=0, prove that the weight-2 triplet modular form exhibits exactly the root-of-unity structure at τ=ω (precision 10⁻¹⁶), and give an eigenrealization of the residual Z₃ symmetry together with the structural exclusion of i and i∞. On the quantitative side (pre-registered criteria A–E): criterion D yields a unique 3σ mixing-angle solution region — the ω band (Re≈0.44–0.5, Im∈[0.826,0.946]) — under the minimal Feruglio A₄ seesaw model, with a random baseline confirming that the region is real; criteria C (selection effect) and E (mass spectrum) are negative results, registered as such. On the deconstruction side: the literature's modulus-potential minima near τ≃ω are reduced to the structural attractor of three-body closure rather than an independent input. On the diagnostic side: a unified diagnosis is given for the DESI cosmology–laboratory tension, the minimal-model mass-spectrum deviation and the NuFIT small deviation, each with a falsifiable criterion. On the prediction side: P1 (τ locked to the ω band) has pre-registered force, while P2 (Σmᵢ≪0.1 eV) and P3 (m_ββ of order meV) are directional commitments.

本记录含中文版与英文版两个文件(Bilingual: Chinese + English DOCX)。