Mass Origin in the State–Relational–Entropy Framework - A Circulant-Operator Formulation on the Three-Ring ℤ₃ Torsor

Background and problem. Within the State–Relational–Entropy (SRE) framework, mass is treated as a projected reading of relational structure rather than an ontological input, and therefore in principle it should not be a number given in advance. However, the question "since mass is an emergent quantity, what exactly is its precise mathematical object?" had not previously been answered head-on. This paper answers that question and places the answer in a recomputable, falsifiable form. Method. This paper takes the SRE nucleon skeleton $Y_3\ltimes\triangle_3$ ($V=12$, $E=18$, $\beta_1=7$, $|\mathrm{Aut}|=36$) as its only structural input. It first proves that the three shared rings of this skeleton form a ℤ₃-torsor under the automorphism group — there is a cyclic order but no absolute origin; then, taking "respecting that structure" as the constraint, it derives the unique admissible form of the mass operator and sets out its spectrum, invariants and external comparisons. Results. The paper gives four mutually independent, verifiable propositions: (1) the object proposition — a mass operator that respects the three-ring structure must commute with the cyclic shift $S$, and "commuting with $S$" holds if and only if it is a circulant matrix; hence $A=c_0\mathbb{1}+c_1S+\bar c_1S^{2}$, and ℤ₃-equivariance compresses 9 matrix entries into 3; (2) the shape proposition (with one correction) — the spectrum of a Hermitian circulant is a set of 120° samples of one cosine, $\lambda_j=c_0+2|c_1|\cos(\theta+2\pi j/3)$, but this form holds trivially for any three generations (three points, three unknowns, always solvable), so the "cosine shape" is a free reparameterisation rather than a constraint; (3) the opening proposition — writing the amplitude multiset as an ordered triple is equivalent to choosing an origin for the DFT characters, and the three choices give exactly the same invariants, so "opening" is a ℤ₃ gauge degree of freedom rather than information; (4) the invariant proposition — the invariant moduli space is two-dimensional, $(\eta,\ \delta\bmod 2\pi/3)$, and the Koide combination $Q=\tfrac13+\tfrac23\eta^{2}$ probes only the single direction $\eta$. Measurement on the charged leptons gives $\eta^{2}=1/2$ to within $3.3\times10^{-6}$. Boundary and conclusion. The paper also states two limitations that must be made explicit: the cosine shape is free (so the entire content of Koide collapses to a single number $\eta=1/\sqrt2$); and absolute mass (the scale $c_0$) does not lie inside SRE, while only renormalization-group-invariant dimensionless combinations qualify as candidate targets. The overall conclusion is registered as case 15 of the discrete-closure law (G12): "three" and the functional form are given (discrete side, closed), while the value of $\eta$ and the scale are not (continuous side, requiring external input). All numerical verification in this paper holds only within the SRE model, and comparisons with charged-lepton masses are always stated as "structural consistency" rather than "numerical prediction". 背景与问题. 在状态—关系熵(SRE)框架中,质量被视作关系结构的投影读数而非本体输入,因此它原则上不应是一个被预先给定的数。然而「质量既然是涌现量,其准确的数学对象究竟是什么」这一问题此前未被正面回答。本文回答该问题,并把答案落在一个可复算、可证伪的形式上。 方法. 本文取 SRE 核子骨架 $Y_3\ltimes\triangle_3$($V=12$、$E=18$、$\beta_1=7$、$|\mathrm{Aut}|=36$)为唯一结构输入,先证明该骨架的三条共享环在自同构群下构成一个 ℤ₃‑挠子(torsor)——存在循环序而无绝对原点;随后以「尊重该结构」为约束,求质量算子的唯一允许形式,并交代其谱、不变量与外部对照。 结果. 全文给出四条彼此独立可验的命题:(1)对象命题——尊重三环结构的质量算子必须与循环移位 $S$ 交换,而「与 $S$ 交换」当且仅当是循环矩阵,故 $A=c_0\mathbb{1}+c_1S+\bar c_1S^{2}$,ℤ₃‑等变性把 9 个矩阵元素压到 3 个;(2)形状命题(含一处修正)——Hermitian 循环矩阵的谱为一条余弦的 120° 采样点 $\lambda_j=c_0+2|c_1|\cos(\theta+2\pi j/3)$,但该形式对任意三代平凡成立(三点三未知,恒可解),故「余弦形状」是免费重参数化而非约束;(3)打开命题——把振幅多重集写成有序三元组等价于为 DFT 特征标选取原点,三种选法给出完全相同的不变量,故「打开」是 ℤ₃ 规范自由度而非信息;(4)不变量命题——不变量模空间为二维 $(\eta,\ \delta\bmod 2\pi/3)$,Koide 组合 $Q=\tfrac13+\tfrac23\eta^{2}$ 只探测 $\eta$ 一个方向。带电轻子的实测给出 $\eta^{2}=1/2$ 成立到 $3.3\times10^{-6}$。 边界与结论. 本文同时给出两条必须写明的限定:余弦形状是免费的(故 Koide 的全部内容收敛为一个数 $\eta=1/\sqrt2$);绝对质量(尺度 $c_0$)不在 SRE 内,且只有重整化群不变的无量纲组合才配当候选靶。全文结论记为离散闭合律(G12)第 15 例:给「三」与函数形式(离散侧,闭合),不给 $\eta$ 的值与尺度(连续侧,须外部输入)。本文全部数值验证仅在 SRE 模型内部成立,与带电轻子质量的对照一律作「结构一致性」而非「数值预言」陈述。