Positioning of Light and Electromagnetic Waves in the SRE Framework -The Involutive ℤ₂ Sector, the Cohomological Ladder, and "One Emission as One Closed→Open Transition"

Background and problem. In prior work, SRE has identified "mass" as a ℤ₃-equivariant circulant operator, and connected it with the ℤ₃-torsor structure of the three-ring skeleton. A natural follow-up question then arises: what is the SRE description of "light and electromagnetic waves", what is its relation to mass, and can it cover at once the radiation of chemical origin and of nuclear origin? This paper answers these three questions head-on. Method. This paper places the project's existing light-side stipulations (the residuality axiom, the Möbius double-cover parameterisation, the involution $P=S^{n/2}$ on the weighted Möbius ladder, the $\mathbb{Z}_2$ holonomy coupling $\delta_A$) and the ℤ₃ structure on the mass side (the three-ring torsor, the circulant operator) under one framework for comparison: the cohomological ladder $\lvert H^{1}(G;\mathbb{Z}_n)\rvert=n^{\beta_1}$. The whole paper needs only one structural input — the involution $P=S^{n/2}$ on the weighted Möbius ladder $M_n$ — and everything else is recomputable linear algebra and group theory. Results. The paper gives four mutually independent, verifiable propositions: (1) the gear proposition — light and mass are two gears of the same machine: light runs on the coefficient group $\mathbb{Z}_2$ (duality), mass on $\mathbb{Z}_3$ (triality), and the two are adjacent rungs of the same cohomological ladder; the root of the difference is a group-theoretic prohibition $\lvert\mathbb{Z}_2\rvert=2<3$ (the light gear structurally cannot hold three generations). (2) the carrier proposition — the light carrier is the odd eigenspace of an involution, with $P^{2}=I$, membership in the automorphism group, commutation with the Laplacian, zero trace, and spectrum exactly $\{\pm1\}$; it is completely determined by the single integer $n$, with zero continuous free parameters, so there is no place to put a source. (3) the source proposition — the spectrum of the weighted Möbius ladder splits into a "sector-blind term + odd-sector constant term" $\mu_k=2c(1-\cos\frac{2\pi k}{n})+2w\cdot\mathbb{1}[k\ \text{odd}]$, in which only the $w$ channel simultaneously satisfies "acts only on the odd sector" and "shifts every mode uniformly"; measurement over $\delta\in[10^{-3},5\times10^{-1}]$ shows that the $n/2$ modes of the odd sector are shifted by strictly the same amount (range $\sim10^{-15}$), that the even sector is always motionless, and that the spectral shape is preserved mode by mode (residual $8.9\times10^{-16}$); hence the source can inject only one scalar on the light side. (4) the transition proposition — "one emission = one closed → open transition": the ledger difference of the transition carries the energy ($E$, $\beta_1$ each $-1$), the spectral invariants of the transition carry the light carrier ($\rho$, the sector structure $P$), and the two are orthogonal; therefore a chemical source (transition at the electronic-molecular level) and a nuclear source (transition at the nuclear level) share the same carrier, their only difference being the level at which the ledger is recorded. Boundary and conclusion. The paper also states four limitations: the light-side readings come from the project's existing axioms, and this paper only puts them into recomputable form without adding any light-side physical content; equating the "open state" with "light" is a structural conjecture, and the paper claims only a consistency of direction; $\mathbb{Z}_3$ is the smallest coefficient group able to carry three gears but not the unique one ($\mathbb{Z}_4$ works as well); the numerical equality $\mathrm{rank}(P_E)=7$ vs. $\beta_1=7$ is a pure coincidence and must not be used as mutual evidence. The overall conclusion is registered as case 18 of the discrete-closure law (G12): it gives "why the gears are two and three", "why the source is a single scalar" and "why energy and carrier are orthogonal" (discrete side, closed), but not the energy scale of light (continuous side, requiring external input). All numerical verification in this paper holds only within the SRE model. 背景与问题. 在前序工作中,SRE 已把「质量」识别为一个 ℤ₃‑等变的循环算子,并把它与三环骨架的 ℤ₃‑挠子结构联系起来。一个自然的追问随之出现:SRE 对「光与电磁波」的描述是什么,它与质量的关系是什么,它能否同时罩住化学来源与核来源的辐射? 本文正面回答这三个问题。 方法. 本文把项目既有的光侧口径(残余性公理、Möbius 双覆盖参数化、加权 Möbius 阶梯上的对合 $P=S^{n/2}$、$\mathbb{Z}_2$ holonomy 耦合 $\delta_A$)与质量侧的 ℤ₃ 结构(三环挠子、循环算子)放在同一个框架下比较:同调阶梯 $\lvert H^{1}(G;\mathbb{Z}_n)\rvert=n^{\beta_1}$。全文只需一个结构输入——加权 Möbius 阶梯 $M_n$ 上的对合 $P=S^{n/2}$——其余均为可复算的线性代数与群论读数。 结果. 全文给出四条彼此独立可验的命题:(1)档位命题——光与质量是同一台机器的两个档位:光走系数群 $\mathbb{Z}_2$(对偶),质量走 $\mathbb{Z}_3$(三力性),二者是同一同调阶梯的相邻两级;差异的根源是一条群论禁令 $\lvert\mathbb{Z}_2\rvert=2<3$(光档在结构上装不下三代)。(2)载体命题——光载体是对合的奇本征空间,$P^{2}=I$、属自同构群、与拉普拉斯算子交换、迹为零、谱恰为 $\{\pm1\}$;它由整数 $n$ 一个参数完全决定,零个连续自由参数,故没有位置可以安放源。(3)源命题——加权 Möbius 阶梯的谱可拆为「扇区盲项 + 奇扇区常数项」$\mu_k=2c(1-\cos\frac{2\pi k}{n})+2w\cdot\mathbb{1}[k\ \text{奇}]$,其中唯 $w$ 通道同时满足「只在奇扇区」与「逐模均匀平移」两条性质;实测在 $\delta\in[10^{-3},5\times10^{-1}]$ 范围内奇扇区 $n/2$ 个模的平移量严格相同(极差 $\sim10^{-15}$)、偶扇区恒不动、谱形逐位不变(残差 $8.9\times10^{-16}$),故源在光侧只能注入一个标量。(4)过渡命题——「一次发光 = 一次闭 → 开过渡」:过渡的账本差承载能量($E$、$\beta_1$ 各 $-1$),过渡的谱不变量承载光的载体($\rho$、扇区结构 $P$),二者正交;因此化学来源(过渡在电子‑分子层)与核来源(过渡在核层)共用同一个载体,其差别只在账本被记在哪一级。 边界与结论. 本文同时写明四条限定:光侧读数来自项目既有公理,本文只把它们做成可复算形式,未新增光侧物理内容;把「开态」等同于「光」是结构性猜想,本文只主张「方向一致」;$\mathbb{Z}_3$ 是能承载三档的最小系数群而非唯一($\mathbb{Z}_4$ 同样可以);$\mathrm{rank}(P_E)=7$ 与 $\beta_1=7$ 的数值相同纯属巧合,不得互证。全文结论记为离散闭合律(G12)第 18 例:给「档位为何是二/三」「源为何只有一个标量」「能量与载体为何正交」(离散侧,闭合),不给光的能量标度(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。