We formalize Domain Algebra as a fibered $\omega$-CPO equipped with a domain-indexed endofunctor whose least fixed point defines semantics.我们将 Domain Algebra 形式化为一个带有域索引自函子的纤维化 $\omega$-CPO，其最小不动点给出语义。
A representation functor into vector spaces preserves this structure, yielding a strict correspondence between symbolic closure and differentiable fixed-point computation.到向量空间的表示函子保持这一结构，从而在符号闭包与可微不动点计算之间建立严格对应。
The homepage states the formal core first: semantic locality, structure-preserving representation, and fixed-point identity across symbolic and differentiable substrates.首页首先陈述形式核心：语义局部性、保持结构的表示，以及符号基底与可微基底之间的不动点同一性。
The @ operator binds a relation to a domain, making the domain constitutive of meaning rather than an external label. Truth is always evaluated locally within a fiber $Q_d$ and never in an abstract global space. $(D, \leq)$ forms a complete Heyting algebra, supporting intuitionistic semantics where “true” means “construction completed within this domain.”@ 算子把关系绑定到一个域上，使域成为意义的构成部分，而不是外加标签。真值总是在纤维 $Q_d$ 内局部判定，而不会在抽象的全局空间中求值。$(D, \leq)$ 构成一个完备 Heyting 代数，支撑直觉主义语义，其中“真”意味着“在该域内构造已经完成”。
A relation carries no meaning independent of its domain. $r@d_1$ and $r@d_2$ are distinct semantic objects whenever $d_1 \neq d_2$. The symbolic layer writes $\langle c, r@d, c' \rangle$ as an admissible quadruple. The differentiable layer preserves this structure through a representation functor $\hat{\varphi}_d$ satisfying $\hat{\varphi}_d \circ T_d = \widetilde{T}_d \circ \hat{\varphi}_d$ so the closure operator commutes with the mapping. Both compute the same fixed point.关系不携带任何独立于域的意义。只要 $d_1 \neq d_2$，那么 $r@d_1$ 与 $r@d_2$ 就是不同的语义对象。符号层将 $\langle c, r@d, c' \rangle$ 写作一个可容许四元组。可微层通过表示函子 $\hat{\varphi}_d$ 保持这一结构，并满足 $\hat{\varphi}_d \circ T_d = \widetilde{T}_d \circ \hat{\varphi}_d$，也就是闭包算子与映射可交换。两层计算得到同一个不动点。
The admissible quadruple is the minimal writable unit of the system. All computation, including closure iteration, fixed-point convergence, admissibility checking, and cross-domain reindexing, is built from this single primitive. Under intuitionistic semantics, writing a quadruple is constructing a proof; truth is not assigned after the fact but realized in the act of construction. Four substrates, Prolog, matrix iteration, attention, and memristive crossbar, implement the same algebraic structure without modification.这个可容许四元组是系统中最小的可写单元。所有计算，包括闭包迭代、不动点收敛、可容许性检查和跨域重索引，都由这一原语构成。在直觉主义语义下，写入一个四元组就是在构造一个证明；真值不是事后赋予的，而是在构造行为中实现的。Prolog、矩阵迭代、注意力机制和忆阻交叉阵列这四种基底都可以在不修改代数结构的前提下实现同一个系统。
The homepage is organized around three mathematical commitments: local semantic order in fibers, structure-preserving representation into vector spaces, and exact agreement of symbolic and differentiable fixed points.首页围绕三个数学承诺组织：纤维中的局部语义序、到向量空间的结构保持表示，以及符号不动点与可微不动点之间的精确一致。
Each domain determines a local semantic order and supports monotone iteration toward a least fixed point. Semantics is therefore generated inside fibers rather than projected from a global truth table.每个域决定一个局部语义序，并支持朝向最小不动点的单调迭代。因此语义是在纤维内部生成的，而不是从一个全局真值表投影下来。
The map into vector spaces is not an approximation layer added later. It preserves the same operator structure, so symbolic closure and differentiable update live in strictly corresponding diagrams.到向量空间的映射不是事后附加的近似层。它保持同一算子结构，因此符号闭包和可微更新位于严格对应的交换图中。
What the symbolic layer reaches by admissible closure is exactly what the differentiable layer reaches by fixed-point convergence. The two substrates differ operationally, but not semantically.符号层通过可容许闭包所达到的结果，正是可微层通过不动点收敛所达到的结果。两种基底在操作上不同，但在语义上并不分离。
The axiom cards below are styled as a compact specification surface: each statement names one invariant, one semantic role, and one algebraic commitment inside the system.下面的公理卡片被组织成紧凑的形式规约界面：每条陈述都对应一个不变量、一个语义角色和一个代数承诺。
“No domain, no independent meaning.”
There exists a Heyting category $\mathcal{C}$ with finite limits, power objects, and internal Heyting structure. Its objects are called concepts.存在一个 Heyting 范畴 $\mathcal{C}$。它有有限极限、幂对象与内部 Heyting 结构，范畴中的对象统一称为 concepts。
There exists a functor $R : \mathcal{C} \times \mathcal{C} \to \mathcal{L}$, where $\mathcal{L}$ is a Heyting algebra. $R(c,c')$ is the truth-value space of relations from $c$ to $c'$.存在函子 $R : \mathcal{C} \times \mathcal{C} \to \mathcal{L}$，其中 $\mathcal{L}$ 为 Heyting 代数。$R(c,c')$ 给出从 $c$ 到 $c'$ 的关系真值空间。
There exists a complete lattice $(D, \leq)$ with a fibration $F : D^{op} \to \mathcal{C}$, where each fiber $Q_d$ is the category of assertions valid in domain $d$.存在一个完备格 $(D, \leq)$，并有纤维化 $F : D^{op} \to \mathcal{C}$，其中每个纤维 $Q_d$ 是在域 $d$ 中成立的断言范畴。
The primitive computation unit is the admissible quadruple $\langle c, r, c', d \rangle$. If two paths lie in the same $Q_d$, they compose inside that domain.原始计算单位是可容许四元组 $\langle c, r, c', d \rangle$。若两段路径都位于同一 $Q_d$ 中，则它们可在域内复合。
The Galois closure $\mathrm{cl}_d$ is closed under path composition. Closure preserves not only points, but also the path structure derived from them.Galois 闭包 $\mathrm{cl}_d$ 对路径复合封闭。闭包不仅保存点，还保存由这些点导出的路径结构。
Admissibility is determined by whether the dependency graph of the closure is acyclic. Equivalently, any endomorphic cycle makes the write inadmissible.可容许性由闭包依赖图是否无环决定。等价地，若存在任何端同态循环，则该写入不可容许。
For each domain $d$, the family of closed sets $L_d$ forms a complete lattice, and the Galois connection $(\alpha_d, \gamma_d)$ induces the closure operator $\mathrm{cl}_d$.对每个域 $d$，闭合集族 $L_d$ 构成完备格，并由 $(\alpha_d, \gamma_d)$ 给出 Galois 连接，诱导闭包算子 $\mathrm{cl}_d$。
Each fiber $Q_d$ carries a valuation morphism $\nu_d : Q_d \to \Omega_d$, and path composition must correspond semantically to meet.每个纤维 $Q_d$ 配备估值态射 $\nu_d : Q_d \to \Omega_d$，并要求路径复合在语义上对应 meet。
Any two realizations preserving Heyting operations, the Galois connection, and valuation must agree on the result of the same computation term.任意两个保持 Heyting 运算、Galois 连接和估值的实现，对同一计算项给出相同结果。
Every computation can be expressed as a finite composition of basic operations, so the model keeps its computational invariant at the minimal path level.每个 computation 都能表达为有限个 basic operations 的复合，因此模型的计算不变量保持在最小的路径层级。
For each domain $d$, $\mathrm{Adm}_d$ is decidable at write-time and compatible with that domain's algebraic structure. Domain-specific rules are instances of A5.对每个域 $d$，$\mathrm{Adm}_d$ 在写入时可判定，并与该域自身的代数结构兼容。域特定规则都是 A5 的实例。
From the homepage, readers can continue into the paper line, the interactive demonstrations, or the author and research context.从首页出发，读者可以继续进入论文线、交互演示线，或作者与研究背景部分。