Term Coding asks: given a finite system of term identities $Γ$ in $v$ variables, how large can its solution set be on an $n$--element alphabet, when we are free to choose the interpretations of the function symbols? This turns familiar existence problems for quasigroups, designs, and related objects into quantitative extremal questions. We prove a guessing-number sandwich theorem that connects term coding to graph guessing numbers (graph entropy). After explicit normalisation and diversification reductions, every instance yields a canonical directed dependency structure with guessing number $α$ such that the maximum code size satisfies $\log_n \Sn(Γ)=α+o(1)$ (equivalently, $\Sn(Γ)=n^{α+o(1)}$), and $α$ can be bounded or computed using entropy and polymatroid methods. We illustrate the framework with examples from extremal combinatorics (Steiner-type identities, self-orthogonal Latin squares) and from information-flow / network-coding style constraints (including a five-cycle instance with fractional exponent and small storage/relay maps).

翻译：项编码问题提出：给定一个包含 $v$ 个变量的有限项恒等式系统 $Γ$，当我们可以自由选择函数符号的解释时，其在 $n$ 元字母表上的解集最大能有多大？这将关于拟群、设计及相关对象的存在性问题转化为定量的极值问题。我们证明了一个猜测数夹逼定理，将项编码与图猜测数（图熵）联系起来。经过显式的归一化与多样化归约后，每个实例均生成一个具有猜测数 $α$ 的规范有向依赖结构，使得最大编码规模满足 $\log_n \Sn(Γ)=α+o(1)$（等价地，$\Sn(Γ)=n^{α+o(1)}$），且 $α$ 可通过熵与多拟阵方法进行界定或计算。我们通过极值组合学（Steiner型恒等式、自正交拉丁方）以及信息流/网络编码风格约束（包括一个具有分数指数与小规模存储/中继映射的五环实例）中的示例来阐释该框架。