Layered Advancement Roadmap for the Unified Connection Structure of the H1–H3–H5 Triple-Gap Framework — Local Anchoring → Intermediate Bridging → Global Closure

This paper is a layered advancement roadmap for the unified connection of the H1–H3–H5 triple-gap framework; it is not a completed construction. It proposes a possible advancement path: local anchoring \(\to\) intermediate bridging \(\to\) global closure; the constructions at each step remain to be completed. Core structural conclusion: for the Type-B candidate phase function \(\Phi_{\mathrm{self}}=\sum_i\arctan2(y_i,x_i)\), the differential \(d\Phi_{\mathrm{self}}\) is not a well-defined 1-form on \(\mathcal{T}^{64}\)—the differential of \(\arctan2\) depends on the coordinate origin and changes under \(\mathbb{Z}^{64}\) translations. Consequently the Type-B \(\delta\) depends on the choice of representative and \(S=\ker\delta\) is not a well-defined subgroup. For Type C (linear phase \(\Phi_{\boldsymbol{\lambda}}(p)=\sum_i(\lambda_i x_i+\mu_i y_i)\)) the coefficients of \(d\Phi_{\boldsymbol{\lambda}}\) are constant, so \(d\Phi_{\boldsymbol{\lambda}}\) is a well-defined closed 1-form on \(T^{2n}\), \(\delta\) is well-defined and \(S=\ker\delta\) is a well-defined subgroup. Type C is the only viable realisation path for SRIHG. Five substantial deepenings: (1) completion of the Kronecker-type argument for Q5—\(\epsilon=1/q\) is the unique “neither-redundant-nor-missing” readout resolution; Q5 is elevated from “candidate” to “proved under the structural hypotheses of Yuanxian H3”; (2) correction of the H3 capacity formula for the general quantum case—\(C_{\mathrm{total}}=\log_2\mathrm{lcm}(q_1,\dots,q_k)\); (3) precise quantification of Type-B path dependence—\(\delta(\gamma_\epsilon)-\delta(\gamma_{-\epsilon})\to-\pi\); (4) precise mathematical statements of the seven lemmas of F1c; (5) reassessment of Type B—infeasible for SRIHG realisation, yet still of candidate value in three contexts (H³ layer, third moment of \(\nu_\infty\), quantisation limit of Type C). Technical completion degree: Type-C parts stand at 2.5; Type-B parts stand at 1.5. Advancement from 2.5 to 3.0 requires Lean compilation of the F1b combinatorial layer, F2 numerical verification (strict Type-C version) and Lean realisation of the F1c analytic layer. --- 本文是 H1-H3-H5 三缺口框架统一联结的分层推进路线图,不是已完成构造的方案。它提出一条可能的推进路径:局部锚定 \(\to\) 中间层搭桥 \(\to\) 全局收束;每一步的构造均为待完成。 核心结构性结论:类型 B 的候选相位函数 \(\Phi_{\mathrm{self}}=\sum_i\arctan2(y_i,x_i)\) 的微分 \(d\Phi_{\mathrm{self}}\) 不是 \(\mathcal{T}^{64}\) 上的良定义 1-形式——\(\arctan2\) 的微分依赖坐标原点,在 \(\mathbb{Z}^{64}\) 平移下改变。因此类型 B 的 \(\delta\) 依赖代表选择,\(S=\ker\delta\) 不是良定义子群。类型 C(线性相位 \(\Phi_{\boldsymbol{\lambda}}(p)=\sum_i(\lambda_i x_i+\mu_i y_i)\))的 \(d\Phi_{\boldsymbol{\lambda}}\) 系数为常数,是 \(T^{2n}\) 上的良定义闭 1-形式,\(\delta\) 良定义,\(S=\ker\delta\) 是良定义子群。类型 C 是唯一可行的 SRIHG 实现路径。 五项实质深化:(一)Q5 的 Kronecker 型论证完整化——\(\epsilon=1/q\) 是唯一「无冗余无遗漏」的读数分辨率,Q5 从「候选」升为「在元宪 H3 结构假设下已证」;(二)H3 容量公式的量子一般情形修正——\(C_{\mathrm{total}}=\log_2\mathrm{lcm}(q_1,\dots,q_k)\);(三)类型 B 路径依赖的精确量化——\(\delta(\gamma_\epsilon)-\delta(\gamma_{-\epsilon})\to-\pi\);(四)F1c 七条引理的精确数学陈述;(五)类型 B 的重新评估——在 SRIHG 实现上不可行,但在 H³ 层、\(\nu_\infty\) 三阶矩、类型 C 的量子化极限三个语境中仍有候选价值。 技术完成度:类型 C 部分为 2.5;类型 B 部分为 1.5。从 2.5 到 3.0 需 F1b 组合层 Lean 编译、F2 数值核验(类型 C 严格版)与 F1c 解析层 Lean 实现。