This paper formalizes the Identity Principle established in Wang 2026 V6 (Zenodo DOI: 10. 5281/zenodo. 20162978) within a K-symmetric graded geometric framework. Main construction: The graded completed geometric state is defined as Xigeo (s) = T (s) ⊕ R (s) ⊕ K (s) on three geometrically orthogonal channels: - T (s): tangential resultant channel - R (s): radial drift channel - K (s): K-odd radial residue channel The classical completed zeta invariant xi (s) is identified as the faithful scalar projection of Xigeo (s). Logical chain (7-theorem structure): - Theorem 0: Classical Anchoring (zetageo equals classical zeta for sigma > 1) - Theorem A: Completed Geometric Identity - Theorem B: K-Symmetry (borrowed from classical functional equation xi (s) = xi (1-s) ) - Theorem E0a: Graded Geometric State Definition - Theorem E0b: Faithful-Zero Condition (★ stated explicitly as Geometric Axiom, source: V6 Identity Principle) - Theorem C: K-odd radial residue obstruction (same-signed Dₙ cannot be canceled by tangential phase interference) - Theorem D: Uniqueness of Critical Line (Dₙ = 0 iff sigma = 1/2) - Theorem E: Pole-Zero Identity Result: Under the Wang 2026 Faithful-Zero Condition, every non-trivial zero of zeta (s) satisfies Re (s) = 1/2. Logical status: This is a CONDITIONAL theorem. The Faithful-Zero Condition is the V6 Identity Principle, here stated explicitly as a geometric axiom — analogous in logical level to Euclid's parallel postulate. It is not derived from prior theorems in classical analysis; it is the geometric foundation of the Wang 2026 framework. Relation to V6: V7 does not replace V6. V6 contains the geometric blueprint (arc accumulation identity, two-frequency spherical helical topology, sphere closure at sigma = 1/2, Identity Principle as geometric statement). V7 provides the formalization scaffolding within K-symmetric graded geometry. The two papers are complementary. This submission contains both English (ENV7) and Chinese (CNV7) versions. 本文 在 K-对称分级几何框架内, 形式化 Wang 2026 V6 (Zenodo DOI: 10. 5281/zenodo. 20162978) 中 建立的 同一性原理。 主要构造: 分级补全几何态 定义为 Xigeo (s) = T (s) ⊕ R (s) ⊕ K (s) 三个 几何正交通道: - T (s): 切向合成通道 - R (s): 径向漂移通道 - K (s): K-奇径向残差通道 经典补全 zeta 不变量 xi (s) 严格 识别为 Xigeo (s) 的 忠实标量投影。 逻辑链 (7 定理结构): - 定理 0: 经典锚定 (zetageo 在 sigma > 1 严格等于 经典 zeta) - 定理 A: 补全几何恒等式 - 定理 B: K-对称 (借用 经典泛函方程 xi (s) = xi (1-s) ) - 定理 E0a: 分级几何态定义 - 定理 E0b: 保零忠实条件 (★ 严格 标为 几何公理, 源于 V6 同一性原理) - 定理 C: K-奇径向残差障碍 (同号 Dₙ 严格 不可被 切向相位干涉抵消) - 定理 D: 临界线唯一性 (Dₙ = 0 当且仅当 sigma = 1/2) - 定理 E: 极点-零点同一性 结果: 在 Wang 2026 保零忠实条件下, zeta (s) 的 每个非平凡零点 严格满足 Re (s) = 1/2。 逻辑地位: 本文 严格 证明 一个 条件性 定理。保零忠实条件 严格 是 V6 同一性原理, 本文 明确 立 为 几何公理 —— 严格 与 欧几里得平行公理 同 逻辑层级。它 严格 不从 经典分析 先前定理 推出;它 严格 是 Wang 2026 框架 的 几何基础。 与 V6 的关系: V7 不替代 V6。V6 含 几何蓝图 (弧累加同一性、双频球面螺旋 拓扑、sigma = 1/2 球面闭合、同一性原理 为 几何陈述) 。 V7 严格 在 K-对称分级几何 内 提供 形式化骨架。两文 严格 互补。 本投稿包含英文 (ENV7) 和中文 (CNV7) 双版本。
Lixin Wang (Thu,) studied this question.