D2c & D2c-C — Watson and Lattice Green-Function Foundations of the Discrete Regime (v1)

Abstract

This combined document establishes the Watson and lattice Green-function foundations of the discrete regime: the standard Watson integrals 𝑊𝑑, the corpus-specific conventions for 𝐺0(ℤ𝑑), the algebraic kernel identity 𝐾8=𝐾42, the Watson kernel-square discrepancy Cov, and the normalized return quantity 𝐶4.Scope boundary (critical): This document covers the lattice identities that are pure mathematical/algebraic facts. It does NOT cover P50-A (𝐺0(ℤ𝐺24)=𝐺0(ℤ4)+Cov/7), the factor 1/7, or the Kreïn gap 𝜆𝑅, which require the 𝐺2Haar measure and belong to a separate document.Self-containment principle: All definitions, proofs, numerical verifications, and conventions necessary to audit this document are reproduced within this file. No external corpus document (discrete, continuous, or cosmological) needs to be consulted. Historical provenance notes are provided for traceability but are never required references.Key results established in this document: 1.The Watson integrals 𝑊𝑑=∫𝑒−𝑑𝑡∞0𝐼0(𝑡)𝑑 𝑑𝑡are defined and their values independently verified by mpmath quadrature at 25-digit working precision. 2.The kernel identity 𝐾8=𝐾42is proven as a pure algebraic fact. 3.The 𝑍2-symmetry convention 𝐺0(ℤ8)corpus:=12𝑊8is defined and motivated. 4.The Watson kernel-square discrepancy Cov=𝑊42−12𝑊8is defined, computed, and its mathematical structure analyzed. 5.The normalized return quantity 𝐶4:=14𝑊4is defined. 6.The Brillouin zone representation of 𝑊𝑑with the normalized lattice dispersion ℎ(𝑘)is proven.