PRE-GHR XXXIX: The Mathematics of δ₀ — Domain-Order Theory, Composition, and the Measurable Structure of the Irreducible Governance Residue

PRE-GHR XXXVIII gave the series its first formal definition of the irreducible governance residue δ₀^μ (P ) := inf P ′ admissible from P ∥residue(P ′ )∥ and proved a bit-level lower bound. This paper builds the property theory. The mathematics is a domain-order theory: every inequality in the paper follows from how the admissible domain D(P ) behaves under enlargement or composition of protocols. We prove monotonicity of δ₀^μ in the erasure surface (a wider admissible domain cannot raise the residue), an exact composition law δ₀^μ (P1 ◦ P2 ) = δ₀^μ (P1 ) + δ₀^μ (P2 ) − |T (P1 ) ∩ T (P2 )| under explicit hypotheses (joint attainability, non-interference, crosssegment cleanliness), and positivity δ₀^μ (P ) > 0 whenever T (P ) ̸= ∅ as a purely normative fact — manufactured by the admissibility constraint, with Landauer's principle confined to the physical interpretation. We then ask what the running system's audit ledgers can measure. The answer is stratified: the ordering structure of δ₀^μ is measurable in principle — conditional on a fixed normalization and full retention — by counting ledger entries and their overlaps, while the absolute value is stated relative to the fixed code-point measure (the multiplicative dependence is eliminated at the definition level of µcp ; within the fixed-schema D5.4 family the residual field-order drift is zero — reordering leaves the serialization length exactly invariant — and the broader L5.2-family discussion retains only the additive N · C width bound as a legacy of the pre-fixed-schema versions) and the aggregate-to-single-protocol bridge remains open. Three open questions are declared: the retention rate, the bridge, and the zero-set of ε (Q5.4); the encoding-family component of the normalization question (Q5.2) is closed — solved modulo explicit normalization form, with the residual drift explicitly bounded. No interception statistic is claimed to equal δ₀^μ ; where the wall cannot be built, the gap is marked, not papered over. v5.4 (2026-08-30): revision batch P2-X39-06, N4, N9, ruled by Kimi K3 red-pen ex-ante review and the author's signature gate: Q5.2 remark duplicate clause removed (the short "unit-agnostic: counting is counting" variant deleted, keeping the measure-invariant long form); Theorem 2's opening sentence aligned with its label and proof (now "Under A1's factorization clause and non-interference"); References [18] DOI aligned to the series convention (version DOI 10.5281/zenodo.21874189 -> concept DOI 10.5281/zenodo.21874188). Directed re-check of the changed lines +/- context: zero residue. No theorem statement, proof, or definition changed. v5.3 (2026-08-30): acknowledgments batch — the review chain of the v5.0–v5.2 documentation batches is recorded in the Acknowledgments section (GPT-5.6 Luna foundation handoff audit; Kimi K3 red-pen adjudication; Zhipu Qingyan GLM-5.3 boundary patrol and string-level verification; Qwen formalization review), with model identities recorded as write-side declarations per system configuration and no internal build identifier fabricated. No theorem statement, proof, or definition changed. v5.2 (2026-08-30): documentation batch closing review tickets P2-X39-02 and P2-X39-03 (GLM boundary patrol) plus notes N1–N3, all ruled by Kimi K3 review and the author: O4.1's "holds exactly under CE" made precise — nonnegativity prefaced, the statement scoped to the T(P)=∅ instances, and the reverse direction (from δ₀^μ(P)=0 to an admissible empty reconstruction) attributed to integer-valuedness (Lemma 1) and D(P)≠∅ (Observation 1); the abstract's hypothesis enumeration aligned with Corollary 2 (joint attainability, non-interference, cross-segment cleanliness); Theorem 4's proof cites Observation 1; Theorem 2's label made precise (A1 factorization clause + A3). No theorem statement, proof, or definition changed. Series: PRE-GHR XXXIX. License CC-BY-4.0.