PRE-GHR XL: The Mathematics of the Gap — Entropy-Weighted Residue and Violation Budgets

The residue δ₀ of XXXVIII/XXXIX counts what must remain — the minimal number of trace units any admissible reconstruction is forced to retain. This paper asks a different question: how much of what remains matters. It develops δ₀^(w), the minimal weighted residue under an admissible weight class, and separates it from the violation budget B — how far practice may drift from the ideal before the budget is exceeded — which lives on the governance layer and is a regime choice, not a consequence of the protocol. The contribution is inheritance, not invention: the paper proves which parts of the XXXIX property layer survive the move from counting measures to non-negative additive set functions, item by item, with each migration condition stated. The weight class is settled at the definition layer: admissibility requires finite additivity on unions, which is exactly what the composition law's subtraction form consumes; submodular weights — the natural home of joint-information measures — are declared open, not admitted, because per-record information sums are neither submodular nor supermodular and the inadditivity has no sign. An empirical section anchors the theory in the Q5.1 retention-rate harvest: the entropy structure of the running ledger and its redundancy signature (91 exact duplicates, all routine probe commands) are reported as measured; retention uniformity is reported as empirically indeterminate at the registered power, with an extension window scheduled; the retention rate itself is withheld from use pending a denominator audit. A budget-feasibility theorem (X5) concludes: on the class of protocols whose erasure surface can crop every non-sender unit, the feasible set {P : δ₀^(w)(P) ≥ H_min, V(P) ≤ B_max} is non-empty if and only if some compliant protocol carries sender-trace weight at least H_min — trace weight, not trace count, is the feasibility currency. v1 (2026-08-20): first release. Three-level annotation discipline for all empirical claims: [measured] entropy structure and redundancy signature (Q5.1 harvest, 2503 events, independently verified); [indeterminate] retention uniformity (45 cells n