IndisputableMonolith.NumberTheory.T1PhaseBudgetBound
This module establishes that a stable ledger budget bounds all finite unresolved phase-cost sums. Number theorists working on phase visibility and Erdős-Straus closure cite it as the T1 budget interface. The argument imports the upstream phase failure cost accumulation and applies it directly to the stable budget definition to cap every finite unresolved sum.
claimLet $B$ be a stable integer ledger budget. Then for every finite collection of unresolved phase costs $c_i$ arising from failed gates, the sum satisfies $0 < c_i$ and $B$ bounds the total.
background
The module resides in the NumberTheory section and imports PhaseFailureCost, whose doc-comment states that failed finite gates contribute positive unresolved phase cost and that finite sums of those costs accumulate. It introduces the stable integer ledger budget as a fixed upper limit on unresolved phase costs together with sibling notions such as uniform failure floor and failed gate count bounds. The local setting is the T1 phase budget interface that prepares controlled unresolved costs for later visibility and RCL arguments.
proof idea
This is a module containing supporting lemmas rather than a single proof. It imports the phase failure cost accumulation result and applies it to the stable budget hypothesis to obtain no_unbounded_unresolved_phase_cost and failed_gate_count_bounded_by_budget. The structure consists of direct applications of the upstream accumulation property.
why it matters in Recognition Science
The module supplies the T1 phase budget bound imported by BoundedPhaseVisibility to prove that finite phase invisibility cannot persist beyond the supplied bound and by PhaseBudgetToErdosStraus to compose the budget with the Erdős-Straus RCL closure chain. It fills the phase-budget interface step that keeps unresolved costs controlled before visibility and number-theoretic closure arguments proceed.
scope and limits
- Does not address infinite phase cost accumulations.
- Does not supply explicit numerical bounds independent of the stable budget parameter.
- Does not connect directly to spatial dimensions or physical constants.
- Does not treat non-finite gate enumerations.