PhaseBudgetEngine
A phase-budget engine packages a bound function from naturals to naturals together with a guarantee that every residual trap admits an admissible hard gate witness inside that bound whose subset-product phase hit is nonempty. Number theorists extending the Recognition Science chain to the Erdős-Straus conjecture cite this structure as the exact interface that converts T1/RCL budget facts into explicit gate witnesses. The declaration is a pure structure definition carrying no proof obligations or computational content.
claimA phase-budget engine consists of a bound function $b : ℕ → ℕ$ such that for every natural number $n$ that is a residual trap there exists $c ≤ b(n)$ with $c$ an admissible hard gate and the subset-product phase hit for the pair $(n, c)$ nonempty.
background
The eight-tick phase is the map sending each Fin 8 index k to kπ/4, supplying the periodic structure used for finite phase separation. Active edge count A equals 1 per fundamental tick and enforces the φ-power balance identity at D = 3, equivalently φ^(-44) · φ^45 = φ, which fixes the coherence energy unit E_coh = φ^(-5). The module composes the phase-budget interface with the already-proved Erdős-Straus RCL closure chain; the structure supplies the remaining physical bridge from finite failure-cost accumulation and the T1/RCL stable-budget interface to an explicit gate witness.
proof idea
This is a direct structure definition with no proof body. The two fields simply record the bound map and the universal supplies_hit property; downstream lemmas such as boundedBalancedSearch_of_phaseBudget and effectivePrimePhaseInput_of_phaseBudgetEngine apply the structure by pattern-matching on these fields.
why it matters in Recognition Science
The structure is the parent for boundedBalancedSearch_of_phaseBudget, effectivePrimePhaseInput_of_phaseBudgetEngine, and erdos_straus_residual_from_phaseBudget inside the same module, and for PhaseBudgetEngineLogic and phaseBudgetEngine in sibling modules. It completes the conditional bridge from T1/RCL budget facts to the Erdős-Straus chain step, leaving open only the concrete construction of an engine satisfying the supplies_hit field.
scope and limits
- Does not construct any concrete bound function.
- Does not prove existence of a PhaseBudgetEngine.
- Does not apply to non-residual traps.
- Does not compute J-cost or defectDist values.
formal statement (Lean)
30structure PhaseBudgetEngine where
31 bound : ℕ → ℕ
32 supplies_hit :
33 ∀ n : ℕ, ResidualTrap n →
34 ∃ c : ℕ, c ≤ bound n ∧ AdmissibleHardGate c ∧ Nonempty (SubsetProductPhaseHit n c)
35
36/-- A phase-budget engine is exactly an effective prime phase input. -/